Question

Given x (t) = 1, |t|<T1; 0, T1<|t|<\(\frac{T_0}{2}\). The DC component of x(t) is ___________

(a) \(\frac{T_1}{T_0}\)

(b) \(\frac{2T_1}{T_0}\)

(c) \(\frac{T_1}{2T_0}\)

(d) \(\frac{T_0}{T_1}\)

This question was addressed to me during an interview.

The origin of the question is Common Fourier Transforms in portion Fourier Transform of Signals and Systems

Answer 1

Correct choice is (b) \(\frac{2T_1}{T_0}\)

The best I can explain: Fourier series the function () can be written as, x(t) = a0 + \(∑_{n=1}^∞ a_n \,cos⁡nωt + b_n  \,sin⁡nωt\)

The DC component is given by, a0 = \(\frac{1}{T_0} \int_{T_0}^ x(t) \,dt\)

Or, a0 = \(\frac{1}{T} \int_{-T_0/2}^{T_0/2} \,x(t) \,dt\)

= \(\frac{1}{T_0}\) [0 + 2T1 + 0]

= \(\frac{2T_1}{T_0}\).