Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 1

9. Chapter: Modulation, demodulation, signal sampling

Time of study: 6 hours

Goals: the student should be able to

• describe basic modulator principles
• describe basic demodulator principles
• describe basic signal sampling principles

Text

1. Amplitude modulation circuits (AM), demodulation

Amplitude modulation of a sine or cosine carrier results in a variation of the carrier am-
plitude that is proportional to the amplitude of the modulating signal. A modulating signal
should produce an AM wave of the form

S (t ) = A0 [1 + m{t}]⋅ cos(2πf c t )

m{t} - modulating signal

A cos(2πf ct ) - carrier signal (A – amplitude; fc – frequency; φ – phase)

Rd

A.cos (2πfct)
Rg

Rg

Square low FET amplitude modulator – FET is nonlinear element

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 2

As an example let us describe FET amplitude modulator.

A square law n-channel FET (Field Effect Transistor) will pass a drain-source current

I ds = K (V g + V p )
2

where Vp is the FET's pinch-off voltage and Vg is the gate voltage.

It is evident that (superposition theorem)

m{t} + A cos(2πf c t )
Vg =
2
thus
I ds =
K
(m{t}+ A cos(2πf c t ) + V p )2
4
I ds =
K 2
4
[ ]
m {t } + V p2 + 2V p m{t } +
K
4
[ ]
2m{t}A cos(2πf c t ) + 2V p A cos(2πf c t ) + [A cos(2πf c t )]
K
4
2

This produces a drain voltage (output) of

VD = Vb − Rd I ds

We know that cos 2 α = (1 + cos 2α ) / 2 . Hence the last term in expression for Ids is a combi-
nation of a steady current and a fluctuation at the frequency – 2fc. For simplicity we can
arrange that the frequencies with which m(t) fluctuate are all << fc. This means that the first
part of the expression consists of a steady current plus some fluctuations at frequencies well
below fc. We can now use a bandpass filter, designed to only pass frequencies fc to strip
away low and high frequencies and obtain an output
− Rd kAV p
Vout =
− Rd k
4
[
2m{t}A cos(2πf c t ) + 2V p A cos(2πf c t ) = ] 2
[ ]
⋅ m{t}/ V p + 1 ⋅ cos(2πf c t )

which we can re-write in the form

Vout = A0′ ⋅ [1 + m ′{t}] ⋅ cos(2πf c t )

where

m′{t } = m{t}/ V p ; A0′ = − Rd kV p A / 2

i.e. the output is a wave whose unmodulated amplitude is A0′ and is amplitude modulated by
an amount, m′(t ) , proportional to the input modulating signal, m(t ) . The circuit therefore
behaves as an amplitude modulator.

If we have now:
N
m{t} = ∑ a m cos(2πf m t + ϕ m ) ,
1

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 3

than
N
m′{t} = ∑ a ′m cos(2πf m t + ϕ m ) , a ′m = a m / V p .

[ ]
1

We easy get ( sin ω1t ⋅ sin ω 2 t = cos(ω1 − ω 2 ) − cos(ω1 + ω 2 ) 2 )

 N

Vout = A0′ ⋅ 1 + ∑ a ′m cos(2πf m t + ϕ m ) ⋅ cos(2πf c t ) =
 1 
A′ N
A′ N
= A0′ ⋅ cos(2πf c t ) + 0 ∑ a ′m cos[2π ( f c + f m )t + ϕ m ] + 0 ∑ a′ cos[2π ( f
m c − f m )t − ϕ m ]
2 1 2 1
CARRIER FR. Upper Sideband - USB Lower Sideband - LSB

For 100% modulation (m´ =1.0 – the degree of modulation), the amplitude of each sideband
will be just one-half of the carrier amplitude (voltage). We must keep 1 + m′{t}>1, always.
If 1 + m′{t}<1 the signal is overmodulated. This signal cannot be recovered well in most
detection systems.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 4

There are various ways to measure or detect the amplitude. We'll consider one of the
simplest, used by most portable radios, the Envelope Detector.

This is just a halfwave rectifier which charges a capacitor to a voltage ≈ to the peak voltage
of the incoming AM waveform. When the input wave's amplitude increases, the capacitor
voltage is increased via the rectifying diode. When the input's amplitude falls, the capacitor
voltage is reduced by being discharged by a ‘bleed’ resistor, R. The main advantage of this
form of AM Demodulator is that it is very simple and cheap! It contains just one diode, and
one capacitor, and one resistor. That's why it is used so often. However, it does suffer from
some practical problems.

All real diodes are non-linear – the current they pass varies with the applied voltage – as
a result, the demodulated signal is slightly distorted. This simple type of AM demodulator
isn't any good if we want the recovered waveform to be an accurate representation of the
original modulating waveform (it is not Hi-Fi!!).

This circuit charges well the capacitor if the input voltage is greater than the capacitor
voltage – it is the behavior of the diode. But this circuit blocks any current when the input
voltage is below the capacitor voltage. The capacitor is discharged only via the resistor R -
Ripple and Negative Peak Clipping. The ripple effect happens because the capacitor will be
discharged a small amount in between successive peaks of the input AM wave.

The illustration shows what happens in the worst possible situation where the modulating
signal is a squarewave whose frequency isn't much lower than the carrier frequency.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 5

The detector time constant (for discharging of capacitor) isτ = RC . The time between
successive peaks of the carrier will be T = 1 / f c . If we have τ >> T we have discharging
current between each peak and the next almost constant ≈ Vmax/R. Now we can determine a
change of charge: ΔQ ≈ T.Vmax/R. Then the change of voltage between successive peaks
(Ripple) is

ΔV ≈ ΔQ/C = T.Vmax/(RC) = Vmax T/τ = Vmax/( τfc)

A sudden, large reduction in the amplitude of the input AM wave means that capacitor
charge isn't being ‘topped up’ by each cycle peak. The capacitor voltage therefore falls ex-
ponentially until it reaches the new, smaller, peak value. To assess this effect, consider
what happens when the AM wave's amplitude suddenly reduces from Vmax to a much
smaller value. The capacitor voltage then declines according to

V (t ) = Vmax ⋅ exp(−t / τ )

This produces the negative peak clipping effect where any swift reductions in the AM
wave's amplitude are ‘rounded off’ and the output is distorted. Here we've chosen the worst
possible case of squarewave modulation. In practice the modulating signal is normally re-
stricted to a specific frequency range. This limits the maximum rate of fall of the AM
wave's amplitude. We can therefore hope to avoid negative peak clipping by arranging that
the detector's time constant τ << tm where

tm = 1/ f m

and fm is the highest modulation frequency used in a given situation.

The above implies that we can avoid negative peak clipping by choosing a small value ofτ .
However, to minimize ripple we want to make τ as large as possible. In practice we should
therefore choose a value

1/fm >> τ >> 1/fc

to minimize the signal distortions caused by these effects. This is clearly only possible if
the modulation frequency fm << fc. Envelope detectors only work satisfactorily when we
ensure this inequality is true.

An example of connection with transistor is in fig. below.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 6

Simple diode modulator - principle

After
Filtering

uN – carrier
us – modulating voltage
U0 – sets quiescent point

Diode modulation consists of a mixing network, a diode rectifier, and an LC tuned circuit,
often. One diode is used as nonlinear element – it “creates” needed frequency spectrum.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 7

Balanced (lattice) modulator - principle

A balanced modulator is a circuit that generates a DSB (double sideband) signal, sup-
pressing the carrier and leaving only the sum and difference frequencies at the output. The
output of a balanced modulator can be further processed by filters or phase-shifting cir-
cuitry to eliminate one of the sidebands, resulting in a SSB (single sideband) signal.

The carrier sine wave is used as a source of forward and reverse bias for the diodes.
- The carrier turns the diodes off and on at a high rate of speed.
- The diodes act like switches that connect the modulating signal at the secondary of T1
to the primary of T2.
- The carrier sine wave is considerably higher in frequency and amplitude than the
modulating signal.

