Amplitude Modulation

Introduction
• Amplitude Modulation is the simplest and earliest form of
transmitters
• AM applications include broadcasting in medium- and high-
frequency applications, CB radio, and aircraft communications
• Once this information is received, the low frequency information must be
removed from the high frequency carrier. This process is known as “
Demodulation”.
 Basic Amplitude Modulation

• The information signal

varies the instantaneous
amplitude of the carrier
Figure Amplitude modulation
 AM Characteristics

• AM is a nonlinear process
• Sum and difference frequencies are created that carry the
information
 Full-Carrier AM: Time Domain

• Modulation Index - The ratio between the amplitudes

between the amplitudes of the modulating signal and
carrier, expressed by the equation:

Em
m=
Ec
 Overmodulation

• When the modulation index is greater than 1,

overmodulation is present
 Modulation Index for Multiple
Modulating Frequencies

• Two or more sine waves of different, uncorrelated

frequencies modulating a single carrier is calculated by the
equation:

2 2
m  m  m   
1 2
Measurement of Modulation Index
Full-Carrier AM: Frequency Domain
• Time domain information can f usb  f c  f m
be obtained using an
oscilloscope
• Frequency domain information
f lsb  f c  f m
can be calculated using Fourier
methods, but trigonometric mEc
methods are simpler and valid Elsb Eusb 
• Sidebands are calculated using
the formulas at the right
2
 Bandwidth
• Signal bandwidth is an important characteristic of any
modulation scheme
• In general, a narrow bandwidth is desirable
• Bandwidth is calculated by:

B 2 f m
 Power Relationships

• Power in a transmitter is
important, but the most
important power measurement is  m 2 
that of the portion that transmits
the information Pt P c
1
 2 
• AM carriers remain unchanged  
with modulation and therefore
are wasteful
• Power in an AM transmitter is
calculated according to the
formula at the right
 Quadrature AM and AM Stereo
• Two carriers generated at the same frequency but 90º out of
phase with each other allow transmission of two separate
signals
• This approach is known as Quadrature AM (QUAM or QAM)
• Recovery of the two signals is accomplished by synchronous
detection by two balanced modulators
Quadrature Operation
 Spectrum and Waveforms
The above are input signals. The diagram below shows the spectrum and
corresponding waveform of the output signal, given by
Vm Vm
vs t V DC cos c t cos c m t cos c m t
2 2
 Double Sideband AM, DSBAM
The component at the output at the carrier frequency fc is shown as a broken line with
amplitude VDC to show that the amplitude depends on VDC. The structure of the
waveform will now be considered in a little more detail.

Waveforms
Consider again the diagram

VDC is a variable DC offset added to the message; m(t) = Vm cos mt

 Double Sideband AM, DSBAM

This is multiplied by a carrier, cos ct. We effectively multiply (VDC + m(t)) waveform
by +1, -1, +1, -1, ...
The product gives the output signal vs t V DC m t cos c t
Double Sideband AM, DSBAM
 Modulation Depth
Consider again the equation v s t = VDC + Vm cosωm t cosωc t  , which may be written as
 V 
v s t = VDC  1+ m cosωm t  cosωc t 
 VDC 
Vm Vm
The ratio is defined as the modulation depth, m, i.e. Modulation Depth m=
VDC VDC
From an oscilloscope display the modulation depth for Double Sideband AM may be
determined as follows:
Vm

VDC 2Emax
2Emin
 Modulation Depth 2

2Emax = maximum peak-to-peak of waveform

2Emin = minimum peak-to-peak of waveform

2 Emax  2 Emin
Modulation Depth m =
2 Emax + 2 Emin
Vm
This may be shown to equal as follows:
VDC

2 Emax = 2VDC +Vm  2 E min 2 V DC V m

2VDC + 2Vm  2VDC + 2Vm 4Vm Vm

m= = =
2VDC + 2Vm + 2VDC  2Vm 4VDC VDC
 Double Sideband Modulation 'Types'
There are 3 main types of DSB

• Double Sideband Amplitude Modulation, DSBAM – with carrier

• Double Sideband Diminished (Pilot) Carrier, DSB Dim C

• Double Sideband Suppressed Carrier, DSBSC

• The type of modulation is determined by the modulation depth, which for

a fixed m(t) depends on the DC offset, VDC. Note, when a modulator is set
up, VDC is fixed at a particular value. In these illustrations we have a fixed
message, Vm cos mt and a varying VDC as per message signal to obtain
different types of Double Sideband modulation.
Graphical Representation of Modulation Depth
and Modulation Types.
Graphical Representation of Modulation Depth
and Modulation Types 2.
 Graphical Representation of Modulation Depth
and Modulation Types 3

Note then that VDC may be set to give

the modulation depth and modulation
type.

