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GATE-2012 ECE Q3 (communication)

Posted By __Krishna Sankar__ On November 1, 2012 @ 10:07 am In __GATE__ | __No Comments__

Question 3 on Communication from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.

To answer this question, am using contents from Section 5.1 of **Digital Communication, third edition, John R. Barry, Edward A. Lee, David G. Messerschmitt** (buy from Amazon.com ^{[1]}, buy from Flipkart.com ^{[2]}).

Let us first try to understand the minimum bandwidth required to transmit the signal at symbol rate with no inter symbol interference (ISI).

The transmitted signal is,

,

where

is the transmitted data symbols,

is the symbol rate and

is the pulse shape.

The sampled signal is,

.

Decomposing the above equation into two parts,

.

The first term is the desired signal and the second term contributes to inter symbol interference.

To ensure that there is no intersymbol interference, the sampled pulse shape should be

.

Taking Fourier transform, this translates to the following criterion

.

**Note : **For convenience, take

.

Note :

The proof for this is discussed in bit more detail in Section 9.2 of **Digital Communications, 4th edition John G. Proakis** (buy from Amazon.com ^{[3]}, buy from Flipkart.com ^{[4]})

Let us assume that pulse shape has a bandwidth . There are three cases to check

**i) When :**

In this case, the plot of will have non-overlapping replicas of the spectrum separated by and there is no choice to meet the criterion.

**ii) When :**

There exists one candidate meeting the criterion,

.

The corresponding pulse shape is

.

Alternately, this can be stated as – for a given single sided bandwidth , the maximum symbol rate which can be achieved for ISI free transmission is . To meet this, the pulse shape has to be a sinc function. Typically usage of the sinc function is not preferred as its tails decay very slowly and a small timing error in the demodulator will result in an infinite series of inter symbol interference components.

**iii) When **

In this case consists of overlapping spectrum of separated by and there exists multiple choices meeting the criterion.

A commonly used pulse shaping filter satisfying the criterion while having a faster decay is the **raised cosine filters **having the following equation,

.

The frequency response is,

.

With a raised cosine pulse shape, the bandwidth is larger than the minimum required pulse shape and is related as,

.

Then term is called the excess bandwidth factor. For example,

- translates to 75% excess bandwidth
- translates to 100% excess bandwidth
- , the raised cosine pulse shape reduces to sinc pulse shape.

**In our question,**

and and the goal is to find the maximum possible symbol rate .

Substituting for , and solving for ,

.

**Matlab example**

% script for plotting the time and frequency response of raised cosine pulse shape % filter with % a) alpha = 0 (sinc pulse) % b) alpha = 3/4 % clear all; close all; fs = 10; % defining the sinc filter sincNum = sin(pi*[-fs:1/fs:fs]); % numerator of the sinc function sincDen = (pi*[-fs:1/fs:fs]); % denominator of the sinc function sincDenZero = find(abs(sincDen) < 10^-10); sincOp = sincNum./sincDen; sincOp(sincDenZero) = 1; % sin(pix/(pix) =1 for x =0 alpha = 0; cosNum = cos(alpha*pi*[-fs:1/fs:fs]); cosDen = (1-(2*alpha*[-fs:1/fs:fs]).^2); cosDenZero = find(abs(cosDen); cosOp(cosDenZero) = pi/4; gt_alpha0 = sincOp.*cosOp; GF_alpha0 = (1/fs)*fft(gt_alpha0,1024); alpha = 0.75; cosNum = cos(alpha*pi*[-fs:1/fs:fs]); cosDen = (1-(2*alpha*[-fs:1/fs:fs]).^2); cosDenZero = find(abs(cosDen); cosOp(cosDenZero) = pi/4; gt_alpha0 = sincOp.*cosOp; GF_alpha0 = (1/fs)*fft(gt_alpha0,1024); alpha = 0.75; cosNum = cos(alpha*pi*[-fs:1/fs:fs]); cosDen = (1-(2*alpha*[-fs:1/fs:fs]).^2); cosDenZero = find(abs(cosDen) cosOp(cosDenZero) = pi/4; gt_alpha_p75 = sincOp.*cosOp; GF_alpha_p75 = (1/fs)*fft(gt_alpha_p75,1024);

**Figure : Time domain plot**

**Figure : Frequency domain plot**

**Based on the above, the right choice is (C) 4000 symbols per second.**

[1] GATE Examination Question Papers [Previous Years] from Indian Institute of Technology, Madras http://gate.iitm.ac.in/gateqps/2012/ec.pdf ^{[5]}

[2] **Digital Communication, Third edition, John R. Barry, Edward A. Lee, David G. Messerschmitt** (buy from Amazon.com ^{[1]}, buy from Flipkart.com ^{[2]})

[3] **Digital Communications, Fourth edition John G. Proakis** (buy from Amazon.com ^{[3]}, buy from Flipkart.com ^{[4]})

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