Receiving

Systems
~~Design

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 Receiving
systems
Design

Stephen J. Erst
Copyright© 1984.
ARTECH HOUSE, INC.
610 Washington St., Dedham, MA
Printed and bound in the United States of America. All rights reserved.
No part of the this book may be reproduced or utilized in any form or by
any means, electronic or mechanical, including photocopying, or by
any information storage and retrieval system, without permission in
writing from the publisher.

International Standard Book Number: 0-89006-135-1

Library of Congress Catalog Card Number: 84-070222
PD SyQhtaham

CONTENTS

PREFACE xi
INTRODUCTION xiii
1 AN OVERVIEW OF SIGNAL CHARACTERISTICS 1
1.1 Receiver Input Power Predictions l
1.2 Free Space Path Loss 2

1.3 Fresnel Zones 5

1.4 Fade Margin 6

2 MODULATION 9

2.1 Amplitude Modulation 9

2.2 Amplitude Shift Keying (ASK) or ON/OFF Keying (OOK) 14

2.3. Single Sideband Suppressed Carrier Signals (SSB SC) 16
2.4 Pulse Modulation 17
2.4.1 Pulse Bandwidth Requirements 19
2.5 Frequency Modulation 21
2.5.1 Carson’s IF Bandwidth Approximation Applied to FM Signal
Spectral Requirements 24
2.5.2 Critical Determination of the Bandwidth Required for FM
Signals by Power Summation 25
2.5.3. Foster-Seeley FM Discrimination Detector 25
2.5.4 Discriminator Detection of FM Signals Using Opposing AM
Detectors 26
2.5.5 Discriminator Output Filtering 26
2.5.6 Phase-Lock FM Demodulation 28
2.5.7 Counter FM Demodulator 28
2.6 Phase Modulation 29
2.6.1 Carson’s IF Bandwidth Approximation Applied to PM Signal
Spectral Requirements 31
 2.6.2 PM Signal IF Bandwidth Requirements Using
Power Summation of the Sidebands
2.7. Phase Shift Signals (PSK)
2.8 Frequency Shift Keying (FSK)
2.8.1 FSK Signal Spectrum and Bandwidth Considerations
2.8.2 Coherent FSK
3 NOISE
3.1 External Noise
3.2 Effective Noise Bandwidth

3.3. Noise Figure and Noise Factor

3.4 Noise Temperature
3.4.1 Cascading Noise Temperature
3.5 Conversion Noise
3.5.1 Ideal Noise Free Signal and Local Oscillator Signals
3.5.2 Noisy Received Signal
3.5.3. Noisy Local Oscillator
3.6 Noise Measurement Techniques
3.6.1 Y Factor Method
3.6.2 Three Decibel Method

4 THE RECEIVER
4.1 The Superheterodyne
4.1.1 Configurations
4.1.1.1 Down Converter
4.1.1.2 Up Converter
4.1.1.3 The Wadley Drift Canceling Local Oscillator
System
“2 Direct Conversion

4.3 The Receiver RF and IF Gain Budget

7.4 Preselector Requirements

4.4.1 Preselector Requirements for Down Converters
4.4.2 Preselector Requirements for Up Converters
4.5 The Need for an Out of Band Intermediate Frequency
4.6 Local Oscillator Frequency
4:1 Automatic Gain Control

4.7.1 Determining the AGC Control Range

vi
 4.7.2 Attenuator AGC
4.7.3 Fast Attack/Slow Decay AGC
4.8 Sensitivity
4.8.1 Measuring Sensitivity Given S/N or (S+N)/N
4.8.2 Measurement of Sensitivity Given SINAD
4.9 Signal to Noise Ratios for Amplitude Modulated Double Side
Band Systems
4.10 FM Carrier to Noise Ratio
4.10.1 FM Output Signal to Noise Ratio Above Threshold
4.10.2 FM Noise Improvement Factor (MNI) Above Threshold
4.10.3 FM Signal to Noise Ratio Below Threshold
4.11 PM Output Signal to Noise Ratio Above Threshold
4.11.1 PM Output Signal to Noise Ratio Below Threshold
4.12 Energy Per Bit to Noise Spectral Density (E,/N,)
4.13 Error Function (erf)
4.14 Complimentary Error Function
4.15 Tangential Sensitivity (TSS)
4.16 Cascade Noise Figure
4.17 Intermodulation Distortion (IM)
4.17.1 Cascade Intercept Point
4.18 Desensitization
4.19 Compression
4.20 Cross Modulation
4.2] Spurious-Free Dynamic Range
4.22 Images
4.23 Higher Order Images
4.24 Selectivity
4.25 Intermediate Frequency (IF) Rejection
4.26 Local Oscillator Radiation
4.27 Predicting Spurious Products
4.28 The Mixer Spur Chart
4.28.1 Spur Chart Limitations 126

4.28.2 Spur Chart Generation by Measurement and Spur

Identification 126
4.29 Local Oscillator Spurious Products 127
4.30 Crossover Frequencies 128

vil ys
4.31 Computing Noise Figure Given TSS 130
5 COMPONENTS 133

Xs| Filters 133

nA | Filter Insertion Loss: Bandpass Case 137
Sliuz Filter Insertion Loss: Low- and Highpass Cases 138
Dik Varactor Tuned Filters 140
5k Crystal Filters 148
9.1.4.1 Monolithic Crystal Filters 149

4Bt
BR Modeling the Butterworth Filter 149

5.1.6 Modeling the Chebyshev Filter 152

5.1.7 Distortion 157

By | Phase Delay 157

Sid2 Delay Distortion 158

38Gi Envelope Delay, Group Delay, or Absolute

Envelope Delay 158

BY OS Relative Envelope Delay or Group Delay Relative 159

Computer Prediction of Group Delay for Butterworth and

Chebyshev Filters 159
5.1.8.1 Butterworth Filter Group Delay 159
5.1.8.2 Chebyshev Filter Group Delay 164
5.2 Mixers 168
5.2.1 Specialized Mixers and Applications 171
52.15) Termination Insensitive Mixers 171
5.2.1.2 Image Rejection Mixers 171
5.2.h.3 Harmonic Mixers 174
5.3 Linear and Non-Linear Amplifiers 175
5.3.1 Limiter 176

5.3.2 Successive Detection Log Amplifiers 177

6 SPECIALIZED RECEIVER APPLICATIONS 181

6.1 Definitions 182
6.2 Scanning Superheterodyne Receiver 183
6.3 Smart Scan Receiver 183
6.4 Instantaneous Frequency Measurement Receivers 184
6.5 Microscan (Compressive) Receiver 186
6.6 Channelized Receivers: Crystal Video 188
6.7 Bragg Cell Receivers 189

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7 DESIGN EXAMPLES 19]
7.1 Example 1 191
7.2 Example 2 201
7.3. Example 3 218

APPENDIX 225
(a) Digital Data Rate 225
(b) Adding in Decibel Notation 225
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 PREFACE

This book is the result of many comments which have expressed a desire for a
text on receiving systems design. Most of the readers have been exposed to the
basics involved but have never put it all together. This text is intended to lead
the reader through typical cases from which variations can be made to suit a
particular need. For those who may desire to refresh themselves in the basics, a
review is presented for reference.

The organization of this book is arranged to address the objective of receiving

systems design, with supporting explanations in the following chapters.

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 INTRODUCTION

This book is intended to assist the reader in the design of receiving systems of
four fundamental types:

Down converter
Up converter
Hybrid up and down converter
Wadley up converter
The text consists of five parts presented in the following sequence:
A basic overview of signal characteristics (Chapter 1)
The superheterodyne (Chapter 4)
Components (Chapter 5)
Specialized receiving systems (Chapter 6)
Design examples (Chapter 7)
Interspersed throughout are computer programs written in the BASIC lan-
guage, to assist the designer in system performance prediction.
The designer should accumulate a library of available components and their
characteristics for ready reference. Generally it is most expeditious to procure
components rather than undergo design and development efforts of these items,
unless the designer has this capability available. This is recommended for
initial modeling, later moving to in-house designs if cost effective.
A final chapter includes examples and the sequence of computations and
considerations leading to the final design. It is almost always a necessity to
revise the structure, as unforseen design faults are found through subsequent
performance analysis.
Experience will provide the designer with an insight into what can be done.
Low noise and high third order intercept performance, almost always specified,
are not simultaneously achievable. A design is usually a compromise of these
characteristics.

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 AN OVERVIEW OF SIGNAL
CHARACTERISTICS

This section is concerned with the reception ofthe signal from a distant emitter.
Considered are the prediction ofthe signal strength and the attenuation due to
free space path loss. The Fresnel zones are defined for link calculation and the
subject of fade margin is addressed. With these basic considerations the link
performance can be predicted. 7
1.1 RECEIVER INPUT POWER PREDICTIONS

To determine the necessary receiver noise figure and sensitivity if it has not
been previously specified, it becomes necessary to estimate the signal strength
at the receiving antenna. Having determined this, the receiver and antenna
requirements can be determined. While this is readily done for line of sight
links, it becomes less defined for ionospheric reflection, troposcatter, knife edge
diffraction systems, ef cetera, and will not be discussed here. Most modern links
are line ofsight limited because of operation at UHF, VHF, and microwave
frequencies, which penetrate the ionosphere and are not, therefore, reflected
back to earth as HF signals are.
To make this calculation the signal strength at the receiving antenna is
P, = P, + A, —path loss (1-1)
where
P, is the power received at the receiving antenna
P, is the transmitter power
A, is the transmitting antenna gain in the receiving direction
Path loss is discussed in section (1.2).
The P, + A, term is the effective radiated power in the direction of the receiving
antenna.
Receiver sensitivity or P, (min) will have been determined from considerations
of SN, C/N, E,/N., et cetera,attainable noise figure, and the receiving antenna
gain requirements A,.
2 RecewingSystemsDesign

A, becomes
A, = P,,ini, ~ P, (dB notation) (1-2)
Example: Find the required receiving antenna gain when given:
Path loss = 170 dB

a! = 100 watts, 50 dBm

A, = 10 dB
Peis) = -100 dBm
then the effective radiated power is:
ERP=P,+A,
= 50 + 10 = 60 dBm (1-3)
and
P.= ERP - Path loss
= 60 dBm - 170 dB = -110 dBm
A,= -100 dBm - (-110 dBm) = 10 dBm (1-4)

The receiving antenna gain must be 10 dB, minimum.

These equations can be manipulated to determine any one parameter knowing
the others.
For a discussion of path loss, see section 1.2.

1.2 FREE SPACE PATH LOSS

Electromagnetic emission from a point source radiates energy equally in all

directions. At any distance d away from the source, this energy is distributed
evenly over a spherical area whose radius is d and its center is the source. It
follows that ifthe transmitted power is P,, the power per unit area ata distance d

is:
P
41d” re1-5

The power received by a receiver with an antenna whose effective area is A is:
P
P=; 47dey 1-6
U6)

Since isotropic antennas are the reference standard upon which antennas are
usually compared, it is convenient to utilize this as the receiving antenna. The
effective area of the isotropic antenna is
d? |
(1-7)
41
Overviewof Signal Characteristics 3

where
locit ta
A is the wavelength, A = ee
frequency
Substituting into (1-6) we have
_ B(A?/4m) i
r 4nd ( r )

at, Ane SA
Yard? 157.9d

The path loss in dB is

P 157.9
L, = 10 log ry =10og( x (1-9)

If disin miles, and A is in centimeters, then equation (1-9) becomes

L,=
10to(157.9
x@amy)
] |
= 10log (+069 - 10" aad (1-10)

In dB notation,
L, = 126.12dB+20 log d - 20log A (1-11)
where
d is in miles

A is in centimeters

Other forms of this basic equation may be obtained by substituting frequency

in GHz for A.
Then

L, =92.45+ 20logf+ 20 logd (1-12)

where
d is in kilometers

J is frequency, in GHz
and
L, =96.58+20 logf +20 logd (1-13)
where
d isin statutemiles
fis in GHz
 4 RecewingSystemsDesign

Note that all of the path loss equations assume isotropic receiving antennas.
_ Where the transmitting or receiving antenna has gain, this must be accounted
for as a reduction of path loss.
Equation (1-13) isshown in graphical form in Fig. (1-1), for reference purpose.

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Fig. 1-1. Free space propagation loss

The reader is cautioned in the use of these equations for link calculations. These
are free space equations with no intervening obstructions or signal reflections,
resulting in multipath situations. If by the use of elevated antennas with
moderate gain a free space situation can be approached, then these equations
are valid.
For frequencies greater than 8 GHz the environmental effects on the signal
must be accounted for. Reference [1] treats this subject in detail.

d; de
d=d,+d2

Fig. 1-2. Physical relationship between transmitter 7 and receiver R where ris
the radius of the first Fresnel zone.
Overviewof Signal Characteristics 5

1.3 FRESNEL ZONES

Fresnel zones describe the phase behavior of a signal originating at a transmit-

ter 7 and appearing at a distant receiving site R. With the aid of Fig. (1-2), T
and R are connected by a line nA long describing the shortest distance between
them. A plane perpendicular to this line is constructed at p and a circle is drawn
on this plane containing all points where the path length has increased by (1 /2)
X to (nt+]/2)X. This is the first Fresnel zone which contains nearly.25% of the
signal power within its boundary. (This is the most important zone.) Similarly,
other circles may be drawn for path length increases of multiples of (1/2) A.
These are successively known as the second, third, fourth, et cetera, zones for
path length increases of A, 1.5A, and 2A, respectively. All odd multiples of
(1 /2)X are in phase at RX,while even multiples, which are in phase with each
other, are out of phase with the odd multiples at RX.The signal contributions of
each zone are nearly equal, diminishing very slightly as the zone numbers
become large. A successive summation of the signal contributions of each zone
(i.e., 1,142, 1+2+3, 1+2+3....n) would show a cyclic behavior until, with a
sufficiently large n, the cyclic amplitude diminishes and the signal at Rbecomes
equal to the free space value.
The first zone is the most important zone and it should be kept clear of
obstructions. The radius ofthis zone at any point along the axis may be found
from
—
7

r=13.16
(42)
Hs
2

(1-14)
where

r is the radius ofthe first zone (feet)

A is the wavelength (cm)
d, is the distance to point pffrom the transmitter (miles)
d, is the distance to point p from the receiver (miles)
d is the straight line distance between transmitter and receiver (miles)
In other units, where d is in miles and F is in MHz:
/
r= 2280 (d,d,/dF) 2 (1-15)

The value of rmaximizes when point f is midway between 7 and R, at which

time r may be found from:

r=1140d/F) > (1-16)

Reflection from the earth will vary in magnitude as a function ofthe reflection
coefficient. Where the angle of incidence is small, this coefficient approaches
6 ReceivingSystemsDesign

unity. The incidence and reflection angles are equal. There is a phase reversal
at the point of reflection for all polarizations. The resulting signal intensity
profile for various clearances is shown in Table 1-1. Shown are the cases of
reflection from highly reflective, relatively smooth ground and water, and are
labeled plane earth and smooth sphere diffraction. The knife edge diffraction
case is applicable to fairly smooth vegetated terrain without atmospheric
disturbances. In plane earth theory, 6 dB signal enhancement is possible at
clearances equal to odd integral multiples of the Fresnel radius.

Table 1-1.
Radio Wave Propagation as Affected by Path Clearance (dB) [2]
aR ecareniceSt Knife Edge Smooth Plane
First FresnelZone Radius Diffraction Sphere Earth
=3 -26 >-70 >-70
=2.5 -24 >-70 >-70
-2 -22 -70 >-70
-1.5 -19 -59 >-70
-] -17 =45 >-70
ae -12 -12 >-70
0 0+] -30 -70
o> 0 0
1.0 +6 +6
1.5 « :
2.0 : r4
2.5 2 :

1.4 FADE MARGIN

A link is subject to degradation ofthe signal because ofphysical changes in the

transmission medium, geometry, or both. An allowance for such changes must
be made to guarantee the communications reliability of the link. This allow-
ance is established in dB and is called the fade margin. Link reliability is
generally expressed in percent values such as 99.9%, which allow an outage of
0.1%.

Multipath fading is a major cause of outages, and is particularly severe in

mobile installations. The mechanism is one of reflection or attenuation of the
signal from: buildings, water, trees, et cetera. The summation of all signals
arriving via different paths at the receiving antenna causes enhancement or
reduction of the signal phases. These variations are largely random and are
called Rayleigh fading because of their distribution. Fig. (1-3) relates link
reliability to relative signal strength and is a theoretical maximum. Using this
 Overviewof Signal Characteristics 7

graph, a system with 99% link reliability would require a design signal strength
18 dB above threshold.

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Example: For a given signal to noise ratio the minimum signal strength is -97
dBm. To meet a link reliability of 99.9%, the system must have a 28 dB fade
margin. This gives the receiver a signal strength of -97+28 = -69 dBm.
Because of the shorter wavelengths, higher frequencies are more prone to
multipath fading which approaches the limit of Fig. (1-3), above 4GHz. The
effect of frequency on Rayleigh fading is shown in Fig. (1-4) and is a percentage
of the maximum shown in Fig (1-3).
Example: For an outage of 0.1% of the time, or a link reliability of 99.9%, a
frequency of | GHz will have a fade depth of 67%of 28dB or 18.76 dB. Moving
the frequency to 4 GHz results in a fade depth of 90% of28 dB or 25.2 dB, or
6.44 dB more.
Other variables are weather related, such as: temperature inversion, diffrac-
tion, scattering or absorption due to moisture, rain, or snow, and temperature
itself.
One solution to fading problems is diversity reception. This is based on redun-
dancy and may involve two or more receivers whose outputs are combined.
The redundancy may involve the use of receiving antennas at different loca-
tions, feeding several receivers tuned to the same signal. The same information
may be transmitted on several different frequencies, each of which is received
by a receiver or a combination of both. Antenna polarization may be utilized,
as well as time, for the system variables.
8 ReceiwingSystemsDesign

Needless to say, diversity is an expensive proposition, and the link configuration

would require careful economic analysis before a decision regarding its use
could be made. oO

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Fig. 1-4. Percent of Raliegh fade maximum versusfrequency [2].

REFERENCES

[1] Skolnik, Merrill I., Introductionto Radar Systems.New York: McGraw-Hill

Book Company, 1962.
[2] Bullington, K., ‘‘Radio Wave Propagation Fundamentals,” B.S. 7.7.,
vol. 36, no. 3, Fig. 8.
 MODULATION

Key to the design of a receiving system is aknowledge of the characteristics of

the signal to be received. The nature of the modulation determines the type of
demodulator which must be employed and whether or not signal limiting is
required. The frequency range of the modulation defines the post-detection
frequency response required. The magnitude of the modulation, together with
the type of modulation and the modulation upper frequency, defines the IF
bandwidth necessary to handle the signal (remember to allow for drift). There-
fore it is vital to have a knowledge of the more popular forms of signals in use
today.

2.1 AMPLITUDE MODULATION

Amplitude modulation or AM, sometimes referred to as ancient modulation,
has been the first practical form of voice transmission by radio waves and has
been the workhorse of communications. AM is still heavily used today and
should not be ignored, despite its inefficiency when compared to other more
modern forms. AM is simply the amplitude variation of a carrier directly
proportional to the magnitude of the intelligence to be sent. This modulation
process is shown graphically in Fig. (2-1).
Mathematically, let the carrier be represented by
A sin (wt + ®) (2-1)
where
A is magnitude
w is 27F, and F is the RF carrier frequency
¢ is an arbitrary angle
The information to be transmitted is selected for this illustration to be a simple
sinusoid represented by:
B cos pt (2-2)
10 ReceivingSystemsDesign

where
B is magnitude
p is 27f, and / is the frequency of the sinusoid to be transmitted

Amplitude

Time

(a) Carrier generated by an oscillator.

Amplitude

Time

(b) Information to be transmitted.

Amplitude

Time X Y

poneenaien
pans
(c) Amplitude modulated carrier.
Fig. 2-1. Graphical representation of amplitude modulation.
Since A represents the magnitude of the carrier and this is to be varied or
modulated in amplitude by 2cos pt ina linear fashion, we add Bcos utto Aand
rewrite the carrier equation (2-1) to its amplitude modulated equivalent:
(A + Bos pt) sin (wi + d)
or
A (1 + B/A cos pt) sin (wt + d) (2-3)
Modulation Il

The ratio of B/A represents the magnitude of the modulation or modulation

factor m. The value of m cannot ever exceed | and still retain a sinusoidal
carrier form. Should m > 1, then the carrier would be interrupted during part
of the modulation cycle and over modulation results, together with the genera-
tion of harmonics, which result in undesired spectral broadening known as
splatter. This splatter causes interference, particularly to adjacent channel
signals.
The percent of amplitude modulation is (m- 100), and is usually measured by
the use of:
modulation meters
spectrum analyzers
oscilloscopes

The modulation meter is a calibrated receiver which uses a meter directly to

indicate the percentage modulation.

Expanding (2-3) we have:

A (1 + mcos pt) sin (wt + d)
= A sin (wt +d) + Amcos pt sin (wi + d)

=Asin(24Fit $)+ + sin [27(F +f) t+]

+= sin [2m(F-f) t+] (2-4)

From these three terms the original carrier plus two additional frequencies
symmetrically spaced about the carrier by fare found. These additional! terms
result from the modulation process and are referred to as sidebands. The
maximum magnitude of these sidebands is found by letting m= | and is found to
be 1/2 that of the carrier.
The total width of the AM signal for this case is 2fand is seen to be independ-
ent of the modulation percentage of carrier power. Thus when a system is
specified to contain frequencies to 10 kHz, the signals spectral width is 20 kHz.
Using a spectrum analyzer or selective voltmeter in the frequency domain, the
y-axis represents magnitude and the x-axis frequency. The AM signal is found
to be as shown in Fig. (2-2) and (2-3).
Most spectrum analyzers display the y-axis in power and the interpretation of
the display must be altered accordingly.
In the time domain, the display of the voltage amplitude versustime of the AM
wave results in the display of Fig. (2-lc) presented earlier. The value of m may
be found from:
Y-X
m% = Ce 100 (upward modulation) (2-5)

“a
12 ReceiwingSystemsDesign

Te
—___. Am/2= a

Fo F F+f

(c) m%=100 A
f= 1kHz

Fig. 2-2.
The AM signal displayed in the frequency domain where F is the carrier, A
is magnitude, the modulating frequency is fixed, and m% is the variable

f=100
Hz
(a)m%=50 eo

F+ ar
(b)mde
ee

(c)m%=50 A

——Am/2=
4
F-t F F+f
Fig. 2-3. The AM signal displayed in the frequency domain where F is the
carrier, A is magnitude, m is constant, and the modulation frequency / is the
variable.
Modulation 13

An alternate form known as downward modulation is:

m% = aVe 100 (2-6)

Complex modulating waveforms represented by g(t) are treated in similar

fashion as that of the sinusoidal case.
The AM wave may be described by:
A (1 + mgt) sin (wt + ) (2-7)

By representing g(t) as a Fourier series and substituting this for g(t), the
resulting spectrum may be determined. This is shown graphically in Fig. (2-4).
In this case the signal spectral width is 2/;. In practical complex cases, a filter is
used to truncate the series, limiting the AM spectral width to a specified or
practical value. The complex spectra is the summation of the AM spectra of
each sinusoid contained in the series.

Amplitude

0 f; fo fs frequency

(a) g(t) = complex modulation waveform

F -fs F -f2 F -f; F F +f, F +f. F+fs

frequency
(b) Spectrum resulting from the modulation of a carrier by complex wave g(t).

Fig. 2-4 (a) complex modulating wave spectrum, (b) the resulting AM
spectrum.
The energy contained in the AM wave is the sum of that in the carrier plus in
the side bands. From Eq. (2-4) it is seen that the energy in the carrier is
unaffected by modulation. It is also seen that additional energy is added to the
AM wave by the side bands. By squaring the magnitudes of Eq. (2-4), the
energy in the wave may be computed for sinusoidal modulation as follows:
I¢ RecewingSystemsDesign

4
7 (1 + am*
2
Table 2-1.
Relative AM Signal Energy versus%m
%om Energy

0 l
10 1.005
20 1.02
30 1.045
40 1.08
50 1.125
60 1.18
70 1.245
80 1.32
90 1.405
100 1.5
From this tabulation we see that the energy contribution to the carrier is 1 /2
that of the carrier itself at 100% modulation. Each of the two sidebands
contributes | /2 of this value or 1 /4 of the energy.

As an example, a 100 watt carrier 100% AM by a sine wave, will have an

average of 50 watts of sideband power, composed of 25 watts from each
sideband.

2.2 AMPLITUDE SHIFT KEYING (ASK) OR ON/ OFF (OOK)

ASK or OOK consists of a carrier which is turned on, for example, by a mark
and off by a space. The carrier takes on the form of an interrupted carrier, such
as in telegraphy, only the data is encoded differently. The signal takes on the
form shown in Fig. (2-5).
Detection of such a signal may be non-coherent or coherent, with the later
being the better performer, although more difficult to achieve.

necanter
—lll Wit All—
Pas a sp ha mek time —____—
Fig. 2-5. The ASK signal in the time domain.
Modulation 15

Non-CoherentDetectionof ASk
Non-coherent detection in its simplest form consists of envelope detection
followed by decision circuitry, as shown in Fig. (2-6). The decision threshold
grossly affects the error probability (P.) for mark and space independently, and
they are therefore not equally probable. This results because the decision
circuitry must distinguish between two signal states which are not equal in all
respects. The mark or carrier on signal consists of carrier plus noise whereas the
space or carrier off signal is noise alone. It has been shown (2-1) that P, mark can
be made equal to P, spacefor a given (C/N),. A threshold of ~ 50% amplitude
achieves this result [1 ].
Minimum probability of error results when a threshold of roughly
7

] -
(>pulse amplitude)-(1+2<.) isused. (2-8)
Where
é, is the pulse energy
N, is the noise density per reference bandwidth
In general, ASK is a poor performer although it is used in non-critical
applications.
For e,/', > 1 and a decision threshold of half the pulse amplitude, the
probability of error for space Bs:
P —@ée
( 2,
* (2-9)

and for mark

] ivan)
ai oa a oe (2-10)
(27 e,/N,) 2
From this, it is seen that the majority of errors are spaces converted to marks.

Fig. 2-6. Non-coherent detection of ASK.

Coherent Detection

Coherent detection requires a product detector with a reference signal which is

phase coherent with the incoming signal carrier (see Fig. (2-7)).
16 RecewingSystemsDesign

ASKinput Bosuinn 3
| circuit Data
output
Phase
coherent
reference pallacm
ofsh
Fig. 2-7. Coherent detection of ASK using synchronous detection.
The product detector is followed by an integrator and a decision circuit timed
to function at the end of | bit, or time rT.
An equivalent performer is the matched filter detector shown in Fig. (2-8).
Here, the output of the matched filter is the convolution of the pulse and the
impulse response of the matched filter. The resulting output is ideally diamond
shaped and ofduration 2 r, with a maximum signal energy at a time of A*/2,
where A is the signal amplitude and 7 is the pulse duration. To complete the
system, the decision circuitry is timed to function at time 7 for optimum
performance.
a Matched Envelope Decision
ASK input filter detector circuit Data output
] End of bit
timing
signal | ;
(a) input 0

(b) impulse A
response 0
of matched T
filter

(c) Convolution
of aandb Y2_ 97

Fig. 2-8. Matched filter detection of ASK signals.

The probability of error for coherent ASK signaling is:
1

P=~erfe(
sf)* (2-11
2.3 SINGLE SIDEBAND SUPPRESSED CARRIER SIGNALS (SSB SC)
An AM signal consists of a carrier and two sidebands and is described by:

Asin(27Ft+) + = sin [27(F+y)t+]

+ ae sin [or (F-p)tt+o] (2-12)
Modulation iv

for a sinusoidal modulation signal of frequency 4 and modulation factor m.

The carrier does not contain the modulation signal and its transmission serves
only as a reference to the sidebands. The sideband components contain the
modulating signal but are redundant. The single sideband principle utilizes
these relationships, and removes the carrier and one of the sidebands from the
AM spectrum, transmitting only the remaining sideband. (Upper or lower
sideband as desired.) At the receiver, the carrier is reinserted and the modula-
tion is recovered.
The principal problem with SSB SC is the accuracy required in the reinsertion
of the carrier, which must be approximately 20 to 80 Hz or less from the missing
carrier frequency, and on the proper side of the sideband. For upper sideband
signals the carrier is reinserted on the low side. For lower sideband signals the
carrier is reinserted on the high side. Failure to do this results in modulation
inversion and unintelligibility. Because of the accuracy required for carrier
reinsertion, SSB SC is seldom used in motional environments where doppler
shifts are uncontrolled or variable. |

The advantages of SSB SC are the narrower spectral occupancy ofthe trans-
mitted signal, the reduced receiver IF bandwidth, and the lower transmitted
power of SSB (compared to AM) for equivalence in (§ + V)/WNat the receiver
output. For an equivalent signal to noise ratio, the SSB signal requires a peak
envelope power equal to 1/2 that of the AM carrier with 100% modulation.
Many comparisons may be made at this point. Comparing total powers
radiated by both methods we have: modulation = 100% with sinusoidal modu-
lating signal.
AM
carrier 1 unit
upper sideband 1/4 unit
lower sideband 1/4 unit

Total 1.5 units

SSB
0.5 units peak envelope power
Power ratio = 3.0 or 4.77 dB. Note that at AM and SSB equivalence, the AM
total sideband power equals that of the SSB signal (PEP). When there is no
modulation in AM the carrier is present, while with SSB there is no SSB signal.
This extreme case shows a marked improvement in power utilized. Thus,
where small, compact, efficient, low cost equipment is called for, SSB could be
the answer. Since SSB SC signals are amplitude varying, they must be ampli-
fied linearly at the transmitter and receiver.

2.4 PULSE MODULATION

When a carrier is turned on and off, a pulse of RF energy is generated. If this is

 18 RecewingSystemsDesign

done repetitively at some rate such as in radar applications, digital data link,
amplitude shift keying (ASK), or on/off keying (OOK), this energy takes on
unique characteristics in the time and frequency domain. These characteristics
are vital to the effective detection and processing of the signal and its contained
data.
The detection of RF low level energy requires that noise be minimized while
the signal is amplified. The noise component is kTB where Bis bandwidth and
is represented in terms of power by -144 dBm for a | kHz bandwidth. As B
increases noise increases. Therefore, it becomes necessary to minimize the
receiver bandwidth, to minimize the noise, and yet be of sufficient bandwidth
to contain the majority of the signal energy.
The signal energy spectral occupancy and the receiver bandwidth can be
computed by use of the Fourier transform on the time domain representation of
the signal envelope.
For a pulse waveform this results in a spectral envelope described by sin x/x.
More specifically, the spectrum is seen to consist of the repetition frequency
and its harmonics; each with differing amplitude described by:
T | sin mj T/tr
A,=2A
: —
t, wiT/tr 2-13
rte)

where

A is amplitude
J is the harmonic of /,
f, is the repetition frequency
t, is the repetition period 1 //,
T is the bit or pulsewidth
The generalized solution of A, is shown in Fig. (2-9).
The energy of the signal is largely contained within the mainlobe of width 2/ T,
which usually represents a sufficiently wide receiver RF and IF bandwidth, to
efficiently process the pulse. Where pulse fidelity is important, a wider band-
width may be necessary.
The detection bandwidth would be 1 /2 of the IF bandwidth with a spectral
content on one side of that of the IF spectrum.

The spectrum, as seen on a spectrum analyzer, is unidirectional in the » or

amplitude axis. All negative spectral lines are seen to be positive. When dealing
with digital data, the bandwidth requirements should be based on a pulse that
__is one bit wide, for a worst case solution. The spectrum of ASK can be that of
one bit, as a limit to all marks or 1's as the other; which is that of an
Modulation 19

unmodulated carrier. The bandwidth, of course, must accommodate the worst

case.

-3/T -2/T -V/T VT 2/T 3/T

(b)Envelope
ofSiN.
X;{.isthecarrier
frequency.

(c) Spectral componentsof b

Fig. 2-9. Pulse modulated carrier and its representative spectrum.

2.4.1 Pulse Bandwidth Requirements

A pulse may be described by a series, consisting of harmonics ofthe repetition
frequency. It follows that the pulse may be faithfully processed by a system with
a bandwidth, which includes all of the terms of that series. In practice, system
bandwidth is minimized, consistant with reasonable pulse fidelity. Apulse may
be described by a (sin x) /xrelationship. From this, the magnitude of the terms
is seen to diminish in cyclic fashion, with increasing harmonic number. The
inclusion of terms with minor magnitude, within a bandwidth window, can
result in increased noise with little or no improvement in the signal. Therefore,
it is necessary to restrict bandwidth to some practical value, determined by rise
time or signal to noise ratio, or by acompromise of both.
Several relationships exist which help to define bandwidth requirements. If the
pulses are treated as normalized Gaussian wave forms, the fractional basewidth
of this wave form, within a specified time slot, may be related to bandwidth by
the following relationships:
 20 RecewingSystemsDesign

p=(2)[2in(4)]*
Geetey oy
where

B is the system bandwidth

T is the width ofthe time slot

kis the fractional base height of the normalized Gaussian pulse within the
time slot

A further definition is shown in Fig. (2-10).

1.0
@
a©|
S
ra
E
a
a©|
®
N
= k
5
za

time
slot
T
Fig. 2-10. A normalized Gaussian pulse bounded by a time slot of width r
where & is less than 1.
The bandwidth (B) computed contains 95.45% of the pulse energy. For pulses
whose normalized magnitude is down to 0.1 within a 1 ywsecond window, a
bandwidth of 1.336 MHz is required as shown:

B=( “=)[2m 1
(+) *=1.336 MHz (2-15)
mw 10 1
In digital systems the value of k determines the adjacent channel spillover and
the lower the value of k, the better the fidelity of the system and the lower the
spillover.
The rise time (t,) of an ideal pulse applied to a band limited circuit may be
approximated from:
ms 0.35 (2-16)
Snes
Modulation 21

where

t, is rise time

Snax18the upper -3 dB frequency response point of the video amplifier

For a fi, Value of 1MHz the pulse rise time is approximately 0.35 ysecond.
For the lower -3 dB response point of the video amplifier, the percentage tilt ofa
pulse train such as a 101010 may be approximated from:

Tilt (%) = 100 * = 628.32 f,, 7 (2-17)

where
Siowis the low frequency -3 dB point of the video amplifier
f=1/2 7 where rf is the bit width
Example: Let 7 = 100 useconds
f=5 kHz
For a tilt of 10%,
10 =0.3183f=
Fite — 159Hz
Thus the frequency response of the video amplifier must extend to 159 Hz at the
low end.
The reader is cautioned to realize that the receiver RF bandwidth must be
twice the high frequency video bandwidth, for on/off keyed carrier signals.

2.5 FREQUENCY MODULATION

A carrier expressed by:
A sin (wt + d)
or
A sin (27 Ft + $) (2-18)
can be described by letting the frequency of the carrier (F) vary linearly with
the modulating frequency. The result of this is frequency modulation. Ma-
thematically, this form is derived from a consideration of phase modulation.
Let (27 Ft + o) = 6 (2-19)
and by definition the time rate of change of phase is frequency. Then:
dé _ d(2rFtt+o) _ d (27 Ft) d(@) = onf
dt dt dt dt
rewriting,
les {Peas (2-20)
22 ReceivingSystemsDesign

Let F be varied or modulated by a sinusoid of the form:

cos ut = cos 27 ft (2-21)
Further, let us restrict the change in F by an amount A F.
We may write:

a mnme
=+ AF cos'2irft (2-22)

Solving for 6 by integration we have:

a W phi

Substituting into (2-18) we have the equation for FM.