DSBAM VDC >> Vm, m  1

DSB Dim C 0 < VDC < Vm,
m > 1 (1 < m < )
DSBSC VDC = 0, m = 

The spectrum for the 3 main types of

amplitude modulation are summarised
 Bandwidth Requirement for DSBAM

In general, the message signal m(t) will not be a single 'sine' wave, but a band of frequencies
extending up to B Hz as shown

Remember – the 'shape' is used for convenience to distinguish low frequencies from high
frequencies in the baseband signal.
 Bandwidth Requirement for DSBAM

Amplitude Modulation is a linear process, hence the principle of superposition applies. The
output spectrum may be found by considering each component cosine wave in m(t)
separately and summing at the output.
Note:

• Frequency inversion of the LSB

• the modulation process has effectively shifted or frequency translated the baseband m(t)
message signal to USB and LSB signals centred on the carrier frequency fc
• the USB is a frequency shifted replica of m(t)
• the LSB is a frequency inverted/shifted replica of m(t)
• both sidebands each contain the same message information, hence either the LSB or USB
could be removed (because they both contain the same information)
• the bandwidth of the DSB signal is 2B Hz, i.e. twice the highest frequency in the baseband
signal, m(t)
• The process of multiplying (or mixing) to give frequency translation (or up-conversion) forms
the basis of radio transmitters and frequency division multiplexing which will be discussed
later.
 Power Considerations in DSBAM
2
 V pk 
Remembering that Normalised Average Power = (VRMS)2 =  
 2
we may tabulate for AM components as follows:
Vm V
v s t = VDC cosωc t + cosωc + ωm t + m cosωc  ωm t 
2 2
Component Carrier USB LSB

Amplitude pk VDC Vmm Vm

22 2
Power
2 2 2 22
VDC  Vm  V  V Vmm  VVmm
2
2
Total Power PT =
  = m   ==
2 2 2 8  22 22  88 Carrier Power Pc
Power
+ PUSB
2
VDC
2 2
m VDC
2
m 2VDC + PLSB
2 8 8
 Power Considerations in DSBAM
From this we may write two equivalent equations for the total power PT, in a DSBAM signal
2 2 2 2 2 2 2 2
V V V V V VDC m 2VDC m 2VDC
PT = DC + m + m = DC + m and PT = + +
2 8 8 2 4 2 8 8

V
2 m2 m2  m2 
The carrier power Pc = DC i.e. PT = Pc + Pc + Pc or PT = Pc  1 + 
2 4 4  2 
Either of these forms may be useful. Since both USB and LSB contain the same information a
useful ratio which shows the proportion of 'useful' power to total power is

m2
Pc
PUSB 4 m2
= =
PT  m2  4 + 2m 2
Pc  1 + 
 2 
 Power Considerations in DSBAM
For DSBAM (m  1), allowing for m(t) with a dynamic range, the average value of m
may be assumed to be m = 0.3

Hence,
m2
=
0.3 = 0.0215
2

4 + 2m 2 4 + 20.32

Hence, on average only about 2.15% of the total power transmitted may be regarded
as 'useful' power. ( 95.7% of the total power is in the carrier!)

m2 1
Even for a maximum modulation depth of m = 1 for DSBAM the ratio =
4 + 2m 2 6

i.e. only 1/6th of the total power is 'useful' power (with 2/3 of the total power in the
carrier).
 Example

Suppose you have a portable (for example you carry it in your ' back pack') DSBAM transmitter
which needs to transmit an average power of 10 Watts in each sideband when modulation depth
m = 0.3. Assume that the transmitter is powered by a 12 Volt battery. The total power will be
m2 m2
PT = Pc + Pc + Pc
4 4
m2 410 40
where Pc = 10 Watts, i.e. Pc = = = 444.44 Watts
4 m 2
0.32

Hence, total power PT = 444.44 + 10 + 10 = 464.44 Watts.

