Communication system 8

GATE Electronics and Communication Topicwise Solved Paper by RK Kanodia & Ashish Murolia 84

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UNIT 8COMMUNICATION SYSTEMS

2013 ONE MARK

8.1 The bit rate of a digital communication system is /kbits sR . The modulation used is 32-QAM. The minimum bandwidth required for ISI free transmission is(A) /10 HzR (B) /10 kHzR

(C) /5 HzR (D) /5 kHzR

2013 TWO MARKS

8.2 Let U and V be two independent zero mean Gaussain random

variables of variances 41 and

91 respectively. The probability

P V U3 2F^ h is(A) 4/9 (B) 1/2

(C) 2/3 (D) 5/9

8.3 Consider two identically distributed zero-mean random variables U and V . Let the cumulative distribution functions of U and V2 be F x^ h and G x^ h respectively. Then, for all values of x(A) F x G x 0#^ ^h h (B) F x G x 0$^ ^h h(C) ( ) ( ) .F x G x x 0#^ h (D) ( ) ( ) .F x G x x 0$^ h

8.4 Let U and V be two independent and identically distributed random

variables such that P U P U1 121^ ^h h . The entropy

H U V^ h in bits is(A) 3/4 (B) 1

(C) 3/2 (D) log 32� � � � � � � � � � � � � � � � � � � � � � � � �Bits 1 and 0 are transmitted with equal probability. At the re-ceiver, the pdf of the respective received signals for both bits are as shown below.

8.5 If the detection threshold is 1, the BER will be

(A) 21 (B)

41

(C) 81 (D)

161

8.6 The optimum threshold to achieve minimum bit error rate (BER) is

(A) 21 (B)

54

(C) 1 (D) 23

2012 ONE MARK

8.7 The power spectral density of a real process ( )X t for positive frequencies is shown below. The values of [ ( )]E X t2 and [ ( )] ,E X t respectively, are

(A) / ,6000 0 (B) / ,6400 0

(C) / ,6400 20/( )2 (D) / ,6000 20/( )2

8.8 In a baseband communications link, frequencies upto 3500 Hz are used for signaling. Using a raised cosine pulse with %75 excess bandwidth and for no inter-symbol interference, the maxi mum possible signaling rate in symbols per second is(A) 1750 (B) 2625

(C) 4000 (D) 5250

8.9 A source alphabet consists of N symbols with the probability of the first two symbols being the same. A source encoder increases the probability of the first symbol by a small amount and decreases that of the second by . After encoding, the entropy of the source(A) increases (B) remains the same

(C) increases only if N 2 (D) decreases

8.10 Two independent random variables X and Y are uniformly distributed in the interval ,1 16 @. The probability that ,max X Y6 @ is less than /1 2 is(A) /3 4 (B) /9 16

(C) /1 4 (D) /2 3

2012 TWO MARKS

8.11 A BPSK scheme operating over an AWGN channel with noise power spectral density of / ,N 20 uses equiprobable signals

( ) ( )sins tTE t2

c1 and ( ) ( )sins tTE

t2

c2 over the symbol interval ( , )T0 . If the local oscillator in a coherent receiver is ahead in phase by 45c with respect to the received signal, the probability of error in the resulting system is

(A) 2QNE0

c m (B) QNE

0c m

(C) QNE

2 0c m (D) Q

NE

4 0c m

8.12 A binary symmetric channel (BSC) has a transition probability of

/1 8. If the binary symbol X is such that ( ) / ,P X 0 9 10 then the

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probability of error for an optimum receiver will be(A) /7 80 (B) /63 80

(C) /9 10 (D) /1 10

8.13 The signal ( )m t as shown is applied to both a phase modulator (with kp as the phase constant) and a frequency modulator (with k f as the frequency constant) having the same carrier frequency.

The ratio /k kp f (in rad/Hz) for the same maximum phase devia-tion is(A) 8 (B) 4

(C) 2 (D)

Statement for Linked Answer Question 14 and 15 :

The transfer function of a compensator is given as

( )G sc s bs a

8.14 ( )G sc is a lead compensator if(A) 1, 2a b (B) 3, 2a b

(C) 3, 1a b (D) 3, 1a b

8.15 The phase of the above lead compensator is maximum at

(A) /rad s2 (B) /rad s3

(C) /rad s6 (D) 1/ /rad s3

2011 ONE MARK

8.16 An analog signal is band-limited to 4 kHz, sampled at the Nyquist rate and the samples are quantized into 4 levels. The quantized levels are assumed to be independent and equally probable. If we transmit two quantized samples per second, the information rate is (A) 1 bit/sec (B) 2 bits/sec

(C) 3 bits/sec (D) 4 bits/sec

8.17 The Column -1 lists the attributes and the Column -2 lists the modulation systems. Match the attribute to the modulation system that best meets it.

Column -1 Column -2

P. Power efficient transmission of

signals

1. Conventional

AM

Q. Most bandwidth efficient

transmission of voice signals

2. FM

R. Simplest receiver structure 3. VSB

S. Bandwidth efficient transmission

of signals with significant dc

component

4. SSB-SC

(A) P-4, Q-2, R-1, S-3

(B) P-2, Q-4, R-1, S-3

(C) P-3, Q-2, R-1, S-4

(D) P-2, Q-4, R-3, S-1

2011 TWO MARKS

8.18 ( )X t is a stationary random process with auto-correlation function

( ) ( )expRX2 . This process is passed through the system

shown below. The power spectral density of the output process ( )Y t is

(A) (4 1) ( )expf f2 2 2 (B) (4 1) ( )expf f2 2 2

(C) (4 1) ( )expf f2 2 (D) (4 1) ( )expf f2 2

8.19 A message signal ( ) 2000 4 4000cos cosm t t t modulates the

carrier ( ) 2cosc t f tc where 1MHzfc to produce an AM signal. For demodulating the generated AM signal using an envelope detector, the time constant RC of the detector circuit should satisfy(A) 0.5 ms < RC < 1 ms (B) 1 s << RC < 0.5 ms

(C) RC << 1 s (D) RC >> 0.5 ms

Common Data For Q. 8.5 & 8.6

A four-phase and an eight-phase signal constellation are shown in the figure below.

8.20 For the constraint that the minimum distance between pairs of signal points be d for both constellations, the radii r 1, and r 2 of the circles are(A) 0.707 , 2.782r d r d1 2 (B) 0.707 , 1.932r d r d1 2

(C) 0.707 , 1.545r d r d1 2 (D) 0.707 , 1.307r d r d1 2

8.21 Assuming high SNR and that all signals are equally probable, the additional average transmitted signal energy required by the 8-PSK signal to achieve the same error probability as the 4-PSK signal is(A) 11.90 dB (B) 8.73 dB

(C) 6.79 dB (D) 5.33 dB

2010 ONE MARK

8.22 Suppose that the modulating signal is ( ) 2 (2 )cosm t f tm and the



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carrier signal is ( ) (2 )cosx t A f tC C C , which one of the following is a conventional AM signal without over-modulation(A) ( ) ( ) (2 )cosx t A m t f tC C

(B) ( ) [1 ( )] (2 )cosx t A m t f tC C

(C) ( ) (2 ) ( ) (2 )cos cosx t A f tAm t f t

4C CC

C

(D) ( ) (2 ) (2 ) (2 ) (2 )cos cos sin sinx t A f t f t A f t f tC m C C m C

8.23 Consider an angle modulated signal

( )x t 6 [2 10 2 (800 )] 4 (800 )cos sin cost t t6#

The average power of ( )x t is(A) 10 W (B) 18 W

(C) 20 W (D) 28 W

8.24 Consider the pulse shape ( )s t as shown below. The impulse response

( )h t of the filter matched to this pulse is

2010 TWO MARKS

Statement for linked Answer Question : 8.10 & 8.11 :

Consider a baseband binary PAM receiver shown below. The additive channel noise ( )n t is with power spectral density

( ) /2 10 /W HzS f Nn 020 . The low-pass filter is ideal with unity

gain and cut-off frequency 1 MHz. Let Yk represent the random variable ( )y tk .

Y Nk k , if transmitted bit b 0k

Y a Nk k if transmitted bit b 1k

Where Nk represents the noise sample value. The noise sample has a probability density function, ( ) 0.5P n eNk

n (This has mean zero and variance 2/ 2). Assume transmitted bits to be equiprob-able and threshold z is set to /2 10 Va 6 .

8.25 The value of the parameter (in V 1) is(A) 1010 (B) 107

(C) .1 414 10 10# (D) 2 10 20

#

8.26 The probability of bit error is(A) . e0 5 .3 5

# (B) . e0 5 5#

(C) 0.5 e 7# (D) 0.5 e 10

#

8.27 The Nyquist sampling rate for the signal

( )(500 ) (700)sin sin

s ttt

tt

# is given by

(A) 400 Hz (B) 600 Hz

(C) 1200 Hz (D) 1400 Hz

8.28 ( )X t is a stationary process with the power spectral density ( )S f 0x

, for all f . The process is passed through a system shown below

Let ( )S fy be the power spectral density of ( )Y t . Which one of the following statements is correct(A) ( )S f 0y for all f

(B) ( )S f 0y for f 1 kHz

(C) ( )S f 0y for , 2 kHzf nf f0 0 kHz, n any integer

(D) ( )S f 0y for (2 1) 1 kHzf n f0 , n any integer

2009 ONE MARK

8.29 For a message siganl ( ) ( )cosm t f t2 m and carrier of frequency fc , which of the following represents a single side-band (SSB) signal ?(A) ( ) ( )cos cosf t f t2 2m c (B) ( )cos f t2 c

(C) [ ( ) ]cos f f t2 c m (D) [ ( ) ( )cos cosf t f t1 2 2m c






2009 TWO MARKS

8.30 Consider two independent random variables X and Y with identical distributions. The variables X and Y take values 0, 1 and 2 with

probabilities ,21

41 and

41 respectively. What is the conditional

probability ( )P X Y X Y2 0 ?

(A) 0 (B) 1/16

(C) 1/6 (D) 1

8.31 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 ?

k 1 2 3 4 5

( )P X k 0.1 0.2 0.3 0.4 0.5

(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

8.32 A message signal given by ( ) ( ) ( )cos sinm t t t21

1 21

2 amplitude - modulated with a carrier of frequency C to generator

( )[ ( )]coss t m t t1 c . What is the power efficiency achieved by this modulation scheme ?(A) . %8 33 (B) . %11 11

(C) %20 (D) %25

8.33 A communication channel with AWGN operating at a signal to noise ration SNR 1 and bandwidth B has capacity C1. If the SNR is doubled keeping constant, the resulting capacity C2 is given by(A) C C22 1. (B) C C B2 1.

(C) C C B22 1. (D) 0.3C C B2 1. +

Common Data For Q. 8.19 & 8.20 :

The amplitude of a random signal is uniformly distributed between -5 V and 5 V.