A sin (2 Ftt AF sin 27 ft + bru)
72 (2-24)

Where A F > f then equation (2-24) reduces to the intuitive form:

A sin (wt +AF sin pt + hey) (2-25)

The amount or degree of modulation is determined by the rate of the deviation

AF to that specified for the equipment, or established by the FederalCommun-
ication Commission (FCC). The ratio of AF /fis termed the modulation index
and is useful for the prediction of the spectral content of the FM signal.
Conversely, the modulation index and deviation may be computed from the
spectrum, for sinusoidal modulation. To establish the spectral prediction,
equation (2-25) can be expanded as follows:
AF
aps = B,w=2mnF, and w= 2rf

A sin (wt + B sin pt + dey) (2-26)

let wt + Diy =
and B sin wt = ¢,
Since sin (¢, + ¢,) = sin @, cos , + cos q, sin g,
Then we have:
A [sin (wt + by) cos (B sin pt)
+ cos (wt + b,,,) sin (B sin pt) | - (2-27)
Using the relationships:
cos (x sin y) =
J, (x) + 2 LF, (x) cos 2y + F, (x) cos 4y + Fg (x) cos Gy+....] (2-28)
Modulation 23

and

sin (x sin y) =
2 [7, (x) sin y + J; (x) sin 3y + J; (x) sin 5y + J, (x) sin THERE] (2-29)
where
J Ax)are Bessel functions of the first kind and are identified by a capital 7
nis the order
x is the argument

Substituting and rewriting (2-27), we have:

A[sin(wt+ber)]+[7.(B)+
2 [72(B)cos2 wt+FZ,
(B)cost wtt+..... ]]+
A [cos (at + br)]* 20,7, (B) sin wt +J; (B) sin3 wet...) (2-30)
Since
oi i gs
sin p Cosg = 2 sin (p+q) + > sin (p - g) (2-31)
and
ee l
Cospfsing = 2 sin (p+q)- ry sin (p - q)
(2-32)
then,
A|7.(B)sin(at+dy)+
2[7.(8)(>si
2 9 sin(wt 2 ut)+ —si
+2ut) 9 sin(wt 2
-2pt))+
Fs(B)> sin(wt+4 wt)+> sin(wt-4 pt)+...
‘10> sin ial(wt
+ut)- >] sin(wt~ut)| +
(4ber
J;(B) ssy
sin(wtt+3 Lighus
pt)- o.00 (wt-3u))a (2-33)
Rearranging in order:
A | 7, (B) sin (wt + Py)
+7, (B) [sin (wt + ut) - sin (wt - wt)]
+ F, (B) [sin (wt + 2 pt) + sin (wt - 2 pt)]
+ 7; (B) [sin (wt + 3 pt) - sin (wt - 3 pt)]
+ 7, (B) [sin (wt + 4 wt) - sin (wt - 4 wt)]
hee } (2-34)
24 ReceiwingSystemsDesign

In general form:
A {7, (B) sin (wt + yy) +
7, (B) [sin (wt +n yt) + sin (wt'-n wt)]} (2-35)

The sign in the 7, term is + for n even, and - for n odd.

For equation (2-34), it can be noted that there are sidebands for the fundamen-
tal modulation frequencies, as well as for the harmonics, spaced about the
carrier. The magnitudes of these terms are easily determined from the Bessel
charts or tables for small values of the modulation index.

Example 2-1: The deviation is given as 5 kHz at 1 kHz.

B=5
J Magnitude (V) Vv’ dB

0 -0.177 031 - 6.93

l -0.327 106 - 1.59
2 0.046 002 -18.84
3 0.364 132 - 0.641
4 0.391 153 0
5 0.261 068 - 3.52
6 0.131 017 - 9.54
7 0.053 .0028 -17.37
8 0.018 00032 -26.8
9 0.005 25E” -37.87

2.5.1 Carson’s IF Bandwidth Approximation Applied to FMSignal

Spectral Requirements
Because the signal sidebands for FM follow the Bessel function curves of the first
kind, it becomes requisite to determine the IF bandwidth necessary to include
the significant spectral terms. A rule which works well for most cases was
developed by J.R. Carson and is known appropriately as Carson’s rule. This
rule takes on the following representative forms:

B, = 2 (AFt+f,), 1> BY 1 (2-36)

where
B;,,isthe IF bandwidth
A F is the peak deviation
Jn is the worst case (highest) modulating frequency
B is the modulation index
Modulation 25

This equation is valid as shown for 1 > B> 1. For those cases where this does not
apply, a more suitable approximation is

B, 2 (AF+2f,),2<B<10 (2-37)
2.5.2 Critical Determination of the IF Bandwidth Required for
FM Signals by Power Summation
The FM process removes energy from a carrier and distributes it within the
modulation sidebands. Modulation does not add energy to the signal as in AM.
The distribution of this energy in the frequency domain is computable for
simple modulation signals as shown in Section 2.5.

The individual sideband energy is proportional to the square of its Bessel

coefficient and the sum of these energies for all sidebands, plus the carrier
component is equal to that of the unmodulated carrier.

Then
7. (8)]?+2(7,(8)) ?+2[7, (B)]?+207,(8)] 7+
217i (BT ee. =] (2-38)

Peak Deviation
Modulating Frequency
Applying this equation to example (2-1), through the eight sidebands, it equals
99.324% of the total power.
Using this relationship it is possible to trade off IF bandwidth against signal to
noise ratio, determining the penalties of signal energy loss. Complex modula-
tions must be handled by computer because of the multiplicity of terms.

2.5.3 Foster Seeley FM Discriminator Detector (Fig. (2-11))

This popular FM discriminator detector consists of a driver-amplifier limiter,
which feeds a constant amplitude FM signal to a specially-wound tuned IF
transformer. This transformer consists of a tuned primary and secondary, with
direct tap and mutual coupling. The result of this coupling is the generation of
a secondary voltage, E,, which is in quadrature to that of the primary voltage,
E>, at resonance. Two rectifier-type detectors are driven by the two outputs of
the secondary. The outputs of the detectors are in opposition. At f, the two
vector inputs to the detectors are equal and opposite, resulting in no output,
since| E,| -| E,| =0. As the deviated carrier shifts frequency, the phase relation-
ship is unbalanced and the detector drive vectors become unequal with the
result| E| -| E,| #0.The transfer characteristic of this circuit is the well-known
S curve.
26 ReceivingSystemsDesign

M
E, E.
FM e, Demodulated
input output
Vec E
+ > = |E.| - |Eo!

(b)

Demodulated
(d) output/

deviation

Fig. 2-11. (a) Foster Seeley discriminator; (b) vector relationships at center
frequency (no deviation); (c) vector relationships with frequency offset /’; (d)
output transfer function.
2.5.4 Discriminator Detection of FMSignals Using Opposing AM
Detectors
Since the carrier of an FM wave is frequency dependent upon the modulating
wave and is of constant amplitude, it follows that detection can be achieved
using two opposed AM demodulators tuned to different frequencies within the
deviation of the carrier.
One implementation of this scheme is shown in Fig. (2-12).
The input FM signalisfedto twoAM demodulatorstuned tof, and/, wheref, -
f, > 2AF and AF is the peak deviation of the FM wave. The Q of the tuned
circuits is such that an output is realized from each demodulator over 2 AF.
This results in each detector being a slope detector to the FM wave. Singly, the
output of each detector has considerable distortion. By using two opposed
frequency offset detectors, this distortion is largely removed. The resulting
output is an § curve typical of discriminators.

2.5.5 Discriminator Output Filtering

The output signal to noise ratio of a limiter discriminator is described by [1}
Modulation 27

where
owe3
(T°)
(zx)
AF is the peak deviation (Hz)
J, is the upper cutoff of the output lowpass filter
C is signal power
N, is the one-sided noise power density in watts /Hz
The above equation is valid for IF signal to noise ratios > 10 dB where full FM
improvement is realized.

Demodulated
output

Detector
#1 Detector
#2
Zenit
me PLL
5 Demodulated
output
A
+- mplitude
Carrier frequency
Fig. 2-12. FM detection using two opposed AM demodulators.
McKay has shown that by using a bandpass filter at the discriminator output,
an improvement factor / results and is described by [3 ]:
]
le =—
1-P”

where
P= Sri
28 RecewingSystemsDesign

represents the ratio of upper to lower cutoff frequency ratio of the ideal
bandpass filter.

This improvement factor increases as the cube of Pand is shown in Table 2-2. It
is therefore desirable to limit the low frequency of the discriminator where such
information is not present or useful.

Improvement Factor Resulting from Bandpass Filtering

the Output of a Discriminator Detector

Ratio offirin/finax Improvement Factor

0 |
1.001

WH
CMOU
1.008
1.027
1.068
1.142
1.275
1.522
2.049
3.69

2.5.66 Phase Lock FM Demodulation

The phase lock demodulator of Fig. (2-13) is a conventional phase lock loop
(PLL) which is locked to the FM carrier. As the FM carrier is deviated, the
PLL error signal is proportional to the shift of the carrier, and may be used as
the demodulated output. The loop bandwidth must include all modulation
terms of interest. This type of demodulator is readily available in integrated
circuit form, and is readily adaptable for FM demodulator applications.

Limiter
6
FM Demodulated
signal output

Voltage
controlled
oscillator
(vCO)
Fig. 2-13. Phase-lock demodulator for FM signals.

2.5.7 Counter FM Demodulator

The counter FMdemodulator of Fig. (2-14) is noted for its wide bandwidth and
excellent linearity. Unfortunately it is useful only for large deviation applica-
tions because of its low sensitivity. For those applications it is excellent.
Modulation 29

WN UL 2 Naar
aim
signal
FM(a) Limiter]
(b)
| Diterentistor
| (c) ta)ife (e)
output
OA Ao FMwave

yA (c)Differentiator
output
tN aR Nd,tel (d)Rectified
output
or (e)Integrator
output
Recovered modulation
Fig. 2-14. Counter-FM demodulator.

Through limiting, the input FM signal results in a flat-topped waveform,

resembling a square wave. This wave form is differentiated usually by an RC
network, and rectified to preserve only one polarity of the differentiated
waveform. Integration of this signal provides the demodulated output.

2.6 PHASE MODULATION

In phase modulation, the phase of a carrier is varied with a modulation signal.
A carrier of the form:
A sin (wt + >)
can be phase modulated by adding a phase variable controlled by a modulat-
ing signal. This phase variable must be limited to a maximum value A@.Fora
simple sinusoidal modulating signal this phase variable becomes:
Aé cos pt

where
w=2rf
fis the modulating frequency.
A@is in radians
Including this term, we have the phase modulated wave described by:

A sin (wt + A6 cos pt + d,) (2-39)

30 RecewingSystemsDesign

Let
x = wt + d,
and
y = A6 cos wt
Expand
sin (x +y) = sin x cos y + cos x sin y
or
sin (wi + @,) cos (A@cos pt)
+ cos (wt + $,) sin (AO cos pt) (2-40)

and
cos (A@cos wt) = 7, (A@)- 2 [7, (A@)cos 2 pt - F, (A@)cos 4 pt
+ F; (AO) cos 6 pt - Fz (A) cos 8 pt.... J
sin (A@cos ut) = 2 [7, (A@)cos wt - 7; (A@)cos 3 pt + J; (A) cos 5 pt
- J; (46) cos 7 pit...... ] (2-41)

Then
A [sin (wt + ,) [7, (A@) - 2 [7, (AO) cos 2 wt - J, (A) cos 4 wt t+..... ]]
+ cos (wt + ,) 2 [7, (A@)cos ut - J; (AO) cos 3 pt+....]] (2-42)
Where 7, (A@) are Bessel functions of the first kind and (A@) is the argument
given in radians.
Rearranging
A [7, (A6) sin (wt + ¢,)
+ F, (A@)cos (wt + ut + b,) + F, (AG) cos (wt - pt + d,)
- F, (A8) sin (wit+ 2 wt + p,)-F» (AO) sin (wt - 2 ut +¢,)
- F; (A@)cos (wt + 3 pt + b,) - Fz (AG) cos (wt - 3 wt + ®,)
+ 7, (A6) sin (wt + 4 ut + h,) + F, (AG) sin (wt - 4 pt + @,)
RT CY sineco ] (2-43)
The phase modulation process does not add power to the signal but redistrib-
utes the carrier energy in the form of sidebands. In other words, the sum of the
powers in the spectrum is equal to that of the unmodulated carrier.

For a fixed AO, the respective sidebands are fixed in magnitude. As the
modulating frequency is allowed to approach 0, the width of the spectrum
collapses to 0. Thus, for phase modulation the spectral width is directly
proportional to the modulating frequency.
Modulation 3]

2.6.1 Carson’s IF Bandwidth Approximation Applied to

PM Signal Spectral Requirements
Carson’s rule for FM may be applied to PM signals by slightly modifing
equations (2-36) and (2-37). This is done by multiplying byf,,/f,, and substitut-
ing M (the modulation index for PM) for AF/f, (the modulation index for
FM). Thus, the IF bandwidth required to accomodate the significant spectral
sidebands for PM is:
B,=2 (M+1)f,, 1>M>1] (2-44)
and
By= 2 (Mt2)f,,, 2<M<10 (2-45)
where
M is the modulation index for PM

J, is the worst case highest modulating frequency

2.6.2 PM Signal IF Bandwidth Requirements Using

Power Summation of the Side Bands
This determination is identical to that of FM and the reader is referred to
section 2.5.2. The only difference is the argument ofthe Bessel function, where
the peak phase deviation in radians is substituted for the FM index B.

2.7 PHASE SHIFT KEYED SIGNALS (PSK)

As implied by the title, the digital modulation is impressed upon the carrier in
such a manner that the phase ofthe carrier is shifted for a mark or space by a
different fixed amount each time either occurs.
In the simplest form, a mark may cause a shift of 7 radians, while a space may
not shift the carrier at all. Thus, the digital modulation is transmitted as a series
of 0 or 7 shifts in carrier phase. Such waveforms are easily generated using a
common doubly balanced mixer, by applying the digital data to the IF port
and the carrier to the LO port. The PSK signal then appears at the RF port.
This signal has the spectral form of (sin x) /x,or more specifically

sin [(o, - w,) tT¢ |

Fiy=ArT (2-46)
[(w, -w,) >T +0]
where
A is the amplitude of the signal
Tis the bit width
w is the carrier radian frequency
6 is an arbitrary phase
32 ReceivingSystemsDesign

The resulting spectrum and receiver IF signal has the spectral form shown in
Fig. (2-15).

!
“~. i ! ~
oe

5
eee eee ee ee eeee ee ee ee eee eee ee
4 reeetaneene?
*2a
a -2 *
‘ -1 4, rsso 3 -,bey 4

r ~ 7 Eee -: T T
T r T

|
Fig. 2-15. PSK RF and IF signal spectrum envelope.
The majority of the signal is contained in the mainlobe and because of the
masking of the minor lobes by noise near the detection threshold, the IF
bandwidth of the receiver seldom exceeds (2 or 3)/7. And the post-detection
bandwidth is 1 /2 of the IF bandwidth.
There is one important spectral consideration and that is for a symmetrical
square wave modulation. For example in a 10101010.... data pattern, the
carrier is suppressed. In fact for random patterns, this suppression will vary and
as a limit will approach the square wave case.
The phase states in PSK, need not be limited to 2, in fact it is feasible to utilize 2”
discrete states for ncommonly up to 4 (16 phase). Where n = 1, the signal is
referred to as biphase (BPSK) and quadriphase where n = 2 (QPSK).
As n increases, the channel can handle more information, but at a sacrifice in
noise immunity. The number of separate data channels a system can handle is
2"/2. Therefore, BPSK can handle one data stream, QPSK can handle two
data streams and eight phase can handle four data streams et cetera.
Modulation 33

The phase states are usually separated equally as follows, to reduce noise
problems:

n Phase Separation (degrees)

] 2 180
2 4 90
3 8 45
4 16 22.5

The phase relationships for aQPSK system are shown in Fig. (2-16).
90°

180° 0°

270°

Fig. 2-16. Phase relationships for a QPSK system, showing a total of four
phase states for a two-signal data capability.
Other angular relationships are usable and often preferred because diametri-
cally opposed phase data results in carrier nulls, making carrier recovery more
difficult. The demodulator process requires a coherent phase reference because
the data is represented as changes in carrier phase. This necessitates the use of
coherent or product detection techniques. A product detector is a three termi-
nal device. It has a signal port, a signal carrier reference port, and an output
port. Any form of mixer, phase detector, multiplier et cetera qualifies as the
detector. The detector is shown in block form in Fig. (2-17).
34 RecewingSystemsDesign

Product Lowpass
detector filter

WwW
PSK Signal

Wwe
Coherent
reference

Fig. 2-17. Basic PSK detection scheme.

There are three popular detection methods:

Coherent (Fig. (2-17) )
Differentially coherent
Delay line method (Fig. (2-18) )
Multiplier loop reference (Fig. (2-19) )
Of these, the later two methods are the most popular. Coherent techniques offer
an improvement of | to 3 dB in e,/N, for a given error probability, but are more
difficult to implement, since a carrier phase reference is required at the receiver.
The differentially coherent and multiplier loop techniques derive the reference
carrier from the signal. The delay line method is shown in Fig. (2-18) and the
multiplier method in Fig. (2-19). The delay line method, also known as the
Kineplex system, compares the | bit delayed signal against the signal without
any delay. The use of a fixed delay limits the system to a specific bit width,
restricting its applications. It also requires differential encoding at the
transmitter.

PSK
input 1 bitdelay
Fig. 2-18. Differentially coherent detection of PSK signals using a delay of 1
bit.
Modulation 35

Fig. 2-19. PSK signal detection using a multiplier loop to recover the carrier for
biphase modulation systems.

The multiplier technique generates the carrier by multiplying the BPSK

carrier shift of zero and 7 by two, resulting in shifts of zero and 27 = 0. This
strips all modulation and leaves a carrier at twice the frequency. The on
frequency carrier is recovered by dividing by two. This technique does not have
the limitations of the delay line method and may be used with any data rate
within its system bandwidth.
The delay line method cannot process data without differential encoding at the
transmitter. The encoding system is shown in Table 2-3 and Fig. (2-20).

Table 2-3.
Differential Encoding DPSK
Transmitter

Message Greve se. hes) oe Oe eee

Encoding eA, ee TO valve) Ae Wid
Transmitter yr 0 .0 0 QO. mies wien
phase
Receiver
Received 7. 0%. ON O2 Op carrey
ies are Oi Oo)
phase
1 bit delay = ee OO OR i eee Oe ee Oe
Phase detector SA Maete P gangs
(Bae eye ont Bedi!
ReedLae | ova
*Arbitrary start bit
36 ReceivingSystemsDesign

DPSK encoded data to

Data (a)
transmitter

—
i— So
oco--
ao-—-
©
—O
Data clock
Fig. 2-20. Differential data encoding for DPSK and delay line detection.

The data is fed into an exclusive NOR circuit, which has as the second input its
own output, shifted by 1 bit. The truth table shows that when inputs a and b are
alike one output results, and conversely, when they differ, a zero becomes the
output.
Signal derived references are corrupted by noise at low signal levels, resulting
in performance degradation. Of the three systems, the performance rating,
based on error probability for a given e,/N,, in decending order, are: coherent,
multiplier, and delay line. However, the performance difference of all three
systems is within a | to 3 dB window.
The probability of error for the three systems has been defined as follows:

Coherent PSK

P,=> erfcrs (2-47)

Signal
Derived
Reference
PSK
P=> A (2-48)
At large e,/N, the performance is less than 1 dB worse than coherent PSK. For
P.= 10" the curves are 3 dB apart, in favor of coherent detection.

2.8 FREQUENCY SHIFT KEYING (FSK)

In FSK signaling, binary data causes the transmission of one frequency for
mark and a different frequency for space. These frequencies can originate from
two discrete oscillators or from a single oscillator, which is frequency modulat-
ed by the two-state digital modulation. These two methods are shown in Fig.
(2-21).
Modulation Sh

Fig. 2-2la. Two-oscillator FSK transmission.

Time ———~-

Carrier frequency

time ————_—~

Fig. 2-21b. FM’d oscillator or binary FM form of FSK.

Reception of FSK has two possible solutions which are:

non-coherent detection
coherent detection
It will be shown that coherent detection is superior (although seldom used)
because it requires a coherent reference signal in the detection process.
Non-coherent detection systems are easily implemented and consist of two
detection channels. Each channel utilizes a bandpass filter and an ordinary
AM envelope detector. One filter is tuned to the mark frequency and the other
to the space frequency. The detector outputs differ in polarity and are summed
to produce the output. This configuration is shown in Fig. (2-22). Each channel
is in reality a single on/off carrier (OOK) system, both of which take turns as
dicatated by the modulation.
The performance of the noncoherent system of FSK reception is given by:

P=>exp
- (=)! (2-49)
38 ReceivingSystemsDesign

where
P. is the probability of error
e, is the energy per bit
N, is the noise density per unit bandwidth
The solution of P, may be found using (2-49) and a series of solutions may be
secured through the BASIC program of Table 2-4.

Fig. 2-22. Non-coherent FSK reception.

Table 2-4.
Probability of Error vs. E,/N, for Non-coherent FSK
Le" SHER TPR
24 PRINT “PROBABILITY OF ERROF
VS EBANG. FOR /bSK *
34 PRINT
456 PRINT “SERRE
REFRES KEKE KAKATE
be Dt oS &HH &thee
FPRIHT
|ee
can PRINT "ESB-NO"; TABC1S); "PE"
Ce
mu DISP "ENTER EB’NO. COBs., MIN,
MAX, STEP"
$6 IHFUT AL: 6i.,51
96 FOR K=A1 TO.B1 STEP Si
168 El=ie-cK-1Bo
119 Pi=.S34EXPC-¢ SAKTDD
{26 PRIHT EK;TABCIS9;P1
128 NExT kK
idé@ EWO

There are several variations of the detection scheme which offer improvement.
These include weighing the two detector outputs and making a decision based
upon which detector has the largest output. For this variation, both detectors
have like polarity outputs and are fed into a differential comparator. Another
variation involves the use of discriminator detection of the FSK signal.
Modulation 39

2.8.1 FSK Signal Spectrum and Bandwidth Considerations

FSK consists of pulses of RF energy alternating between two frequencies such
that RF output is present at one frequency or the other. The duration of one
such pulse or bit possesses the characteristics typical of any, and also represents
the widest spectral occupancy.
For an example let a mark bit be represented by:
f (t) =A sin (w, t + 6)
where
- T/2<t<r/2 (2-50)
and
A is magnitude
w, is 27 * (mark frequency)
6 is an arbitrary phase angle
Tis the bit width
In the frequency domain this becomes:
sin(w-w,)T/2
F (w) - A r/2 aoa .
(w-w,) 7/2
sin(wtw,) 7/2
+ pHlO-#/2), (2-51)
(w+ w,) 7/2
Thus the spectrum is seen to have a (sinx) /x response as depicted below:

sb
tte tet.+hat het fbb Here. bat

oo nw -
“a| ee HH

Fig. 2-23.
T

Envelope
T T

of FSK
epee
ac:
ginny
34
f
for 1 bit mark
T

of duration
T

T.
T T
40 ReceivingSystemsDesign

There exists a second such spectrum at the space frequency. The total spectral
occupancy of the signal is the sum of the two. In many cases the mark and space
spectrums partially overlap reducing the total bandwidth.
It can be noted from Fig. (2-23) that the majority of the signal energy is
contained in a frequency bandwidth of 2/r MHz, (where frequency is the
reciprocal! of time). The total spectrum of the FSK signal includes two spectra
separated by a guard frequency. In the interest of spectral conservation and
receiver carrier to noise ratio, the guard frequency is made necessarily small,
resulting in spillover of mark and space energy into each others filter band-
widths. As a rule, the guard band is made equal to 2/7 MHz, using mark and
space bits of 7 seconds.
It has been shown that the mark and space filter bandwidths of 1.5/7 are
good choice from a noise and intersymbol interference standpoint [ 4 ].
The RF and IF bandwidths are approximately:
By, 2 (D+fn) (2-52)
where
D is the shift of the carrier from its mean value, and

T-1/%

2.8.2 Coherent FSK

The optimum demodulator utilizes matched filtering. This can be realized by
utilizing product detection (coherent multipliers) where each of the signals for
mark and space are multiplied by a coherent reference, followed by lowpass
filtering. This coherent demodulated output produces a baseband output,
which is processed to determine the dominant output.
The disadvantage of this signal recovery mean is the need of knowledge of the
exact frequency of the mark and space frequencies. This can be accomplished
through the use of phase-locked techniques tuned to the respective frequencies.
The advantage of coherent detection is the improved error probability given
by:

where
erfc is the complimentary error function (see section 4.14)
é, is the energy per bit
N, is the noise density
Modulation 4]

MARK REFERENCE

DECISION
CIRCUIT

| BANDPASS FILTER
SPACE

SPACE REFERENCE

Fig. 2-24. Coherent FSK demodulator block diagram.

This form of demodulator is shown in block form in Fig. (2-24). Because of the
additional complexity, this form of signal recovery is seldom utilized except in
critical cases.

REFERENCES

[1] Panter, P.F., ModulationandNoiseandSpectralAnalysis.New York: McGraw-

Hill, 1965.
[2] Noel, F. and Kolodzey, J., ““Nomograph Shows Bandwidth for Specified
Pulse Shape,”’ Electronics,April 1, 1976, p. 102.
[3] Kay, G.A., “Signal to Noise Ratio in the Band Pass Output of a Discrimi-
nator,” JEEE Transactions on Aerospaceand Electronic Systems, vol. AES-6,
no. 3, p. 340.
[4] Farone, J.N., and J.R.Feldman, AnAnalysisofFrequencyShift KeyingSystems.
Armour Research Foundation Technology Center, Chicago, Illinois.
 30“278
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 NOISE

3.1 EXTERNAL NOISE

A communications link is affected by external noise which degenerates the

performance of that link. It is not fruitful to have a low noise receiver capable of
signal reception of -115 dBm signals in the screen room when it is later placed
in service where the external noise level is -90 dBm. In such a situation a less
costly receiver design could suffice, but the link specifications would not be met
except in the screen room. In such a situation either more transmitting power,
higher antenna gain, or both are the only answer, short of relocation.
There are several sources of noise, including:
Galactic origination outside the earth’s atmosphere
Atmospheric disturbance and storms
Manmade
The characteristics of these noise sources are shown graphically as a function of
frequency in Fig. 3-1. The average noise intensity is shown on a relative basis
referenced to kTB. Considerable global and seasonal variance exists for these
characteristics, and serious concern for these noise sources should involve a
specific site investigation. Generally, it can be concluded that noise diminishes
with increasing frequency and distance from urban environments.
External noise may be referred to the antenna effective temperature by:

tre ep (3-1)
where
T, = 290°K
J, is the effective antenna noise factor
44 | ReceivingSystemsDesign

and
=_ ir
*" ETB (3-2)

where
P, is the noise power at the antenna, loss-free
k = 1.38 +10 joules /°Kelvin
T, = 290° Kelvin
B is the bandwidth in Hz

See reference [1] for more information.

yams.Noles Poe:Ht:
usingOmni Directional|,|.
Antenna
ae aa it aTFAAP
FAC
| (Aad
Go
Lilioil i!ili
Mle ‘|"i

Fig. 3-1. Average noise power using omni-directional antenna [1 ].

3.2 EFFECTIVE NOISE BANDWIDTH

A filter response is often difficult to describe mathematically because of its

non-ideal characteristic. Amore useful convention results when the filter is
idealized by an equivalent rectangular shape of equal area. This is done by
integrating the power response of the filter or circuit. Do not confuse this with
Noise 45

dB or voltage responses, or erroneous results will be obtained. Next, compute

an equivalent area rectangle whose magnitude is equal to the original response.
These two areas, being equal, result in an ideal equivalent bandpass. The
width of this rectangle is the equivalent noise bandwidth. This is also sometimes
referred to as ENB, or simply, noise or ideal bandwidth. As an illustration, a
much used application is the determination of noise power after a filter. Here,
the noise is usually white with a given spectral density (D) in convenient units
such as power in milliwatts or dBm per unit bandwidth. Multiplying the noise
spectral density by the equivalent noise bandwidth gives the noise power.
The response of Fig. (3-2a) is readily measured and integrated by using: the
polar planimeter, the rectangular, the trapazoidal, or Simpson’s rule. Convert-
ing to an equivalent height rectangle gives (3-2b), which is the idealized
equivalent noise bandwidth (f/, -f,) where the heights and areas of both are
equal.

Frequency
Actual filter response (a)

Pmax

3
a
f fr
Frequency
idealized equivalent filter response (b)
Fig. 3-2. An example of an actual filter response and its equivalent rectangular
bandwidth or effective noise bandwidth. Both responses are of equal height and
area.
46 RecewingSystemsDesign

3.3. NOISE FIGURE AND NOISE FACTOR

By definition noise figure is noise factor expressed in dB.
NF = 10 log,,F (3-3)
where
NF is noise figure
F is noise factor

Furthermore,
S/N;
= —
SiAN: 3-4
Sag

where
S; is signal input power applied to the circuit
S, is the signal output power from the circuit
N;, is the input noise power
N, is the output noise power
An input stimulus, consisting of a signal 5; plus associate noise V,, applied toa
circuit will be processed by that circuit. The output which results will consist of
an output signal S, with its associated noise V,. The input ratio S;/N, will not be
equal to the output ratio S,/N,, because ofnoise generated by the circuit itself.
The ratio of these ratios, ($;/V) / (S,/N,), is a measure of the circuit and is
called the noise factor F.

All circuits have gain which may be greater or less than 1. An amplifier would
be an example ofgain greater than 1, whereas a mixer ofthe diode type would
have a conversion loss of typically 6 or 7 dB.
The gain G is the ratio of the output signal divided by the input signal. Or:
S
a =_— 3-5
(3-5)

Substituting into equation (3-4)

S;NV, N
SO)3
aNS
e, Se GN 3-
co)

The output noise, consists of the input noise multiplied by the gain plus the
noise generated by the circuit itself. It is represented by:
N,=GN, +N, (3-7) ~
where
NN,is the noise generated internally in the circuit appearing at its output
terminals
Noise 47

Then
GN, +N, N,
GN, =] + GN, (3-8)

and

N,=kTB (3-9)
where
k isBoltzmann’sconstant and isequal to (1.38044+ 0.00007)10™joules/°
Kelvin
T is the source temperature, usually taken as 290° Kelvin
B is the effective noise bandwidth (Hz)
Equation (3-8) becomes:
Z ppaodes. (3-10)
GkTB
Solving for the circuit noise WV,we have:
N,=(F-1) GkTB (3-11)
The value of B is the equivalent rectangular bandwidth ofarea equal to that of
the device.
From equation (3-6)
N
F=>—GN 3-1]
(3-12)

then
N.
N.F=
Feo — 3-13
(3-13)

Since the output noise ofa circuit divided by gain must be the equivalent noise
input to the circuit, then V;F is the equivalent circuit, input noise power. From
this we have the important relationship:
Nvwio,=NF
iequiv.
=KTBF (3-14)
In receiving applications, kTBF represents the noise power at the input of the
receiver, which the signal level must overcome. If the input signal power is
equal to kTBF then §/N = 1. For computation ease, the log form of kTBF is
suggested as follows:
10 log kT = -204 dBw = -174 dBm (3-15)
where

dBw refers to dB relative to 1 watt

48 ReceivingSystemsDesign

and

dBm to dB relative to 1 milliwatt

Then

WN
=2bn
ne
eal
oueis==
nououw
et
we
had Se
—=EEE
erts
TUTNor
ay
6p
N <<}
fa}
Ae--0--0-
&
This readily shows the importance of minimizing bandwidth to reduce noise.
Fig. (3 -3) relates kKTBin dBm to bandwidth B for quick reference and is of
cfc
sufficient accuracy for most computations.
Example: It is specified that a receiver must process a -97 dBm unmodulated
carrier witha S/N of 10 dB. The receiver bandwidth is given as 50 kHz. What is
the maximum noise figure the receiver can have?

From Fig. (3 3) kTB 127 dBm

Then NF 127- 97 - 10 = 20 dB
A noise figure of 20 dB is the maximum allowable value.

ieee
SOS
ts:
PME
Sted
he
SRG
mE
=e
to
Hiswl
ALES
Sw

Fig. 3-3. Plot of kTB for T=290°K.

 Notse 49

3.4 TEMPERATURE

Noise temperature is related to noise factor (F). This relationship may be

derived by modeling an actual amplifier by using an ideal (noiseless) amplifier
preceded by a summing junction with two inputs. The two inputs are the input
noise and the noise generated within the real amplifier. This is shown below:

Output noise
Input G (kT.B + kT enB)
Noise

kT .B Noiseless
amplifier
Amplifier
noise
kT
errB
Fig. 3-4. Model of an amplifier using a summing junction and an ideal
amplifier plus two noise sources. :

input noise = kTB: amplifier noise = kT,B (3-17)

where
k is Boltzmann’s constant (1.38044 + 0.00007) + 10° 7 watts/° Kelvin
T, is 290° Kelvin (unless specified otherwise)
B is bandwidth in Hz
Tis the effective noise temperature of the amplifier
The amplifier has a gain G, therefore the output noise is:

N,=G(kT, B+ kT 7B) (3-18)

Since noise factor F is the ratio of signal to noise in, to signal to noise out, or
J
S;N, he JN;
Fo— S = ——S
SN; = —
G N; 3-19
ond)

1] (AT, Bt+kT,,B)G
a Caer nana: sa

Ty
F=1+— (3-21)
50 RecewingSystemsDesign

Knowing the noise factor, the effective noise temperature may be found from:

Pray la (3-22)
Note that these relationships are gain independent.
Although an amplifier was used in these derivations, the resulting equations
apply equally well to other devices such as mixers, filters, et cetera.

3.4.1 Cascading Noise Temperatures

The noise factor equation for cascading two stages is:
F,-1
|ICT IRRmn, (3-23)

where
Subscripts 1 and 2 refer to the first and second stages respectively. The
first stage is the input stage.
Substituting equation (3-21) into the above
Te |
, T, (pea
F.=1 He=}+-“ + 1 +—
T, WIe G,
or,
FF
Tate lame pFe (3-24)
G;
Example: A filter with a 3 dB loss (and a 3 dB noise figure) is placed ahead of an
amplifier with an effective noise temperature of 864°K. What is the new overall
noise temperature?
The 3 dB noise figure is converted to effective noise temperature.
290(1.995 -1) = 288.55°K
then
Tp = 288.55 + (864/.501) = 2013°K
This is a noise factor of (using equation (3-21))
F = 1 + (2013/290) = 7.94

The noise figure is:

NF = 10 log,, 7.94 = 8.998 dB
Where more than two stages are involved the stages are taken two at a time and
the process is repeated as often as necessary. This piecewise solution avoids
lengthy equations and allows stage by stage examination.
Noise 51

3.5 CONVERSION NOISE

3.5.1 Ideal Noise Free Signal and Local Oscillator Signals

The ouput noise from a mixer is found from the definition of noise factor.

From
oepee ee Se VE
ES SOWA GN, ETBG we?)
where
F is noise factor
S; is input signal
S, is output signal
kTB is the noise in a bandwidth B
G is gain
The signal output of the mixer 1s: |
S, = GS; (3-26)

Then for the ideal case of Fig. (3-5) we have:

G = -7.5 dB
-10 dBm NF = -7.5 dB -17.5 dBm

120 dBm R xX 112.5 dB

KTB = -130 dBm

17 dBm

-130 dBm

R is the received port

L is the local oscillator port
X is the intermediate frequency port

Fig. 3-5. Ideal signal case of mixer operation where the bandwidth is 25 kHz.

N, = kTBFG = -130 dBm + 7.5 dB - 7.5 dB = -130 dBm

S, = §,G =-10 dBm - 7.5 dB = -17.5 dBm
Then
S/N, = -17.5 dBm -(-130 dBm) = 112.5 dB
Note that the input signal to noise ratio is -10 dBm - (-130 dBm)=120 dB. This
illustrates that kTB noise cannot be attenuated bya loss such as negative gain,
and that the mixer output signal to noise ratio is degraded by the mixer noise
figure.
52 RecewingSystemsDesign

3.5.2 Noisy Received Signal

Let the input received signal of Fig. (3-5) be corrupted by noise such that the
previous signal of -10 dBm now has a noise floor of -90 dBm, all other
conditions remaining the same. This case is shown in Fig. (3-6).
G=-7.5 dB
NF=7.5 dB -17.5 dBm
-10 dBm R x

ihe ee -90 dBm .: -97.5 dBm

Le MPRA oe -130 dBm

B=25 kHz
kTB=-130 dBm

-130 dBm

Fig. 3-6. Mixer output for the case of an ideal local oscillator signal and a
non-ideal received signal.

From (3-25) and (3-26)

S, = -10 dBm -7.5 dB = -17.5 dBm
N, (ideal received signal) = -130 dBm
NV,(noise due to the signal) = -90 dBm + (-7.5 dB) = -97.5 dBm
It can be seen that there are two noise contributors. One being kTBGF and the
other being due to the noise component ofthe signal, multiplied by gain. Since
NV,(noise due to the signal) is dominant, the mixer output noise floor is -97.5
dBm, rather than the -130 dBm of the ideal case.

