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COMMUNICATION SYSTEMS Ashish Dixit
Prashant Singh
M.Tech. (V.L.S.I. Design) C-DAC, Mohali 2-Times GATE Qualified & ISRO-2004 Qualified Assitant Professor (ECE Department) AMITY University, Lucknow
3-Times GATE Qualified (Design Engineer Taiwan Semiconductor Manufacuring Company)
Te TECHGURU EDUCATIONALS
M.Tech. (IIT Bombay)
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CONTENT: COMMUNICATION SYSTEMS CHAPTER NAME / TOPIC NAME CHAPTER -1 : FUNDAMENTALS OF COMMUNICATION SYSTEMS INTRODUCTION TO BASICS OF COMMUNICATION SYSTEMS FREQUENCY RANGES OF VARIOUS SPECTRUM SOME MOST WIDELY SPECTRUM WITH THEIR FREQUENCY RANGE FOURIER SERIES COMPLEX EXPONENTIAL FOURIER SERIES FOURIER TRANSFORM PALEY-WIENER CRITERION GATE FUNCTION / RECTANGULAR PULSE SAMPLING / INTERPOLATING / SINC FUNCTION POWER SPECTRUM CROSS-CORRELATIONS FUNCTION AUTOCORRELATION FUNCTION PROBLEMS BASED ON GATE/IES/PSUs CHAPTER -2 : RANDOM VARIABLES & RANDOM PROCESS RANDOM SIGNALS PROPERTIES OF RANDOM VARIABLE PROBABILITY DENSITY FUNCTION CUMMULATIVE DISTRIBUTIVE FUNCTION PROPERTIES OF P.D.F. fX (x) MARGINAL PROBABILITY FUNCTION TWO-DIMENSIONAL DISTRIBUTION FUNCTION EXPECTATION OF A RANDOM VARIABLE COVARIANCE SOME COMMONLY OCCURRING PDFS PROBLEMS BASED ON GATE/IES/PSUs MEAN AND VARIANCE OF THE SUM OF RANDOM VARIABLES SOLVED EXAMPLES SPECIAL RANDOM PROCESS CLASSIFICATION OF RANDOM PROCESSES CORRELATION TRANSMISSION OF RANDOM PROCESS THROUGH LINEAR SYSTEMS SOLVED EXAMPLES PROBLEMS BASED ON GATE/IES/PSUs CHAPTER -3 : MODULATION NEED OF MODULATION DISTORTIONLESS TRANSMISSION TYPES OF DISTORTIONS CONCEPT OF MODULATION AND DEMODULATION GENERATION OF AM WAVE DEMODULATION CHAPTER - 4 : AMPLITUDE MODULATION INTRODUCTION TO AMPLITUDE MODULATION (AM) BLOCK DIAGRAM OF THE AMPLITUDE MODULATOR POWER CALCULATION OF AM WAVE AM DEMODULATION GENERATION OF AM SIGNALS ADVANTAGE OF A BALANCED MODULATOR OVER A SIMPLE NONLINEAR CIRCUIT
Page No. 1 3 4 4 5 6 6 7 7 8 9 10
11 12 13 13 14 15 16 16 17 18 20 22 23 26 28 29 32 33 40 51 53 54 56 57 58 59 60 62 64 68 69
CONTENT: COMMUNICATION SYSTEMS SUMMARY OF DIFFERENT POSSIBLE AMPLITUDE MODULATED SYSTEM MIXER DOUBLE-SIDEBAND SUPPRESSED CARRIER (DSB-SC) MODULATION SINGLE-TONE MODULATION OF DSB-SC GENERATION OF DSB-SC SIGNALS DIODE-BRIDGE MODULATOR RING MODULATOR OR CHOPPER TYPE BALANCED MODULATOR SYNCHRONOUS OR COHERENT OR HOMODYNE DETECTION EFFECT OF PHASE AND FREQUENCY ERRORS IN SYNCHRONOUS DETECTION SINGLE SIDEBAND (SSB) MODULATION HILBERT TRANSFORM PROPERTIES OF HILBERT TRANSFORM CONCEPT OF PRE-ENVELOP OF ANALYTIC SIGNAL GENERATION OF SSB SIGNALS (I) Frequency Discrimination Method (II) Phase Discrimination Method or Phasing Method VESTIGIAL SIDEBAND (VEB) MODULATION SYSTEMS Generation and Detection of VSB Signal SUMMARY: modulators and demodulators used by various AM systems. PROBLEMS BASED ON GATE/IES/PSUs CHAPTER - 5 : AM TRANSMITTERS AND RECEIVERS INTRODUCTION TO AM TRANSMITTERS AND RECEIVERS BLOCK DIAGRAM OF AM-TRANSMITTER USING LOW-LEVEL MODULATION BLOCK DIAGRAM OF AM-TRANSMITTER USING HIGH-LEVEL MODULATION MASTER OSCILLATOR (MO) SOME FACTS REGARDING TO THE STABILITY OF MASTER OSCILLATOR FREQUENCY AM RECEIVER TYPE OF AM RECEIVER TUNED RADIO FREQUENCY RECEIVER (TRF) SUPERHETERODYNE RECEIVER MAIN FUNCTIONS OF RF AMPLIFIER FREQUENCY CONVERSION OR MIXING SOME FACTS ABOUT CHOICE OF QUALITY FACTOR (Q) OF IF AMPLIFIER TRACKING OF A RECEIVER TYPES OF AGC PROBLEMS BASED ON GATE/IES/PSUs CHAPTER - 6 : FREQUENCY MODULATION INTRODUCTION TO ANGLE (FREQUENCY OR PHASE) MODULATION IMPORTANT DIFFERENCES BETWEEN AM AND FM/PM SOLVED EXAMPLES TYPES OF FM INTERNATIONAL REGULATION FOR FREQUENCY MODULATION Performance Comparison of FM and PM Systems Performance Comparison of FM and AM System FM GENERATION PRACTICAL ARMSTRONG METHOD FOR FM GENERATION CHARACTERISTICS OF A WBFM SIGNAL
70 71 72 72 73 74 75 77 78 79 80 82 83 84 84 85 86 87 89 90 105 105 106 106 107 107 107 108 109 109 110 112 112 114 115 117 117 121 125 126 127 128 129 131 134
CONTENT: COMMUNICATION SYSTEMS SOLVED EXAMPLES FOSTER-SEELEY (CENTRE-TUNED) DISCRIMINATOR CONCEPT OF PRE-EMPHASIS AND DE-EMPHASIS PRE-EMPHASIS DE-EMPHASIS PROBLEMS BASED ON GATE/IES/PSUs CHAPTER - 7 : NOISE INTRODUCTION TO NOISE ERRACTIC NOISE MAN MADE NOISE POWER DENSITY SPECTRUM OF SHOT NOISE IN DIODE WHITE NOISE NOISE BANDWIDTH NOISE-TEMPERATURE NOISE-FIGURE FIGURE OF MERIT NOISE IN ANALOG MODULATION NOISE IN FM PROBLEMS BASED ON GATE/IES/PSUs CHAPTER - 8 : SAMPLING THEOREM SAMPLING THEOREM SAMPLING OF BANDPASS SIGNALS PROOF OF SAMPLING THEOREM SOLVED EXAMPLES RECONSTRUCTION FILTER (LOW-PASS FILTER) PROBLEMS BASED ON GATE/IES/PSUs CHAPTER - 9 : DIGITAL COMMUNICATION ADVANTAGES OF DIGITAL COMMUNICATION OVER ANALOG COMMUNICATION PULSE CODE MODULATION (PCM) QUANTIZER WORKING PRINCIPLE OF QUANTIZER BANDWIDTH OF THE PCM SYSTEM DM (DELTA MODULATION) COMPANDING NOISE IN DM (DISADVANTAGES OF DM) CONDITION TO AVOID SLOPE OVERLOAD NOISE DIFFERENTIAL PULSE-CODE MODULATION (DPCM) ADAPTIVE DELTA MODULATION (ADM) S- ARY SYSTEM SOLVED EXAMPLES PROBLEMS BASED ON GATE/IES/PSUs CHAPTER - 10 : DIGITAL COMMUNICATION DIGITAL CARRIER MODULATION PROBABILITY OF ERROR (PE) CHAPTER - 11 : INFORMATION THEORY & CODING INTRODUCTION TO INFORMATION UNIT OF INFORMATION ENTROPY H(X) RATE OF INFORMATION OR INFORMATION RATE (R) SHANNON - HARTLEY LAW
CODING
136 138 139 140 140 142 153 153 153 155 156 158 161 163 164 165 167 168 173 174 175 178 182 184 194 195 196 197 200 201 203 205 206 207 210 212 213 219 227 217 230 230 231 232 233 235
CONTENT: COMMUNICATION SYSTEMS CODING EFFICIENCY SHANNON-FANO CODING HUFFMAN CODING PROBLEMS BASED ON GATE / PSUS / IES CLASSROOM PRACTICE SHEET PROBLEMS BASED ON RANDOM VARIABLES ANSWER KEY PROBLEMS BASED ON RANDOM VARIABLES ANSWER KEY PROBLEMS BASED ON AMPLITUDE MODULATION ANSWER KEY PROBLEMS BASED ON FREQUENCY MODULATION ANSWER KEY PROBLEMS BASED ON QUANTIZATION , PCM, DPCM ANSWER KEY PROBLEMS BASED ON SAMPLING THEOREM, FILTERS, CHANNEL CODING, PLL ANSWER KEY PROBLEMS BASED ON MATCHED FILTER RECIEVER, BANDWIDTH, PROBABILITY OF ERROR, TDMA, FDMA, CDMA, GSM ANSWER KEY PROBLEMS BASED ON DIGITAL MODULATION TECHNIQUES ANSWER KEY PROBLEMS BASED ON INFORMATION THEORY & NOISE ANSWER KEY
235 236 239 241 245 246 247 254 255 267 268 282 283 292 293 296 297 300 301 305 306 313
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. 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.
