The GATE Grind

GATE 2026 EC – Question 37

Networks, Signals and Systems · Continuous-time Signals · 2 marks · Multiple choice

The continuous time signal $x(t)$ is real, periodic with period $T$ and satisfies the Dirichlet conditions. The Fourier series representation of $x(t)=\sum_{-\infty}^{\infty}a_ne^{j(2\pi nt/T)}$ and $x(t)$ satisfies the following: $x(t-\frac T2)=-x(t)$. For any integer $m$, which of the following options is correct?

  1. $a_{2m}=0$
  2. $a_{2m}=1$
  3. $a_{2m}=a_{2m+1}$
  4. $a_{2m}=-1$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $a_{2m}=0$

Explanation

Time shift by $T/2$ multiplies $a_n$ by $e^{-j\pi n}=(-1)^n$. The condition requires $(-1)^na_n=-a_n$, so all even-indexed coefficients vanish: $a_{2m}=0$ (half-wave symmetry).