Question

The Fourier series representation of an impulse train denoted by s(t) = \(∑_{n=-∞}^∞ δ(t-nT_0)\) is given by _____________

(a) \(\frac{1}{T_0} ∑_{n=-∞}^∞ exp⁡(-\frac{j2πnt}{T_0}) \)

(b) \(\frac{1}{T_0} ∑_{n=-∞}^∞ exp⁡(-\frac{jπnt}{T_0})\)

(c) \(\frac{1}{T_0} ∑_{n=-∞}^∞ exp⁡(\frac{jπnt}{T_0}) \)

(d) \(\frac{1}{T_0} ∑_{n=-∞}^∞ exp⁡(\frac{j2πnt}{T_0}) \)

I have been asked this question in an interview.

Question is from Miscellaneous Examples on Fourier Series in portion Fourier Series of Signals and Systems

Answer 1

The correct option is (b) \(\frac{1}{T_0} ∑_{n=-∞}^∞ exp⁡(-\frac{jπnt}{T_0})\)

The explanation: s (t) = \(∑_{n=-∞}^∞ C_n e^{jnω_0 t}\), where ω0=(2T/T0)

And Cn = \(\frac{1}{T_0} \displaystyle\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} δ(t) e^{-jnω_0 t} \,dt\)

= \(\frac{1.e^{-jnω_0 t}}{T_0}\)

So, Fourier series representation = \(\frac{1}{T_0} ∑_{n=-∞}^∞ exp⁡(-\frac{jπnt}{T_0})\).