physical message
(300-3.5 kHz) Voice signal) Trans(20-20 kHz) mitter
medium
Receiver
PERSONAL REMARK :
physical message
(Audio signal)
Information Source Voice/Speech : Bulk of communication TV : Transmission of Pictures Data : Between Computers
,
Kind of communications system which we want to design will depends upon the type of information source which we want to transmit
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.
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. ,
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.
PERSONAL REMARK :
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. ,
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.
PERSONAL REMARK :
S Therefore, SNR = S/N. In dB, SNR = 10 log10 0 N0 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
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Red
Orange
Yellow
Green
Blue
Indigo
Violet
Ultraviolet
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. ,
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
PERSONAL REMARK :
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 1016 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
| x(t) | dt 0
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. 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.
PERSONAL REMARK :
Then x(t) can be expanded into terms of various possible fourier series n
,
Fourier series of x(t) = a 0
(a n cosn 0 t b n sinn 0 t)
n
1 where , T = , a0 = T
t0
2 x(t) dt , an T
t0 T
x(t) cos ω
0
t dt
t0
t0 T
& bn = T ,
t0 T
x(t) sin n ω 0 t dt t0
Trigonometric fourier series may also be represented by
C n cos (n ω0 t n )
f(t) = C0 +
n 1
Where, C0 = a0 and Cn =
a 2n b 2n
–1 b and n tan n an
The coefficient Cn are called spectral Amplitudes i.e. Cn is the amplitudes of spectral components Cn 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) =
Fn e
jn 0 t
n –
where,
1 Fn = T0
t0 T
f(t) e
– jn 0 t
dt
t0
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 e jn 0 t 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 –n0t i.e. e –jn 0 t = cosn 0t – j sin n 0t
e jn LUCKNOW 0522-6563566
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0t
and
= cosn 0t + j sin n 0t
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. FOURIER TRANSFORM ,
PERSONAL REMARK :
Fourier transform is the extension of the fourier series to the general –i2 f t
x(t)e
class of signals (periodic and non peroidic) X (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 f1(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
–
,
n | H() | 1 2
d
If H() does not satisfy this condition, it is unrealizable. IMPULSE SIGNAL (DIRAC DELTA FUNCTION) (t) 1
at t 0 (t) = 0 other wise t 0 unit impulse signal (t) = o other were
Properties of Impulse function x(t) (t) = x (0) (t)
} Product property
x(t) (t – ) = x() (t – )
x(t) (t) dt x(0) – t Shifting Property – LUCKNOW LUCK NOW 0522-6563566
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. PERSONAL REMARK :
x(t) (t – ) dt x() –
(t) dt –
Ex.1
–
Sol.
and (dt) =
(t) }scaling property ||
3t dt is 2
(t) cos
(a)
1
[GATE-EC-2001]
(b)
–1
(c)
0
(d)
1
Ex.2 Convolution of x (t + 5) with (t – 7) is equal to (a) Sol.
x(t – 12)
(b) x (t + 12)
(c) x (t – 2) (d) x (t + 2)
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)
–T T t A for x(t) 2 2 0 otherwise
A
T/2
0
+T/2
t A rect = T 0
–T T t 2 2 otherwise
for
SAMPLING / INTERPOLATING / SINC FUNCTION ,
The function
sin x is the "sine over argument" function and it is x
denoted by "sinc(x). It is also known as "filtering function"
1
–3 –2
,
Sinc (x) or sa(x)
– 0
2 3
x
Fourier transform of rectangular pulse
–T T t A for F. T. of x(t) = 2 2 0 other were LUCKNOW 0522-6563566
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.
i.e.
X() =
x(t) e
Ae
dt
–
or X() =
PERSONAL REMARK :
T/2 – j t
– j t
dt
– T/2
– A – j t e jω
T/2 – T/2
=
2A ω
e j T/2 – e –j T/2 2A ωT sin 2j 2 ω
ωT or X() = AT sinc 2 |X()| AT
– 6 T
2 T
–2 T
– 4 T
4 T
6 T
Energy Spectrum (for Non periodic signal) / Parseval's theorem for Energy Signals
1 | X (ω)|2 dω = Ex = 2π – ,
–
2
2
|X (f)| df = 2 |X (f)| df = 0
| x (t) |
2
dt
–
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.
1 1 | X (ω)|2 dω = ESDT = 2π – 2π
ψ (ω) dω
–
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
1 as , P = T LUCKNOW 0522-6563566
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T/2
2
|x (t)| dt where , x(t) =
n –
–T/2
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. ,
Parseval's theorem for Power signals
1 T ,
T/2
PERSONAL REMARK :
n
|x (t)|2 dt
|Cn |2
n –
–T/2
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 x1(t) and x2(t) is defined as R12 ()
,
Lim T
1 T
T/2
x1(t) x2 (t – τ) dτ
–T/2
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 ( Lim
T
,
1 T
T/2
x(t) x (t – ) d – T/2
The autocorrelation function is defined separately for energy signals and for the power signals.
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. PROBLEMS BASED ON GATE/IES/PSUs
PERSONAL REMARK :
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) 1.
Let the transfer function of a network be H(f) =|H(f)|ej(f)=2e–j4f. 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) 2.
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) 3.
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 .
B.
t
2.
C. Speech Signal
3.
t
D.
f
f=0
f
f=0
f
f=0
4.
f
f=0
Codes. A
B
C
D
(a) 1
3
2
4
(c) 2
3
1
4
A
B
C
D
(b)
2
4
1
3
(d)
1
4
2
3
Sol.(c)
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. CHAPTER-2:RANDOM VARIABLES & RANDOM PROCESS RANDOM SIGNALS
PERSONAL REMARK :
Conditional Probability P(B/A) denotes the Probability of event B when it is known that event A has already occurred. i.e. and
P(A/B) =
P(A B) P(B)
.... (I)
and
P(A B) P(A)
P(B/ A)
.... (II)
Bayes Rule By using Bayes rule one conditional probability can be expressed in terms of the reversed conditional probability. P (A / B)
and
P (A )
. P (B / A )
Bayes, theorem P (B ) P (A / B) . P ( A / B ) P (A )
P (B )
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 Random Variable
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HH
HT
TH
TT
2
1
1
0
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. A random variable is a function X() with domain s and range (, )
PERSONAL REMARK :
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 X1 and X2 are random variable and c is a constant then c x1, x1 + x2, x1x2 are also random variable If x is a random variable then
1 , Where x
X + (ω) = maximum
X () min imum 0, X()
1 (w ) , if X() = 0 x
0,
X(ω) ,
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...xn... then the function p(x) defined as: p(x xi ) px (x) 0
if x xi
if x xi ;i 1, 2..
