GATE 2021 EE – Question 53

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

Consider a continuous-time signal $x(t)$ defined by $x(t)=0$ for $|t|>1$, and $x(t)=1-|t|$ for $|t|\le1$. Let the Fourier transform of $x(t)$ be defined as $X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt$. The maximum magnitude of $X(\omega)$ is ________.

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Correct answer: 1

Explanation

$x(t)$ is a non-negative triangle, so $|X(\omega)|\le\int x(t)\,dt=X(0)$. The area of the triangle is $\frac12\times2\times1=1$, so the maximum magnitude is 1.