The GATE Grind

GATE 2017 EC – Question 41

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

Let $x(t)$ be a continuous time periodic signal with fundamental period $T = 1$ seconds. Let $\{a_k\}$ be the complex Fourier series coefficients of $x(t)$, where $k$ is integer valued. Consider the following statements about $x(3t)$:

I. The complex Fourier series coefficients of $x(3t)$ are $\{a_k\}$ where $k$ is integer valued

II. The complex Fourier series coefficients of $x(3t)$ are $\{3a_k\}$ where $k$ is integer valued

III. The fundamental angular frequency of $x(3t)$ is $6\pi$ rad/s

For the three statements above, which one of the following is correct?

  1. only II and III are true
  2. only I and III are true
  3. only III is true
  4. only I is true

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Show answer and explanation

Correct answer: (B) only I and III are true

Explanation

Compressing time by 3 makes the period $\frac{1}{3}$ s, so the fundamental angular frequency is $2\pi \times 3 = 6\pi$ rad/s, which makes III true. The same set of harmonics appears with the same amplitudes, so the coefficients are still $\{a_k\}$ with respect to the new fundamental. Statement I is true and II is false.