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Ashish Dixit

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Assitant Professor (ECE Department)

AMITY University, Lucknow

Prashant Singh

M.Tech. (IIT Bombay)

3-Times GATE Qualified

(Design Engineer

Taiwan Semiconductor

Manufacuring Company)

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CHAPTER NAME / TOPIC NAME

Page No.

CHAPTER -1 : FUNDAMENTALS OF COMMUNICATION SYSTEMS

INTRODUCTION TO BASICS OF COMMUNICATION SYSTEMS 1

FREQUENCY RANGES OF VARIOUS SPECTRUM 3

SOME MOST WIDELY SPECTRUM WITH THEIR FREQUENCY RANGE 4

FOURIER SERIES 4

COMPLEX EXPONENTIAL FOURIER SERIES 5

FOURIER TRANSFORM 6

PALEY-WIENER CRITERION 6

GATE FUNCTION / RECTANGULAR PULSE 7

SAMPLING / INTERPOLATING / SINC FUNCTION 7

POWER SPECTRUM 8

CROSS-CORRELATIONS FUNCTION 9

AUTOCORRELATION FUNCTION

PROBLEMS BASED ON GATE/IES/PSUs 10

CHAPTER -2 : RANDOM VARIABLES & RANDOM PROCESS

RANDOM SIGNALS 11

PROPERTIES OF RANDOM VARIABLE 12

PROBABILITY DENSITY FUNCTION 13

CUMMULATIVE DISTRIBUTIVE FUNCTION 13

PROPERTIES OF P.D.F. f_X (x) 14

MARGINAL PROBABILITY FUNCTION 15

TWO-DIMENSIONAL DISTRIBUTION FUNCTION 16

EXPECTATION OF A RANDOM VARIABLE 16

COVARIANCE 17

SOME COMMONLY OCCURRING PDFS 18

MEAN AND VARIANCE OF THE SUM OF RANDOM VARIABLES 22

SOLVED EXAMPLES 23

SPECIAL RANDOM PROCESS 26

CLASSIFICATION OF RANDOM PROCESSES 28

CORRELATION 29

TRANSMISSION OF RANDOM PROCESS THROUGH LINEAR SYSTEMS 32

SOLVED EXAMPLES 33

CHAPTER -3 : MODULATION

NEED OF MODULATION 51

DISTORTIONLESS TRANSMISSION 53

TYPES OF DISTORTIONS 54

CONCEPT OF MODULATION AND DEMODULATION 56

GENERATION OF AM WAVE 57

DEMODULATION 58

CHAPTER - 4 : AMPLITUDE MODULATION

INTRODUCTION TO AMPLITUDE MODULATION (AM) 59

BLOCK DIAGRAM OF THE AMPLITUDE MODULATOR 60

POWER CALCULATION OF AM WAVE 62

AM DEMODULATION 64

GENERATION OF AM SIGNALS 68

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SUMMARY OF DIFFERENT POSSIBLE AMPLITUDE MODULATED SYSTEM 70

MIXER 71

DOUBLE-SIDEBAND SUPPRESSED CARRIER (DSB-SC) MODULATION 72

SINGLE-TONE MODULATION OF DSB-SC 72

GENERATION OF DSB-SC SIGNALS 73

DIODE-BRIDGE MODULATOR 74

RING MODULATOR OR CHOPPER TYPE BALANCED MODULATOR 75

SYNCHRONOUS OR COHERENT OR HOMODYNE DETECTION 77

EFFECT OF PHASE AND FREQUENCY ERRORS IN SYNCHRONOUS DETECTION

78

SINGLE SIDEBAND (SSB) MODULATION 79

HILBERT TRANSFORM 80

PROPERTIES OF HILBERT TRANSFORM 82

CONCEPT OF PRE-ENVELOP OF ANALYTIC SIGNAL 83

GENERATION OF SSB SIGNALS 84

(I) Frequency Discrimination Method 84

(II) Phase Discrimination Method or Phasing Method 85

VESTIGIAL SIDEBAND (VEB) MODULATION SYSTEMS 86

Generation and Detection of VSB Signal 87

SUMMARY: modulators and demodulators used by various AM systems. 89

CHAPTER - 5 : AM TRANSMITTERS AND RECEIVERS

INTRODUCTION TO AM TRANSMITTERS AND RECEIVERS 105

BLOCK DIAGRAM OF AM-TRANSMITTER USING LOW-LEVEL MODULATION

105 BLOCK DIAGRAM OF AM-TRANSMITTER USING HIGH-LEVEL

MODULATION

106

MASTER OSCILLATOR (MO) 106

SOME FACTS REGARDING TO THE STABILITY OF MASTER OSCILLATOR FREQUENCY

107

AM RECEIVER 107

TYPE OF AM RECEIVER 107

TUNED RADIO FREQUENCY RECEIVER (TRF) 108

SUPERHETERODYNE RECEIVER 109

MAIN FUNCTIONS OF RF AMPLIFIER 109

FREQUENCY CONVERSION OR MIXING 110

SOME FACTS ABOUT CHOICE OF QUALITY FACTOR (Q) OF IF AMPLIFIER

112

TRACKING OF A RECEIVER 112

TYPES OF AGC 114

CHAPTER - 6 : FREQUENCY MODULATION

INTRODUCTION TO ANGLE (FREQUENCY OR PHASE) MODULATION 117

IMPORTANT DIFFERENCES BETWEEN AM AND FM/PM 117

SOLVED EXAMPLES 121

TYPES OF FM 125

INTERNATIONAL REGULATION FOR FREQUENCY MODULATION 126

Performance Comparison of FM and PM Systems 127

Performance Comparison of FM and AM System 128

FM GENERATION 129

PRACTICAL ARMSTRONG METHOD FOR FM GENERATION 131

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SOLVED EXAMPLES 136

FOSTER-SEELEY (CENTRE-TUNED) DISCRIMINATOR 138

CONCEPT OF PRE-EMPHASIS AND DE-EMPHASIS 139

PRE-EMPHASIS 140

DE-EMPHASIS 140

CHAPTER - 7 : NOISE

INTRODUCTION TO NOISE 153

ERRACTIC NOISE 153

MAN MADE NOISE 153

POWER DENSITY SPECTRUM OF SHOT NOISE IN DIODE 155

WHITE NOISE 156

NOISE BANDWIDTH 158

NOISE-TEMPERATURE 161

NOISE-FIGURE 163

FIGURE OF MERIT 164

NOISE IN ANALOG MODULATION 165

NOISE IN FM 167

CHAPTER - 8 : SAMPLING THEOREM

SAMPLING THEOREM 173

SAMPLING OF BANDPASS SIGNALS 174

PROOF OF SAMPLING THEOREM 175

SOLVED EXAMPLES 178

RECONSTRUCTION FILTER (LOW-PASS FILTER) 182

CHAPTER - 9 : DIGITAL COMMUNICATION

ADVANTAGES OF DIGITAL COMMUNICATION OVER ANALOG COMMUNICATION

194

PULSE CODE MODULATION (PCM) 195

QUANTIZER 196

WORKING PRINCIPLE OF QUANTIZER 197

BANDWIDTH OF THE PCM SYSTEM 200

DM (DELTA MODULATION) 201

COMPANDING 203

NOISE IN DM (DISADVANTAGES OF DM) 205

CONDITION TO AVOID SLOPE OVERLOAD NOISE 206

DIFFERENTIAL PULSE-CODE MODULATION (DPCM) 207

ADAPTIVE DELTA MODULATION (ADM) 210

S- ARY SYSTEM 212

SOLVED EXAMPLES 213

CHAPTER - 10 : DIGITAL COMMUNICATION

DIGITAL CARRIER MODULATION 227

PROBABILITY OF ERROR (PE) 217

CHAPTER - 11 : INFORMATION THEORY & CODING

INTRODUCTION TO INFORMATION 230

UNIT OF INFORMATION 230

ENTROPY H(X) 231

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CODING EFFICIENCY 235

SHANNON-FANO CODING 236

HUFFMAN CODING 239

PROBLEMS BASED ON GATE / PSUS / IES 241

CLASSROOM PRACTICE SHEET

PROBLEMS BASED ON RANDOM VARIABLES 245

ANSWER KEY 246

PROBLEMS BASED ON RANDOM VARIABLES 247

ANSWER KEY 254

PROBLEMS BASED ON AMPLITUDE MODULATION 255

ANSWER KEY 267

PROBLEMS BASED ON FREQUENCY MODULATION 268

ANSWER KEY 282

PROBLEMS BASED ON QUANTIZATION , PCM, DPCM 283

ANSWER KEY 292

PROBLEMS BASED ON SAMPLING THEOREM, FILTERS, CHANNEL CODING, PLL

293

ANSWER KEY 296

PROBLEMS BASED ON MATCHED FILTER RECIEVER, BANDWIDTH, PROBABILITY OF ERROR, TDMA, FDMA, CDMA, GSM

297

ANSWER KEY 300

PROBLEMS BASED ON DIGITAL MODULATION TECHNIQUES 301

ANSWER KEY 305

PROBLEMS BASED ON INFORMATION THEORY & NOISE 306

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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.

PERSONAL REMARK :



CHAPTER-1 : INTRODUCTION TO BASICS OF COMMUNICATION SYSTEMS

, _{Electronic communication involves the transmission of information} from one point to another point through a communication channel by means of electronic signals.