Amplitude modulator with an analog multiplier

The AD633 can be used as a linear amplitude modulator with no external components.
Figure below shows the circuit. The carrier and modulation inputs to the AD633 are multi-
plied to produce a double-sideband signal. The carrier signal is fed forward to the
AD633’s Z input where it is summed with the double-sideband signal to produce a double-
sideband with carrier output.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 8

AM with analog multiplier

It is evident that

sin ωt ⋅ sin ω M t = [cos(ω − ω M )t − cos(ω + ω M )t ] 2

Thus all is clear, now.

2. Angular modulation –Frequency (FM) and Phase (PM)

The usual expression for a harmonic signal of angular frequency ω C is

S (t ) = cos(Φ(t )) = cos(ωC t + ϕ 0 )
We define the instantaneous radian frequency ω i to be the derivative of the angle Φ (t ) as a
function of time:
dΦ d
ωi = = (ω C t + ϕ 0 )
dt dt
If Φ (t ) varies in some manner with the modulating signal f (t ) the result is angular modu-
lation.

Phase modulation – we vary the phase of the carrier linearly with the modulation signal; k1
is a constant of the system:
Φ (t ) = ωC t + ϕ0 + k1 ⋅ m(t )
The phase of the carrier (PM) varies with the modulating signal (directly).

Frequency modulation – we wary the frequency of the carrier linearly with the modulation
signal; k2 is a constant of the system:
ω (t ) = ωC + k2 ⋅ m(t )
Thus

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 9

t t
dΦ(t ) =ω i ⋅dt ⇒ φ (t ) = ∫ (ω C + k 2 ⋅ m(t )) ⋅ dt = ω C ⋅ t + k 2 ⋅ ∫ m(t ) ⋅ dt + ϕ 0
0 0
The phase of the carrier (FM) varies with the integral of the modulating signal.

Let us analyze FM first. We assume a harmonic modulation signal at the frequency f m :

m(t ) = a ⋅ cos(ωmt )
Let us have the instantaneous radian frequency ω i
ω i = ω C + ∆ω PEAK ⋅ cos(ω m t ); ∆ω PEAK 〈〈ω C
∆ω PEAK is a constant depending on the amplitude, a, of the modulating signal and on the
properties of the modulating system.
Now we easy get relation
∆ω PEAK
Φ (t ) = ∫ ωi dt = ωC t + sin( ω m t ) + ϕ 0
ωm
We can take ϕ 0 as zero. The frequency modulated carrier is than expressed by formula
 ∆ω PEAK 
S (t ) = A ⋅ cos ω C t + sin( ω m t )  = A ⋅ cos(ωC t + m ⋅ sin(ω m t ) )
 ωm 
where
∆ω PEAK ∆f PEAK
m= =
ωm fm
is the modulation index (represent the maximum phase shift of the carrier);
∆f PEAK is the maximum frequency deviation of the carrier

For simplicity we assume that m << π/2 (usually m < 0, 2) – narrow band FM. We get

S (t ) = A ⋅ cos(ω C t + m ⋅ sin( ω m t ) ) = A[cos(ω C t ) ⋅ cos(m ⋅ sin(ω m t )) − sin( ω C t ) ⋅ sin( m ⋅ sin(ω m t ))]

For m → 0 is true that m ⋅ sin( ω m t ) → 0. Thus we can use approximate formulas:

cos(m ⋅ sin(ω m t ) ≈ 1 ; sin( m ⋅ sin(ω m t ) ≈ m ⋅ sin( ω m t )

Thus we get

S (t ) = A[cos(ω C t ) − m ⋅ sin(ωC t ) ⋅ sin(ω m t ))]

Now we can easy get sideband form:

Am Am
S (t ) = A cos(ωC t ) + cos(ω C + ω m )t − cos(ωC − ω m )t
2 2

You can see spectra of AM and narrowband FM below.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 10

Spectra of AM and narrowband FM

If m is not small – wideband FM – mathematic will be more complicated. We can find

out that:
cos(m ⋅ sin(ω m t ) = J 0 (m) + 2 ⋅ J 2 (m) ⋅ cos 2ω m t + 2 ⋅ J 4 (m) ⋅ cos 4ω m t + ...
sin( m ⋅ sin(ω m t ) = 2 ⋅ J 1 (m) ⋅ sin ω m t + 2 ⋅ J 3 (m) ⋅ sin 3ω m t + ...
J n (m) is the nth – order Bessel function of the first kind.

Then we get

S (t ) = A ⋅ cos(ω C t + m ⋅ sin(ω m t ) ) = A[cos(ω C t ) ⋅ cos(m ⋅ sin(ω m t )) − sin(ωC t ) ⋅ sin(m ⋅ sin(ω m t )) ] =

= J 0 ( m) ⋅ cosω C t − J 1 ( m) ⋅ [cos(ω C − ω m )t − cos(ω C + ω m )t ] +
+ J 2 (m ) ⋅ [cos(ω C − 2ω m )t + cos(ω C + 2ω m )t ] −
− J 3 ( m) ⋅ [cos(ω C − 3ω m )t − cos(ω C + 3ω m )t ] + ...

We thus have a time function consisting of a carrier and an infinite number of sidebands –
amplitudes are proportional to J n (m) . The spectrum of the wideband FM you can see be-
low.

Spectrum of the wideband FM

The wideband FM wave has spectral components at all the frequencies, ω C ± nω m , where n
can be any integer from − ∞ to + ∞ . This result is rather startling since it means that, strictly
speaking, a system has to provide an infinite bandwidth to carry an accurate FM (or PM)
signal! Fortunately, there is a general tendency for J n (m) → 0 as n → ∞ . Moreover the
higher order Bessel function values fall quickly with n when the modulation index m is
small. In many practical situations we can arrange that and when this is true we find that

J 0 (m) = 1; J 1, (m) ≈ m / 2; and J n ≈ 0 for n ≥ 1

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 11

We get narrowband FM again.

Narrowband FM only requires a bandwidth of 2fm. A “high-m; wideband” FM can be

thought as a carrier whose frequency is varied over a relative wide range. It will therefore
require a bandwidth of at least 2 ⋅ ∆f PEAK

Combining these results with leads to Carson's Rule, that minimum practical bandwidth
required to transmit an FM/PM signal will be

B = 2 ⋅ ( f m + ∆f PEAK )

This rule is a useful guide when we have to choose a system to carry an FM signal. It
should be remembered, however, that in theory FM signals require an infinite bandwidth if
we want to avoid any signal distortion during transmission.

As an example of FM modulator let us describe Voltage Controlled Oscillator (VCO)

based upon the phase shift oscillator.

Voltage Controlled Oscillator (VCO) based upon

the phase shift oscillator.

In the VCO two of the original oscillator's capacitors have been replaced with Varactor
diodes. When we apply a Forward bias voltage to a diode it will conduct. When we apply a
Reverse bias voltage a Depletion Zone forms at the diode junction. Charge carriers can't
cross this zone, so the diode won't conduct in the ‘reverse’ direction. The reverse biased
diode therefore has a capacitance of C d = εA / d where d is the width of the depletion zone
and A is the area of each of diode junction. Increasing the applied reverse bias pulls the
carriers in the two halves apart, widening the depletion zone - the capacitance goes down.
As a result the reverse biased diode has a capacitance which depends upon the applied volt-
age.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 12

Basic demodulation FM principle

An FM demodulator or frequency discriminator is essentially a frequency-to-voltage

converter (F/V). An F/V converter may be realised in several ways, including for example,
tuned circuits and envelope detectors, phase locked loops etc. Demodulators are also called
FM discriminators.

We define Vo as the output when fIN = fc, the nominal input frequency. The gradient
K = ∆V / ∆f is called the voltage conversion factor. Then VOUT = V0 + K.fIN , where V0
represents a DC offset in VOUT.

The basic concept of FM demodulation is shown below. The FM input signal is input to
a frequency selective network – it is any electronic circuit or element that gives a response
(output) that varies with frequency. This circuit converts the frequency modulation into an
amplitude modulation – it is then demodulated using any standard AM demodulator. Al-
ways must be ensured a constant amplitude of the FM input signal – it is used input limiter.
It eliminates the effects of any parasitic amplitude modulation of the FM signal.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 13

Tuned Circuit – One method (used in the early days of FM) is to use the slope of a tuned
circuit in conjunction with an envelope detector.