8.34 If the signal to quantization noise ratio required in uniformly quantizing the signal is 43.5 dB, the step of the quantization is approximately(A) 0.033 V (B) 0.05 V

(C) 0.0667 V (D) 0.10 V

8.35 If the positive values of the signal are uniformly quantized with a step size of 0.05 V, and the negative values are uniformly quantized with a step size of 0.1 V, the resulting signal to quantization noise ration is approximately(A) 46 dB (B) 43.8 dB

(C) 42 dB (D) 40 dB

2008 ONE MARK

8.36 Consider the amplitude modulated (AM) signal

cos cos cosA t t t2c c m c . For demodulating the signal using envelope detector, the minimum value of Ac should be(A) 2 (B) 1

(C) 0.5 (D) 0

2008 TWO MARKS

8.37 The probability density function (pdf) of random variable is as shown below

The corresponding commutative distribution function CDF has the form

8.38 A memory less source emits n symbols each with a probability p . The entropy of the source as a function of n(A) increases as log n (B) decreases as ( )log n

1

(C) increases as n (D) increases as logn n

8.39 Noise with double-sided power spectral density on K over all frequencies is passed through a RC low pass filter with 3 dB cut-off frequency of fc . The noise power at the filter output is(A) K (B) Kfc

(C) k fc (D) 3

8.40 Consider a Binary Symmetric Channel (BSC) with probability of error being p . To transmit a bit, say 1, we transmit a sequence of three 1s. The receiver will interpret the received sequence to represent 1 if at least two bits are 1. The probability that the transmitted bit will be received in error is(A) ( )p p p3 13 2 (B) p3

(C) ( )p1 3 (D) ( )p p p13 2



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8.41 Four messages band limited to , ,W W W2 and W3 respectively are to be multiplexed using Time Division Multiplexing (TDM). The minimum bandwidth required for transmission of this TDM signal is(A) W (B) W3

(C) W6 (D) W7

8.42 Consider the frequency modulated signal

[ ( ) . ( )]cos sin sint t t10 2 10 5 2 1500 7 5 2 10005# # #

with carrier frequency of 105 Hz. The modulation index is(A) 12.5 (B) 10

(C) 7.5 (D) 5

8.43 The signal .cos cos sint t t0 5c m c is(A) FM only (B) AM only

(C) both AM and FM (D) neither AM nor FM

Common Data For Q. 8.29, 8.30 and 8.31 :

A speed signal, band limited to 4 kHz and peak voltage varying between +5 V and 5 V, is sampled at the Nyquist rate. Each

sample is quantized and represented by 8 bits.

8.44 If the bits 0 and 1 are transmitted using bipolar pulses, the minimum bandwidth required for distortion free transmission is(A) 64 kHz (B) 32 kHz

(C) 8 kHz (D) 4 kHz

8.45 Assuming the signal to be uniformly distributed between its peak to peak value, the signal to noise ratio at the quantizer output is(A) 16 dB (B) 32 dB

(C) 48 dB (D) 4 kHz

8.46 Assuming the signal to be uniformly distributed between its peak to peak value, the signal to noise ratio at the quantizer output is(A) 1024 (B) 512

(C) 256 (D) 64

2007 ONE MARK

8.47 If ( )R is the auto correlation function of a real, wide-sense stationary random process, then which of the following is NOT true(A) ( ) ( )R R

(B) ( ) ( )R R 0#

(C) ( ) ( )R R

(D) The mean square value of the process is ( )R 0

8.48 If ( )S f is the power spectral density of a real, wide-sense stationary random process, then which of the following is ALWAYS true?(A) ( ) ( )S S f0 # (B) ( )S f 0$

(C) ( ) ( )S f S f (D) ( )S f df 03

3

-

�8.49 If E denotes expectation, the variance of a random variable X is

given by

(A) [ ] [ ]E X E X2 2 (B) [ ] [ ]E X E X2 2

(C) [ ]E X2 (D) [ ]E X2

2007 TWO MARKS

8.50 A Hilbert transformer is a(A) non-linear system (B) non-causal system

(C) time-varying system (D) low-pass system

8.51 In delta modulation, the slope overload distortion can be reduced by(A) decreasing the step size (B) decreasing the granular

noise

(C) decreasing the sampling rate (D) increasing the step size

8.52 The raised cosine pulse ( )p t is used for zero ISI in digital communications. The expression for ( )p t with unity roll-off factor is given by

( )p t ( )

sinWt W t

Wt

4 1 164

2 2

The value of ( )p t at tW41 is

(A) .0 5 (B) 0

(C) 0.5 (D) 3

8.53 In the following scheme, if the spectrum ( )M f of ( )m t is as shown, then the spectrum ( )Y f of ( )y t will be

8.54 During transmission over a certain binary communication channel,

bit errors occur independently with probability p . The probability of AT MOST one bit in error in a block of n bits is given by(A) pn (B) p1 n

(C) ( ) ( )np p p1 1n n1- (D) ( )p1 1 n

8.55 In a GSM system, 8 channels can co-exist in 200 kHz bandwidth using TDMA. A GSM based cellular operator is allocated 5 MHz

bandwidth. Assuming a frequency reuse factor of 51 , i.e. a five-cell

repeat pattern, the maximum number of simultaneous channels that can exist in one cell is(A) 200 (B) 40

(C) 25 (D) 5






8.56 In a Direct Sequence CDMA system the chip rate is .1 2288 106#

chips per second. If the processing gain is desired to be AT LEAST 100, the data rate(A) must be less than or equal to .12 288 103

# bits per sec

(B) must be greater than .12 288 103# bits per sec

(C) must be exactly equal to .12 288 103# bits per sec

(D) can take any value less than .122 88 103# bits per sec


Two 4-array signal constellations are shown. It is given that 1 and 2 constitute an orthonormal basis for the two constella-tion. Assume that the four symbols in both the constellations are

equiprobable. Let N20 denote the power spectral density of white

Gaussian noise.

8.57 The if ratio or the average energy of Constellation 1 to the average energy of Constellation 2 is(A) a4 2 (B) 4

(C) 2 (D) 8

8.58 If these constellations are used for digital communications over an AWGN channel, then which of the following statements is true ?(A) Probability of symbol error for Constellation 1 is lower

(B) Probability of symbol error for Constellation 1 is higher

(C) Probability of symbol error is equal for both the constellations

(D) The value of N0 will determine which of the constellations has a lower probability of symbol error

Statement for Linked Answer Question 8.44 & 8.45 :

An input to a 6-level quantizer has the probability density function

( )f x as shown in the figure. Decision boundaries of the quantizer are chosen so as to maximize the entropy of the quantizer output. It is given that 3 consecutive decision boundaries are’ '.' '1 0 and

' '1 .

8.59 The values of a and b are

(A) a61 and b

121 (B) a

51 and b

403

(C) a41 and b

161 (D) a

31 and b

241

8.60 Assuming that the reconstruction levels of the quantizer are the mid-points of the decision boundaries, the ratio of signal power to quantization noise power is

(A) 9

152 (B) 364

(C) 376 (D) 28

2006 ONE MARK

8.61 A low-pass filter having a frequency response ( ) ( )H j A e ( )j does not produce any phase distortions if(A) ( ) , ( )A C k3 3 (B) ( ) , ( )A C k2

(C) ( ) , ( )A C k 2 (D) ( ) , ( )A C k 1-

2006 TWO MARKS

8.62 A signal with bandwidth 500 Hz is first multiplied by a signal ( )g t where

( )g t ( ) ( . )t k1 0 5 10k

R

4#

3

3-

=-

�The resulting signal is then passed through an ideal lowpass filter

with bandwidth 1 kHz. The output of the lowpass filter would be(A) ( )t (B) ( )m t

(C) 0 (D) ( ) ( )m t t

8.63 The minimum sampling frequency (in samples/sec) required to

reconstruct the following signal from its samples without distortion

( )x t 5 7sin sintt

tt2 100 2 1003 2` `j j would be

(A) 2 103# (B) 4 103

#

(C) 6 103# (D) 8 103

#

8.64 The minimum step-size required for a Delta-Modulator operating at 32k samples/sec to track the signal (here ( )u t is the unit-step function)

( )x t 125[ ( ) ( 1) (250 )[ ( 1) ( 2)]u t u t t u t u t= +

so that slope-overload is avoided, would be

(A) 2 10- (B) 2 8-

(C) 2 6- (D) 2 4-

8.65 A zero-mean white Gaussian noise is passes through an ideal lowpass filter of bandwidth 10 kHz. The output is then uniformly sampled with sampling period .t 0 03s msec. The samples so obtained would be(A) correlated (B) statistically independent

(C) uncorrelated (D) orthogonal

8.66 A source generates three symbols with probabilities 0.25, 0.25, 0.50 at a rate of 3000 symbols per second. Assuming independent generation of symbols, the most efficient source encoder would have average bit rate is(A) 6000 bits/sec (B) 4500 bits/sec

(C) 3000 bits/sec (D) 1500 bits/sec

8.67 The diagonal clipping in Amplitude Demodulation (using envelop detector) can be avoided it RC time-constant of the envelope detector satisfies the following condition, (here W is message bandwidth and

is carrier frequency both in rad/sec)

(A) RCW1 (B) RC

W1



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(C) RC 1 (D) RC 1

8.68 A uniformly distributed random variable X with probability density function

( )f xx ( ) ( )]pu x u x101 5 5

where (.)u is the unit step function is passed through a transfor-

mation given in the figure below. The probability density function

of the transformed random variable Y would be

(A) ( ) [ ( . ) ( . )]f y u y u y51 2 5 2 25y

(B) ( ) . ( ) . ( )f y y y0 5 0 5 1y

(C) ( ) . ( . ) . ( . ) ( )f y y y y0 25 2 5 0 25 2 5 5y

(D) ( ) . ( . ) . ( . ) [ ( . ) ( . )]f y y y u y u y0 25 2 5 0 25 2 5101 2 5 2 5y

8.69 In the following figure the minimum value of the constant " "C , which is to be added to ( )y t1 such that ( )y t1 and ( )y t2 are different , is

(A) 3 (B) 23

(C) 12

23 (D)

L3

8.70 A message signal with bandwidth 10 kHz is Lower-Side Band SSB modulated with carrier frequency f 10c1

6 Hz. The resulting signal is then passed through a Narrow-Band Frequency Modulator with carrier frequency f 10c2

9 Hz.The bandwidth of the output would be(A) 4 104

# Hz (B) 2 106# Hz

(C) 2 109# Hz (D) 2 1010

# Hz


Let ( ) ( )*( )g t p t pt , where * denotes convolution &

( ) ( ) ( ) limp t u t u t 1z

�3

with ( )u t being the unit step function

8.71 The impulse response of filter matched to the signal

( ) ( ) ( )* ( )s t g t g t1 2 is given as :(A) ( )s t1 (B) ( )s t1

(C) ( )s t (D) ( )s t

8.72 An Amplitude Modulated signal is given as

( )x tAM [ ( ) . ( )]cosp t g t t100 0 5 c

in the interval t0 1# # . One set of possible values of modulating

signal and modulation index would be(A) , .t 0 5 (B) , .t 1 0

(C) , .t 2 0 (D) , .t 0 52


The following two question refer to wide sense stationary stochas-tic process

8.73 It is desired to generate a stochastic process (as voltage process)

with power spectral density ( ) 16/(16 )S 2 by driving a Linear-Time-Invariant system by zero mean white noise (As voltage process) with power spectral density being constant equal to 1. The system which can perform the desired task could be(A) first order lowpass R-L filter

(B) first order highpass R-C filter

(C) tuned L-C filter

(D) series R-L-C filter

8.74 The parameters of the system obtained in previous Q would be(A) first order R-L lowpass filter would have R 4 L H1

(B) first order R-C highpass filter would have R 4 .C F0 25

(C) tuned L-C filter would have L H4 C F4

(D) series R-L-C lowpass filter would have R 1 , L H4 ,

C F4


Consider the following Amplitude Modulated (AM) signal, where

f Bm

( )X tAM ( . )sin cosf t f t10 1 0 5 2 2m c

8.75 The average side-band power for the AM signal given above is(A) 25 (B) 12.5

(C) 6.25 (D) 3.125

8.76 The AM signal gets added to a noise with Power Spectral Density

( )S fn given in the figure below. The ratio of average sideband power to mean noise power would be :

(A) N B825

0 (B)

N B425

0






(C) N B225

0 (D)

N B250

2005 ONE MARK

8.77 Find the correct match between group 1 and group 2.Group 1 Group 2P. { ( ) ( )}sinkm t A t1 c W. Phase modulationQ. ( ) ( )sinkm t A tc X. Frequency modulationR. { ( )}sinA t km tc Y. Amplitude modulation

S. ( )sinA t k m t dtc

t

3-; E

� Z. DSB-SC modulation

(A) P Z, Q Y, R X, S W

(B) P W, Q X, R Y, S Z

(C) P X, Q W, R Z, S Y

(D) P Y, Q Z, R W, S X

8.78 Which of the following analog modulation scheme requires the minimum transmitted power and minimum channel bandwidth ?(A) VSB (B) DSB-SC

(C) SSB (D) AM

2005 TWO MARKS

8.79 A device with input ( )X t and output ( )y t is characterized by:

( ) ( )Y t x t2 . An FM signal with frequency deviation of 90 kHz and modulating signal bandwidth of 5 kHz is applied to this device. The bandwidth of the output signal is(A) 370 kHz (B) 190 kHz

(C) 380 kHz (D) 95 kHz

8.80 A signal as shown in the figure is applied to a matched filter. Which of the following does represent the output of this matched filter ?

