The GATE Grind

GATE 2022 EC – Question 23

Communications · Random Processes · 1 mark · Multiple choice

The frequency response $H(f)$ of a linear time-invariant system has magnitude as shown in the figure.

Statement I: The system is necessarily a pure delay system for inputs which are bandlimited to $-\alpha\le f\le\alpha$.

Statement II: For any wide-sense stationary input process with power spectral density $S_X(f)$, the output power spectral density $S_Y(f)$ obeys $S_Y(f)=S_X(f)$ for $-\alpha\le f\le\alpha$.

Which one of the following combinations is true?

Diagram for GATE 2022 EC question 23
  1. Statement I is correct, Statement II is correct
  2. Statement I is correct, Statement II is incorrect
  3. Statement I is incorrect, Statement II is correct
  4. Statement I is incorrect, Statement II is incorrect

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Show answer and explanation

Correct answer: (C) Statement I is incorrect, Statement II is correct

Explanation

Only $|H(f)|=1$ is given; the phase is unspecified, so the system need not be a pure delay (which would also need a linear phase). Statement I is false. For a WSS input, $S_Y(f)=|H(f)|^2S_X(f)=S_X(f)$ in $|f|\le\alpha$, regardless of phase, so Statement II is true.