Question

The Fourier transform of the signal \(∑_{m=0}^∞ a^m δ(t-m)\) is _____________

(a) \(\frac{1}{1+ae^{-jω}}\)

(b) \(\frac{1}{1+ae^{jω}}\)

(c) \(\frac{1}{1-ae^{jω}}\)

(d) \(\frac{1}{1-ae^{-jω}}\)

This question was addressed to me in unit test.

I would like to ask this question from Common Fourier Transforms in portion Fourier Transform of Signals and Systems

Answer 1

Correct answer is (d) \(\frac{1}{1-ae^{-jω}}\)

The best explanation: X (jω) = \(\int_0^∞ ∑_{m=0}^∞ a^m δ(t-m) e^{-jωt} \,dt\)

= \(∑_{m=0}^∞ (ae^{-jω})^m \)

= \(\frac{1}{1-ae^{-jω}}\).