Question

What is the energy density spectrum of the signal x(n)=a^nu(n), |a|<1?

(a) \(\frac{1}{1+2acosω+a^2}\)

(b) \(\frac{1}{1-2acosω+a^2}\)

(c) \(\frac{1}{1-2acosω-a^2}\)

(d) \(\frac{1}{1+2acosω-a^2}\)

I had been asked this question during an interview for a job.

Query is from Properties of Fourier Transform  for Discrete Time Signals topic in portion Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

The correct choice is (b) \(\frac{1}{1-2acosω+a^2}\)

The explanation: Given x(n)= a^nu(n), |a|<1

The auto correlation of the above signal is

rxx(l)=\(\frac{1}{1-a^2}\) a^|l|, -∞< l <∞

According to Wiener-Khintchine Theorem,

Sxx(ω)=F{rxx(l)}=\([\frac{1}{1-a^2}]\).F{a^|l|} = \(\frac{1}{1-2acosω+a^2}\)