Communication Theory
(EC 2252)
Prof.J.B.Bhattacharjee
K.Senthil Kumar
ECE Department
Rajalakshmi Engineering College
1
Review of Spectral characteristics

Periodic and Non-periodic Signals: A signal is said to
be periodic, if it exhibits periodicity. i.e.,
x(t +T)=x(t) , for all values of t.
Periodic signal has the property that it is unchanged
by a time shift of T. A signal that does not satisfy the
above periodicity property is called a non-periodic
signal.

Periodic signals can be represented using the
Fourier Series. Non-periodic signals can be
represented using the Fourier Transform.

Both Fourier series and Fourier Transform deal with
the representation of the signals as a combination of
sine and cosine waves.
Fourier Series

Fourier series: a complicated waveform analyzed
into a number of harmonically related sine and
cosine functions

A continuous periodic signal x(t) with a period T
may be represented by:
∞
 x(t)=Σ k=1 (Ak cos kω t + Bk sin kω t)+ A0

Dirichlet conditions must be placed on x(t) for the
series to be valid: the integral of the magnitude of
x(t) over a complete period must be finite, and the
signal can only have a finite number of
discontinuities in any finite interval
Fourier Series Equations
The
Fourier series represents a periodic
signal Tp in terms of frequency components:
x(t) 

ikω 0 t
X
e
 k ,
k  
where ω0  2 / T p
We
get the Fourier series coefficients as
follows:
1
ikω t
Xk 
The
Tp
 x(t)e
0
dt
Tp
complex exponential Fourier coefficients
are a sequence of complex numbers
representing the frequency component ω0k.

Periodic signals represented by Fourier Series
have Discrete spectra.
The Fourier Transform
 Fourier
transform is used for the nonperiodic signals. A Fourier transform
converts the signal from the time domain
to the spectral domain.
 Continuous Fourier Transform:
Hf

   ht e  2ift dt

ht    H  f e 2ift df


Non-periodic signals represented by Fourier
transform have Continuous spectra.
Fourier Transform Pairs
Note: Π stands for rectangular function. Λ stands for triangular function.
Introduction to Communication
Systems
– Basic process of
exchanging information from one location
(source) to destination (receiving end).
 Refers – process of sending, receiving and
processing of information/signal/input from
one point to another point.
 Communication
Flow of information
Source
Destination
Figure 1 : A simple communication system
9
Communication System –
defined as the whole mechanism of
sending and receiving as well as
processing of information electronically
from source to destination.
 Example – Radiotelephony, broadcasting,
point-to-point, mobile communications,
computer communications, radar and
satellite systems.
 Electronic
10
Objectives
System – to produce an
accurate replica of the transmitted
information that is to transfer information
between two or more points (destinations)
through a communication channel, with
minimum error.
 Communication
11
NEED FOR COMMUNICATION

Interaction purposes – enables people to
interact in a timely fashion on a global level in
social, political, economic and scientific areas,
through telephones, electronic-mail and video
conference.

Transfer Information – Tx in the form of audio,
video, texts, computer data and picture through
facsimile, telegraph or telex and internet.

Broadcasting – Broadcast information to
masses, through radio, television or teletext.
12
Terms Related To Communications

Message – physical manifestation produced by the
information source and then converted to electrical
signal before transmission by the transducer in the
transmitter.

Transducer – Device that converts one form of energy
into another form.

Input Transducer – placed at the transmitter which
convert an input message into an electrical signal.

Example – Microphone which converts sound energy to
electrical energy.
Message
Input
Transducer
Electrical
Signal
13
Output Transducer – placed at the receiver
which converts the electrical signal into the
original message.
 Example – Loudspeaker which converts
electrical energy into sound energy.

Electrical
Signal

Output
Transducer
Message
Signal – electrical voltage or current which
varies with time and is used to carry message or
information from one point to another.
14
Elements of a Communication
System
 The
basic elements are : Source,
Transmitter, Channel, Receiver and
Destination.
Information
Source
Transmitter
Channel
Transmission
Medium
Receiver
Destination
Noise
Figure : Basic Block Diagram of a Communication System
15
Function of each Element.

Information Source – the communication system
exists to send messages. Messages come from
voice, data, video and other types of information.