8.81 Noise with uniform power spectral density of N0 W/Hz is passed though a filter ( ) ( )expH j t2 d followed by an ideal pass filter of bandwidth B Hz. The output noise power in Watts is(A) N B2 0 (B) N B4 0

(C) N B8 0 (D) N B16 0

8.82 An output of a communication channel is a random variable v with

the probability density function as shown in the figure. The mean square value of v is

(A) 4 (B) 6

(C) 8 (D) 9

8.83 A carrier is phase modulated (PM) with frequency deviation of 10 kHz by a single tone frequency of 1 kHz. If the single tone frequency is increased to 2 kHz, assuming that phase deviation remains unchanged, the bandwidth of the PM signal is(A) 21 kHz (B) 22 kHz

(C) 42 kHz (D) 44 kHz

Common Data For Q. 8.69 and 8.70 :

Asymmetric three-level midtread quantizer is to be designed as-suming equiprobable occurrence of all quantization levels.

8.84 If the probability density function is divide into three regions as shown in the figure, the value of a in the figure is

(A) 31 (B)

32

(C) 21 (D)

41

8.85 The quantization noise power for the quantization region between

a and a in the figure is

(A) 814 (B)

91

(C) 815 (D)

812

2004 ONE MARK

8.86 In a PCM system, if the code word length is increased from 6 to 8 bits, the signal to quantization noise ratio improves by the factor

(A) 68 (B) 12

(C) 16 (D) 8

8.87 An AM signal is detected using an envelop detector. The carrier frequency and modulating signal frequency are 1 MHz and 2 kHz respectively. An appropriate value for the time constant of the envelop detector is(A) sec500 (B) sec20



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(C) . sec0 2 (D) sec1

8.88 An AM signal and a narrow-band FM signal with identical carriers, modulating signals and modulation indices of 0.1 are added together. The resultant signal can be closely approximated by(A) broadband FM (B) SSB with carrier

(C) DSB-SC (D) SSB without carrier

8.89 In the output of a DM speech encoder, the consecutive pulses are of opposite polarity during time interval t t t1 2# # . This indicates that during this interval(A) the input to the modulator is essentially constant

(B) the modulator is going through slope overload

(C) the accumulator is in saturation

(D) the speech signal is being sampled at the Nyquist rate

8.90 The distribution function ( )F xx of a random variable x is shown in the figure. The probability that X 1 is

(A) zero (B) 0.25

(C) 0.55 (D) 0.30

2004 TWO MARKS

8.91 A 1 mW video signal having a bandwidth of 100 MHz is transmitted to a receiver through cable that has 40 dB loss. If the effective one-

side noise spectral density at the receiver is 10 20- Watt/Hz, then the signal-to-noise ratio at the receiver is(A) 50 dB (B) 30 dB

(C) 40 dB (D) 60 dB

8.92 Consider the signal ( )x t shown in Fig. Let ( )h t denote the impulse response of the filter matched to ( )x t , with ( )h t being non-zero only in the interval 0 to 4 sec. The slope of ( )h t in the interval t3 4 sec is

(A) sec21 1- (B) sec1 1-

(C) sec21 1- (D) sec1 1-

8.93 A source produces binary data at the rate of 10 kbps. The binary

symbols are represented as shown in the figure.The source output is transmitted using two modulation schemes, namely Binary PSK (BPSK) and Quadrature PSK (QPSK). Let

B1 and B2 be the bandwidth requirements of the above rectangular pulses is 10 kHz, B1 and B2 are

(A) B 201 kHz, B 202 kHz (B) B 101 kHz, B 202 kHz

(C) B 201 khz, B 102 kHz (D) B 101 kHz, B 102 kHz

8.94 A 100 MHz carrier of 1 V amplitude and a 1 MHz modulating signal of 1 V amplitude are fed to a balanced modulator. The ourput of the modulator is passed through an ideal high-pass filter with cut-off frequency of 100 MHz. The output of the filter is added with 100 MHz signal of 1 V amplitude and 90c phase shift as shown in the figure. The envelope of the resultant signal is

(A) constant (B) ( )sin t1 2 106#

(C) ( )sin t45 2 106 (D) ( )cos t

45 2 106

#

8.95 Two sinusoidal signals of same amplitude and frequencies 10 kHz and 10.1 kHz are added together. The combined signal is given to an ideal frequency detector. The output of the detector is(A) 0.1 kHz sinusoid (B) 20.1 kHz sinusoid

(C) a linear function of time (D) a constant

8.96 Consider a binary digital communication system with equally likely 0’s and 1’s. When binary 0 is transmitted the detector input can lie between the levels .0 25 V and .0 25 V with equl probability : when binary 1 is transmitted, the voltage at the detector can have any value between 0 and 1 V with equal probability. If the detector has a threshold of 0.2 V (i.e., if the received signal is greater than 0.2 V, the bit is taken as 1), the average bit error probability is(A) 0.15 (B) 0.2

(C) 0.05 (D) 0.5

8.97 A random variable X with uniform density in the interval 0 to 1 is quantized as follows : If .X0 0 3# # , x 0q

If . X0 3 1# , .x 0 7q

where xq is the quantized value of .XThe root-mean square value of the quantization noise is(A) 0.573 (B) 0.198

(C) .2 205 (D) .0 266

8.98 Choose the current one from among the alternative , , ,A B C D after matching an item from Group 1 with the most appropriate item in Group 2.Group 1 Group 2






1. FM P. Slope overload2. DM Q. -law3. PSK R. Envelope detector4. PCM S. Hilbert transform T. Hilbert transform U. Matched filter(A) 1 - T, 2 - P, 3 - U, 4 - S (B) 1 - S, 2 - U, 3 - P, 4 - T

(C) 1 - S, 2 - P, 3 - U, 4 - Q (D) 1 - U, 2 - R, 3 - S, 4 - Q

8.99 Three analog signals, having bandwidths 1200 Hz, 600 Hz and 600 Hz, are sampled at their respective Nyquist rates, encoded with 12 bit words, and time division multiplexed. The bit rate for the multiplexed. The bit rate for the multiplexed signal is(A) 115.2 kbps (B) 28.8 kbps

(C) 57.6 kbps (D) 38.4 kbps

8.100 Consider a system shown in the figure. Let ( )X f and ( )Y f and denote the Fourier transforms of ( )x t and ( )y t respectively. The ideal HPF has the cutoff frequency 10 kHz.

The positive frequencies where ( )Y f has spectral peaks are(A) 1 kHz and 24 kHz (B) 2 kHz and 244 kHz

(C) 1 kHz and 14 kHz (D) 2 kHz and 14 kHz

2003 ONE MARK

8.101 The input to a coherent detector is DSB-SC signal plus noise. The noise at the detector output is(A) the in-phase component (B) the quadrature - component

(C) zero (D) the envelope

8.102 The noise at the input to an ideal frequency detector is white. The detector is operating above threshold. The power spectral density of the noise at the output is(A) raised - cosine (B) flat

(C) parabolic (D) Gaussian

8.103 At a given probability of error, binary coherent FSK is inferior to binary coherent PSK by.(A) 6 dB (B) 3 dB

(C) 2 dB (D) 0 dB

2003 TWO MARKS

8.104 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 probability that ( )z 1# is

(A) zero (B) 61

(C) 31 (D)

121


( )X t is a random process with a constant mean value of 2 and the

auto correlation function ( ) ( )R e4 1.xx

0 2- .

8.105 Let X be the Gaussian random variable obtained by sampling the process at t ti and let

( )Q e dy

21 x

2

23

�The probability that x 1#6 @ is(A) ( . )Q1 0 5 (B) ( . )Q 0 5

(C) Q2 2

1c m (D) Q12 2

1c m8.106 Let Y and Z be the random variable obtained by sampling ( )X t at

t 2 and t 4 respectively. Let W Y Z . The variance of W is(A) 13.36 (B) 9.36

(C) 2.64 (D) 8.00

8.107 A sinusoidal signal with peak-to-peak amplitude of 1.536 V is quantized into 128 levels using a mid-rise uniform quantizer. The

quantization-noise power is

(A) 0.768 V (B) V48 10 6 2#

-

(B) 12 10 V6 2#

- (D) .3 072 V

8.108 Let ( ) ( ) ( ). ( )cos cosx t t x t2 800 1400 is sampled with the rectangular pulse train shown in the figure. The only spectral components (in kHz) present in the sampled signal in the frequency range 2.5 kHz to 3.5 kHz are

(A) 2.7, 3.4 (B) 3.3, 3.6

(C) 2.6, 2.7, 3.3, 3.4, 3.6 (D) 2.7, 3.3

8.109 A DSB-SC signal is to be generated with a carrier frequency f 1c MHz using a non-linear device with the input-output characteristic

V a v a vi i0 0 13 where a0 and a1 are constants. The output of the

non-linear device can be filtered by an appropriate band-pass filter.Let ( ) ( )cosV A f c m t2i c

i i t is the message signal. Then the value of fc

i (in MHz) is(A) 1.0 (B) 0.333

(B) 0.5 (D) 3.0


Let ( ) [( ) ]cosm t t4 103# be the message signal &

( )c t 5 [(2 10 )]cos t6#= be the carrier.

8.110 ( )c t and ( )m t are used to generate an AM signal. The modulation index of the generated AM signal is 0.5. Then the quantity

Carrier powerTotal sideband power

is



by RK Kanodia

Now in 3 Volume






(A) 21 (B)

41

(C) 31 (D)

81

8.111 ( )c t and ( )m t are used to generated an FM signal. If the peak frequency deviation of the generated FM signal is three times the transmission bandwidth of the AM signal, then the coefficient of the term [ ( )]cos t2 1008 103

# in the FM signal (in terms of the Bessel coefficients) is

(A) ( )J5 34 (B) ( )J25 38

(C) ( )J25 48 (D) ( )J5 64

8.112 Choose the correct one from among the alternative , , ,A B C D after matching an item in Group 1 with most appropriate item in Group 2.Group 1 Group 2P. Ring modulator 1. Clock recoveryQ. VCO 2. Demodulation of FMR. Foster-Seely discriminator 3. Frequency conversion

S. Mixer 4. Summing the two inputs 5. Generation of FM 6. Generation of DSB-Sc(A) ; ; ;P Q R S1 3 2 4 (B) ; ; ;P Q R S6 5 2 3

(C) ; ; ;P Q R S6 1 3 2 (D) ; ; ;P Q R S5 6 1 3

8.113 A superheterodyne receiver is to operate in the frequency range 550 kHz - 1650 kHz, with the intermediate frequency of 450 kHz. Let

/R C Cmax min denote the required capacitance ratio of the local oscillator and I denote the image frequency (in kHz) of the incoming signal. If the receiver is tuned to 700 kHz, then(A) . ,R I4 41 1600 (B) . ,R I2 10 1150

(C) . ,R I3 0 600 (D) . ,R I9 0 1150

8.114 If Eb , the energy per bit of a binary digital signal, is 10 5- watt-sec and the one-sided power spectral density of the white noise,

N 1006- W/Hz, then the output SNR of the matched filter is

(A) 26 dB (B) 10 dB

(C) 20 dB (D) 13 dB

8.115 The input to a linear delta modulator having a step-size .0 6283 is a sine wave with frequency fm and peak amplitude Em . If the sampling frequency f 40x kHz, the combination of the sine-wave frequency and the peak amplitude, where slope overload will take place is Em fm(A) 0.3 V 8 kHz(B) 1.5 V 4 kHz(C) 1.5 V 2 kHz(D) 3.0 V 1 kHz

8.116 If S represents the carrier synchronization at the receiver and represents the bandwidth efficiency, then the correct statement for the coherent binary PSK is