, _{Block diagram of electrical communication signal is shown below.}

Recei-ver physical message Trans-mitter medium physical message (300-3.5 kHz) Voice signal) (20-20 kHz) (Audio signal) Information Source

Voice/Speech : Bulk of communication TV : Transmission of Pictures

Data : Between Computers

, _{A communication system has three basic components namely} (i) Transmitter

(ii) Transmission media, and (iii) Receiver

, _{The function of a transmitter is to process the electrical signal from} different aspects. For example in radio broadcasting the electrical signal obtained from sound signal is processed to restrict its range of audio frequencies (20 Hz – 20 kHz)

, _{However in the long distance radio communication or broadcasting} signal amplification is necessary before modulation.

, _{Inside a transmitter, signal processing such as}  Restriction of range of audio frequencies  Restriction of range of video frequencies  Amplification

 Modulation etc. are achieved.

, _{Transmission media or communication channel means the medium} through which message travels from transmitter to receiver. , _{The main function of receiver is to reproduce the message signal in}

electrical form, from the distorted received signal.

, _{The reproduction of the original signal is accomplished by a process} known as the demodulation or detection.











Kind of communications system which we want to design will depends upon the type of information source which we want to transmit

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PERSONAL REMARK :



LUC K NO W_0522-6563566LUCKNOW GORAKHPUR_9919526958 _9451056682AGRA ALLAHABAD_9919751941 _9919751941PATNA _9919751941NOIDA SUMMER CRASH COURSE_{WINTER CRASH COURSE} _{OFF-LINE TEST SERIES}ONLINE TEST SERIES 2

, _{Normally used transmission media of communication channels are} twisted pair, coaxial cable, fiber optic cable and free space. , _{Depending on the transmission media, communication is}

divided into two groups

(i) Line communication or Wireline Communication (ii) Radio communication or Wireless Communication

 Line communication uses a pair of conductors called transmission line. Each transmission line can normally convey only one message at a time.

 In radio communication a wireless message is transmitted through open space by electro-magnetic waves called radiowave, and communication is referred as radio communication.

, _{The two primary communication resources are transmitted} power and channel bandwidth.

 The transmitted power is the average power of the transmitted signal while the channel bandwidth is defined as the band of frequencies allocated for the transmission of the message signal.  The most important system design objectives is to use these two resources as efficiently as possible. In most communication channels one resource may be considered more important than other. Because of this, we may classify communication channels as power limited or band limited. , _{There are many reasons for distortion in the received signal.}

The signal may be distorted mainly due to following reasons-(i) Insufficient channel bandwidth.

(ii) Random variations in the channel characteristics, (ii) External interference, and

(iv) Noise.

, _{Communication systems, as a subject, covers the study of all} aspects of message transmission with particular emphasis on the following

-(1) Reliability of the system (2) Accurary (i.e. least error) (3) Speed of Transmission (4) Bandwidth requirement (5) Power requirement (6) Circuit complexity (7) Cost

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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.

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

, _{When the spectrum of a message signal extends down to zero or} low frequencies, we define the bandwidth of the signal as that upper frequency above which the spectrum content of the signal is negligible and therefore, unnecessary for transmitting information. The important point is unavoidable presence of noise in a communication system. , _{Noise refers to unwanted waves that tend to disturb the transmission}

and processing of message signals in a communication system. The source of noise may be internal or external to the system.

, _{A quantitative way to account for the effect of noise is to introduce} signal-to-noise ratio (SNR) as a system parameter. We may define the SNR at the receiver input as the ratio of the average signal power to the average noise power, both being measured at the same point. Therefore, SNR = S/N. In dB, SNR = 10 log₁₀ _       0 0 N S Where, S = signal power, N = noise power

, _{Table given below shows frequency ranges of various spectrum} S.No. Frequency Range Band Designation

1. 3 Hz - 30 Hz Ultra Low Frequency (ULF)

2. 30 Hz - 300 Hz Extra Low Frequency (ELF)

3. 300 Hz - 3000 Hz Voice Frequency (VF)

4. 3 kHz - 30 kHz Very Low Frequency (VLF)

5. 30 kHz - 300 kHz Low Frequency (LF)

6. 300 kHz - 3000 kHz Medium Frequency (MF)

7. 3 MHz - 30 MHz High Frequency (HF)

8. 30 MHz - 300 MHz Very High Frequency (VHF) 9. 300 MHz -3000 MHz Ultra High Frequency (UHF) 10. 3 GHz - 30 GHz Super High Frequency (SHF) 11. 30 GHz - 300 GHz Extreme High Frequency (EHF) 12. 300 GHz - 900 THz Infra Red Frequencies

Visible Spectrum  Red  Orange  Yellow  Green  Blue  Indigo  Violet 13. 900 THz-30000 THz Ultraviolet

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, _{Table given below shows some most widely spectrum with} their frequency range

S. No. Spectrum Frequency Range

1. Voice frequency 300 Hz to 3.5 kHz 2. Audio spectrum 20 Hz to 20 kHz 3. Radio spectrum 20 kHz to 20 MHz 4. Video spectrum 0Hz to 6.5 MHz 5. Long wave 150 kHz to 285 kHz 6. Medium wave 350 kHz to 1500 kHz 7. Short wave 6 MHz to 25 MHz 8. AM Bandwidth 1100 kHz 9. FM Bandwidth 20 MHz 10. Bandwidth of 3 kHz telephone channel

11. Frequency band for 8 GHz to 16 GHz

Mobile communication 12. Frequency band 800 MHz to 1800 MHz for WLL 13. Optical fiber 1012_{Hz to 10}16_Hz communication FOURIER SERIES

, _{The analysis of signal and linear systems in frequency domain is} based on representation of signals in frequency variable and is done through employing fourier series and fourier transform.