Transmitter – Transmit the input message into
electrical signals such as voltage or current into
electromagnetic waves such as radio waves,
microwaves that is suitable for transmission and
compatible with the channel. Besides, the
transmitter also do the modulation and encoding
(for digital signal).
16
Block Diagram of a Transmitter
Transmitting
Antenna
Modulating
Signal
Audio
Amplifier
Modulator
RF
Amplifier
Carrier
Signal
5 minutes exercise;
Describe the sequence of events that happen at
the radio waves station during news broadcast?
17
– is the link or path over
which information flows from the source to
destination. Many links combined will
establish a communication networks.
 There are 5 criteria of a transmission
system; Capacity, Performance, Distance,
Security and Cost which includes the
installation, operation and maintenance.
 2 main categories of channel that
commonly used are; line (guided media)
and free space (unguided media)
 Channel/Medium
18

Receiver – Receives the electrical signals or
electromagnetic waves that are sent by the
transmitter through the channel. It is also
separate the information from the received
signal and sent the information to the
destination.

Basically, a receiver consists of several stages
of amplification, frequency conversion and
filtering.
19
Block Diagram of a Receiver
Receiving Antenna
RF
Amplifier
Mixer
Intermediate
Frequency
Amplifier
Demodulator
Audio
Amplifier
Destination
Local
Oscillator

Destination – is where the user receives the
information, such as loud speaker, visual
display, computer monitor, plotter and printer.
20
Analog Modulation
 Baseband



Voice
Transmission
Baseband signal is the information either in a
digital or analogue form.
Transmission of original information whether
analogue or digital, directly into transmission
medium is called baseband transmission.
Example: intercom (figure below)
Microphone
Audio
Amplifier
Audio
Amplifier
Voice
Speaker
Wire
21
Baseband signal is not suitable for
long distance communication….

Hardware limitations



Requires very long antenna
Baseband signal is an audio signal of low frequency.
For example voice, range of frequency is 0.3 kHz to
3.4 kHz. The length of the antenna required to
transmit any signal at least 1/10 of its wavelength (λ).
Therefore, L = 100km (impossible!)
Interference with other waves

Simultaneous transmission of audio signals will cause
interference with each other. This is due to audio
signals having the same frequency range and
receiver stations cannot distinguish the signals.
22
Modulation

Modulation – defined as the process of modifying a
carrier wave (radio wave) systematically by the
modulating signal.

This process makes the signal suitable for transmission
and compatible with the channel.

Resultant signal – modulated signal

2 types of modulation; Analog Modulation and Digital
Modulation.

Analogue Modulation – to transfer an analogue low pass
signal over an analogue bandpass channel.

Digital Modulation – to transfer a digital bit stream the
carrier is a periodic train and one of the pulse parameter
(amplitude, width or position) changes according to the
audio signal.
23
Purpose of Modulation Process in
Communication Systems


To generate modulated signal that is suitable for
transmission and compatible with the channel.
To allow efficient transmission – increase transmission
speed and distance, eg;
1.
By using high frequency carrier signal, the information
(voice) can travel and propagate through the air at
greater distances and shorter transmission time
2.
Also, high frequency signal is less prone to noise and
interference. Certain types of modulation have the useful
property of suppressing both noise and interference
3.
For example, FM use limiter to reduce noise and keep
the signal’s amplitude constant. PCM systems use
repeaters to generate the signal along the transmission
path.
24
Amplitude Modulation (AM)

Objectives:





Recognize AM signal in the time domain, frequency
domain and trigonometric equation form
Calculate the percentage of modulation index
Calculate the upper sidebands, lower sidebands and
bandwidth of an AM signal by given the carrier and
modulating signal frequencies
Calculate the power related in AM signal
Define the terms of DSBSC, SSB and VSB
Understand the modulator and demodulator
operations
25

Modulation




Information signal, v m  Vm sin 2fmt
Modulated Wave


Sinusoidal wave, v c  Vc sin 2fc t
Modulating Signal/Base band


The alteration of the amplitude, phase or frequency of an
oscillator in accordance with another signal.
Input signal is encoded in a format suitable for transmission
A low frequency information signal is encoded over a higher
frequency signal
Carrier Signal


Introduction
Higher frequency signal which is being modulated
Modulation Schemes

To counter the effects of multi path fading and time-delay
spread
26
Modulation Schemes
Carrier Signal,
Vc
Modulating Signal,
Vm
Modulated Signal
VAM
VPM
VFM
27
Amplitude Modulation

Time Domain

Frequency Domain
28
AM Modulator
Information Signal
v m  Vm sin 2fmt
Modulator
Output
VAM  Vc sin 2fc t

Vm sin 2fm t (sin 2fc t )
Carrier Signal
v c  Vc sin 2fc t
29
Amplitude Modulation
Vc
- Vc
Vm
- Vm
Vam
- Vam
30
Modulation Index