(A) . ,S0 5 is required (B) . ,S1 0 is required

(C) . ,S0 5 is not required (D) . ,S1 0 is not required

8.117 A signal is sampled at 8 kHz and is quantized using 8 - bit uniform quantizer. Assuming SNRq for a sinusoidal signal, the correct statement for PCM signal with a bit rate of R is(A) R 32 kbps, .SNR 25 8q dB

(B) R 64 kbps, .SNR 49 8q dB

(C) R 64 kbps, .SNR 55 8q dB

(D) R 32 kbps, .SNR 49 8q dB

2002 ONE MARK

8.118 A 2 MHz sinusoidal carrier amplitude modulated by symmetrical square wave of period 100 sec. Which of the following frequencies will NOT be present in the modulated signal ?(A) 990 kHz (B) 1010 kHz

(C) 1020 kHz (D) 1030 kHz

8.119 Consider a sample signal ( ) ( ) ( )y t t t nT5 10 s

n

6# #

3

3-

=-

+�where ( ) ( )cosx t t10 8 103

# and T 100s sec.When ( )y t is passed through an ideal lowpass filter with a cutoff frequency of 5 KHz, the output of the filter is

(A) 5 10 (8 10 )cos t6 3# #

- (b) ( )cos t5 10 8 105 3# #

-

(C) ( )cos t5 10 8 101 3# #

- (D) ( )cos t10 8 103#

8.120 For a bit-rate of 8 Kbps, the best possible values of the transmitted frequencies in a coherent binary FSK system are(A) 16 kHz and 20 kHz (C) 20 kHz and 32 kHz


8.121 The line-of-sight communication requires the transmit and receive antennas to face each other. If the transmit antenna is vertically polarized, for best reception the receiver antenna should be(A) horizontally polarized

(B) vertically polarized

(C) at 45c with respect to horizontal polarization

(D) at 45c with respect to vertical polarization

2002 TWO MARKS

8.122 An angle-modulated signal is given by

( )s t ( )cos sin cost t t2 2 10 30 150 40 1506# .

The maximum frequency and phase deviations of ( )s t are(A) 10.5 kHz, 140 rad (B) 6 kHz, 80 rad

(C) 10.5 kHz, 100 rad (D) 7.5 kHz, 100 rad

8.123 In the figure ( ) , ( )sin cosm ttt s t t2 2 200 and ( ) sinn t

tt199

.The output ( )y t will be






(A) sintt2 (B) sin sin cos

tt

tt t2 3

(C) . .sin sin costt

tt t2 0 5 1 5 (D) .sin sin cos

tt

tt t2 0 75

8.124 A signal ( ) ( )cosx t t100 24 103# is ideally sampled with a sampling

period of sec50 ana then passed through an ideal lowpass filter with cutoff frequency of 15 kHz. Which of the following frequencies is/are present at the filter output ?(A) 12 kHz only (B) 8 kHz only


8.125 If the variance x2 of ( ) ( ) ( )d n x n x n 1 is one-tenth the

variance x2 of stationary zero-mean discrete-time signal ( )x n , then

the normalized autocorrelation function ( )R k

x

xx

2 at k 1 is

(A) 0.95 (B) 0.90

(C) 0.10 (D) 0.05

2001 ONE MARK

8.126 A bandlimited signal is sampled at the Nyquist rate. The signal can be recovered by passing the samples through(A) an RC filter

(B) an envelope detector

(C) a PLL

(D) an ideal low-pass filter with the appropriate bandwidth

8.127 The PDF of a Gaussian random variable X is given by

( )p x e3 2

1 ( )

x

x

18

4 2

--

. The probability of the event { }X 4 is

(A) 21 (B)

3 21

(C) 0 (D) 41

2001 TWO MARKS

8.128 A video transmission system transmits 625 picture frames per second. Each frame consists of a 400 400# pixel grid with 64 intensity levels per pixel. The data rate of the system is(A) 16 Mbps (B) 100 Mbps

(C) 600 Mbps (D) 6.4 Gbps

8.129 The Nyquist sampling interval, for the signal ( ) ( )sin sinc t c t700 500 is

(A) sec3501 (B) sec

350

(C) sec7001 (D) sec

175

8.130 During transmission over a communication channel, bit errors occur independently with probability p . If a block of n bits is transmitted, the probability of at most one bit error is equal to(A) ( )p1 1 n (B) ( )( )p n p1 1

(C) ( )np p1 n 1- (D) ( ) ( )p np p1 1n n 1-

8.131 The PSD and the power of a signal ( )g t are, respectively, ( )Sg and

Pg . The PSD and the power of the signal ( )ag t are, respectively,(A) ( )a Sg

2 and a Pg2 (B) ( )a Sg

2 and aPg

(C) ( )aSg and a Pg2 (D) ( )aSg and aPs

2000 ONE MARK

8.132 The amplitude modulated waveform ( ) [ ( )]coss t A K m t t1c a c is fed to an ideal envelope detector. The maximum magnitude

of ( )K m t0 is greater than 1. Which of the following could be the detector output ?(A) ( )A m tc (B) [ ( )]A K m t1c a

2 2

(C) [ ( ( )]A K m t1c a (D) [ ( )]A K m t1c a2

8.133 The frequency range for satellite communication is(A) 1 KHz to 100 KHz (B) 100 KHz to 10 KHz

(C) 10 MHz to 30 MHz (D) 1 GHz to 30 GHz

2000 TWO MARKS

8.134 In a digital communication system employing Frequency Shift Keying (FSK), the 0 and 1 bit are represented by sine waves of 10 KHz and 25 KHz respectively. These waveforms will be orthogonal for a bit interval of(A) sec45 (B) sec200

(C) sec50 (D) sec250

8.135 A message ( )m t bandlimited to the frequency fm has a power of Pm

. The power of the output signal in the figure is

(A) cosP2

m (B) P4m

(C) sinP4

m2

(D) cosP4

m2

8.136 The Hilbert transform of cos sint t1 2 is(A) sin cost t1 2 (B) sin cost t1 2

(C) cos sint t1 2 (D) sin sint t1 2

8.137 In a FM system, a carrier of 100 MHz modulated by a sinusoidal signal of 5 KHz. The bandwidth by Carson’s approximation is 1 MHz. If ( )y t (modulated waveform)3, than by using Carson’s approximation, the bandwidth of ( )y t around 300 MHz and the and the spacing of spectral components are, respectively.(A) 3 MHz, 5 KHz (B) 1 MHz, 15 KHz

(C) 3 MHz, 15 KHz (D) 1 MHz, 5 KHz

1999 ONE MARK

8.138 The input to a channel is a bandpass signal. It is obtained by linearly modulating a sinusoidal carrier with a single-tone signal. The output of the channel due to this input is given by

( )y t (1/100) (100 10 ) (10 1.56)cos cost t6 6

The group delay ( )tg and the phase delay ( )tp in seconds, of the channel are(A) , .t t10 1 56g p

6 (B) . ,t t1 56 10g p6

(C) , .t t10 1 56 10g p8 6

# (D) , .t t10 1 56g p8

8.139 A modulated signal is given by ( ) ( ) (2 ) ( ) (2 )cos sins t m t f t m t f tc c1 2



by RK Kanodia

Now in 3 Volume






where the baseband signal ( )m t1 and ( )m t2 have bandwidths of

10 kHz, and 15 kHz, respectively. The bandwidth of the modulated

signal, in kHz, is(A) 10 (B) 15

(C) 25 (D) 30

8.140 A modulated signal is given by ( )s t [( ) ] ( )cose t u tatc ,

where a andc are positive constants, and c . The complex envelope of ( )s t is given by(A) ( ) [ ( ) ] ( )exp expat j t u tc

(B) ( ) ( ) ( )exp expat j t u t

(C) ( ) ( )exp j t u t

(D) [ ) ]exp j tc

1999 TWO MARKS

8.141 The Nyquist sampling frequency (in Hz) of a signal given by

( ) * ( )sin sinc t c t6 10 400 10 1004 2 6 3# is

(A) 200 (B) 300

(C) 500 (D) 1000

8.142 The peak-to-peak input to an 8-bit PCM coder is 2 volts. The signal power-to-quantization noise power ratio (in dB) for an input of

. ( )cos t0 5 m is(A) 47.8 (B) 49.8

(C) 95.6 (D) 99.6

8.143 The input to a matched filter is given by

( )s t ( )sin sec

otherwise

t2 10 0 1 10010 6 4

#"The peak amplitude of the filter output is(A) 10 volts (B) volts5

(C) millivolts10 (D) millivolts5

8.144 Four independent messages have bandwidths of 100 Hz,

Hz and Hz200 400 , respectively. Each is sampled at the Nyquist rate, and the samples are time division multiplexed (TDM) and transmitted. The transmitted sample rate (in Hz) is(A) 1600 (B) 800

(C) 400 (D) 200

1998 ONE MARK

8.145 The amplitude spectrum of a Gaussian pulse is(A) uniform (B) a sine function

(C) Gaussian (D) an impulse function

8.146 The ACF of a rectangular pulse of duration T is(A) a rectangular pulse of duration T

(B) a rectangular pulse of duration T2

(C) a triangular pulse of duration T

(D) a triangular pulse of duration T2

8.147 The image channel selectivity of superheterodyne receiver depends upon(A) IF amplifiers only

(B) RF and IF amplifiers only

(C) Preselector, RF and IF amplifiers

(D) Preselector, and RF amplifiers only

8.148 In a PCM system with uniform quantisation, increasing the number of bits from 8 to 9 will reduce the quantisation noise power by a factor of(A) 9 (B) 8

(C) 4 (D) 2

8.149 Flat top sampling of low pass signals(A) gives rise to aperture effect (B) implies oversampling

(C) leads to aliasing (D) introduces delay dis-tortion

8.150 A DSB-SC signal is generated using the carrier ( )cos te and modulating signal ( )x t . The envelope of the DSB-SC signal is(A) ( )x t (B) ( )x t

(C) only positive portion of ( )x t (D) ( )cosx t

8.151 Quadrature multiplexing is(A) the same as FDM

(B) the same as TDM

(C) a combination of FDM and TDM

(D) quite different from FDM and TDM

8.152 The Fourier transform of a voltage signal ( )x t is ( )X f . The unit of

( )X f is(A) volt (B) volt-sec

(C) volt/sec (D) volt2

8.153 Compression in PCM refers to relative compression of(A) higher signal amplitudes (B) lower signal amplitudes

(C) lower signal frequencies (D) higher signal frequencies

8.154 For a give data rate, the bandwidth Bp of a BPSK signal and the bandwidth B0 of the OOK signal are related as

(A) B B4p

0 (B) B B

2p0

(C) B Bp 0 (D) B B2p 0

8.155 The spectral density of a real valued random process has

(A) an even symmetry (B) an odd symmetry

(C) a conjugate symmetry (D) no symmetry

8.156 The probability density function of the envelope of narrow band Gaussian noise is(A) Poisson (B) Gaussian

(C) Rayleigh (D) Rician

1997 ONE MARK

8.157 The line code that has zero dc component for pulse transmission of random binary data is






(A) Non-return to zero (NRZ)

(B) Return to zero (RZ)

(C) Alternate Mark Inversion (AM)

(D) None of the above

8.158 A probability density function is given by ( )p x Ke x/x 22

3 3

. The value of K should be

(A) 21

(B) 2

(C) 2

1 (D)

21

8.159 A deterministic signal has the power spectrum given in the figure is, The minimum sampling rate needed to completely represent this signal is

(A) 1 kHz (B) 2 kHz

(C) 3 kHz (D) None of these

8.160 A communication channel has first order low pass transfer function. The channel is used to transmit pulses at a symbol rate greater than the half-power frequency of the low pass function. Which of the network shown in the figure is can be used to equalise the received pulses?

