Communication System Jto Lice Study Material Sample

7/25/2019 Communication System Jto Lice Study Material Sample

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JTO LICE Communication System

2015 ENGINEERS INSTITUTE OF INDIA .All Rights Reserved JTO LICE : Classroom , POSTAL, All India TEST Series 28-B/7, Jia Sarai, Near

IIT, Hauz Khas, New Delhi-110016. Ph. 011-26514888. www.engineersinstitute.com

1

LICE - JTO

Limited Internal Competi t ive Examinat ion

STUDY MATERIAL

COMMUNICATION SYSTEM




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CONTENTS

Chapte r-1 Random signals & noise 3 -16

Chapte r-2 M odulat ion Tec hniques 16-49

Chapte r-3 Digital Communicat ion 50-65

Chapter-4 Stored program control (SPC) 65 -72

Chapte r-5 Propagat ion of w aves 72 -75

CHAPTER-6 I ntroduct ion to microwa ve engineering 75 -84

Chapte r-7 Satellite communicat ions 84 -109

Chapter-8 Optical communication 110-115

Frequently asked MCQ 115-131

Chapte r-w ise Solved Problems

SYLLABUS : Communication Systems

Amplitude, frequency and phase modulation, their generation and demodulation,

Noise. PCM, basic principles of SPC Exchanges. Quantization & Coding; Time

division and frequency division multiplexing; Equalization; Optical communication-in

free space & fibre optic; Propagation of signals at HF,VHF,UHF and Microwave

frequency; Satellite Communication.


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Chapter-1 RANDOM SIGNALS & NOISE

Random Signals and Noise

Signals can be classified as deterministic and random. Deterministic signals can be

completely specified as functions of time. Random signals are those whose value cannot be

predicted in advance. Radio communication signals are random signals, which are invariably

accompanied by noise. Noise like signals, also called random processes can be predicted

about their future performance with certain probability of being correct.

Probability:

The concept of probability occurs naturally when we contemplate the possible outcomes

of an experiment whose outcomes are not always the same. Thus the probability of

occurrence of an event A, that is

P(A) =number of possible favourable outcomes

total number of possible equally likely outcomes... (i)

Two possible outcomes of an experiment are defined as being mutually exclusive if the

occurrence of one outcome precludes the occurrence of the other. Hence,

P (A1 or A2) = P (A1) + (A2) ... (ii)

Where A1 and A2 are two mutually exclusive events.

The joint probability of related and independent events is given as:

P(A, B)P(B/A)

P(A) ... (iii)

Where P(B/ A) is the conditional probability of outcome B, given A has occurred and

P(A, B) is the probability of joint occurrence of A and B.

Random Variables:

The term random variable is used to signify a rule by which a real number is assigned

to each possible outcome of an experiment.

A random experiment with sample space S. A random variable X()is a single valued

real function that assigns a real number called the value of to each sample point of

S.


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The sample space S is termed as domain of random variable X and the collection of

all numbers [values of] is termed as the range of random variable X.

The random variable X induces a probability measure on the real line

If X can take on only a countable number of distinct values, then X is called a discrete

random variable.

If X can assume any values within one are more intervals on the real line, then X is called a

continuous random variable.

Cumulative Distribution Function (CDF):

The cumulative distribution function associated with a random variable is

defined as the probability that the outcome of an experiment will be one of the

outcome for which X(A) x, where x is any given number. If Fx(x) is the

cumulative distribution function then

Fx (x) = P (X x)

where P (X x) is the probability of the outcome.

Fx (x) has the following properties:

1. 0 F (x) 1

2. F () = 0, F () = 1

3. F (x1) F (x2) if x1 < x2.

Probability Density Function (PDF):

The probability density function fx(x) is defined in terms of cumulative distribution

function Fx (x) as:

fx (x) = xd

F (x)dx

The properties of PDF are:

1.fx (x)0, for all x


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2. xf (x)dx 1

3.

x

x xF (x) f (x) dx

The relation between probability and probability density can be written as:

P(x1 xx2) =2

1

x

x

f(x)dx

Example 1:

Consider the probability f (x) = ae-b|x|, where X is a random variable whose Allowable values

range from x = to x = +.

