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GATE-2012 ECE Q15 (communication)

by Krishna Sankar on January 25, 2013

Question 15 on communication from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.

Q15. 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

Solution

Entropy of a random variable  is defined as ,

 .

Refer Chapter 2 in Elements of Information Theory, Thomas M. Cover, Joy A. Thomas (Buy from Amazon.comBuy from Flipkart.com)

Let us consider a simple case where  can take two values 1 and 0 with probability  and respectively, i.e.

.

The entropy of  is,

.

The plot of the entropy versus the probability is shown in the figure below.

clear all; close
p = [0:.001:1];
hx = -p.*log2(p) - (1-p).*log2(1-p);
plot(p,hx);
xlabel('probability, p'); ylabel('H(X)');
title('entropy versus probability, p'); 
axis([0 1 0 1]);grid on;

entropy_versus_probability

Figure : Entropy versus probability for binary symmetric source

It can be see that the entropy (also termed as uncertainty) is maximum when  and for other values of , the entropy is lower. The entropy becomes 0 when  i.e. when the value of  becomes deterministic. If we extend this to a source with more than two symbols,  when probability of one of the symbols  becomes more higher than the other, the uncertainty decreases and hence entropy also decreases.

Based on the above, the right choice is (D) decreases

 

References

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

[2] Elements of Information Theory, Thomas M. Cover, Joy A. Thomas (Buy from Amazon.com, Buy from Flipkart.com)

 

 

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