Digital Communications Exam paper

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Semester- V

Academic Year 2015-16

The LNM Institute of Information Technology, Jaipur, Rajasthan Mid Semester Exam ECE325, Digital Communications Time : 1:30 Hours Maximum Marks : 50 Weightage : 25% ———————————————————————————————————————————  Instructions: •

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

 4 .5 MHz. This signal is sampled, quantized, 1) A television television signal signal (video and audio) has bandwidth bandwidth of  4 and binary coded to obtain a PCM signal a) Determ Determine ine the sampling sampling rate rate if the signal signal is to be sample sampled d at a rate 25%   above the Nyquist rate. b) If the the sampl samples es are are quant quantiz ized ed into into L   = 2048   levels, determine the number of binary digits required to encode each sample. c) Determ Determine ine the number number of binary binary digits per second second (bits/s) (bits/s) required required to encode encode this signal signal.. d) Find the minimum bandwidth bandwidth required required to transmit this signal. signal. [2+2+2+2=8 [2+2+2+2=8 marks] 2) The output SNR (signal-to-quanti (signal-to-quantizatio zation-noise n-noise ratio) ratio) of a 10-bit PCM was found to be 30dB. The desired SNR is 48dB. It was decided to increase the SNR to the desired value by increasing the number of quantization levels  L . Find the fractional increase in the transmission bandwidth required for this increase in L. [2 marks] 3) The following following four waveforms waveforms are used for signaling signaling in a digital digital communication communication system: 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), where u(·)  is the unit step function. a) Determine Determine the signal space space representation representation of the four signals s k (t), k  = 1, 2, 3, 4  by using two basis functions defined as

f 1 (t) =  u (t) − u(t − 1), f 2 (t) =  u (t − 1) − u(t − 2) b) Plot Plot the signal signal space space diagram diagram and also using using Gray encodi encoding, ng, label the signal points points with the corresponding 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 be transmitted at baseband by a line coding scheme, described by ∞

−∞

 ∞

X (t) =

an p(t − nT  − ∆),

n=−∞

where  ∆  is a random variable that is independent of the value of  an  and uniformly distributed over 0  ≤  ∆  < T  and p(t)  is shown in Fig. P4.  p(t) 1

0

T  2

T 

t

Fig. P4 a) b) c) d) e)

Write name of this line coding scheme. Derive the autocorrelation function of  X (t). Derive the power spectral density S X (f ) of  X (t). Roughly sketch this S X (f ). Determine the first null bandwidth (FNB) of the signal X (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), where u(·)   is the unit step function. a) Determine a set of orthonormal functions for the signals by using Gram-Schmidt Orthogonalisation starting with s1 (t)  and going in sequence. b) Determine the dimensionality of the signals. [4+1=5 marks] 6)

a) b) c) d) e)

Explain slope-overload distortion and granular noise of the delta modulation system. What is the function and purpose of the digital modulator? Sketch the signal space diagram for a π4 -QPSK signal with Gray encoding. Differentiate source coding and channel coding. 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 process X (t) and output random process Y  (t): d Y (t) =  X (t) Y  (t) + dt Assume that the input random process X (t)  is a real WSS process with autocorrelation function RX (τ ) = 1000 exp(−10|τ |) . a) Determine the frequency response H (ω )  and the impulse response h(t)   of the system. b) Find the power spectral density of the output random process Y (t). c) Find the power of the output random process Y(t). [2 + 3 + 4 = 9  marks]