is called the probability mass function. The set of ordered pairs x i , p( x i ) ; i 1, 2, 3,...n... or
x1 , p1 , x 2 , p 2 ,.....x n , p n ..... , specifies the probability distribution of the random variable X. Discrete Distribution Function
pi 0 ,
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p , such that i
i
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F( x )
p
i
i:x x1
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.
px1 px x i Fx1 Fx i 1 , where F is the distribution function of X
Cummulative Distributive Function (cdf) for a discrete random variable X can be defined as Fx ( x ) P ( X x )
f (u ) ,
PERSONAL REMARK :
x
ux
If X can take on the values x1, x2, x3, ...... xn then the distribution function is given by x x1 0 x1 x x 2 f (x1) Fx ( x ) f (x ) f (x ) x 2 x x3 2 1 f (x ) f (x ) f (x ) .....f (x ) x x 2 3 n n 1
Domain of Fx(x) is , and its range is [0,1]. Properties of F(x) Fx ( x ) 0 Fx () 1 Fx ( ) 0 FX(x) is a non-decreasing function, i.e., monotonically increasing function FX ( x1 ) FX ( x 2 ) for x1 x 2
PROBABILITY DENSITY FUNCTION
fx (x)dx
x
x+dx
X
Consider the small interval (x, x + dx) of length dx around the point x. Let fx (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). Px x x dx f x x dx
or f x x lim
x 0
Px x x dx 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
P x c 0 c . which is not possible in discrete case.
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.
Probability Density Function (PDF) for a continuous random variable is defined as f X ( x)
d FX ( x ) dx
The pdf [i.e. fX (x)] is the first derivative of the probability distribution function FX(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 FX(x) is a continuous function
PERSONAL REMARK : 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) DRDO-EC-2008)
x
FX ( x ) P X x
f X ( x ) dx
However, P X x
fX (x)dx
x
x2
f
x
( x ) dx Fx ( x 2 ) Fx ( x1 ) P ( x1 X x 2 )
x1
Properties of P.D.F. fX (x)
PDF is non-negative function
Area under the pdf curve is unity
f X ( x ) dx 1
The probability of X lying between a and b is given by b
P (a x b )
f
X (x)
....(A)
dx
a
For a continuous case, the probability of x being equal to any particular value is zero. Hence equation (A) can be written as
P( a x b ) P( a x b ) P( a x b ) P( a x b )
Let fx(x) or f(x) be the pdf of a random variable X, where X is defined from a to b. Then b
Arithmetic Mean =
x f (x ) dx a
b
1
x f (x ) dx
Harmonic Mean =
a
b
Geometric Mean =
log x f (x) dx a
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. PERSONAL REMARK :
b
x
r (about origin ) =
r
f ( x ) dx
a
b
r (about the point x A)
r
x A f (x) dx a
b
r (about mean )
r
x mean f ( x ) dx a
Median : It is the point which divide the entire distribution into two equal parts. M
b
f x dx f x dx
a
M
1 2
Mean deviation Mean deviation about mean b
M.D = x mean f x dx
a
Mean deviation about any point A b
M.D about ‘A’ x A f x dx
a
Q
1 i Q Quartilies : 1 f x dx , i 1, 2, 3, 4 4 a
Di
Deciles : D i f ( x ) dx a
i , i 1, 2..., 8, 9 10
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 R 2 R R is called two dimensional random variable. MARGINAL PROBABILITY FUNCTION m
p x x
p x , y x, y
i
i
j1
n
p x y
p x , y x, y
i
i
j1
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. Two-dimensional distribution function
PERSONAL REMARK :
Fxy x,y P X x, Y y
Marginal Distribution Function Fx x P X x, Y Fxy x, (in discrete case) x
dx fXY x, y dy (in continous case)
FY y P X , Y y Fxy , y y
dy fXY x, y dx
Marginal Density Function
fx x
p x, y for XY
discrete case
y
f xy
f (X) x
XY
dy(for continous case)
fy (y)
p
XY
x
(xy) fXY x, ydx
Condition for independence Two random variables are independent if and only if
fXY x, yfX xfY y FXY x, y FX xFY y
Two statistical averages that are most commonly used for characterizing a random variable, X are its mean ( x ) and variance 2x . Expectation of a Random Variable It is the average value of a random phenemenon.For random variable X, expectation is defined as EX
x f (x)
(for discrete random variable)
x
EX x f (x ) dx (for continous random variable)
Expectation value of a random variable g(x) is defined as LUCKNOW 0522-6563566
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. Eg x
g x f x
PERSONAL REMARK :
x
Eg x g ( x ) f ( x ) dx
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) b f ( x ) dx b f ( x ) dx b
If g(x) is non-linear
1 1 E X 2 E ( x ) 2
E 1 1 E ( x ), x
Elog x log E( x ),
If X and Y are independent random variables, then
E X 2 E(x ) 2
E h x . k Y E h x E k Y
Variance: Variance of a random variable X with mean x is defined
as E X x 2 E X 2 x 2 2 x X
or E X 2 x 2 2 x 2 E X 2 2x Properties of Variance V aX b a 2 Vx
If b = 0, then V(ax) = a2V(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. V X1 X 2 V X1 V X 2 2 Cov X1 , X 2
If X1 and X 2 are independent V X1 X 2 V X1 V X 2
Covariance Covariance between random variable X and Y is defined as Cov (X, Y) EX EX Y EY Cov(X, Y) EX, Y EX EY LUCKNOW 0522-6563566
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG. 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)
XX YY 1 Cov X, Y Cov , x y x y
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 ) .
PERSONAL REMARK :
Some Commonly Occurring PDFS (i)
1
Uniform pdf : fX ( x ) b a ,
x (a, b )
f x(x)
1 (b a )
a
(ii)
b
t
Gaussian or Normal pdf : A random variable X is called normal or Gaussian pdf if its form like. fx(x)
1 2x
x
f X x
1 2 . x
x
x x 2
,e
22 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. x fX ( x ) e x
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1 x 2 2 x
; x0
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CHAPTER-1 (BASICS OF COMMUNICATION SYSTEMS) : COMMUNICATION ENGG.
Gaussian or normal pdf occurs in so many application because of remarkable phenomenon called CENTRAL 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
FX (x) P(x a)
a
or
1 e 2 x
( x x )2 2 2x
x x FX (x) P(x a) 1 Q x
a
or FX ( x ) P ( X x)
1 1 2 2 x
x
e
PERSONAL REMARK :
x x Q x
dx
( x x )2
1 2 x
.e
2 2x
.dx
( x x ) 2 2 2x
. dx
o
If we assume Z be the standarized random variable corresponding to X. Thus if Z
Hence, f Z ( z)
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x x x
1 2
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. Then mean of Z is zero and its variance is 1.
.
z2 e 2
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION 1.
PROBLEMS BASED ON GATE/IES/PSUs The PDF of a Gaussian random variable X is given by
Px (x)
1 3 2π
PERSONAL REMARK :
(x 4)2 e 18 .The probability of the event { X = 4} is
(GATE-EC-2001) (a)
1 2
1
(b)
(c)
0
(d)
3 2π Sol.(c) pdf of the gaussian distribution function is given by ( x 4) 1 e 18 Px(x) = 3 2
1 4
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
σ 2x of a stationary zero-mean discrete-time signal x(n), then the normalized autocorrelation function R xx (k) | 2 x at k 1 is (GATE-EC-2002) (a) 0.95 (b) 0.90 (c) 0.10 (d) 0.05
`
Sol.(a) 2d =E[x2 (n)]=E[x[n] x(n 1)]2
2 d =E[x 2 (n)]+E[x 2 (n 1)] 2E[x(n)x(n 1)] 2 d = 2 x + 2 x 2R xx (1) or 19 2 x or 2Rxx(1) = 10
or
2x =2 2 x 2R xx (1) 10
R xx (1) 19 = =0.952 2 x 20
Common Data for Questions 3 and 4. Let X be the Gaussian random variable obtained by sampling the
y2 1 α) e 2 dy .Autocorelation process at t = ti and let Q() 2π α
0.2 τ 1 and mean = 0 function R xx (τ) 4 e 3.