SIMPLE TUNED CIRCUIT

The tuned circuit is tuned so the fc, the nominal input frequency, is on the slope, not at
the centre of the tuned circuits. As the FM signal deviates about fc on the tuned circuit
slope, the amplitude of the output varies in proportion to the deviation from fc. Thus the
FM signal is effectively converted to AM. This is then envelope detected by the diode
etc to recover the message signal. A better method is to use 2 similar circuits, known as
a Balanced Discriminator

Balanced Discriminator

Circuit Operation of a Foster-Seeley Discriminator

View (A) of figure below shows a typical Foster-Seeley discriminator. The collector cir-
cuit of the preceding limiter/amplifier circuit (Q1) is shown. The limiter/amplifier circuit is
a special amplifier circuit which limits the amplitude of the signal. This limiting keeps in-
terfering noise low by removing excessive amplitude variations from signals. The collector
circuit tank consists of C1 and L1. C2 and L2 form the secondary tank circuit. Both tank
circuits are tuned to the center frequency of the incoming fm signal. Choke L3 is the dc
return path for diode rectifiers CR1 and CR2. R1 and R2 are not always necessary but are
usually used when the back (reverse bias) resistance of the two diodes is different. Resistors
R3 and R4 are load resistors and are bypassed by C3 and C4 to remove rf. C5 is the output
coupling capacitor.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 14

FOSTER-SEELEY DISCRIMINATOR

The operation of the Foster-Seeley discriminator can best be explained using vector dia-
grams [above, view (B)] that show phase relationships between the voltages and currents in
the circuit. Let's look at the phase relationships when the input frequency is equal to the
center frequency of the resonant tank circuit. The input signal applied to the primary tank
circuit is shown as vector ep. Since coupling capacitor C8 has negligible reactance at the
input frequency, rf choke L3 is effectively in parallel with the primary tank circuit. Also,
because L3 is effectively in parallel with the primary tank circuit, input voltage ep also ap-
pears across L3. With voltage ep applied to the primary of T1, a voltage is induced in the
secondary which causes current to flow in the secondary tank circuit. When the input fre-
quency is equal to the center frequency, the tank is at resonance and acts resistive. Current
and voltage are in phase in a resistance circuit, as shown by is and ep. The current flowing
in the tank causes voltage drops across each half of the balanced secondary winding of
transformer T1. These voltage drops are of equal amplitude and opposite polarity with re-
spect to the center tap of the winding. Because the winding is inductive, the voltage across
it is 90 degrees out of phase with the current through it. Because of the center-tap arrange-
ment, the voltages at each end of the secondary winding of T1 are 180 degrees out of phase
and are shown as e1 and e2 on the vector diagram. The voltage applied to the anode of CR1
is the vector sum of voltages ep and e1, shown as e 3 on the diagram. Likewise, the voltage
applied to the anode of CR2 is the vector sum of voltages ep and e 2, shown as e4 on the
diagram. At resonance e3 and e4 are equal, as shown by vectors of the same length. Equal
anode voltages on diodes CR1 and CR2 produce equal currents and, with equal load resis-

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 15

tors, equal and opposite voltages will be developed across R3 and R4. The output is taken
across R3 and R4 and will be 0 at resonance since these voltages are equal and of apposi-
tive polarity. The diodes conduct on opposite half cycles of the input waveform and pro-
duce a series of dc pulses at the rf rate. This rf ripple is filtered out by capacitors C3 and
C4.

A phase shift occurs when an input frequency higher than the center frequency is ap-
plied to the discriminator circuit and the current and voltage phase relationships change.
When a series-tuned circuit operates at a frequency above resonance, the inductive reac-
tance of the coil increases and the capacitive reactance of the capacitor decreases. Above
resonance the tank circuit acts like an inductor. Secondary current lags the primary tank
voltage, ep. Notice that secondary voltages e 1 and e2 are still 180 degrees out of phase with
the current (iS) that produces them. The change to a lagging secondary current rotates the
vectors in a clockwise direction. This causes el to become more in phase with ep while e2 is
shifted further out of phase with ep. The vector sum of ep and e2 is less than that of ep and
e1. Above the center frequency, diode CR1 conducts more than diode CR2. Because of this
heavier conduction, the voltage developed across R3 is greater than the voltage developed
across R4; the output voltage is positive.

When the input frequency is lower than the center frequency, the current and voltage
phase relationships change. When the tuned circuit is operated at a frequency lower than
resonance, the capacitive reactance increases and the inductive reactance decreases. Below
resonance the tank acts like a capacitor and the secondary current leads primary tank volt-
age ep. This change to a leading secondary current rotates the vectors in a counterclockwise
direction. From the vector diagram you should see that e2 is brought nearer in phase with
ep, while el is shifted further out of phase with ep. The vector sum of ep and e2 is larger than
that of e and e1. Diode CR2 conducts more than diode CR1 below the center frequency.
The voltage drop across R4 is larger than that across R3 and the output across both is nega-
tive.

Disadvantages - these voltage outputs can be plotted to show the response curve of the
discriminator discussed earlier. When weak AM signals (too small in amplitude to reach the
circuit limiting level) pass through the limiter stages, they can appear in the output. These
unwanted amplitude variations will cause primary voltage ep to fluctuate with the modula-
tion and to induce a similar voltage in the secondary of T1. Since the diodes are connected
as half-wave rectifiers, these small AM signals will be detected as they would be in a diode
detector and will appear in the output. This unwanted AM interference is cancelled out in
the ratio detector (to be studied next in this chapter) and is the main disadvantage of the
Foster-Seeley circuit.

Ratio detector

It uses a double-tuned transformer to convert the instantaneous frequency variations of the

fm input signal to instantaneous amplitude variations. These amplitude variations are then
rectified to provide a dc output voltage which varies in amplitude and polarity with the in-
put signal frequency. This detector demodulates fm signals and suppresses amplitude noise
without the need of limiter stages.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 16

Ratio detector

Circuit Operation

Figure above shows a typical ratio detector. The input tank capacitor (C1) and the primary
of transformer T1 (L1) are tuned to the center frequency of the fm signal to be demodu-
lated. The secondary winding of T1 (L2) and capacitor C2 also form a tank circuit tuned to
the center frequency. Tertiary (third) winding L3 provides additional inductive coupling
which reduces the loading effect of the secondary on the primary circuit. Diodes CR1 and
CR2 rectify the signal from the secondary tank. Capacitor C5 and resistors R1 and R2 set
the operating level of the detector. Capacitors C3 and C4 determine the amplitude and po-
larity of the output. Resistor R3 limits the peak diode current and furnishes a dc return path
for the rectified signal. The output of the detector is taken from the common connection
between C3 and C4. Resistor RL is the load resistor. R5, C6, and C7 form a low-pass filter
to the output.

This circuit operates on the same principles of phase shifting as did the Foster-Seeley
discriminator. In that discussion, vector diagrams were used to illustrate the voltage ampli-
tudes and polarities for conditions at resonance, above resonance, and below resonance.
The same vector diagrams apply to the ratio detector but will not be discussed here. Instead,
you will study the resulting current flows and polarities on simplified schematic diagrams
of the detector circuit.

When the input voltage ep is applied to the primary in figure above it also appears across
L3 because, by inductive coupling, it is effectively connected in parallel with the primary
tank circuit. At the same time, a voltage is induced in the secondary winding and causes
current to flow around the secondary tank circuit. At resonance the tank acts like a resistive
circuit; that is, the tank current is in phase with the primary voltage ep. The current flowing in
the tank circuit causes voltages e1 and e2 to be developed in the secondary winding of T1.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 17

These voltages are of equal magnitude and of opposite polarity with respect to the center
tap of the winding. Since the winding is inductive, the voltage drop across it is 90 degrees
out of phase with the current through it.

Figure below is a simplified schematic diagram of a ratio detector at resonance. The

voltage applied to the cathode of CR1 is the vector sum of e1 and ep. Likewise, the voltage
applied to the anode of CR2 is the vector sum of e2 and ep. No phase shift occurs at reso-
nance and both voltages are equal. Both diodes conduct equally. This equal current flow
causes the same voltage drop across both R1 and R2. C3 and C4 will charge to equal volt-
ages with opposite polarities. Let’s assume that the voltages across C3 and C4 are equal in
amplitude (5 volts) and of opposite polarity and the total charge across C5 is 10 volts. R1
and R2 will each have 5 volts dropped across them because they are of equal values. The
output is taken between points A and B. To find the output voltage, you algebraically add
the voltages between points A and B (loop ACB or ADB). Point A to point D is −5 volts.
Point D to point B is + 5 volts. Their algebraic sum is 0 volts and the output voltage is 0 at
resonance. If the voltages on branch ACB were figured, the same output would be found
because the circuit branches are in parallel.