Modulation Index, m



Indicates the amount that the carrier signal is
modulated.
It is an expression of the amount of power in the
sidebands.
Modulation level ranges = 0-1 where
• 0 = no modulation
• 1 = full modulation
• >1 = distortion
Vm
m
Vc
V max  V min
m
V max  V min
31
Modulation Index
Vm
m
Vc
32
Modulation Index
Vmax
Vmin
Vmax (p-p)
Vmin (p-p)
m
V max  V min
V max  V min
33
Modulation Index
m=0
m = 0.5
m=1
34
Bandwidth
VC
mVc
2
mVc
2
fc-fm

fc
fc+fm
Bandwidth for AM signal,
B  (fc  fm)  (fc  fm)
B  2fm
35
Power Distributions
fc-fm

fc
fc+fm
Total transmitted power, PT
PT  PC  PLSB  PUSB

If R= 1,
 m2 

PT  PC 1 


2


36
Double Side Band Suppressed Carrier (DSBSC)


It is a technique where it is transmitting both the
sidebands without the carrier (carrier is being
suppressed/cut)
Characteristics:

Power content less

Same bandwidth

Disadvantages - receiver is complex and expensive.
37
Single Side Band
(SSB)

Improved DSBSC
and standard AM,
which waste
power and
occupy large
bandwidth


SSB is a process
of transmitting
one of the
sidebands of the
standard AM by
suppressing the
carrier and one of
the sidebands

Advantages:

Saving power

Reduce BW by 50%

Increase efficiency,
increase SNR
Disadvantages

Complex circuits for
frequency stability
38
Vestigial Side Band (VSB)





VSB is mainly used in TV broadcasting for
their video transmissions.
TV signal consists of

Audio signal – transmitted by FM

Video signal – transmitted by VSB
A video signal consists a range of frequency
and fmax = 4.5 MHz.
If it transmitted using conventional AM, the
required BW is 9 MHz (BW=2fm). But
according to the standard, TV signal is
limited to 7 MHz only
So, to reduce the BW, a part of the LSB of
picture signal is not fully transmitted.
39
Vestigial Side Band (VSB)

The frequency spectrum for the TV signal / VSB:
Video
Audio
Carrier
Carrier
Total TV signal bandwidth = 7 MHz
4.5 MHz
Upper
Video
Bands
Lower
Video
Bands
Lower
Audio
Bands
Upper
Audio
Bands
f (MHz)
0
1.25
5.75
6.25
6.75
7.0
40
Modulator Circuits
B
Carrier
R1
Modulating
Signal
A
D
C
Output
Diode
R2
R3
E
C
L
41
Modulator Circuits
A. Modulating Signal
B. Carrier
C. Sum of carrier and
modulating signal
D. Diode current
E. AM output across
tuned circuit
42
Demodulator
A
B
C
Diode
C’
AM
Signal
R1
C1
R’
43
Demodulator
A. AM signal
B. Current pulses
through diode
C. Demodulating signal
D. Modulating signal
44
Frequency Modulation (FM)

Objectives:




Recognize FM signal in the time domain, frequency
domain and trigonometric equation form
Calculate the percentage of modulation index
Calculate the upper sidebands, lower sidebands and
bandwidth of an FM signal by Carsons’s Rule and
Bessel Function Table
Calculate the power related in FM signal
Understand the modulator and demodulator of FM
45
Introduction




FM is the process of varying the frequency of a
carrier wave in proportion to a modulating signal.
The amplitude of the carrier is kept constant while its
frequency is varied by the amplitude of the
modulating signal.
In all types of modulation, the carrier wave is varied
by the AMPLITUDE of the modulating signal.
FM signal does not have an envelope, therefore the
FM receiver does not have to respond to amplitude
variations  it can ignore noise to some extent.
46
Frequency Modulation
47
Frequency Modulation

The importance features about FM waveforms
are:




The frequency varies
The rate of change of carrier frequency changes is
the same as the frequency of the information signal
The amount of carrier frequency changes is
proportional to the amplitude of the information
signal
The amplitude is constant
48
Frequency Modulation

Carrier Signal



v c  Vc sin 2fc t
Modulating Signal/Base band



Sinusoidal wave
Information signal
v m  Vm sin 2fmt
Modulated Wave



Higher frequency signal which is being modulated
v FM  Vc cos (2fc t   sin 2fm t )
Where

KVm
2fm
49
Frequency Modulation

Time Domain

Frequency Domain
50
FM Modulator
51
FM Modulator
Information Signal
v m  Vm sin 2fmt
Modulator
Output
v FM  Vc cos (2fc t   sin 2fmt )
Carrier Signal
v c  Vc sin 2fc t
52
Frequency

Carrier Frequency


As in FM system, carrier frequency in FM systems
must be higher than the information signal frequency.
Maximum Frequency
fma x  fc  f