464.44
Hence, battery current (assuming ideal transmitter) = Power / Volts= amp
12
i.e. a large and heavy 12 Volt battery.

Suppose we could remove one sideband and the carrier, power transmitted would be
10 Watts, i.e. 0.833 amps from a 12 Volt battery, which is more reasonable for a
portable radio transmitter.
Suppressed-Carrier Signal
 Suppressed-Carrier AM

• Full-carrier AM is simple but not efficient

• Removing the carrier before power amplification allows
full transmitter power to be applied to the sidebands
• Removing the carrier from a fully modulated AM systems
results in a double-sideband suppressed-carrier
transmission
 Single Sideband Amplitude Modulation

One method to produce signal sideband (SSB) amplitude modulation is to produce

DSBAM, and pass the DSBAM signal through a band pass filter, usually called a
single sideband filter, which passes one of the sidebands as illustrated in the diagram
below.

The type of SSB may be SSBAM (with a 'large' carrier component), SSBDimC or
SSBSC depending on VDC at the input. A sequence of spectral diagrams are shown
on the next page.
 Single-Sideband AM

• The two sidebands of an AM signal are mirror images of

one another
• As a result, one of the sidebands is redundant
• Using single-sideband suppressed-carrier transmission
results in reduced bandwidth and therefore twice as many
signals may be transmitted in the same spectrum
allotment
• Typically, a 3dB improvement in signal-to-noise ratio is
achieved as a result of SSBSC
DSBSC and SSB Transmission
 Power in Suppressed-Carrier Signals

• Carrier power is useless as a measure of power in a DSBSC

or SSBSC signal
• Instead, the peak envelope power is used
• The peak power envelope is simply the power at
modulation peaks, calculated thus:

2
Vp
PEP 
2 RL
Demodulation of Amplitude Modulated Signals

There are 2 main methods of AM Demodulation:

• Envelope or non-coherent Detection/Demodulation.

• Synchronised or coherent Demodulation.

 Envelope or Non-Coherent Detection

An envelope detector for AM is shown below:

This is obviously simple, low cost. But the AM input must be DSBAM with m << 1, i.e.
it does not demodulate DSBDimC, DSBSC or SSBxx.
 Large Signal Operation
For large signal inputs, ( Volts) the diode is switched i.e. forward biased  ON, reverse
biased  OFF, and acts as a half wave rectifier. The 'RC' combination acts as a 'smoothing
circuit' and the output is m(t) plus 'distortion'.

If the modulation depth is > 1, the distortion below occurs

 Small Signal Operation – Square Law Detector

For small AM signals (~ millivolts) demodulation depends on the diode square law
characteristic.

The diode characteristic is of the form i(t) = av + bv2 + cv3 + ..., where

v = VDC + mt cosωc t  i.e. DSBAM signal.

 Small Signal Operation – Square Law Detector

aVDC + mt cosωc t + bVDC + mt cosωc t  + ...

2
i.e.


= aVDC + amt cosωc t + b VDC + 2VDC mt + mt  cos ωc t + ...
2 2 2

= aV DC + am t cos ωc t + bVDC
2
+ 2bV DC m t + bm t 2 1
 + 
1
cos 2ω t
c 
2 2 
2bVDC mt  bmt 2
2 2
bVDC VDC
= aV DC + am t cos ωc t + + + + b cos2ωc t + ...
2 2 2 2
'LPF' removes components.
2
bVDC
Signal out = aV DC + + bVDC mt  i.e. the output contains m(t)
2
 Synchronous or Coherent Demodulation

A synchronous demodulator is shown below

This is relatively more complex and more expensive. The Local Oscillator (LO) must be
synchronised or coherent, i.e. at the same frequency and in phase with the carrier in the
AM input signal. This additional requirement adds to the complexity and the cost.