Current flow and polarities at resonance

When the input signal reverses polarity, the secondary voltage across L2 also reverses.
The diodes will be reverse biased and no current will flow. Meanwhile, C5 retains most of
its charge because of the long time constant offered in combination with R1 and R2. This
slow discharge helps to maintain the output.

When a tuned circuit operates at a frequency higher than resonance, the tank is induc-
tive – see figure below. The secondary current is lags the primary voltage ep. Secondary
voltage e1 is nearer in phase with primary voltage ep, while e2 is shifted further out of
phase with ep. The vector sum of e1 and ep is larger than that of e2 and ep. Therefore, the
voltage applied to the cathode of CR1 is greater than the voltage applied to the anode of
CR2 above resonance.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 18

Current flow and polarities above resonance

Assume that the voltages developed above resonance are such that the higher voltage on
the cathode of CR1 causes C3 to charge to 8 volts. The lower voltage on the anode of CR2
causes C4 to charge to 2 volts. Capacitor C5 remains charged to the sum of these two volt-
ages, 10 volts. Again, by adding the voltages in loop ACB or ADB between points A and
B, you can find the output voltage. Point A to point D equals -2 volts. Point D to point B
equals +5 volts. Their algebraic sum, and the output, equals +3 volts when tuned above
resonance. During the negative half cycle of the input signal, the diodes are reverse biased
and C5 helps maintain a constant output.

When a tuned circuit operates below resonance (figure below), it is capacitive. Secon-
dary current is leads the primary voltage ep and secondary voltage e2 is nearer in phase
with primary voltage ep. The vector sum of e2 and ep is larger than the sum of e1 and ep.
The voltage applied to the anode of CR2 becomes greater than the voltage applied to the
cathode of CR1 below resonance.

below resonance

Assume that the voltages developed below resonance are such that the higher voltage on
the anode of CR2 causes C4 to charge to 8 volts. The lower voltage on the cathode of CR1
causes C3 to charge to 2 volts. Capacitor C5 remains charged to the sum of these two volt-

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 19

ages, 10 volts. The output voltage equals −8 volts plus +5 volts, or −3 volts, when tuned
below resonance. During the negative half cycle of the input signal, the diodes are reverse
biased and C5 helps maintain a constant output.

Advantage of a Ratio Detector - the ratio detector is not affected by amplitude variations
on the fm wave. The output of the detector adjusts itself automatically to the average ampli-
tude of the input signal. C5 charges to the sum of the voltages across R1 and R2 and, be-
cause of its time constant, tends to filter out any noise impulses. Before C5 can charge or
discharge to the higher or lower potential, the noise disappears. The difference in charge
across C5 is so slight that it is not discernible in the output. Ratio detectors can operate with
as little as 100 millivolts of input. This is much lower than that required for limiter satura-
tion and less gain is required from preceding stages.

Often is used for FM/PM demodulation Double Balanced Mixer (DBM) as a phase
detector. Describe it once again. It consists of four diodes linking two transformers. There
are three ways or ports by which signals can get in/out of the DBM. It is conventional to
call these the RF, LO- local oscillator, and IF ports since DBM's are often used as mixers
in heterodyne systems. To see how the system works we can start by considering the LO
port. This lets us apply a voltage between the points A and B in the circuit.

When we apply a voltage which makes A positive with respect to B (which we'll assume
occurs when Vlo ≥ 0 ) the diodes, D1, D2 will conduct and D3, D4 will not. This means that
D1 and D2 will present a very low resistance to any signals and D3; D4 will present a very
high resistance. As a result, the circuit will behave like below. The two transformers will be
‘directly’ connected together via the conducting diodes. Using 1:1 transformers and low-
loss diodes we therefore find that, when Vlo ≥ 0 - is positive, Vout = Vin (for a.c. signals, of
course!).

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 20

A positive referred to B

When we apply an LO voltage such that A is negative compared to B we can use a simi-
lar argument to show that, in this case, the circuit behaves like below. Hence when Vlo ≤ 0 -
is negative, Vout = −Vin . We can therefore use the DBM as a sort of ‘switch’ to control
whether we pass on the signal unaffected or invert it by applying the required polarity of
Vlo .

A negative referred to B

There is an example below.

NO AMPLITUDE MODULATION

Box “delay” defines group delay (time delay) on carrier frequency ωC : it must be just
T
τ ⋅ ωC = π / 2 ⇒ τ = π /(2 ⋅ ωC ) = π /(2 ⋅ 2πf C ) = C
4
In the ideal case we suppose ideally limited signals at inputs LO and RF. Then we can easy
derive picture below:

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 21

TC/4 = τ
TC
VLO = Vin

TC VRF -delayed Vin

0
fin= fC

If fin= fC then average voltage of Vout is just zero.

0

Tin > TC VLO = Vin

VRF -delayed Vin

fin< fC
0

If fin< fC then average voltage of Vout is positive.

0

Tin < TC
VLO = Vin
0

VRF -delayed Vin fin> fC

0

time
If fin> fC then average voltage of Vout is negative.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 22

Let us suppose that Vout takes values ± 1 , only. It is evident it is enough to investigate
just one half of period.

Vout = - 1 in the time interval τ = π /(2ωC )

Vout = + 1 in the time interval Tin / 2 − τ = Tin / 2 − π /(2ωC )

Now we can easy determine the average value in the output:

(−1) ⋅ π /(2ω C ) + 1 ⋅ (Tin / 2 − π /(2ω C ))

VoutAV = = ω in = 2πf in = 2π / Tin ⇒ Tin = 2π / ω in
Tin / 2

− π /(2ω C ) + (π / ω in − π /(2ω C ))
VoutAV = = 1 − ω in / ω C
π / ω in

If is valid ω in = ω C ± ∆ω we get formula

ω C ± ∆ω
VoutAV = 1 − ω in / ω C = 1 − = m ∆ω
ωC

This average voltage is separated by means of the lowpass filter.

Delay circuit converts FM modulation into PM but preserves FM.

If we can use some multiplier we can construct circuit as below – quadrature

demodulator:

quadrature demodulator

The constant delay gives a phase shift of π/2 (quadrature) at the carrier frequency. The dif-
ference term from the multiplication
cos[(ω c t + ϕ (t )) + (ω c (t − τ ) + ϕ (t − τ ))]
cos(ω c t + ϕ (t )) ⋅ cos(ω c (t − τ ) + ϕ (t − τ )) = +
2
cos[(ω c t + ϕ (t )) − (ω c (t − τ ) + ϕ (t − τ ))]
+
2
is

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 23

cos[(ω c t + ϕ (t )) − (ω c (t − τ ) + ϕ (t − τ ))] cos[ω c t + ϕ (t ) − ω c t + ω cτ − ϕ (t − τ ))]

= = ω cτ = π / 2 =
2 2
cos[π / 2 + ϕ (t ) − ϕ (t − τ ))] sin [ϕ (t ) − ϕ (t − τ ))]
= = cos(π / 2 + a) = − sin a = −
2 2
This component passes through the low pass filter. If we suppose very small phase differ-
ence over the period τ then we can write

sin [ϕ (t ) − ϕ (t − τ ))]
− ≈ −(ϕ (t ) − ϕ (t − τ ))
2
and it is valid

ϕ (t ) − ϕ (t − τ ) dϕ (t ) dϕ (t )
≈ ⇒ − (ϕ (t ) − ϕ (t − τ ) ) ≈ −τ ⋅
τ dt dt
The sum term from the multiplication gets twice the carrier frequency and it is removed by
the low pass filtering. Then the output signal is

τ dϕ ( t ) τ
t
− ⋅ = FM − ϕ (t ) = k 2 ⋅ ∫ m(t ) ⋅ dt + ϕ 0 = − ⋅ k 2 ⋅ m(t )
2 dt 0
2

- it approximates the original message.