8.161 The power spectral density of a deterministic signal is given by

[ ( )/ ]sin f f 2 where f is frequency. The auto correlation function of this signal in the time domain is(A) a rectangular pulse (B) a delta function

(C) a sine pulse (D) a triangular pulse

1996 ONE MARK

8.162 A rectangular pulse of duration T is applied to a filter matched to this input. The out put of the filter is a(A) rectangular pulse of duration T

(B) rectangular pulse of duration T2

(C) triangular pulse

(D) sine function

8.163 The image channel rejection in a superheterodyne receiver comes

from(A) IF stages only (B) RF stages only

(C) detector and RF stages only (D) detector RF and IF stages

1996 TWO MARKS

8.164 The number of bits in a binary PCM system is increased from n to

n 1. As a result, the signal to quantization noise ratio will improve by a factor

(A) nn 1

(B) 2( )/n n1

(C) 2 ( )/n n2 1 (D) which is independent of n

8.165 The auto correlation function of an energy signal has(A) no symmetry (B) conjugate symmetry

(C) odd symmetry (D) even symmetry

8.166 An FM signal with a modulation index 9 is applied to a frequency tripler. The modulation index in the output signal will be

(A) 0 (B) 3

(C) 9 (D) 27



by RK Kanodia

Now in 3 Volume






SOLUTIONS

8.1 Option (B) is correct.

In ideal Nyquist Channel, bandwidth required for ISI (Inter Symbol

reference) free transmission is

W 2Rb

Here, the used modulation is QAM32 (Quantum Amplitude modulationi.e., q 32or 2v 32 v 5 bitsSo, the signaling rate (sampling rate) is

Rb R5

(R" given bit rate)

Hence, for ISI free transmission, minimum bandwidth is

W kHzR R2 10b


Given, random variables U and V with mean zero and variances 41

and 91

i.e., U V 0

u2

41

and v2

91

so, P U 0$^ h 21

and P V 0$^ h 21

The distribution is shown in the figure below

f uu ^ h 2

e1

u

u2 2 u

2

f vv ^ h e21

v

v2 2 v

2

We can express the distribution in standard form by assuming

X 2uYu U0

u 2

and Y vYv

V0 3v 3

for which we have X U2 0 Y V2 0

and X2 4U 12

also, Y2 9V 12

Therefore, X Y is also a normal random variable with X Y 0Hence,

P X Y 0$^ h P X Y 021

#^ hor, we can say

P U V2 3 0#^ h 21

Thus, P V U3 2$^ h 21

8.3 Option (C) is correct.

The mean of random variables U and V are both zeroi.e., U V 0Also, the random variables are identicali.e., f uU ^ h f vV ^ hor, F uU ^ h F vV ^ hi.e., their cdf are also same. So, F uU ^ h F v2V2 ^ hi.e., the cdf of random variable V2 will be also same but for any instant V2 U$

Therefore, G x^ h F x^ hbut, x G x^ h xF x$ ^ hor, F x G x x^ ^h h6 @ 0#


Given, P U 1^ h P U 121^ h

where U is a random variable which is identical to V i.e.,

P V 1^ h P V 121^ h

So, random variable U and V can have following values U ,1 1; ,V 1 1

Therefore the random variable U V can have the following values, U V

1

1, 1 1, 1

1

When

When or

When

U V

U V u v

U V

2

0

2*

Hence, we obtain the probabilities for U V as follows

U V P U V^ h

221

21

41

#

021

21

21

21

21

# #b bl l2

21

21

41

#

Therefore, the entropy of the U V^ h is obtained as

H U V^ h logP U VP U V

12^ ^h h' 1/

4 2log log log41

21

41 42 2 2

42

21

42






23

8.5 Option (D) is correct.

For the shown received signal, we conclude that if 0 is the transmitted

signal then the received signal will be also zero as the threshold is

1 and the pdf of bit 0 is not crossing 1. Again, we can observe that

there is an error when bit 1 is received as it crosses the threshold.

The probability of error is given by the area enclosed by the 1 bit

pdf (shown by shaded region)

P (error when bit 1 received) .21 1 0 25

81

# #

or transmitted

receivedP

01b l 8

1

Since, the 1 and 0 transmission is equiprobable:

i.e., P 0^ h P 121^ h

Hence bit error rate (BER) is BER

transmittedreceived

transmittedreceivedP P P P

10 0

01 1b ^ b ^l h l h

081

21

#

161


The optimum threshold is the threshold value for transmission as

obtained at the intersection of two pdf. From the shown pdf. We

obtain at the intersection

(transmitted, received) ,54

51b l

we can obtain the intersection by solving the two linear eqs x y 1 pdf of received bit 0

y .x

20 5 pdf of received bit 1

Hence for threshold 54 , we have

BER

1transmitted

receivedtransmitted

receivedP P P P

00

10 1b ^ b ^l h l h

21

51

21

21

21

54

51

21

# # # ## #b bl l

201 <(BER for threshold 1)

Hence, optimum threshold is 54

8.7 Option (A) is correct.

The mean square value of a stationary process equals the total area

under the graph of power spectral density, that is

[ ( )]E X t2 ( )S f dfX3

3#or, [ ( )]E X t2 ( )S d

21

X3

3#or, [ ( )]E X t2 2 ( )S d

21

X0

#3# (Since the PSD is even)

[ ]intarea under the triangle egration of delta function1

21 1 10 61 2 4003# # #b l; E

1 6000 4006 @ 6400

[ ( )]E X t is the absolute value of mean of signal ( )X t which is also

equal to value of ( )X at ( )0 .

From given PSD

( )SX0 0

( )SX ( )X 02

( )X0

2 0

( )X0 0


For raised cosine spectrum transmission bandwidth is given as

BT (1 )W " Roll of factor

BT ( )R2

1b Rb" Maximum signaling rate

3500 ( . )R2

1 0 75b

Rb .1 753500 2 4000#


Entropy function of a discrete memory less system is given as

H logPP1

kkk

N

0

1 b l/where Pk is probability of symbol Sk .

For first two symbols probability is same, so

H log log logPP

PP

PP

1 1 1k

kk

N

11

22 3

1b b bl l l/

log log logP P P P P Pk k

k

N

1 1 2 2

3

1e o/

log logP P P P2 k k

k

N

3

1e o/ ( )P P P1 2

Now, P1 ,P P P2

So, Hl ( ) ( ) ( ) ( )log log logP P P P P Pk k

k

N

3

1= G/By comparing, Hl H , Entropy of source decreases.


Probability density function of uniformly distributed variables X

and Y is shown as

[ ( , )]maxP x y21& 0



by RK Kanodia

Now in 3 Volume






Since X and Y are independent.

[ ( , )]maxP x y21& 0 P X P Y

21

21b bl l

P X21b l shaded area

43

Similarly for Y : P Y21b l 4

3

So [ ( , )]maxP x y21& 0 4

343

169

#

Alternate Method:

From the given data since random variables X and Y lies in the

interval [ , ]1 1 as from the figure X , Y lies in the region of the

square ABCD .

Probability for , 1/2max X Y6 @ : The points for , /max X Y 1 26 @

will be inside the region of square AEFG .

So, ,maxP X Y216 @& 0 Area of square

Area ofABCD

AEFG4

2 223

23

169

#

#


In a coherent binary PSK system, the pair of signals ( )s t1 and ( )s t2

used to represent binary system 1 and 0 respectively.

( )s t1 sinTE t2

c

( )s t2 sinTE t2

c

where t T0# # , E is the transmitted energy per bit.

General function of local oscillator

( )t1 ( ),sinT

t2

c t T0#

But here local oscillator is ahead with 45c. so,

( )t1 ( )sinT

t2 45c c

The coordinates of message points are

s11 ( ) ( )s t t dtT

1 10#

( )sin sinTE

tT

t dt2 2 45c

T

c0

c# ( ) ( )sin sin

TE t t dt2 45c c

T

0c#

[ ( )]sin sinTE

Tt dt

2 221 45 2 45c

T

0c c#

( )sinTE dt

TE t dt1

21 1 2 45

T

c

T

0 0

0

c

1 2 34444444 4444444# #

E2

Similarly, s21 E2

Signal space diagram

Now here the two message points are s11 and s21.

The error at the receiver will be considered.

When : (i) s11 is transmitted and s21 received

(ii) s21 is transmitted and s11 received

So, probability for the 1st case will be as :

transmitted

receivedPss

11

21b l ( )P X 0 (as shown in diagram)

/P E N2 0_ i /P N E 2_ iTaking the Gaussian distribution as shown below :






Mean of the Gaussian distribution /E 2

Variance N2

0

Putting it in the probability function :

P NE2b l

Ne dx

22

1/

/

N

x E

0

0

2 2

2

0

2

3

` j#

Ne dx

1 /

N

x E

0

20

0

2

3

` j#

Taking, /

/

N

x E

2

2

0

t

dx Ndt

20

So, /P N E 2_ i e dt21

/

t

E N

2

0

23# Q

NE

0c m

where Q is error function.

Since symbols are equiprobable in the 2nd case

So,

transmitted

receivedPss

21

11b l QNE

0c m

So the average probability of error

transmitted

receivedtransmitted

receivedPss

Pss

21

11

21

21

11b bl l; E Q

NE Q

NE Q

NE

21

0 0 0c c cm m m= G

8.12 Option ( ) is correct.


General equation of FM and PM waves are given by

( )tFM ( )cosA t k m d2c c f

t

0; E#

( )tPM [ ( )]cosA t k m tc c p

For same maximum phase deviation.

[ ( )]k m t maxp ( )k m d2max

f

t

0; E#

2kp# 2 [ ( )]k x t maxf

where, ( )x t ( )m dt

0#

[ ( )]x t max 4

So, 2kp# 2 4k f #

kk

f

p 4


( )G sC s bs a

j bj a

Phase lead angle tan tana b

1 1a ak k tan

ab

a b

1

12

J

L

KKK

N

P

OOO

( )tan

ab

b a12c m

For phase-lead compensation 0

b a 0

b a

Note: For phase lead compensator zero is nearer to the origin as

compared to pole, so option (C) can not be true.


tan tana b

1 1a ak k

dd

/ /

a

a

b

b

1

1

1

10

2 2a ak k

a ab1

2

2

b b a1 1

2

2

a b1 1

ab a b1 12 b l

ab / secrad1 2 2#


Quantized 4 level require 2 bit representation i.e. for one sample

2 bit are required. Since 2 sample per second are transmitted we

require 4 bit to be transmitted per second.


In FM the amplitude is constant and power is efficient transmitted.

No variation in power.

There is most bandwidth efficient transmission in SSB- SC. because

we transmit only one side band.

Simple Diode in Non linear region ( Square law ) is used in

conventional AM that is simplest receiver structure.

In VSB dc. component exists.


We have ( )S fx { ( )} { ( )}expF R Fx2

2fe

The given circuit can be simplified as

Power spectral density of output is

( )S fy ( ) ( )G f S fx2

2

2( ( ) )

j f

f

2 1

2 1

f

f

2

2 2e

e

or ( )S fy 2

(4 1)f f2 2e


Highest frequency component in ( )m t is /f 4000 2 2000m Hz

Carrier frequency fC 1 MHz

For Envelope detector condition

/f1 C 1/RC fm

1 s 0.5 msRC


Four phase signal constellation is shown below



by RK Kanodia

Now in 3 Volume






Now d2 r r12

12

d2 r2 12

r1 / 0.707dd 2

M2

82

4

Applying Cooine law we have

d2 cosr r r242

222

22

2 2 1/r r 222

22 ( )r2 2 2

2

or r2 1.3065dd

2 2


Here Pe for 4 PSK and 8 PSK is same because Pe depends on d .

Since Pe is same, d is same for 4 PSK and 8 PSK.

Additional Power SNR

( ) ( )SNR SNR2 1

log logNoE

NoE10 10S S2 1b bl l

logEE10S

S

1

2b l

0.7071.3065log log log

dd

rr

rr10 20 20

1

2 2

1

2&a ak k

Additional SNR 5.33 dB


Conventional AM signal is given by

( )x t [ ( )] ( )cosA m t f t1 2C C

Where 1, for no over modulation.

