GATE 2022 EE – Question 47

Signals and Systems · Applications of Fourier Transform for continuous and discrete time signals, Laplace Transform and Z transform · 2 marks · Multiple choice

Let an input $x(t)=2\sin(10\pi t)+5\cos(15\pi t)+7\sin(42\pi t)+4\cos(45\pi t)$ is passed through an LTI system having an impulse response,

$$h(t)=2\left(\frac{\sin(10\pi t)}{\pi t}\right)\cos(40\pi t).$$

The output of the system is

$2\sin(10\pi t)+5\cos(15\pi t)$
$5\cos(15\pi t)+7\sin(42\pi t)$
$7\sin(42\pi t)+4\cos(45\pi t)$
$2\sin(10\pi t)+4\cos(45\pi t)$

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Correct answer: (C) $7\sin(42\pi t)+4\cos(45\pi t)$

Explanation

$\frac{\sin(10\pi t)}{\pi t}$ has a flat spectrum of height 1 over $|\omega|<10\pi$. Multiplying by $2\cos(40\pi t)$ shifts it to $\pm40\pi$ with gain 1, so the system is a band-pass filter passing $30\pi<|\omega|<50\pi$ rad/s with unit gain. The input components at $42\pi$ and $45\pi$ pass unchanged, while $10\pi$ and $15\pi$ are blocked. The output is $7\sin(42\pi t)+4\cos(45\pi t)$.