GATE 2014 EC (EC3) – Question 59

Communications · Random Processes · 2 marks · Multiple choice

A WSS process X(t) has PSD S_X(f). For Y(t) = X(2t−1), find S_Y(f).

$\frac12S_X(f/2)e^{-j\pi f}$
$\frac12S_X(f/2)e^{-j\pi f/2}$
$\frac12S_X(f/2)$
$\frac12S_X(f/2)e^{-j2\pi f}$

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Correct answer: (C) $\frac12S_X(f/2)$

Explanation

For Y(t)=X(2t−1), the time shift cancels when forming autocorrelation, giving RY(τ)=RX(2τ). Fourier scaling then gives **SY(f)=(1/2)SX(f/2),C**. A PSD has no deterministic time-shift phase factor; such a factor belongs to a waveform transform.