In option (C)

( )x t ( ) ( )cosA m t f t141 2C C: D

Thus 141 and this is a conventional AM-signal without over-

modulation


Power P ( )

18 W26 2


Impulse response of the matched filter is given by

( )h t ( )S T t


Let response of LPF filters

( )H f 1, 1

0,

MHz

elsewhere

f*Noise variance (power) is given as

P 2 ( )H f N df2f

o0

2

2

o# (given)

df2 10 20

0

1 106

### 2

2

2 10 1020 6# # 2

2

2 1014

or 107


Probability of error is given by

Pe [ ( / ) ( / )]P P21 0 1 1 0

( / )P 0 1 . .e dn e0 5 0 5/

n a2 10

3#

where a 2 10 V6# and V107 1

( / )P 1 0 . e dn0 5/

n

a 2

3# . e0 5 10

Pe . e0 5 10


( )S t ( ) ( )sin sinc t c t500 700

( )S f is convolution of two signals whose spectrum covers 250 Hzf 1

GATE Electronics and Communication Topicwise Solved Paper by RK Kanodia & Ashish Murolia 3





and 350 Hzf 2 . So convolution extends

f 25 350 600 Hz

Nyquist sampling rate

N f2 2 600# 1200 Hz


For the given system, output is written as

( )y t [ ( ) ( . )]dtd x t x t 0 5

( )y t ( ) ( . )dtdx t

dtdx t 0 5

Taking Laplace on both sides of above equation

( )Y s ( ) ( )sX s se X s. s0 5

( )H s ( )( )

( )X s

Y ss e1 . s0 5

( )H f ( )jf e1 . f0 5 2# ( )jf e1 f

Power spectral density of output

( )S fY ( ) ( )H f S fX2 ( ) ( )f e S f1 f

X2 2

For ( )S f 0Y , e1 f 0

f ( )n f2 1 0

or f0 1 KHz


(2 ) (2 )cos cosf t f tm c � DSB suppressed carrier

( )cos f t2 c � Carrier Only

[ ( ) ]cos f f t2 c m � USB Only

[ ( ) ( )]cos cosf t f t1 2 2m c � USB with carrier


We have

( )p X 0 ( )p Y 021

( )p X 1 ( )p Y 141

( )p X 2 ( )p Y 241

Let X Y A2 �and X Y B0 �Now

( )P X Y X Y2 0 ( )

( )P B

P A B+

Event ( )P A B+ happen when X Y 2 and X Y 0. It is only

the case when X 1 and Y 1.

Thus ( )P A B+ 41

41

161

#

Now event ( )P B happen when

X Y 0 It occurs when X Y , i.e.

X 0 and Y 0 or

X 1 and Y 1 or

X 2 and Y 2

Thus ( )P B 21

21

41

41

41

41

166

# # #

Now ( )

( )P B

P A B+

//

6 161 16

61


The mean is

X ( )x p xi i

. . . . .1 0 1 2 0 2 3 0 4 4 0 2 5 0 1# # # # #

. . . . .0 1 0 4 1 2 0 8 0 5 .3 0

X2 ( )x p xi i2

. . . . .1 0 1 4 0 2 9 0 4 16 0 2 25 0 1# # # # #

. . . . .0 1 0 8 3 6 3 2 2 5 .10 2

Variance x2 X X2 2^ h

. ( )10 2 3 2 .1 2


( )m t cos sint t21

21

1 2

( )s tAM [ ( )]cosm t t1 c

Modulation index ( )

V

m tmax

c

m 21

21

212 2` `j j

%m

m

2100

2

2

#

% %2

100 202

1 2

21 2

#``jj


We have C1 logBNS12` j

logBNS

2. ` j As NS 1

If we double the NS ratio then

C2 logBNS2

2. ` j log logB B

NS22 2. B C1.


We have SNR . n1 76 6

or .43 5 . n1 76 6

n6 . .43 5 1 76

n6 . n41 74 7� .

No. of quantization level is

27 128

Step size required is

V V128H L

( )128

5 512810

.078125 .0667.


For positive values step size

s+ .0 05 V

For negative value step size

s- .0 1 V

No. of quantization in ive is

.s

50 055 100

+

Thus 2n+ n100 7� +

No. of quantization in ve



by RK Kanodia

Now in 3 Volume






Q1 .s5

0 15 50

-

Thus 2n-

n50 6� -

NS

+` j . n1 76 6 + . .1 76 42 43 76 dB

NS

-` j . n1 76 6 - 1.76 .36 37 76= + dB

Best NS

0` j .43 76 dB


We have ( )x tAM cos cos cosA t t2c c m c

cos cosAA

t t1 2C

cm cc m

For demodulation by envelope demodulator modulation index

must be less than or equal to 1.

Thus A2c 1#

Ac 2$

Hence minimum value of A 2c


CDF is the integration of PDF. Plot in option (A) is the integration

of plot given in question.


The entropy is

H logpp1

iii

m

2

1=

� bits

Since p1 ...p pn1

n2

H logn

n1

i

n

1=

� logn


PSD of noise is N20 K ...(1)

The 3-dB cut off frequency is

fc RC21 ...(2)

Output noise power is

RCN

40 N

RC2 210c m K fc


At receiving end if we get two zero or three zero then its error.

Let p be the probability of 1 bit error, the probability that

transmitted bit error is

= Three zero + two zero and single one

( )C p C p p3 133

32

2

( )p p p13 2


Bandwidth of TDM is

21 (sum of Nyquist Rate)

[ ]W W W W21 2 2 4 6 W7


We have i ( ) . ( )sin sint t t2 10 5 2 1500 7 5 2 10005

2 10 10 1500 (2 1500 ) 15 1000 (2 1000 )cos cosdtd

t tii 5

Maximum frequency deviation is

max3 ( . )2 5 1500 7 5 1000# #

fmax3 15000

Modulation index is f

f

150015000 10max

m

3



fm 4 KHz

fs f2 8m kHz

Bit Rate Rb nfs 8 8 64# kbps

The minimum transmission bandwidth is

BW R2

32b kHz


NS

0

0c m . n1 76 6 dB

.1 76 6 8# .49 76 dB We have n 8


As Noise L

12

\

To reduce quantization noise by factor 4, quantization level must

be two times i.e. L2 .

Now L 2 2 256n 8

Thus L2 512


Autocorrelation is even function.


Power spectral density is non negative. Thus it is always zero or

greater than zero.


The variance of a random variable x is given by

[ ] [ ]E X E X2 2


A Hilbert transformer is a non-linear system.


Slope overload distortion can be reduced by increasing the step size

Ts3

$ slope of ( )x t


We have ( )p t ( )

( )sin

Wt W t

Wt

4 1 16

42 2

at tW41 it is

00 form. Thus applying 'L Hospital rule

)p( W4

1

[ ]

( )cos

W W t

W Wt

4 1 48

4 42 2






( )cos

W t

Wt

1 48

42 2

.cos1 3

0 5


The block diagram is as shown below

Here ( )M f1 ( )M ft

( )Y f1 ( )M f e e2

j B j B2 2-

c m ( )Y f2 ( )M f e e

2

j B j B

1

2 2-

c m ( )Y f ( ) ( )Y f Y f1 2

All waveform is shown below


By Binomial distribution the probability of error is

pe ( )C p p1nrr n r-

Probability of at most one error

Probability of no error + Probability of one error

( ) ( )C p p C p p1 1n n n n0

0 01

1 1- -

( ) ( )p np p1 1n n 1-


Bandwidth allocated for 1 Channel 5 M Hz

Average bandwidth for 1 Channel 55 1 MHz

Total Number of Simultaneously Channel 1 8 40Mk200# Channel


Chip Rate RC .1 2288 106# chips/sec

Data Rate Rb GRC

Since the processing gain G must be at least 100, thus for Gmin we

get

R maxb GR

min

C .100

1 2288 106# .12 288 103

# bps


Energy of constellation 1 is

Eg1

( ) ( ) ( ) ( ) ( )a a a a0 2 2 2 2 22 2 2 2 2

a a a a2 2 2 82 2 2 2 a16 2

Energy of constellation 2 is

Eg2 a a a a2 2 2 2 a4 2

Ratio E

E

a

a

416 4

g

g

2

1

2

2


Noise Power is same for both which is N20 .

Thus probability of error will be lower for the constellation 1 as it

has higher signal energy.


Area under the pdf curve must be unity

Thus a a b2 4 4 1

a b2 8 1 ...(1)

For maximum entropy three region must by equivaprobable thus

a2 b b4 4 ...(2)

From (1) and (2) we get

b 121 and a

61

8.60 Option (*) is correct.


A LPF will not produce phase distortion if phase varies linearly

with frequency.

( ) \

i.e. ( ) k


Let ( )m t is a low pass signal, whose frequency spectra is shown

below

Fourier transform of ( )g t

( )G t .

( )f k0 5 10

1 20 10k

43

##

3

3

/Spectrum of ( )G f is shown below



by RK Kanodia

Now in 3 Volume






Now when ( )m t is sampled with above signal the spectrum of

sampled signal will look like.

When sampled signal is passed through a LP filter of BW 1 kHz,

only ( )m t will remain.


The highest frequency signal in ( )x t is 1000 3 3# kHz if expression

is expanded. Thus minimum frequency requirement is

f 2 3 10 6 103 3# # # Hz


We have

( ) 125 [ ( ) ( 1)] (250 125 )[ ( 1) ( 2)]x t t u t u t t u t u t

The slope of expression ( )x t is 125 and sampling frequency fs is

32 1000# samples/sec.

Let 3 be the step size, then to avoid slope overload

Ts3 $ slope ( )x t

fc3 $ slope ( )x t

320003# 125$

3 32000125

$

3 2 8-


The sampling frequency is

fs 0.031 33

m kHz

Since f f2s m$ , the signal can be recovered and are correlated.


We have .p 0 251 , .p 0 252 and .p 0 53

H logpp1

i

1 211

3

=

� bits/symbol

log log logpp

pp

pp

1 1 11 2

12 2

23 2

3

..

..

..

log log log0 250 251 0 25

0 251 0 5

0 51

2 2 2

. . .log log log0 25 4 0 25 4 0 5 22 2 2

. .0 5 0 521

23 bits/symbol

Rb 3000 symbol/sec

Average bit rate R Hb

23 3000# 4500 bits/sec


The diagonal clipping in AM using envelop detector can be avoided

if

1c RC

W1

But from RC1

cossinWt

W Wt

1$

We can say that RC depends on W , thus

RC W1



When /23 is added to ( )y t then signal will move to next quantization

level. Otherwise if they have step size less than 23 then they will be

on the same quantization level.

8.70 Option (C) is correct.After the SSB modulation the frequency of signal will be f fc m i.e.

1000 10 kHz 1000. kHz

The bandwidth of FM is

BW ( ) f2 1 3

For NBFM 1, thus

BWNBFM f23. ( )2 10 10 2 109 6 9#.


We have ( )p t ( ) ( )u t u t 1

( )g t ( )* ( )p t p t

( )s t ( ) ( )* ( )g t t g t2 ( ) ( )g t g t 2

All signal are shown in figure below :

The impulse response of matched filter is

( )h t ( )s T t ( )s t1

Here T is the time where output SNR is maximum.


We have ( )x tAM [ ( ) . ( )]cosP t g t t10 0 5 c

where ( )p t ( ) ( )u t u t 1

and ( )g t ( ) ( ) ( )r t r t r t2 1 2

For desired interval t0 1# # , ( )p t 1 and ( )g t t , Thus we

have,

( )x tAM ( . )cost t100 1 0 5 c






Hence modulation index is 0.5


We know that ( )SYY ( ) . ( )H SXX2

Now ( )SYY 16

162 and ( )S 1XX white noise

Thus 16

162 ( )H 2

or ( )H 16

42

or ( )H s s4

4

which is a first order low pass RL filter.


We have R sLR

s44

or sL

RLR

s4

4

Comparing we get L 1 H and R 4


We have ( )x tAM ( . )sin cosf t f t10 1 0 5 2 2m c

The modulation index is 0.5

Carrier power Pc ( )

210

502

Side band power Ps ( )

210

502

Side band power Ps m P

2c

2

( . ) ( )

.2

0 5 506 25

2


Mean noise power Area under the PSD curve

BN4

21

2o

# #; E BNo

The ratio of average sideband power to mean noise power is

Noise Power

Side Band Power .N B N B6 25

425o0


{ ( )} ( )sinkm t A t1 c � Amplitude modulation

( ) ( )dm t A tsin c � DSB-SC modulation

{ ( )}sin cosA t km t � Phase Modulation

[ ] ( )sinA k m t dtct t

3- � Frequency Modulation


VSB f fm c�

DSB - SC f2 m� SSB fm

� AM f2 m�Thus SSB has minimum bandwidth and it require minimum pow-

er.


