The GATE Grind

GATE 2022 ME (ME2) – Question 11

Engineering Mathematics · Calculus: Partial derivatives, Taylor series, maxima and minima, Fourier series · 1 mark · Multiple choice

F(t) is a periodic square wave function as shown. It takes only two values, 4 and 0, and stays at each of these values for 1 second before changing. What is the constant term in the Fourier series expansion of F(t)?

a square wave alternating between 4 and 0, with period 2 seconds.
  1. 1
  2. 2
  3. 3
  4. 4

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) 2

Explanation

The constant term (the $a_0/2$ term) of a Fourier series is the **average of the function over one period**.

The wave takes the value 4 for 1 s and 0 for 1 s, so the period is $T=2$ s.

$$\text{Constant term}=\frac1T\int_0^T F(t)\,dt=\frac12\big(4\times1+0\times1\big)=\mathbf{2}.$$