The constant delay circuit τ we can see below, its small signal model:

SMALL SIGNAL MODEL OF THE „DELAY τ“ BLOCK

The ratio of V2 over V1 is the ratio of impedances Zp (it is parallel combination of L, Rp and
Cp) over (Zp +1/(pCs)), p = jω – harmonic steady state – it is the impedance divider, simply.
Simplifying this ratio we get

Cs p2 Cs p2
V2 / V1 = ⋅ = ⋅
Cs + C p 1 1 Cs + C p ω
p2 + p ⋅ + p 2 + p ⋅ 0 + ω 02
R p (C s + C p ) L(C s + C p ) Q

From this formula is evident that

ω0 1 Rp
ω 0 = 1 / L(C s + C p ) and = ⇒ Q=
Q R p (C s + C p ) ω0 L

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 24

If we substitute p = jω we get

Cs −ω2
V2 / V1 = ⋅
Cs + C p ω0
ω 02 − ω 2 + jω ⋅
Q

Now we can derive phase responses as

ωω 0
Θ(ω ) = −π − arctg
Q (ω 02 − ω 2 )
Next we can get time delay of the circuit:
d  ωω 0  d
τ = − dΘ(ω ) / dω =  arctg 2 
= (arctgx ) = 1 2 =
dω  Q(ω 0 − ω )  dx
2
1+ x
1 d  ωω 0  ω 03 + ω 2ω 0
= ⋅   = ... =
 ωω 0 
2
dω  Q(ω 02 − ω 2 )  Q((ω 02 − ω 2 ) 2 + ω 02ω 2 / Q
1+  2 
 Q (ω 0 − ω ) 
2

If we suppose ω = ω 0 + ∆ω = ω 0 (1 + ∆ω / ω 0 ) = ω 0 (1 + x) we can get

τ=
ω 03 + ω 2ω 0
=
[
ω 03 1 + (1 + x) 2 ] = x→0 ≅
[ ]
Q((ω 02 − ω 2 ) 2 + ω 02ω 2 / Q ω 04 Q (1 − (1 + x) 2 2 + ω 04 (1 + x) 2 / Q
2Q 1 + x
≅ ⋅
ω0 1 + 2 x

Similarly we can determine the modulus

V2 Cs ω2 Cs Q(1 + 2 x)
= ⋅ = x→0 = ⋅
V1 C s + C p ω0 2 Cs + C p 1+ x
(ω 02 − ω 2 ) 2 + (ω ⋅ )
Q

So if we have ∆ω / ω 0 << 1 the delay and modulus are constant nearly:

2Q V2 Q ⋅ Cs
τ ≅ ≅
ω0 V1 C s + C p

Phase-Locked Loop (PLL)

− Commonly used for carrier synchronization and indirect FM demodulation
− Consists of - VCO – performs frequency modulation

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 25

- Multiplier
- Loop Filter – low-pass kind, removes the high-frequency components con-
tained in the multiplier output.
− To explain the operation, let assume that the VCO frequency is initially set to the carrier
frequency of the incoming FM signal and the VCO’s output has a 90- degrees phase-shift.

We have FM signal

s (t ) = Ac sin [ω c t + ϕ1 (t )]; ϕ1 (t ) = 2πk1 ∫ m(t )dt ; m(t) – modulation signal.

r (t ) = Av cos[ω c t + ϕ 2 (t )] - it describes the 90 –degree phase shift and the carrier

frequency, too.

But VCO is frequency modulator, too – so it must be ϕ 2 (t ) = 2πk 2 ∫ v(t )dt .

e(t ) = k m ⋅ Ac sin [ω c t + ϕ1 (t )] ⋅ Av cos[ω c t + ϕ 2 (t )] =

= k m ⋅ Ac ⋅ Av
[sin( 2ω c t + ϕ1 + ϕ 2 ) + sin(ϕ1 − ϕ 2 )]
2
km defines a multiplier constant.
Now we can determine a phase difference, low-pass loop filter removes the high-frequency
components contained in the multiplier output:
∆Θ = ϕ1 (t ) − ϕ 2 (t ) = 2πk1 ∫ m(t )dt − 2πk 2 ∫ v(t )dt

If we can ensure ∆Θ → 0 , we get ∆Θ = ϕ1 (t ) − ϕ 2 (t ) = 2πk1 ∫ m(t )dt − 2πk 2 ∫ v(t )dt = 0 thus
2πk1 ∫ m(t )dt − 2πk 2 ∫ v(t )dt = 0 thus the output voltage v(t ) = (k1 / k 2 ) ⋅ m(t ) - the output
voltage approximates the original message m(t).

Now we differentiate the phase difference

d
∆Θ = 2πk1 m(t ) − 2πk 2 v (t ) (*)
dt
and will investigate the problem in more detail.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 26

Let we have

v(t ) = k LP ⋅ e(t ) = low frequencies only = k LP k m Ac Av

[sin ( 2ωc t + ϕ1 + ϕ 2 ) + sin(ϕ1 − ϕ 2 )]
2
k LP k m Ac Av k k AA
v(t ) = ⋅ sin(ϕ1 − ϕ 2 ) = ∆Θ → 0 ≅ LP m c v ⋅ ∆Θ ,
2 2
k LP - defines a transfer constant of the loop (lowpass) filter. Hence for the small phase
difference

d k k AA
∆Θ = 2πk1 m(t ) − 2πk 2 k LP v (t ) = 2πk1 m(t ) − 2πk 2 ⋅ LP m c v ⋅ ∆Θ
dt 2

d
We can rewrite this formula as below ∆Θ = 2πk1 m(t ) − K ⋅ ∆Θ
dt

k LP k m Ac Av
where K = 2πk 2 ⋅ is Phase Lock Loop Gain.
2
Now is valid too

k LP k m Ac Av 2πk 2 k LP k m Ac Av K ⋅ ∆Θ
v(t ) ≅ ⋅ ∆Θ = ⋅ ⋅ ∆Θ =
2 2πk 2 2 2πk 2

It is supposed the “signal” output voltage of v(t ) = Vo sin( ω m t ) , now. We easy determine
corresponding (needed) phase difference

K ⋅ ∆Θ 2πk 2
v(t ) = Vo sin( ω m t ) = ⇒ ∆Θ = ⋅ Vo sin(ω m t )
2πk 2 K

Now from eq. (*) we determine

d 2π k 2
2π k 1 m ( t ) = ∆Θ + 2 π k 2 v ( t ) = ⋅ V o ω m cos( ω m t ) + 2 π k 2 V o sin( ω m t )
dt K

k 2 ⋅ ωm k
Thus m(t ) = Vo ⋅ cos(ω m t ) + 2 ⋅ Vo ⋅ sin( ω m t )
k1 ⋅ K k1

If the K is infinite (large enough) we get again

k2 k
m(t ) = ⋅ Vo ⋅ sin(ω m t ) = 2 ⋅ v (t )
k1 k1

For the K large enough, very small phase difference can generate appropriate output
signal v(t) – this approximates original message.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 27

The PLL has some interesting features. The most useful of these is that it largely ignores
fluctuations in the amplitude of the input FM wave. This means that the PLL FM demodu-
lator tends to ignore any unwanted interference which appears alongside the input signal.
The loop is said to have the property of AM Rejection.
Of course, the system can't work perfectly for any input signal size, no matter how
small. The loop gain does depend upon Ac, so if the FM wave's amplitude is too small we
can't assume that the gain is very large. For this reason we usually try to ensure that the
input wave's amplitude is ‘big enough’ to avoid these problems.

The above analysis assumed that the VCO could always adjust its output to ‘track’ any
changes in the FM wave's frequency. This is the same as assuming that the low pass filter
isn't affecting the DBM's output. In reality, a low pass filter will tend to attenuate (and
phase shift) any swiftly changing signals. As a result, the above arguments only strictly
apply for a system which is locked (the signal and VCO maintain a steady phase relation-
ship) when the modulation frequency is ‘low’ (i.e. passes through the filter without being
affected).