3.5.3 Noisy Local Oscillator

The effect of a noisy local oscillator signal on a mixer with an ideal received
signal may be found by considering the illustration of Fig. (3-7).
G=-7.5 dB
NF=7.5 dB
-10 dBm R x op bea
: L ofl -97.5 dBm

-130 dBm 17 dBm Aen aan

-63 dBm B = 25 kHz

Fig. 3-7. An illustration of mixer behavior with a noisy local oscillator signal,
and with an ideal received signal.
 Noise 53

From (3-25) and (3-26)

S, = -10 dBm - 7.5 dB = -17.5 dBm
N, (noise due to signal) = -130 dBm + 7.5 dB - 7.5 dB = -130 dBm

Because the LO signal is not noise free, the local oscillator signal may be
considered to be the sum of many LO signals. Therefore, conversion of the
received signal would be expected over the whole of the LO signal. The result is
a mixer output whose spectral profile is an emulation of the LO signal. The
result ofthis is a raising of V, to the S/N of the LO, which in this case is 80 dB.
Once again there exist two noise sources, at the output port, of -130 dBm
(ideal) and -97.5 dB, due to the LO noise (the latter dominates). A second
consequence is the conversion of unwanted signals near the desired received
signal by the broad noisy LO signal.
The effective noise figure of the mixer in this example becomes (using equation
(3-25)):
F= N./GN,
= -97.5 dBm -(-7.5 dBm)-(-130 dBm)
= 40 dB
Note, as the level of the input signal S;diminishes, the mixer output signal noise
floor also moves down GB for dB and eventually the LO noise floor no longer
affects the output signal, which becomes dominated by kTBFG. Also, the
effective noise figure of the mixer approaches that of the mixer with ideal
signals. ‘Thus low level operation of a mixer is not affected by LO noise. This
bold statement must be qualified by requiring the LO signal be band limited to
exclude any frequencies at the IF frequency. This prevents LO noise from
entering the IF output of the mixer through the mixer’s L to X port leakage.

3.6 NOISE MEASUREMENT TECHNIQUES

There are several methods of noise figure measurement. All are based on the
injection of a known amount of noise into a receiver and observing the receiver
output behavior. The source of this noise is almost universally white and is
thermal noise. Thermal noise is a result of random electron motion. This noise
exhibits a uniform energy distribution in the frequency domain and therefore
has a constant power density with a normal or Gaussian level distribution.

A noise source should be capable of producing a significant noise power

exceeding that of the device being measured. There are several popular noise
sources, including: forward biased semiconductor diodes, temperature limited
diodes, gas discharge tubes, and hot / cold resistive sources. The sources most
often used are the diode and gas discharge types. They are useful to several
hundreds of MHz and tens of GHz, respectively.
54 ReceivingSystemsDesign

The temperature limited diode noise power may be predicted from:

P,=kTB+ (elBR) /2 (3-27)

where
kTB is termination resistance noise power for T = 290°K
e is the charge of an electron (1.59 - 10° coulomb)
I is the average diode current in amperes
B is the effective noise bandwidth in Hz

The noise power of the gas discharge source is:

P,=kTB+k(T,- T) B, (3-28)
where
B, is the overall source bandwidth
T, is the effective temperature of the gas discharge in degrees Kelvin
All other terms have been previously defined.

Noise factor may be computed from:

rT
————_- - ]
290° Kelvin (constant)
if SE Y-] Y-] oe
3-
where

T, is the effective temperature of the noise source

Y is the ratio of the output power of the device under test (with the noise
source connected to the input) to the output of the device, with the input
properly terminated at 290°Kelvin.
The only variable in the above equation is Y. Since the noise source is term-
inated with a load equal to that of the device under test,

prey ean
Y= P. off (3-30)

and
Pon P,on- P, off
Y-l= P, off -]| = — P, off eo-3)

From this, noise factor may be measured by measuring Y-1 at the test unit’s
output. The measurement of noise power must be made using a true rms meter,
such as one using a bolometer or barretter detector.
Automatic noise figure measurement is made by gating the noise source on and
off, and measuring the noise power at the output of the device under test witha
meter calibrated directly in noise figure. The implementation ofan automated
measurement system is shown in Fig. (3-8).
Noise 55

Noise source

Square
wave
source

Fig. 3-8. Automatic noise figure measurement system.

3.6.1 Y Factor Method

One of the most accurate methods of noise figure measurement is the Y factor
methods of Fig. (3-9).

Amplifier
detector
and meter

Termination
P,, off

Fig. 3-9. Y factor method of noise figure measurement.

The attenuator is adjusted to provide the same output for P, on and P, off. The
differential value of attenuation required to do this is 7.Then
F = 10 log (( T,/290° Kelvin ) -1 ) -10 log ( Y-1) (3-32)

where Y is a ratio.
This method provides accuracies of .1 to .2 d#.

3.6.2 3 dB Method

The 3 dB method determines noise figure by measuring the output power of the
unit under test at properly terminated conditions, as shown in Fig. (3-10).

Fae (3-33)
The noise source is connected and a 3 dB attenuator is inserted before the power
meter. The noise attenuator A, is adjusted to provide a reading equal to that
obtained earlier. The noise power into the receiver, or P, -A,, is equal tokT BF.
Measurement equipment is generally of fixed frequency and requires conver-
sion to measure noise figure at other frequencies. Since the noise sources are
broad band, a mixer, when tested for noise figure, will produce erroneous
results because the image noise power will add to the desired noise power and
provide double sideband noise figures. This may be corrected for by adding 3
dB to the number obtained, which results in the single sideband value.
“56 RecewingSystemsDesign

When using mixers in the test setup, it is vital that the local oscillators have
spectral purity or the results obtained will be erroneous.

ena Punit
teat| 3;
Step 1. The output
terminated.
power is monaedT es with the unit under test properly

Step 2. The noise source is connected and the 3 dB attenuator is inserted before
the power meter. The attenuator is adjusted to produce an indication equal to
that of step 1.

Fig. 3-10. The 3 dB method of noise figure measurement.

REFERENCE

[1] World Distribution and Characteristicsof AtomosphericRadio Noise, CCIR Re-

port 322, 10th Plenary Assembly, Geneva, Switzerland, 1963.
 THE RECEIVER

While there are several forms of receiving systems, none are more widely used
than the superheterodyne. This chapter deals with the various forms of the
superheterodyne, the constraints and considerations of the system’s critical
functional blocks, and the characteristics by which the system’s performance is
measured.

4.1 THE SUPERHETERODYNE

Practically all receivers in use today are of the superheterodyne configuration.

As the name implies, a heterodyne or mixing process is involved. In this process
the input signal is mixed with the output of an oscillator to produce an output
frequency.
Mathematically this process is described by:
IF =| f,+f,| (4-1)
where
IF is the output or intermediate frequency
f, is the received frequency
J, is the local oscillator frequency
The process results in principal output terms at two IF frequencies. These terms
are the absolute value of the sum (+) and the difference (-) of f, and f,. In
practice only one is used and the desired output frequency is selected by a filter.
The other is called an image frequency.
The major advantage of the superheterodyne configuration is that the bulk of
the receiver’s gain is placed at a single intermediate frequency (IF), rather than
at a variable frequency as used in early tuned radio frequency receiving
systems. The output of the IF amplifier is followed by detection and
amplification.
58 RecewingSystemsDesign

4.1.1 Configurations
The superheterodyne has two basic forms which are:
down converter and
up converter

Each ofthese configurations may involve one to three or more conversions. As

will be shown, the down converter can be accomplished usually in one conver-
sion, but up converters utilize at least two in the usual case.

The basic superheterodyne configuration is shown in Fig. (4-1).

Intermediate
frequency
amplifier

oscillator
1
baseband output
Fig. 4-1 The basic superheterodyne configuration.
The input signal f, is selected by a filter and fed to a mixer (M1) where it is
mixed with the local oscillator signal (/;), to produce the intermediate frequen-
cy (IF). The output of the IF amplifier is detected to recover the modulation or
baseband signal which is amplified and furnished as an output.

4.1.1.1 Down Converter

The down converter was the original realization of the superheterodyne receiv-
ing configuration, and it is almost exclusively used in home entertainment
products, as well as in some high performance designs. The limit to its useful-
ness is preselection difficulties in wide band systems, because the preselector
must be tunable. Tunability of the preselector also becomes a problem at the
higher frequencies.

Tunable Pre-
preselector amplifier

88 to 108 MHz

Fig. 4-2. Basic down converter configuration.

The Receiver 59

An example of a down converter design is shown in Fig. (4-2). This represents a

typical FMhome entertainment receiver. The input tuning range is 88 to 108
MHz and the intermediate frequency is 10.7 MHz. The advantage of down
conversion is low cost and simplicity. However, it is not always the best choice.
4.1.1.2 Up Converter
By definition up conversion is the conversion of a frequency or band of
frequencies to a higher intermediate frequency. This process results when the
intermediate frequency is greater than the received frequency. As an example,
a 2 to 30 MHz receiver using an IF of 120 MHz represents an up conversion
configuration.
Up conversion is the trend in modern wide band high-performance systems.
Hardware and component breakthroughs make it practical well into the GHz
range. An example of up conversion is shown in Fig. (4-3).

F, Preselector Pre- 2nd IF

90 to Amplifier Mxr 1.5 GHz Mxr 30 MHz
1200 MHz

IF>F,
Detector

Fig. 4-3. An illustration of an up converter receiving system.

4.1.1.3 The Wadley Drift Canceling Local Oscillator System

Consider the up conversion receiver of Fig. (4-4). An input signal in the range of
30 to 400 MHz is converted by mixer M1 to a first IF of 1300 MHz, by a
variable synthesized first local oscillator F;,, whose frequency covers the range
of 1330 to 1700 MHz. A second conversion to the second intermediate frequen-
cy of 110 MHz is provided by mixer M2 and a second local oscillator F;,.,
operating at a frequency of 1190MHz. The first local oscillator being synthe-
sized is based on a temperature compensated crystal oscillator (TCXO), whose
accuracy is 1 ppm (part per million). The frequency error of F;,, can then be
expected to be as high as 1700-10" + 1+10° or 1700 Hz. Assuming that the cost
is a major consideration, it would then be desirable not to unduly complicate
the design by deriving the second local oscillator from the TCXO witha second
phase-locked loop. Using a crude oscillator, for the second local oscillator, a
drift of 25ppm is expected. This results in a maximum frequency error ofthe
second local oscillator of 25 «1190= 29,750 Hz. The total maximum error of the
receiver is 29,750 + 1,700 = 31,450 Hz. Errors of such magnitude can be an
appreciable part of the receiver bandwidth. Should the final IF bandwidth be
60
ReceivingSystemsDesign

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25 kHz and the receiver tuned toa particular frequency; a desired sign, even
though present in the RF spectrum, would not be received. One solution is to
phase lock the second local oscillator to the TCXO. This reduces the second
local oscillator error to 1.19 kHz, but at considerable expense.

A second solution is the Wadley drift canceling local oscillator system. This is
shown in Fig. (4-5). The system is the same as that ofFig. (4-4), except the first
local oscillator frequency has been reduced to 140 to 510 MHz and is mixed
with the second local oscillator frequency of 1190 MHz in mixer M,. The
output of this mixer, as before, is the sum of the two inputs or 1330 to 1700
MHz. The drift canceling of the second local oscillator results as follows:

The first local oscillator frequency is:

Fi, re Fin2+ F, (4-2)

The first intermediate frequency is

IF, = F,, - F, (4-3)

The second intermediate frequency is
IF, = IF, - F,,2 (4-4)

Combining,

IF, = (Fi —FE)—\Fi

IF, = Fg +tFy- F, - F,,2 (4-5)
IF, = F,,- F.

It can be seen that the second intermediate frequency (IF,) is completely

independent ofthe frequency of the second local oscillator (IF,,). Clearly any
frequency errors resulting from the second local oscillator are cancelled com-
pletely. There is a limit to the allowable error of the second local oscillator,
which is defined by the first intermediate frequency filter bandwidth. This
bandwidth must accommodate: its own drift, the desired signals bandwidth,
the frequency error of the incoming signal and the errors ofthe first and second
local oscillators. This is seldom a problem, but it should be considered in the
design.

Note, the tuning range of the two systems synthesized local oscillators, differ.
The conventional approach tuned over a ratio of 1700/1330 = 1.278, while for
The Recewer 63

the Wadley system this ratio is 510/140 = 3.64. While the 1.278 ratio can be
accommodated by a single voltage tuned oscillator (VTO), the 3.64 tuning
ratio requires switching between several VTOs. This results in additional
complexity.

The number of VTOs can be reduced by allowing the Wadley mixer M, to

operate in both the sum and difference modes. The design modifications are as
follows:

Make the second intermediate frequency equal to the receiver’s tuning range
divided by four, or in this example (400-30) /4 = 92.5 MHz.
The first mixer is allowed to operate in both the sum mode for the lower half of
the receiver tuning range and in the difference mode for the upper half of that

range.

Example:
30 to 215 MHz
Mixer ™, operates in the sum mode
215 to 400 MHz
Mixer M, operates in the difference mode.

The first local oscillator frequencies are:

Band F (MHz) F,,, (MHz)

] 30 to 215 1270 to 1085
2 215 to 400 1515to 1700

The second local oscillator frequency is the sum of the first and second inter-
mediate frequencies.

Example: 1300 + 92.5 = 1392.5 MHz = F,,

The VTO frequencies are:
Band b Fiz ~ Fi
or 1392.5 - 1270 to 1085 = 122.5 to 307.5 MHz

Band 2 F,, - Fi.

0.

1515 to 1700 - 1392.5 = 122.5 to 307.5 MHz

The revised system is shown in Fig. (4-6). Note a trade off between the number
of VTOs (reduced tuning range) and filters was made.
64 RecewingSystemsDesign

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 The Receiver 65

4.2 DIRECT CONVERSION RECEIVERS

This is a special class of receiver which employs superhetrodyne principles but

utilizes a zero IF. The local oscillator frequency is made equal to that of the
signal.
Since IF = F, + F, for a superheterodyne receiver and for a direct conversion
receiver IF = 0, then:
O=-F-F, (4-6)

or
FP, (4-7)
where
IF is the intermediate frequency
F, is the signal frequency
F,, is the local oscillator frequency
The implementation of such a receiver is shown in Fig. (4-7).

Local oscillator

Fig. 4-7. Basic direct conversion receiver.

The signal from the RF amplifier is fed to a mixer (or product detector) whose
local oscillator frequency is made equal to that of the incoming signal. The
output of the mixer is the demodulated RF input signal and does not require
any other detection. A simple lowpass filter of the RC variety provides the
required IF selectivity. The cutoff frequency is slightly above the highest
demodulated frequency of interest. This baseband signal is amplified and
provided to the output terminals.
Such low cost simple receivers are generally not used because of several serious
limitations:

The local oscillator stability requirements restrict its usefulness to fre-

quencies below 20 MHz.
They can demodulate forms of AM signals only such as on/off keying,
SSB, and AM DSB. The latter can only be demodulated if care in tuning
is exercised. The detection of FM and PM is not feasible using direct
conversion techniques.
66 ReceivingSystemsDesign

Direct conversion must not be used without RF amplification in order to

reduce local oscillator levels present at the receiver’s antenna terminals
to tolerable limits.

4.3 THE RECEIVER RF AND IF GAIN BUDGET

The receiver detector serves as a convenient point of reference for the deter-
mination of the gain distribution of the receiver. After selecting the input signal
requirements of the detector, the combined RF and IF gain needed to amplify
the input signal at the receiver antenna terminals may be computed.
For diode detection, an input power level of -10 to 0 dBm is generally
adequate. For high sensitivity receivers where IF gain becomes critically high,
-10 dBm is a good choice.
For example, if the minimum signal power is -98 dBm, and a detector input
power level of - 10 dBm, allowing a 10 dB margin, the required combined RF
and IF gain must be:
(-98 - (-10) -10) = 98 dB
This value of gain consists of the total sum ofall gains and losses between the
detector and the antenna terminals of the receiver. This gain will be split
between the signal frequency and the intermediate frequency (or frequencies),
with the bulk of the gain being contained in the latter. Depending upon the
circuit designer’s skill, the gain at any one single frequency should not exceed
100 dB. Where greater gain is required, it is preferred to split it between several
amplifiers at different frequencies. If the required gain at any signal IF is found
to be unmanageable, an extra conversion to another frequency is usually a
good solution.

4.4 PRESELECTOR REQUIREMENTS

The preselector must reduce the receiver’s response to the image frequency,
which is twice the intermediate frequency away from the desired frequency.
The preselector must also reduce the radiation of the receiver’s local oscillator
out of the receiving antenna (which is only the intermediate frequency away
from the tuned receiver frequency). Additionally, it must reduce the receiver’s
response to the intermediate frequency and its subharmonics, as well as provid-
ing sufficient rejection to spurious responses. Such responses are caused by the
mixing process, while not attenuating the desired input frequency. The specific
needs of preselection differ between up and down conversions, as explained in
the following sections.

4.4.1 Preselection Requirements for Down Converters

Because a down converter uses as the first IF a frequency lower than that being
received, wide band receivers using a single conversion usually require a
The Receiver 67

tunable preselector. Where multiple down conversions are used, such as for
UHF, VHF or microwave bands, or for narrow reception band applications,
the design becomes a border line case. In such cases a complete performance
analysis must be made. This includes the response of the receiver to unwanted
signals such as spurious response images and IF rejection, et cetera.
Tunable preselection is usually required in down conversion applications. This
is a result of the fact that the LO and image frequencies are usually in band.
Because the IF must be out of band, IF rejection is not a serious problem. The
subharmonics of the received signal, mixing with the fundamental of the LO, is
a serious problem.
Consider a receiver operating in the 30 to 200 range and down conversion is
selected, although this may not be the best solution. The first IF must be less
than 30 to be out of band. A popular frequency where IF filters are readily
available is 21.4 MHz. Using this frequency, the LO frequency range is
computed using high side injection (LO above signal).
LO = F. + IF , (4-8)
where
F is the received frequency
then
LO = 30 + 21.4 to 200 + 21.4
= 51.4 to 221.4 MHz

It can be seen that the LO is in band over a considerable portion of the receiver
tuning range. It must not be allowed to radiate out through the receiver
antenna causing interference to other receivers and perhaps, in some Cases,
allowing direction finders to locate the receiver.

The image frequencies (in this case) are two times (the IF) above the receiver
frequency, and may be found from:
Poe = fet 21 (4-9)
or
Frage= LO + IF (4-10)
from (4-10)
Foage = 51.4 + 21.4 to 221.4 + 21.4 MHz

which is largely in band.

Low order subharmonics of F,, mixing with the LO, will cause serious spurious
products and must be attenuated by the preselector. This applies also to the
case where the IF subharmonics fall into the preselector passband. ‘The most
serious case results at the 1 /2IF and the IF itself.
68 ReceiwingSystemsDesign

The summary of frequencies which must be at the preselector’s ultimate

attenuation zone are (where n is an integer):
Intermediate Frequency / n
Received Frequency / n
Local Oscillator Frequency
Image Frequency
It can be seen that the IF determines the width of the preselector filter’s
ultimate attenuation bandwidth, as does the received frequency.

Considering the IF alone, the preselection type is a function offrequency. Low

IF's are best filtered by tunable means. Higher frequencies are filterable by
either fixed tuned or tunable filters, except at high RF input signals, where the
fixed tuned filter is best.
It is most economical to utilize tuned preselection which tracks the receiver
tuning. Tuned preselection is jimited in tuning range to less than 3 to 1;
therefore several tuned preselectors are required. The number of preselection
bands would be:
Pi = ee) (4-11)
then
30 K"= 200
K =\/6.666, where K < 3
n = 2 gives K = 2.58
where

K is the tuning range

n is the number of filters

Therefore two to three switched tunable preselection filters are required. While
some designs use mechanical tuning of capacitance or inductance, the trend, in
modern equipment, is to use varactor or digital tuned electronic tuning.

4.4.2 Preselection Requirements for Up Converters

Because of a high intermediate frequency, the up converter has the advantage
of simpler preselection requirements. The first constraint is the m = 1 andn = 2
spurious responses which for a doubly balanced mixer is typically -67 to -83
dBc. This 1 by 2 spur is the desired receiver frequency divided by 2; mixing with
the fundamental of the local oscillator. Should -67 to -83 dBc be considered
satisfactory the designer must consider the | by 3 case, where the response is
typically -49 to 90 dBc, depending on the mixer used and the local oscillator
power. In either case, the preselector will have to make up any additive
attenuation plus margin. If the 1 by 2 case is the limiting case, the preselector
The Receiver 69

filter bandwidth must be less than 2 to 1 (or less than 3 to 1 for the 1 by 3 case). A
simple fixed tuned filter will usually suffice for upconversion receiver designs of
high performance. A value of 1.5 to 1 is generally sufficient for n = 4, 0.1 dB
ripple Chebyshev preselector filtering. Ifa more detailed analysis is desired; the
attenuation at the high end ofthe band to the response at | /2 of this value can
be computed and added to the value given in a mixer spur table. A similar
procedure is used if the 1 by 3 case is the limit. When the | by 2 case is the limit,
the 1 by 3 case is automatically satisfied.
For example, consider a receiving system which covers a tuning range of 90 to
450 MHzand the one by two case is the limit. Here, the number offixed tuned
preselector filters is computed from
(frrin)K” =fax Where K * 1.5 (4-12)
and
n is the integer number of filters
Then
KL
foul frin (4-13)
In thisexample,
K=x/450/ 90=x/5_
fromwhichK= 1.49ifn=4
Therefore four filters are required and are defined as follows:

Band MHz.

] 90 to 1.49 +»90 = 90 to 134

2 134 to 1.49 + 134 = 134 to 200
3 200 to 1.49 + 200 = 200 to 298

4 298 to 450

Local oscillator radiation supression by the preselector for up conversion is

generally limited by the ultimate attenuation of the preselector. As in the case of
the first IF frequency, being reasonably higher than the highest frequency to be
received. The attenuation of the local oscillator frequency by the preselector
must be examined on an individual basis to determine the effectiveness of the
preselector.
The IF rejection of the preselector is usually limited by the highest preselector
band filter. The filter characteristic must be examined at the IF to determine its
attenuation, which should be at the filter’s ultimate value. It is not uncommon
to add more sections to the filter to achieve this goal.
Image rejection is usually not a problem because of the high first IF. To verify
70 ReceivingSystemsDesign

this, the designer should compute the image frequency band and compare it to
the preselector filter characteristics. The preselector should present its ultimate
attenuation to the image frequencies.
The spurious response of the receiver should be examined on a band by band
basis. The fact that the LO and IF are high results and that most of the
troublesome spurs are out of band, they therefore suffer the ultimate attenua-
tion of the preselector.

Most preselectors offer only about 50 or 60 dB isolation between bands because

of switching imperfection which effectively limit the ultimate attenuation of the
filters to that value. It is usually found necessary to add an additional lowpass
filter, to secure the desired level of performance.
An example of an up converter is shown in Fig. (4-8).

90 to
134 Frequencies in MHz
134 to
200
200to bes lowpass
filter to mixer
[LLPe 450
\ 450 \

Fig. 4-8. A typical up converter preselector configuration.

4.5 THE NEED FOR AN OUT OF BAND

INTERMEDIATE FREQUENCY

The conversion process where the signal frequency is changed to the interme-
diate frequency is described by:
F,= |F,+F,| (4-14)
where
F;, is the intermediate frequency and is greater than 0
F is the receiver frequency
Ff,is the local oscillator frequency
Note: The associated sign may be either + or —but not both for any single
frequency. Ifthe received signal was equal to that of the intermediate frequency
amplifier then:
F,= F,
The Receiver 71

and
F,= |¥,+F,| =0,2F,
It is not practical to let F,= 0 inany tunable re¢eiving system. The reason being,
at some other value of F, a non-zero value of F, results, and the required local
oscillator tuning ratio becomes:
R = F, / 0, which is infinity and is not realizable
Further, a mixer operated with F,= 0 is no longer a mixer but a switch, which is
on all of the time and may as well be deleted. The receiver then reduces to a
tuned amplifier, followed by a detector, which is suitable for fixed frequency
operation. The exception to this discussion is the direct conversion receiver
which uses a 0 IF value described elsewhere.
Considering the case where F, = 2F,, there are two signal outputs from the
mixer at F,; one being the converted signal and the other the mixer leakage
signal, which is unconverted. The levels of these signals for a good quality
double balanced mixer are -6 dB and -20 to 25 dB, respectively. Because of
s,ight frequency inaccuracies of the transmitted frequency and the receiver
local oscillator frequency, F, will not be exactly equal to F,,. Therefore a very
serious heterodyne results.
A ratio of 0.5 between the received frequency and the local oscillator should be
avoided, because of the very serious multiplicity of spurious products created
with this ratio (see section 4.27). Therefore, it is good design practice to avoid
having an intermediate frequency which is included within the tuning range of
the receiver.

4.6 THE LOCAL OSCILLATOR FREQUENCY

The local oscillator frequency F,, is dependent upon the input frequency to the
mixer. The intermediate frequency i¥ as defined by:
F,= |IF+F] (4-15)
The sign is dependent upon the choice of the sum or difference mixing. The
difference mode is always employed for down converters, with the local oscilla-
tor being either above or beiow the received signal. In up conversion, either
sum or difference mixing may be employed.
It is mandatory that the local oscillator (LO) low order harmonics must not be
equal to the first IF. If equal, serious beats or birdies will result and AGC
take-over may desensitize the receiver at these points.

4.7 AUTOMATIC GAIN CONTROL

Where the modulated signal is represented by variations in signal magnitude

such as: amplitude modulated carriers, double sideband suppressed carriers,
72 RecewingSystemsDesign

and single sideband suppressed carrier signals, the amplitude of the signal
components must be retained and linear amplification must be utilized. Fail-
ure to maintain the amplitude variations through the receiver will result in
distortion or total loss of the modulation.
Because the signal must be amplified 100 dB or more, and signal strength may
vary equally, it is not possible for an amplifier to cope with the situation for all
cases. To illustrate, assume an amplifier was designed to process a signal of -93
dBm and has a 100 dB gain in order to provide the required output level of 7
dBm or 1/2 volt. The amplifier power supply is 12 volts dc. Should a strong
signal of 0dBm be encountered, the output would have to be 0 dBm +100 dB or
100 dBm, which is in excess of 10,000 volts. If it were possible to accomplish
such a feat, an output level control would have to be adjusted every time a
signal level changed, due to fading or a different signal with a different signal
level.
In the usual case the output of an amplifier is limited by its power supply
voltage. An input signal level which causes the last stage of the amplifier to
become non-linear, because it can no longer duplicate the input signal varia-
tions, results in distortion caused by flattening out the signal peaks.
As the input signal level is increased the distortion increases until all signal
amplitude variations are totally lost, and the intelligence to be transmitted is no
longer recoverable. This effect is often called compression blocking or limiting,
and is undesirable for amplitude modulated signals.
A technique which can overcome this problem does so by automatically
adjusting the gain of the amplifier directly proportional to that of the signal
strength. This technique is called Automatic Gain Control or AGC. The gain
of the receiver is controlled by the signal strength at the receiver output.
Basically, the output of the detector is compared toa reference. Any differential
is amplified and fed back to the previous stages as a control signal to vary the
gain of the receiver such that the output is constant over the range of expected
signal intensity. This is illustrated in Fig. (4-9).

Pre-amplifier IF amplifier

Amplifier

Reference
Fig. 4-9. An illustration of automatic gain control of the receiver.
The Receiver 73

4.7.1 Determining the AGC Control Range

To determine the dynamic range of the AGC system it is necessary to establish:
the minimum signal power level (S.,,,) and
the maximum signal power level (5,,,,)
The ACG control (A ,,,.) is therefore
Sets
max Taki
min Quce (4-16)

4.7.2 Attenuation AGC

The use of an attenuator for AGC overcomes the problem of dynamic range
variation due to operating point shift of the devices used in the amplifier. This
shift of operating point from high gain to low gain in either the forward or
reverse AGC modes reduces the output swing of the amplifier for strong signal
cases. While this is of noconsequence in most cases, it is ofvital concern to the
designer striving for larger values of instantaneous dynamic range. For the
latter case, an amplifier of fixed gain is used with one or more voltage variable
attenuators. The location of such attenuators requires careful planning. The
basic concept of attenuation AGC is shown in Fig. (4-10).

Fig. 4-10. Basic concept of AGCusing attenuation provided by voltage varia-

ble attenuators.
The attenuator is placed ahead of the RF-IF amplifier to prevent the input
signal from ever saturating the amplifier. The AGC detector provides an
output voltage proportional to signal strength, which is fed to a DC differential
amplifier. The second input of the amplifier is provided with a reference voltage
which establishes the output of the detector. The dc amplifier’s output is
lowpass filtered with a time constant long enough to prevent instability of the
loop and destruction of the low modulation frequencies of the signal, which the
AGC voltage must not be allowed to follow. This filtered voltage is the AGC
voltage, whose magnitude is related to signal strength. It is applied to the
attenuator to control its attenuation, providing attenuation directly propor-
tional to signal strength. The result is the amplifier input voltage is nearly
constant with a flatness depending upen the AGC loop gain.
74 RecewingSystemsDesign

While Fig. (4-10) illustrates the concept, it is not the most desirable implemen-
tation. Since the signal input power to the amplifier is held constant, and the
noise generated in the amplifier is constant, the output signal to noise ratio is
also constant, regardless of signal strength. This is unacceptable except for low
quality links.
By distributing the attenuation within the amplifier in two or more blocks, the
overall noise figure, which with the configuration of Fig. (4-10) increased
directly with the signal power dB for dBm, can be buffered by gain and held
nearly fixed. The output signal to noise ratio can be made to increase with
input signal power. Fig. (4-11) illustrates this preferred arrangement.

Fig. 4-11. Preferred distributed form of attenuation AGC.

The gainG, must be such that the gain maximum times signal product must be
less than the compression point of the amplifier G,.The same consideration
applies to amplifiers G, and G;. The distribution of attenuation must satisfy the
following rules:
S;max+ G; ri A = Si : (4-17)

Sta tG.ek eee | (4-18)

S, +0, + GBS, SSS (4-19)

or generally
Suvt a, + Gis 7 S, n+l= S, “an (4-20)
where
S', is the maximum input power to the first stage (dBm)
G, is the gain of the input amplifier (dB)
S,, is the output signal power from the first stage (dBm)
S,,' is the -1 dB compression point of the first stage
G, is the gain of the second amplifier (dB)
a, is the maximum attenuation of the first attenuator (dB)
n is the nth stage, et cetera
The Receiver 75

To result in the lowest noise output or maximum quieting with increasing

signal strength, the AGC should be applied beginning with the attenuator
closest to the output. The sequence of attenuation call up should be @,...... Q;,
@,,a,. This is accomplished by introducing delay in the AGC buss to the earlier
attenuators. Generally two to three attenuation sections are adequate for most
requirements. This procedure maintains the lowest overall noise figure with
increasing signal power and it prevents the compression of any stage.

4.7.3 Fast Attack/Slow Decay AGC

The. fast attack /slow decay AGC from involves circuit modifications, which
cause the AGC voltage to build up rapidly with the application ofa signal and
down slowly upon removal of the signal. This form of AGC is generally used
with signal sideband suppressed carrier signals. In special cases it is used with
conventional AM.
There are many ways of implementing this form of control. Some of the more
popular forms are shown in Fig. (4-12). Essentially, the time constant of the
normal lowpass AGC filter is replaced with a dual time constant lowpass filter,
where the charge time (t,) is less than the discharge time (t,).
R To R; To
controlled controlled
AGC voltage C stages AGC voltage C 2R2_ stages

(a) Normal AGC (b) Fast attack slow decay

t=t AGC R2>R;
Re
ran To
AGC voltage controlled
iP stages

(c) Fastattack/slowdecayAGCR2> R:”

Fig. 4-12 AGC filter forms: (a) normal AGC low pass filter used in average
AGC system; (b) and (c) fast attack /slow decay AGC filter forms.
Since AGC utilizes negative feedback, it is important to choose the attack time
constant so that the desired baseband signals do not appear on the AGC buss. If
they do, they will be removed or mutilated at the receiver’s output.
Because of differing attack and decay time constants, the system tends toward
peak detection of the signal and is no longer an averaging AGC (which refers to
the carrier and not the modulation peaks). Fast attack/slow decay AGC
establishes its reference to the signal peaks; thus, this form of AGC is amplitude
modulation dependent. Suppressed carrier systems have little or no carrier but
contain sideband energy whenever modulation is present. At all other times,
76 RecewingSystemsDesign

for all practical purposes, there is no carrier. Thus, fast attack /slowdecay AGC
is mandatory for such applications. It is optional for all other forms.
4.8 SENSITIVITY

Sensitivity is a measure of how weak a signal can be and still be received

satisfactorily. The limit to the sensitivity ofa receiver is noise, both external and
internal. Since external or site noise is beyond the control of the designer, only
internal receiver noise is considered in the sensitivity specification. The usual
forms of specifications for this parameter are given as:
n, microvolts input for an, (dB) S/N
n, microvolts input for a n, (dB) (S +N) /N
n, microvolts input for an, (dB) SINAD.
In some cases the input signal strength may be given in dBm rather than
microvolts.
Where

S/N is the output signal to noise power ratio.

(S+N) /N is the.output signal plus noise to noise ratio.
SINAD is an abbreviated form ofthe output signal plus noise and distortion to
noise and distortion power ratio. The unit dBm is the signal power level relative
to 1 milliwatt. It is also necessary to specify the following:
receiver input impedance (usually 502)
modulating frequency
amount of modulation
type of modulation
S/N and (S + N)/N are related by:
StN S
eras W +1.(power
ratios) (4-21)
Table 4-1 tabulates this relationship for convenience, and a graph of this
relationship is shown in Fig. (4-13).
4.8.1 Measuring Sensitivity Given S/N or (S +N)/N
A test setup suitable for the measurement of sensitivity is shown in Fig. (4-14).
The signal is tuned to the receiver frequency of interest and is modulated at a
frequency and degree given in the test specifications. The modulation is turned
on and off while the output of the receiver is measured on a true RMS
voltmeter. The RF signal amplitude of the signal generator is adjusted until the
required (§ +.) /N ratio is observed.
Note that the true RMS voltmeter measures (5 +.) /N which if required must
be converted to S/N. Also it is mandatory that a true RMS meter is employed
for this measurement, to give an accurate noise component reading.
The Receiver 77

Table 4-1.
S/N as Related to (S§+ N)/N
S/N, dB S/N, ratio (S +N)/N, ratio (S +N) /N, dB

] 1.25 2.25 3.52

2 1.58 2.58 4.12
3 1.99 2.99 4.76
4 2.51 3.51 5.45
5 3.16 4.16 6.19
6 3.98 4.98 6.97
7 5.01 6.01 7.79
8 6.3 7k 8.63
9 7.94 8.94 9.51
10 10 111 10.41
15 31.62 32.62 15.13
20 100 101° 20.04

==eocaee
ae= Seg=eases7
Spas es
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= = =

Fig. 4-13. Conversion of (S+N )/N to S/N, dB and conversely.

78 RecewingSystemsDesign

TRUE RMS
METER

Fig. 4-14. Sensitivity test set up given S/N or (S+N) AN.

4.8.2 Measurement of Sensitivity Given SINAD

SINAD isa ratio ofsignal plus noise, distortion to noise and distortion in dB.
This ratio may be given in numerical or dB form and is a power ratio. The only
difference between the numerator and the denominator is that portion pro-
vided by the signal. Since this is a demodulated output measurement, it
requires the removal of the fundamental of the modulating frequency; while
retaining the noise and distortion components in the denominator. The test set
up used for this measurement is shown in Fig. (4-15).

DEMODULATED
OUTPUT

DISTORTION
ANALYZER

L., Seema ae
Fig. 4-15. Sensitivity test set up given SINAD requirements.