Minimum Frequency
fmin  fc  f

Carrier Swing
fcs  2 f
53
Modulation Index

Modulation Index, m @ β




Indicates the amount that the carrier signal is
modulated.
It is an expression of the amount of power in the
sidebands.
Modulation level ranges = 0 – 
Where
• Δf = fd = frequency deviation
• fm = modulating frequency
• Vm = amplitude of modulating signal
f
m
fm
kVm
f 
2
54
Modulation Index
β
=1
β
=5
55
Modulation Index
β
= 25
56
Modulation Index
57
Bandwidth

Using Bessel Function, the bandwidth for
FM signal,
BW  2nfm
n = number of pairs of the significant sidebands
fm = the frequency the modulating signal
58
Bandwidth

Using Carson’s Rule, to estimate the
bandwidth for an FM signal transmission.
BW  2( f  f
m (max)
)
Δf = peak frequency deviation
fm(max) = highest modulating signal frequency
59
Power Distributions

FM transmitted power, PFM
2
2
Vrms
PC
PFM 

R
2R
where
Vrms
V

2
60
Narrowband FM and Wideband FM

Narrowband FM has only a single pair of significant
sidebands. The value of modulation index β <1.

Wideband FM has a large number (theoretically
infinite) number of sidebands. The value of
modulation index β >=1.
Generation of Narrowband FM (NBFM)
_
INTEGRATOR
PRODUCT
MODULATOR
Σ
NBFM
WAVE
+
MODULATING
WAVE
-90 PHASE
SHIFTER
CARRIER
WAVE
v FM  Vc cos (2fc t   sin 2fm t )
If   1, then we have
vNBFM  Vc cos (2f ct )  Vc sin( 2f ct ) sin( 2f mt )

The modulator splits the carrier into two paths. One path is
direct. The other path contains a -90 degree phase shift unit
and a product modulator. The difference between the signals
in the two paths produces the NBFM signal.
Frequency Modulators

A frequency modulator is a circuit that varies carrier
frequency in accordance with the modulating signal.

There are two types of frequency modulator circuits.

(1) Direct FM: Carrier frequency is directly varied by the
message through voltage-controlled oscillator.


Eg: Varactor diode modulator.
(2) Indirect FM: Generate NBFM first, then NBFM is
frequency multiplied for targeted Δf.

Eg: Armstrong modulator
FM Varactor Modulator
64
The Operation of the Varactor Modulator

The info signal is applied to the base of the input
transistor and appears amplified and inverted at the
collector.

This low freq signal passes through the RF choke
(L1) and is applied across the varactor diode.

Varactor diode behaves as voltage controlled
capacitor.

When low reverse biased voltage is applied, more
capacitance is generated and thus decrease the
frequency.

When high reverse biased voltage is applied,
less capacitance is generated and thus increase
the frequency.

The varactor diode changes its capacitance in
sympathy with the info signal and therefore
changes the total value of the capacitance in the
tuned circuit.

The changing value of capacitance causes the
oscillator freq to increase and decrease under
the control of the information signal.

The output is therefore an FM signal.
Armstrong of indrect FM generation

In this method the message signal is first
subjected to NBFM modulator using a crystalcontrolled oscillator for generating carrier.

Crystal control provides frequency stability.

The NBFM wave is next multiplied in frequency by
using a frequency multiplier so as to produce the
desired wideband FM.
Frequency Demodulator
 The
FM demodulating circuits used to recover
the original modulating signal.
 Any
circuit that will convert a frequency
variation in the carrier back into a proportional
voltage variation can be used to demodulate or
detect FM signals.
A
popular method used for FM demodulation
is the Frequency discriminator.
Frequency discriminator
Output of the Frequency discriminator

The Frequency discriminator circuit consists of
the slope ciruit followed by the envelope
detector.

The slope circuit converts the instantaneous
frequency variations of the FM input signal to
instantaneous amplitude variations.

These amplitude variations are rectified by the
envelope detector to provide a DC output
voltage which varies in amplitude and polarity
with the input signal frequency.
FM vs AM:
Advantages
Better
noise
immunity
Rejection of
interfering signals
because of capture
effect
Better transmitter
efficiency
Disadvantages
Excessive
use of
spectrum
More complex and
costly circuits
71
Review of Probability



1.
2.
3.


