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GATE 2019 EE – Question 38

Signals and Systems · Fourier series representation of continuous and discrete time periodic signals, Sampling theorem · 2 marks · Multiple choice

A periodic function $f(t)$, with a period of $2\pi$, is represented as its Fourier series,

$$f(t)=a_0+\sum_{n=1}^{\infty}a_n\cos nt+\sum_{n=1}^{\infty}b_n\sin nt.$$

If

$$f(t)=\begin{cases}A\sin t,&0\le t\le\pi\\0,&\pi<t<2\pi\end{cases}$$

the Fourier series coefficients $a_1$ and $b_1$ of $f(t)$ are

  1. $a_1=\frac A\pi;\ b_1=0$
  2. $a_1=\frac A2;\ b_1=0$
  3. $a_1=0;\ b_1=\frac A\pi$
  4. $a_1=0;\ b_1=\frac A2$

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Correct answer: (D) $a_1=0;\ b_1=\frac A2$

Explanation

$a_1=\frac1\pi\int_0^\pi A\sin t\cos t\,dt=\frac A{2\pi}\int_0^\pi\sin2t\,dt=0$ and $b_1=\frac1\pi\int_0^\pi A\sin^2t\,dt=\frac A\pi\cdot\frac\pi2=\frac A2$.