, _{Fourier series is applied to periodic signals whereas the fourier} transform can be applied to periodic and non periodic signals. , _{Let the signal x(t) be a periodic signal with period T. If the following}

contitions (Known as Dirichlet Conditions) are satisfied. 1. x(t) is absolutely integrable over its period i.e.



 T 0 dt | x(t) |

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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.

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2. The number of maxima and minima of x(t) in each period is finite. 3. The number of discountinuities of x(t) in each period is finite.

Then x(t) can be expanded into terms of various possible fourier series , _{Fourier series of x(t) =}



        n n 0 n 0 n 0 (a cosn t b sinn t) a where , T =     , a₀ = 0 0 0 0 t T t T n 0 t t

1

2 x(t) dt , a

x(t) cos ω t dt

T

 





& b_n= T 



 T t t 0 0 0 dt t ω n sin x(t)

, _{Trigonometric fourier series may also be represented by}

f(t) = C₀+



    1 n n 0 ncos(n ω t ) C

Where, C₀ = a₀ and C_n = a 2_n b2_n and _         n n 1 – n a b tan

The coefficient C_n are called spectral Amplitudes i.e. C_n is the amplitudes of spectral components C_n cos (n ₀t – _n) having a frequency n f₀ whereas _n specifies the phase information of the spectral components n f₀.

COMPLEX EXPONENTIAL FOURIER SERIES

As the exponential form of fourier series is simpler and more compact it has extensive application in communication theory.

f(t) =



    – n t jn ne F 0 _{where, F} n =



  T t t t jn – 0 0 0 0 _dt e f(t) T 1

Note : The trigonometric series and the complex exponential series are two ways of representing the same series and one series can be derived from the other.

 The complex function _ejn 0t_{can be seen as a vector of unit length}

and angle n _t.

 Similarly _e–jn 0t_{can be viewed as a vector of unit length and}

angle –n₀t i.e._e–jn 0t_{= cosn }

0t – j sin n 0t and

t jn ₀

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PERSONAL REMARK :



FOURIER TRANSFORM

, _{Fourier transform is the extension of the fourier series to the general}

class of signals (periodic and non peroidic) X (f) =

x(t)e

–i2 t

  



f dt CONVOLUTION

, _{Convolution is a mathematical operation and is useful for describing} the input/output relationship in a LTI system.

, _{The convolution of two time functions f}

1(t) and f2(t) is defined by the

following integral. f(t) = f₁(t)  f₂(t) =

_



 –

f₁(  ) f₂(t –  ) d  SPECTRAL ESTIMATION : INTRODUCTION

, _{The signal processing methods which characterise the frequency} content of a signal corresponds to spectral analysis is called spectral estimation.

, _{Spectral analysis is useful in variety of disciplines like astronomy,} communication engineeering etc.

, _{In communication engineering, spectral estimation is helpful in} detecting the signal component (carrier) which has the noise component in it.

PALEY-WIENER CRITERION

, _{The necessary and sufficient condition for the amplitude response}

|H()| be realizable is    



  d 1 | ) H( | n – 2 

, _{If H() does not satisfy this condition, it is unrealizable.} IMPULSE SIGNAL (DIRAC DELTA FUNCTION)

(t) =        other wise 0 0 at t (t) 1

unit impulse signal (t) =        other were o 0 t

Properties of Impulse function

x(t) (t) = x (0) (t) } Product property x(t) (t – ) = x() (t – )      



  t – x(0) dt (t) x(t) – Shifting Property

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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.

PERSONAL REMARK :





      – ) x( dt ) – (t x(t)



     – dt (t)  _{and  (dt) =} | |   (t) }scaling property Ex.1

_

   – dt 2 3t cos (t) is [GATE-EC-2001] (a) 1 (b) –1 (c) 0 (d)   Sol. 1

Ex.2 Convolution of x (t + 5) with  (t – 7) is equal to

(a) x(t – 12) (b) x (t + 12) (c) x (t – 2) (d) x (t + 2)

Sol. x (t + 5) ×  (t – 7) [GATE-EC-2002]

from convolution property we get (t) = x (t + 5 – 12) = x (t – 7) GATE FUNCTION / RECTANGULAR PULSE

, _{Let us consider a rectangular pulse as shown in figure}

x(t) A T/2 0 +T/2 _        otherwise 0 2 T t 2 T – for A x(t) =              otherwise 0 2 T t 2 T – for T t rect A

SAMPLING / INTERPOLATING / SINC FUNCTION , _{The function}

x sin x

is the "sine over argument" function and it is denoted by "sinc(x). It is also known as "filtering function"

Sinc (x) or s (x)_a 1

–3 _–2 – 0  2 3 x

, _{Fourier transform of rectangular pulse}

F. T. of x(t) =        other were 0 2 T t 2 T – for A

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PERSONAL REMARK :

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i.e. X() =

_

    _ – T/2 T/2 – t j – t j – _dt _A_e _dt e x(t) or X() =





T/2–T/2 t j – e jω A –  =                  2 ωT sin ω 2A 2j e – e ω 2A j T/2 –j T/2 or X() = AT sinc       2 ωT |X( )| AT – 6 T   – 4 T   –2 T   2 T   4 T  6 T  

Energy Spectrum (for Non periodic signal) / Parseval's theorem for Energy Signals

E_x = 2 –

1 | X (ω) | dω

2π

 



= 2 –

|X (f)| df

 



= 2 0

2 |X (f)| df





=



  – 2_dt | (t) x |

, _{This theorem states that energy of a signal x(t) may be obtained} with the help of its fourier transform i.e. without knowing its time domain form.

, _{x(t) is an energy signal if 0 < E <  and P = 0}

, _{"Energy Spectral Density" or "Energy Density Spectrum" is the} energy contribution per unit Bandwidth of a signal. It is denoted by

ESD = () = |X ()|2

, _{So, the total energy of signal may be obtained by integrating } over bandwidth of a signal i.e.

ESD_T = 2 – –

1

1 | X (ω) | dω =

ψ (ω) dω

2π

   



POWER SPECTRUM (for Periodic Signal) , _{x(t) is a "power signal" if 0 < P <  and E = }

Note: Almost all the practical periodic signals are power signals. , _{The power of a periodic signal spectrum x(t) in time domain is defined}

as , P = T/2 2 –T/2

1 |x (t)| dt

T



where , x(t) =



    – n t jn ne C

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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.

