The GATE Grind

GATE 2017 EC – Question 18

Networks, Signals and Systems · Continuous-time Signals · 1 mark · Multiple choice

A periodic signal $x(t)$ has a trigonometric Fourier series expansion

$$x(t) = a_0 + \sum_{n=1}^{\infty}(a_n\cos n\omega_0 t + b_n\sin n\omega_0 t)$$

If $x(t) = -x(-t) = -x(t - \pi/\omega_0)$, we can conclude that

  1. $a_n$ are zero for all $n$ and $b_n$ are zero for $n$ even
  2. $a_n$ are zero for all $n$ and $b_n$ are zero for $n$ odd
  3. $a_n$ are zero for $n$ even and $b_n$ are zero for $n$ odd
  4. $a_n$ are zero for $n$ odd and $b_n$ are zero for $n$ even

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Correct answer: (A) $a_n$ are zero for all $n$ and $b_n$ are zero for $n$ even

Explanation

The condition $x(t) = -x(-t)$ says the signal is odd, so all the cosine terms and $a_0$ vanish. The condition $x(t) = -x(t - \pi/\omega_0)$ says the signal changes sign after half a period, which means it has half-wave symmetry, so only odd harmonics are present. Hence $b_n = 0$ for even $n$.