Question 52 on communication from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.

## Q2. The power spectral density of a real process for positive frequencies is shown below. The values of and , respectively are

## (A)

## (B)

## (C)

## (D)

## Solution

For a wide sense stationary function, the auto-correlation with delay is defined as,

From the Weiner-Kinchin theorem, the auto-correlation function and power spectral density are Fourier Transform pairs, i.e.

Expressing in terms of , where

.

When the delay , the above equations simplifies to

.

Applying this to the problem at hand,

.

Further, since the power spectral density does not have any dc component, the mean of the signal

**Based on the above, the right choice is (B) **

## References

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

[3] Wide sense stationary function

[4] Auto-correlation

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