The probability that x 1 is :
(a)
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1 – Q (0.5)
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(GATE-EC-2003)
1 1 (d) 1 – Q 2 2 2 2
(b) Q(0.5) (c) Q
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION Sol.(d) The pdf for Gaussian random variable is p x (x)=
1 x 2
e (x x )
2
/ 2
For zero mean, p x ( x )
PERSONAL REMARK :
2 x
1 x
2
ex
2
/ 2 x2
2 x R xx (0) 8 or x 2 2 or P[x 1] 1 P(x 1)
P[x 1] 1 p x (x)dx or P[x 1] 1 1
or P[x 1] 1
1 2 2
or P[x 1] 1
4.
1
1 2
e 2
x 2 /16
dx , Put
1
1 2 2
1 x 2 x 2 2
e x
2
y or
/ 2 2x
dx
dx 2 2
dy
2 1 e y / 2 dy or P[x 1] 1 Q 2 2
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 W E[W 2 ] E[Y Z]2 or 2 W E[Y 2 ] E[Z2 ] 2E[YZ]
2 W 2 Y 2 z 2R YZ ( ) Here, t = 2, since Y sampled at t = 2 and Z sampled at t = 4 2 W 8 8 2 4(e 0.4 1) or 2 W 2.64
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION Mean and Variance of the Sum of Random Variables
PERSONAL REMARK :
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
z x y
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
2z ( x y) 2
2
(x y)
f(x, y) dx dy
x 2 f ( x ) dx . f ( y )dy
2
or
2
x y 2 x y
y. f ( y ) dy
(
f ( x ) dx
or
2z
2x
2y
f ( x )dx
2
x f ( x ) dx
2z
y 2 f ( y )dy
f( y) dy 1)
2 x y
Special Case : If either x or y or both are zero, then resultant variance becomes , 2z 2x 2y
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 y Z X independently of the value of X i.e. for x . This probability is FZ ( z) P ( Z z ) P X , Y z X Y
Region where Y Z–X
Y=Z– X
X
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION PERSONAL REMARK :
zx
or FZ ( z )
dx
f x, y dy
.....(A)
and the probability density function of Z is given as
d FZ (z) FZ (z) f (x, z x).dx dz
.....(B)
When X and Y are independent, f ( x, y ) fx . fy and equation (B) may be
written FZ ( 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 fXY (x, y) is given, and given Z = g (X,Y) and W = h (X, Y). Then fZ W (z,w) can be determined as, n
FZ, W ( z, w )
F
xy ( xi, yi )
| Ji |
i1
where, Ji is the Jacobian of the transformation defined as. xi z Ji = Ji yi z
xi w yi w
Theorem 2 :To Determine fY (y) when f X ( x ) is given.Solve the equation y = g ( x ), and find its real roots say x1, x2..... xk. we have Y g(x1) g(x 2 ) ........... g(xk ) k
Then f Y ( y ) k 1
fX ( xk ) | g' ( x k ) |
SOLVED EXAMPLES Example 1 : Given Y = 2 X + 3. If random variable X is uniformly distributed over [– 1, 2] find fY( y ). fX(x) 1/3
–1
x
2
1 1 x 2 f x 3 Solution : We have x 0 otherwise y g ( x ) 2x 3
....(1)
and g' ( x ) 2
The range of y is LUCKNOW 0522-6563566
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION y1 = 2 x (–1) + 3 = 1 to y2 = 2 x 2 + 3 = 7.
PERSONAL REMARK :
y3 x1 2
Equation (1) has a single solution i.e.
1 y7 fY (y) 6 0 otherwise
f ( x ) 1/ 3 1 So f Y ( y ) X 1 or | g' ( x 1 ) | 2 6
Example 2 : Let Y = sin x, where X is uniformly distributed with 1 0 x 2 , find fY (y) fX (x) = 2 0 otherwise Solution : y g(x) sin x
From equation (A) it is clear that for
| y | 1,
....(A) the equation, y sin x has
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
x1
–1
i.e., x1 sin 1 y
x2
x
and x2 x1 sin1 y
g' ( x1) cos x1 cos ( sin1 y ) 1 y 2 g' ( x 2 ) cos x 2 cos ( x1 ) cos x1 1 y 2
fY(y) =
fx(xK) f (x) f (x ) = x 1 + x 2 +....... | g' (xK) | | g' (x1 ) | | g'(x2) | 1 2π
or f Y (y)
1/ 2π
+ 2
=
2
1 y
1 y
1 π 1 y2
, | y | 0, then equation y = x2 has two solution i.e. x y or x1 y , x 2 y Since, X = N ( 0 : 1 ) given means 0 mean and 1- variance. fX ( x )
Now
or
e
2
f Y (y)=
fY ( y ) =
LUCKNOW 0522-6563566
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1
x2 2
i.e., f X ( x ) is an even function.
f x (x k ) f (x ) f (x ) = x 1 + x 2 g'(x k ) | g'(x1 ) | | g'(x 2 ) |
fx
y + f y
2
x
y
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2
y
y>0
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION PERSONAL REMARK : y y 1 1 e 2 e 2 2 or f Y ( y ) 2 2 y 2 y 1 y / 2 or fY ( y ) .e .u ( y ) 2y
u ( y)
Example 4 : Consider X has a uniform probability density function given 1 fX (x) = 2 0
0 x 2 otherwise
Determine x, E [ X2 ], x 2
x
Solution :
x.fX (x)dx
0
2
2
E [x ]
2
x.
x .f (x) dx X
0
1 2
1 x2 2 2
1 x3 x . .dx 2 2 3 2
1
4 2 ( )2 3
x E [ x 2 ] 2x
.dx
2
0
2
0
1 4 2 . 2 2
4 2 3
2 Ans. 3 3
Example 5 : Given a random process X ( t ) = A ( t – ) where
cos
is a random variable, and A and are deterministic. 1
Assume a uniform distribution f () = 2 [0,2], find x and 2x .
Solution: x E[X(t)]
x(t)fX (x)dx
A 2
A cos ( t ) . f ( ) d
2
cos ( t ) d
with as a random variable
0
2
A sin (t A sin ( t ) 02 2 1 2 0
A sin ( 2 t ) sin ( 0 t ) A sin t sin t 0 2 2
2x E [X (t) x ]2 E [X2] 2x [ A cos (t ) ]2 f ( ) d 0.2
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LUCKNOW
A2 2
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2 2
cos 0
2 2 ( t ) d A . 2 A 2 2 2
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TECHGURU CLASSES for ENGINEERS (Your Dedication + Our Guidance = Sure Success)
CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION 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 S we assign a real valued function of time X(t, ). This real valued function of time is called Random Process.
PERSONAL REMARK :
For fixed say 1 , we have a function of time X(t, i ) x i (t) called sample function. Set of sample function is called ensemble For fixed t say t j , X(t j , ) X j called a number
x1(t)
x1(t1)
1
t1 t2
x1(t2) t 3 x1(t3)
2 2
2 3
t
x2 (t)
2 outcomes
n
2 1
t1 t 2
t
t3
n
t1
t2
t3 t
n 1
n 3
x1(t), x2(t) .........xn(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 )] & the set of autocorrelations
i = 1, .........n
R xx (t i , t j ) E[X(t i )X(t j )] i, j = 1 ......n
If the set of random variables X(ti) i = 1 , .....n is unocorrelated i.e. Cij = 0 then X(ti) are independent.
If a gaussian process X(t) is WSS then X(t) is SSS.
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION
B.
If a gaussian process X(t) of a linear system is gaussian then the output process Y(t) is also gaussian. White Noise S
XX
S xx ( )
2
(t) 2 Mean of white noise is zero. Band-limited white noise
2
R xx ()
C.
PERSONAL REMARK :
Sxx ( ) 2 0
| | B | | B
R XX ( )
B sin B 1 B j e d 2 B 2 2 B
Sxx () RXX
n 2
–
D.
B
B
–
B
0 B
Figure : Band Limited white noise Narrowband Random Process A WSS process x(t) with zero mean & its PSD Sxx() 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.