Find:

a. The cumulative distribution function F (x).

b. The relationship between a and b

c. The probability that the outcome x lies between 1 and 2.

Solution:

a) The cumulative distribution function is :

F (x) = P (X x)

x

x

x

b|x|

bx

bx

f (x) dx

ae

ae , x 0

b

a(2 e ) x 0

b

f (x)dx 1

b|x| 2aae dx 1b

a 1

b 2

c) The probability that x lies in the range between 1 and 2 is

2

b|x| b 2b

1

b 1P(1 x 2) e dx (e e )2 2


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The average value of a random variable is given as:

i i

i

X E(x) m m x P (x )

where X or E(x) or m represents the average, expected or mean value of random variable

x.

The variance (2) x is given as

2= E (x2)m2

where is the standard deviation of x.

If the average valuem = 0 then

The covariance of two random variables x and y is defined as: = E {(xmx) (ymy)}

where mi denotes the corresponding mean. For independent random variables = 0.

The correlation coefficient between the variables x and y, serves as a measure of the

extent to which X and Y are dependent. It is given by

( )

x y x y

E xy

falls within the range1 1.

When the random variables X and Y are said to be uncorrelated. When random variables are

independent, they are uncorrelated but vice versa is not true.

Gaussian PDF:

The Gaussian PDF is of great importance in the analysis of random variables. It is defined as

2

2

( )

2

2

1( )

2

x m

f x e

And is plotted in figure below, m and2 are the average value and variance of f(x).

2 2E(X)


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The Gaussian density Function

Thus, x m xf (x)dx and

2 2 2E[(x m) ] (x m) f (x) dx

It can be deduced from figure that when x - m = , f (x) has fallen to 0.606 of its peak

value. When x = m 2, f (x) falls to 0.135 of the peak value.

The Rayleigh PDF:

The Rayleigh PDF is defined by

2

2

r,

22

re 0 r

f(r)

0, r < 0

A plot of f (r) as a function of r is shown in figure below.


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The Rayleigh PDF

The mean value is given by R ,2

, the mean squared value = 2a2

, and the variance

r2 =

22 .2

Auto carrelation and Power Spectral Density (PSD):

A random process n(t), being neither periodic nor of finite energy has an autocorrelation

function defined by

1

2

TT

2

1R( ) lim n(t) n(t )dt

T

Where is known as time lag parameter. For a random process the PSD G(f) is given as:

jwG(f ) F[R( )] R ( )e dr

Thus,F

G(f ) R( ) they constitute a Fourier Transform pair.

Consider a deterministic waveform v (t), extending from - to. Let a section of

waveform V(t) be Selected such that VT (t) = v(t) forT T

to2 2

and zero otherwise. It vT (f)

is the Fourier Transform of vT(t), |vT(f)|2 is the energy spectral density. Hence over the

interval T the normalized power density is2

T| v (f ) |

.T

As T, VT (t)V(t)

Thus, the physical significance of PSD G (f), is that

21( ) lim | ( ) |TT

G f V f T

Also, we can write the total energy of the signal V (t) is

2| ( ) |E G f df

This is known as Rayleighs Energy Theorem.

Properties of Autocorrelation function, R():

1. R() =R(-), i.e. it is an even function of time.


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2. R(0) = E[X2(t)]

3. max|R()| = R (0), i.e. R () has maximum magnitude at zero time lag.

Properties of PSD, G(f):

1. G(f) is a real function of frequency f for a random process.

2. G(f) = G(-f), i.e., it is an even function of f.

3. G(f) for all f.

Noise:

Noise is an unwanted signal of an unpredictable nature. It can either be man made

ofrnatural. The internally generated noise due to circuit components is of two types shot noise

and thermal noise.