The signal generator is tuned to the receiver frequency of interest and is

modulated at a frequency and degree given in the specificiations. The distor-
tion analyzer is tuned to null out the modulation signal fus:rdamental. Switch-
ing to the total input power position, the meter reads signal plus noise and
distortion. Switching to the null position, the signal is nulled out, leaving the
noise plus distortion components. The signal generator output attenuator is
adjusted until the required SINAD is observed.
The Receiver 79

4.9 SIGNAL TO NOISE RATIOS FOR AMPLITUDE MODULATED

DOUBLE SIDEBAND SYSTEMS

In section 2.1 it was shown that an amplitude modulated double sideband

signal consists of a carrier and two sidebands. The power in each component of
the wave which contributes to the total power is:

A‘m? A*m?
P,=A* + —— +
4 4 (4-22)
where
P, is the total power in the signal
A is the magnitude of the carrier
m is the modulation factor
or

P=A°+ t ySe:
Aim =a'(1+7)2 (4-23)
2 2
The portion of the signal containing the modulation is:

SF
Atoe: 5 (4-24)
Let P. = A’ = the carrier power. The output signal to noise ratio becomes:

N kIBF 2kTBF ($25)

Where

P. is the carrier power

m is the modulation factor
1 = 100%, .5 = 50%, et cetera
k is Boltzmann’sconstant (1.38+10~°)joules / °K
T is 290° Kelvin
B is the post-detection effective noise bandwidth
K is degrees Kelvin
F is noise factor

In a more convenientdB notation,

S
W =P.+m’-(3+kT+BtNF) (4-26)
Where

NF is noise figure = 10 log F

10 log 2 =3 dB
80 RecewingSystemsDesign

Example.4-1:
Given:
The signal strength is 3 wV across 50 2)
The modulation percentage is 30
The post-detection effective noise bandwidth is 4 kHz (see section 3.2 for
a discussion of ENB)
The signal plus noise to noise ratio is 10 dB.
Find the receiver’s required noise figure.
The signal power is
\2

p= ae = -127.44 dBw= -99.744 dBm

The value of S/N is found from

ret+ = 10 dB=10 ratio

S+N=10N

or
S = 9N

and

S/N =9

10 log 9= 9.54 dB
A modulation percentage of 30 percent is a modulation factor of 0.3 = m
10 log m*= -10.46 dB
kKTBis in dB
10 log (1.38-10 + 290-410’) = -168 dBw
= -138 dBm.
Rewriting (4-26) and solving for VF we have:
NF = -(S/N) +P, +m? -kTB - 3
Substituting,

NF = -9.54 - 97.44 - 10.46 + 138 -3

= 17.56 dB

This represents the maximum receiver noise figure which will satisfy the
requirements. Because of production variance the designer should allow a
comfortable margin and design for a lesser value.
The Receiver 81

4.10 FM CARRIER TO NOISE RATIO

The carrier to noise ratio of a received signal is defined by:

3sA in ad ae (4-27)
Nose N;, KT BF
where
P. is carrier power
kT is -144 dBm/kHz (usedasa ratio)
B is the effective IF bandwidth (see section 3.2)
F is the receiver noise factor
NV;is the noise measured at the input of the limiter
In dB notation this ratio may be expressed by:
P.-kT-B-NF (4-28)
Example 4-2:
P.= -60 dBm
kT = -144 dBm/kHz
B= 25 kHz = 14 dB
NF = 12 dB

The carrier to noise ratio is:

-60 - (-144) - 14-12 =58 dB

4.10.1 FM Output Signal to Noise Ratio Above Threshold

The FM outpt signal to noise ratio is related to the FM carrier to noise ratio
by a term known as the modulation noise improvement ratio (MNJ), as
shown below:

(S7N ou = (MNT) (P./N;) (4-29)

where
Sis signal
N is noise
P. is carrier power
NV,is noise measured at the input to the limiter
The modulation noise improvement factor is a term resulting from a trade-off
between bandwidth and noise.

4.10.2 FM Noise Improvement Factor (MNI) Above Threshold

For carrier to noise ratios > © 10 dB, the modulation noise improvement factor
82 RecewingSystemsDesign

is defined by:
Ae 3..
2 ( (AF
B \*iy(8B, a
= jens
a

Where

B, is the one-sided audio noise bandwidth

A Fis the peak deviation of the carrier
B,,is the effective IF bandwidth
Example 4-3:
In a digital system, the IF bandwidth is 240 kHz and the data rate is 120
kbits/sec with a peak deviation of 300 kHz

MNI= a (2) 2 fea = 18.75= 12.73dB

2 120 120
The value of MNIcan be improved by increasing deviation and IF bandwidth
for a given upper modulation frequency. Reducing the modulation frequency
for a given deviation and bandwidth improves MNI.
For a given deviation, the IF bandwidth must be changed to include all
significant signal spectral terms. The necessary bandwidth is given by Carson’s
rule:
B,=2 (AF +f), AF> 1 (4-31)
where
f,, isthe modulationfrequency
4.10.3 FM Signal to Noise Ratio Below Threshold

Where the carrier to noise ratio is less than 10 but more than 3, the output
signal to noise ratio may be calculated by modifying the above threshold
relationship as follows:

cima= (F)(4 (F) a a

iURANO.)
t/
1+0.9 BUN
(5*) Tete F (4-32)

where all of the above terms were defined in the previous associated sections.
See reference [1] for additional information.
The Receiver 83

4.11 PM OUTPUT SIGNAL TO NOISE RATIO

ABOVE THRESHOLD

The PM output signal to noise ratio is related to the carrier to noise ratio by:
(S7M) uu= M° (By/(2 B,)) (P./N;) (4-33)
where

S/N is the output signal to noise ratio

B,,is the IF noise bandwidth
B, is the one-sided audio noise bandwidth
M is the peak phase deviation
P/N. is the carrier to noise ratio in the IF noise bandwidth

This differs from the equation for FM by M°*replacing 3(AF/B,)* but is

otherwise identical.
See reference [2] for additional information.

4.11.1 PM Output Signal to Noise Ratio below Threshold

Where the carrier to noise ratio is less than approximately 10, the output signal
to noise ratio is defined by:

(SPM)
ou= 2B,/ \N; (4-34)
1+09(Fe) (PLN)ere @PIN,
B (1aef Niy? a

All of the above terms have been defined in the preceding section.

Example 4-4:
Given:
B,, = 21 kHz
B,=3 kHz
M=5
Compute the relationship between (S/.V),, and (P./.V,) over the range of 0 to
30 dB for (P./.N;).
The short computer program of Table 4-2 was executed and the print-out is
shown in Table 4-3. The threshold of 10 dB for (P./N,) is seen as the break point
between the below and above threshold regions. Above this threshold the
denominator of equation (4-34) is unity and may be omitted.
A second program with a plot routine is shown in Table 4-4 and the plot
obtained for Example 4-4 is shown in Fig. (4-16).
84 ReceiwingSystemsDesign

P./Ni, (dB)
Fig. 4-16. Plot of Example 4-4 using the program of Table 4-4.

Table 4-2.
Computer Program for the Calculation of §/N as a Function of the Carrier
to Noise Ratio and the Phase Modulation Index

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The Receiver 85

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the Carrier to Noise Ratio and Phase Modulation Index
(The plot is shown in Fig. 4-16.)

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The Receiver 87

4.12 Energy per Bit to Noise Spectral Density (E,,/N,)

In digital or quasi-analog situations it is often necessary to determine the
signal to noise ratio (S/N) relationship to the bit error rate (BER).
The standard method used establishes S/N first by measuring the signal e,
and the noise e¢,using a true RMS voltmeter. The true RMS voltmeter must
have a sufficient bandwidth to include all of the signal and noise components
within the system bandwidth. Having established e,/e, in dB it is necessary
to add a value 4(dB) for conversion to E/N,
E, is the energy per bit which is equal to the power level of that bit.

or
Rise eee (4-35
°-z_—sCbit rate, (b,) a

where
z is the impedance across which e, 1smeasured.
Similarly
= cy cee 4-36
om, ican ENB aid

where
ENB is the effective noise bandwidth.
The ratio of E, AN, is
E, e, ENB
N,
— iti e,
— —b, (4- 37)

In log,) notation:
E
=+ (dB)=20log + +10 log ENB (4-38)
N, e, }

— (dB)=k+20log = (4-39)
N, e,
where

20 log — = — , (dB)

and

k=10 log

or

k =ENB -b,, in dB.

 88 ReceivingSystemsDesign

or

k = ENB - b,, in dB.

Example 4-5:
Let
6, = 2400 bits per second (bps)
ENB = 2730 Hz
k = 34.36 - 33.8 = 0.56 dB

This value k must be added to each of e,/e, measured (dB).

4.13 ERROR FUNCTION (erf)

By definition:
De ag
erff(x)=
(x) ne J é dt (4-40 )

The limiting condition x > © results in erf(x) = 1. Thus we may say 0 Serf(x) S
la0<x<@,
Tables of erf(x) are available for use (see reference [3]).
Solutions of erf(x) may be computed for small x (< 2) by the use of the
Maclaurin series.
Re:
CHAS Sar
cee (s 3°25OA anys
tea
307 roe
se
Larger values ofx require the use of the asymptotic expansion oferf (x )which 1s:

ere) =7 a(—~= += + +++) (4-42)

Forlarge x
erf (x)~1 - “; “ (4-43)

A second variation of erf (x) for large x is:

erf (x)~ 1 -2k(./2 x) (4-44)
Wherek (a) is
IO i (4-45)
V
27a
and

a= V/2«x (4-46)
The Receiver 89

Example 4-6:
let
x=2
using equation (4-44)
then
erf(x) =1-2k( 2-2)
=] -2k (2.828)
and
] 2.8287 /2
k (2.888) ~ Tom 2.898 é

= .141+ .01833 ~ .00259

“. erf (x) © 1-2 (.00259) = 0.99482
This compares with
Example 4-7:
Let
x = 2, using equation (4-43)
1
erf (2) =] - ed e*

0.994833
erf(2) = 0.995322 given by table reference.
It should be noted that the use of the asymptotic series solution does not
necessarily result in decreasing error as the series is lengthened, since the
solution is cyclic. Where high accuracy is required, the tables are suggested.

4.14 COMPLIMENTARY ERROR FUNCTION (erfc)

The complimentary error function erfc(x) is given by:

erfc (x) = 1 - erf (x) (4-47)

The solution of erf(c) is explained in detail in 4.13. The compliment of this value
is found first by finding erf(x).

4.15 TANGENTIAL SIGNAL SENSITIVITY (TSS)

The TSS definition resulted from the visual observation of a pulse in the
presence of noise. When the top of the pulse contains noise, and when the pulse
90 3 , RecewingSystemsDesign

is of such magnitude that the bottom of this noise coincides with the top of the
no signal noise, the TSS point has been reached. Fig. (4-17) serves to illustrate
this condition. Since a visual observation requires an operator’s judgement,
some variability in measurement results.
It is generally accepted that TSS corresponds to an output signal to noise ratio
of 8 dB which isa power ratio of 6.3 or a voltage ratio of 2.5. Referring this to the
input, the detector characteristics must be considered. With seldom-used linear
detection, the output signal to noise ration can be transferred directly to the
input. This is not the case with square law detectors. To produce an output
signal to noise ratio of 2.5 (voltage), its input ratio is 2.5 '“”which when referred
to the input in GB is 4 dB.

Signal + Noise

Noise Noise

Fig. 4-17. An illustration of the definition of TSS and the display obtained
on an oscillograph.
4.16 CASCADE NOISE FIGURE
Two or more stages, each with its own internally generated noise, when
connected in series, will contribute to the overall noise of that group. The
overall noise factor of n stages connected in series is described by:

CES DR ie? it Porch) ye

F=f :+ G, + G,G, teecccce GG, Greens (4-48)
-

where

F is the overall noise factor

F, is the noise factor of the first stage (input)
F,, is the noise factor of the second stage
F’, is the noise factor of the third stage
F’ is the noise factor of the nth stage
G, is the power gain of the first stage
G, is the power gain of the second stage
G; is the power gain of the third stage
G, is the power gain of the nth stage
Note: all values are ratios and are not in dB.
The Receiver 9]

1st 2nd 3rd 4th

stage stage stage stage

F, Fo Fs F,

Gi; Ge Gs Ga,

(Fs - 1)
—$—————

F=F,+(Fe-0
Fig. 4-18. The calculation ofnoise factor by pairing, which allows the exami-
nation of individual stage effects.

When computing the overall noise figure which is 10 log (VF), this calculation
becomes cumbersome and does not allow a detailed examination of each stage.
In cases where a particularly low noise figure is desired, it is helpful to know the
effect of each stage.
A more practical calculation results by pairing and using successive application
ofthe noise factor calculation for two stages, starting with the last stage. This
process is shown in Fig. (4-18).
An example showing the calculation of the overall noise figure of a receiver,
using the pairing ofstages, is shown in Fig. (4-19). Note that attenuation adds
directly to the noise figure ofthe following stage. That is, the noise figure ofthe
first IF amplifier is increased by the noise figure of the IF crystal filter and that
92 ReceivingSystemsDesign

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a

JaMOd
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The Receiver 93

of the passive mixer in a simple sum. This calculation is simply 6.5 + 3.5 +7 =17
dB. This fact reduces the computation task, which may be done by inspection
up to that point.
The computer program of Table 4-5 is useful for the calculation and analysis of
cascaded stages. The user need only answer the input requirements. The
program begins with the last stage and proceeds toward the input stage.

Table 4-5.
Computer Program for the Calculation of Cascade Noise Figure

i6 FRINT #9 CRSCADE NOSE FACURE

24 PRINT ° STARTS bEDITSH (OAST A

GE AND WORKS UP TO THE INPUT

a3 PRIMT " HFT=1a#loT¢Fit+e cre

1>/G1)9;DE" |
40 PRINT " WHERE”
5@ PRINT "“ NFT=TOTAL NOISE FIGU
RECO)"
6@ PRINT " FL=PRECEDING STAGE N
OISE FIGURECRATIC3"
7 PRINT " PZ=FOLLOWING STAGE H
GISE FIGURECRATIO?"
75 PRINT " Gi=1ST STAGE GAIN. OB

ut o « PRINT “ALL PROGRAM ENTRIES A

RE IN CB"
94 PRINT
1HS PRINT “FPRRRREPRRRESRRRERAKE
rae Be ey ge tp th 4 ae
110 PRINT "HF G CAS NF
STAGE "
120 PRINT "De Oe ne "
124 “JT,
1fPs
taf
e PRINT
144 DISP "IDENT LAST STAGEC<11 ¢c
Hk
Fes
te
pe CIM AFLisa
ia! INPUT? AZ
| BIi6P EM TERRLas Danee Oa NCD)
mt)
el105
|lb+im _
— INPUT F2,G2
"3 PRINT TAB C13; F2;TABC?2iG2:TA
_
BCZ20 AF
ts ~~EN OISP "IDENT NEXT STAGE<<11 C
HAR" 3
94 RecewingSystemsDesign

|a
— OIM &f0C154

hy
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MRL ABS
MOOTSe “TREN, NEX) S4 AGE RHR
INC DBD";
POP NPIS TE Pet
268 wSLa-cP le ie
fre U=le*¢Fe-1es
Leh VeiBen.Gl- {Hs
29M N= LE@ALGTCA+CU-La-
295 Y=IHTiW#16+. S710,
280 PRINT a ee LPT TS TABCLIO GIS ta
Be 1SI;3;
pet
aa
Dat
Sad = Y;TAREC22);8¢
Jfo]
fadesoe
faa
ta)
a il
2PhoO
mo 4 ei

4.17 INTERMODULATION DISTORTION (IM)|4]

When two signals whose frequencies F, and F, are applied to a non-linear
device, other frequencies are generated at:
F,'=F,-FA
and
FF,’=F,+FA
where
FA= F,-F, and F,>F,
This shown in Figure (4-20).

FA FA FA

F,’ F, Fo F.'

Fig. 4-20. Spectrum due to intermodulation distortion of signals at F, and F,

Assume that a receiver is tuned to a frequency F; = F,’. Whenever signals

appear simultaneously at frequencies F, and F,,, an unwanted signal F,’ appears
at F;, which for all practical purposes appears to be a legitimate signal. This
spurious signal may cause serious interference for a desired signal.
The Receiver 95

Consider a 1000 channel system with 25 kHz channel spacing. Intermodula-

tion products can result for any pair of signals whose separation from each other
is V-F A and they are spaced V*FA and 2 NFA respectively, from a desired
channel, where FA = 25 kHz. The possibilities of interference resulting from
intermodulation distortion are very great.
All signal processors are non-linear to some degree and will cause the genera-
tion of intermodulation distortion terms.

When two signals

[A, cos w, t and A, cos (w, +6) ]=p, (4-49)

are simultaneously applied to a non-linear processor whose transfer function is
of the form
K, p, +Kk,p, +K, p, tees KnPp, (4-50)
output termsresultwhichincludeall possibleproducts.The mostsignificantof
theseare |
K, [A, cos w, t + A, cos (w,t + 8) | (4-51)
which represents the two signals,
0.5 A,’ K, cos 2w, t
0.5 A,” K, cos 2 (w,t + 6)
A, A,K, cos [w,t - w, (t + 6)]
A, A, K, cos [w,t + w, (t + )]
representing the second order terms and
0.75 K, A,’ A, cos [ 2w, t + w, (t + 6)]
0.75 K, A, A,’ cos [ 2w, (t + 6) + w, t]
which represent the third order terms of interest.
Since these products have different slopes, they will cross at some point for a
given input level. This intersection is termed the intercept point (Fig. (4-21)).
Once this point is determined it is possible to predict the intermodulation
product magnitudes, with good accuracy. The charts of Fig. (4-22) and (4-23)
may be used to obtain the absolute or relative magnitudes of the second or third
order products, given the second or third order intercept point, respectively.
It is also possible to calculate the distortion product magnitudes by noting that
for a 0 dBm two-tone signal input, the third order intercept is equal to -] /2 of
the third order product magnitude for a unity gain device. For example, given
a third order intercept point of +20 dBm, the third order distortion product
magnitudes would be 20 + (-1 /2) = -40 dBm, for a two-tone input signal with
0 dBm magnitude. This relationship may also be used in reverse. Given a third
 96 RecewingSystemsDesign

order product magnitude of -60 dBm for 0 dBm two-tone input signal and
unity gain device, the intercept point is -60 x (-1 /2) = 30 dBm.
All intercept information is referenced to the output unless otherwise specified.
This includes the two-tone magnitude as well as the distortion product level.
Assume an amplifier has gain of 10 dB and the two-tone input magnitude is -10
dBm. The amplifier output will consist of the two-tone signals whose magni-
tude is now 0 dBm; assuming that the third order products are -40 dBm, the
third order output intercept point is +20 dBm. To relate this output intercept
value to the input, simply subtract the gain.

50 4
x
2ndorder
intercept
point ‘s
slope
2 Lf
VA
a
us
3rdorder
intercept
point / ea yy
30 slope
3 rt V /
E ie /1 ie
@ y. y 4 Output saturation
[= i
=
iS
a Fundamental
slope
1 ie
fo)
10

-20 -10 0
Input power dBm
10 20 30
Fig. 4-21. Device outputs showing the fundamental and 2nd and 3rd order
distortion products together with the extrapolated respective intercept points.
When using other than 0 dBm two-tone signals, normalize the levels to 0 dBm
remembering that the third order distortion products increase 3 dB for every
_ GB increase in the two-tone level.
The Receiver 97

Intercept Point
(dbm)
+40

+35
Signal Level
(dbm)
+30 +40

+25 +30 Spurious Response Level

(db Down)
2nd Order 3rd Order
+20 +20 0

+15 +10 “oe 10

+10 0 Te) 20

+5 -10 5 30

@) -20 20 40

a -30 25 50

-10 -40 30 60

=-5 -—50 35 70

-20 -60 80
Fig.4-22..Relative
levelspuriousresponsenomograph.(Based
onnomographs
fromAvantek,Inc.,SantaClara,California.
)
For a two-tone level of -30 dBm, at the input of an amplifier whose gain is}20
dB, we have an output two-tone level of -10 dBm. Assume that the third order
distortion products have a magnitude of -50 dBm. To normalize the two-tone
output signal level of -10 dBm, add 10 dBm. Also add 3 + 10 dBm to the third
order products (-50 + 3+10= -20 dBm). The output third order intercept point
98 ReceivingSystemsDesign

Intercept Point
(dbm)
+40 Spurious Response Level
Signal Level 2nd Order Srd Order
(dbm) (-dbm) (-—dbm)
+30 +10 10 30

+20 0 20 40

+10 -10 30 50

@) -—20 40 60

-10 -30 50 70

—20 —40 60 80

-30 ~50 70 90

-40 —60 80 100

90 110

100 : 120
Fig. 4-23. Absolute level spurious response nomograph. (Based on nomo-
graphs from Avantek, Inc., Santa Clara, California.)
is -20 dBm - (-1 /2) = 10 dBm.
When dealing with relative magnitudes, proceed as above, except note that
there is a 2 dB/ dBm relationship between the output two-tone signals and the
The Receiver 99

distortion products.
For the case where the two-tone signals are unequal in magnitude, simply
subtract 1/3 the difference between them from the larger.
Given:
Signal | +18 dBm
Signal 2 0 dBm
The equivalent equal magnitude two-tone signal has a power level of +18 dBm
- 1/3 (18 dBm - 0 dBm) = 12 dBm. Then proceed as before, using either the
charts of calculation.
Second order intercepts are seldom considered because those products are
generally farther removed from the desired frequency. Should it be of interest,
second order terms may be related by the intercept point, as before.

4.17.1 Cascade Intercept Point

The system designer may be called upon to predict the intermodulation
performance of many stages in cascade. One such application may involve a
receiver where the intermodulation performance is specified. Assume that the
specification reads: “The intermodulation distortion products resulting form a
two-tone -30 dBm signal, seperated 2 and 4 MHz (respectively) from the
desired frequency, and on the same side, shall not exceed -100 dBm.” From
Fig. (4-22) or (4-23), or by calculation, the third order intercept point must not
be less than +5 dBm at the receiver input.
Further, assume that the receiver of Fig. (4-24) is being considered for this
application. Since the two tones require 4 MHz of bandwidth, they will be
processed by all stages, up to the first IF filter, which has a 50 kHz bandwidth
and does not allow them to pass. This is the intermodulation distortion trunca-
tion point. All calculations must include all stages between the antenna and
this filter which heavily attenuates the two-tone signal. Further intermodula-
tion distortion contributions are negligible.
Norton’s equation (4-52) may now be applied successively stage by stage,
beginning at the antenna, up to the determined truncation point and the
output intercept point determined. The gain to this point must be subtracted
from this value to obtain the input intercept point, which may now be com-
pared to the requirements.
3

3 3 ] 1,
L=ie= 10log Pit a *: a (4-52)
82 3
rf
100 ReceivingSystemsDesign

3
I, is the third order cascade output intercept point (dBm),
3
I, is the second stage third order output intercept point (dBm),
g, is the power gain of the second stage, and
3
I, is the first stage third order output intercept point.

Note: The terms in the brackets are not in dB notation. Use the numerical
equivalent. (See reference [5] ).
intermodulation
truncation point

oscillator

Output Intercept Cascade Output

PointdBm GaindB Intercept Point
Tuner N/A -3.0 _
Preamp 20 15.0 20.0
Mixer 15 -7.0 10.875
5.0 Input intercept point =
10.875 - 5.0 =5.875 dBm

Fig. 4-24. Receiver example showing the input intercept point calculation.

The previous simplified example illustrates the cascade calculation method

which indicates an input intercept point of 10.875 - 5 = 5.875 dBm. This
numerical value barely meets the requirement. However, it must be kept in
mind that Norton’s equation assumes coherence of the intermodulation prod-
ucts through the various stages, which is not likely. The calculation is therefore
considered pessimistic.
The second order intercept point may be computed in similar fashion using the
equation:
The Receiver 101

2
2 ae l Pp
1,=1,-20log | 14/ —* = (4-53)
82 7
1
2 .

I, is the second order cascade output intercept point (dBm)

2 . .

I, is the second stage second order output intercept point (dBm)

g, is the power gain of the second stage

2
I, is the first stage second order output intercept point

Note: the terms in the brackets are not in dB notation.

The procedure is to begin at the input as the first stage and compute the cascade
with the following stage. This value becomes the first stage and the next stage
(third in this case) becomes the next or second stage, et cetera.

The input intercept point becomes the final cascade intercept point minus the
preceding gain in dB notation, as shown in Fig. (4-24). This computation can
be tedious for complex systems and it is suggested that the program of Table 4-6
be utilized for that purpose.
Table 4-6.
Computer Program for the Calculation of Cascade Intercept Point
im PRIN”:
mr CASCADE. TINTEREERAS
ft PRINT " COMPUTES DEGRADATIO
NGF THE) INTERCEPT POINT: DUE
TO A PRECEQING STAGE"
PRIRT
t+l0S PRINS eet Lee, INR, INTORTALS
THE DOUTPUT: ZNTLRT-GATH?
|a 5ts DTSP “CHODSE "2HD2 c2ra2OReSRO
So) (ORDER * 3
64 INPLT W
7U PRINT “CASCADE INTERCEPT FGI
MT”
S@ IF N=2 THEN 14h
S46 IF H=3 THEH lek
146 PRIHT “SECOND URDER"
1te COTO 136
124 PRIWHT "“THIRO OFROGER”
120 FRIAT "EERFRERSRREPHEEERERERS
Re ak he ea
146: PRINT -eLet, 2 G@ bas. I GT Th
LP. STAGE!
i154 PRINT "OBM DEB OB De: Cl
BM"
102 RecewingSystemsDesign

1 PRINT
a O1Se toh deeb déa7 STHGE« CHAR 3

greet
jE
bent QIM ACSI
byt
mt
itt INPUT Fg
1eas
4 \) tS CISe
Pe bhmi a cg Lm
CTEM)"
fa
fadi 7,Face?
ood”
, [NPT %
) i
Pe ISP "ENTER 1ST
ry Tyo STAGE GAINED
Ei % 7. s
238 [MPUT ®
“46 PRIWT TABCLs3d 51,.P3TABS
2o2GAF
SSH PRIWT
2S Popa pr! Tt FrEoR ND rs CHAR
“4ti ,
al A ig 8 ayWe=. 3Bs
“S08 INFUT &F
cee Uber (ons Chenu INT FTCDBM:?.
GHINGCDE";
SUM TMP: 2
218 K=2h-¢H-13
226 Lela c2-1s
33R MELGecyeid:
S48 J=10°¢% tas
358 D=F+2
SEO CHTHTccvY-KELGT¢
Trtqd-Hi tas at), 8+. 5° 5a.

478 E=€-O

258 eaon aeTRB

aAricci
Wee VYiTAHBees,
C153;0; THE Cee
beriB3
Te
fo
fa
taj
on0
ooh
me
e
Mt
Hej
tet

The most frequent error in the computation ofthe second order intercept point
is the determination ofthe truncation point. To determine this point, it is first
necessary to develop an understanding of the second order distortion terms.
These have been shown to be of the form:

Second harmonic terms, 2F,, 2F,

The Receiver 103

Sum and difference terms:

Fit,

Where

F, and F, are the two frequencies which cause the second order distortion.

There can be two cases which must be considered.

These are:

In band and
Out of band

The in band case results when F; and F, are within the receiver’s tuning range.
The out of band case results when these two signals are out of the receiver’s
tuning range.
The only barriers to second order distortion generation are the preselector
attenuation, the conversion of the first mixer, and the use of high second order
intercept components.
Many of the distortion products will not be converted by the first mixer and the
remainder will suffer preselector attenuation. Where the conversion is the
truncator the designer need not look beyond this point for further additive
distortion. The attenuation ofthe preselector of the signal F, and (or) F,, serves
to increase the systems intercept point, which permits the use of lower intercept
point components, if desired. Seldom is it ever necessary to extend the intercept
analysis to the detector.
The following is a good approach to second order analysis:
Determine the range of the signals involved
Analyze the effect of the preselector on these signals
Examine the conversion of these distortion terms coordinated with preselector
tuning.
A good design will attenuate the signals F, and (or) F; to livable levels and also
reject through conversion the remainder.
Note: A simple relationship permits the determination of intercept point
knowing the level of the distortion products.
This relationship is as follows:
m

R
ee ee
(m=1)

where
104 ReceivingSystemsDesign

m is the order
S is the signal (dBm)
R is the intermodulation ratio (dB)

Example 4-7:
The requirements call for a suppression of second order products of 60 dB,
resulting from signals of -50 dBm.
Then

I= -50 + 60/(2-1) = 10 dBm

The equation is equally applicable to the third order case.
4.18 DESENSITIZATION

Desensitization ofa receiver is the reduction of gain of that receiver to a desired

frequency by a second unwanted signal. This can result from the presence of a
stronger signal within the IF bandwidth of an FM receiver, the compression of
stages in an AM receiver or from AGC takeover in AM and FM receivers, if
used for the latter case.

Poorly designed receivers are subject to AGC takeover if the final selectivity is
placed before the AGC detector, instead of at the front of the IF amplifier. An
example of this is amultibandwidth multifunction receiver which must provide
simultaneous outputs at several bandwidths. The filter following the final IF
a-
converter must be wider than the widest final filter bandwidth preceding the
detectors. The IF strip is exposed to signals in bandwidth window possibly
wider than that of the AGC system. The result of this is potential blocking or
overload of the receiver.
A second possibility is AGC derived in a wide bandwidth with narrow band
detection. Here a strong unwanted signal developes AGC, resulting in a weak
desired output. To reduce desensitization effects, the designer must provide
proper filtering throughout the receiver and eliminate sources of broadband
noise (external and internal) to the receiver.
4.19 COMPRESSION

All linear systems, when given a sufficiently strong input signal, depart from a
linear relationship between input and output. Sucha system is said to be going
into compression or saturation. An example of this situation would be an
amplifier whose performance is shown in Fig. (4-25). The performance is linear
up to an input signal level in excess of 0dBm. Beyond this point, the output falls
below the linearly extrapolated input/output characteristic and the amplifier
is no longer linear.
The Receiver 105

Compression, or the -1 dB compression point, is generally taken as that point of

-1 dB departure from linearity. In the example this is 10 dBm input or 19 dBm
output. The input signal level which causes this departure is the value of
compression usually specified. Where the output compression point is given,
the gain between input and output (in dB) is subtracted from the output, to
secure the input compression point.
Non-linear processes which provide a linear output amplitude, such as a mixer,
are subject to compression as well. In this case the mixer is operating with a
fixed local oscillator drive level and the input signal is increased while monitor-
ing the output. Care must be exercised in this measurement to exclude all but
the desired converted signal component. This may be readily accomplished by
using a spectrum analyzer and examining the desired spectral line, or by using
a RF voltmeter preceded by an appropriate IF filter. As a rule of thumb for
double balanced diode mixers, the output compression point is the conversion
loss below the local oscillator drive level. The input compression point is the
output compression point increased by the amount of the conversion loss
through the mixer.

|
EEUU
ALE
AUTEN
vofste
bea
ed
i

=
pe —+--+
GonSi +
puwen —
—ho ee epee

===BoaoHes
Amplifier
SS gain
i s1048 ===
SS SS SSSS

Fig. 4-25. An example showing the compression of a linear amplifier. Where

the departure from linearity is -] dB, this is the -1 dB compression point.
106 Recewing SystemsDesign

Example 4-8:
Given an output compression point of 9.5dBm and a conversion loss of 6.5 dB,
the input compression point 1s
9.5 + 6.5 = 16 dBm

4.20 CROSS MODULATION

When two signals appear simultaneously at a receiver’s input, and one is

modulated and the second is not, non-linearities in the receiver will impart
modulation to the unmodulated signal from the modulated one. This process is
called cross modulation and it is related to the third order intercept point by
3
I 1
M.=
isl + —
AP lee 2 (4-54)

where
3
[ is the receiver third order intercept point
P, is the power level of the stronger modulated signal
Note that the signal strength of the lesser unmodulated signal does not enter
into the computation.
In most cases specifications define M, and P,; then the designer must solve for
_
the required intercept point / from

p=
3 (0,- =|
l 4P, (4-55
Example 4-9:
Given:
Cross modulation must not exceed 20 dB
The interfering signal P, is -10 dBm
Find the required intercept point.

hatSO oege oOe]

Leen
TG
= 5.79 dB
For convenience, a chart of the cross modulation ratio and the intercept point
minus the power of the stronger modulated signal is given in Fig. (4-26) and is
based upon:
3

—
I
Aeeh ( ai
=4|M -
])
Paha
—] WhereM,=5 tik m,m
— Peat
-

mis the modulationindex

m, is the effective modulation index
The Receiver 107

Example 4-11:
Find the solution to the previous problem using the chart of Fig. (4-26).
Enter the chart at m/m, = 20 dB and read't- P,= 16. Since P, = -10,1 = 6 dBm.
References [6] and [4] contain additional information on this subject.

( sono

Fig. 4-26. Graph of cross modulation versusintercept point [6].

108 RecewingSystemsDesign

4.21 SPURIOUS FREE DYNAMIC RANGE

It is informative to determine the dynamic signal operation range of a system

which 1sfree of spurious signals resulting from third order intermodulation
products. In this definition spurious free means that these spurious signals are
equal to the noise level. The third order intermodulation product signal levels
are related to the level of the two signals (producing them, respectively, on a
three for one dB relationship). The spurious free dynamic range is related to the
third order system intercept point by the following relationship:
3
SFDR (dB) = 0.67 (J - kT /MHz - 10 logB - NF) (4-57)

where
SFDR is the spurious free dynamic range

/ ; 1sthe system third order input intercept point

kT is the thermal noise level in a 1MHz bandwidth
= -114dBm / MHz
B is bandwidth in MHz
NF is the system noise figure ( dB)
An example of the application of this relationship follows.
Example 4-11:
Given:
Third order input intercept point = 10 dBm (Note: Given the output
intercept point, the input intercept point is the output intercept point
minus the gain in dB notation.)
Bandwidth is 10 kHz
Noise figure is 5 dB
Find the spurious free dynamic range.
Solution:
SFDR = 0.67 (10 - (-114) - (-20) - 5)
= 92.67 dB

4.22 IMAGES
In the mixing process it was shown that when two input signals comprised of
an RF signal and a local oscillator signal are applied to a mixer, inter-
mediate frequency signals are produced at the mixer’s output. More
specifically:

ie We,
~MF,| (4-58)
The Receiver 109

where

Fis the intermediate frequency

Fis the receive frequency
F,, is the local oscillator frequency
M and WN are intergers
The primary or desired outputs result when M = NV= 1|

Then
equation
(4-58)
results
in:
F,=Ft F,,| (4-59)
Since F;,is a constant, then for a given LO frequency there exist two values of F,
which satisfy the relationship.
EXAMPLE 4-13:

LetF,=160.7MHz
F,,=10.7MHz
thenF,= [Fy F,|
or |10.7+ 160.7|= 150and171.4
MHz.
Thus, the mixer is equally responsive to two frequencies, both of which are
twice the IF apart. Of these, one is the desired response and the other is called
the image frequency. The receiver must reject the image term to provide
satisfactory performance. This is one of the reasons mixers are always preceded
by preselection filtering of some form. The selectivity requirements of the
preselector are governed by the frequency separation between the image
frequency and the desired frequency. The amount of image frequency rejection
is strictly a function of the attenuation, provided by the preselector filter alone
(unless an image rejection mixer is used).
To avoid extreme selectivity requirements from the preselector, the ratio of F, to
F,, should not exceed 10 or 20 to 1 for a first conversion in a down conversion
superheterodyne receiver.
In up-conversion systems, the high value of intermediate frequency effectively
removes the image frequency out of the preselector bandwidth. Here, simple
fixed tuned filters often suffice as preselector filters, eliminating tracking and
tuning problems. Fig. (4-27) is an example illustrating the relative image
frequency behavior between up and down conversion.
4.23 HIGH ORDER IMAGES

When a receiver design utilizes more than one conversion, the image problem
becomes more complex. For every mixer in a receiver there will exist an image
frequency. A double conversion receiver will have primary. and secondary
110 ReceivingSystemsDesign

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The Receiver 111

images. A triple conversion receiver will have primary, secondary, and tertiary
images, and so on.
The treatment for high order images is the same as for the primary case. The
only means of image rejection is that of the filter, which precedes the mixer. For
this reason, in multiple conversion systems, the IF filter preceding the mixer in
second and higher conversions must provide attenuation at the respective
image frequency. In some cases, additive attenuation may be provided by
filters and/or preselection from earlier conversions. For example, a secondary
image may also be attenuated by the preselector. A tertiary image could be
rejected by the second IF filter, et cetera. In difficult cases other sources of
attenuation may be of value.
To illustrate the situation, an example of a triple conversion receiver is pro-
vided in Fig. (4-28). To illustrate the situation, the parameters used in the
illustration are not necessarily optimized for a real-world design.
The higher the order of an image frequency the more stages are involved, and
the likelihood of securing additional attenuation is generally greater. In the
illustration, had the third IF been lower and the receiver tuning range greater,
the tertiary image would have been in band and the preselector would not have
had any effect in the worst case.
A summary of the attenuators for the respective images follows.