– c
–
–
X(t)=V(t)cos[ct +(t)] S XX
0
+c
V(t) = envelop function , (t) = Phase function x(t) V(t)cos (t) cos c t V(t)sin (t)sin c t
= X c (t) cos c t X s (t) sin c t Xc (t) V(t)cos (t) (in-phase component) X s (t) V(t)sin (t)
(quadrature component)
X (t) V(t) X 2c (t) X 2s (t), (t) tan 1 s Xc (t) LUCKNOW 0522-6563566
LUCKNOW
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TECHGURU CLASSES for ENGINEERS (Your Dedication + Our Guidance = Sure Success)
CHAPTER-2 (RANDOM VARIABLES ) : COMMUNICATION ENGG. 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.
PERSONAL REMARK :
Random variable wide sense stationary (WSS) Strict Sense Stationary (SSS) Ergodic
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 x( t ) Lim
T
1 2T
T
x ( t ) dt E [ x ( 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. i.e.
X(t) x(t)
1 E x(t) = Lim T T
T/2
x(t)dt x
T / 2
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. i.e.
E X ( t1) E X ( t 2 ) ....... E X( t )
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 t2 and LUCKNOW 0522-6563566
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CHAPTER-2 (RANDOM VARIABLES ) : COMMUNICATION ENGG. depends upon the time difference i.e. ( t 2 t1 ) only i.e.
PERSONAL REMARK :
x (t) x Constant
2x (t) 2x 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 )
Rxx () ....(A)
Special Case (i)
(ii)
By setting 0, equation (A) becomes E[X2(t)] , thus the average power of a WSS prosess is independent of time and equals to Rxx(0). 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 R xx ( ) E X ( t ) , X ( t ) Lim
T
1 T
T/2
x ( t ). x ( t ) dt
T / 2
Properties of R xx ()
R xx ( ) R xx ( ) for real signal
RXX(t ) = R*XX(-t ) for complex signal
| R xx ( ) R xx (0) i.e.Rxx(0) is the maximum value of R xx ( )
and
occurs at the origin.
R xx (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) = Lim
T
Note :
LUCKNOW 0522-6563566
LUCKNOW
1 T
+T / 2
x(t) * y(t τ) dt
T /2
The conjugate symbol * is removed if the functions are real.
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CHAPTER-2 (RANDOM VARIABLES ) : COMMUNICATION ENGG.
If the correlation is defined for energy signal, then
RXY()
PERSONAL REMARK :
x(t).y(t )d
x(t ).y*(t)d
Properties of R xy ( )
RXY () RXX()
| RXY ()| RXX(0). RYY(0)
| RXY ()|
1 2
R
XX
(0) RYY(0)
Autocovariance, CXX () RXX() 2X
Cross covariance, CXY () 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, CXY () 0 i.e., RXY() X Y i.e., cross correlation functions RXY () 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.,
S xx ( )
R
xx ( ) . e
j
d
R xx ( ) from the given signal can be calculated as T 2
1 x t .x t .dt T T T 2
R xx Lim
1 and R xx ( ) 2
S
xx
(). e j . d
Properties of S xx ()
SXX () is real i.e. SXX () 0 for all .
SXX () SXX () i.e. Power spectral density of a random
process X(t) is an even function of frequency .
LUCKNOW 0522-6563566
LUCKNOW
SXY () SYX() S*YX () GORAKHPUR 9919526958
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CHAPTER-2 (RANDOM VARIABLES ) : COMMUNICATION ENGG.
Relation between input and output spectral densities
PERSONAL REMARK :
2
SYY () |H () | . SXX()
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
E
1 X().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). P1,2 of the combined function is given by. T 2
P1, 2
2 1 f1 t f 2 t dt T T 2
T2
T2
1 1 2 2 2 f1 t dt f 2 t dt T T 2 T T 2 T
T 2
f t .f t dt 1
2
T 2
or P1, 2 P1 P2 2R 1, 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 crosscorrelation between them. If the functions f1 (t) and f2 (t) are uncorrelated, i.e., R1, 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.
Note : R XX ( ) some times can be written as RX (t ) LUCKNOW 0522-6563566
LUCKNOW
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION Transmission of Random Process through Linear Systems A.
PERSONAL REMARK :
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) LTI System 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
h()E[X(t )]d
h( )
X
(t )d h(t) * x (t)
For wide sense stationary random process, E[X(t )] x
E[Y(t)]
h() d h()d H(0) x
x
x
Thus for WSS, Y(t) is constant H(0) is the frequency response of filter at = 0 RYY(t1,t2) = E[Y(t1)Y(t2)] E h()X(t 1 )h()X(t 2 )dd
h()h()E[X(t
1
)X(t 2 )]d d
h()h()R
XX
(t1 , t 2 )dd
XX
(t 2 t1 )dd
For WSS
R YY ( )
h( )h()R
C.
Power spectral Density of the output
SYY () | H() |2 SXX ()
R YY ( )
1 2 j | H() | SXX ()e d 2
E[Y 2 (t)] R YY (0) LUCKNOW LUCKNOW
0522-6563566
GORAKHPUR 9919526958
1 2 | H() | SXX ()d 2
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION SOLVED EXAMPLES Ex.1 X(t) = Acos(t + ), where A &are constants & is a uniform random variable over [ ] show that X(t) is WSS.
PERSONAL REMARK :
1 , , f () 2 0 otherwise
Sol.
WSS expectation is constant
E[X(t)] =
A cos(t )f
()d
A A Acos(t )f ()d sin(t ) 2 2
A A sin(t ) sin(t ) [sin(t) sin t] 0 2 2
R XX
( ) E[X(t)X(t )]
A cos(t )Acos(t )f
()d
A2 cos(t )cos(t )d 2
A2 cos(2t 2 )d cos d 4
A2 A2 | sin(2t 2 | cos 2 cos 4 2
Ex.2 Consider a random process X(t) = Acos(t + ) where & are constants & A is a random variable. Determine whether X(t) is WSS. Sol.
E[X(t)] E[A cos(t )] cos(t )E[A] A cos(t ) = constant
R XX ( ) E[A 2 cos(t )cos(t )] A2 E cos(2 t 2 ) cos 2
1 cos(2t 2) cos cos E[A2 ] 2
Comment : X(t) is not WSS. Ex.3 X(t) A cos t Bsin t ,where is constant & A & B are random variables. (a) Show that the condition E[A] = E[B] = 0 is necessary for X(t) to be stationary. LUCKNOW LUCKNOW
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION (b) Show that X(t) is WSS if the random variables A&B are uncorrelated with equal variable i.e.
PERSONAL REMARK :
Sol. (a) X(t) to be stationary E[X(t)] = constant E[Acost + B sint] = constant cos t E[A] + sin t E[B] = constant For E[A] = E[B] = 0, E[X(t)] = 0 = constant (b) R XX (t1 , t 2 ) E[X(t1 )X(t 2 )] = [(Acost1+Bsint1)(Acost2+Bsint2)] 2
=E[A cost1cost2+ABcost1sint2+ABsint1cost2+B2sint1sint2] = cost1cost2E[A2]+cost1sint2E[AB] + sint1cost2 E[AB] + sint1 sint2E[B2 ] If E[AB] = 0 and E[A2] = E[B2] = 2 RXX (t1,t2) = 2cost1cost2 +2sint1sint2 RXX(t1, t2) = 2 cos(t1 t2) RXX(t1, t2) = 2cos = RXX() Ex.4 A random process X(t) is said to be covariance stationary if the covariance of X(t) depends only on the time difference Z = t1 t2 i.e. CXX(t. t + ) = CXX() For X(t) = (A+1)cost + Bsint, show that X(t) is not WSS but is covariance stationary where A & B are independent random variables for which E[X(t)] = E [(A+1)cost + B sint] E[A] = E[B] = 0 & E [A2] = E [B2] = 1 Sol. E[X(t)] = costE [A + 1] + sin t E[B] = cos t which is not constant CXX(t, t + ) = E[X(t) X(t +)] E [X(t)] E[X(t + )] = E[{(A+1)cost + B sint}{(A+1) cos(t + )+ B sin (t +)] E[(A + 1) cost + B sint]E[(A+1)cos(t+) + B sin (t + )] = E [(A+1)2 cos t cos (t +) + (A +1) B cost sin(t + ) + (A +1) B sin t cos (t + ) + B2 sin (t +) sin t] cost cos (t + ) = cos t cos(t + )E[A2 +2A +1] + E (AB + B)cos t sin(t + + E[AB+B]sin t cos(t ++ sin (t + ) sin t E [B2] cost cos (t + ) = 2 costcos(t +) + sin (t +) sint cos t cos (t + ) = cos (t + )cost + sin (t +) sin t = cos (t + t) = cos Ex.5 Show that X(t) = Acos (t +) is ergodic in both the mean & the autocorrelation T/2
Sol. < X(t) Tlim
LUCKNOW LUCKNOW
0522-6563566
1 A cos(t )dt T T/ 2
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION PERSONAL REMARK :
T/2
A sin(t ) lim T T T / 2
T T sin sin 2 A 2 lim T T
lim
A sin( ) sin( ) T
lim
A sin sin 0 T
T
T
T /2
< X(t) X(t +)> Tlim
1 A 2 cos(t )cos(t )dt T T / 2
T /2 A2 T / 2 A2 lim cos(2 t 2 )dt cos dt T 2T 2T T / 2 T / 2
=
A2 cos 2
We have proved already E[X(t)] = 0 RXX ()
A2 cos 2
Ex.6 If X(t) is WSS then E[{ X(t + )] X(t)2}] = 2[RXX(0) RXX()]
where RXX() is the autocorrelation of X(t) Sol.