PERSONAL REMARK :



, _{Parseval's theorem for Power signals}

T/2 n 2 2 n n – –T/2

1 |x (t)| dt

|C |

T

  







, _{Power Spectral Density (PSD) may be treated as average power}

per unit Bandwidth. It is generally denoted by S() i.e. S() =   d ) ( p d CROSS-CORRELATIONS FUNCTION

, _{The cross-correlation between two different waveforms or two signals} may be defined as the measure of match or similarity between one signal and time delayed version of another signal.

, _{This means that cross-correlation between two signals explains how} much one signal is related to the time delayed version of another signal.

, _{Cross correlation between two signals x}

1(t) and x2(t) is defined as R₁₂ () T/2 1 2 T –T/2

1 Lim

x (t) x (t – τ) dτ

T





_

, _{From the above expression it is clear that cross-correlation represent} the over lapping area between the two signals.

AUTOCORRELATION FUNCTION

, _{Autocorrelation function gives the measure of similarity, match or} coherence between a signal and its delayed replica. This means that autocorrelation function is a special form of crosscorrelation function.

R (

_

   T/2 T/2 – T T x(t) x 1 Lim (t – ) d

, _{The autocorrelation function is defined separately for energy signals} and for the power signals.

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PERSONAL REMARK :



10

ONLINE TEST SERIES SUMMER CRASH COURSE

AGRA GORAKHPUR

LUCKNOW ALLAHABAD PATNA NOIDA

PROBLEMS BASED ON GATE/IES/PSUs

1. Let x(t) be a real signal with the Fourier transform X(f). Let X*(f) denote the complex conjugate of X(f). Then (IES-EE-2002) (a) X(–f) = X*(f) (b) X(–f) = X(f)

(c) X(–f) = –X(f) (d) X(–f) = –X*(f) Sol.(a)

2. Let the transfer function of a network be H(f) =|H(f)|ej(f)₌₂e–j4f_{. If}

a signal x(t) is applied to sush a network, the output Y(t) is given by (IES-EE-2002) (a) 2x(t) (b) x(t–2) (c) 2x(t – 2) (d) 2x (t – 4) Sol.(c)

3. Power spectral density of a signal is (IES-EE-2003) (a) Complex, even and nonegative(b) Real, even and non negative (c) Real, even and negative (d) Complex, odd and negative Sol.(b)

4. Match List I (Signal) with List II (Spectrum) and seletct the correct answer using the code given below the lists: (IES-EE-2005)

List I List II A. _t 1 . f=0 f B. t _2. f=0 f C. Speech Signal _3. f=0 f D. t 4. f=0 f Codes. A B C D A B C D (a) 1 3 2 4 (b) 2 4 1 3 (c) 2 3 1 4 (d) 1 4 2 3 Sol.(c)

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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.

PERSONAL REMARK :



CHAPTER-2:RANDOM VARIABLES & RANDOM PROCESS RANDOM SIGNALS

 Conditional Probability

P(B/A) denotes the Probability of event B when it is known that event A has already occurred.

i.e. P(A/B) =P(A B) and

P(B)  _{.... (I)} and P(B/ A) P(A B) P(A)   .... (II)  Bayes Rule

By using Bayes rule one conditional probability can be expressed in terms of the reversed conditional probability.

theorem Bayes, ) B / A ( P . ) A ( P ) B ( P ) B / A ( P and ) A / B ( P . ) B ( P ) A ( P ) B / A ( P               Independent Events

If one coin is tossed and one dice is thrown, then these two events are called independent events.

Two events are said to be independent when conditional probability i.e.

P(A/B) = P (A) or P(B/A) = P (B) Thus for two independent events, A and B

P(AB)P(A).P(B)

 For two marginal probability, P(A/B) = P(B/A) = 1

 An experiment whose outcome cannot be predicted exactly, is called a random experiment (e.g. tossing of a coin, drawing of a card from a deck of playing cards).

 The collective outcomes of a random experiment form a sample space. A particular outcome is called a sample point or sample collection of outcomes is called an event.

 A random variable is a real valued function defined over the sample space of random experiment is known as stochastic variable or random function.

RANDOM VARIABLE

From random variable we mean, a real number connected with the outcome of random experiment.

Let W be the outcome of random experiment then X() is a real number associated with the event W.

Let w be the event of tossing two coins. X() is the number of heads.

Outcome HH HT TH TT

Random

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12

AGRA GORAKHPUR

PERSONAL REMARK :



A random variable is a function X() with domain s and range (,) such that for every real number a, the event

_

w : X( ) a

_

B S : Sample space

B : event of sample space

Properties of Random Variable

 A function x(w) from S to R (,) is a random variable if and only if for real a,



w:x(w)a



B

 If X₁and X₂are random variable and c is a constant then c x₁, x₁ + x₂, x₁x₂are also random variable

 If x is a random variable then  x 1 , Where        ) w ( x 1 , if X() = 0  X (ω) = maximum+



0, X(ω)



,  X ( )_   min imum 0, X( )







 X  random variable

 If X₁ and X₂ are random variable then max [x₁, x₂] and min [x₁, x₂] are also random variable.

 If X is a random variable and f ( X ) is a continous or/and increasing function, then f(x) is a random variable.

Discrete Random Variable

A real valued function defined on a discrete sample space is called a discrete random variable. Examples are marks obtained in a test, telephone calls per unit time, number of successes in n trials.

Probability Mass Function

If X is a discrete random variable with distinct values x₁, x₂...x_n... then the function p(x) defined as:

           .. 2 , 1 i ; x x if 0 x x if ) x x ( p ) x ( p i i i x

is called the probability mass function.

The set of ordered pairs



xi,p(xi);i1,2,3,...n...



or



 









x1,p1 , x2,p2 ,...xn,pn ...



, specifies the probability distribution

of the random variable X.

Discrete Distribution Function  pi0, p ,such that i i







  1 x x : i i p ) x ( F

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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.

PERSONAL REMARK :



 p

 

x1 p



xxi



F

 

x1 F



xi1



, where F is the distribution function

of X

 Cummulative Distributive Function (cdf) for a discrete random variable X can be defined as

       



 x , ) u ( f ) x X ( P ) x ( F x u x

If X can take on the values x₁, x₂, x₃, ... x_n then the distribution function is given by

                         x x ) (x ...f ) (x f ) (x f ) (x f x x x ) (x f ) (x f x x x ) (x f x x 0 ) x ( F n n 3 2 1 3 2 2 1 2 1 1 1 x

Domain of F_x(x) is



,



and its range is [0,1]. Properties of F(x)

 F_x(x)0  F_x()1  F_x()0

F_X(x) is a non-decreasing function, i.e., monotonically increasing function

2 1 2 X 1 X(x ) F (x ) for x x F  

PROBABILITY DENSITY FUNCTION

x

f (x)dxx

x+dx X

Consider the small interval (x, x + dx) of length dx around the point x. Let f_x (x) be any continous function of x so that f(x) dx represents the probability that X falls in the infinitesimal interval (x, x + dx).



x x x dx



f

 

x dx P     x or

 





x dx x x x P lim x f 0 x x       

The curve f(x) is called the probability density function for continous distribution function