Primary Image
The attenuation is the ultimate attenuation of the preselector at the image
frequency. In this case the image frequency is far removed from the preselector
tuning range and the ultimate attenuation may rebound from its floor value.
For this reason the preselector must be specified, designed and tested at the
image frequency to ensure predictable performance. For this illustration a
value of -70 dB is assumed. In some cases this may not be sufficient and an
image rejection (lowpass) filter would be added ahead of mixer M..

SecondaryImage
The preselector provides its ultimate attenuation to the secondary image
frequency, (the previous comments made in the primary image’s case still
apply.) Additionally, the first IF filter will provide attenuation. For a four-
section Chebyshev filter with a 0.1 dB ripple the value is 95.25 dB. This is well
beyond the usual ultimate attenuation, soa value of 70 dB will be assumed. The
total attenuation is 140 dB.

Tertiary Image
The rejection of the tertiary image frequency is the sum of the attenuation of
112 RecewingSystemsDesign

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The Receiver 113

the preselector, plus the first and second IF filters at the image frequency.
An attenuation value of 33.7 dB is computed for the second IF filter. The first IF
filter supplies its ultimate value of 70 dB by inspection (large frequency offset).
The preselector is assumed to have the following characteristics:
Unloaded Q
Q, = 90
Loaded Q
Q ,= 200/10 = 20
Since the tertiary image frequency is 168.6 MHz, the worst case preselection
attenuation results when the preselector is tuned to 200 MHz. A bandwidth of
5% at 200 MHz is 10 MHz.
Therefore
Q |= center frequency /bandwidth
Using Table 5-4 for critically coupled tuned circuits, the preselector is found to
provide 11.4 dB of attenuation. |
The total tertiary image rejection due to filtering alone is:
33.7 + 70 + 11.4 = 115.1 dB
In real situations, additional sources of attenuation are provided by mixers and
amplifiers. For example, it would not be economical to use a second IF
amplifier (which would be so broad as to pass the tertiary image frequency) or
for that matter any frequency other than those required. Mixers are also
relatively narrowband. They range from narrowband to several octaves at high
frequencies, or to more than a decade at low frequencies.
In an image performance analysis the response of every stage should be
considered. Often, after having determined the required rejection the analysis
may stop at a point, providing the minimal attenuation value plus margin.
As a typical case, a specification may define a signal environment of -10 dBm,
representing the maximum value. Signals of this magnitude could be encoun-
tered at the image frequencies. The specification may also dictate that image
responses must be equal to noise. Assume that in this case it is found that the
noise floor is -120 dBm. The required image attenuation must be -120 - (-10) =
-110 dB minimum.
For the system illustrated, it can be seen that the primary image rejection is
only 70 dBand an additional 40 dB-plus margin must be secured. Ina real case,
the preamplifier would usually not pass the primary image frequency and
additional attenuation would be provided by this stage. In addition, the
mixer’s R port response may be down, providing more primary image attenua-
tion. All of these sources should not be overlooked in difficult cases. Should the
 114 Recewing SystemsDesign

image rejection still be inadequate, then additional filtering could be added at

the primary image frequency. Having completed the analysis and made the
necessary circuit refinements the image performance predictions give confi-
dence to the design adequacy.
The previous discussion assumes that the image frequencies are applied at the
receiver input connector and many specifications may so state. Although
meeting the letter of the design specification is legally correct it is not ethical to
overlook internal-image frequency generation, even though no specification
may cover the case. In Fig. (4-28) it can be seen that the first local oscillator
ranges from 1200 to 1400 MHz and the secondary image frequency is 1220
MHz. When the receiver is tuned to 380 MHz, the first local oscillator will
generate the secondary image frequency of 1220 MHz. This can be a serious
problem.
Modern high performance receivers may utilize first local oscillator power
levels of 20 dBm or more. Assuming that this is the case, the secondary image
frequency signal level at the X port of the first mixer. M, would be 20 -70
(ultimate attenuation ofthe first IF) - 20 (mixer L to Xport isolation) = -70 dBm
at the first IF filter output. This level must be equal to noise level at the IF filter
output to meet specifications. The noise level at the first IF filter output is the
input noise level at the attenna connector plus the intervening gain (or loss). In
Fig. (4-28) assume:
Preselector loss 2.18 dB
Preamplifier gain 15 dB
Mixer M, conversion - 7 dB
First IF filter -3 dB
total 2.82 dB

The input noise level was -120 dBm. Therefore the noise level at the output of
the first IF filter is -120 dBm + 2.82 dB = -117.18 dBm. To meet the specifica-
tions the magnitude of the image frequency from the local oscillator must be
less than -117.18 dBm. The required additional filtering must be:

-117.18 dB - (-70 dBm) = 47.18 dB min

An additional filter could be added following the first IF amplifier, so that the
requirements are met. Here it was assumed that the first IF amplifier was
equally responsive to the first IF frequency and the secondary image frequency.
Where this is not the case, the first IF amplifier attenuation would contribute to
the requirements.
The best solution is to avoid such situations by relocating the frequencies
involved. Here the second local oscillator frequency could have been moved to
1790 MHz, thereby shifting the secondary image frequency to 1980 MHz.
 The Receiver 115

This case was used to illustrate the pitfalls in analysis. It is good practice to
examine the design for internal sources of image frequencies and take necessary
action.

4.24 SELECTIVITY

The selectivity of a superheterodyne receiver is determined solely by the

selectivity of the IF amplifier. Here the selectivity may be obtained by the use of
selective networks such as tuned circuits, crystal filters, or both.
The need for selectivity results when several signals are presented to the
receiver simultaneously. Assuming that these signals are in close proximity to
the desired signal, as in adjacent channels in a channelized network, the
receiver must reject all but the desired channel.
A typical communications network will assign specific frequencies to each
included station, and each frequency will be spaced from the frequencies of
adjacent stations by a fixed amount. In a high density environment, and for
maximum utilization of the RF spectrum, this spacing is minimized and
determined by:
the spectral width of the transmitted signal
the frequency inaccuracies of the transmitter
the frequency inaccuracy ofthe receiver
plus margin
The situation is illustrated in Fig. (4-29).

Ah CRIES Fern eae Poe

Desired
channel
N-1 N N+1

Fig. 4-29. Channelized spectrum with a typical receiver selectivity curve

superimposed.

The receiver selectivity curve’s solid line is shown to include the desired
channel (V) within a reasonably flat part of its response. Being non-ideal, the
response falls off but is not totally exclusive to channels +] andN -1, et cetera.
Therefore, signals within these adjacent channels could cause interference to
116 | ReceivingSystemsDesign

those of the desired channel WNand under certain conditions cause complete
communications failure.

Frequency management avoids this problem by allowing guard bands of one or

more channels on either side of achannel within a geographical zone. This zone
is determined bya signal intensity profile which considers path losses, transmit-
ted power and antenna directivities, to ensure that adjacent channel signal
intensity is low enough to minimize serious interference.
The selectivity of the filter is not usually achievable at the generally high
receive frequency and is therefore applied at a lower intermediate frequency.
From this discussion, we develop the following rule:
Toensure optimum selectivity, it is good design practice to restrict the
receiver IF bandwidth to that ofthe received signal’s spectral width plus
the sum of the drift allowance for both the transmitter and receiver.
Additionally, the shape of the filter should be made as nearly rectangular
as possible, constrained only by permissibie distortion.

4.25 INTERMEDIATE FREQUENCY (IF) REJECTION

A mixer will pass signals present at its input (R port) to its output (IF port),
without conversion through its imperfect isolation, between these two ports.
This is illustrated in Fig. (4-30). Typically, this isolation value for a double
balanced mixer is 20 dB. This unwanted signal path can result in desensitiza-
tion and heterodyne problems, or both.

Received signal at the IF a ;

eed throughor
ig een isolation (R to X) 20 dB typ.

—\—————___ Conversion path -6 GB typ.

Input port R X IF (output) port

L
Local Oscillator Port

Fig. 4-30. An illustration ofa double balanced mixer’s typical output through
the conversion path (desired) and the leakage path (undesired).
Since it was shown in section 4.5 that the IF must be out of band, the only other
source of IF signal attenuation is presented by the preselector filter’s and
preamplifier’s frequency response.

The designer must provide a preselector and preamplifier (if used) response
which, when added to the isolation ofthe mixer, results in a IF rejection level
which meets the specifications.
The Receiver b1Z

Example 4-14:
Specification:
When the receiver is presented with a signal level of -25 dBm at the IF, the
resulting receiver output shall not exceed the noise level.
For a receiver with a noise level of -110 dBm, the receiver front end must
provide an attenuation greater than 65 dB.

|-110-(-20 -25)| = 65 dB min where the L to R isolation is 20 dB.

Achieving IF rejection is generally not difficult unless the designer selects an IF
too close to the received frequency.
4.26 LOCAL OSCILLATOR RADIATION

A local oscillator (LO) or any oscillator is a transmitter if provided with a

suitable radiator, conduction, or leakage path. This can be troublesome within
a receiver itself or to other co-located receivers.
Internal oscillator problems within a receiver are a design problem and can be
cured by appropriate shielding, isolation, layout et cetera. The co-location
problem can be severe within an installation and for this reason specifications
may be applied which govern the amount of local oscillator radiation that can
be tolerated.
A typical specification may dictate that the local oscillator signal present at
the receiver antenna connector be equal to or less than noise by x dB. It may
also specify in band and out of band radiation. The design treatment in any
case is the same.
From the discussion of mixers it was shown that even with the best double
balanced mixers available, local oscillator to receive port isolation is only 20 to
30dB. Assuming a LO power level of 10 dBm, a signal due to the local oscillator
will appear at the mixer input receive port. This signal, with a magnitude of 10
- (20 to 30) = -10 to -20 dBm, would cause severe interference to other receivers
in the vicinity ifit were coupled to an antenna and tuned to this frequency. This
This problem is particularly severe with down conversion receivers (low IF),
where the local oscillator is the IF away from the tuned frequency.
To reduce local oscillator radiation, preselection and preamplification are
employed. The local oscillator radiation requirements generally dictate the
selectivity requirements of a preselector, more so than image rejection. This is
due to the fact that the local oscillator is one IF away from the received
frequency and the image frequency is two IFs away, making it easier to filter.
Because of this, low intermediate frequencies with high RF frequencies can
result in failure to meet specifications.
An example is provided illustrating the local oscillator radiation problem
and its treatment.
118 RecewingSystemsDesign

where
F is the receive frequency
F,, is the local oscillator frequency
IF is the intermediate frequency
a, is the reverse gain of the preamplifier
a, is preselector attenuation at F,, when the receiver is tuned to F,
a, is the mixer isolation (L to R port)
The local oscillator signal power level at the antenna connector is
P,, +a, + a, + a, = P,, (ant)
Illustration:
a, = -20 to -30 dB
a, = -20 dB (measured or from data sheet)
a, is from the preselector curve at F,, when tuned to F,
P,,= 10 dBm
Pra) = ~10 -(20 to 30) -20 + a, = -30 to -40 dBm + a,
Find the necessary preselector attenuation required to reduce the local oscilla-
tor signal level at the antenna to -110 dBm.
Solution:
a, =-110 + (30 to 40) = 70 to 80 dB
Such a high value of preselector attenuation demands that the local oscillator
frequency be separated from the tuned frequency by an intermediate fre-
quency that is sufficiently high toensure the necessary attenuation is secured.
4.27 PREDICTING SPURIOUS PRODUCTS

The spurious performance of paper designs are easily verified through the use of
The Recetver 119

computers.
The mixer equation was given as
F,,=| NF, + MF, | (4-60)
where
F,,is the intermediate frequency
F is the receive frequency
F,, is the local oscillator frequency
M and WNare integers
Since mixer generated spurious products must have an input stimulus, F, is
made the variable, F, is fixed by design, and F’, is solved where M and WNare
equal to I.
Then

Given F,,,, and F,,,,

Choose F;,and F,,. where F;,, is an incremental change of F,
Compute F,, for M= N=] for every value of F, ranging from F,,,,, to F,,,,, using
equation (4-60) and the appropriate sign (where F, = F,,,;, + Fin.) < Fomax:
Solve for F, where:

ee F,+MF, (4-61)
and M and N take on all values from 9 to -9 in all combinations. This may be
accomplished in two loops. For example, let M = 9 and let Vrun from 9 to -9
then decrement M to 8 and run N from 9 to -9, et cetera.

Compute DELF =| F, - F |
Compute ORD =| M|+| |
Print
F, F,, F, DELE, M, N, ORD
This represents a minimal program which requires reference to a mixer table.
By including a look-up table in the program, as well as the preselector attenua-
tion characteristics, all necessary information will be included. The inclusion of
the preselector characteristics lets the designer explore performance against out
of band signals. These computer print-outs provide a tremendous insight to
spurious performance and provide the designer, management, and customer
with reasonable confidence regarding performance.
A program in BASIC which is suitable for spurious prediction is shown in Table
4-7. This program has the following features:

Interactive
Prompting
Built in preselection options
(120 Recewing
Systems
Design

Outputs
Receive frequency (F,)
Local oscillator frequency (F,,)
Spur identity
harmonic of F. (to 7th)
harmonic of F;, (to 8th)
Mixer spur level up to 15th order
Total preselector attenuation at the spurious frequency
Total spurious level in dB
Spurious frequency
The program also features a search floor which is selected by the user and
excludes all spurious responses below that floor.

Table 4-7.
Spurious Response Program for the Prediction
of Receiver Spurious Responses
18 PRINT "SPUR SEARCH PROGRAM"
20 mm PRINT “RE INPUT ~-1@DBM.LO 17
OeN"
29° PRINTso" FROM”
40 PRINT Ser Sech lee axes
2H PRIWT “WHERE”
BM PRINT " FS=SPUR FRESUERCY"
49 PR IB LS FIFSINTERMEDIATE F
REGQHUENC
~ |
hon] PRIWT
oo)
Lae " FLO=LOCAL OS FRED
7i a)we al ot as MEM ARE fTHTEGERS oO
F BOTH SIGNS”
149 PRIMT “COMPUTES UP To 15TH Oo
ROE R"
114 PRIHT
128 PRINT "GEFINITIONS" |
12@ PRINT " ORDERSORQ=ABSt¢M+hH>"
144 PRINT " FR=TUNED FRESUENCY"
15@ FRIHT "“ FMIN=MIN LIMIT GF F |
RM
iS@ PRINT " FMAX=MAM LIMIT OF F :
FM
17S FRINT
184 SHORT LY |
194 DISP OSEMLEE seLgiberieies ok uc &
The Receiver 121

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IF AS="TUNED" THEN 466
CISP "“CHEBISHEY FILTER <FIXE
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INPUT AZ. A2,HI-RL.R2
PRINT"FIXED TUNEQ
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GOTO 496
DISP “ENTER GL ANC NUMBER OF
CRITICALLY COUPLED TRANSFOR
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ATION"
IMPLIT Q1.M1.R4
122 Recewing SystemsDesign

400 PRINT "TUNABLE TE TER VOUS ING

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FoR -=0 To 2
READ Wed. kK
86 DATA’ 27; 31036 0477R3E, 82372 1 me
G27 28/0739 941 dallte, 6c) eose
186,75,84,75,86,74787,74,84 Yo ®
7i@ DATA 87.77.87, 78,90, 75,85.77
/28.90,90,90.90,98,99, 50,98,
96,90,.90,90,90,90.98,98,.98,9
eI
7270 CATA 90,50,.90,98,90,98,98. 98
94,.90,98,990,90,90,90,46, 40
an
720 HE:
740 MET J
750 G=WEW. LS
766 RESTORE
7708 IF Q°AL THEN 626
726 IF BS=""" THEN 798 ELSE sap
The Receiver 123

79a
s IF M>B2 THEN 82a ELSE sea
a0 @ J2=6
a1 A GOTO $26
a2 GQ FE=ABSCCB2* agaa1-x-H-B2>
§ GS=C2eLOGCPS+(F2%2-19".5)
33
s4 BMHZ=CEXPCG29+EXPC-G2) 9-2
S35 @ l2=1a*(02-183-1
S65 ® JZ=16#LGT(1+12eH2*24F 2)
ma IF J22E2 THEM J2=E2
a0 IF A¢="FIXED" THEN 946
eo9 GOTO 1448
aan IF “<AZ THEN 934
a{@ IF 8>A2 THEN 928
926 GOTG 1866
A3QME=ABSCCASHASSR-K Ie CAZ-AB»
944 Y=HLELOGCRZ+¢N2°2-19°.93
958 CL=CEXPCY2+EKPC-Y2 272
960 Psia*cRi-1@3-1
O74 MU=1G# LGTCit+ReCis2en2s
gg0 IF “3>R2 THEN ¥3=R2
298 GOTO 1414
iGaa x3=6
1418 Ti=XZ+a4+I2
{G24 IF Ti>At THEN 628
{a3 GOTO 614
1949 MI=POELGTCCL+¢O1*2KCA-K IAD
ede Avo th ie 2s
{458 IF %3>R4 THEN XZ=R4
1966 Tisx3+H+J2
1974 IF Ti>Ai THEN 528
1480 GOTO 614
{99a END

4.28 THE MIXER SPUR CHART

Several manufacturers in the mixer field have published charts describing

mixer performance. Much of the-data is computer generated in lieu of actual
measurement. This can be appreciated when a measurement approach is
attempted. Although direct measurement provides a realistic overview of
mixer performance, particularly because such performance is governed by
mixer termination, this approach is seldom used for economic reasons. The
data provided is useful; however, it should be realized that these data are made
under conditions which may vary drastically from those of the design to be
analyzed. The designer must allow some margin for such variance.
Mixer spur tables assume ideal broadband terminations of usually 50 12, since
it is impossible to cover all other possibilities and because the 50 1 broadband
124 RecewingSystemsDesign

termination offers the best overall performance. An example is shown in Table

4-8. The data contained can be broken down into three distinct groups. These

are:

harmonics of F,
harmonics of F,,
spurious products due to mixing
The table includes values of Vfrom 0 to 7 and M of 0 to 8. Thus, the highest
order spurious term included in the table is 15. The order of any spurious term
is|M| +|V|. For example, the fourth harmonic of F, and the fifth harmonic ofF,,
will result in a spurious product whose predicted magnitude is 76 dB below the
desired output (where M=V= | (for a mixer signal input level of0 dBm anda
local oscillator level of +7dBm). The order ofthis spur is 4+5=9. Note the effect
of mixer input signal level. As the magnitude of F, is decreased, spurious
performance improves. Also, as the local oscillator drive is increased, spurious
performance improves. Therefore, for optimum spurious performance from a
mixer, operate it at low input signal levels, with a high local oscillator drive
level.
Harmonics of F, go straight through a mixer without benefit of conversion by
the local oscillator. Where the harmonic ofthe input signal F, is ] =.N, this level
is the mixer isolation or through-put and appears on most data sheets. Where
the harmonic numbers are greater than one, the through-put suffers increased
attenuation. This fact is important to up converter designs where it is possible
for a harmonic of F, to be equal to the first intermediate frequency.
For example, the first IF is selected to 610 MHz. The input signal F. ranges
from 50 to 250 MHz. By dividing 610 MHz by increasing interger values; note
the values which fall into the range of F, where 50 < F, < 250.
We find

-dB
610/1 = 610 MHz
610/2 = 305
610/3 = 203.33 51
610/4 = 152.5 80
610/5 = 122 72
610/6 = 101.66 90
et cetera (where F, = -10 dBm and F,, = 7 dBm)
Such performance would not be acceptable in any designs but low grade. The
poorest acceptable spurious performance level is generally 60 dB below the
desired level (and 80 dB below the desired level for a quality system). The only
option here is to raise the IF so that 3F, or N=3 is excluded. Where cost is a
factor, customers often will waive specifications for a few such isolated spurious
responses.
The Receiver 125

Similarly, F,, and its harmonic will appear at a mixer’s output without regard
to the input frequency /. Where the harmonic of F), = 1 or M = 1, this value is
called the F,, to IF mixer isolation and appears in the mixer data sheet. This
value is indicative of the balance of a mixer.
Where the harmonic number ofF,, is greater than one, these will appear at the
mixer’s IF output port with increased attenuation. It is undesirable to have
harmonics of F,, falling within the IF. This can cause heterodyne whistles,
birdies, or desensitization of a receiving system at these frequencies.
Example 4-15:
50 < F< 250 MHz
IF = 610 MHz

We find

F,,= IF -F,
= 610 - (50< F< 250)
then

560 > F,, > 360

solving for
KF,, = IF or K = IF /F,,, where K is an integer
610
560 > F,, > 360
We see for this case there are no local oscillator harmonic problems. Here it is
avoided by parameter choice using up conversion. Should the IF have been 403
MHz, then

353> F,,> 153

K(353> F,,> 153)=403
the results would have been:
K IF/K dB

] 403 (out of F range)

2 201.5 45
3 134.33 52
4 100.75 63
5 80.6 45

where F,, = 7 dBm.

This represents poor performance and the designer should reconsider the IF
selection. Note that increasing the level of F,,, while reducing spurious product
126 RecewingSystemsDesign

levels, where M > 1 and V > 1, increases the F,, harmonic problem. For more
information on IF selection see Section 4-5.

4.28.1 Spur Chart Limitations

A spur chart assumes ideal broadband terminations seldom realized in prac-
tice. Further, it is derived at a single value of intermediate frequency. The chart
rarely totally applies to the design for which it is used as an analysis tool. It does
serve as a valuable aid, a step better than a mixer spur graph, but the ultimate
test is an actual receiver measurement. The designer should allow a design
margin when using spur chart values in analysis.

4.28.2 Spur Chart Generation by Measurement

and Spur Identification

F, IF
Signal generator Spectrum analyzer

Signal generator

Fig. 4-31. Test setup for spur chart measurement and spur identification. ©

This technique, though tedious, is easily implemented as shown in Fig. (4-31).

The IF and a value of F, are selected. The spectrum analyzer serves as the IF
filter and provides spur product magnitude and identification. A value of F,, is
computed where:
F,, =| IF + F,| (sign as applies)
The mixer under test is provided the F/, input signal at the desired level. This
remains fixed for the spur search duration.
The spectrum analyzer is tuned to the IF and remains fixed. The signal
generator providing the F, signal is set to the desired test level and the spectrum
analyzer is calibrated to a reference level on the signal where the relationship
IF =| F,, + F,| is satisfied. The frequency F, is then varied over the spur search
frequency range while observing the spectrum analyzer for any other re-
sponses. When one is found, note the magnitude relative to the desired signal.
The spur may be identified by incrementing either F, or F,, first, by aknown
amount, noting the frequency shift of the spur and then repeating the process
The Receiver 127

for other F, or F,, values.

Example 4-17:
A spur is found and its magnitude is 62 dB below the desired reference output.
The F source is shifted by asmall convenient amount AMHz and the spectrum
analyzer display of the spur is seen to shift 3 A. This is a third harmonic of F,, so
N= 3.

Shifting the frequency of F,, by A’. Therefore, this is a fourth harmonic F,, and
m= 4. The spur is then identified as resulting from a third harmonic of F,anda
fourth harmonic of F,,. The order is seventh. The -62 dB level is entered in the
3F. by 4F,, location of the chart. This process is repeated until all slots are filled.
Where no spurs are found, enter a level equal to or greater than the search
threshold or sensitivity. In addition to spur chart generation, this process also
serves to identify troublesome spurs on a systems level.

4.29 LOCAL OSCILLATOR SPURIOUS PRODUCTS

Local oscillators are seldom totally pure and.at the least contain s#me LO
harmonic signals. Synthesized sources will contain clock spurs, which are clock
fundamental and harmonic frequencies, located on both sides of the main
signal, in addition to the main signal harmonics. Where mixers are employed in
the output of the local oscillator chain, mixer generated spurs can be added to
those previously mentioned resulting in further spectral contamination. The
result of all this is aworsening of the receiving system’s spurious performance,
because of LO impurity.
The LO signal, spurious input terms, and their relative magnitudes presented
to a mixer, together with a pure RF input signal, will produce an IF output
which is a replica of the LO signal. However, the IF output is scaled to the
magnitude of the RF input signal less conversion loss of the mixer. The
replication will be reversed spectrally if the LO is below the received signal.
An illustration of this effect is shown in Fig. (4-32).
Gain -6 dB
-0 dBm
R Cx X, IF output

Received signal +10 dBm

Contaminated LO signal

Fig. 4-32. An illustration of IF output contamination due to LO impurity,

excluding harmonic effects which behave similarly but are not shown.
128 ReceiwingSystemsDesign

Harmonics of the LO will cause spurious mixer outputs scaled to the main LO
signal, as before. If they are particularly high they can cause serious additional
spurious outputs. Consider as a limit the configuration of Fig. (4-33). Here the
LO source contains an unwanted spur which is equal to that of the desired
signal. The result is a set of spurious outputs which satisfy the relationship
IF - mF, + nF, + pF, (spur)
where m, n, and p are integers
It is not beneficial to pursue the solution of this relationship because of the
three-dimensional behavior. Spur tables are available only up to two dimen-
sions making it impossible to make any useful spurious predictions. The best
approach is direct measurement.
A partial solution can be had by first letting = 0 and solving for F, for all values
ofm andn which is the ideal case, and then letting m= 0 and solve again for all m
and p values.
The use of impure LO sources should be avoided unless the design is of very low
performance intentionally.

0 dBm Gain -6 dB -6 dBm -6 dBm

| 1 X,
IF
oBeets
utput
Received
signal
gai
dBm =pd
EIihsono BmFadaishan
ta
o utput
Fig. 4-33. An illustration
Contaminated
LO
of mixer
Signal
output for the case of an LO spur equal to
the LO signal itself.

Local oscillator spurs can cause receiver outputs as the receiver is tuned, even
though the receiver antenna input is terminated and the receiver is placed ina
screen room. This is a very simple test to perform and will indicate the presence
of internally generated spurs caused by LO impurity. This test does not show all
spurious terms. It should be followed by a sweep using a fixed received signal
while the receiver is tuned, followed by a swept received signal using a fixed LO
frequency for all frequencies.
For a spur identification method see section 4.28.2

4.30 CROSSOVER FREQUENCIES

Spur search programs have a resloution limit which often prevent a direct
observation of that spurious frequency, which is exactly equal to that at which
the receiver is tuned. In such programs the incrementation of the analysis
The Receiver 129

results in offsets between the receive and spurious frequencies. While it is

possible to predict such points of frequency equality by extrapolation of com-
puter data it becomes a tedious task.
Spurious frequencies are migratory with the exception of harmonics of the local
oscillator, harmonics of the received frequency, and subharmonics of the inter-
mediate frequency. As the receiver frequency is tuned, all other spurious terms
will change their spectral positions. Some spurious terms will diverge from the
receive frequency, others will converge, while some will actually cross through
the receive frequency. The latter are called crossover spurious frequencies.
They are important in that the point where the spurious and receive frequen-
cies are equal or within the IF passband, the receiver will see two signals
simultaneously. The result of this will be heterodynes (or beats), desensitiza-
tion, or both, depending upon the type of receive system. It is desirable to find
these frequencies and determine their magnitude since no preselection filtering
will be effective in their removal.
A simple relationship determines the crossover spurious frequencies.
Case 1 Difference mode.

F,,=le,-F,| (4-62)
and

F,,=me’,
+nF| (4-63)
substituting
(4-62)
into
(4-63)
F,=|mFi, Pye | (4-64)
but
F = F, to be a crossover term.
then
Fi, (1-m) = F, (n-m) (4-65)

ah |(n-m). (4-66)
Case 2 Sum mode

F,,=|F,+F, (4-67)
and
F,= |mF,,
+n (4-68)
substituting
(4-67)
into(4-68)
F,=|mk,,+
mk,+nF| (4-69)
but
=F.
130 RecewingSystemsDesign

then
Fi, (1-m)= F, (mtn) (4-70)
.eeFo= | Fyaa
eas, :
(4-71)

Where
Fis the desired receive frequency
F. is the spurious frequency
F,, is the local oscillator frequency
F,,is the intermediate frequency
m and n are integers of either sign
(a useful range of mand n is 9, to -9 or 18th order maximum )
The solution of these simple equations is solvable on hand-held calculators but
is somewhat slow and should be solved by desk top or better machines.

4.31 COMPUTING NOISE FIGURE GIVEN TSS

From the definition of TSS as 8 dB output $AN, noise figure may be computed
from:

NF = TSS -8-kTB (4-72)

Example 4-18:
Given

TSS = -90 dBm

kTB = -114 dBm, (B = 1 MHz)

Solution:
NF = -90-8 - (114) = 16 dB

REFERENCES
[1] Frutiger, “Noise in FM Receivers with Negative Feedback,” JEEE, Vol.
54, Nov. 1966.

[2] Schwartz, M., InformationTransmissionandNoise.NewYork:McGraw-Hill,

ee ps)
[3] Emde, Jahnke, and Losh, Tables of Higher Functions, 6th ed. New
York: McGraw-Hill, 1960.

[4] McVay, Franz C., ‘“‘Don’tGuess the Spurious Level,” ElectronicDesign,Vol.

3. Feb. 1, 1967.
The Receiver 131

[5] Norton, David E., ‘The Cascading of High Dynamic Range Amplifiers,”
Waltham, MA: Anzac Electronic.

[6] Goldberg, Harold, ‘‘Predict Intermodulation Distortion’’, Electronic De-

sign, Vol., 10, May 10, 1970.
 .

132 Rece wing SystemsDesi1gNn

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 COMPONENTS

Vital to the characterization of a receiving system and the prediction of its

performance is a knowledge of the components used and their capabilities. This
chapter addresses this area in a practical sense. The goal is not to design
components but to assist the designer in the selection of acomponent and assign
achievable performance values to that component.
Several computer programs are included which allow the user to determine the
theoretical performance of some of the more popular forms. It is strongly
recommended that the reader develop a library ofcomponents from manufac-
turers specializing in their design. This is invaluable for reference purposes and
future procurement.

5.1 FILTERS

The filtering problem is solvable through the use ofseveral popular filter types.
Included are:
LC
Crystal
Helical
Tubular
Cavity
It is not the intent of this text to examine this subject in any detail except to
point out the limits of the operating frequency of each type together with the
range of achievable bandwidths. The data is necessarily approximate but
serves as a guide for the designer. Where requirements are out of these bounds it
is best to contact a supplier to see if the art has been extended and the design is
feasible.
The LC filter is capable of operating over a range of roughly 100 Hz to several
GHz with bandwidths ranging from approximately 1/2 to 130%. The particu-
lar range of realizable bandwidth versusfrequency is shown in Fig. (5-1).
134 ReceivingSystemsDesign

a——_—__——._ Lowpass Limits —————_->

(Bandwidth does not apply)

(Bandpass
only)
B
%andwidth
10

Da Wvtoghratelbetat
slagebboeid eee a
O44) ) 30)| 1004°9) 19D) 100. 4< ae
——_——kHz——_——___}_ MHz + GHz—
Fig. 5-1. Approximate range of bandwidths available for LC filters versus
frequency (©1982, K&L Microwave, Inc.)

Helical resonator filters are easily realized over a frequency range of approxi-
mately 20 MHz to 1 GHz, with bandwidths of 0.2 to 3, or more (typically 15%).
Tubular filters, while bulky, are useful in some applications. This class of filter
is available in either lowpass or bandpass designs. The lowpass filter is available
from 10 MHz to 18 GHz and the bandpass type is available in the 30 MHz to
12.4 GHz range. Bandpass filter bandwidths are in the | to 80% area.
Typical helical and tubular filter characteristics are shown in Fig. (5-2).
 Components 135

rf |
-al___towpass___o]
Tubular
Filter

.
&

F 1
3 typ 15 max
2Apr
SEE
Eee
||
co
Cc
qe
a

s
:
z
oe
a
=
8
é 0.1

| | |
10 100 1 10 100
EN Ee Semen ealSO 2 a
Fig. 5-2. Percent bandwidth and frequency range for tubular and helical
bandpass filters. For the lowpass tubular filters, the percent bandwidth does
not apply (©1982, K&L Microwave, Inc. and an April 1976 TelonicAltair
catalog)
Cavity filters, which are popular for the first conversion filter in up conversion
receiver designs, are capable of narrow bandwidths (typically 0.1 to 3.5%) in
the frequency range of 30 MHz to over 40 GHz. Wideband filters of this class
are available with bandwidths of 5 to 50% at frequencies ranging from 200
MHz to over 18 GHz. Fig. (5-3) illustrates these characteristics.
136 RecewingSystemsDesign

Wide Band

Bandpass
Filters
Coaxial
for
Bandwidth
Percent
a
as
mf
w
won
Narrow Band

| | |
10 100 1 10 100
MHz———_——- GHz ——
Fig. 5-3. Percent bandwidth and frequency range for cavity filters (©1982,
K&L Microwave, Inc.)
The characteristics of crystal filters are discussed in section 5.1.4. Several other
filters are available and are worth mentioning. This includes the interdigital
filter, which, when using strip line techniques and the air dielectric, a 3 to 30%
bandwidth is achievable over the frequency range of 1 to 5GHz.
A second important filter class is the comb-line, which is capable of 1 to 15%
bandwidths over the frequency range of 1.3, to greater than 20 GHz with the
air dielectric.
Both of these filters are made in other forms and utilize dielectrics other than
air. Generally these filters are relatively high frequency types. The size of the
Components 137

filter is a function of the number of elements as well as the type of dielectric. It is

suggested that suppliers be contacted for details.

One of the most popular filter types is the Chebyshev because of its steep
selectivity. This is achieved at a sacrifice of phase linearity. These filters have an
inherent ripple, which results in distortion of certain waveforms. Where this is a
concern, other filter types should be considered. The Bessel and the Gaussian
filter have a good phase linearity characteristic but with a sacrifice in skirt
selectivity. Here the response is parabolic. The Gaussian filter has a rounded
group delay characteristic where the Bessel filter has a flat group delay.
In some applications it may be necessary to match the phase of filters. Where
this is the case, such performance is typified by Fig. (5-4). This is asomewhat
generalized illustration and it must be kept in mind that such matching is more
difficult in complex designs.

bp
S
-
Qa
~
— 3 ~
=, al £
3 tlh nae
a an Was ‘ oe

0 10 20 30 40 50 60 70 #80 90 100
Percent -3 dB Bandwidth
Fig. 5-4." Typically achievable phase match in degrees, as a function of a 3 dB
bandwidth (from a TelonicAltair catalog dated April 1976).
Phase linearity is often important. This can be approximated by reference to
filter charts for particular types. In general, for a 1.3/1 VSWR condition, the
phase linearity illustrated in Fig. (5-5) is achievable. Again reference should be
made to filter curves for specific cases.

5.1.1 Filter Insertion Loss, Bandpass Case

While there are equations for filter insertion losses, the most practical ap-
proach, except for special cases, is the use of the loss constant (L,). For any filter
the insertion loss (J;) is simply:
(N + 0.5)
fat %B ae: (5-1)
138 RecewingSystemsDesign

H4 / /
3 % /
2o /
Q3 ty =
£: fr
S Spat
£
a
®
2 VSWR
1.3/1
<1
a

0 10 20 30 40 50 60 70 80
Percent of the -3dB Bandwidth

Fig. 5-5. Typical phase linearity which is achievable as a function of the -3 dB

bandwidth of the filter at a VSWR of 1.3/1 (from a TelonicAltair catalog
dated April 1976).
where

N is the number of sections

L, is the loss constant

B is the 3 dB bandwidth

It is readily apparent that a narrow bandwidth has the highest insertion loss,
and the converse it is also true.
Typical values of loss constants are given in Table 5-1. These values are
approximate, but sufficiently accurate for most system uses. For special filters,
or where fractions of a dB are of concern, filter specialists should be consulted.