E[X{(t + ) X(t)}2] = E[X2(t + ) 2 X(t +)X(t) + X2(t)] = E[X2 (t + )] 2 E[X(t + )X(t)] + E[X2(t)] = RXX (0) 2 RXX() + RXX(0) = 2 [RXX(0) RXX ()]
Ex.7 For WSS |RXX ()| < RXX (0) E [{X(t) + X(t + )2] > 0 Sol.
E[X2 (t)] + 2E [X(t) X(t + )]+ E[X2(t + )] > 0 RXX(0) + 2 RXX () + RXX (0) > 0 2 RXX () + 2RXX () > 0 RXX () + RXX () > 0 RXX(0) > |RXX()|
LUCKNOW LUCKNOW
0522-6563566
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TECHGURU CLASSES for ENGINEERS (Your Dedication + Our Guidance = Sure Success)
CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION Ex.8 Show that power spectrum of a real random process x(t) is real SXX(-) = SXX ()
PERSONAL REMARK :
Sol.
SXX() =
R
XX
( )e j dt
or SXX()
R
XX
( )(cos jsin )d
or SXX ( )
R XX ( ) cos d j R XX ( )sin d
and SXX ( )
R XX ( ) cos d j R XX ( )sin d
or real random process
SXX ( )
R
XX
( ) cos d SXX ( )
Ex. 9 A class of modulated random signal Y(t) is defined by Y(t) = AX(t) cos (ct + ) where X(t) is the random message signal & A cos (ct + ) is the carrier. The random message signal X(t) is a zero mean stationary random process with autocorelation RXX() & power spectrum SXX(). The carrier amplitude A & frequency c are constants & phase is a random variable uniformly dirtributed over [0, 2]. Assuming X(t) & are independent , find mean, autocorelation &power spectrum of Y(t). Sol. E[Y(t)] = E[AX(t) cos( ct +)] = AE [X(t)]E[cos(ct + )] E[X(t)] = 0, E [Y(t)] = 0 RYY (t1, t2) = E [Y(t1)Y(t2)] = E[AX(t1)cos(ct1+) AX (t2) cos (ct2 +)] = A2 E[X(t1)X(t2)cos(ct1+)cos (ct2 +)]
A2 E[X(t1 )X(t 2 )]E[cos{ c (t1 t 2 ) 2}cos c (t 1 t 2 )] 2
A2 E[X(t1 )X(t 2 )]{E[cos{ c (t1 t 2 ) 2}] E[cos c ]} 2
A2 R XX ( ) cos c 2 2
E[cos c {(t1 t 2 ) 2}]
LUCKNOW LUCKNOW
0522-6563566
GORAKHPUR 9919526958
1 cos{ c (t1 t 2 ) 2}d 2 0
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TECHGURU CLASSES for ENGINEERS (Your Dedication + Our Guidance = Sure Success)
CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION 2
PERSONAL REMARK :
1 sin c (t1 t 2 ) 2 0 2 2 0
F.T. SYY RYY ()
SYY
A2 A2 j R ( )cos e d F{R XX ( ) cos c ) c 2 XX 2
or SYY
A2 A2 F{R XX ( )e j } F{R XX ( )e c } 22 22
or SYY
A2 [SXX ( c ) SXX ( c ) 4
Ex.10Let X(t) & Y(t) be both zero-mean & WSS random process consider the random process Z(t) defined by Z(t) = X(t) + Y(t) (a) Determine autocorrelation & power spectrum of Z(t) if X(t) & Y(t) are joint WSS. (b) Repeat part (a) if X(t) & Y(t) are orthogonal. (c) Show that if X(t) & Y(t) are orthogonal, then the mean square of Z(t) is equal to the sum of the mean squares of X(t) & Y(t). Sol. RZZ (t, t + ) = E [Z(t) Z(t + )] = E [{X(t) + Y(t)}{X(t + ) + Y (t + )}] = E [X(t)X(t + )] +E[X(t)Y(t + )] + E[Y(t)X(t +)] +E[Y(t)Y(t+)] RZZ() = RXX () + RXY() + RYX () + RYY() SZZ() = SXX() + SXY() + SYX() + SYY() (b) If X(t) & Y(t) are orthogonal E[X(t) Y (t + )] = E [Y(t)X(t + )] = 0 RZZ () = RXX () + RYY() (c) E[Z2(t)] = E[X2(t) + 2X(t)Y(t) + Y2(t)]
= E[X2(t)] + 2E [X(t)Y(t)] + E[Y2(t)] = E[X2(t)] + E [Y2(t)] Ex.11Two random processes X(t) & Y(t) are given by X(t) = A cos (t +), Y(t) = A sin (t + ) where A & are constants & is a uniform random variable over [0, 2]. Find the cross-correlation of X(t) & Y(t). Sol. RXY(t1, t2) = E[X(t1) Y(t2)] = E[Acos(t1 + )] A sin (t2 + )] = A2 E[cos (t1+)sin (t2 + )]
LUCKNOW LUCKNOW
0522-6563566
A2 E[sin(t1 t 2 2) sin(t 1 t 2 )] 2
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION
A2 A2 A2 E[sin(t1 t 2 2) E[sin ] sin 2 2 2
PERSONAL REMARK :
Ex.12: Let X(t) = A cos t + Bsin t and Y(t) = B cost A sin t ,where is constant & A & B are independent random variable both having zero mean & variance 2.Find the crosscorrelation of X(t) & Y(t) Sol. RXY (t, t + ) = E [X(t)Y (t + ] = E[Acost + B sint) (Bcos (t +) A sin(t + )] = E [AB cos tcos(t + ) A2 costsin (t + ) + B2 sint cos (t + ) AB sin tsin(t +)] = cos t cos(t + ) E[AB] cos t sin (t + ) E[A2] + sin cos (t + ) E[B2] sin t (t + ) E[AB] sin t cos(t + ) cost sin (t +) = sin Ex.13: A WSS random process X(t) is applied to the input of an LTI system with impulse response h(t) = 3e 2tu(t). Find the mean value of the output Y(t) of the system if E[X(t)] = 2 Sol. E[Y(t)] E[X(t) * h(t)] E h( )X(t )d
2 h()E[X(t )]d 2 3e u()d = 6 e2 d 6 e2 0
3 0
Ex.14Let X(t) & Y(t) be the WSS random input process & random output process respectively of a quadrature phase - shifting filter.Show that RXX() = RYY(), RXY() = RXX() Sol. For Hilbert-transform, h(t)
1 , H( j) jsgn() t
H ( ) 1
SYY () | H( ) |2 SXX () SXX () so, R YY () R XX () RXY() = h()* RXX() = RXX()
Ex.15A WSS random process X(t) with autocorrelation RXX() = Ae-a|| where A & a are real positive constants is applied to an input of an LTI system with impulse response h(t) = e-btu(t), where b is a positive constant. Find the autocorrection of the output Y(t) of the system.
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TECHGURU CLASSES for ENGINEERS (Your Dedication + Our Guidance = Sure Success)
CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION Sol.