Let ( )x t be the input signal where

( )x t ( )cos cos cost tm1

( )y t ( )x t2 ( )cos cost t

21

22 2c m1

Here 2 1 and f

f

590 18

m1

3

BW ( ) f2 1 m ( )2 2 18 1 5# # 370 kHz


The transfer function of matched filter is

( )h t ( ) ( )x t t x t2

The output of matched filter is the convolution of ( )x t and ( )h t as

shown below


We have ( )H f e2 j td-

( )H f 2

( )G f0 ( ) ( )H f G fi2

N4 o W/Hz

The noise power is N B4 o #


As the area under pdf curve must be unity

( )k21 4 # k1

21�

Now mean square value is

v2 ( )v p v dv2

3

3

-

+�

v v dv8

2

0

4

` j�

as ( )p v v81

v dv8

3

0

4 c m�

8


The phase deviation is

f

f

110 10

m

3

If phase deviation remain same and modulating frequency is

changed

BW ( ) f2 1 'm ( )2 10 1 2 44 kHz


As the area under pdf curve must be unity and all three region are

equivaprobable. Thus are under each region must be 31 .

a241

# a31

32�


Nq ( )x p x dxa

a2

-

+�

2 x dx41a 2

0$

� x a

21

3 6

a3

0

3

; ESubstituting a

32 we have

Nq 814



by RK Kanodia

Now in 3 Volume







When word length is 6

NSN 6=

` j 2 22 6 12#

When word length is 8

NSN 8=

` j 2 22 8 16#

Now NSN

NSN

6

8=

=^^

hh

22 2 16

12

164

Thus it improves by a factor of 16.


Carrier frequency fc 1 106# Hz

Modulating frequency

fm 2 103# Hz

For an envelope detector

f2 c Rc

f1 2 m

f2

1c RC

f21m

f2

1c RC

f21m

2 10

16 RC

2 101

3#

.1 59 10 7#

- .RC 7 96 10 5#

-

so, 20 sec sec best lies in this interval.


( )S tAM [ . ]cos cosA t t1 0 1c m m

( )s tNBFM [ . ]cos sinA t t0 1c c m

( )s t ( ) ( )S t S f tAM NB m

[ . ] ( . )cos cos cos sinA t t A t t1 0 1 0 1c m c c c m

0.1cos cos cosA t A t tc c c m c= +

(0.1 ) . ( . )cos cos sin sin sin sinA t t A t t0 1c c m c c m+

As . sin t0 1 m .0 1, to .0 1

so, ( . )cos sin t0 1 m 1.

As when is small cos 1. and sin , , thus

( . )sin sin t0 1 m

0.1 sincos cos cost t A tc m c c= +

0.1 sin sinA t tc m c

. ( )cos cosA t A t2 0 1cosec

c c c c m

USB

� � �� Thus it is SSB with carrier.


Consecutive pulses are of same polarity when modulator is in slope

overload.

Consecutive pulses are of opposite polarity when the input is

constant.


( )F x X x1 2# ( ) ( )p X x P X x2 1

or ( )P X 1 ( 1 ) ( 1 )P X P X= = =+ -

. .0 55 0 25 .0 30


The SNR at transmitter is

SNRtr BPN

tr

10 100 10

1020 6

3

# #-

-

109=

In dB SNRtr log10 10 909 dB

Cable Loss 40 db

At receiver after cable loss we have

SNRRc 90 40 50 dB


The impulse response of matched filter is

( )h t ( )x T t

Since here T 4, thus

( )h t ( )x t4

The graph of ( )h t is as shown below.

From graph it may be easily seen that slope between t3 4 is

1.


The required bandwidth of M array PSK is

BW nR2 b

where M2n and Rb is bit rate

For BPSK, M n2 2 1n �Thus B1

R1

2 2 10 20b# kHz

For QPSK, M n4 2 2n �Thus B2

R2

2 10b kHz


We have fc 100 MHz 100 106# and f 1m

MHz

1 106#

The output of balanced modulator is

( )V tBM [ ][ ]cos cost tc c

[ ( ) ( ) ]cos cost t21

c m c m

If ( )V tBM is passed through HPF of cut off frequency f 100 10H6

#

, then only ( )c m passes and output of HPF is

( )V tHP ( )cos t21

c m






Now ( )V t0 ( ) ( )sinV t t2 100 10HP6

# #

[2 100 10 2 1 10 ] (2 100 10 )cos sint t21 6 6 6

# # # # #

[ ] ( )cos sint t21 2 10 2 10 2 108 6 8

[ (2 10 ) (2 10 )] [2 10 (2 10 ) 2 10 ]cos cos sin sin sint t t t t t21 8 6 8 6 8

( )cos cos sin sint t t t21 2 10 2 10 1

21 2 10 2 106 8 6 8` j

This signal is in form

cos sinA t B t2 10 2 108 8

Now

GATEELECTRONICS & COMMUNICATION

by

RK Kanodia & Ashish Murolia

in 3 Volumes

Fully Revised

Every Unit Contain All Questions of Last 15 Year Papers

Purchase from Nodia Online Store at Maximum Discount with free

Shipping

The envelope of this signal is

A B2 2

( ) (cos sint t21 2 10 1

21 2 106 2 6 2` `j j

( ) ( ) ( )cos sin sint t t41 2 10 1

41 2 10 2 102 6 2 6 6

( )sin t41 1 2 106

( )sin t45 2 106


( )s t

[ ] [ . ]cos cosA t A t2 10 10 2 10 1 103 3# #

Here T1 sec10 10

1 1003

#

and T2 .

sec10 1 10

1 993

#

Period of added signal will be LCM [ , ]T T1 2

Thus T [ , ] secLCM 100 99 9900

Thus frequency f .9900

1 0 1 kHz


The pdf of transmission of 0 and 1 will be as shown below :

Probability of error of 1

( . )P X0 0 2# # .0 2

Probability of error of 0 :

( . . )P X0 2 0 25# # .0 05 2# .0 1

Average error ( . ) ( . . )P X P X

20 0 2 0 2 0 25# # # #

. . .0

0 2 0 1 0 15


The square mean value is

2 ( ) ( )x x f x dxq2

3

3

-

� ( ) ( )x x f x dxq

2

0

1�

( ) ( ) ( . ) ( )x f x dx x f x dx0 0 7.

..2 2

0 3

0 1

0

0 3��

.x x x x3 3

0 49 142

.

.

3

0

0 3 3 2

0 3

1; ;E Eor 2 .0 039

RMS 2 .0 039 .0 198


FM � Capture effect

DM � Slope over load

PSK � Matched filter

PCM � law


Since f f2s m , the signal frequency and sampling frequency are as

follows

fm1 1200 Hz 2400� samples per sec



Thus by time division multiplexing total 4800 samples per second

will be sent. Since each sample require 12 bit, total 4800 12# bits

per second will be sent

Thus bit rate Rb .4800 12 57 6# kbps


The input signal ( )X f has the peak at 1 kHz and 1 kHz. After

balanced modulator the output will have peak at f 1c ! kHz i.e. :

10 1! 11� and 9 kHz

( )10 1! 9� and 11 kHz

9 kHz will be filtered out by HPF of 10 kHz. Thus 11 kHz will re-

main. After passing through 13 kHz balanced modulator signal will

have 13 11! kHz signal i.e. 2 and 24 kHz.

Thus peak of ( )Y f are at 2 kHz and 24 kHz.



by RK Kanodia

Now in 3 Volume







The input is a coherent detector is DSB - SC signal plus noise.

The noise at the detector output is the in-phase component as the

quadrature component ( )n tq of the noise ( )n t is completely rejected

by the detector.


The noise at the input to an ideal frequency detector is white. The

PSD of noise at the output is parabolic


We have Pe E

21

2erfc d

= c mSince Pe of Binary FSK is 3 dB inferior to binary PSK


The pdf of Z will be convolution of pdf of X and pdf of Y as shown

below.

Now [ ]p Z z# ( )f z dzZ

z

3-

� [ ]p Z 2# ( )f z dzZ

2

3-

-�

Area [ ]z 2#

21

61 1

121

# #


We have ( )RXX ( )e4 1.0 2-

( )R 0XX ( )e4 1 8.0 2 0 2-

or 2 2 Given

mean 0

Now ( )P x 1# ( )F 1x

1 QXc m at x 1

1 Q2 21 0c m 1 Q

2 21c m


W Y Z

[ ]E W2 [ ]E Y Z 2 [ ] [ ] [ ]E Y E Z E YZ22 2

w2

We have [ ( )]E X t2 ( )R 10x

[ ]e4 1.0 2 0- [ ]4 1 1 8

[ ]E Y2 [ ( )]E X 2 82

[ ]E Z2 [ ( )]E X 4 82

[ ]E YZ ( )R 2XX 4[ 1]e . ( )0 2 4 2 .6 68

[ ]E W2 .8 8 2 6 68w2

# .2 64


Step size . .L

m2128

1 536 0 012p V

Quantization Noise power ( . )

12 120 0122 2

12 10 6# V2


The frequency of pulse train is

f101

3- 1 k Hz

The Fourier Series coefficient of given pulse train is

Cn TAe dt

1/

/

o

jn t

T

T

2

2o

o

o#

TAe dt1

/

/

o

j t

T

T

6

6o

o

o#

( )[ ]

T jA e /

/

o o

j tTT

66o

o

o

( )

( )j nA

e e2

/j t j T 6o o o

( )j nAe e

2/ /j j3 3

or Cn sinnA n

3` jFrom Cn it may be easily seen that , , , ,1 2 4 5 7, harmonics are present

and , , , ,..0 3 6 9 are absent. Thus ( )p t has 1 kHz, 2 kHz, 4 kHz, 5 kHz,

7 kHz,... frequency component and 3 kHz, 6 kHz.. are absent.

The signal ( )x t has the frequency components 0.4 kHz and 0.7

kHz. The sampled signal of ( )x t i.e. ( )* ( )x t p t will have

.1 0 4! and .1 0 7! kHz

.2 0 4! and .2 0 7! kHz

.4 0 4! and .4 0 7! kHz

Thus in range of 2.5 kHz to 3.5 kHz the frequency present is

.2 0 7 .2 7 kHz

.4 0 7 .3 3 kHz


vi ( ) ( )cosA f t m t2c c1

v0 a v avo i i3

v0

[ ( ) ( )] [ ( ) ( )]cos cosa A f t m t a A f t m t2 2' ' ' 'c c c c0 1

3

(2 ) ( ) [( 2 )cos cosa A f t a m t a A f t' ' ' 'c c c c0 0 1

3= + +






( (2 ) ) ( ) ( ) ( ) ( )]cos cosA f t m t A f t m t m t3 2' ' ' 'c c c c

2 2 3+

(2 ) ( ) ( 2 )cos cosa A f t a m t a A f t' ' ' 'c c c c0 0 1

3= + +

3( )

( )cos

a Af tm t

21 4'

'

cc

12

++; E

( ) ( ) ( )cosa A f t m t m t3 2' 'c c1

2 3

The term 3 ( ) ( )a A m t' 4cosc

f t1 2

'

c is a DSB-SC signal having carrier

frequency 1. MHz. Thus f2 1'c MHz or .f 0 5'

c MHz


PT P 12c

2

c m

Psb ( . )P P

2 20 5c c

2 2

or PP

c

sb 81


AM Band width f2 m

Peak frequency deviation ( )f f3 2 6m m

Modulation index f

f66

m

m

The FM signal is represented in terms of Bessel function as

( )x tFM ( ) ( )cosA J n tc n c n

n 3

3

=-

� nc m ( )2 1008 103

#

n2 10 4 106 3# ( ),n2 1008 10 43

#

Thus coefficient ( )J5 64


Ring modulation � Generation of DSB - SC

VCO � Generation of FM

Foster seely discriminator � Demodulation of fm

mixer � frequency conversion


fmax 1650 450 2100 kHz

fmin 550 450 1000 kHz

or f LC2

1

frequency is minimum, capacitance will be maximum

R ( . )CC

f

f2 1

min

max

min

max2

22

or R .4 41

fi f f2c IF ( )700 2 455 1600 kHz


Eb 10 6- watt-sec

No 10 5- W/Hz

(SNR) matched filler .E

2 1010 05

No

25

6

o

#-

(SNR)dB ( . )log10 10 0 05 13 dB


For slopeoverload to take place Ef

f

2mm

s3$

This is satisfied with .E 1 5m V and f 4m kHz


If s carrier synchronization at receiver

represents bandwidth efficiency

then for coherent binary PSK .0 5 and s is required.