5.1.2 Filter Insertion Loss: Lowpass and Highpass Cases

The insertion loss (/,) equation for the lowpass and highpass cases is:
I, =NL, (5-2)

where
N is the number of sections
L, is the loss constant
Table 5-2 lists some typical values for various filters. These values serve as a
guide in system design. Where the design is critical, or special filters are used,
filter experts should be consulted.
Components 139

Table 5-1.
Approximate Loss Constants for Bandpass Filters
Frequency (MHz)
Type & Ripple 30 50 65 100 400 600 900 1300 1800 3000 10000
50 65 100 400 600 900 1300 1800 3000 10000 12000
Tubular
.05 dB
.25”"dia. 5 4.) 450 35 Sb) Bb cerZ
.375” dia. 4 ae a 2 1.6
5” dia. CP Se,ShYR OP. M4 a 4 |
.75" dia. rE Rehe agi Ls ae he 1.2
1.25” dia. Bad Zed ch BnekkeLe als?
LC0.1dBbas to6—
Micro miniature 6.8to 4.9 4to 3.25 3.0
0.1 dB ripplt 6.8 5.7 to 43.25

Cavity 1.716 1.5 1.4 ltolto.35to .3to 3to .45 35

a Te «IR
Miniature 3to 2
Cavity yt

Table5-2.
Approximate Loss Constants for High- and Lowpass Filters
Frequency (MHz)
Type & Ripple 10 25 50 100 250 500 1000 2000 4000 6000
25 50 100 250 500 1000 2000 4000 6000 18000

Tubular
.05 dB
.25” dia. oo 1.250425 2 18 a 1
375” dia. Pride
Wt a 18 16
5” dia. i Dakts Geek(pin (Recs, 1] oa
.75" dia. ovale ho vio cle ohi!
1.25” dia. a SF2" "091.08 +206" 107
LC .01 dB bi to L4—

Micro min. LC 45 eybe 1 ] ]

.1 dB ripple to.2 to.15 to.15 to.15

SA
140 RecewingSystemsDesign

5.1.3 Varactor Tuned Filters

The varactor tuned or voltage tuned filter is an LC filter, where C is the

variable supplied by a varactor tuning diode.
These filters are usually two or four pole designs, or cascades of two pole filters.
Tracking becomes a problem as more poles are used. These filters are most
often critically coupled tuned circuits and are available in constant Q, or
constant bandwidth designs.
Typical performance from a two pole design is shown in Table 5-3. These filters
are used with a tuning linearizer for remote tuning purposes. In manually
tuned applications, a simple potentiometer serves as the control element.

Table 5-3.
Typical Performance of Two-Pole Varactor-Tuned Filters
Available frequency range 10 MHzto2 GHz
Tuning range 1 octave
Bandwidth 4 to 12%
Insertion loss =2 dB
Shape factor 3/30 dB a 5
Input signal power 2 watts
Tuning time microseconds
Tuning voltage 0 to 20 or 60 volts
Size 1” by 1” by 0.5”
Weight 1.5 oz

The varactor tuned filter is often designed in house because of its relative
simplicity. The design usually takes on the form of two tuned circuits coupled
together. Where this configuration does not provide the necessary selectivity,
two or more (usually not more than three) tuned circuit pairs are placed in
series, with amplifier isolation between each pair.
There are several popular configurations in use. They are shown in Figs. (5-6)
through (5-8).
Coupling into the filter from the input to the primary tuned circuit can take on
any of the impedance transformation configurations. This applies equally
well to the output tuned circuit coupling. Of particular interest is tap or link
coupling. If this is accomplished through a series inductance, the coupling loss
of the filter can be made nearly flat. This is illustrated in Figs. (5-9) and (5-10).
While the previous discussion has ignored the varactor diode, or tuning diode,
there is little significant alteration of the filter, except for a degradation of Q
power handling, and intermodulation distortion of the end configuration.
Components 141

C, C,

L, Cm L; k= VOpCs

A. M=-O

eee
p =s

B.

C, L, M M. L, C, = M:Mz
= CHIL

C. Li’ Ls’
Fig. 5-6. Low impedance coupling forms of two coupled tuned circuits.

Lp L, L

Fig. 5-7. High impedance coupling examples of two coupled tuned circuits.
142 RecewingSystemsDesign

Fig. 5-8. Complex coupling methods which reduce the bandwidth dependen-
cy on frequency.

Cstray

Fig. 5-9. An illustration of the use of series inductance when coupling to the
filter, to flatten losses over a wide range of frequency.
 Components 143

Amplitude
LOG FREQUENCY

Amplitude
LOG FREQUENCY
L, = FINITE VALUE
Fig. 5-10. An illustration of the effect of L, on loss as a function offrequency.
The tuning diode is one where the diode capacity, as a function ofreverse bias,
is controlled and enhanced. The operating range of such a diode is from cutoff
to below reverse breakdown. In its equivalent form it consists of a series parallel
arrangement of R, C and L as shown in Fig. (5-11).
Data sheets will generally specify the values of the parameters of Fig. (5-11).
The reverse bias is usually given at four or six volts in a circuit application at 1
MHz. Diode capacitance will be given as C,,,where v is the bias potential. This
value gives the designer a measure of the device’s capacitance for comparative
use.
The Q of the diode is often neglected by the designer resulting in selectivity
degradation. The data sheet will often provide Q. values (at a bias of v volts) or
provide C,,,and R,,, at some frequency / such as 50 MHz. Then:

: I
Co ie Ree. relates these parameters (5-3)
144 ReceivingSystemsDesign

High Q is more easily obtained with diodes which have lower breakdown
potentials. The penalty for this is a lower capacitance ratio given by:
CG, _ __ Capacitanceat 0 volts bras (5-4)
Cy, Capacitanceat reversebreakdown

Where high Q and high dynamic range are mandatory, varicap diodes should
be used in groups of two in a back to back arrangement, as shown in Fig. (5-12).
The program of Table 5-4 may be used to evaluate the performance of critically
coupled transformers. This program is valuable in the initial design stages of
system design and serves to inform the designer of the capabilities of this type of
filter. For more exact solutions of a particular design, particularly where
complex coupling is utilized in a circuit design phase, a circuit analysis pro-
gram such as COMPACT should be utilized.

R” 10° OHMs Typ.

R’ 10OHM Typ.

Fig. 5-11. Equivalent circuit of a varactor diode where C” and L’ are the
parasitic capacity and inductance, respectively; R’ is the sum of the bulk
resistances; R” is the leakage resistance of the junction; and C is the capacit-
ance. The latter three parameters are a function of reverse bias.
Components 145

Ee tuning
voltage

tf nw tuning
voltage

Fig. 5-12. The preferred arrangement of tuning diodes for lower distortion,
higher Q, and power handling capability.

Table 5-4.
Computer Program for Critically Coupled Transformer Response
iY LN te uce eae
e4 SHORT F
3G 01S “CRITICAL COUPLED
FORMER
©SRESPOUSE"
44 DISP “GEFINE CENTER FREGQUENE
VYORHSo.0L.N4UNBER OF

90 INFUT F1.G,4H
ee: Dgsr “HEF INE FREGHIEMIC
Y
2TART. STGP, AHO INCRE!
CMHE 3"
70
26 THPLUT
O1SP ALE,
“ENTER ULTIMATE ATTENUA

TION COB"
99 INPUT UJ
1G@@ DISE “GRAFH? YES nl"
11 INPUT Ag
146 ReceivingSystemsDesign

26 1F AS="VES" THEN 24 ELSE 13

MA

a PRINT "CRUTI Gers COUPLEDTRA

SPORMERSS«RESBOHSE
oe 9'sHae
=4Sak Oe a medSE FREGUENC Y="; KI
CaaSytyfte
158 PRINT “GQL="i0he "N="5H
LO Pi LAT Oo BS ee ee Se ee ee ee
by, de ee te AeaeSea>Be
76 "PRINT “FRED ORES ELITE
ND
UrBoebtEP ed
Pel+cuete ee Fob
ell THEW aaeli
="NO" THEN PRINT TABt12
Bcl9*: X3-ELSEMORAW
FF. +X
ea
MwA
To
220
Ty
oN
ed
de
fat NEXT F

rs “4 I as »x

TSP "ENTER ¥ a imfi MAX ATTNOD

=ate!
Lire
|rhWt
od E H.B.-K. &

|hom
diol “t te
pel
net
w+. iT—{fi
as.
eae
iwa) i
— TSFim "ENTER
rm<4o YARIS TIC. MARKS <
DES!
IHPUT Me
eS
oie
mw
Whee AARRIS -kK.M1
Bo
Bes
OS
ae
Soe
OS)
eS =

AXIS As Me
REM LABEL #AKIS
LOIR 3A
FOR X=A+2eM1 TO B STEP Mi
MOVE Rue CKE. SS
LABEL VWALE¢K>
IN
Be
BY
ce NEXT X
REM LABEL ‘AX
LOIF
FOR Y=-K TO 8 STEP M2
445 MOWE AFtCB-Aoe 2a, Y
450 LABEL VWALSEcyo
46@ HEXT YY
47@ MOVE At+CB-AaeS, 2ee-k 3
$56 LABEL "CRITICAL COUPLED"
 Components 147

$98 MOVE A+CE-AD75S. .25%*¢-K>

2H LABEL " TRAWSBayt

S18 MOVE A+CB-AdeS. . Sk0-K >
228 LABEL “GQL="SVALE¢o>
234 MOVE Htc B-Ab’) So. 2kC-K)
"48 LABEL “H="SBVALECH 4
S20 MOVE AtCB-AI 4? 2, 97#¢-K35
meh
tSlg i 9| 3aif Rohan
ed9fpSa ee
oF MOVE AtCR-Ade?, SkE-K 3
one & AEE Lot obLe
ore i ae
SH GOTO 186
Bel EWC
Example 5-1:
Center frequency? 90 MHz
Qi? 40
No. of Transformers? 9
Sweep Start? 70 MHz
Sweep Stop? 110 MHz
Increment? 1 MHz
Ultimate Attenuation? 60 dB
Graph? Yes
(It is suggested that a No be used first to establish the scaling from a print out
table)
Y axis max attenuation? 70 dB
X axis TIC 5 MHz
Y axis TIC 10 dB
The result of these inputs is shown in Fig. 5-13. It is not required to enter the
scaling such as MHz or dB. These are a part of the program scale factors.

| /\as.\
|
-16
‘. TPANS
AL!; e oMPLED
CORLITTE
SFORNERAS
-20@ aL=4@ H \ N=s
|| a3a 'A ''
{ 4
LoBS ! \
-49 : \

-5 ; ‘
a i = uw ™
=A! re bs u a on
nea
CMH!
ar A ET OP LeSea, eae,meee S DS. eheee
Fig. 5-13. The computer output for the graph condition for Example 5-1. This
graph is valuable in the determination of attenuation capabilities for critical
frequencies such as: IF rejection, IMAGE rejection, LO leakage to the anten-
na, et cetera.
148 ReceivingSystemsDesign

5.1.4 Crystal Filter

The crystal filter is the one most widely used in IF applications because of its
inherent stability, low cost and good performance. The crystal filter exhibits a
typical insertion loss of 3 dB, a phase linearity of 10% over a 75% bandwidth,
with shape factors of 3 to 5.5 and for special cases (with poorer phase linearity)
bandwidths of 200 Hz with shape factors (3 to 60 dB) of 1.1 to 1.
A graph of typically attainable performance is shown in Fig. 5-14. Shown, are
bandwidths obtainable as a function of center frequency. ‘The graph serves as a
guide to performance. Borderline cases should be verified with manufacturers,
since breakthroughs are possible.
10 —

1.0 —

£
=
s
a
r=
©
ise]

0.1 —

0.01 +
1 10 100 1 10
|} kz —_______—_—}—-
MHz
Fig. 5-14. Approximate range of available crystal filter bandwidth as a func-
tion of frequency.
 Components 149

Because of the very high Qof crystals, the usual filter consists of several crystals
staggered in frequency throughout the bandpass (resulting in ripple). Ripple
may be of concern to the designer since distortion can result, particularly with
FM systems.

5.1.4.1 Monolithic Crystal Filters

The monolithic crystal filters are a type of crystal filter where the input and
output resonators are deposited on opposite sides of a piezo electric wafer.
These very small, low cost devices are capable of operation in the 5 to 350 MHz
range with inductorless bandwidths ranging from 0.3 to 2% of the center
frequency.
Because of their low cost and small size, monolithic crystal filters can be used to
good advantage in channelized receiver front ends, as well as in intermediate
frequency applications.

5.1.5 Modeling the Butterworth Filter

The Butterworth filter response is described by:
A (dB) = 10 log,( 1+ (k)”) (5-5)
where
k = b,/b; for low- and bandpass filters
k = b,/6, for highpass and notch filters
b, is the 3 dB bandwidth of the filter

6, is the bandwidth at the frequency x

nis the number ofresonators for band and notch filters and is the number
of reactances for the high and lowpass cases

This equation accurately describes the response of any of the four filter classes.
The response predicted by this equation is approached in practice, and for this
reason a margin must be allowed when going from theory to practice.
When making calculations of a Butterworth response, a finite value ofultimate
attenuation should be selected to truncate the calculated attenuation. Typical-
ly, a value of 60 dB is attainable without much difficulty and values to 100 dB
are approached.
To assist the reader the Butterworth equation has been programmed in the
BASIC language and is shown in Table 5-5. This program has a print or plot
option, is prompting and interactive, requiring the operator to only answer the
questions asked. Some modification may be necessary in the plot routine to
accommodate the particular computer used. (This program was written for the
HP 85 machine.)
150 RecewingSystemsDesign

Table 5-5.
Butterworth Filter Program for Low-, High-, Bandpass, and Reject Filters
19 SHORT F
a DISF "BUTTERWORTH FILTER RES
PONSE™
3Q DISF “SELECT FILTER TYPE <L>
OW, (BYANO. CHIT PASS.OR cRIES
ECT"
40 INPUT Bf
S@ OISE "ENTER 308 POINTS FMIN,
FMAM"
INPUT F1-F2
ots
Motte
Nad
ty!
Hla)
Nat OISE “ENTER WK"

te,
iHPUT
DiSP
Hi
“OEFIME FREQUENCY SHEEP
START. STOP, ANDO INCREMENT
CMHES"
1@G INFUT ALE. I
110 DISP "ENTER ULTIMATE ATTENUA
TION <OBo"
1206 THPUT U
130 DIS "GRAPH? YES No"
140 INPUT Ag
i598 IF AS="YES" THEN 398 ELSE 16
t-M4
168 PRIHT "BUTTERWORTH FILTER RE
SPONSE"
178 PRINT. "BAND WTO" Pit to:
SCO 43BA
12H PRINT "“H=";Hi
T9OQ PRINT #®At Sete eee hee
tw hk See Se
2G PRINT "“FRE®Q cCMHe> ATE
WN: BB it
FOR, F=F, TO BOSTEP. wf
Hh
ts
fae
PoP
Th
Pl
tal
fe
A AEN
EN
baIF.
IF
Bs="6".
8S="L"
THEN
THEN
GOTO.
GOTO.
2
2
IF BS$="H”" THEN LOTU 3
on B$="F"THEN GOTO sea
Jonas
MMthe
ifoot
|'
re

cro-.
om
SERBS CF -OCRA-FbAH2eFi
=FE-Fi @ H2=18#LGT*t1+¢bBle-
a» 8

}
mn
waee @ BA=FS 2 AS=LAELGTO1+¢
Components 151

268 BLi=2S*ABSCF-C¢CF2-Fide2+Fias @
BS=F2-Fi @ HS=1G#LGTC1+¢ Bae
BLS G2aN ho
14 GOTO 344
2 Bi F @ B2=F1 @ XS=1GKLGT<C1+¢

340 IF ¥3>U THEW X3=U

oe SE=INTCXS*106+
537100
B® IF At="HO" THEN PRINT TABI
sFSTRBCI95;5;43 ELSE ORAW F.,-

HEXT F
O
c~mmS
te) com
me
JOo
GoTo 74e
GRAPH
GCLEAR
CISP "ENTER YAMIS MAX RTTHCO
By"
Je
fi A
te)ae THPUT
OU K
SCALE fees ke
CIS “ENTER MAMIS TIC MARES

MANOA
ee
OO2H
6b
c MHZ"
ITHMPUT ft
DISP
CE >”
"ENTER YAXIS TIC MARKESS

INPUT M2
Wt
He
ist
Cyt
tlt
Ot SARIS -K.M1
oN
Sr
SCO
Oe
oe
cc WARTS AMZ
FEM LABEL XAKIS
LOIFP 36
FOR A=A+2S¢M1 TO B STEP fi
MOVE: Ke —-CKK.OD
LABEL VALS(¢XKo
=
mI HEXT &

0S
eS
ieam
S REM
LOHIR
FOR
MOVE
LABEL
&
y=-K
‘AXIS

TO 6
A+Cb-Ao-2o.
STEF
7
Me

TT
at LABEL
ey NES Th c¥ VWALECY 3

oT
=)
ad
Fe
itSe MOVE AtCB-AIeS, 2#¢-k
(orLHBEL "“BLUTTERHORTH FILTER"
ce
ee 5

MOVE A+CB-AdeS. 259#¢€-k 4

LABEL “FREGUENCY RESPONSE"
MOVE At+(CB-AD 41 5S. . SKC—-K>
LABEL “"H="SVALECH1 3
152 ReceivingSystemsDesign

634 MOVE A+CB-AdH2. 2, . 97° KC-KS

699. (LABEL SVLenhs 2
(06 MOVE HtCB—H) Af soe C4K 2
BAO ee ce Std.
(fee MOVE “Ark
738 GOTO 216
r46 ENG

Example 5-2:
Computer Inputs:
Select Filter (Reject
-3 dB Points (MHz)
FMIN 300
FMAX 330
N? 4
Sweep (MHz)
Start 250
Stop 350
Inc. 2
Ultimate Attenuation (dB) 60
Graph ? YES
Y axis max dB 70
X axis TIC (MHz) 10
Y axis TIC (dB) 10
The result ofthese computer inputs are shown in Fig. (5-15). The program will
provide responses for all of the popular forms of this class of filter.

5.1.6 Modeling the Chebyshev Filter

This popular form of filter may be modeled in its theoretical form by the
equation shown below:

A(dB)
=10log,E+[(log”
<= - ]Jcosh’
[ncosh
(#)]
where (5-6
A.,,,, 18the ripple in dB
n is the number of resonators for the bandpass and notch filter cases

a
Components 132

n = the number of reactances in the low- and highpass cases

k = 6,/6, for the highpass and band reject cases
and
k = b,/b, for the low- and bandpass cases
(6; is the 3 dB bandwidth and 3, is the bandwidth at frequency x. While this
definition is not exact it is useful for k values of two or more.)
25ethanesentmet
athemnar
ssaes

| :
ra 1 i ‘ ;
BUTTERWORTH) FILTER!
..) FREQUENCY a Ry,

| 1
fiaak i
a
COB |
~48 |

-~5a | |
pir
= me mt em) Ts! we! ot me cys is
“65 rt on 7; roe cs ty mm i um
Morag «Go °t0. To 1° . Bs” 4
ae CMHZ >
eel: &GRR 2bee ae AO ae eee 2
Fig. 5-15. The computer graphic output for the notch filter specified in
Example 5-2.
This equation is useful for low-, high-, and bandpass, as well as the notch filter
cases. It is not useful to carry out this computation beyond the expected
ultimate attenuation. This value is typically 60 dB, with 100 dB approachable
(using care).
The sharpness of the response is a function of the value of and ripple. As these
values increase, the shape factor is reduced and approaches unity. The re-
sponses predicted are approachable in practice; although some allowance must
be made for departure in the practical sense.
This equation has been programmed in BASIC and is shown in Table 5-6. The
program is interactive and prompting, with print or plot options. The user
needs only to answer the questions asked. Some modifications may be necessary
for machine variances.
154 RecewingSystemsDesign

Table 5-6.
BasicProgramforthe Print or Plot of theChebyshevFilter ResponseforLow-,
High-, Bandpassand Notch Filter
HES Poulos1B}
ThBaie
cy DISFE. “CHEB PSHEVMOPTOTERPRESFU
ens
RipdehESfae9ciate | ahONREI opeh ashg 3) ad
OM. CBoAnO. CHsTGH. PASS OF CR
Fetes 17
46 INPUT 6¢
324 OSPF “OQEFINE FILTER BAND HID
THiFMIN. FAAS. CMHS2?"
ete INPUT
reeth Fi.Fe
S
i
my t
ee OISP WENTeER!RIPPLES cen ans
ye
IMPUT Fi.
—_
OISP "DEG TNE FREQUENCY, SHEEP
i START. STOP. ANO THCREMNENT
(MHZ
oh iI I —”*INPUT
peem AE. 1
«Ce QDOISP"ENTER ULTIMATE ATTEHMUA
TIQN «OB >"
fe
CF
PsSt
tad 1aS IHPUT
adak
7)
Se
ee BISP*’ SGRAEH? TES ANE
INFUT AF
IF AS=2VES™ THEN 320 ECSET IS
~
o
ty —
Ls
. he ERIN) "CHEBLSHEY- FLCLER REESE
OME"
if] FRINT "“GAND WIDTH="; Fi: ’ TO";
PS tibe: &
TSO PRINT ARLEPLESRS hala ie Nie
196 PRIHT “FREEEPRAEREEEERERREER SE
KEKEKEKRARHK"
“64 PRIWT “FRED «MHZ, HT TE
euler Be=de
fea
fis
PAPI
Ma
ha Pil
fe
Ty
isd
ee FOR Shi TOs eer an
ea
ho
ee IF
Te BS="E" THEH GOTO hed
PH
ed
Cod
Past
Ty
mJ
+ ol
TF BS="L" THEN GOTO et
ee
TY
Ee
TF B$="H" THEN GOTO
IF B#="R" THEN GOTO
Bi=2SrtABS CE Cer e—-hil 2 jou.
ie fe
BS=FS-Fil &@X2=B1-BS
te OPA. Mel eee eee coe
M401
mJ TRAST AEN Soe wel ors toe
i GOTO 488
oo i=F & Ba=Fe @ K2=BI-ES
Components 155

Cad
taj
tl ts)
Poe THEH 468
The
7De
TDineon
eo
ov
TH
mat
ee

U3)
5)
Po

oon
Ss
Oo
aS

Fi) @#2= 3B 2e9R1a7ve

BRP
hPHhHh
HH
ot
ol Bi= F CyB= 1 Oo oR:
IF F>Fi THEN 466
Toe
-)
CO
be
oj
Mo
oo
mo
es
CS
ee
eS BOT 4h
Y=HLi*KLOGCRKS+¢X%2-
“e-13
CL=CEXPCYI+EXPC-Y5 d+
R=1O*¢R1i-1@)5-1
2=1864¢LGT¢1+RKC1L*2¢ X25
¥3==RBS*1BG#K3+55-108
GOTO47h
“2=8
TF RSPu THEH #3=U
IF A="HO" THEH PRINT THEC1>
iF TABCISISSS ECSE DRAW F.-

Neeste

hel
ac
eS
Pte
ALA
Sm
mo
"ENTER YARIS MAS ATTNSO

a " il
TNPUT KF
AeA
mes
SCALE
onDISP
Ty ii
A. B:-K;@
"EWTER KARIS PIL
0 obPE Na
lo
J IHPUT ft
Tt ~oven DISP “ENTER YARPFS LC
LIE 4 at
A
yy THPUT Me
Tt “AAIS -K. M1
Ro
aeed
mo
mg
iAe
Pl
eT
fe
Tey
Ts
oy mo
TySahelYARISA:RES
ekREMLABEL
Sad ok Ss

LOIRa
FOR
MOVE R=A+S4M1
y axl SeaT O B STEP Mi

LABELWALECK >
Sa
Sr
Sy
i
NEO
REM
TLABEL YAXRIS
LOIR
PUR. Yeh LOSS STEER Me
156 RecewingSystemsDesign

MOVE A+CB-Aav2e.7
LABEL VALECY2
CONT
Le
OhNESW! F
Jmd
m
od
md)
JEy
Se
eT)
Sy
eeMOVESARtCBSAD435i S24 9=K9
CHBbEL “CREB YOHEYVSAIE)Gk’
MOVE A+CB-Ao-S. . 25k C-K 3
LHEEL “FREQUENCY RESPONSE"
MOVE A+CR-Aaed, . GREE-K 3
798 LABEL "“RIPPLE="VALECR 15
S60 MOVE At+CB-AD/ 1.SHtSEC SK
S16 LABEL "“"N="8VALSCH1
S20 MOVE A+CB-AI72 2; 97 KC-K>
a4 SoLock. (MNEs
is MOVE A+CB-Ade PR. SKC-K3
S28 LABEL * (OB a?
S66 MOVE A, -K
S78 GOTO 216
S388 DRAW F.KS
$38 END

Example 5-3:
Computer inputs for the Chebyshev filter program:
Filter (H)ighpass
Bandwidth (MHz)
F min 400
F max 10,000
Ripple (dB) 0.1
N 5
Sweep (MHz)
Start 0
Stop 600
Increment 6
Ultimate Attenuation (dB) 65
Graph ? YES
Y axis max (dB) 80
X axis TIC (MHz) 100
Y axis TIC (dB) 10
The computer output for this filter is shown in Fig. (5-16).
Components 157

nee |

Fig. 5-16. Computer print-out for Example 5-3. This isa highpass Chebyshev
filter with a ripple of 0.1 dB and N=5. The program is usable for the other
popular filter types.

5.1.7. Distortion

Since nothing is perfect, the output of a device, circuit, or network differs from
the input stimulus. This difference or distortion has several forms and defini-
tions, which include:
Phase delay
Delay distortion
Envelope distortion, group delay, or envelope delay
Each of these will be described in the following paragraphs.

5.1.7.1 Phase Delay

Phase delay results from the time it takes a sinusoidal stimulus to pass througha
circuit network or device. Because of this time delay (which is almost always
frequency dependent) the output has a phase shift, which is a function of
frequency. This characteristic is shown in a typical example, in Fig. (5-17).
Here the phase delay may be computed from:
pp phase shift in radians _—9
seconds (5-7)
w ( radians/ second)
158 RecewingSystemsDesign

where
w=2r f
f is frequency of interest
5.1.7.2 Delay Distortion
Delay distortion is the phase delay difference between two frequencies. Refer-
ring to Fig. (5-17), the delay distortion between two frequencies f,; and /, is:
6, 6,
t= eo eens) (5-8)
W» W
5.1.7.3 EnvelopeDelay, Group Delay, or AbsoluteEnvelopeDelay
The derivative ofthe phase shift versusfrequency curve at a particular frequen-
cy of interest results in the rate of change of phase or group delay at that
frequency. A plot of group delay, for various frequencies, provides the designer
with information regarding the linearity of that network, circuit, or device.
Where linearity is critical, for example + 10%, reference to the group delay
curve will define the frequency limits available (within this percentage). Given
group delay, the phase shift at any frequency may be computed from:
Phase shift = group delay (sec) *w
d6
0 = ‘ow (5-9)
dw

Phase
Shift
(7
radians)

p=

0 1 2 3
Frequency in MHz
Fig. 5-17. An illustration of the various forms of definitions of distortion and
delay.
Components 159

Example: The group delay at 2MHz is given as 1.825 yseconds; find the phase
shift at this frequency.
1.825 -10°+-2m+2x 10” radians/seconds =
7.3 radians = 7.3 +57.3°/radian = 418.3°
This example was taken from Fig. (5-17). (Note the correlation in the result.)

5.1.7.4 Relative EnvelopeDelay or Group Delay Relative

The group delay between two frequencies, where one is the reference, provides
a measure of relative group delay in seconds.

Example 5-4 (Ref. Fig. (5-17) ):

Phase delay at 1 MHz.
res 3.6 1 Rad
= 1.8 psec
"Qa °1+10° Rad/sec

Delay distortion from | to 2.5 MHz.

td=t»( 1 MHz) -t,( 2.5 MHz )

Gs 9.5 m Rad 0.1

me Bee ——_————
=
Dr 2.5 10° Rad/ sec. -0.1 ap sec

Group delay at 2 MHz.

Ag gg
d0 ___ (8-654)
8 - 6.54 wRad
Rad = 1.825 psec.
S1y das (2.2-1.8 ) 2 710° Rad/sec
5.1.8 Computer Prediction of Group Delay for Butterworth
and Chebyshev Filters
The designer may wish to have a print-out or plot of group delay for particular
frequencies and bandwidths, without the laborious process of translation from
the normalized curves contained in filter handbooks. Two sets of equations are
presented for two popular filter types, the Butterworth and the Chebyshev. The
equations are applicable for lowband and highpass responses and may be used
for the readers’ own predictions. For those who have access to machines
programmable in BASIC, two programs are included. It may be necessary for
the reader to modify these programs to cope with machine variances.
5.1.8.1 Butterworth Filter Group Delay
Group delay for the Butterworth class of filters may be approximated from:
: | : | of|
m
160 ReceivingSystemsDesign

where
T, is the group delay in seconds
F,,,, is the upper -3 dB frequency (Hz)
Fi, 1sthe lower -3 dB frequency (Hz)
F is the frequency of computation (Hz)
n is the number of resonators for bandpass cases or the number of
reactances for the lowpass case

J is 0 for lowpass and | for bandpass filters

0, = cos 8,
w, = sin 6,

2k
6, = ————+n-1 90°
n

l ae F aie
oe esSas F ee}.Subtnmonssnanbuavanst
Fax 7 Fimin F

For lowpass, enter a non-zero value for F,,,, such as 0.000001.

For bandpass filters enter the -3 dB points. The above equations are contained
in the BUTTERWORTH GROUP DELAY PREDICTION program of
Table 5-7. The program language is BASIC. Prompts are included in interac-
tive form, with print or plot options.
Example: Two examples using this program are included for reader reference.
The first example is a 100 MHz lowpass filter with five sections. The following
parameters were entered:

L for lowpass filter

N=5

F,,,, = 100
INC = 2 for the increment

YES for PLOT ?

10 ns for Y axis

10 for X axis TIC marks

1 for Y axis TIC marks

Components | 161

Table 5-7.
Program for the Prediction of Group Delay for Low- and Bandpass
Butterworth Filters
18 FRIWHTER [5 1
2u SHORT FFL F2.FS. 71 at eX
34 ODISP "BUTTERWORTH FILTER GRO
UF DELRY®
48 DISP "CHOOSE LOW ¢L> OR BAND
CEo (PASS RITTER?
74 .INPUT .-S
oo IF F¢="L" THEN J=4 ELSE J=1
f) DISP “ENTER NUMBER OF SECTIO
Ns "
Ss IHPUT Hil
98 DISP "ENTER FMIN.FMAK.F IWCR
EMENT ¢fH2>"
1pe
PO
tJ INPUT FIsF2.r¢
cae
dl
ee iT eo DISP “GRAPH? YES NO"
INFUT BS

aPRINT6$=**TES”" THEN.31@
"BUTTERNQRTH
ELSE
GROUP
-14
DEL
t—~
uy
fo ‘o—
G' , ur
PRINT "H="GNi
— @ PRINT TABC13;. "MHZ"; TABCISo;"
Hsa il
17@ FOR F=F1 TO F2 STEP F3
ise@ DEG
i94@ Ti=G
208 FOR N=1 TO Ni STEP 1
S16 A=C2HN+H1-1)-N1¥#94
220 O=ABS¢i-CF2-F13¥¢F-F2eF ie Fd?
234 SZ=SINCA?
244 C1=COSCA
250 TH18G8/¢2XPI¥CF2-F1) RCL +INE
ABS (Cid ¢C1*2+¢0-S2)%25
Ped
faa
Tit
cTsSy
-sJ
o
= m™x af =
IF BS="NO" THEN PRINT TABEL)
(FS TABC15S3;7T1 ELSE DRAW F.T1
NEXT F
to
Pl
oy
hm
BoTS
cS
eye
GOTO 67a
GRAPH
GCLEAR
162 RecewingSystemsDesign

ha]helht] DOisSP “ENTER YARIS MAK VALUE

HAHNDSECOWOS"
248 IHMPUTFE
e348 SUCALEcELake ais &
26% DISP "ENTER ARIS TIC MARKS¢
MH22"
eS SE ok Bored
450 JO0 SF CEN ei Tee Pee oe
HAHOSECONOS2"
236 INPUT Me
406 #AAIS &.Ml
416 TAXIS FL.Me
426 REM LABEL “AXIS
436 LOIR 3&
$40 FUR Soe) tt) Oe eee eee
456 WOVE #.E “26
456 LABEL VALECK:
HEAT «a
REM LHBEERTRATS
SS
0 LO Le.
CO
O
—1
eo
me
oe
Oo
am
1Saal
N
y
_ae POR: ¥=%2 TU. KwSTEP Me
ACA
MM
BMA
MMMM
A
AeA
LA
fpLAMOVE Fit+«Fe-Fl: ar
Sa
eal

aat
Uae
eS LABEL VALCOYo
HEAT ¥
NOVESR IACh eek Df)Anak
LABRb CRON Lia fk Lede
MOVE EBLAtGPe-Flo/5. Soak
ae
Ty
oe
ys
a LABEL “GROUP DELAY"
HBOVEMFist Ch P=Fi) 43 oc SEE
LABEL NHE"SMALS CNT3
MOVe>FFLSCE ASF lee fae an
LABEL. = (BHZ.2°
Ed
Sy
ee
a BUMEor EUR ePID Ie fe od ate
LABEL TiN.
MOVE Fi.@
GOTO i746
iy ORAM
mr F.T1
ENO

The result of this program is shown in Fig. (5-18). Asecond example is shown
for a bandpass case. This example retains the same filter bandwidth of 100
MHz; to show the doubling of the delay for a bandpass filter (for a given
bandwidth as compared to the lowpass case). The entries are as follows:
B for bandpass filter
Ness ae
F min
nin= 300
Components 163

F nar = 400

INC = 2 for the increment in frequency

YES for PLOT ?

20 ns for the Y axis

10 for X axis TIC marks

2 for the Y axis TIC marks

The results are shown in Fig. (5-18). Note the doubling of the lowpass values for
the bandpass filter with the same bandwidth.

HUTTERWOR
TH FILTER
ROUFQOELAY

4 “MHED
=e aon] mi Dat ™ Si ot] iS tS) ™
efi MM Row oo on Oe
TE RO weMe a ty Pe
Be filter ee ol

Fig. 5-18. Examples of lowpass and highpass filter group delay predictions
using the computer program of Table 5-7. Note the filter bandwidths are the
same, and the bandpass filter group delay is twice that of the lowpass case. Also
note the program inaccuracies near zero frequency for the lowpass case.
164 ReceivingSystemsDesign

Please note that the program for the lowpass case is inaccurate near zero
frequency. The true characteristics would be an extrapolation of the curve
from the value (from right to left). In other words, the characteristic is flat near
the zero frequency point.
It is suggested that a print be selected (NO to PLOT?) before a PLOT is
attempted, to secure the scale factors. The print output is not constrained and
the actual values will be output. Once this is secured, a plot can be executed
with the appropriate scale factors.
The program statement 10 PRINTER IS 1, should be deleted for the print
option. This statement was only included to save paper in the print-out. Once a
satisfactory solution is achieved, a print condition can be exercised. Alterna-
tively, COPY could be used for a hard copy ofthe screen.

5.1.8.2 ChebyshevFilter Group Delay

Chebyshev filter group delay may be computed from:

wyek aela ee ia eae

Lae Poee ay F
;> | o,| (5-11)
where
T, is group delay in seconds
F,,, 1sthe upper -3 dB frequency (Hz)
F inISthe lower -3 dB frequency (Hz)
m

F is the frequency of computation (Hz)

o,=sinh
ie] sinh”
+]sin
] 0,
n a

w,=cosh
| +sinh”
] =Jeo
] 6,
n a
Components 165

sinh x = In[x + (x? +1)'?]

where
n is the number of resonators (bandpass and notch) and the number
of reactances (low- and highpass)
ah e@-e
sinh x =
2
eox + ¢ -X%
cosh x =
2
This program approximates the actual value at mid-band and the band edges
but tends to be low in other areas. It is still useful in approximate terms.

Aj, 18ripple (dB)

For lowpass filters F,,;, = 0
For highpass filters F> F,,,,

The above equations which are quite similar to those of the Butterworth class,
are contained in the Chebyshev Group Delay Prediction Program of Table
(5-8), and include prompts. The program is interactive and allows the selection
of print or plot options.

The references for this section are [1] and [2].