F.T. SXX () RXX ()
SYY ( )
R YY ()
A2a 2 a 2
2
1 1 2 and | H() | j b b2 2
2aA 1 2 2 2 2 2 2 ( a )( b ) b a
PERSONAL REMARK :
2aA a 2 bA 2 2 2 2 ( a ) ( b )
A |a| ae |b| e u(t) (b 2 a 2 ) b
Ex.16 Consider a WSS process X(t) with autocorrelation RXX() and power spectrum SXX(). Let X’(t) = dx(t)/dt. Show that R XX' () d 2 and R X'X' () d
Sol.
X(t)
R XX () d
R XX () dt 2
Differentiation j
Y(t)=dX(t) = X'(T) dt
SXX () | H() | SXX () jSXX () R XX' () d
R XX () d
similarly, SX'X' (ω) =| H(jω)|2 SXX (ω) =ω2SXX(ω) =-(jω)2SXX (ω) so, RX'X' () d2
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION PROBLEMS BASED ON GATE/IES/PSUs
PERSONAL REMARK :
In a communcation system, a process for which statistical averages and time averages are equal, is called. (IES-EE-2003) (a) Stationary (b) Ergodic (c) Gaussian (d) BIBO stable Sol.(b) 1.
Consider the following statements : (IES-EE-2009) The thermal noise power generated by a resistor is proportional to 1. The value of the resistor 2. The absolute temperature 3. The bandwidth over which it is measured 4. The Boltzmann’s constant. Which of the above statements is / are correct? (a) 1,2 & 3 (b) 2 only (c) 2 & 3 only (d) 1, 2, 3 & 4 Sol.(d) 2.
A random process obeys Poisson’s distribution. It is given that the mean of the process is 5. Then the variance of the process is : (IES-EC-2003) (a) 5 (b) 0.5 (c) 25 (d) 0 Sol.(a) The mean and variance of Poisson’s distribution is same. 3.
Match List-I (Type of Random Process) with List-II (Property of the Random Process) and select the correct answer using the code given below the lists: (IES-EC-2005) List-I A. Stationary process B. Ergodic process C. Wide sense stationary process D. Cyclostationary process List-II 1. Statistical averages are periodic in time 2. Statistical averages are independent of time 3. Mean and autocorrelation are independent of time 4. Time averages equal corresponding ensemble average Codes: A B C D A B C D (a) 3 1 2 4 (b) 2 4 3 1 (c) 3 4 2 1 (d) 2 1 3 4 Sol.(b)Stationary process - statistical averages are independent of time. Ergodic process - Time averages equal corresponding ensemble average. Wide sense stationary process - mean and autocorrelation are independent of time. Cyclostationary process - statistical averages are periodic in time. 4.
5.
Match the List-I(Type) with List-II (Application) and select the correct answer using the codes given below the lists: (IES-EC-2006) List-I A. Speech signal B. Non-stationary C. Random signal D. Chaotic signal
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TECHGURU CLASSES for ENGINEERS (Your Dedication + Our Guidance = Sure Success)
CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION List-II 1. The received signal of a radar system monitoring variation in prevalent weather condition 2. One dimensional signal where amplitude varies with time 3. Signals of coupled system of non-linear difference 4. Ensemble of unpredictable waveforms Codes: A B C D A B C D (a) 2 1 4 3 (b) 4 3 2 1 (c) 2 3 4 1 (d) 4 1 2 3 Sol.(c)
PERSONAL REMARK :
Source S1 produces 4 discrete symbols with equal probability. Source S2 produces 6 discrete symbols with equal probability. It H1 and H2 are the entropies of sources S1 and S2 respectively, then which one of the following is correct? (IES-EC-2008) (a) H1 is always less than H2 (b) H1 is always greater than H2 (c) H1 is always equal to H2 (d) H2 is 1.5 times H1 only Sol.(a) Entropy of a source S is given by H = log2n where n = number of equiprobable symbols. H1 = log2 4 = 2 and H2 = log26 = 2.59 6.
The outputs of two noise sources each producing uniformly distributed noise over the range -a to +a are added. What is the p.d.f. of the added noise? (IES-EC-2008) (a) Uniformly distributed over the range -2a to +2a (b) Triangular over the range -2a to +2a (c) Gaussian over the range to (d) None of the above Sol.(b) The random variable Z which is expressed as Z = X + Y has pdf which is convolution of pdfs of individual random variable X and Y. So, fz(z) = fx(x) * fY(y) 7.
8.
Let X and Y be two statistically independent random variables uniformly distributed in the ranges (–1,1) and (–2,1) respectively. Let Z = X + Y. Then the pobability that (z – 2) is (GATE-EC-2003) (a)
zero
(b)
1 6
(c)
1 3
(d)
1 12
Sol.(d) The pdf of random variable X and Y are fx(x) =
1 , 1< x < 1 2
and
fY(y) =
1 , 2 < y < 1 3
P(z 2)=P(X+Y 2) The shaded region satisfies the condition, x y 2, 1 x 1 and 2 y 1 0 2 x
P(z 2)=P(X+Y --2)=
-1 2
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1 1 1 dxdy or P(z 2)= 2 3 12
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TECHGURU CLASSES for ENGINEERS (Your Dedication + Our Guidance = Sure Success)
CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION The distribution function Fx(x) of a random variable X is shown in the figure. The probability that X = 1 is (GATE-EC-2004) (a) zero 1.0 (b) 0.25 (c) 0.55 0.55 (d) 0.30 0.25 Sol.(d)From the given figure, the probability at X = 1 is given by 9.
–2
0
Fx (x) F x (x) = 0.55 0.25 = 0.30 10.
1
PERSONAL REMARK :
3
A random variable X with uniform density in the interval 0 to 1 is quantized as follows: (GATE-EC-2004) If 0 X 0.3 xq = 0 If 0.3 X 1 xq= 0.7 where Xq is quantized value of X. The root mean square value of the quantization noise is (a) 0.573 (b) 0.198 (c) 2.205 (d) 0.266
Sol.(b) Noise - power
2
2 E[(x x q ) ] or x 2
x
(x x
q
) 2 p x (x)dx
The variable is distributed uniformly for interval 0 to 1 0.3
1
2 x dx
(x 0.7)
2 0 x 1 and x
p x (x) 1,
0
0.3
x 2 dx
0
1
1
2 x dx
0.49dx 1.4 xdx
0.3
2
dx
0.3
1
0.3
0.3
1 +0.343 0.567=0.39 3 RMS value of noise -power x 0.39=0.198 11.
An output of a communication channel is a random variable V with the probability density function as shown in the figure. The mean p(v) square value of V is: (GATE-EC-2005) (a) 4 p(V) (b) 6 k (c) 8 (d) 9 0
Sol.(c)From given figure,
v
4
p(v)dv 1
1 0 4 k = 1 2
and
or
k=
1 2
or p(v) =
and p(v) = mv
k .v 4
p(v) =
(m = slope =
k 4
1 v 8 4
1 v4 v Now, mean square value = v p(v)dv = 0 v dv = 8 8 4 0 8
2
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2
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TECHGURU CLASSES for ENGINEERS (Your Dedication + Our Guidance = Sure Success)
CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION Common Data for Questions 12 and 13
PERSONAL REMARK :
Asymmetric three-level midtread quantizer is to be designed assuming equiprobable occurence of all quantization levels. 12.
If the probability density function is divided into three regions as shown in the figure, the value of a in the figure is: (GATE-EC-2005) 1 4 Region 1 1 8 Region 2 –3
(a)
1/3
Region 3
–1
–a
(b)
+a
2/3
+1
+3
(c)
1/2
X
(d) 1/4
p(x)
Sol.(b) We know that
1
f (x)dx 1
1
-3
-1-a
+3
+a+1
x
Since given that the three regions are divided into equiprobable region
1
a
Therefore,
or
f (x)dx 3 a
2a
a
or
a
1 1 = 4 3
1 1 dx 4 3 or
a =
2 3
The quantization noise power for the quantization region between -a and + a in the figure is (GATE-EC-2005) (a) 4/81 (b) 1/9 (c) 5/81 (d) 2/81 Sol.(a)Quantization noise power is given as 13.
x 3 a 1 2 2 = a f (x).x dx = a x dx = 3 a 4 a
a
3 1 a 4 1 2a 3 a3 = = = = 6 6 81 4 3 6
A zero-mean white Gaussian noise is passed through an ideal lowpass filter of bandwidth 10kHz. The output is the uniformly sampled with sampling period ts = 0.03m sec. The samples so obtained would be. (GATE-EC-2006) (a) correlated (b) statistically independent (c) uncorrelated (d) orthogonal Sol.(a) Since sampling frequency fs > 2fm The samples obtained will be look alike so, samples are correlated 14.