 P



axb



P



axb



P



axb



P



axb



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14

AGRA GORAKHPUR

PERSONAL REMARK :



 Probability Density Function (PDF) for a continuous random variable is defined as ) x ( F dx d ) x ( f_X  _X

The pdf [i.e. f_X (x)] is the first derivative of the probability distribution function F_X(x). The first derivative of probability distribution may not exist at all points because the probability distribution function may be discontinous function for discrete random variables. Here we assume that F_X(x) is a continuous function



X x



f (x)dx P ) x ( F _X x X



     However,





X x P X x f (x)dx   

_



  2 1 x x 1 x 2 x x(x)dx F (x ) F (x ) f P(x₁X x₂)  Properties of P.D.F. f_X (x)  PDF is non-negative function  Area under the pdf curve is unity

1 dx ) x ( f_X 



  

 The probability of X lying between a and b is given by P(a x b) fX(x)dx b a



   ....(A)

 For a continuous case, the probability of x being equal to any particular value is zero. Hence equation (A) can be written as

) b x a ( P ) b x a ( P ) b x a ( P ) b x a ( P           

Let f_x(x) or f(x) be the pdf of a random variable X, where X is defined from a to b. Then Arithmetic Mean =



b a dx ) x ( f x Harmonic Mean =

_

b a dx ) x ( f x 1 Geometric Mean =



b a dx ) x ( f x log

Ex. A probability density function is of the form p(x) = Ke-a|x|, x (–).

The value of K is

(a) 0.5 (b) 1 (c) 0.5  (d) 

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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.

PERSONAL REMARK :



) origin about ( r  = x f(x)dx b a r





x A



f(x)dx ) A x int po the about ( b a r r  



 



x mean



f(x)dx ) mean about ( b a r r 



 

Median : It is the point which divide the entire distribution into two equal parts.

 

_

 



  b M M a 2 1 dx x f dx x f Mean deviation

Mean deviation about mean

M.D = x meanf

 

x dx

b

a





Mean deviation about any point A

M.D about ‘A’ x Af

 

x dx b a



  Quartilies :

 

1 Q 1 a

i

Q

f x d x

, i 1, 2, 3, 4

4 

_



Deciles : 

_

  i D a i ,i 1,2...,8,9 10 i dx ) x ( f D

Mode: It is the value of x for which f(x) is maximum. Two-Dimensional Random Variables

Let X and Y be two random variables defined on the same sample space then the function (x, y) that assigns a point in R2

_

RR

_

is called two dimensional random variable.

MARGINAL PROBABILITY FUNCTION

 

_





  m 1 j i i y , x x x p x ,y p

 

_





  n 1 j i i y , x x y p x ,y p

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16

AGRA GORAKHPUR

PERSONAL REMARK :



Two-dimensional distribution function

 





 



xy

F x,y P X x, Y y

Marginal Distribution Function

 





   









x xy

F x P X x, Y F x, (in discrete case)





   

_{ }

x XY

dx f x, y dy (in continous case)

 





  











Y xy F y P X , Y y F , y





  



_{ }

y XY

dy f

x, y dx

Marginal Density Function

 













x XY

y

f x

p

x, y

for discrete case

f_x(X)

 







_

f_XY xy dy(for continous case)





 







_

y XY XY x

f (y)

p (xy)

f

x, y dx

Condition for independence

Two random variables are independent if and only if



x

,

y



f

   

x

f

y

f

_XY



_X _Y



x

,

y



F

   

x

F

y

F

_XY



_X _Y

 Two statistical averages that are most commonly used for characterizing a random variable, X are its mean (_x) and variance 2_x.

Expectation of a Random Variable

It is the average value of a random phenemenon.For random variable X, expectation is defined as

 



_

x ) x ( f x X

E (for discrete random variable)

 

X xf(x)dx

E

_



 

 (for continous random variable)

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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.

PERSONAL REMARK :



 







_

   

x x f x g x g E

 





_

    g(x)f(x)dx x g E Properties of Expectation  E (X + Y) = E(X) + E(Y)

 E (X Y) = E(X) E(Y) [when X and Y are independent ]  E(a X + b) = a E (X) + b  E(b)



bf(x)dxb



f(x)dxb        If g(x) is non-linear

 

_        ₁_E₍_x_), _E _X12 _E₍_x₎12 x 1 E  _E



_log

 

_x



__log



_E₍_x₎



_, _E

  

_X2 _ _E₍_x₎2



 If X and Y are independent random variables, then

   



h x.k Y



E



h

 

x



E



k

 

Y



E 

Variance: Variance of a random variable X with mean  is defined_x as E

_

X_x

_

2E



X2 _x22_xX



or

 

2 x 2 2 x 2 x 2 X E 2 X E        Properties of Variance



aX b



a V

 

x V   2 If b = 0, then V(ax) = a2_V(x)

 Variance depends on change of scale If a = 0, then V (b) = 0

 Variance of a constant is zero If a = 1, then V (X + b) = V (X)

 Variance is independent of change of origin.



X1 X2



V

 

X1 V



X2



2Cov



X1,X2



V    

If X₁andX₂ are independent



X1 X2



V

 

X1 V



X2



V   

Covariance

Covariance between random variable X and Y is defined as

 







 





X EX Y EY



E ) Y , X ( Cov   



X,Y



E

   

X EY E ) Y , X ( Cov  

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18

AGRA GORAKHPUR

PERSONAL REMARK :



For independent random variable. X and Y, E(X, Y) = E(X) E(Y) Cov (X, Y) = 0

Important Points Regarding Covariance  Cov (aX, bY) = ab Cov(X, Y)

 Cov (X + a, Y + b) = Cov (X, Y)  Cov X X,Y Y 1 Cov



X,Y



y x y x               

 Cov (X + Y, Z) = Cov (X, Z) + Cov (Y, Z)

 The positive root of variance is called standard deviation (_x).  The variance or a standard deviation is a measure of the ‘spread’ of

the value of random variable, X, from its mean (_x).

Some Commonly Occurring PDF_S

(i) Uniform pdf : , x (a,b) a b 1 ) x ( fX    f (x)x t b a ) a b ( 1 

(ii) Gaussian or Normal pdf : A random variable X is called normal or Gaussian pdf if its form like.

 2 1 x x f (x)x _x

 

              x , e , . 2 1 x f 2x 2 x 2 x x X

where, _x  mean of random variable. 2x  variance of random variable.

(iii) Rayleigh pdf : Used for describing the peak values of random process. 0 x ; e x ) x ( f 2 x x 2 1 x X             

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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.

PERSONAL REMARK :



 Gaussian or normal pdf occurs in so many application because of

remarkable phenomenon called C

ENTRAL LIMIT THEOREM.  As we know that electrical noise in communication systems is often due to cumulative effects of a large number of randomly moving charged particles and hence the instantaneous value of noise will have a Gaussian distribution.