Example 5-5:
Ripple(dB) 0.5
N (sections)
~-3dB Points (MHz)
Min 850
Max 950
Increment 2

GRAPH ? YES

Y axis max value ns 24

X axis TIC MHz 10

Y axis TIC ns 2

The results of these inputs to the computer, using the Chebyshev program for
group delay, results in the output of Fig. (5-19).
166 RecewingSystemsDesign

Table 5-8.
Program for the Prediction of Chebyshev Filter Group Delay
LY SHORT Far i ahe ease) a ee ee
2a°OUISPr “CHEBYSHEY, FIL GER GROUP
BELA
36 OISPF "ENTER RIPPLECOE?.WHUMBE
Re GhoS&e hiGuse
44 INPUT F.N1
“6 UISP “ENTER FRINGEMANSE INER
EMENT CMH"
a)
Lal
i INPUt em dy heaikss
Oey
aory—
UESP. 7ORAPH?) J ESano"
IHPUT 8
wy
= (Se if BS="VEs" THER 294 SLSE 16
(a
7"* _ —_ PRINT "“"CHEBYSHEY GROUP DELAY
~ t {

[nh
tathe
A) x PRIHT “RIPPLE="GR."°N="GN1
ce Ee LAT TRB 1 OS @MHZ* PEAR hs) an

1.28 For Pe) LL eee

148 QEG
134 T1=68
i6@
Hot
MM
ftoew
he
lv @-ESOL. Reh
FOR H=1 TO
eer ore
Wi STEP 1

1S) SISTANISCLOGGT ECCLES 4eta

194 A=C(2HEN-L1I-Ni+¢s90
298 C=RBSCi-CF2-FL eC P-F2EEL oF 3%
21M S@=(ENPCS1I-EXPC-S1 903-2451
Fi
S2M CLH=CESPCSLI+EXR Pe Siac eesecase
A
SEA ToLGRG-CSEPLECPE-Fiv sbi se ze
FivF2 *ABS(S2946924274+°0-C1)
rn—t
sd TL=T14+T
2S HEXT H
S64 IF BS="HO" THEN FRINT TABE13
27 PeTARCISx;TL
HEXAifs ELSE DRAW ET
236 GOTo 8Pre
Components 167

Pej
[4
cegiA
eeG
ed
yesade GRAPH
iad
GLLEAR
OISF "ENTER YAXIS MRE VALUE
HANUSECONOS"
CJ
tel
tad IHPUT FE
ON
was
m
>J PCALE F1,F2:8,K
i DISF "ENTER SARKIS TIC MARKSC
ila aeoe
Un]
Wad INPUT [41
et
i
OISP “ENTER YARIS TIC MARKS¢
NANOSECONDS 3"
INPUT M2
SAIS w.Mt

Snap
DMloc
Oe
omMe
NUP
oOo
oo
oe
oy YARISFi. Me
AION
AMAAATAA
OHM
BER
HL
Pj
i!
PRE
i)AI
VINOD
VIO
VO
UII
Vo
goD
sHPD REM
LOTR

MOVE
LABEL
LABEL

FORSK=FA
96
th
A.K AEE
SARIS

WALECRS
TOSFS STEP M1

NEMT &
REM LABEL YARIS
LOTR ©
FOR Y=M2 TO K STEP He
NOVE Fir tre lL yee.
LABEL VAL#¢Y>
NEAT ¥
MOVER PSOE SHF baxcds. OK
LABEL “CHEBYCHEYVFILTER”
MOVE} Robt OPreatoPhLov Sins SHEK
LABEL “GROUP OELAY"
NOVEs dE eek dD 7 SR aK
LABEL "“RIFPLE="8VALECR)
Uv tt Orie Lake to. teh
LABEL "“H="B.VALECHL
MOVErE ie Ch2chilavia2iiin. 2kK
1 GLA
st
Uys Bele +/,0MNHZ do
,a MOVED LKOr Cer Lr» aia
SLAB EL = SONS2%
MOVE Fi.
im GOTO
cS)
me 1268
DRAW F.T 4
EHD
168 RecewingSystemsDesign

pore SHebT CHE Vere TER

Lee BROUPF OELA‘ 7
RIPPLE=.5 H=4
I| = hi
' i
Lé i
'
1\3 ! !

io Se TRE ee.

6
(MHZ
_ rey mi ml im me my im ™ mi mi
Peas
© i m iT; = al mJ a b> uy
mo Ww co mm Ty uy iT) Ty it ivy
Sciavenaapnaps
happen
cenennllinsiepianentnalioennntnageabdiedne
RE RT Pa RE SG Fan
Fig. 5-19. The computer prediction of group delay for a filter of the Chebyshev
class with a ripple of 0.5 dB and four resonators. See Example 5-5.

5.2 MIXERS

The mixer is the key to superheterodyne design. It may be considered a three

port component. There are two input ports. These consist of the received signal
and local oscillator ports. The output is the intermediate frequency port. The
mixer (sometimes referred to as a multiplier, though it is not in the strict sense)
outputs two principal signals, which consist of the sum and the difference of the
two inputs. In practice, only one of these is selected as the desired signal. This
selection is done by appropriate filtering of the mixer’s output. Either output is
acceptable as the signal, depending on the mixer mode (sum or difference).
Mixers may take on several different forms, which may employ: diodes, transis-
tors of bipolar or field effect types, et cetera. The three basic mixer forms include:

single ended
balanced
double balanced
These configurations are shown in Fig. (5-20). Variations include the use of
transistors, several diodes in series, either of these in series with resistors and
capacitor balancing, or any combination of these. Higher performance bal-
anced mixers usually employ baluns at the RF and LO ports when the inputs
are unbalanced.
Components 169

(b) Balanced mixer

Fi. F,

a
(c) Double balanced mixer
Fig. 5-20. The three basic mixer forms: (a) single ended, (b) single balanced,
(c) double balanced.
170 ReceivingSystemsDesign

The single ended mixer is the simplest of the various forms and is used in low
performance radios, such as in the home entertainment and hobby field. It is
the poorest performer from the spurious product point of view. Spurious
products are those mixer outputs which result in something other than the
desired converted signal. A mixer will generate outputs as defined by:
|M F,,+NF|
where
M and WN
are integers including 0
and
F’, is the local oscillator frequency
Fis the received frequency
The desired output occurs where M = V= | using the appropriate sign for the
mixer mode ( + for the sum and - for the difference).
There can exist a multiplicity of products, including harmonics of F,and F,,. To
avoid as many of these products (which could fall into the intermediate
frequency passband, and appear as legitimate signals even though all but one of
the M= N= | are not), balancing is utilized to reduce the magnitude of some of
these terms.
The simplest of the balanced mixers is shown in Fig.(5-20b). Here the spurious
population is reduced through balance by decreasing the magnitude of some of
the terms by 20 dB or so. By using the doubly balanced configuration of(c), a
further reduction of the spurious population is secured. This is the most
popular class of mixer which is used in quality products.
A further consequence of balance is the increase of the power handling capabil-
ity of the mixer, because two or more diodes or active devices are used. The
result of this is an increase of the local oscillator drive capability which, if
exploited, increases the intercept point and -1 dB compression point of the
mixer.
The relative merits of the three configurations is tabulated in Table (5-9).
Mixers are characterized by using purely resistive terminations matched to the
mixer’s impedance. Any other non-resistive matching results in an increased
level of the spurious population. Therefore, it is desirable to provide the mixer
with broadband resistive matching. This idealized condition is seldom realized
in practice because it is usually ignored.
The mixer, local oscillator, and input signal ports are easily matched using
broadband amplifier drivers. Filters’ driving mixers provide a match only
within their bandpass and should be avoided when high performance is
desired.
 Components 171

Table 5-9.
Relative Mixer Performance

Class
Single Single Double
Ended Balanced Balanced

Spurious Density | 4X 2X ]

L.O. Drive Typ. dBm 7 7 to 10 10 to 27 or more

Port to Port
Isolation dB ey 10 to 20 20 to 50

-1 dB Compression dBm 0 0 to 4 4 to 23

Intercept Point dBm 10 10 to 15 20 to 33

The same holds true for the IF output port. The IF output port should see a
broadband match such as is provided by a broadband amplifier’s input impe-
dance. An alternative solution is to use two filters with parallel inputs. The IF
bandpass filter’s output is fed to the IF amplifier. This filter provides a match to
the mixer within its bandpass. Elsewhere the impedance is not matched. By
adding a notch filter in parallel at the input and terminating this filter resistive-
ly, the input impedance of this pair of filters will be essentially flat over a
broadband. This filter can provide a match to the mixer IF port everywhere
except within the notch, where it has a higher impedance. By making the notch
bandwidth equal to the desired IF bandwidth, a broadband match is achieved.
These mixer matching methods are shown in Fig. (5-21).

5.2.1 Specialized Mixers and Applications

5.2.1.1 Termination Insensitive Mixers

A termination insensitive mixer results when the configuration of Fig. (5-22) is

employed.
Mixers ofthis type are popular in high performance designs. They are capable
of handling strong received signals with good spurious performance without
exercising Care in impedance matching.

5.2.1.2 Image Rejection Mixers

In certain wideband applications it may not be practical to utilize preselection
filtering of the RF input signals. In such cases, the image response could appear
as a legitimate signal. A degree of image rejection (20 to 30 dB) is achievable by
using the configuration of Fig. (5-23).
 172 RecewingSystemsDesign

IF Filter

Ai, Ao, A3 are broad-

band amplifiers

(a) Broadband amplifier matching of a mixer.

IF Filter

Fe aie ——-—To IF Amplifier

Zo

z5 fa fr

Frequency Frequency
IF Filter Impedance Notch Filter Impedance

Zo
Frequency
Composite Filter Impedance

(b) IF Port Matching Using a Notch Filter.

Fig. 5-21. Methods of broadband matching of a mixer IF Port.

Components 173

Fig. 5-23. Image rejection mixer configuration capable of 20 to 30 dB of

rejection.
The degree of rejection depends upon circuit balance. Not only must the
amplitudes of the converted signals at the final combiner be equal, but the
phase must also be matched. The severity of this problem is addressed in
reference [3]. Amplitude balance is given as:
(1+ Q/(I-Q)
Example 5-6:
If7=1.1Q
then the balance would be 21 or 26.44 dB and the two signals would have to be
amplitude matched to .83 dB.
174 ReceivingSystemsDesign

The phase balance severity is shown to be:

cos 9 = 1 -c’/2a,” assuming ideal amplitude balance
Where
a and dare the two converted signals at the combiner
and
a = b and cis the third side of the triangle.
Example 5-7:

Assume a= 1 andc=.1 then

cos 6 = .995 or 5.73°
and the balance is 2a/c or 20 or 26 GB.

One ofthe difficulties encountered with this configuration is maintaining gain

and phase balance over broad frequency bands. However, for certain relatively
narrow band applications, this represents an ideal solution.
See reference [3] for more information.
5.2.1.3 Harmonic Mixers

Often at microwave frequencies a mixer is operated at product orders other

than 2. That is, M+N# 2. More specifically, MF,, + F,= IF where V= 1 and
the sign as applies. Thus the mixer utilizes harmonics of the local oscillator to
combine with the fundamental of the received frequency to generate the
intermediate frequency. While effective, the conversion loss is necessarily high,
directly increasing with M for single ended mixers. Balanced mixers are seldom
used especially where the LO port is balanced, unless only odd harmonics of F,,
are used.

Typically LO harmonic numbers of 2 through 10 are common and in some

special cases up to 60. In general, a good mixer is a poor harmonic mixer.
Therefore the balanced mixer is a poor choice because harmonic responses are
suppressed.
Biasing also has the effect of reducing the required LO drive which may be
reduced to 0 or -10 dBm in starved operation.
The conversion loss of the harmonic mixer is on the order of:
HARMONIC OF LO CONVERSION LOSS, -dB
2 : 12
ST
OO
Po
SR
i

~I
—
 Components 175

The noise figure is the conversion loss. Consequently, harmonic mixers are not
used in a high sensitivity design.

5.3 LINEAR AND NON-LINEAR AMPLIFIERS

A linear amplifier faithfully reproduces all amplitude variations of the input

signal at its output. A non-linear amplifier reproduces the input signal ampli-
tude variations in a non-linear manner. The non-linearity may be a threshold
which must be exceeded by the input signal before an output occurs. It may
take the form ofa linear amplifier operated in saturation, or an amplifier witha
non-linear input/output relationship (such as a log amplifier).
There is definite need for each category of amplification. The linear amplifier
can, of course, process any and all signals; however, it may not be economical to
do so.
The application rules are as follows: where the information to be processed is
contained in amplitude variations of the signal, the signal must be amplified
linearily. (This does not include on/off keyed signals.) Failure to follow this
rule will result in severe distortion or complete loss of information. Examples of
some signals requiring linear amplification are shown in Table 5-10.

Where information to be processed is contained in phase, frequency, a combi-

nation of the two, pulsewidth, or position, non-linear amplification may be
used.

Table 5-10.
Popular Signal Forms Requiring Linear Amplification
Double sideband amplitude modulation (DSB AM or AM)
Double sideband suppressed carrier amplitude modulation
(DSB-SC AM)
Single sideband amplitude modulation
(SSB-AM)
Multilevel pulse amplitude modulation
(M PAM-AM)
Amplitude shift keying
(ASK)

In certain applications, such as limiting amplification, the use of the non-linear

form is mandatory. Some of the more popular forms of signals which may be
amplified by non-linear means are shown in Table 5-11.
176 RecewingSystemsDesign

Table 5-11.
Popular Signal Forms Which may be Processed by Non-Linear Amplification
Continuous wave interrupted carrier (CW)
On-off keying (OOK)
Pulsewidth or duration modualtion (PWM or PDM)
Phase modulation (PM)
Multiphase modulation (digital includes biphase, four or eight phase, etc.)
Frequency modulation (FM)
Frequency shift keying (FSK)

5.3.1 Limiter

A limiter is a signal processor which has upper and lower bounds that limit the
excursions of a function of time in the amplitude axis. The idealized transfer
function of a limiter is shown in Fig. (5-24).

input amplitude

Fig. 5-24. Idealized limiter transfer characteristic.

This transfer function is known as a hard limiter because the action is definite
and invariant. Once the limiting threshold is reached, any increase of input
amplitude does not allow any further increase in output above that threshold.
Departure from this characteristic results in what is known as soft limiting. The
distinction has no well-defined boundary.

The behavior of a limiter is illustrated in Fig. (5-25).

Limiters are used to remove amplitude perturbations from a signal and in no
way do they disturb the crossings of a signal in the time axis. Limiters should
not be used to process signals where the message is modulated onto the carrier
in the amplitude axis, such as AM, because the modulation would be removed.
Where the message is modulated onto a carrier in quadrature to the amplitude
axis, such as FM, limiting will not disturb the message. The limiting process
Components 177

causes the generation of harmonics of the signal frequency because the sinu-
soidal carrier is converted to a form of the square wave. Limiting also is effective
in the removal of AM noise from a signal, as is obvious.
Many forms of FM demodulators require limiting of their input signals to
maintain constant output with signal strength. Examples of this are the dis-
criminator and the slope detector.
Limiters are simply devices which go into saturation on overload. One simple
configuration is two diodes in parallel with one being reversed in series, with a
current limiting resistor. The voltage across the diode would be limited to + 0.6
volts for silicon diodes. Other forms are amplifiers with low supply voltage. The
amplifier output cannot exceed the supply voltage. Most limiters are of this
form.

Amplitude
Input signal Output

Fig. 5-25. An illustration of a limiter processing an amplitude variant signal.

5.3.2 Successive Detection Log Amplifiers
A logarithmic amplification characteristic may be desirable in applications
where signal level information is required, or where instantaneous outputs are
required from a wide dymanic range of signals. Linear amplifiers designed for
threshold signals with gains of 100 dB (such as an IF amplifier) may have an
output capability of several volts at threshold and a saturation level several.
times that. Therefore, only a slight increase in input level will cause saturation
of the amplifier. Most such amplifiers have a headroom ofonly 3 to6 dB. AGC
is one solution for saturation but because it is generally non-linear and has a
finite time constant, it cannot be used with any accuracy where instantaneous
outputs are required or where signal strength measurements are required.
The use of a cascade of amplifiers, where the output of each stage is detected
and all detector outputs are summed, results in a quasi-logarithmic, character-
istic of the form:

e,=k log eé,,

where &kis a scale factor.
The configuration is shown in Figures (5-26) and (5-27), together with a
computer-simulated output which assumes ideal linear detection. The result
178 ReceivingSystemsDesign

clearly shows the logarithmic characteristic with the deviation from the ideal.
Because of the non-linear input/ output characteristic, linear amplitude mod-
ulated signals will suffer from amplitude distortion. The main application of
logarithmic amplifiers are in: pulsed signal systems, spectral analysis, filter
characteristic measurements, ef cetera.

k log input

Fig. 5-26. Successive detection logarithmic amplifier.

LOG AMPLIFIER ANALYSIS
NUMBER OF STAGES IS 5
STAGE GAIN IS 18
STAGE SATURATED OUTPUT IS 18
VOLTS

TOTAL GAIN IS 186 De

INPUT SIGNAL IS .841 TO 1646

MILIVOLTS

=
< ptae
®
Sie
a <
|pao
| w
= ~
is
Oi baad P

fs

fetadinit abewwbyians
Yleathgaadud,be—— eae
.01 1 1 .01 .01 1
Millivolts Volts
Fig. 5-27 A computer-simulated output for a successive detection log
amplifier.
Components 179

REFERENCES

[1] Schreiber, Heinz H., ‘“‘Phase and Time Delay of Butterworth and Cheby-
shev Filters,” Microwaves, March 1965, page 14.
[2] Nicholson, B.F., ““The Practical Design of Interdigital and Comb-line
Filters,’’ The Radio and Electronic Engineer, July 1967, page 39.
[3] Gorwara, Asok K., “‘Phase and Amplitude Balance: Key to Image Rejec-
tion,” Microwaves, October 1972, page 64.
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 SPECIALIZED RECEIVER
APPLICATIONS

Other than the reception of communication signals for information exchange,

the second most important application is the surveillance ofsections of the RF
spectrum in search of signals with particular characteristics. This is done
largely in electronic warfare applications. The major variables which are
involved are one or more of the following:
Frequency
Direction of the emitting source or signal spatial position
Time or duty cycle
Modulation characteristics
Signal strength, et cetera
These receiving systems consist of three major parts:
Antenna(s) and distribution system(s)
The receiver(s)
The signal processor
The antenna system may be called upon to provide directional information to
the emitter and could consist ofa single movable array, or a multiplicity offixed
antennas whose signal strengths (and/or times of signal arrival) are compared
for signal positional computation. The latter may use one or more receivers.
The receiver may be called upon to provide the frequency of the emitter for
jamming or logging purposes. Additionally, information regarding the signal
strength of the emitter (at the receiving site) plus the nature and content of the
modulation may be required.
All of this information is fed to a processor which may store and/or compare
these data, with a reference for decision making purposes or analysis.

There are two major categories of receivers used for this purpose; included are
narrowband (less than one spectral octave) and the wideband (covering one or
more octaves). Included in the narrowband group are:
Scanning superheterodyne
Linear
182 ReceivingSystemsDesign

Smart scan
Microscan or compressive receiver
The wideband group includes:

Channelized receivers
Crystal video
Tuned radio frequency
Instantaneous frequency measurement (IFM) receivers
The principal advantage of the wideband receiver group is their non-scanning
operation, which permits instantaneous handling of frequency agile signals. A
disadvantage is their spectral resolution. Each ofthe above will be discussed in
the following sections.
6.1 DEFINITIONS: SCANNING SUPERHETERODYNE

Linear

The spectrum analyzer is an example of a superheterodyne structured receiver,

whose local oscillator and preselector are tuned, tracked, and linearly swept
over a range of frequencies.
Smart Scan

This is a special case ofthe linear type where the desired signal population is
known and where the sweep is restricted to spectral zones. The advantage of
this technique is the lower scan time.

Compressweor Microscan Receiver

This is another variation of the swept superheterodyne with an extremely fast
sweep time (typically one microsecond). To resolve a signal without amplitude
degradation, a wide IF bandwidth is required. The output of the wideband
filter isan FM chirp which is fed to a compressive delay line, whose time delay is
frequency dependent and matched to the sweep. This compresses the signal,
concentrating the signal power while the noise is not concentrated, resulting in
an enhanced signal to noise ratio.

Instantaneous FrequencyMeasurement (IFM) Receiver

This is a receiver which utilizes a wideband discriminator usually one or more
GHz wide to measure frequency. Since it is a non-scanning system, its opera-
tion is instantaneous.

Channelized Receiver

The desired spectrum to be monitored is broken into cells, usually contiguous,

though not necessarily so. Each of these cells is provided with receivers whose
detection bandwidths are equal to that of the cells. Such a system is non-
SpecializedReceiverApplications 183

scanning and is therefore an instantaneous wideband type. Frequency meas-

urement resolution is necessarily restricted to one cell width.

6.2 SCANNING SUPERHETERODYNE RECEIVER

A conventional superheterodyne receiver (when provided with a swept local

oscillator and a wideband or tracking preselector) becomes a scanning super-
heterodyne receiver.
The scan speed is limited by the IF bandwidth. Narrow IF bandwidths require
a slower sweep speed, or a loss of sensitivity will result. This relationship is:
AF’ 0.25
NE) cwuesermaecaranel
sae|
5.282 7B" ie
where

A F is the sweep width

t is the time of one sweep
B is the 3 dB bandwidth of the receiver

Bandwidth governs the noise floor level or threshold of detection for positive
signal to nosie ratios. It also governs the resolution of the system. An appro-
priate selection of parameters for specific performance must be made. A typical
scanning superheterodyne receiving system is shown in Fig. (6-1).

Fig. 6-1. A basic scanning superheterodyne receiver.

6.3 SMART SCAN RECEIVER

The smart scan receiver is a variation of the scanning superheterodyne receiver,

which speeds up the system by omitting those portions of the spectrum which
are of no interest. To illustrate, reference is made to Fig. (6-2). Shown is the
local oscillator waveform, which is swept only through the frequency ranges of
F2 to F3 and F4 to F5. The remainder ofthe range (such as F/ to F2, F3 to F4
and F5 to F6) is skipped by increasing the sweep speed as much as possible. The
result is that there will be no output of useful signals in these unwanted zones.
The principal advantage of this technique is a lower acquisition time of the
desired signals.
184 RecewingSystemsDesign

Sweep
Frequency
F;
eee) ee rs ta
TimingSweepWaveform.(One Cycle)
Fig. 6-2. An illustration of smart scanning.

6.4 INSTANTANEOUS FREQUENCY MEASUREMENT

RECEIVERS (IFM)
This class of receivers is designed to provide the frequency of an incoming signal
on an instantaneous basis. Such performance precludes the use of swept tech-
niques. One realization of such a receiver is shown in Fig. (6-3). The signal is
band limited to that of the wideband discriminator, amplified, and fed to two
channels. These are the signal presence and the frequency measuring channels.
The signal presence channel serves to detect the presence of a signal, and then
keys a signal processor. The second channel consists of a delay line and a phase
discriminator, driven from a power divider. The discriminator is sometimes
called a correlator network. The arrangement is shown in Fig. (6-4).

The phase discriminator consists of two input ports which are a 180° and a 90°
hybrid, which in turn drive four diode detectors through two 90° hybrids, as
shown in Fig. (6-5). The outputs of the four detectors produce four outputs
whose magnitude and phase are dependent upon the phase difference of the
two inputs to the correlator. These are represented by:
SpecializedReceiverApplications 185

Fig. 6-3. An illustration of an instantaneous frequency measuring receiver.

FZb

PaZa Phase
Discriminator

Fig. 6-4. The discriminator used in the IFM receiving system.

V,=(A’+B’)+2ABcos6
V,=(A’ +B’)-2ABcos 0
V,=(A’+ B’)+2ABsin 6
V,=(A’+ B’)- 2ABsin6
By differentially comparing V,and V,, and also V,and V,, the term (A’ + B’) is
eliminated, leaving two signals:
I= Scos 6
Q=Ssin 6
These signals are furnished to a processor which vectorially sums the Jand Q
signals, producing a magnitude component proportional to signal power and
an angle proportional to phase. The delay line shown in Fig. (6-4) establishes a
phase @,which is a function of frequency, since time and frequency are related.
This relationship is linear, resulting in the capability of measuring frequency
with good accuracy. To avoid ambiguity 6 must be less than 360° over the
frequency band of interest.
The delay line is theoretically errorless, however the correlator is not (because
of its complexity and tracking errors). Typically, a+6° error is realized. Fora |
to 2 GHz system this results in a + 17 MHz error possibility.
The correlator output is usually digitally processed. Should greater accuracy
be required, four correlators may be used with 4" weighting to provide 10 bit
186 RecewingSystemsDesign

AZ@

0 1/2 1 3/2 2n

Fig. 6-5. Correlator configuration and output phase relationships.

accuracy, as shown in Fig. (6-6). The delay lines are ¢for D,; 4¢for D,; 16t for D,
and 64t for D,. The respective resolutions are 1, 1/4, 1/16, and 1/64. Thus, the
system’s resolution is 64 times better than that of one correlator system. The
ambiguity is still determined by the delay line associated with D,. A four-
correlator system will have a resolution of one part in 1024. Therefore, for a 1 to
2 GHz band, the resolution is ~1 MHz. The dynamic range of the IFM is
typically limited by the correlator to 30 dB.

6.5 MICROSCAN (COMPRESSIVE) RECEIVER

This variation of the swept superheterodyne (Fig. 6-7)) results in fast signal
acquisition. To accomplish this, the sweep speed is dramatically increased to
cover the entire tuning range in a microsecond or so. As pointed out in section
6.2, the signal amplitude loss would be very great unless a very wide IF
bandwidth is used. To overcome this, the signal (which is aFM chirp) is fed toa
compressive filter which has a time delay matched to the sweep. This delay isa
function of frequency. For a positive or increasing frequency IF chirp, the time

S
SpecializedReceiverApplications 187

10 Bit Binary Output

Fig. 6-6. Multiple discriminator arrangement for improved resolution.

Fig. 6-7. Compressive (microscan) receiver block diagram.

delay of the compressive filter is negative with increasing frequency. This
characteristic causes the low frequency end ofthe chirp (the first to arrive in this
case) to be delayed most, while the higher frequencies are progressively delayed
less. The result is acompression ofthe IF signal energy into a narrow pulse. The
associated noise is not correlated and is therefore unaffected in density, whereas
the signal energy is enhanced. The result is a compression gain of the signal,
which is the time bandwidth product of the compressive filter.
Example 6-1:

Given

Tuning range 1 to 2 GHz

Time of one sweep 1 microsecond
Then

B =|1-2GHz| = 10’ Hz
The compression factor is:
10° x 10° = 10° = 30 dB

The output pulsewidth is:

l microsecond / 10° = 1 nanosecond
 188 RecewingSystemsDesign

The performance ofthis class of receiver is limited by the side lobe level of the
compressive filter, which is typically 30 dB, restricting its dynamic range.
Sensitivity and frequency resolution are good, and acquisition time is ~ 10 ys.

6.6 CHANNELIZED RECEIVERS (CRYSTAL VIDEO)

The crystal video receiver consits of an input bandpass filter followed by
optional RF gain and a detector. A typical configuration is shown in Fig. (6-8).

NY, Optional
PreAmplifier
BandPass Ms
Filter L- DiodeDetector =
ToVideo
Amplifier
Fig. 6-8. Basic crystal video receiver.
Three of the most popular detectors are the:

Schottky diode
Tunnel diode
Point contact diode

Each diode has unique features which make it useful for a particular
application.
The Schottky diode has the highest tangential signal sensitivity, output, and
burnout rating.
The tunnel diode does not require bias. It has the best response time, lowest
output video resistance, and the best thermal stability.
The point contact detector provides the best match, requires no bias, and has
the best frequency response.
Of the three, the Schottky diode is the most popular, largely because of its higher
sensitivity. Typically this diode is capable of a TSS of -50 to -52 dBm ina 2
MHz video bandwidth, and of a video amplifier noise figure of 3 dB. This can
be improved through the addition of an RF amplifier.
The crystal video receiver is most often used at microwave frequencies because
of its broadband nature. It is ideal for channelized applications where several
receivers are used to cover adjacent frequency bands. The response is instan-
taneous for signal presence because of its non-scanning nature; it is also capable
of a degree of frequency measurement resolvable to its bandwidth.
An example of channelized receiving is shown in Fig. (6-9). Here the 1 to2 GHz
frequency range is divided into 10 bands, each of which is covered by a crystal
video receiver (preceded by an appropriate bandpass filter). The frequency
resolution would be:
 SpecializedReceiverApplications 189

(1 - 2GHz)/10 = 100 MHz

Because the bandpass filters are not ideal, there is a zone of overlap at the band
edges with adjacent filters. Should this be of concern, adjacent channel output
amplitude comparison would resolve this problem.
The preamplifier is usually of bipolar, tunnel diode, or GaAs FET design.
Where low power consumption is a major concern, tunnel diode and, more
recently, GaAs FETamplifiers are used.
The function of the preamplifier is to provide sufficient gain to overcome
detector noise and become as dominant as the noise source. Typically, 40 dB of
gain is used for this purpose.
Because of the non-AGC’d design, the crystal video receiver has a low dynamic
range. This is particularly important when preamplification precedes the
detector.

1 to 1.1 GHz
sent 7
Filter

NY, 1.1 to 1.2 GHz

sea :
Filter
1 to 2 GHz

Band
Pass
Filterhana Ere :
1.9 to 2.0 GHz
Channel 10
Filter

Fig. 6-9. Basic 10 channel channelized crystal video receiver.

6.7 BRAGG CELL RECEIVER

The Bragg cell receiver is an electrooptical spectrum analyzer. The basic

configuration is shown in Fig. (6-10). The Bragg cell is a processor which is
furnished with two inputs, one of which is from a wideband receiver through an
RF acoustic coupler and the second, an optical beam from a laser source. The
cell is acoustically and optically transparent with photon-phonon relationships
directly related to its inputs. This results in a variation of the refractive index of
the cell, which is linearly related to signal frequency.
190 ReceivingSystemsDesign

Laser
Illuminator

one deat ga“ F2

\ ee ge
‘xs >
ash e §
sh ae Nee
Sit 3 8
\ 5 0
\ \\ /
ey,
snl
4
Fig. 6-10. An illustration of a basic Bragg cell receiver.

With no signal input the laser beam remains at A, which is off the detector array
surface. An increasing RF input signal frequency causes the beam to deflect
toward B&in a linear relationship.
The detector consists of an array of optical detector cells. The resolution of
frequency is directly related to the number of detector cells. High resolution
capabilities are available through the use of TV pickup devices such as CCD
optical detectors.
The Bragg cell receiver is capable of multiple signal handling, which produces
individual outputs in the Fourier plane.
Summarizing, the Bragg cell receiver is an instantaneous processor capable of
multiple signal handling, which can provide (as outputs) the frequency ofthe
input frequencies, as well as their distribution. The frequency resolution of the
system is directly related to the number of cells used in the detector array. The
sensitivity of this type of system is in the vicinity of -80 dBm.
The limitation ofthis approach is the processing speed (which is limited by the
photodetector response time and its inability to output signal characteristics for
modulation analysis).
 DESIGN EXAMPLES

Three examples are presented here, covering the most frequently encountered
design problems. These are treated as they would be in practice. It is often
necessary to modify a system design one or more times (as problems are
encountered) in subsequent analysis. This is the advantage of performance
analysis. Changes can be made on paper inexpensively before hardware is
started. It is foolhardy to start hardware for a system, without prior perform-
ance analysis.
These examples are useful guides to the designer, who should follow the design
sequence presented, after the configuration is selected. They assume that the
initial system selection has been made. The reader is referred to the section on
IF (4.3); which serves as background for these examples.
All of the computer programs are written in the BASIC language for the HP
series 80 computers. Some minor modifications may be necessary for other
machines.

7.1 EXAMPLE 1

Down Conversion Receiver

Specifications: Required Predicted

Frequency Range, MHz 30 to 80 30 to 80

IF Rejection, dB > 60 > = 80*
Image Rejection, dB > 55 60
LO Radiation, dBm <-80 -88
Detection Mode FM FM
Spurious Response, -dB a > 65 71
Intercept Point, dBm (3rd order, input) > 10 15.6
IF Bandwidth, kHz 30 30
Sensitivity, FMpv
502 source open ckt. 14 Gl
S+N/N = 20 dB
Deviation + 8kHz at 1 kHz
Frequency Response, Audio 30 to 15 kHz Same
192 RecewingSystemsDesign

* TheIF rejection is the sum ofthe preselector attenuation, 60 dB, plus the
mixer’s RF to IF isolation, which is 20 or more dB, for a total of 80 dB.
** The truncation point for the intercept point is, in this case, the IF filter.
This filter will stop the two tones if their spacing is greater than the filter
bandwidth.
Selectingthe IntermediateFrequency(IF)
For down conversion the following rules apply:
The IF must be out of band
The preselector must present its ultimate attenuation to the:

Intermediate frequency
Image frequency
Local oscillator frequency
To minimize the LO tuning range, high side injection is selected.
This is described by:
IF = F,, - F,
where
F,, is the local oscillator frequency
and
F, is the received frequency
Pictorially these conditions are shown in Fig. (7-1).

5 Frequency ———™

Fig. 7-1. Pictorial representation of the IF, LO and preselector relationships.

 Design Examples 193

From Fig. (7-1) and the specifications, the IF must be less than 30 MHz (in the
worst case, half the -60 dB bandwidth) of the preselector. Using two varactor-
tuned critically coupled transformers in cascade for the preselector, the pro-
gram of Table 5-4 is executed for the typical 4 to 12% bandwidth range. After
several iterations it can be concluded that a 4% bandwidth is required for the
preselector. This represents a loaded Q , of F,/B, =25. The results are shown in
Fig. (7-2). At 80 MHz, F,, =95 MHz, from which:
IF, =95- 80=15MHz
Examining the low frequency end of F, = 30 MHz, the highest IF which can be
used is 25 MHz. From these limits an IF of 21.4is selected and all of the rules are
satisfied. (Note that the image frequency is above the local oscillator frequency
by an amount equal to the IF.)
The Block Diagram
Having selected an IF and preselector design, the rough block diagram is
drawn (as shown in Fig. (7-3)), using further inputs from the following consid-
erations and calculations. |

Sensitwity
The receiver must match the 50 Qsource impedance for best performance. The
loaded input signal voltage to the receiver is 14volts / 2 = 7 pvolts. Convert-
ing this to dBm we have:
. 107%2
10 log,, Sy + 30 = -90 dBm
50
Calculating the signal to noise ratio
va? vsAF\, Bate N
2B,
= —— Fad14a

AF =8 kHz peak
B, =15kHz
B, = 40 kHz(ENB)
C =-90 dBm
N =kTB,=-128 dBm
S/N = (StN)/N& 20 dB

or
SiMe
5(a) \?sett)asf
8+10°/40-10°
\_
S/N = 1.138C/N
194
ReceivingSystemsDesign

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Design Examples 195

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196 RecewingSystemsDesign

In log from:
S/N = 20 dB = 0.561 + (-90 dBm) - (-128 dBm) - VF
Solving for the noise figure:
NF = 18.561dB max
This represents the maximum noise figure the receiver can have and still meet
the sensitivity specification.
The noise figure to the mixer input is 13.5 dB.
The preselector will have a loss of:

L (dB) = 20 log, ake (7-1)

where
Q, is the unloaded Q (assume 100)
and
Q, is the loaded Q = 25
Then
L (dB) = 2.5 dB
In the block diagram the preselector is shown to be in two parts; this was done
for better matching. Each ofthese is allocated 2.5 dB loss for margin.
The noise figure of the preselector is equal to its loss and is 2.5 dB per
transformer. Selecting a single stage amplifier for the preamplifier with moder-
ate gain, low noise figure, and high intercept point, we have the following:
Gain = 9dB

NF = 7dB

I output intercept point = 27 dBm.