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TECHGURU CLASSES for ENGINEERS (Your Dedication + Our Guidance = Sure Success)
CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION A uniformly distributed random variable X with probability density function
15.
PERSONAL REMARK :
1 FX ( x ) (u ( x 5) u ( x 5)) 2 where u(.) is the unit step function is passed through a transformation given in the figure below. The probability density function of the transformed random variable Y would be (GATE-EC-2006) (a)
1 FY ( y ) (u ( y 2.5) u ( y 2.5)) 5
(b)
FY ( y ) 0.5 ( y ) 0.5 ( y 1)
(c)
FY ( y )0.25 ( y 2.5)0.5 ( y2.5)5 ( y )
(d)
FY ( y ) 0.25 ( y 2.5) 0.25 ( y 2.5)
1 10
(u ( y 2.5) u ( y 2.5))
Sol.(a) If E denotes expectation, the variance of a random variable X is given by (GATE-EC-2007) (a) E[X2] – E2[X] (b) E[X2] + E2[X] (c) E[X2 ] (d) E2[X] Sol.(a) Variance of a random variable X is given by E[(X )2] = E[X2 2X + 2] = E[X2] E[2X]+E [] E[(X )2] = E[X2] 2E[X] + 2 E[(X )2] = E[X2] 22 + 2 = E[X2] 2 = E[X2] {E[X]}2 16.
17.
If R ( ) is the autocorrelation function of a real, wide-sense stationary random process, then which of the following is NOT true? (a) R( ) = R ( ) R( ) = R( )
(c)
(b)
R( ) R(0) (GATE-EC-2007)
(d)
The mean square value of the process is R (0) Sol.(c)For real function f(t) autocorrelation is given by
1 T/ 2 1 T/ 2 f (t )f (t)dt and R( ) = T/ 2 f (t )f (t)dt T/ 2 T T Let t = p or dt = dp
R() =
which gives R ( ) = R() i.e. even function. From this result, we conclude that option (c) is wrong. 18. If S(0) is the power spectral density of a real, wide sense stationary random process, then which of the following is ALWAYS true? (GATE-EC-2007)
(a) S(0) >S(f) (b) S(f) 0
(c) S( f) = S(f) (d)
S(f )df 0
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TECHGURU CLASSES for ENGINEERS (Your Dedication + Our Guidance = Sure Success)
CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION PERSONAL REMARK :
| X( j) |2 Sol.(b) Power spectral density, S() = lim
Therefore, for wide-sense stationary random process, power spectral density is greater than or equal to zero, i.e. S(f) 0. Consider two independent random variables X and Y with identical distributions. The variables X and Y take values 0,1 and 2 with
19.
1 1 1 , and respectively. What is the conditional 2 4 4 probability P(X + Y = 2 | X–Y = 0)? (GATE-EC-2009)
probabilities
(a)
0
1
(b)
(c)
16
1
(d)
6
1
Sol.(c) P(X Y 2 | X Y 0) P[X Y 2] (X Y 0)] P(X Y 0)
P(1,1) Since, X & Y are independent P(0, 0) P(1,1) P(2, 2)
Required Probability
20.
P(1)P(1) P(0)P(0) P(1)P(1) P(2)P(2)
1/ 4 1/ 4 1 6 1 1 1 1 1 1 2 2 4 4 4 4
A discrete random variable X takes values from 1 to 5 with probabilities as shown in the table. A student calculates the mean X as 3.5 and her teacher calculates the variance of X as 1.5. Which of the following statements is true? (GATE-EC-2009) K
1
2
3
P(X=k)
0.1
0.2
0.4
(a) Both the student and the teacher are right (b) Both the student and the teacher are wrong (c) The student is wrong but the teacher is right (d) The student is right but the teacher is wrong Sol.(b)Also product of eigen values = det A.
k k k k
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION s = Mean calculated by student = 3.5
PERSONAL REMARK :
2
T = Variance calculated by teacher = 1.5 5
Now, mean =
P
K
k 0.1 0.4 1.2 0.8 0.5 = 3
k 1
5
Variance =
P
K
k 2 2
k 1
2 2 = (0.1 + 0.8 + 3.6 + 3.2 + 2.5) (3) = 10.2 9 = 1.2
Both students and teacher are wrong. 21.
X(t) is a stationary process with the power spectral density Sx(f) > 0 for all f. The process is passed through a system shown below-
d dt
X(t)
Y(t)
Delay = 0.5 ms
Let SY(f) be the power spectral density of Y(t). Which one of the following statements is correct? (GATE-EC-2010) (a) SY(f) > 0 for all f (b) SY(f) = 0 for |f| > 1 kHz (c) SY(f) = 0 for f =n f0, f0 = 2 kHz, n any integer (d) SY(f) = 0 for (2n + 1) f0, f0 = 1 kHz, n any integer Sol.(d)Given PSD input Sx(f)>0 f or = 0.5 ms m(t) = x(t) +x(t ) or M(s) = X(s) [1 +e-s] or Y(s) = sM(s)
Y( j) = j[1 + e-] or H(j) = j[1+cos jsin ] X( j) SY(f) = |H(j)|2 Sx(f) and at f = 0 H(f) = 0 Hence SY(f) 0 for all f at f | > 1 kHz|H(f)|> 0 and SX(f) > 0 Hence SY(f) 0 SY(f) = 2f |[1+cos 2f sin 2f]| f = nf0 f0 = 2 103 kHz SY(f) = 2n 2 103 | [1 + cos 2 n 2
1 1 j sin 2 2n ]| 2 2
SY(f) = 4n103|[1 + cos2n jsin2n]| for n - integer cos2 n= 1 sin2n= 0 and hence SY(f) 0. and SY(f) = 2(2n +1) f0 |[1+cos 2(2n + 1)
1 1 - j sin 2(2n+1) ]| 2 2
for every n cos(2n+1) = - sin2(2n+1) = 0. Hence Sy(f) = 0 for every f. LUCKNOW GORAKHPUR 9919526958 0522-6563566
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TECHGURU CLASSES for ENGINEERS (Your Dedication + Our Guidance = Sure Success)
CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION 22. The power spectral density of a real process X(t) for positive frequencies
PERSONAL REMARK :
2
is shown below. The values of E[X (t)] and | E[X(t)]| , respectively,, 4 Sx () are 400 ( -10 ) (a) 6000/, 0 (GATE-EC-2012) 6 (b) 6400/, 0 (c) 6400/, 20/( 2 ) (d) 6000/, 20/( 2 ) 3 10 11 (10 rad/s) 9 0 Sol.(b) Power spectral density is the Fourier Transform of autocorelation function. or R xx ( )
F
R xx ( ) S xx ( )
Also, Rxx(t ) = E [X(t)X (t + t )]
1 Sxx ()e j d 2
or Rxx (0) = E [X2(t)]
1 Sxx ()d 2
For real two-sided random process R xx (0)
1 1 Sxx ( )d Sxx ( )d 2 0
S
xx
( )d is the area under the psd for positive frequency so,
xx
( )d =
0
S 0
R xx (0)
1 2 6 103 + 400 = 6400 2
6400
For most random process R xx ( ) 2 where is the mean or E [x(t)] and 0 if the psd contain impulse at 0 Two independent random variables X and Y are uniformly distributed in the interval [ 1, 1]. The probability that max[X,Y] is less than 1/ 2 is (GATE-EC-2012) (a) 3/4 (b) 9/16 (c) 1/4 (d) 2/3 Sol.(b) The region of the xy plane such that max (x,y) z is the set of points such that x z and y z. 23.
y z
z
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CHAPTER-2 (RANDOM VARIABLES & RANDOM PROCESS) : COMMUNICATION 1 2
The pdf of X and Y is px(x) = 1 2
pY(y) =
0
PERSONAL REMARK :
, 1