 In our studies on the effect of Gaussian noise on digital signal transmission, we shall often be interested in probabilities such as

2 x 2 x ( x ) 2 X x a 1 F (x) P(x a) e dx 2          



x x x Q     _ _    or x X x x F (x)P(x a) 1 Q_   _            



2 x 2 x ( x ) a 2 x 1 .e .dx 2 or e .dx 2 1 2 1 ) x X ( P ) x ( F 2x 2 x 2 ) x ( x o x X    



      

If we assume Z be the standarized random variable corresponding to X. Thus if x x x Z   

 . Then mean of Z is zero and its variance is 1.

Hence, 2 z Z 2 e . 2 1 ) z ( f   

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PERSONAL REMARK :



20

AGRA GORAKHPUR

PROBLEMS BASED ON GATE/IES/PSUs 1. The PDF of a Gaussian random variable X is given by

18 2 4) (x e 2π 3 1 (x) x P  

 .The probability of the event { X = 4} is (GATE-EC-2001) (a) 2 1 (b) 2π 3 1 (c) 0 (d) 4 1

Sol.(c) pdf of the gaussian distribution function is given by

P_x(x) = 2 ( x 4) 18 1 e 3 2   

Probability of the event at X = 4 P(X = 4) = P (X 4)  P (X< 4) or P (X = 4) = 1  P (X > 4)  P (X< 4) or P (X = 4) = 1  (P(X> 4) +P (X < 4)) or P (X = 4) = 1  1 = 0 2. If the variance 2 x

 of d(n) = x(n)–x(n–1) is one-tenth the variance 2

x

σ of a stationary zero-mean discrete-time signal x(n), then the normalized autocorrelation function 2

xx x R (k) | at k1 is (GATE-EC-2002) ` (a) 0.95 (b) 0.90 (c) 0.10 (d) 0.05 Sol.(a) 2 2 2 d =E[x (n)]=E[x[n] x(n 1)]    2 2 2

d =E[x (n)]+E[x (n 1)] 2E[x(n)x(n 1)]

    2 2 2 d = x+ x 2R (1)xx     or 2 2 x x xx =2 2R (1) 10    or 2R_xx(1) = 2 x 19 10  or xx2 R (1) 19 = =0.952 x 20  Common Data for Questions 3 and 4.

Let X be the Gaussian random variable obtained by sampling the

process at t = t_i and let Q()

2 1 ₂ α) e dy 2π α

y

  



.Autocorelation

function R_xx(τ)4 e



0.2 τ 1



and mean = 0

3. The probability that

_

x



1 _

is : (GATE-EC-2003) (a) 1 – Q (0.5) (b) Q(0.5) (c)       2 2 1 Q (d)       2 2 1 Q – 1

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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION

PERSONAL REMARK :



Sol.(d) The pdf for Gaussian random variable is

2 2 x x ( x ) / 2 x x 1 p (x)= e 2     

For zero mean, x2/ 2 2x

x x 1 p ( x ) e 2      2 x R (0)xx 8    or  x 2 2 or P[x1] 1 P(x  1) x 1 P[x 1] 1 p (x)dx    

_

or x / 22 2x 1 x 1 P[x 1] 1 e dx 2        



or x /162 1 1 P[x 1] 1 e dx 2 2 2       



, Put x y 2 2  or dx dy 2 2 or y / 22 1 2 2 1 P[x 1] 1 e dy 2      



or 1 P[x 1] 1 Q 2 2      _ _  

4. Let Y and Z be the random variables obtained by sampling X(t) at t = 2 and t = 4 respectively. Let W = Y – Z. The variance of W is (GATE-EC-2003) (a) 13.36 (b) 9.36 (c) 2.64 (d) 8.00

Sol.(c) 2 2 2

W E[W ] E[Y Z]

    or 2 2 2

W E[Y ] E[Z ] 2E[YZ]

   

2 2 2

W Y z 2RYZ( )

      

Here, t = 2, since Y sampled at t = 2 and Z sampled at t = 4 2 0.4 W 8 8 2 4(e 1)        or 2 W 2.64  

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PERSONAL REMARK :



22

AGRA GORAKHPUR

Mean and Variance of the Sum of Random Variables  Let X and Y be two random variables with means x andy.

Let Z = X + Y with mean Z given as

 

     







_z

(

x

y

)

2

f

(

x

,

y

)

dx

.

dy

y x z  

 _{i.e., equal to the sum of the means.}

Note : This result holds whether the variables X and Y are independent or not.

 Variance (or the second moment) of Z = X + Y is given as 2 2 z



( 

x

y

)



_(x _y)2 _{f(x, y) dx dy}     

_{ }





              x2f(x)dx. f(y)dy y2f(y)dy f(x)dx



      2 xf(x)dx y.f(y)dy or _z2x2y22x y



        f(y)dy 1) dx ) x ( f ( or z22x 2y 2xy

Special Case : If either x or y or both are zero, then resultant

variance becomes , 2y 2 x 2 z  

 Probability density of Z = X + Y (i.e., sum of random variables) Here we want to calculate the probability density f_Z (Z) of Z = X + Y in terms of joint density f (x, y). Assume an arbitrary value of Z and call it z. Then the region YZX is shown as shaded

region.

Hence the probability that _{Z }_z is the same as the Probability that

X Z

y  independently of the value of Xi.e. for x. This probability is F_Z(z)P(Zz)P



X,YzX



Y Y=Z– X Region where Y Z–X X

(29)

CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION

PERSONAL REMARK :



or F (z) dx f



x,y



dy x z Z



       ...(A) and the probability density function of Z is given as

Z Z d F (z) F (z) f (x, z x).dx dz    

_

 ...(B) When X and Y are independent, f(x,y)fx.fyand equation (B) may be

written F_Z(z)

_

f(x).f(z x)dx     

Theorem Based on Transformation of Random Variables  Theorem 1 :Let X and Y be continuous random variables whose joint

pdf f_XY(x, y) is given, and given Z = g (X,Y) and W = h (X, Y). Then f_{Z W}(z,w) can be determined as,

| J | ) y , x ( F ) w , z ( F _i _i _i n 1 i xy W , Z



 

where, J_i is the Jacobian of the transformation defined as.

J_i = i i i i i x x z w J y y z w         

 Theorem 2 :To Determinef (y)_Y when f_X(x)is given.Solve the equation y = g ( x ), and find its real roots say x₁, x₂... x_k. we have

1 2 k Yg(x )g(x ) ...g(x ) Then

_

  k 1 k k k X Y | ) x ( ' g | ) x ( f ) y ( f SOLVED EXAMPLES

Example 1 : Given Y = 2 X + 3. If random variable X is uniformly distributed over [– 1, 2] find f_Y( y ).

f (x)X 1/3 –1 2 x Solution : We have

 

         otherwise 0 2 x 1 3 1 x f_x 3 x 2 ) x ( g y   ....(1) and g'(x)2 The range of y is

(30)

PERSONAL REMARK :



24

AGRA GORAKHPUR

y₁ = 2 x (–1) + 3 = 1 to y₂ = 2 x 2 + 3 = 7. Equation (1) has a single solution i.e.