Using the program in Table 4-5, we secure the print-out of Table 7-1. The
receiver noise figure is computed to be 12.5 dB, leaving a 6 dB margin.
The system intercept point is computed using the program of Table 4-6. The
intercept points of the components are based on catalog or prior design
information. An initial selection is made and the intercept point is computed.
Adjustments through alternative component selections are made until a suit-
able fit results. The print-out is shown in Table 7-2. The input intercept point of
the receiver is 15.6 dBm, as compared to the specified value of 10 dBm, leaving
a 5 dB margin. It should be remembered that this prediction assumes that all
intermodulation terms are additive, which is not always the case. The predic-
tion is therefore pessimistic.
Design Examples 197

Table 7-1.
Computer Print-Out of the Noise Figure of the Down Converter Receiver

CASCADE NOISE FIGURE

STARTS WITH LAST STAGE AHO WORK
S UP TO THE INPUT
NFT=16*¥LGTCFit+¢ CF2-159/6155; 06
WHERE
HFT=TOTAL NOISE FIGURECDE:
FIi=PRECEDING STAGE HOISE FIGURE
CRATIO®
FS=FOLLOWING STAGE NOISE FIGURE
fRATIO®
fS$1=15T STAGE GAIN
ALL PROGRAM ENTRIES ARE IN DE

KEEKEEEREREREEPREREEESEEKERER ESE
HF GS CAS NF... STAGE

OB Oe DB

3 LA IF AMF
3 - 3 5 IFi FEIR
F i at re ee" MIXER
ee -2.9 16 PRE SEL I!
f 3 18 PRE HAP
25 a2 .5 a FRE SEL 2

Table 7-2.
Computer Print-Out of the Third Order Input Intercept Point of the
Down Converter Receiver

CASCADE INTERCEPT
COMPUTES DEGRADATION OF THE IW
TERCEPT POINT DUE TO A PRECEDING
STAGE

2G SNP GNP Pe IS THE OUTPUT

INT PT-GAINW
CASCADE INTERCEPT POINT
THIRO OROQER
198 | RecewingSystemsDesign

KRERHKEREEEKEFEFEEREREKAERERERERSE
PRbionG AAS odto hit st TRE wGPAGE
DEN ODE OB8M DE DEM

SoM 2S PRE $S

6.0. 26.4 F AMP

SIR HS SD
24 4 cl PRES

1 i _/ 3
) OXEF
Bo ice MIF

Spurious ResponsePredictions
These computations are best left toa computer. Utilizing the program ofTable
4-7 and entering the appropriate parameters, a series ofprint-outs are obtained
which show the spur frequency order and magnitude. To reduce the clutter, a
spur search floor of 10 dB better than the specifications is used.
From the print-out of Table 7-3, we see that the worst spurious response is the
3 x 1 = -71 dB, resulting in a 6 dB margin and an image response of 60 dB
(which results in a 5 dB margin).
The specifications are for a low performance receiver and are not generally
satisfactory unless the receiver is operated in a weak signal environment. To
improve the performance it is necessary to improve the preselector selectivity
and ultimate attenuation by using traps.
The receiver noise floor is kTBF or
-128 + 18.56 = -109.4 dBm (specified or implied)
For the received signal to be below the noise floor, the image signal cannot
exceed -49.4 dBm and the spurious signal input must be below -38.4 dBm.
Stronger signals will cause these signals to be above the noise floor.

LF Rejection
From the print-out of Table 7-3, the IF rejection is better than the search floor
of 70 dB. This response would have had an M = 0 and N = 1 identity.

IF Gain

To determine the gain required in the IF amplifier, the noise floor should be
Design Examples 199

amplified up to the required detector level. Foran FM application there are

several detection forms to choose from, some of which require limiting. It
remains to select one best suited to the task and then proceed with the
calculation.
To illustrate, assume the detector requires an input of -10 dBm. The noise floor
is -109.4 dBm. The difference is the required gain, which in this case is 99.4 dB.
The pre-IF amplifier gain is -6.5 dB. Therefore the required IF gain is 106 dB.
This is a rather high gain and it should be broken up into two frequencies, or
the preamplifier gain should be increased by about 7 dB. An alternative
solution is the use of a detector with a lower detection threshold. A good design
rule is to at least break even in the pre-IF gain chain.

Table 7-3.
Spurious Prediction Print-Out for the Down Conversion Receiver of
Example 7-]

SPUR SERRCH PROGRAM

FF INPUT -16D6M.LO0 17 CEM
FROM
FS=CPIF-M#FLOO+H
WHERE
FS=SPUR FREQUENCY
FIF=INTERMEDIATE FREQUENCY
FLO=LOCAL OSC FRESG
M&M ARE INTEGERS OF BOTH SIGN

TOMPLITES
UP TO 15TH ORDER
VEFINITIONS
OROER=ORD=AESCM+H 2
FR=TUNEOD FREGUEHCY
FMIN=MIN LIMIT OF FR
FMAS=NMAx LIMIT OF FR

FMIH= 28 FMRAX= a ~—
o

IF= 21.4
MPTIQN= 1
SPUR FLOOR= aH DOE
TUHABLE FILTER USING mt
TRANSFORMERS WITH A LOADED @ OF
2 AMD. ULT.. AT TH,.OF. 68
200 RecewingSystemsDesign

EKEELEEEKALAELAKAKEAHKELEKAKERAHS
TUNED FREG@= 38 FLOS 14
FSPUR " Bes iy ltt dl eae LOR
338.4 i. i 19 «66 ripe.
233.6 a 1 id 68 ,
L32VS iS 1 ii «568 ri
38 ae 1 a 4 a
SxS 1 i a 68 bE
Lf 2.58 3 1 ng ES ri
ort. 4 va 1 i¢ 668 i4
$31.2 P 1 1 ae Oe
£XEKHEAS AEAAAAEA AKERS ED EEAEER ESS

TUNED FREQ= 535 FLO= *6 4

FSFUR M NSU Peo ee
313.4 at 1 LF) ae ne,
368 6 ad 1 i¢ «68 r4
2H? 5 —. 1 11 5664 bon
a =< 1 5) 4 a
oT 1 1 a 68 ba
258 6 3 1 Si ee a
4H3 4 a 1 i¢ 68 rg
wba}= ap = r i iS 64 i,
£RERARARAARRAAS RARER REE

TUNED FREG= 34 FLO= 141.4

FSPUR iM he =e Poy Ser
be8 .4 os i r2. “6G ‘9
4353.6 = 1 14-468 +4
ese .¢ met } 12° vee 74
oy a i a a 8
122.8 1 1 a bG 64
320.6 3 1 il *“6¢6 vil
223.4 > l if 68 4
ree rf 1 13ers a
Liew g o a a4 34

Local Oscillator Radiation

The local oscillator signal path to the antenna is shown in Fig. (7-4). The local
oscillator signal, whose magnitude is 10 dBm goes through the mixer L to R port
isolation (-25 dB); the preselector (- 30 dB); reverse isolation of the preamplifi-
er (-13.5 dB); and the input preselector (-30 dB). This results in a LO level at
the antenna of:
10 -25 -30 -13 -30 = -88 dBm
Where receivers are co-located with antennas in close proximity or in common,
this may not be sufficient. In this case it would be necessary to improve the
preselectivity. For this requirement the specifications are met with an 8 dB
margin.
 Design Examples 201

This example was presented to illustrate the method. The final design is left to
the reader.

Fig. 7-4. The local radiation path for the receiver of Example 7.1.
7.2 EXAMPLE 2

Design a receiver covering the 50 to 1200 MHz.frequency range, which meets

the specifications of Table 7-4, using a hybrid up and down conversion scheme.

Table 7-4.
Specifications for EXAMPLE 2
Specifications: Required Predicted

Tuning Range, MHz 50 to 1200 50 to 1200

Sensitivity, dBm -90 ~94.6

30 AM at | kHz
(S+N)/N=10 dB
Bandwidth, kHz
Shape Factor, 3 to 60 dB
LO Radiation, dBm

Image Rejection, dB
IF Rejection, dB
Spurious Responses, dB
Intermodulation Distortion
3rd order
2 tones, -35 dBm spaced
1 MHz

AGC Cut On, dBm

AGC Range, dB 105

*F,+ 2F, Low end of band B

202 Recewing SystemsDesign

Sensitwity
(S+V)/N = 10dB= 10 ratio
S/N =9 ratio = 9.54 dB
and
Pm’
S/Nal = ———
2k TBF

where
P.= -90 dBm
m = 0.3, m* = 0.09 = -10.46 dB
kT = -144 dBm/kHz

B= By ueyg= 31 kHz = 14.9 dB

F is the unknown
In dB notation:
9.54 = -90 -3 + (-10.46) - (-144) -14.9 - NF
NF = 16.1 dB

This is the maximum system noise figure. The maximum noise floor level is
k TBF or:

-144 + 14.9 + 16.1 =-113 dBm

SystemConfiguration
The tuning range of 50 to 1200 MHz is split into two parts, one of which will be
down converted and the other up converted.

This is defined as follows:

Band A
Fina) =90 MHz
Fyaxa)= 20 + (1200 - 50) /2 = 625 MHz
and
Band B

Fing) = 829 MHz

Faxb) = 1200 MHz
This spectral division is shown in Fig. (7-5).
Band A will be up converted into an IF in band B, and band B will be down
converted into an IF in band A. By symmetric spacing of the IF's the first LO
tuning range is reduced by 1/2. This spacing must be made equal to the width
Design Examples 203

of the bands, 1.e., 575 MHz. Then

Band A IF = 625 + (625 - 50)/2 = 912.5 MHz
and
Band B IF = 625 - (625 - 50) /2 = 337.5 MHz

BAND A BAND B

0 50 625 1200
Frequency in MHz
Fig. 7-5. RF band plan for the frequency range of 50 to 1200 MHz.

With this selection, the frequency relationships are tabulated in Table 7-5.
Table 7-5.
Receiver Frequency Relationships (MHz)
Band Receive Frequency IF Ist LO Mode

A 50 912.5 962.5 IF=F,,-F,

A 625 912.5 1537.5 IF=F,-F.
B 625 337.5 962.5 IF=F,.-F.
B 1200 337.5 1537.5 IF=F,.-F.

Note that the same LO is used in both bands.

Preselector

Because band A is up converted, it is acceptable to use a fixed tuned filter bank.

The number of filters required is:
50k" = 625
k =V/ 12.5

An n value of 6 gives a k = 1.52, which is close to the 1.5 rule for four section
filters. Thus, six filters will be used. These are listed in Table 7-6.

A tentative filter selection is made at this point to be a 0.1 dB ripple Chebyshev

filter with an ultimate attenuation of 60 dB.

Band B (although down converted) is a border line case and fixed tuned
filtering will be used. This results because of the relatively high band AIF. For
the preselector to be effective in suppressing LO radiation, the filter’s ultimate
attenuation must be presented to the LO. With a fixed &value, preselector
204 RecewingSystemsDesign

bandwidth increases with frequency. The = 1.5 rule does not apply here. The
worst case would occur at 1200 MHz.

Table 7-6.
Band A Preselector Specifications (MHz)

(Fto 1.52F) Width F, %B

50 to 76 26 63 41
76 to 116 40 96 4]
116to 176 60 146 4]
176 to 269 93 225.5 41
269 to 410 141 339.5 4]

410to 625 215 517.5 41

Fmin Fmax

Permissible
LO Frequency
Zone

Fig. 7-6. Desired preselector frequency relationships for band B (625 to 1200
MHz).

Executing the program of Table 5-6, and iterating until the LO is related to the
bandwidth, as shown in Fig. (7-6), it is found that the bands can be defined as
follows:
Design Examples 205

Table 7-7.
Band B Preselector Specifications (MHz)
Frequency (MHz) Ripple (dB) %B
Min Max AF N F.
1060 1200 140 5 1130 0.1 12.4
915 1060 145 5 987.5 0.1 14.6
770 915 145 5 842.5 0.1 17.2
625 770 145 5 697.5 0.1 20.0

Preselector filter definitions for the down conversion of 625 to 1200 MHz
frequencies are given in Table 7-7. A Fis the bandwidth ofthe filter in MHz, V
is the number of sections, F, is the center frequency, and %B is the bandwidth in
percent.
The system block diagram may now be drawn and the related information
computed and added. This is shown in Fig. (7-7). The preselector filter bank
shown in the system block diagram of Fig. (7-7) is expanded in Fig. (7-8). It
consists of a 10 PST pin diode switch feeding 10 filters whose outputs are
selected by a second 10 PST switch. All filters are 0.1 dB ripple Chebyshev
types using four or five sections as shown. The worst case loss and noise figure is
3.5 dB and a third order intercept point is 35.5 dB.
Preamplifier
A single preamplification stage is selected with a gain of 12 dB, a noise figure of
2 dB and an output intercept point (third order) of 20dBm. This is an initial
choice and the final selection will be contingent upon the overall system
performance. Experience will serve as a guide in this initial selection. Where a
high intercept point is called for, a low gain power type of stage is required.
These generally have a poorer noise figure. In this case the selection made
should be adequate.
This amplifier is placed between the preselector and the lowpass filter. The
amplifier serves as a termination to both filters and results in a good VSWR.
The Mixers

Both mixers are double balanced for good isolation and spurious performance.
In all but low performance receivers, the double balanced mixer should be
used. For best spurious and intermodulation performance this mixer should be
of the termination insensitive type, operating with high LO drive (17 to 23 dBm
typical). For this design a drive level of 17 dBm is selected for both mixers,
which are diode double balanced, termination insensitive types.
206 RecewingSystemsDesign

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Design Examples 207

50 to 76 MHz
N=4, L=.75 dB

76 to 116 MHz
N=4, L=.75 dB

116 to 176 MHz

N=4, L=.75 dB

176 to 269 MHz

N=4, L=.75 dB

269
to4
pra N=4, 10
MHz
L=.75
dBnen
410
to625
MHz N=5,L=.75dB
625 to 770 MHz
N=5, L=1.1 dB

770 to 915 MHz

N=5, L=1.18 dB

915 to 1060 MHz

N=5, L=1.22 dB

Worst Case
3
Gain = -3.33 dB
1,=50 dBm
NF = 3.33 dB
3
I, =35.5 dBm
3
| = 38.83 dBm

Fig. 7-8. Preselector configuration and performance data.

The Second Local Oscillator

The second local oscillator is selected for low side injection because lower
frequencies are less noisy. The second LO frequencies are related to the receive
and first intermediate frequencies, as shown in Table 7-8.

For the best frequency accuracy, these frequencies should be phase locked to
the first LO reference standard.
208 RecewingSystemsDesign

Table 7-8.
Frequency Relationships Selected for the Hybrid Up-Down
Conversion Receiver
ReceiveFrequency
aaah eat hobicaied
AEBweaee IstOIF GEYSOchibi 2nd IF aD 2nd
SeBenceLO
50 to 625 912.5 21.4 891.1
625 to 1200 337.5 21.4 316.1
All values are shown in MHz.

First LF Amplifier
The first IF amplifier is driven by the first mixer. ASPDT switch drives either of
two IF filters, whose outputs are selected by a second SPDT switch. These
filters are made as narrow as practical, to improve intermodulation perform-
ance and local oscillator feed through. The critical requirements result from
the two local oscillator frequencies, which must be in the first IF filter ultimate
attenuation zone. Pictorially this is shown in Fig. (7-9).
From this, the critical first IF filter requirements are determined as follows:
316.1 MHz = first IF band B filter ultimate attenuation

and
891.1 MHz and 962.5 = first IF band A filter ultimate attenuation.

Input Frequency
SOMHZ BandA 625MHZ _ BandB 1200 MHz

1st LO Frequency

i ba 962.5 MHz 1537.5| MHz

337.5 MHz
1st IF Band B 912.5 MHz
1st IF BandA
316.1 MHz 891.1 MHz
2nd LO 2nd LO
Band B BandA

Fig. 7-9. Receiver frequency relationships for RF, IF, and LO frequencies.
Design Examples 209

For the best system noise figure, this filter should have low loss. This is achieved
by minimizing the number of sections. By either referring to a catalog or the
program in Table 5-6, the filter can be defined. For a four section 0.05 dB
Chebyshev filter the 70 dB bandwidth (of the filter) is about eight times the 3 dB
bandwidth. Then, four bandwiths must equal to << 21.4 MHz or B™5 MHz.
This represents a 1.48% bandwidth. Such a narrow bandwidth dictates the use
of a cavity-type filter. There the insertion loss varies with size. An approximate
value of loss constant of 1.4 to 1.2 will be used.

Then
Insertion loss (337.5 MHz filter) =

(1oss
constant)
(No.of 1
sections
+om)
% 3 dB Bandwidth + ie

_ (1.4 (440.5)
+ 0.2 = 4.7 dB
1.48
Similarly for band A (the 962.5 MHz filter), the 3 dB bandwidth is ~5 MHz or
0.5% and the insertion loss is:

L=
0.35(4+0.5) + 0.2 = 3.35 dB
0.5
The IF filters are followed by two amplifiers in cascade separated by an AGC
attenuator. The gain of these amplifiers should be sufficient to cancel any
pre-second IF amplifier loss.
Computing the required gain between the detector and the antenna we have:
Detector Signal = Total Gain + Noise Floor (dB notation)
-10 dBm = Total Gain + kTBF
From which

Total Gain = 113 dBm - 10 dBm = 103 dB

For good intermodulation performance it is necessary to minimize the gain

ahead ofthe truncating IF filter, which in this case is the 21.4 MHz second IF
filter. Therefore the second IF gain should be maximized (but it should be less
than the practical limit of 100 dB). In this example the pre-second IF gain must
be 3 dB. Summing the losses and gains to the second IF filter we have:
210 RecewingSystemsDesign

Table 7-9.
Tabulation of Gains and Losses of the Pre-second IF Amplifier to Compute the
Gains of Al, A2, and A3

Stage Loss (dB) Gain (dB)

Preselector 3.33
Preamplifier Al (12)
Lowpass Filter 0.8
Ist Mixer ype
Ist IF switch 0.5
Ist IF Filter 4.7
Ist IF Switch 0.5
Ist IF Amplifier A2 (14)
AGC Block 1.0
lst IF Amplifier A3 (9)
2nd Mixer 6.0
2nd IF Filter 3.0

27.33 3 +27.33,-0+10
(Required gain)

Amplifiers AJ, A2, and A3 must total a gain of30.33 -0 + 10 dB. Amplifier A/ is
12 dB and amplifiers A2 and A3 are budgeted at 14 dB and 9 dB respectively,
giving a 5 dB margin.

The SecondIF Amplifier

This amplifier is driven through the second mixer and the IF crystal filter. The
AGC attenuators are ofthe PIN diode variety. Each ofthese provide up to >35
dB of attenuation for a total of > 105 dB (including the one located in the first
IF chain). They are distributed in the gain chain, such that at no time does any
stage go into compression on modulation peaks with the maximum signal to be
encountered. Because the gain was computed to noise threshold, the AGC
system will begin to function with any noise, as required. The second IF
amplifier contains the bulk of the IF gain of 100 dB. The remaining 3 dB plus
margin is secured in pre-second IF gain.

Image Rejection
There are two image cases to consider, both of which are defined by;
IF =| F,,-F|
For the 50 to 625 MHz receive band:
912.5 =| (962.5 to 1537.5) - F,|
Design Examples 211

There are two solutions:

Desired F, = 50 to 625 MHz and

Image F, = 1875 to 2450 MHz
The undesired image frequencies are out of band and attenuated by the
preselector and lowpass filter.
For the 625 to 1200 MHz receive band:
337.5 =| (962.5 to 1537.5) - F.|
The solutions are:
Desired F, = 625 to 1200 MHz
Image F,= 1300 to 1865 MHz
This undesired image response is also out of band and attenuated by the
preselector and lowpass filters. The preselector can supply 60 dB of image
attenuation which by itself is not enough. Therefore the lowpass filter must be
added to the preselector.
For the 50 to 625 MHz range the lowpass filter will provide its ultimate
attenuation of > 70 dB. For the 625 to 1200 MHz range, the lowpass filter is less
effective toward the low edge of this band. However, only 20 dB ofadditional
attenuation is needed at 1300 MHz. A sharp cutoff 1200 MHz lowpass filter
will have to be used to secure this value. An eight section filter will provide this
performance with only 0.8 dB thus of insertion loss. Therefore image rejection
will range from 130 dB to > 80 dB, thus meeting the requirements.

IF Rejection
The first IF is always out of the preselector band. Therefore the preselector will
provide its ultimate attenuation of 60 dB to this frequency. Additionally, since
the response at the IF is not a converted response, the first mixer will provide an
additonal attenuation of20 to 30 dB; resulting from its input /output (R to X)
port leakage. The total will be in excess of 80 dB. The configuration will meet
the requirements of > 80 dB.
LO Radiation

The LO radiation path is shown in Fig. (7-10). The LO, whose magnitude is 17
dBm, goes through the LO to #, port leakage of 20 to 30 dB. The following
lowpass filter, whose cutoff is 1200 MHz, is ineffective in suppressing LO
frequencies between 962.5 and 1200 MHz. The signal is then attenuated by the
preamplifier reverse isolation of 25 dB and fed to the preselector, which by
design provides 60 dB of additional loss. The net result is aLO level of -88 dBm
at the antenna (which meets the requirements with an 8 dB margin).
212 RecewingSystemsDesign

P,. =17 dBm-20 dB-0 dB -25 dB-60 dB = -88 dBm

Fig. 7-10. The local oscillator leakage path with a worst case P,, level of -88
dBm.

Noise Figure
The computer program ofTable 4-5 was utilized with the inputs taken from the
block diagram of Fig. (7-7). The result is shown in the print-out of Table 7-10.
The design is predicted to have an overall noise figure of 11.5dB, compared to
the required value computed to be 16.1 dB. The sensitivity of the design is
therefore -90 - (16.1 - 11.5) = -94.6 dBm.

Intermodulation Distortion

The specification calls for the intermodulation distortion products to be > -75
dB below the two tone input level of -35 dBm, where the tones are separated by
1 MHz. Because of 1MHz spacing, the two-tone truncation point is the second
IF filter.
The required input intercept point is from Figures (4-22) and (4-23), or
computation, to be 2.5 dBm. Exercising program Table 4-6 and entering the
values from Fig. (7-7), the computer print-out of Table 7-11 indicates a third
order system intercept point of 5dBm, which meets the requirements witha 2.5
dB margin.

SpuriousResponse
The computer program of Table 4-7 is executed using inputs from Fig. (7-7)
and (7-8). For a complete prediction a minimum of 20 runs should be made.
(There are 10 preselector filters which should be examined, as a minimum, at
each band edge.) By using a floor 5 dB below the requirements, the print-out is
less cluttered.
Design Examples 213

Table 7-10.
Computer Print-Out of the System Noise Figure for the
Hybrid Conversion Receiver
CASCADE NOISE FIGURE
STARTS WITH LAST STAGE ANDO WORK
> Wie Qua pM Nt
NFT=1@4LGTCFI+¢ CF2-19-61995D8
WHERE
NFT=TOTAL NOISE FIGURECOBs
FI=PRECEDING STAGE NOISE FIGURE
CRATIOD
F2=FOLLOWING STAGE NOISE FIGURE
CRATIO?
81=15T STAGE GAIN
ALL PROGRAM ENTRIES ARE IN DB

KEERREREKAREEEHEERERKEEEEEKERALRE
MF G CAS NF 7 STAGE

Oe DE DB

4 1b ZNO IF AMP
3 =\§ ra 2NO IF FLTR
6, nd= 13 ZNO MyF
7 or AMP AS
l =j —Oe AGO ATTH
4.5 id ms AMP Ae
iw - 5 =a aft= IF Sid
4.77 VeGeF? igs2 IF FLULTR
os =.5 Lae iF Si
(i3 - =f is 2 157T MKR
a= - $ i3 LOW PASS
2 iz &.1 FRE AMP
SUES ATStSS wlio PRE SELECT

Table 7-11.
Computer Print-Out of the Cascade Third Order Input Intercept Point
for the Hybrid Conversion Receiver
CASCADE INTERCEPT
COMPUTES DEGRADATION OF THE IN
TERCEPT POINT DUE TO A PRECEDING
STAGE
214 Recewwing
SystemsDesign

THE INPUT INT PT IS THE OUTPUT

INT PT-GAIN
CASCADE INTERCEPT POINT
THIRO ORDER
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CBM OB DBM OB DEM

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7. 26. AD
19.67
Design Examples 215

For this example only four filters are examined (eight runs) with a floor of -80
dB. These are shown in Table 7-12. There appears to be a problem in the 625 to
770 MHz band on the image and the | X 2 spur. The | x 2 spur is right at
specifications and is in band as a consequence of the system, with no solution for
improvement. This should be flagged and verified on the bench. The image is
actually in spec (since it was pointed out that a fast fall filter will be used which
is not in the program). Actually the fast fall filter will add ~ 30 dB loss rather
than the seven shown on the run, totaling ~ 90 dB down.

Table 7-12.
Computer Print-Outof the SpuriousPerformanceof theHybrid Conversion
Receiver
SPUR SEARCH PROGRAM
RF INPUT -1G08M,L0 17 DBM
FROM
FS=CFIF-M#FLO3/N
WHERE
FS=SPUR FREQUENCY
FIF=INTERMEQIATE FREQUENCY
FLO=LOCAL OSC FREG
M&N ARE INTEGERS OF BOTH SIGN
c
FOMPUTESUP TO 15TH ORDER
CEFINITIONS
ORDER=ORO=ABS¢CM+N>
FR=TUNED FREQUENCY
FMIN=MIN LIMIT OF FR
FMAX=MAX LIMIT OF FR

FMIN= 5G FMSK= 76
ire oie S
NP TION= 1}
SPUR FLOOR= 36 O6
LOW PASS FILTER CUT OFF= 1268
NUMBER OF ELEMENTS = §&
RIPPLE= .1°08
ULTIMATE ATTN= 7&4 DB
FIMED TUNEOQ CHEBYSHEY FILTER FMI
N= 56 FMAX= 76 N= 4 RIPPLE= 1
ULT ATN= 6&8
EE SKEFKERELESLSREREFERESHSERERERE
TUNED FRE@= 5&8 FLO= 962.5
FSPUR iM NDS Fay | =FOT
38 ma 1 8 a e
216 RecewingSystemsDesign

EKEKSEPRAEE
RARER RARER ERREREA ESS
TUNED FREG= 76 FLO= 385.5
FSPUR iM a, Se ee Ree
? ae. 1 a 4 a)

FMIN= 416 FMAX= 625

IF= 912.5
OPTION= 1
SPUR FLOOR= su OB
LOW PHSS FILTER CuT OFF= 1206
NUMBER OF ELEMENTS = 8
Rifrces..4 Le
ULTIMATE ATTN= 76 OB
FIXEQ TUNEOQ CHEBYSHEY FILTER FMI
N= 416 FMAX= 625 N= 5 RIPPLE=
.1 ULT ATN= 68
KEKKAREEREKKERERKAER KARR KKKARA
TUNED FREG= 4164 FLO=
1322.5
FSPUR M Nn Ue. Akei ee
416 = 1 5) 8 3)
¥EKEKEKEKAEEKERKERER ERR ERRER ERS
TUNED FREGQ=625 FLO=
1537.5
FSPUR M N. ~UB." kL. l Ue
625 = 1 a a) Ss)

SPUP SEARCH PROGRAM

RF INPUT -18DEM,.LQ 17 DBM
FROM
FS=CFIF-M#FLO “NN
WHERE
FS=SPUR FRESUENHCY
FIF=INTERMEDIATE FREQUENCY
FLO=LOCAL OSC FREG
M&M ARE INTEGERS OF SOTH SIGN
Ww
rOMPUTES UP TO 15TH ORDER

NEFIHITIONS
OROER=OROQ=RESCM+N >
FR=TUNMED FREGUENHCY
FMIN=HIN LIMIT OF FR
FMAR=MAN LIMIT OF FR
Design Examples 217

FMIN= 625 FMAX= ~J |a Ase

OPT TOH= 1

SPUR FLOOR= 86 OG
LOW PASS FILTER CUT OFF= 1266
NUMBER OF ELEMENTS = 8
RIPPLE? 1).11..OB
ULTIMATE ATTN= 76 DB
FISEQ TUNEO CHEBYSHEY FILTER FMI
N= 625 FMAX= 776 N= 5S RIPPLE=
.1 ULT ATN= 68
EREKEEEEEEERERRKEREKEAKEEAERKEREKE
TUNED FREGQ= 625 FLO= 962.5
FSPUR M Roepe FRE oofoT
625 ae. 1 a 4 8
1386 i j a bu or
ied 1 2 fy 8 eS
FREED SELES ER RELL EEKERERE
TUNED FREQ= 776 FLOQ=
1147.5
FSPUR M nN. 6 §$DG FG eae
776 4 1 & a) 8
(22.3 i 2 fo 86 fe)

FMIN= 1966 FMAX= 1288

IF= 337.38
OPTIOQN= 1
SPUR FLOOR= S@ O86
LOW PASS FILTER CUT OSOFF= 1266
NUMBER OF ELEMENTS = 8
RIPPLE= .1 DB
ULTIMATE ATTN= 76 OB
FIMEOQ TUNED CHEBYSHEY FILTER FMI
H= 1666 FMAX= 1266 N= 5 RIPPLE=
.1 ULT ATN= 68
EREKEEKRARKEAEAK KERKAKRERKAEEKEKE
TUNED FREGQ= 1668 FLO=
1397.5
FSPUR iM Nie UB 4s JL ee OL
1Ge8 pe! 1 a 4 6
KEEKKRELKARHEREEKAKSRERAK KEKKEKE
TUNED FRE@G=1266 FLO=
1537.5
FSPUR M Me. -D8:o ELT Arpt
1266 saa | 1 8 a iS
218 RecewingSystemsDesign

7.3 EXAMPLE 3

Design a receiver using an up conversion system covering the frequency range

of 30 to 250 MHz. Present designs using standard up conversion and the
Wadley system. Since the details of computing performance are presented in
Example 2, limit this effort for a receiver with spurious performance of -80 dB
with inputs of -10 dBm to only the preselector and IF choice.
Preselector

Either a fixed or varactor tuned filter bank is acceptable. Rough calculations

indicate that six fixed tuned filters are required for the fixed filter band, while
only three are necessary for the varacter tuned filter bank. Also the varacter
tuned filter provides superior spurious performance. Choosing the latter, the
tuning ranges are:
30k" = 250
k=4/250 /30
where
k™ <2, n=3 and k= 2.027
The filters are defined as follows:

Frequency (MHz) Number oftransformers (N) Bandwidth %

30to 61 2 5
61 to 123 2 5
123 to 250 2 5

IF Selection

The first IF must be > 3 times the higest tuned frequency to prevent 3 F, = IF,
from being < 80 dB down. Then IF, > 3.250 = > 750 MHz. A tentative value is
790 MHz which allows a 40 MHz guard band between these frequencies within
which the first IF filter will be at its ultimate attenuation value at 3F.,,..

Second IF Selection

Standard Up Converter
The second IF is selected to be the popular frequency of 21.4 MHz where filters
are readily available and the frequency is low enough to allow a high gain
amplifier to be used. Any IF less than that of the first IF could have been used,
Design Examples 219

but it is good practice to get toa workable frequency with as few conversions as
possible. Other possibilities are 30, 70, 120, et cetera. A 10.7 MHz value would
demand a very narrow first IF filter which would have higher loss and poorer
stability.
The First Local Oscillator

The first LO frequency range is

F,, =| IF, + F|
where
IF, = 790 MHz
F = 30 to 250 MHz
There are two choices:
F,,, = 790 + (30 to 250) = 820 to 1040 MHz
or
F, = 790 - (30 to 250) = 760 to 540 MHz
le,

Of these the lower frequency range is selected.

The Second Local Oscillator

The second LO could be placed on either side of the first IF without serious
conflict. However, for maximum isolation this frequency is placed on the high
side of the first IF. (Low side injection would have had a worst case separation of
only 8.6 MHz from the maximum frequency ofthe first LO.)
Then
%, = IF, + IF,
= 790 + 21.4 = 811.4 MHz

The First 1F Filter

The principal constraint is that the first IF filter be non-responsive (present its
ultimate attenuation) to frequencies removed from its center frequencies, by an
amount equal to the second IF. Fig. (7-11) illustrates this case.
From Fig. (7-11) the ultimate attenuation bandwidth must be less than 2+ IF,=
42.8 MHz. A four section cavity filter has a -70 dB bandwidth of ~ + 4( -3dB
bandwidths). Then the 3 dB bandwidth is:
B, = 42.8/8 = 5.35 MHz max. = 0.68%

A 0.5% bandwidth is therefore selected, with a ripple of .05 dB.

220 ReceivingSystemsDesign

540 MHz
heMHz
760 MHz
max
—}-<21.4
ap - 1st IF
790 MHz 811.4 MHz

Frequency
Fig. 7-11. Critical frequency relationships for Example 3.

Basic Block Diagram of the UpConverterReceiwer

At this point the basic block diagram may be drawn; it is shown in Fig. (7-12).

The WadleyDesign
The preselector and first IF are unchanged from that of Fig. (7-12). Defined
are:

Preselector, same as Fig. (7-12)

First IF = 790 MHz
First IF Filter =790 MHz, V= 4, B= 0.5%, 0.05 dB ripple cavity
The basic block diagram ofFig. (7-13) isdrawn and completed as the following
are considered.

The SecondIntermediate Frequency

It is good practice to keep the variable frequency oscillator (VFO) in the
ultimate attenuation zone of the preselector. The preselector is widest at 250
MHz. Then with a 5% bandwidth and an ultimate attenuation bandwidth of
+4 (-3 dB bandwidths), the closest this oscillator can be to the receive frequen-
cy is 50 MHz. Therefore, the second IF must be > 50 MHz. Since 70 MHz isa
frequency where filters are available, this value is selected.

WadleyOscillator Frequency(SecondLocal Oscillator)

The Wadley oscillator frequency is:
F’,, = first IF + second IF = 790 + 70 = 860 MHz
Design Examples 221

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Design Examples 223

Note that the difference frequency was not used because it would fall into the
first LO frequency range of 540 to 760 MHz. This avoids potential beats and
whistles.

The VariableFrequencyOscillator (VFO)

The VFO must satisfy the relationship:
first LO frequency = Wadley frequency - VFO frequency
then
(540 to 760) = 860 - VFO frequency
and
VFO frequency = 100 to 320 MHz
Note that the + sign was not used because the result would have been aVFO
frequency in the GHz range, which becomes more difficult to realize, particu-
larly if phase locking is used.

The First Local Oscillator

The VFO and Wadley oscillator frequencies are combined in a mixer and the
frequency difference is selected by an appropriate filter, producing the first LO
frequencies of 540 to 760 MHz. Note that where it is desirable to reduce the
VFO tuning range by 50%, the second IF would have been
|F-F,,] _ [30-250|
= 55 MHz
4 } 4
Then by using both the sum and difference modes in the first LO mixer, the
VFO range would have been as follows:

The input RF band would be split into two equal parts.

Part |
( | abd tut
F to F. + Se a
min min 2
and
Part 2
G + Lei
min mar to F
mun 2 max
Or
30 to 140 and 140 to 250 MHz

The frequency relationships would be as shown in Table 7-13.

224 ReceivingSystemsDesign

Table 7-13.
Frequency Relationships for aWadley Configuration (MHz)

Receiver VFO Ist LO Mixer Oscillator

Frequency Frequency Frequency Mode Frequency 2nd IF
30 85 760 - 845 55
140 195 650 = 845 55
>140 85 930 + 845 55
250 195 1040 + 845 55

From this it can be seen that the VFO range has been reduced by 50%. The
penalty paid is usually a need for three conversions or a non-standard IF. Also
two sideband filters must be switched in the LO chain.

The Wadley technique results in the generation of a first LO which is contami-

nated by spurious frequencies, because of the mixing process involved. This
results in a receiver which has poorer spurious performance than obtained with
standard up conversion.
 APPENDIX

(a)

Digital Data Rate

Two terms are used to describe the data transmission speed of a digital system.
These are bits per second and baud. A bit represents the smallest piece of
information transmitted. The baud is one signal element per unit time, usually
taken as one second.
Data to be transmitted must be encoded to expand the system’s vocabulary and
provide some means of identifying start and stop. Thus, as an example, to
transmit n useful bits, n + & bits would have to be used. The number of useful
message bits per unit time is the system speed in bits per unit time. The total bits
transmitted are baud rate or bauds per unit time.

(b)

Adding Numbers (dB) Notation

To add numbers in dB notation it is first necessary to take the anti-logarithm of
the numbers, add them and then convert that sum to dB form. While not
difficult, this can be time consuming, especially if many numbers are involved.
Table b-1 and the curve of of Fig. (b-1) may be used to perform the task of
adding in dB notation.

To use the table or curve, enter at the dB difference and read the add to larger
value, then add it to the larger dB number.

Example b-1:
Add two powers of 20 and 26 dBm. The difference is 6 dB. From the table or
chart find .96 dB, and add it to the larger. The sum is then 26 + .96 or 26.96
dBm.
226 ReceiwingSystemsDesign

Table b-1.
Addition of Units in dB Notation

DIFFERENCE AOD TO LARGER

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