Shot Noise:

This type of noise is predominant in valves and transistors as current in these devices is

due to flow of discrete electrons of diffusion of electrons and holes. The mean square value

of shot noise current is:

2SNI = 2eI0f amp.

Where, I0- mean value of current (dc)

f-bandwidth of device in Hz.

Thermal Noise:

The noise in a conductor of resistor due to random motion of electrons is called thermal a

noise. The mean sqare value of thermal noise voltage appearing across the terminals of the

resistor R is

ETN2 = 4KTRf

Where,

T - the absolute temperature

K - Boltzmanns constant,

Noise Figure:

Noise figure is defined as

=Total available output power (due to device and source) per unit bandwidth

Total available output noise power due to the source alone

= 10 log10 F, gives the Noise Factor


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Terms of equivalent noise temperature T, we can write.

Te = To(F1)

Where Te= equivalent noise temperature of the amplifier or receiver whose Noise Figure is F.

T0= 17 C = 290 K

We can also write,input

output

(SNR)

(SNR)

Example 2:

A pentode having an equivalent noise resistance of 5 k is selected for Equation as an

amplifier with a signal source of emf 100V and an internal resistance 125 k connected

across the input. The grid leakage resistance is 175 k. The width may be taken as 10KHz

and the temperature as 27oC. Calculate the signal to --ratio at the output of the amplifier and

express the value in dB. (Given 1.38x1023 J/K).

Solution:

Give input resistances g

s g

R R 125 17572.92k

R R 125 175

175100 58.33 V

300

Voltage,

= 4(Ra + Reff) KTB

= 4(5 x 103 + 72.92 x 103) x 1.38 x 10-23 x 300 x 104

= 1290.36 x 10-34 volts2

= 3.592V

22

n4

2

pentode n

VS 58.33263.7

N V 3.592


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10log10 263.7 = 24.2dB

Example: A receiver connected to an antenna whose resistance is 50has an equivalent noise

temperature of Calculate the receivers noise figure in decibels and its equivalent noise temperature.

Sol. eq

0

R 30F 1 1 1 0.6 1.6R 50

10 log 1.6 = 2.04 dB

Teq = T0 (F1)290(1.61) = 290 0.6 = 174 K

Information Theory

Information

The amount of information contained in an event is closely related to its uncertainty. For

a discrete memory less source (DMS), denoted by X, with symbols {x1, x2, ... xm}, the

information context of a symbol is given by

1

1( ) log

( )iI x

P x

Unless mentioned take base as 2 since we are dealing with binary digits

2( ) log ( )i iI x P x bits

Average Information or Entropy:

Entropy of a source is given by

m m

i i i i

i 1 i 1H(x) P log(P ) P(x ) I (x ) bits / symbol

When all the input digits have equal probability of occurrence we have maximum un-certainty and

hence and hence maximum entropy.

For example say all them events have equal probability of occurrence

1 2 3

max 2

1( ) ( ) ( ) ....... ( )

log /

mP x P x P x P xM

H m bits symbol


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Example: A source is generating 3 possible symbols with probability1 1 1

, , .4 4 2

Find the information

associated with each of the symbol.

Solution:1 2

1 1( ) ( )

4 4P x P x

We know that 21

1( ) log( )

ii

xP x

Using above formula

1 2 2 2 3 2( ) log 4 2 ( ) log (4) 2 ( ) log (2) 2I x bits I x bits I x bits

The probability of occurrence ofx2 is high hence information associated with it will be less

Information Rate:

If the time rate at which source x emits symbols is r (symbols/s), the information rate R of

the source is given by

R = rH b/s

Example:A source is generating 3 possible symbols with probabilities of1 1 2

, , .4 4 4

Find entropy and

information. Rate if the source is generating one symbol per millisecond.

Solution: We have

2

2 2 2

H P( ) log P( )

1 1 1 1 1 1log 4 log 4 log 2

4 4 2 2 2 2

i ix x

3

2H bits/symbols r= 1 symbol/millisecond

Information rate (R) = H r

3

3 1

2 10 bits/second

3

2R kbps

Channel Capacity of a lossless channel:

C = rCs = r log2m, m-number of symbols in x.

