The GATE Grind

GATE 2025 EC – Question 55

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

Let $f(t)$ be a periodic signal with fundamental period $T_0>0$. Consider the signal $y(t)=f(\alpha t)$, where $\alpha>1$.

The Fourier series expansions of $f(t)$ and $y(t)$ are given by

$$f(t)=\sum_{k=-\infty}^{\infty}c_k\,e^{j\frac{2\pi}{T_0}kt}\quad\text{and}\quad y(t)=\sum_{k=-\infty}^{\infty}d_k\,e^{j\frac{2\pi}{T_0}\alpha kt}.$$

Which of the following statements is/are TRUE?

  1. $c_k=d_k$ for all $k$
  2. $y(t)$ is periodic with a fundamental period $\alpha T_0$
  3. $c_k=d_k/\alpha$ for all $k$
  4. $y(t)$ is periodic with a fundamental period $T_0/\alpha$

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Show answer and explanation

Correct answer: (A) $c_k=d_k$ for all $k$; (D) $y(t)$ is periodic with a fundamental period $T_0/\alpha$

Explanation

Replacing $t$ by $\alpha t$ in the series for $f$ gives $y(t)=\sum_kc_ke^{j\frac{2\pi}{T_0}k\alpha t}$, so matching with the given form, $d_k=c_k$ (A true, C false). The fundamental angular frequency of $y$ is $\alpha\cdot2\pi/T_0$, so its fundamental period is $T_0/\alpha$ (D true, B false).