The GATE Grind

GATE 2026 EE – Question 36

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

A time-limited waveform $g(x)$ is specified as follows:

$$g(x)=\begin{cases}-k,&-\pi<x\le0\\+k,&0<x\le\pi\\0,&\text{otherwise}\end{cases}$$

A new waveform $f(x)$ is constructed from $g(x)$ as follows:

$$f(x)=\sum_{m=-\infty}^{\infty}g(x+2\pi m),\quad\text{for all }x\in R$$

The sum of the coefficients of the third harmonics of the sine and cosine terms in the trigonometric Fourier series expansion of $f(x)$ is $\dfrac{2}{3\pi}$.

What is the value of $k$?

  1. 1
  2. $\dfrac12$
  3. $\dfrac13$
  4. $\dfrac14$

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Show answer and explanation

Correct answer: (B) $\dfrac12$

Explanation

$f(x)$ is a $2\pi$-periodic odd square wave of amplitude $k$: $f(x)=\dfrac{4k}{\pi}\sum_{n\text{ odd}}\dfrac{\sin nx}{n}$. There are no cosine terms, and the third-harmonic sine coefficient is $\dfrac{4k}{3\pi}$. Setting $\dfrac{4k}{3\pi}=\dfrac2{3\pi}$ gives $k=\dfrac12$.