Question

The Fourier series coefficient of time domain signal x (t) is X[k] = \((-\frac{1}{3})^{|k|}\). The fundamental frequency of signal is ω=1. The signal is _____________

(a) \(\frac{4}{5 + 3 sin⁡}\)

(b) \(\frac{5}{4 + 3 sin⁡t}\)

(c) \(\frac{5}{4 + 3 cos⁡t}\)

(d) \(\frac{4}{5 + 3 sin⁡t}\)

The question was asked in my homework.

Question is taken from Fourier Series Coefficients topic in portion Fourier Series of Signals and Systems

Answer 1

Right choice is (d) \(\frac{4}{5 + 3 sin⁡t}\)

To explain I would say: \(x (t) = ∑_{k=-∞}^∞ X[k]e^{jkt}\)

Or, x (t) = \(∑_{k=-∞}^{-1} (-\frac{1}{3})^{-k} e^{jk} + ∑_{k=0}^∞ (-\frac{1}{3})^k e^{jkt}\)

= \(\frac{\frac{-1}{3} e^{-jt}}{1+\frac{1}{3} e{-jt}} + \frac{1}{1 + \frac{1}{3} e^{jt}}\)

= \(\frac{4}{5 + 3 sin⁡t}\).