Sample Space：the space of all possible outcomes (δ)
Event：a collection of outcomes：subset of δ
Probability：a “measure” assigned to the events of a
sample space with the following properties：
for all event A in S
P(A)  0
)  1B are mutually exclusive,
IfPA( Sand
P( A  B)  P( A)  P( B)
Theorem:
P( A  B )  P( A)  P( B )  P( A  B )
The Conditional probability of an event A given the
occurrence of event B is
P( A  B)
P( A | B) 
P( B)
 Two
events A and B are independent if
P( A  B)  P( A)  P( B)
 Random
Variables
 A rule which assigns a numerical value to
each possible outcomes of a chance
experiment.
 If the experiment is flipping a coin. Then a
random variable X can be defined as :
S1
H
X(S1)=1
S2
T
X(S2)=-1
 Cumulative Distribution
 FX (x )

≜
Function (CDF)
Prob{X  x}
Properties of CDF：
1. 0  FX ( x)  1, FX ()  1, FX ()  0
2. F ( x) is continuous from right, i.e. lim
3. FX ( x) is a nondecreas ing function of x.
X
x  x0 
FX ( x)  FX ( x0 ).
 Probability Density Function (PDF)
x
dFX ( x)
f
(
x
)
FX ( x )   f X (t )dt
 X
≜
dx
 Properties
of PDF： f X ( x )  0 , 


f X ( x )dx  1
P( x1  X  x2 )  FX ( x2 )  FX ( x1 )  x f X ( x )df
x2
1
,
 Random
Processes: A random process is
a mapping from the sample space to an
ensemble of time functions.
Sample function
X1(t)
The totality of all sample
functions is called
an ensemble
X2(t)
For a specific time
X(tk) is a random variable
XN(t)
t
Gaussian process

A random process X(t) is a Gaussian process if
for all n and for all (t1 t2 ... tn), the sequence of
random variables { X(t1), X(t2)... X(tn) } has a
jointly Gaussian density function.

Central limit theorem
 The sum of a large number of independent
and identically distributed(i.i.d) random
variables getting closer to Gaussian
distribution.

Thermal noise can be closely modeled by
Gaussian process.
 Property

For Gaussian process, knowledge of the
mean(m) and covariance(C) provides a
complete statistical description of process.
 Property

1
2
If a Gaussian process X(t) is passed through
a LTI system, the output of the system is also
a Gaussian process. The effect of the system
on X(t) is simply reflected by the change in
mean(m) and covariance(C) of X(t).
Noise Theory

Shot noise: It results from the shot effect in the
amplifying devices and active device. It is
caused by random variation in the arrival of
electrons (or holes) at the output of the devices.

For diode, the rms shot noise current is given by:
i n  2ei p δ f
i n  rms shot noise
e  charge of electron
i p  direct diode current
δ f  bandwidth of system

Thermal noise is the electrical noise arising from
the random motion of electrons in a conductor.
The noise power generated by a resistor is given
by:
Pn  kTδ f
Pn  noise power
k  Boltzmann' s constant
T  absolute temperatur e
δ f  bandwidth of system
 White
noise: It is the idealized form of noise,
whose spectrum is independent of the
operating frequency. The power spectral
density of white noise w(t) is Sw(f)=N0 /2. The
autocorrelation Rw(t) of white noise is an
impulse as shown below.
Sw(f)
N0
2
f
Rw()
N0
2
 ( )

Narrow band noise (Ideal case)
w(t)





BPF
n(t)
filtered noise is narrow-band noise
n(t) = nI(t)cos(2fCt) - nQ(t)sin(2fCt)
•
where nI(t) is inphase, nQ(t) is quadrature component
 filtered signal x(t)
x(t) = s(t) + n(t)
- Average Noise Power = N0BT
81
Noise Figure

Consider a signal source. The signal to noise
ratio (SNR) available from the source is given by:
(S/N) in  Psi /kTδ f
Psi  signal power from the source
k  Boltzmann' s constant
T  absolute temperatur e
δ f  bandwidth of system

Consider that the source is connected to an
amplifier with gain G. Since all amplifiers
contribute noise, the available output SNR will be
less than the SNR of the source.

The noise power at the output of the amplifier will
be Pno  GkT  f

The noise factor F is defined as :
available S/N power ratio at input
F
available S/N power ratio at output
Psi
Pno
Pno
F


kT f GPsi GkT  f

When noise factor is expressed in decibels, it is
called noise figure.
Noise figure = (F) dB = 10logF

The noise power expressed in terms of a
temperature is callled Noise Temperature.

If the amplifier noise is Pna , then the equivalent
noise temperature Te of the amplifier is given by
the equation Te  Pna / k f
Since Pna  (F - 1)kT0 f
The noise temperatu re can be written as
Te  Pna / k f  (F - 1)kT0 f / k f  (F - 1)T0
 Te  (F - 1)T0
AM SUPERHETERODYNE RECEIVER

RF section: It generally consists of a pre-selector
and an amplifier stage. The pre-selector is a
broad tuned band-pass filter with adjustable
center frequency that is tuned to the desired
carrier frequency. The other functions of the RF
section are detecting, band limiting and
amplifying the received RF signals.