Consider for a moment what happens when we start with the PLL not locked to an input
signal. The input (fs) and VCO (fo) frequencies will differ by some amount ∆f = f s− f o so
the output voltage (averaged over a few cycles of carrier) will be
v(t ) ≈ (k1 / k 2 ) ⋅ sin(2π∆f )

i.e. the output voltage tends to oscillate at the difference frequency,. Provided that ∆f is low
enough this variation can pass through the low pass filter the loop can use this to help it
lock onto the input wave. This brings the two frequencies together, and removes this ‘beat-
ing’ effect. However, if the difference frequency is too high to get through the low pass
filter the multiplier's output won't be communicated to the VCO. The PLL then can't lock
onto the input FM wave and essentially ignores it! As a result, the system can only lock
onto an input and demodulate it if the signal frequency is such that ∆f ≤ B where B is the
bandwidth of the filter. The system is said to have a Lock in range of ± B set by the choice
of the loop filter. A loop ‘free running’ at a frequency, f o will therefore ignore signals out-
side the range f o ± B .

Now we investigate PM

The usual expression for a harmonic signal is

S (t ) = cos(Φ(t )) = cos(ωC t + ϕ 0 )
The instantaneous radian frequency ω i is
dΦ d
ωi = = (ω C t + ϕ 0 )
dt dt
t
Frequency modulation: ω (t ) = ωC + k2 ⋅ m(t ) ; φ (t ) = ω C ⋅ t + k 2 ⋅ ∫ m(t ) ⋅ dt + ϕ 0
0

d
Phase modulation: Φ (t ) = ωC t + ϕ0 + k1 ⋅ m(t ) ; ω (t ) = ω C + k1 ⋅ m(t )
dt

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 28

The phase of the carrier (PM) varies with the modulating signal (directly).
The phase of the carrier (FM) varies with the integral of the modulating signal.

Comparing formulas above we find that — unless we know something about the modu-
lation in advance — it may not be obvious whether the signal is FM or PM modulated. In
both cases the wave's frequency and phase vary from moment to moment. Mathematically
speaking, FM & PM are almost identical twins. The only difference is that one corresponds
to a modulation pattern which is the differential of that produced by the other. The ‘good
news’ is that this means we can mix FM & PM arguments and most of our conclusions
about one apply to the other. The ‘bad news’ is that, in practice we often have to know in
advance which type of modulation is being used if we want to recover the modulated in-
formation correctly. In general, however, both FM & PM waves are obviously different to
an AM wave.

Generally is valid for modulation:

Integrator PM
vm(t) vFM(t)
∫ dt Modulator

Generation of FM

Differentiator
d FM
vm(t) dt Modulator
vPM(t)

Generation of PM
Analogously for demodulation we can get

Differentiator
vFM(t) PM d vm(t)
Demodulator dt

FM Demodulator

vPM(t) FM Integrator
vm(t)
Demodulator ∫ dt

PM Demodulator

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 29

The earliest analog phase modulators are of type below. A crystal oscillator (XO) gener-
ates the desired frequency and voltage controlled phase shifter does the modulating. Typi-
cally, the phase shifter is a tuned circuit with voltage controlled capacitors as the tuning
element.

PHASE MODULATOR

Figure below shows the amplitude and phase of a single tuned resonant circuit. The
plot is normalized so units are independent of Q. The plot shows the phase shift of 45 de-
grees (0.785 radians) at the –3 dB points on the amplitude curve. The curve for phase is a
tangent curve and goes from +90 to – 90 degrees with zero degrees at resonance. The
modulation of the voltage to the varicap shifting the resonant frequency results in phase
shifting the output following the phase response curve.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 30

PM demodulator
The basic concept upon which phase detection rests is that the application of two identi-
cal frequencies, constant amplitude signals to a mixer results in a dc output which is pro-
portional to the phase difference between the two signals. While it is true that even a single
diode can be used as a mixer, most phase detectors involve the use of double balanced mix-
ers – see description of the DBM above.

Very often the Gilbert-cell (see figure below) is used as a multiplier circuit (modulators,
demodulators, mixers). It includes a tail current source (Q7), a differential transconductance
stage (Q1, Q2), and a switching quad (Q3-Q6). The output can be driven to a resistive or
reactive tuned load.

BJT Gilbert cell

If we suppose small signals only we can write (see small signal BJT model, Chapter 2):

U th U th
re 3 ≅ re 4 ≅ ; re5 ≅ re 6 ≅
IT + I s IT − I s
I T + I s VLO / 2 I + I s VLO / 2
I3 = − ; I4 = T +
2 re3 2 re4
I T − I s VLO / 2 I − I s VLO / 2
I5 = + ; I4 = T −
2 re5 2 re 6
I + I s VLO I T − I s VLO
= I T + LO ⋅ (− 1 / re3 + 1 / re5 )
V
I O1 = I 3 + I 5 = T − + +
2 2re 3 2 2re 5 2
I + I s VLO I T − I s VLO
= I T + LO ⋅ (1 / re 4 − 1 / re 6 )
V
IO2 = I 4 + I 6 = T + + −
2 2re 4 2 2re 6 2

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 31

An output current we define as a difference

 
I o = I O1 − I O 2 = I T + LO ⋅ (− 1 / re3 + 1 / re5 ) −  I T + LO ⋅ (1 / re 4 − 1 / re6 ) ≅ VLO (1 / re 5, 6 − 1 / re 3, 4 )
V V
2  2 
It is evident that for small signals is I s ≅ VRF /(2 ⋅ re ) where re ≅ U th / I T .
Now is valid
 
 
 I + I s IT − I s   2 ⋅ Is 
I o ≅ V LO (1 / re 5,6 − 1 / re3, 4 ) = VLO  1 1 
− = V LO  T −  = VLO  
 U th U th 
   U th U th   th 
U
 T
I + I s I T − I s 

 2 ⋅ VRF /( 2 ⋅ re )  VLO ⋅ V RF
= VLO   =
 U th  reU th

We can investigate large signal properties, too. The mixing principle of a Gilbert cell is
based on the so called controlled transconductance mixer; see simplified (basic) circuit in
the figure below. For a down-conversion application, the information carrying high fre-
quency (RF) signal is applied as a voltage that modulates the current source with quiescent
current 2Io = I1 + I2. First, assuming no RF signal, the current is divided up in the currents I1
and I2 depending on the applied voltage VLO through
2I 0 2I 0
I1 = I2 =
1 + e −VLO / U th 1 + eVLO / U th
from the exponential voltage to current relation of the bipolar transistor. Here, Uth = kT/q
denotes the thermal voltage (Uth = 26mV at room temperature).

BOX
I E ≅ I E 0 ⋅ exp(U BE / U th ) → I E1 ≅ I E 0 ⋅ exp(U BE1 / U th ) ; I E 2 ≅ I E 0 ⋅ exp(U BE 2 / U th ) →
U BE1 = U th ⋅ ln( I 1 / I E 0 ) ; U BE 2 = U th ⋅ ln( I 2 / I E 0 )
It is evident that: VLO = U BE1 − U BE 2 →
VLO = U BE1 − U BE 2 = U th ⋅ ln( I 1 / I E 0 ) − U th ⋅ ln( I 2 / I E 0 ) = U th ⋅ ln( I 1 / I 2 ) →
I1
= exp(VLO / U th ) →
I2
I 1 = I 2 ⋅ exp(VLO / U th ) ; I 2 = I1 ⋅ exp( −VLO / U th )

It is evident that: I 1 + I 2 = 2I 0 →
2I 0
I 1 + I 2 = I1 + I 1 ⋅ exp( −VLO / U th ) = 2 I 0 → I1 =
1 + e −VLO / U th
2I0
I 1 + I 2 = I 2 ⋅ exp(VLO / U th ) + I 2 = 2 I 0 → I2 =
1 + eVLO / U th