Channel Capacity of Additive White Gaussian Noise Channel (AWGN):

The capacity of the AWGN channel is given by

2log 1 S

C BN

b/s........Shannon-Hartley law

where B-bandwidth


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S

N-SNR at the Channel output

Assuming Bandwidth to be infinite

Then

Shannon-Hartley law becomes.

Each of the frequency component transmitted through a channel affected by same amount of white

noise power.

2log 1 S

C BN

Replace N = N0 where N0Power spectral density of white noise

2SB log 1

N Bo

C BBandwidth of white noise

As,S

C 1.44N

o

B

Proof: For AWGN channel as B channel capacityS

C 1.44N

o

2

S

B log 1 N BoC

Multiply and divide byS

No

2

NS S. B log 1

N S N B

o

o o

C

AssumeN B

[as B , ]S

o x x

2

S 1C lim log 1

N xo

xx

We know, (1 )lim log1x

x e

x

where, nB = N

As bandwidth increase, noise power also increases.

So no channel can have infinite capacity.

B 0

n s slim .B. log 1

S n nB


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B 0

S nB s, lim , log 1

n s, nB

Now using formula

X1lim x log 1 logex

2B

Slim C log e 1.44s / n bits / sec

n

For efficient channel capacity there is trade off between bandwidth and signal to noise

ratio.

Example 3:

Consider a telegraph source having two symbols, dot and dash. The dot duration is 0.2 s. the

dash duration is 3 times the dot duration. The probability of the dots occurring is twice that

of the dash, and the time between symbols is 0.2 s. Calculate the information rate of the

telegraph source.

Solution:

Given, p (dot) = 2p (dash)

Now, P(dot) + P(dash) = 3P(dash) = 1

P(dash) =1

3, P(dot) =

2

3

Using2

1 1

i 1

H(x) P(x )I (x )

We have, H (x) = P(dot) log2 P(dot)P(dash)log2 P(dash)

= 0.667 (0.585) + 0.333 (1.585)

= 0.92 b/symbol

The average time per symbol is

Ts = P(dot)tdot+ P(dash) tdash + tspace= 0.533 s/symbol

The average information rate is

s

H(x) 0.92R rH(x) 1.725 b / s

T 0.533

Example 4:


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Consider an AWGN channel with 4 KHz bandwidth and the noise psd12

102

w/Hz. The

signal power required at the receiver is 0.1mw. Calculate the capacity of this channel.

Solution:

B = 4000 Hz, S = 0.1 103 W

N =B = 2 1012 4000 = 8 109 W

3

4

9

S 0.1x101.25 10 W

N 8x10

Capacity, 4 32 2S

C Blog 1 4000log (1 1.25 10 ) 54.44 10 b / sN

Mutual Information:

The mutual information I (x; y) of a channel is defined by

I (x; y) = H (x)H(x/y) b/symbol.

Where H (x) represents the uncertainty about the channel input before the channel output

is observed and H(x/y) represents the uncertainty about the channel input after the channel

output is observed.

For a binary symmetric channel

I (x; y) = H (y) + plog2 p + (1-p) log2 (1-P)

Source Coding:

A conversion of the output of a DMS into a sequence of binary symbols is called source

coding.

An objective of source coding is to minimize the average bit rate required for

representation of the source by reducing the redundancy of the information source and also to

achieve power saving.

Code Length and Efficiency:

The average code word length,m

1 1 1

l 1

L P (x ) n , n

length in bits of symbol x,


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Code efficiency: minL H(x)

L L

Code redundancy: y = 1

For discrete source, two methods of source coding

1. Shannon Fanno coding

2. Huffman 1 coding

When the source is transmitting message with unequal probabilites variable length source

coding is efficient than fixed lengh source coding.

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Communication System Jto Lice Study Material Sample