However, the AM input may be any form of AM, i.e. DSBAM, DSBDimC, DSBSC or
SSBAM, SSBDimC, SSBSC. (Note – this is a 'universal' AM demodulator and the
process is similar to correlation – the LPF is similar to an integrator).
 Synchronous or Coherent Demodulation

If the AM input contains a small or large component at the carrier frequency, the LO
may be derived from the AM input as shown below.
 Synchronous (Coherent) Local Oscillator

If we assume zero path delay between the modulator and demodulator, then the ideal
LO signal is cos(ct). Note – in general the will be a path delay, say , and the LO
would then be cos(c(t – ), i.e. the LO is synchronous with the carrier implicit in the
received signal. Hence for an ideal system with zero path delay

Analysing this for a DSBAM input = VDC + mt cosωc t 

Synchronous (Coherent) Local Oscillator

VX = AM input x LO

= VDC + mt cos 2 ωc t 

= VDC + mt cosωc t  cosωc t 
= VDC + mt  1 + 1 cos2ωc t 
2 2 

VDC VDC mt  mt 

Vx = + cos2ωc t + + cos2ωc t 
2 2 2 2
We will now examine the signal spectra from 'modulator to V x'
 DSB-AM at frequency domain
• Take FT
– U ( f )  F [ Ac m(t ) cos(2 f c t )] 
Ac A
M ( f  fc )  c M ( f  fc )
2 2
• Transmission Bandwidth: BT
– BT = 2W

DSB-AM U(f)
M(f) 2W
A
AAc2/2
f f
-W 0 W -fc fc
 Power of modulated signal
• If m(t) is lowpass signal with frequency
contents much less than 2fc
1 T /2 2 1 T /2 2 2
– Pu  lim  u (t )dt  lim  Ac m (t ) cos 2 (2 f c t )dt
T T  T /2 T T  T /2

1 T / 2 2 2 1  cos(4 f c t )
 lim  Ac m (t ) dt
T T  T /2 2 0
2
Ac 1 T /2 2 1 T /2 2
 {lim  m (t )dt  lim  m (t ) cos(4 f c t ) dt}
2 T T  T /2 T T  T /2

Ac2
 Pm
M(f) 2 U(f)
Ac/2
Pm A P /2
2
c m f
-fc fc
 SNR for DSB-AM
• Equal to baseband SNR
– (
S P
)0  R
N N 0W

U(f) R(f)
Ac2Pm/2
PR
Transmit
- Distortion
fc fc
- Loss
N(f)
White Gaussian Noise N0/2 WN0

2W
 Brain Treasure Work
• What happens if the duration of message
signal t0 changes? What is the effect on the
BW and SNR ?
 Demodulation of AM signals
• Demodulation
– The process of extracting the message signal from
modulated signal
• Type of demodulation
– Coherent demodulation
• Local oscillator with same frequency and phase of the carrier at
the receiver
• DSB – AM , SSB – AM
– Noncoherent demodulaion
• Envelope detector which does not require same frequency and
phase of carrier
• Easy to implement with low cost : Conventional AM
 DSB – AM demodulation
• Coherent demodulation
u (t )  Ac m(t ) cos(2 f c t ) Lowpass Ac
m(t )
Filter 2

cos(2 f c t )
y (t )  Ac m(t ) cos(2 f c t ) cos(2 f c t )
Ac Ac
 m(t )  m(t ) cos(4 f c t )
– Local oscillator 2 2
• How do we generate ?
cos(2 f c t )
– Frequency and phase should be synchronized to incoming
signal
– PLL or FLL
 DSB – AM demodulation
• Frequency domain
– Y( f ) 
Ac A A
M ( f )  c M ( f  2 fc )  c M ( f  2 fc )
2 4 4
M(f) DSB-AM U(f)
Modulation Ac/2
f
0 W -fc fc
Lowpass Filter
With BW=W Y(f)

f
Demodulation
-2fc 0 2fc
 Effect of phase error on DSB – AM
• In practice, it is hard to synchronize phase
u (t )  Ac m(t ) cos(2 f c t ) Lowpass Ac
m(t ) cos( )
Filter 2

cos(2 f c t   )
y (t )  Ac m(t ) cos(2 f c t ) cos(2 f c t   )
Ac A
 m(t ) cos( )  c m(t ) cos(4 f c t   )
• Power in lowpass 2 2
2
– A
Pdem  c Pm cos 2 ( )
4
– 3 dB power loss when   4  cos 2 ( )  1
2
– Nothing can be recovered when
  2  cos 2 ( ) 0
 More on Demodulation
• Coherent demodulation requires carrier
replica generated at LO(Local Oscillator)
– Frequency and phase should be synchronized to
carrier
• Generally, 2 types of carrier recovery loop
– Costas loop
– Squaring loop
– Noise performance of 2 types are equivalent
• Implementation is depends on cost and accuracy
 Squaring loop
• Recover frequency using squaring
u (t )  Ac m(t ) cos(2 f c t ) Lowpass Ac A0
m(t )
Filter 2