Mixer/converter section: It is the stage of downconverts the received RF frequencies to
intermediate frequencies (IF) which are simply
frequencies that fall somewhere between the RF
and information frequencies, hence the name
intermediate. This section also includes a local
oscillator (LO).

IF Section: IF or intermediate frequency section
is the stage where its primary functions are
amplification and selectivity.

AM detector Section: AM detector section is the
stage that demodulates the AM wave and
converts it to the original
 information signal.

Audio section: Audio section is the stage that
amplifies the recovered information.
Performance of CW Modulation
Systems
 Introduction

- Receiver Noise (Channel Noise) :
additive, White, and Gaussian
 Receiver

Model
1. RX Model
Sw(f)
N0
2
f
Rw()
N0 = KTe where K = Boltzmann’s constant
Te = equivalent noise Temp.
Average noise power per unit bandwidth
N0
 ( )
2

88
SNR

The signal x(t) available for demodulation is defined by
x(t )  s (t )  n(t )



The output signal-to-noise ratio (SNR)O is defined as the
ratio of the average power of the demodulated message
signal to the average power of the noise, both measured
at the receiver output.
The channel signal-to-noise ratio, (SNR)C is defined as the
ratio of the average power of the modulated signal to the
average power of the channel noise in the message
bandwidth, both measure at the receiver input.
For the purpose of comparing different CW modulation
systems, we normalize the receiver performance by
dividing (SNR)O by (SNR)C. This ratio is called figure of
merit for the receiver and is defined as
Figure of merit 
( SNR) O
( SNR) C
Noise in DSB-SC Receivers
DSB-SC
signal s(t)
+
BPF
x(t)
Noise
w(t)
Product
modulator
v(t)
LPF
y(t)
cos(wct)
Local
Oscillator
Coherent
detector
Let’s consider the case of DSB-SC. The expression for the
modulated signal is given as s(t )  AC cos( 2f ct )m(t )
The carrier wave is statistically independent of the message
signal. The average power of DSB-SC modulated
component of s(t) is
Ac2 Pm
2
90

With a noise PSD of N0/2 the average noise power in the
message bandwidth W equals WN0 (baseband
scenario).

Pm is the power of the message. Hence we have
Ac2 Pm
(SNR) C 
2WN 0

Finding an expression for (SNR)O, we have
x(t )  s (t )  n(t )
 Ac cos2f c t m(t )  nI (t ) cos2f c t   nQ (t ) sin 2f c t 
v(t )  x(t ) cos2f ct 
Ac
1
1
1

m(t )  nI (t )  Ac m(t )  nI (t )cos4f ct   nQ (t ) sin 4f ct 
2
2
2
2
1
1
y (t )  Ac m(t )  nI (t )
2
2

Output of the LPF is

The power of the signal component at the
receiver output is A2 Pm / 4 . The average power of
the filtered noise is 2WN0.
C
S N ( f  f c )  S N ( f  f c ),
S N I ( f )  S NQ ( f )  
0,

W  f  W
elsewhere
The average noise power at the receiver output
is  1  2
1
  2WN 0  WN 0
2
2

Hence we have,
Ac2 Pm / 4 Ac2 Pm
(SNR)O,DSB-SC 

WN 0 / 2 2WN 0
( SNR)O
Figure of merit 
1
( SNR)C
Noise in AM receiver using envelope detection

The expression for AM signal is given as
s(t )  Ac 1  ka m(t )cos2f ct 
where it is assumed that
ka m(t )  1
AM signal
s(t)
+
BPF
x(t)
Envelope
Detector
y(t)
Noise
w(t)
AC2 / 2.
The average power of the carrier in the AM signal s(t) is
The average power of the information bearing component
2 2
Ac ka m(t ) cos2f ct  is AC k a Pm / 2
2
2
A
(
1

k
a Pm ) / 2
Average power of the full AM signal s(t) is C

Hence, the channel signal to noise ratio for AM is
( SNR) C , AM


AC2 1  k a2 Pm

2WN 0

Finding an expression for (SNR)O, we have
x(t )  s (t )  n(t )
x(t )  AC  AC k a m(t )  nI (t )cos( 2f c t )  nQ (t ) sin( 2f c t )
y (t )  envelope of x(t )
y(t )  AC ka m(t )  nI (t )
( SNR)O , AM 
AC2 k a2 Pm
2WN 0
( SNR) O
Figure of Merit
( SNR) C

AM
k a2 Pm
1  k a2 Pm
Threshold Effect

 (t )
r(t)
AC 1  k a m(t )cos (t )
AC 1  k a m(t )sin (t )
1 
ka m
y(t)
AC
R
ant
e su l t
(t )

