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Semester- V Academic Year 2015-16

The LNM Institute of Information Technology,Jaipur, Rajasthan

Mid Semester ExamECE325, Digital Communications

Time : 1:30 Hours Maximum Marks : 50Weightage : 25%———————————————————————————————————————————Instructions:

• All questions are compulsory.• Make suitable assumptions if necessary; write them with your answer.

1) A television signal (video and audio) has bandwidth of4.5MHz. This signal is sampled, quantized,and binary coded to obtain a PCM signal

a) Determine the sampling rate if the signal is to be sampled at a rate25% above the Nyquistrate.

b) If the samples are quantized intoL = 2048 levels, determine the number of binary digitsrequired to encode each sample.

c) Determine the number of binary digits per second (bits/s)required to encode this signal.d) Find the minimum bandwidth required to transmit this signal.

[2+2+2+2=8 marks]

2) The output SNR (signal-to-quantization-noise ratio) ofa 10-bit PCM was found to be30dB. Thedesired SNR is48dB. It was decided to increase the SNR to the desired value by increasing thenumber of quantization levelsL. Find the fractional increase in the transmission bandwidth requiredfor this increase inL. [2 marks]

3) The following four waveforms are used for signaling in a digital communication system:

s1(t) = u(t)− 1.5u(t− 1) + 0.5u(t− 2),

s2(t) = −0.5u(t) + 1.5u(t− 1)− u(t− 2),

s3(t) = −u(t− 1) + u(t− 2),

s4(t) = 0.5u(t) + 0.5u(t− 1)− u(t− 2),

whereu(·) is the unit step function.a) Determine the signal space representation of the four signalssk(t), k = 1, 2, 3, 4 by using two

basis functions defined as

f1(t) = u(t)− u(t− 1),

f2(t) = u(t− 1)− u(t− 2)

b) Plot the signal space diagram and also using Gray encoding, label the signal points with thecorresponding data bits. [4+2=6 marks]

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4) The information sequence{an}∞n=−∞is a sequence of independent and identically distributed (iid)

random variables, each taking values+1 and −1 with equal probability. This sequence is to betransmitted at baseband by a line coding scheme, described by

X(t) =∞∑

n=−∞

an p(t− nT −∆),

where∆ is a random variable that is independent of the value ofan and uniformly distributed over0 ≤ ∆ < T andp(t) is shown in Fig. P4.

TT

2

0

1

t

p(t)

Fig. P4

a) Write name of this line coding scheme.b) Derive the autocorrelation function ofX(t).c) Derive the power spectral densitySX(f) of X(t).d) Roughly sketch thisSX(f).e) Determine the first null bandwidth (FNB) of the signalX(t). [1+3+2+2+2=10 marks]

5) Consider the four waveforms defined as:

s1(t) = u(t)− u(t− 1) + u(t− 2)− u(t− 3),

s2(t) = u(t− 1)− u(t− 2) + u(t− 3)− u(t− 4),

s3(t) = u(t− 1)− u(t− 3),

s4(t) = u(t− 1)− u(t− 2)− u(t− 3) + u(t− 4),

whereu(·) is the unit step function.a) Determine a set of orthonormal functions for the signals by using Gram-Schmidt Orthogonal-

isation starting withs1(t) and going in sequence.b) Determine the dimensionality of the signals. [4+1=5 marks]

6) a) Explain slope-overload distortion and granular noiseof the delta modulation system.b) What is the function and purpose of the digital modulator?c) Sketch the signal space diagram for aπ

4-QPSK signal with Gray encoding.

d) Differentiate source coding and channel coding.e) Sketch the input-output characteristic of two-level quantizer. [3 + 2 + 2 + 2 + 1 = 10 marks]

7) Consider the continuous-time LTI system with input random processX(t) and output random processY (t):

Y (t) +d Y (t)

dt= X(t)

Assume that the input random processX(t) is a real WSS process with autocorrelation functionRX(τ) = 1000 exp(−10|τ |) .

a) Determine the frequency responseH(ω) and the impulse responseh(t) of the system.b) Find the power spectral density of the output random processY (t).c) Find the power of the output random process Y(t). [2 + 3 + 4 = 9 marks]