GATE 2014 EC (EC3) – Question 60
A real band-limited random process has two-sided PSD $S_X(f)=10^{-6}(3000-|f|)$ W/Hz for |f| ≤ 3000 Hz, and zero otherwise. It modulates $\cos(16000\pi t)$; the result passes through a unity-gain ideal bandpass filter centered at 8 kHz with bandwidth 2 kHz. Find output power (W).
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Show answer and explanation
Correct answer: 2.4 to 2.6
Explanation
Multiplication by cos(2π8000t) produces shifted PSD copies with factor1/4. The2kHzbandpass centred at8kHz retains offsets−1000through1000at each of±8kHz. Hence output power=2×(1/4)×∫₋₁₀₀₀¹⁰⁰⁰10⁻⁶(3000−|f|)df=(1/2)×5=**2.5W**.