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 32

BASIC
GILBERT CELL
PRINCIPLE

T1 T2

VLO

T3
VRF

The differential output current i0 is then given by

V 
iout = I 1 − I 2 = 2 I 0 tanh LO 
 2U th 

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 33

BOX
2I 0 2I 0 VLO 1 + e +α − (1 + e −α )
iout = I1 − I 2 = − = α = = 2 ⋅ I ⋅ =
1 + e −α 1 + e +α (1 + e −α )(1 + eα )
0
U th
e +α − e −α (e +α / 2 ) 2 − (e −α / 2 ) 2 ( e +α / 2 ) 2 − ( e −α / 2 ) 2
= 2 ⋅ I0 ⋅ = 2 ⋅ I0 ⋅ = 2 ⋅ I 0 ⋅ +α / 2 2 =
1 + e α + e −α + 1 e α + 2 + e −α (e ) + 2 + ( e −α / 2 ) 2
+α / 2 −α / 2 (e +α / 2 ) 2 − (e −α / 2 ) 2
=e ⋅e = 1 = 2 ⋅ I 0 ⋅ +α / 2 2 =
(e ) + 2 ⋅ (e +α / 2 ⋅ e −α / 2 ) + (e −α / 2 ) 2
(e +α / 2 + e −α / 2 ) ⋅ (e +α / 2 − e −α / 2 ) e +α / 2 − e −α / 2 V 
= 2 ⋅ I0 ⋅ +α / 2 −α / 2 2
= 2 ⋅ I ⋅ +α / 2 −α / 2
= 2 ⋅ I 0 ⋅ tanh LO 
+e +e
0
(e ) e  2U th 

If VLO is small (<< U th ), tanh (VLO / 2U th ) ≅ VLO / 2U th and so the output current i0 is
approximately linearly proportional to VLO (the mixer is said to be working in
the”multiplication region”; in the opposite case tanh → ±1 – it means switching region):

VLO
iout ≅ I 0 ⋅
U th

Now, if VRF is a small signal voltage and the transconductance of the current source (T3) is
gm, 2I0 is replaced by 2I0 + gmVRF and

VLO V g V V
iout ≅ (2I 0 + g mVRF ) ⋅ = I 0 ⋅ LO + m RF LO
2U th U th 2U th

The first term is called LO leakage or feed-through and the second term is the wanted
one. The first term can be cancelled by employing a second, identical, circuit, driven by
−VLO, which output current is

VLO g mVRF VLO

iout 2 ≅ I 0 ⋅ −
U th 2U th

and the output current is taken as the difference between iout and iout 2 :

g mVRF VLO
i IF = iout − iout 2 =
U th

See figure below – and compare with Gilbert cell described above.

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Although the Gilbert-cell was initially designed with bipolar transistors, its operation
principle is similar using CMOS technology – see figure below.

MOSFET Gilbert cell

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 35

3. Pulse modulations
Another class of modulation, called pulse modulation, is applicable in sampled wave-
form scenarios. The information in a sampled signal can be represented (or encoded) by
varying the amplitude, duration (ore width), position (relative to a time reference), or repe-
tition rate of the transmitted pulses. The way in which information is encoded by several
types of pulse modulation is illustrated in figure below. The most common forms of pulse
modulation are essentially forms of amplitude modulation. However, frequency or polariza-
tion modulation could also be used to encode the signal amplitude modulation.

Types of pulse modulations

As an example let us see a PDM (PWM) modulator principle. In the analogue domain a
PWM signal can be generated by comparing the signal to a triangle or sawtooth waveform.
This technique, called natural sampling, is the basis of almost all analogue modulators. See
Figure below.

PDM PRINCIPLE

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 36

When the momentary value of the input signal is larger than the triangle, the output of
the switch is high. It is easy to see that in this way the pulse width at the output is propor-
tional to the input voltage.

Related to but not a true form of pulse modulation is pulse code modulation (PCM).
PCM is a truly digital modulation technique. In PCM the signal is sampled, the sampled
signal is quantized into two more discrete levels, and the quantized level is transmitted as a
unique string of bits, the bits having been encoded into pulses. Pulse amplitude, position,
duration, pulse rate, frequency, polarization, phase, and polarity among others are capable
of being used to encode the bits.

Analog-over-digital methods:
Pulse-code modulation (PCM)
Differential PCM (DPCM)
Adaptive DPCM (ADPCM)
Delta modulation (DM)
Sigma-delta modulation
Continuously variable slope delta modulation (CVSDM), also called Adaptive-delta modu-
lation (ADM)
Pulse-density modulation (PDM)

As an example let us see a delta modulator – see figure below. The analog input is a
voice signal with amplitude of a few volts, while the output signal is a stream of digital
ones and zeros. A comparator decides which has the greater voltage, the incoming analog
signal, or the voltage stored on the capacitor. This decision, in the form of a digital one or
zero, is applied to the input of the latch. At each clock pulse, typically at a few hundred
kilohertz, the latch transfers whatever digital state appears on its input, to its output. This
latch insures that the output is synchronized with the clock, thereby defining the sampling
rate, i.e., the rate at which the 1 bit output can update itself.

DELTA MODULATOR

Block diagram of a delta modulation circuit. The input voltage is compared with the voltage
stored on the capacitor, resulting in a digital zero or one being applied to the input of the latch. The
output of the latch is updated in synchronization with the clock, and used in a feedback loop to
cause the capacitor voltage to track the input voltage.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 37

Figure below illustrates the signals produced by this circuit. At time equal zero, the ana-
log input and the voltage on the capacitor both start with a voltage of zero. As shown in (a),
the input signal suddenly increases to 9.5 V on the eighth clock cycle. Since the input signal
is now more positive than the voltage on the capacitor, the digital output changes to a one,
as shown in (b). These results in the switch being connected to the positive charge injector,
and the voltage on the capacitor increasing by a small amount on each clock cycle. Al-
though an increment of 1 V per clock cycle is shown in (a), this is only for illustration, and
a value of 1 mV is more typical. This staircase increase in the capacitor voltage continues
until it exceeds the voltage of the input signal. Here the system reached equilibrium with
the output oscillating between a digital one and zero, causing the voltage on the capacitor
to oscillate between 9 V and 10 V.

Example of signals produced by the delta modulator

In this manner, the feedback of the circuit forces the capacitor voltage to track the voltage
of the input signal. If the input signal changes very rapidly, the voltage on the capacitor
changes at a constant rate until a match is obtained. This constant rate of change is called
the slew rate, just as in other electronic devices such as op amps.

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 38

Now, consider the characteristics of the delta modulated output signal. If the analog input
is increasing in value, the output signal will consist of more ones than zeros. Likewise, if
the analog input is decreasing in value, the output will consist of more zeros than ones. If
the analog input is constant, the digital output will alternate between zero and one with an
equal number of each. Put in more general terms, the relative number of ones versus zeros
is directly proportional to the slope (derivative) of the analog input.

Output information of the delta modulator is related to the derivative of the input signal.
It does not contain “DC information". The delta-sigma converter, shown below, eliminates
these problems by cleverly combining analog electronics with DSP algorithms. Notice that
the voltage on the capacitor is now being compared with ground potential. The feedback
loop has also been modified so that the voltage on the capacitor is decreased when the cir-
cuit's output is a digital one, and increased when it is a digital zero. As the input signal in-
creases and decreases in voltage, it tries to raise and lower the voltage on the capacitor.
This change in voltage is detected by the comparator, resulting in the charge injectors pro-
ducing a counteracting charge to keep the capacitor at zero volts.

Block diagram of a delta-sigma analog-to-digital converter. In the simplest case, the pulses from a delta
modulator are counted for a predetermined number of clock cycles. The output of the counter is then
latched to complete the conversion. In a more sophisticated circuit, the pulses are passed through a digital
low-pass filter and then resampled (decimated) to a lower sampling rate.

If the input voltage is positive, the digital output will be composed of more ones than
zeros. The excess number of ones is needed to generate the negative charge that cancels
with the positive input signal. Likewise, if the input voltage is negative, the digital output
will be composed of more zeros than ones, providing a net positive charge injection. If the
input signal is equal to zero volts, an equal number of ones and zeros will be generated in
the output, providing an overall charge injection of zero.

The relative number of ones and zeros in the output is now related to the level of the
input voltage, not the slope as in the previous circuit. This is much simpler. For instance,

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 39

you could form a 12 bit ADC by feeding the digital output into a counter, and counting the
number of ones over 4096 clock cycles. A digital number of 4095 would correspond to the
maximum positive input voltage. Likewise, digital number 0 would correspond to the
maximum negative input voltage, and 2048 would correspond to an input voltage of zero.
This also shows the origin of the name, delta-sigma: delta modulation followed by summa-
tion (sigma).