2 3 y x1   So 6 1 2 3 / 1 | ) x ( ' g | ) x ( f ) y ( f 1 1 X Y    or fY (y) 1 y 7 6 0 otherwise       

Example 2 : Let Y = sin x, where X is uniformly distributed with

         Y X 1 , find f (y) 0 x 2 f (x) = 2 0 otherwise

Solution : yg(x)sin x ....(A) From equation (A) it is clear that for |y|1, the equation, y sinxhas no solution. Hence f_Y(y)0. If |y|1. Then y = sin x, has two solution

in the interval 0x2. y=sin x 2 –1 x  x1 x2

i.e., x₁sin1y and x2x1sin1y

2 1

1

1) cosx cos(sin y) 1 y

x ( ' g      2 1 1 2

2) cosx cos( x ) cosx 1 y

x ( ' g       x K x 1 x 2 Y K 1 2 f (x ) f (x) f (x ) f (y) = = + +... | g' (x ) | | g' (x ) | | g'(x ) | or

f (y)

_Y



₂ 2 2 1 1/ 2π 1 2π ₊ ₌ _{, | y | <1} π 1 y 1 y  1 y 

Example 3 : Given, Y = X2_{, find f}

Y (y) for x = N (0 : 1)

Solution : Y = g ( x ) = X2 _....(A)

form equation (A) we conclude that if y < 0, then the equation Y = X2

has no real solution, hence f_Y(y) = 0. However if y > 0, then equation y = x2 _{has two solution i.e.}_x _y _or _x _y_,_x _y

2

1 

 

Since, X = N ( 0 : 1 ) given means 0 mean and 1- variance.

2 x X 2 e 2 1 ) x ( f  

 i.e.,fX(x) is an even function.

Now Y x k x 1 x 2 k 1 2 f (x ) f (x ) f (x ) f (y)= = + g'(x ) | g'(x ) | | g'(x ) | or x

 

x





Y f y f y f ( y ) = + y > 0 2 y 2 y 

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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION

PERSONAL REMARK :



or u(y) y 2 e 2 1 y 2 e 2 1 ) y ( f 2 y 2 y Y                     or .e .u(y) y 2 1 ) y ( f_Y y/2  

Example 4 : Consider X has a uniform probability density function given          X 1 0 x 2 f (x) = 2 0 otherwise Determine _x, E [ X ],2 _x Solution : 2 2 ₂ ₂ x X 0 0 1 1 x 1 4 x.f (x)dx x. .dx . 2 2 2 2 2       _           __ __ 



2 2 ₃ 2 2 2 2 X 0 0 1 1 x 4 E [x ] x .f (x) dx x . .dx 2 2 3 3               _{ }_ __



3 3 ) ( 3 4 ] x [ E 2 2 2 2 x 2 x             _Ans.

Example 5 : Given a random process X ( t ) = A cos ( t – ) where  is a random variable, and A and  are deterministic.

Assume a uniform distribution      1 f ( ) = [0,2 ], 2   find x and 2x. Solution:    x E[X(t)]

_

x(t)f (x)dxX         



Acos( t ).f ( )d     

_

 d ) t ( cos 2 A 2 0

with  as a random variable

             2 0 1 t ( sin 2 A





      sin( t) 2₀ 2 A



sin(2 t) sin(0 t)



2 A        



sin t sin t



2 A       _₀  2  2 2 2 x E [X (t) x] E [X ] x        



2 2 _f ₍ ₎_d ₀_. ] ) t ( cos A [  _



    d ) t ( cos 2 A 2 0 2 2 2 2 . 2 A2    2 A2 

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PERSONAL REMARK :



26

AGRA GORAKHPUR

RANDOM PROCESS

A probalistic description of a collection of function of time is called random process.

Consider a random experiment having sample space S and outcomes  for each outcome   we assign a real valuedS function of time X(t,  ). This real valued function of time is called Random Process.

For fixed  say 1 , we have a function of time X(t, i) x (t)i

called sample function.

Set of sample function is called ensemble For fixed t say t , X(t , )j j  Xj called a number

outcomes ₁ 2 n x (t)₁ x (t )_{1 1} t1 t2 t3 x (t )1 2 x (t )1 3 1 2 2 22 2 3 t3 t2 t1 n t1 t₂ _t₃ n 1 _{n 3} x (t)2 t t t

x₁(t), x₂(t) ...x_n(t) are the sample function. SPECIAL RANDOM PROCESS

A. Gaussian Random Process

 A gaussian process X(t) is completely specified by the set of means  _i E[X(t )]_i i = 1, ...n

& the set of autocorrelations

xx i j i j

R (t , t )E[X(t )X(t )] i, j = 1 ...n

 If the set of random variables X(t_i) i = 1 , ...n is unocorrelated i.e. C_ij = 0 then X(t_i) are independent.

(33)

CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION

PERSONAL REMARK :



 If a gaussian process X(t) of a linear system is gaussian then the output process Y(t) is also gaussian.

B . White Noise x x S ( ) 2    xx R ( ) (t) 2    

Mean of white noise is zero. C. Band-limited white noise

B xx B | | S ( ) 2 0 | |           _{  }  B B j B B XX B sin 1 R ( ) e d 2 2 2            



    R XX n 2   – B B 0 _B  _B  –   S ( )_xx

Figure : Band Limited white noise D. Narrowband Random Process

A WSS process x(t) with zero mean & its PSD S_xx() is non - zero only in some narrow frequency if bandwidth 2W that is very small compared to a center frequency _c, as shown in fig. The process X(t) is narrowband random process.

S XX   – _c 0 _c X(t)=V(t)cos[ t + (t)]_c  – – + 

V(t) = envelop function , (t) = Phase function

c c

x(t)V(t) cos (t) cos  t V(t) sin (t)sin  t = X (t) cos_c _ctX (t) sin_s _ct

X (t)_c V(t) cos (t) (in-phase component) X (t)_s V(t)sin (t) (quadrature component)

2 2 1 s c s X (t) V(t) X (t) X (t), (t) tan X (t)        _ _    S XX 2

(34)

LUCKNOW ONLINE TEST SERIES 28

OFF-LINE TEST SERIES SUMMER CRASH COURSE

WINTER CRASH COURSE AGRA 9451056682 GORAKHPUR 9919526958 LUCKNOW 0522-6563566 PATNA 9919751941 NOIDA 9919751941 ALLAHABAD 9919751941 PERSONAL REMARK :



CLASSIFICATION OF RANDOM PROCESSES

 A random process X(t, S) represents an ensemble or a set of family of time functions where t and s are variables. In place of x(t, s) and X(t, s) the short notations x(t) and X(t) are often used.

 Figure shows the classification of random process.

Ergodic

Strict Sense Stationary (SSS)

wide sense stationary (WSS) Random variable

Fig. : Classification of random process.

 The mean value of X(t) = E[X(t)] is known as ensemble average.  However if sample function say x(t) over the entire time scale, then

] ) t ( x [ E dt ) t ( x T 2 1 Lim ) t ( x T T T      

_

  

called time average, which is expected value of all mean values.

 Ergodic Process

‘’Ensemble averages is equal to time average’’ i.e. when all statistical ensemble properties are equal to statistical time properties, then the process is known as ergodic process.

T / 2 x T T / 2 X(t) x(t) 1 E x(t) Lim x(t)dt T             



i.e.

Note : An ergodic processes is necessary stationary processes, but the reverse is not true.