When carrier-to-noise ratio is small as compared
to unity the noise term dominates the
performance of the envelope detector and is
completely different. Representing the
narrowband noise n(t) in terms of its envelope and
phase, we have n(t )  r (t ) cos2f ct  (t )
 The phasor diagram for x(t) = s(t) + n(t) becomes

The noise envelope is used as a reference here due to its
dominance. Here it is assumed that Ac is small as
compared to r(t). If we neglect the quadrature component
of the signal with respect to the noise we have
y(t )  r (t )  AC cos(t )  AC ka m(t ) cos(t )

Hence, when carrier-to-noise ratio is small the detector
has no component that is strictly proportional to the
message signal m(t). Recalling that  (t ) is uniformly
distributed over radians. Hence, it follows that we have a
complete loss of information at the detector output (as
expected value will be zero). This loss of information m(t)
at the output of the envelope detector is called the
threshold effect.
Pre-emphasis and De-emphasis



FM results is an unacceptably low SNR at the high
frequency end of the message spectrum. To offset this
undesirable occurrence, pre-emphasis and de-emphasis
technique is used.
Pre-emphasis consists in artificially boosting the spectral
components in the higher part of the message spectrum.
This is accomplished by passing message signal m(t) ,
through the pre-emphasis filter, denoted Hpe(f) . The preemphasized signal is used to frequency modulate the carrier
at the transmitting end.
In the receiver, the inverse operation, de-emphasis, is
performed. This is accomplished by passing the
discriminator output through a filter, called the de-emphasis
filter, denoted Hde(f ) .
Pre-emphasis and de-emphasis in FM
P.S.D. of noise at FM Rx output
P.S.D. of typical message signal
H de (f ) 
1
,
H pe (f )
-W f W
P.S.D of noise nd (t) at the discrimina tor output
N0 f 2

SNd (f)   A C2
 0

f 
BT
2
otherwise
98
Information theory
 What is information theory ?
 Information theory is needed to enable the
communication system to carry information
(signals) from sender to receiver over a
communication channel
• it deals with mathematical modelling and analysis
of a communication system
• its major task is to answer to the questions of
signal compression and data transfer rate.

Those answers can be found and solved by
entropy and channel capacity

Information is a measure of uncertainty. The less
is the probability of occurrence of a certain
message, the higher is the information.

Since the information is closely associated with
the uncertainty of the occurrence of a particular
symbol, When the symbol occurs the information
associated with its occurrence is defined as:
I k  log (
1
)  - log(P k )
Pk
where Pk is the probabilit y of occurrence of symbol ' k'
and I k is the informatio n carried by symbol ' k'.
Entropy
 Entropy
is defined in terms of probabilistic
behaviour of a source of information
 In information theory the source output
are discrete random variables that have a
certain fixed finite alphabet with certain
probabilities

Entropy is an average information content for
the given source symbol. (bits/message)
K 1
1
H   pk log 2 (
)
pk
k 0
 Rate
of information:
 If
a source generates at a rate of ‘r’
messages per second, the rate of
information ‘R’ is defined as the average
number of bits of information per second.
 ‘H’
is the average number of bits of
information per message. Hence
R = rH
bits/sec
Source Coding

Source coding (a.k.a lossless data
compression) means that we will remove
redundant information from the signal prior
the transmission.

Basically this is achieved by assigning short
descriptions to the most frequent outcomes
of the source output and vice versa.

The common source-coding schemes are
prefix coding, huffman coding, lempel-ziv
coding.
Source Coding Theorem

Source coding theorem states that the output of
any information source having entropy H units per
symbol can be encoded into an alphabet having N
symbols in such a way that the source symbols
are represented by code words having a weighted
average length not less than H/logN.

Hence source coding theorem says that encoding
of messages from a source with entropy H can be
done, bounded by the fundamental information
theoretic limitation that the Minimum average
number of symbols/message is H/logN.
Source coding example
 Prefix
coding has an important
feature that it is always uniquely
decodable and it also satisfies KraftMcMillan (see formula 10.22 p. 624)
inequality term
 Prefix codes can also be referred to
as instantaneous codes, meaning
that the decoding process is achieved
immediately

Shannon-Fano Coding: In Shannon–Fano
coding, the symbols are arranged in order from
most probable to least probable, and then
divided into two sets whose total probabilities
are as close as possible to being equal. All
symbols then have the first digits of their codes
assigned; symbols in the first set receive "0" and
symbols in the second set receive "1".

As long as any sets with more than one member
remain, the same process is repeated on those
sets, to determine successive digits of their
codes. When a set has been reduced to one
symbol, of course, this means the symbol's code
is complete and will not form the prefix of any
other symbol's code.

