Question

The Fourier series coefficient for the periodic rectangular pulses of height 2A is ____________

(a) \(\frac{2A}{jkπ} \,sin\,⁡ \frac{π}{2} k\)

(b) \(\frac{2A}{jkπ} \,cos⁡\, \frac{π}{2} k\)

(c) \(\frac{2A}{kπ}  \,sin⁡\, \frac{π}{2} k\)

(d) \(\frac{2A}{kπ}  \,cos⁡\, \frac{π}{2} k\)

This question was addressed to me by my college director while I was bunking the class.

I'm obligated to ask this question of Fourier Series Coefficients topic in portion Fourier Series of Signals and Systems

Answer 1

The correct option is (c) \(\frac{2A}{kπ}  \,sin⁡\, \frac{π}{2} k\)

Easy explanation: \(X[k] = \frac{1}{T} \displaystyle\int_{-\frac{T}{2}}^{\frac{T}{2}} x(t)e^{-jkωt} \,dt\)

\(= \frac{1}{T} \displaystyle\int_{-\frac{T}{4}}^{\frac{T}{4}} Ae^{-jkωt} \,dt\)

\(= \frac{2A}{T}[\frac{e^{-jkωt}}{-jkω}]\) (from –\(\frac{T}{4}\) to \(\frac{T}{4}\))

\(=\frac{2A}{kπ}  \,sin⁡\, \frac{π}{2} k\).