The ones and zeros produced by this type of delta modulator are very easy to transform
back into an analog signal. All that is required is an analog lowpass filter, which might be
as simple as a single RC network. The high and low voltages corresponding to the digital
ones and zeros average out to form the correct analog voltage.

This method of transforming the single bit data stream back into the original waveform
is important for several reasons. First, it describes a slick way to replace the counter in the
delta-sigma ADC circuit. Instead of simply counting the pulses from the delta modulator,
the binary signal is passed through a digital low-pass filter, and then decimated to reduce
the sampling rate. For example, this procedure might start by changing each of the ones and
zeros in the digital stream into a 12 bit sample; ones become a value of 4095, while zeros
become a value of 0. Using a digital low-pass filter on this signal produces a digitized ver-
sion of the original waveform, just as an analog lowpass filter would form an analog recrea-
tion. Decimation then reduces the sampling rate by discarding most of the samples. This
results in a digital signal that is equivalent to direct sampling of the original waveform.

This approach is used in many commercial ADC's for digitizing voice and other audio
signals. An example is the National Semiconductor ADC16071, which provides 16 bit ana-
log-to-digital conversion at sampling rates up to 192 kHz. At a sampling rate of 100 kHz,
the delta modulator operates with a clock frequency of 6.4 MHz. The low-pass digital filter
is a 246 point FIR, such as described in Chapter 16. This removes all frequencies in the
digital data above 50 kHz, ½ of the eventual sampling rate. Conceptually, this can be
viewed as forming a digital signal at 6.4 MHz, with each sample represented by 16 bits.
The signal is then decimated from 6.4 MHz to 100 kHz, accomplished by deleting every 63
out of 64 samples. In actual operation, much more goes on inside of this device than de-
scribed by this simple discussion.

Another very simple example of Sigma Delta Circuit you can see below:

In a ΔΣ converter, the analog input voltage signal is connected to the input of an inte-
grator, producing a voltage rate-of-change, or slope, at the output corresponding to input
magnitude. This ramping voltage is then compared against ground potential (0 volts) by a
comparator. The comparator acts as a sort of 1-bit ADC, producing 1 bit of output ("high"
or "low") depending on whether the integrator output is positive or negative. The compara-
tor's output is then latched through a D-type flip-flop clocked at a high frequency, and fed
back to another input channel on the integrator, to drive the integrator in the direction of a 0
volt output. The basic circuit looks like this

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The leftmost op-amp is the (summing) integrator. The next op-amp the integrator feeds
into is the comparator, or 1-bit ADC. Next comes the D-type flip-flop, which latches the
comparator's output at every clock pulse, sending either a "high" or "low" signal to the next
comparator at the top of the circuit. This final comparator is necessary to convert the single-
polarity 0V / 5V logic level output voltage of the flip-flop into a +V / -V voltage signal to
be fed back to the integrator.

If the integrator output is positive, the first comparator will output a "high" signal to the
D input of the flip-flop. At the next clock pulse, this "high" signal will be output from the Q
line into the noninverting input of the last comparator. This last comparator, seeing an input
voltage greater than the threshold voltage of 1/2 +V, saturates in a positive direction, send-
ing a full +V signal to the other input of the integrator. This +V feedback signal tends to
drive the integrator output in a negative direction. If that output voltage ever becomes nega-
tive, the feedback loop will send a corrective signal (-V) back around to the top input of the
integrator to drive it in a positive direction. This is the delta-sigma concept in action: the
first comparator senses a difference (Δ) between the integrator output and zero volts. The
integrator sums (Σ) the comparator's output with the analog input signal.

Functionally, this results in a serial stream of bits output by the flip-flop. If the analog
input is zero volts, the integrator will have no tendency to ramp either positive or negative,
except in response to the feedback voltage. In this scenario, the flip-flop output will con-
tinually oscillate between "high" and "low," as the feedback system "hunts" back and forth,
trying to maintain the integrator output at zero volts:

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 41

If, however, we apply a negative analog input voltage, the integrator will have a ten-
dency to ramp its output in a positive direction. Feedback can only add to the integrator's
ramping by a fixed voltage over a fixed time, and so the bit stream output by the flip-flop
will not be quite the same:

Sampling theorem

Frequently this is called the Shannon sampling theorem, or the Nyquist sampling theo-
rem, after the authors of 1940s papers on the topic. The sampling theorem indicates that a
continuous signal can be properly sampled, only if it does not contain frequency compo-
nents above one-half of the sampling rate. For instance, a sampling rate of 2,000 sam-
ples/second requires the analog signal to be composed of frequencies below 1000 cy-
cles/second. If frequencies above this limit are present in the signal, they will be aliased to
frequencies between 0 and 1000 cycles/second, combining with whatever information that
was legitimately there.

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Basic texts

Other text

Questions

' Answers you find in this text

1. Define amplitude and frequency modulations - basic properties.
2. Write the general mathematical expression for modulation.
3. Draw the frequency spectrum of AM.
4. How do you get FM using PM system?
5. Describe basic demodulation principles (AM, FM, PM) – circuitry.
6. Define delta and sigma delta modulations – basic properties.
7. What is aliasing – sampling theorem?
8. What is PCM?

Problems

Recommendation

If you can solve and answer more than circa 60 % of the problems and questions, you
may continue your study.

16. 11. 2009

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 Punčochář, Mohylová: TELO, Chapter 9, Modulation, demodulation, signal sampling 44

APPENDIX
1. Expression for AM & its Power and Efficiency calculation:
AM – Definition

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3. What is the bandwidth of AM? - Bandwidth =2fm

4. Why is FM superior to AM in performance?

i). In AM system the bandwidth is finite. But FM system has infinite number of sidebands
in addition to a single carrier.
ii). In FM system all the transmitted power is useful whereas in AM most of the transmitted
power is used by the carrier.
iii). Noise is very less in FM, hence there is an increase in the signal to noise ratio.

5. Define demodulation.
Demodulation or detection is the process by which modulating voltage is recovered from
the modulated signal. It is the reverse process of modulation.

6. What are the degrees of modulation?

Under modulation. m<1
Critical modulation m=1
Over modulation m>1

7. What is a limiter?
A limiter a circuit that produces a constant amplitude output for all input signals above a
prescribed minimum input level called the threshold, quieting or capture level.

8. How will you obtain PM demodulator from FM demodulator?

PM demodulator is obtained from FM demodulator by placing an integrator followed by
FM modulator.

9. How will you obtain PM modulator from FM modulator?

PM modulator is obtained from FM modulator by placing an differentiator followed by an
FM modulator

10. How will you obtain FM demodulator from PM demodulator?

FM demodulator is obtained from PM demodulator by placing a differentiator after the PM
modulator.

11. How will you obtain FM modulator from PM modulator FM modulator is obtained
from PM modulator by placing an integrator followed by a PM modulator

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12. Define angle modulation

Angle modulation is defined as the process by which the frequency or phase of the carrier
wave is changed in accordance with the instantaneous value of the message signals.

13. Differentiate Digitizer and modulator.

• Digitizer converts analog data to digital data. This process is called digitization.
• Modulator converts analog / digital data to analog signal. This process is called modula-
tion

14. What is PCM?

PCM (Pulse Code Modulation) is a process used to convert analog signal to digital data. In
PCM, the analog signal is first sampled then quantized then each sample is replaced with n
bits binary data.

14. Define instantaneous phase?

15. What is aliasing?

The phenomenon of a high-frequency in the spectrum of the original signal seemingly tak-
ing on the identity of a lower frequency in the spectrum of the sampled signal is called
aliasing or foldover.

16. What is meant by PCM?

Pulse code modulation (PCM) is a method of signal coding in which the message signal is
sampled, the amplitude of each sample is rounded off to the nearest one of a finite set of
discrete levels and encoded so that both time and amplitude are represented in discrete
form.. This allows the message to be transmitted by means of a digital waveform.

17. Define quantizing process.

The conversion of analog sample of the signal into digital form is called quantizing process.

18. Define Pulse Width modulation (PWM)?

The width of a constant-amplitude pulse is varied proportional to the amplitude of the ana-
log signal at the time the signal is sampled. This is PWM. PWM is also called as pulse du-
ration modulation (PDM) or pulse length modulation (PLM).

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