A0 cos(2 f c t )

Squaring Frequency
Device Divider
1 2 2
Ac m (t )[1  cos(4 f c t )] A0 cos(4 f c t )
2
Bandpass Limiter
Filter 1 2 2
Ac m (t ) cos(4 f c t ) (or PLL)
2
 Costas loop(or Costas PLL)
• Goal of Costas loop: e0
Ac A0
m(t ) cos( e )
Baseband 2
LPF

u (t )  A0 cos(2 f c t   e )
Ac m(t ) cos(2 f c t )
VCO LPF
1 Ac A0
[ m(t )]2 sin(2 e )
2 2
-90 K sin(2 e )
, for small  e
Phase shift
A0 sin(2 f c t   e )
Baseband
Ac A0
LPF m(t ) sin( e )
2
 What if ?
• What happens if –m(t) instead of m(t) is used
– Both Costas loop and Squaring loop have a 180
phase ambiguity
– They don’t distinguish m(t) and –m(t)
• A known test signal can be sent after the loop is turned
on so that the sense of polarity can be determined
• Differential coding and decoding may be used
 More on PLL
• PLL(Phase Locked Loop)
– Tracks the phase (and frequency) of incoming
signal

PD(Phase Detector) x(t ) sin( i   0 )  sin(4 f c t   i   0 )

u (t )  Ac cos(2 f c t   i ) Loop filter e0 (t )

H(f)
A0 sin(2 f c t   o ) v0  A0 cos(2 f c t   o )

-90
VCO
output Phase shift
VCO(Voltage Controlled Oscillator)
• An oscillator whose frequency can be
controlled by external voltage

eo (t )
VCO cos(2 f c t  ceo (t ))

Free running frequency

(frequency when eo(t)= 0)

Constant of VCO
 PLL tracks Phase or Frequency ?
• All that is needed is to set the VCO free
running frequency as close as possible to the
incoming frequency

• If the VCO output is v (t )  A cos(2 f t   )
o 0 c o

– We can express it as

vo (t )  A0 cos(2 f c t   0 )

 A0 cos(2 f c t  2 ( f c  f c )t   0 )

 A0 cos(2 f c t   0 )

– Note that d
 (t ) 2 f
dt
 How the PLL works ?
• Output of PD
– x(t )  Ac A0 sin(2 f c t   o ) cos(2 f c t   i )
1
 Ac A0 [sin( i   o )  sin(4 f c t   i   o )]
2
• Output of LPF
– Loop Filter is lowpass narrow band filter
• 1 1
eo (t )  Ac A0 sin( i   o )  Ac A0 sin( e )
2 2
1
 Ac A0 e , for small  e
2
 How the PLL works ?
• At steady state:  e  i   o eo (t ) 0
• If input changes to:

Ac cos(2 ( f c  k )t   i )  Ac cos(2 f c t  (kt   i ))  Ac cos(2 f c t   i )

• It causes increasing of phase error

– Or increasing of eo(t): 1
eo (t )  Ac A0 sin( e )
2

• It causes increasing of VCO output

– The PLL tracks the phase(or frequency) of incoming signal
 More on PLL
• Hold-in(or Lock) range
– A PLL can track the incoming frequency over a finite range
of frequency shift
– If initially input and output frequency is not close enough,
PLL may not acquire lock
• If Doppler shift exists, Acquisition is needed
• Pull-in(or Capture) range
– The frequency range over which the input will cause the
loop to lock
– If input frequency changes too rapidly, PLL may lose lock
 PLL used in frequency synthesizer
• Generate a periodic signal of frequency
Oscillator Vin Ve
Frequency LPF
Frequency Standard
Divider, M
f=fx Vo

In steady state Frequency

VCO
Ve = 0, Vin = Vo Divider, N
N
fx f f out  fx
 out M
M N
By choosing M,N
We can generate desired frequency
 Oscillator
• What happens if frequency standard is
incorrect ?
– Errors of Crystal Oscillator
• More than 50ppm
• Drift : Sensitive to temperature
– TCXO
• Temperature Compensated Crystal(X-tal) Oscillator
• Less than 5ppm