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GATE 2022 IN – Question 13

Signals and Systems · Fourier analysis and sampling · 1 mark · Multiple choice

A periodic function of period 2 is $f(x)=\begin{cases}-1-x,&-1\le x<0\\1-x,&0<x\le1\end{cases}$. Its Fourier series contains ______.

  1. Both $\cos(n\pi x)$ and $\sin(n\pi x)$, $n=1,2,3,\ldots$
  2. Only $\sin(n\pi x)$, $n=1,2,3,\ldots$
  3. Only $\cos(n\pi x)$, $n=1,2,3,\ldots$
  4. Only $\cos(2n\pi x)$, $n=1,2,3,\ldots$

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Correct answer: (B) Only $\sin(n\pi x)$, $n=1,2,3,\ldots$

Explanation

For positive x, f(x)=1−x, while f(−x)=−1+x=−f(x). The function is odd over its symmetric period. Consequently its constant and cosine coefficients vanish, leaving **sine terms only, B**.