 Stationary or Strict Sense Stationary (SSS) Process

If all the statistical properties of a random process are independent of time, then it is stationary processes or S.S.S. process.



X(t )



E



X(t )



... E



X(t)



E ₁  ₂   i.e.

which indicates that if X(t) is S.S.S. process, the joint density of random variable X(t) and X(t + ) is independent of actual time t₁ and t₂and

(35)

CHAPTER-2 (RANDOM VARIABLES ) : COMMUNICATION ENGG.

PERSONAL REMARK :



depends upon the time difference i.e. (t₂t₁) only

i.e. _x(t)  _x Constant

2 2

x(t) x Constant

   

 Wide Sense Stationary (WSS) Process A random process will be WSS if

(i) Its mean is constant i.e. E X(t)_ _  _x Constant, and

(ii) Its autocorrelation depends only on the time difference,

i.e. E

_

X (t) . X (t)

_

R ( )xx....(A)

Special Case

(i) By setting 0, equation(A) becomes E[X2_{(t)] , thus the average}

power of a WSS prosess is independent of time and equals to R_xx(0).

(ii) Two process X(t) and Y(t) are called joint WSS if each is WSS and their cross-correlation depends only on the time difference,. i.e., RXY_t,t )_ E _X(t)Y(t )_RXY( )

AUTO-CORRELATION

 The correlation is similarity between one waveform and time delayed version of the other waveform. An analogy case may be stated as “comparison of your present photograph and the photograph taken 10 years back.’’

 Autocorrelation function is given as





x(t).x(t )dt T 1 Lim ) t ( X , ) t ( X E ) ( R 2 / T 2 / T T xx    



    Properties of R_xx()

 R_xx()R_xx() for real signal  R_XX(t) = R*_XX(-t ) for complex signal

 | R ( )_xx   R (0)_xx i.e.R_xx(0) is the maximum value of R_xx() and occurs at the origin.

 Rxx(0)E[X2(t)]

AUTO-CORRELATION

For real signa (or non-periodic signal)Cross correlation function is given as , RXY(τ) = E X(t), Y(t + τ)  + T / 2 T T / 2 1 = Lim x(t) y(t τ) dt T    



*

(36)

LUCKNOW ONLINE TEST SERIES 30

OFF-LINE TEST SERIES SUMMER CRASH COURSE

WINTER CRASH COURSE AGRA 9451056682 GORAKHPUR 9919526958 LUCKNOW 0522-6563566 PATNA 9919751941 NOIDA 9919751941 ALLAHABAD 9919751941 PERSONAL REMARK :



 If the correlation is defined for energy signal, then

       

_

  

_

  XY R ( ) x(t).y(t )d x(t ).y*(t)d Properties of R_xy()  R_XY ( ) R ( )  _XX 

| R

_XY

( )|

 

R (0). R (0)

_XX _YY  XY  



XX  YY



1 | R ( )| R (0) R (0) 2  Autocovariance, 2 XX XX X C ( ) R ( )      Cross covariance,C XY ( ) R ( )  XY  X Y

 Two process, X (t)and Y (t) are called orthogonal or incoherent if, R_xy()0

 Two process, X (t)and Y (t) are uncorrelated if, C_XY ( ) 0 

i.e., R ( )_XY  _X _Y i.e., cross correlation functions R_XY ( ) are equal to the multiplication of mean values.

 Power spectral density of a random process X(t) is a Fourier transform of autocorrelation function for a periodic or aperiodic signal i.e.,

       



d e . ) ( R ) ( Sxx xx j ) (

R_xx  from the given signal can be calculated as

 

x

  

t.x t



.dt T 1 Lim R 2 T 2 T T xx



      and         



d . e . ) ( S 2 1 ) ( R_xx _xx j Properties of S_xx()

 SXX ( ) is real i.e. SXX ( ) 0 for all  .

 SXX ( ) S ( )XX  i.e. Power spectral density of a random

process X(t) is an even function of frequency.

(37)

CHAPTER-2 (RANDOM VARIABLES ) : COMMUNICATION ENGG.

PERSONAL REMARK :



 Relation between input and output spectral densities

2

YY XX

S ( ) |H ( ) | . S ( )

 



 Energy Spectral Density )

(

X 

 is a measure of density of the energy contained in random process X (t) in joules per hertz.

Since the amplitude spectrum of a real-valued random process X(t) is an even function of  , the energy spectral density of such a system is symmetrical about the vertical axis passing through the origin.

Total energy of the random process X(t) is defined as

      



X 1 E ( ).d 2

Note: Autocorrelation function of a pulse type signal (i.e. energy signal) gives energy density spectrum

     

X i.e. F RXX  X

  _    _

 Power of correlated function

Let us consider a function f₁(t) with power P₁and another function f₂(t) with power P₂. The normalized power (r.m.s. value). P_1,2of the combined function is given by.

 







f t f t



dt T 1 P 2 2 T 2 T 2 1 2 , 1



    

 





T 2 dt t f T 1 dt t f T 1 T2 2 T 2 2 2 T 2 T 2 1    

_

 

  





   2 T 2 T 2 1 t.f t dt f or P₁_,₂P₁P₂2R₁_,₂

 

 ....(A) Following conclusion are drawn from equation (A)

 The power of two correlated function is equal to the sum of powers of each individual function plus twice the cross-correlation between them.

 If the functions f₁ (t) and f₂ (t) are uncorrelated, i.e.,

R_{1, 2} () = 0, then the powers of the combined functions is equal to the sum of the powers of each individual function, and  Functions correlated by dc components are considered as

uncorrelated.

(38)

PERSONAL REMARK :



32

ONLINE TEST SERIES SUMMER CRASH COURSE

AGRA GORAKHPUR

LUCKNOW ALLAHABAD PATNA NOIDA

Transmission of Random Process through Linear Systems A. System Response : In the given figure X(t) is the input random

process and Y(t) is the output random process of a system having impulse response h(t) Y(t) = X(t) * h(t) = h(t) *X(t) = h( )X(t )d      



B . Mean and Autocorrelation of Output

E[Y(t)] E[h(t) * X(t)] E h( )X(t )d       _    _ 



 X x h( )E[X(t )]d h( ) (t )d h(t) * (t)                



For wide sense stationary random process, E[X(t  )] _x

x x x

E[Y(t)] h( ) d h( )d H(0)

 

 



_

    

_

   

Thus for WSS, Y(t) is constant H(0) is the frequency response of filter at  = 0 R_YY(t₁,t₂) = E[Y(t₁)Y(t₂)] E h( )X(t1 )h( )X(t2 )d d        _       _ 

 

 h( )h( )E[X(t1 )X(t2 )]d d     

_{ }

       h( )h( )RXX(t1 , t2 )d d     

_{ }

       For WSS YY XX 2 1 R ( ) h( )h( )R (t t )d d      

_{ }

        

C. Power spectral Density of the output

2 YY XX S ( ) | H( ) | S   ( ) 2 j YY XX 1 R ( ) | H( ) | S ( )e d 2         



2 2 YY XX 1 E[Y (t)] R (0) | H( ) | S ( )d 2        



X(t) _LTI System Y(t)