Huffman Coding: Create a list for the symbols, in
decreasing order of probability. The symbols with
the lowest probability are assigned a ‘0’ and a ‘1’.

These two symbols are combined into a new
symbol with the probability equal to the sum of
their individual probabilities. The new symbol is
placed in the list as per its probability value.

The procedure is repeated until we are left with 2
symbols only for which 0 and 1 are assigned.

Huffman code is the bit sequence obtained by
working backwards and tracking sequence of 0’s
and 1’s assigned to that symbol and its
successors.

Lempel-Ziv Coding: A drawback of Huffman
code is that knowledge of probability model of
source is needed. Lempel-Ziv coding is used to
overcome this drawback.

while Huffman’s algorithm encodes blocks of
fixed size into binary sequences of variable
length, Lempel-Ziv encodes blocks of varying
length into blocks of fixed size.

Lempel-Ziv coding is performed by parsing the
source data into segments that are the shortest
subsequences not encountered before.
Mutual Information

Source
X
Channel
Receiver
Y

Consider a communication system with a source of entropy
H(X). The entropy on the receiver side be H(Y).

H(X|Y) and H(Y|X) are the conditional entropies, and H(X,Y)
is the joint entropy of X and Y.

Then the Mutual information between the source X and the
receiver Y can be expressed as:
I(X,Y) = H(X) - H(X|Y)

H(X) is the uncertainty of source X and H(X/Y) is the
uncertainty of X given Y. Hence the quantity H(X) - H(X|Y)
represents the reduction in uncertainty of X given the
knowledge of Y. Hence I(X,Y) is termed mutual information.
Channel Capacity

Capacity in the channel is defined as a
intrinsic ability of a channel to convey
information.

Using mutual information the channel
capacity of a discrete memoryless channel is
the maximum average mutual information in
any single use of channel over all possible
probability distributions.

Thus Channel capacity C=max( I(X,Y) ).
Shannon’s Channel Coding theorem

The Shannon theorem states that given a noisy channel
with channel capacity C and information transmitted at a
rate R, then if R < C there exist codes that allow the
probability of error at the receiver to be made arbitrarily
small. This means that theoretically, it is possible to transmit
information nearly without error at any rate below a limiting
rate, C.

The converse is also important. If R > C, an arbitrarily small
probability of error is not achievable. All codes will have a
probability of error greater than a certain positive minimal
level, and this level increases as the rate increases. So,
information cannot be guaranteed to be transmitted reliably
across a channel at rates beyond the channel capacity.
Shannon-Hartley theorem or Information
Capacity Theorem

An application of the channel capacity concept to
an additive white Gaussian noise channel with B
Hz bandwidth and signal-to-noise ratio S/N is the
Information Capacity Theorem.

It states that for a band-limited Gaussian channel
operating in the presence of additive Gaussian
noise, the channel capacity is given by
C = B log2(1 + S/N)
where C is the capacity in bits per second, B is the
bandwidth of the channel in Hertz, and S/N is the
signal-to-noise ratio.
Band width and SNR tradeoff




As the bandwidth of the channel increases, it
is possible to make faster changes in the
information signal, thereby increasing the
information rate.
However, as B  , the channel capacity
does not become infinite since, with an
increase in bandwidth, the noise power also
increases.
As S/N increases, one can increase the
information rate while still preventing errors
due to noise.
For no noise, S/N   and an infinite
information rate is possible irrespective of
bandwidth.
Implications of the Information Capacity
Theorem
Rate distortion theory


Rate distortion theory is the branch of information
theory addressing the problem of determining the
minimal amount of entropy or information that
should be communicated over a channel such
that the source can be reconstructed at the
receiver with a given distortion.
Rate distortion theory can be used for the given
below situations:
 1. Source coding in which the coding alphabet
cannot exactly represent the source information.
 2. when the information is to be transmitted at a
rate greater than channel capacity.
Lower the bit rate R by allowing some
acceptable distortion D of the signal
 Rate
Distortion Function:
 The functions that relate the rate and
distortion are found as the solution of the
following minimization problem.
 In
the above equation, I(X,Y) is the Mutual
information.
Rate distortion function for Gaussian
memory-less source
Px(X) is Gaussian, variance is 2 and if
we assume that successive samples of the
signal x are stochastically independent, we
find the following analytical expression for
the rate distortion function.
 If
A Plot of the Rate distortion function for
Gaussian source
Lossy Source Coding

Lossy source coding is the representation of the
source in digital form with as few bits as possible
while maintaining an acceptable loss of
information.

In lossy source coding, the source output is
encoded at a rate less than the source entropy.

Hence there is reduction in the information content
of the source.

Eg: It is not possible to digitally encode an analog
signal with a finite number of bits without producing
some distortion.