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GATE 2018 EC – Question 24

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

Let $x(t)$ be a periodic function with period $T=10$. The Fourier series coefficients for this series are denoted by $a_k$, that is,

$$x(t)=\sum_{k=-\infty}^{\infty}a_ke^{jk\frac{2\pi}{T}t}$$

The same function $x(t)$ can also be considered as a periodic function with period $T'=40$. Let $b_k$ be the Fourier series coefficients when the period is taken as $T'$. If $\sum_{k=-\infty}^{\infty}|a_k|=16$, then $\sum_{k=-\infty}^{\infty}|b_k|$ is equal to

  1. 256
  2. 64
  3. 16
  4. 4

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Show answer and explanation

Correct answer: (C) 16

Explanation

When the same signal is viewed with period $T'=4T$, the fundamental is four times lower, so the old harmonics $a_k$ become the coefficients $b_{4k}=a_k$ and all other $b_k$ are zero. So $\sum|b_k|=\sum|a_k|=16$.