Bit Rate k8 8 64# kbps

(SNR)q . . n1 76 6 02 dB

. . .1 76 6 02 8 49 8# dB


The frequency of message signal is

fc 1000 kHz

1 The frequency of message signal is

fm 100 10

1 106

#-

kHz

Here message signal is symmetrical square wave whose FS has only

odd harmonics i.e. 10 kHz, 30 kHz 50 kHz. Modulated signal con-

tain f fc m! frequency component. Thus modulated signal has

f fc m! ( )1000 10! kH 1010 kHz, 990 kHz

f f3c m! ( )1000 10! kH 1030 kHz, 970 kHz

Thus, there is no 1020 kHz component in modulated signal.


We have ( )y t ( ) ( )x t t nT5 10 s

n

6#

3

3-

=-

+� ( )x t ( )cos t10 8 103

#

Ts sec100

The cut off fc of LPF is 5 kHz

We know that for the output of filter

( ) ( )T

x t y t

s

( )cos t

100 10

10 8 10 5 106

3 6

#

# # #

-

-

( )cos t5 10 8 101 3# #

-


Transmitted frequencies in coherent BFSK should be integral of bit

rate 8 kHz.


For best reception, if transmitting waves are vertically polarized,

then receiver should also be vertically polarized i.e. transmitter and

receiver must be in same polarization.


( )s t ( )cos sin cost t t2 2 10 30 150 40 1506#

{ ( )}cos sint t4 10 100 1506

Angle modulated signal is

( )s t { ( )}cos sinA t tc m

Comparing with angle modulated signal we get

Phase deviations 100

Frequency deviations

f3 .f 1002150 7 5m # kHz



by RK Kanodia

Now in 3 Volume






8.123 Option (*) is correct.

We have ( ) ( )m t s t ( )y t1

( ) ( )sin cos

t

t t2 2 200

( ) ( )sin sin

t

t t202 198

( ) ( )y t n t1

( ) sin sin siny ttt t

tt202 198 199

2

( ) ( )y t s t2 ( )u t

[ ]sin sin sin cost

t t t t202 198 199 200

[ (402 ) (2 ) { (398 ) (2 )}sin sin sin sint t t t21

= +

(399 ) ( )]sin sint t+

After filtering

( )y t ( ) ( ) ( )sin sin sin

t

t t t

22 2

( ) ( . ) ( . )sin sin cos

t

t t t

22 2 0 5 1 5

. 1.5sin sin costt

tt t

22 0 5


The signal frequency is

fm 2

24 10 123

kHz

Ts sec fT

50 1501 10 20s

s

6 #

kHz

After sampling signal will have f fs m! frequency component i.e. 32

and 12 kHz

At filter output only 8 kHz will be present as cutoff frequency is

15 kHz.


( )d n ( ) ( )x n x n 1

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

or [ ( )]E d n 2

[ ( )] [ ( )] [ ( ) ( )]E x n E x n E x n x n1 2 12 2

or d2 ( )R2 1x x xx

2 2 as k 1

As we have been given d2

10x2

, therefore

10x2

( )R2 1x x xx2 2

or ( )R2 1xx 1019

x2

or R

x

xx2

.2019 0 95

8.126 Option (A) is correct.An ideal low - pass filter with appropriate bandwidth fm is used to recover the signal which is sampled at nyquist rate f2 m .


For any PDF the probability at mean is 21 . Here given PDF is

Gaussian random variable and X 4 is mean.


We require 6 bit for 64 intensity levels because 64 26

Data Rate = Frames per second # pixels per frame # bits per pixel

625 400 400 6 600# # # Mbps sec


We have

( ) ( )sin sinc t c t700 500 ( ) ( )sin sint

t

t

t

700700

500500

Here the maximum frequency component is f2 700m i.e.

f 350m Hz

Thus Nyquist rate fs f2 m

( )2 350 700 Hz

Thus sampling interval sec7001


Probability of error p

Probability of no error ( )q p1

Probability for at most one bit error

Probability of no bit error

+ probability of 1 bit error

( ) ( )p np p1 1n n 1-


If ( ) ( )g t GFT

then PSD of ( )g t is

( )Sg ( )G 2

and power is

Pg ( )S d21

g3

3

-

�Now ( )ag t ( )aG

FT

PSD of ( )ag t is

( )Sag ( ( ))a G 2

( )a G2 2

or ( )Sag ( )a Sg2

Similarly Pag a Pg2


The envelope of the input signal is [ ( )]k m t1 a that will be output

of envelope detector.


Frequency Range for satellite communication is 1 GHz to 30 GHz,


Waveform will be orthogonal when each bit contains integer number

of cycles of carrier.

Bit rate Rb ( , )HCF f f1 2

( , )HCF k k10 25






5 kHz

Thus bit interval is Tb R k1

51

b .0 2 msec 200 sec


We have Pm ( )m t2

The input to LPF is

( )x t ( ) ( )cos cosm t t to o

( )

[ ( ) ]cos cosm t

t2

2 o

( ) ( ) ( )cos cosm t t m t

22

2o

The output of filter will be

( )y t ( )cosm t

2

Power of output signal is

Py ( )y t2 ( )cosm t41 2 2 cosP

4m

2


Hilbert transformer always adds 90c to the positive frequency

component and 90c to the negative frequency component.

Hilbert Trans form

cos t sin t sin t cos t Thus cos sint t1 2 sin cost t1 2


We have ( )x t { }cos sinA t tc c m

( )y t { ( )}x t 3

( ) ( )cos sin cos sinA t t t t3 3 3c c m c m2

Thus the fundamental frequency doesn’t change but BW is three

times.

BW ( ') ( )f f2 2 3 33 3 # MHz



This is Quadrature modulated signal. In QAM, two signals having

bandwidth. &B B1 2 can be transmitted simultaneous over a

bandwidth of ( )B B Hz1 2

so . .B W (15 10) 25 kHz


A modulated signal can be expressed in terms of its in-phase and

quadrature component as

( )S t

( ) ( ) ( ) ( )cos sinS t f t S t f t2 2c Q c1

Here ( )S t

[ ] ( )cos sin sine cps t t e t t tatc

atc

[ ] [ ]cos cos sin sine t f t e t f t2 2atc

atc

( ) ( )cos sinS t f t S t f t2 2c Q c1

Complex envelope of ( )s t is

( )S t ( ) ( )S t jS tQ1

cos sine t je tat at

[ ]cos sine t j tat

( ) ( ) ( )exp expat j t t


Given function

( )g t

( ) ( )sin sinc t c t6 10 400 10 1004 2 6 3# )

Let ( )g t1 ( )sinc t6 10 4004 2#

( )g t2 ( ) ( )sinc t10 1006 3

We know that ( ) ( ) ( ) ( )g t g t G G1 2 1 2?) occupies minimum of

Bandwidth of ( ) ( )orG G1 2

Band width of ( )G1

2 400 800 / 400secrad or Hz#

Band width of ( )G2

3 100 300 / 150secrad or Hz#

Sampling frequency 2 150 300 Hz#


For a sinusoidal input ( )dBSNR is PCM is obtained by following

formulae.

( )dBSNR . n1 8 6 n is no. of bits

Here n 8

So, ( )dBSNR .1 8 6 8# .49 8


We know that matched filter output is given by

( )g t0 ( ) ( )g g T t d03

3# at

t T0

( )g tmax

06 @ ( ) ( )g g d3

3# ( )g t dt2

3

3# [10 (2 10 ) ]sin dt

1 106 2

0

4

###

[ ( )]g t max0 21 100 10 4# # 5 mV


Sampling rate must be equal to twice of maximum frequency.

f s 2 400# 800 Hz


The amplitude spectrum of a gaussian pulse is also gaussian as

shown in the fig.

( )f yY expy

21

2

2c m




by RK Kanodia

Now in 3 Volume






Let the rectangular pulse is given as

Auto correlation function is given by

( )Rxx ( ) ( )Tx t x t dt

1/

/

T

T

2

2

#When ( )x t is shifted to right ( ), ( )x t0 will be shown as dotted

line.

( )Rxx T

A dt1 2

2

2

T

T

#

TA T T

TA T

2 2 2

2 2: :D D( ) can be negative or positive, so generalizing above equations

( )Rxx TA T

2

2: D( )Rxx is a regular pulse of duration T .


Selectivity refers to select a desired frequency while rejecting all

others. In super heterodyne receiver selective is obtained partially

by RF amplifier and mainly by IF amplifier.


In PCM, SNR 2 n2

so if bit increased from 8 to 9

( )( )SNR

SNR

2

1 22 2

41

2 9

2 82

#

#

so SNR will increased by a factor of 4


In flat top sampling an amplitude distortion is produced while

reconstructing original signal ( )x t from sampled signal ( )s t . High

frequency of ( )x t are mostly attenuated. This effect is known as

aperture effect.


Carrier ( )C t ( )cos te

Modulating signal ( )x t

DSB - SC modulated signal ( ) ( )x t c t ( ) ( )cosx t te

envelope ( )x t


In Quadrature multiplexing two baseband signals can transmitted

or modulated using I4 phase & Quadrature carriers and its quite

different form FDM & TDM.


Fourier transform perform a conversion from time domain to

frequency domain for analysis purposes. Units remain same.


In PCM, SNR is depends an step size (i.e. signal amplitude) SNR

can be improved by using smaller steps for smaller amplitude. This

is obtained by compressing the signal.


Band width is same for BPSK and APSK(OOK) which is equal to

twice of signal Bandwidth.


The spectral density of a real value random process symmetric about

vertical axis so it has an even symmetry.



It is one of the advantage of bipolar signalling (AMI) that its

spectrum has a dc null for binary data transmission PSD of bipolar

signalling is


Probability Density function (PDF) of a random variable x defined

as






( )P xx e21 /x 22

so here K 21


Here the highest frequency component in the spectrum is 1.5 kHz

[at 2 kHz is not included in the spectrum]

Minimum sampling freq. .1 5 2# 3 kHz


We need a high pass filter for receiving the pulses.


Power spectral density function of a signal ( )g t is fourier transform

of its auto correlation function

( )Rg ( )SgF

( )here Sg ( )sinc f2

so ( )R tg is a triangular pulse.

[ .]triangf ( )sinc f2


For a signal ( )g t , its matched filter response given as

( )h t ( )g T t

so here ( )g t is a rectangular pulse of duration T .

output of matched filter

( )y t ( ) ( )g t h t)

if we shift ( )g t for convolution ( )y t increases first linearly then

decreases to zero.


The difference between incoming signal frequency ( )fc and its image

frequency ( )fc is I2 f (which is large enough). The RF filter may

provide poor selectivity against adjacent channels separated by a

small frequency differences but it can provide reasonable selectivity

against a station separated by I2 f . So it provides adequate suppression

of image channel.


In PCM SNR is given by

SNR 23 2 n2

if no. of bits is increased from n to ( )n 1 SNR will increase by a

factor of 2 ( )/n n2 1


The auto correlation of energy signal is an even function.

auto correlation function is gives as

( )R ( ) ( )x t x t dt3

3

#

put ( )R ( ) ( )x t x t dt3

3

#Let t

dt d

( )R ( ) ( )x x d3

3

#

change variable t"

( )R ( ) ( ) ( )x t x t dt R3

3

#

( )R ( )R even function


Communication system 8

Engineering

Transcript of Communication system 8