Question

What is the Fourier transform of the signal x(n) which is defined as shown in the graph below?

(a) Ae^-j(ω/2)(L)\(\frac{sin⁡(\frac{ωL}{2})}{sin⁡(\frac{ω}{2})}\)

(b) Ae^j(ω/2)(L-1)\(\frac{sin⁡(\frac{ωL}{2})}{sin⁡(\frac{ω}{2})}\)

(c) Ae^-j(ω/2)(L-1)\(\frac{sin⁡(\frac{ωL}{2})}{sin⁡(\frac{ω}{2})}\)

(d) None of the mentioned

I have been asked this question in an online interview.

The doubt is from Frequency Analysis of Discrete Time Signal in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right answer is (c) Ae^-j(ω/2)(L-1)\(\frac{sin⁡(\frac{ωL}{2})}{sin⁡(\frac{ω}{2})}\)

To explain I would say: The Fourier transform of this signal is

X(ω)=\(\sum_{n=0}^{L-1} Ae^{-jωn}\)

=A.\(\frac{1-e^{-jωL}}{1-e^{-jω}}\)

=\(Ae^{-j(ω/2)(L-1)}\frac{sin⁡(\frac{ωL}{2})}{sin⁡(\frac{ω}{2})}\)