For every real , - g(t)e-j t\,dt= e-22. Let y(t)= -t g()\,d. Evaluate - y(t)\,dt.

GATE 2014 EC (EC1) – Question 37

Networks, Signals and Systems · Continuous-time Signals · 2 marks · Multiple choice

For every real $\omega$, $\int_{-\infty}^\infty g(t)e^{-j\omega t}\,dt=\omega e^{-2\omega^2}$. Let $y(t)=\int_{-\infty}^t g(\tau)\,d\tau$. Evaluate $\int_{-\infty}^\infty y(t)\,dt$.

$0$
$-j$
$-j/2$
$j/2$

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Correct answer: (B) $-j$

Explanation

The supplied Fourier transform is G(ω)=ωexp(−2ω²), so G(0)=0. Since y is the integral from−∞ to t, its transform is Y(ω)=G(ω)/(jω)=−jexp(−2ω²); the possible impulse term vanishes because G(0)=0. Therefore ∫y(t)dt=Y(0)=**−j,B**.