Question

What is the value of |X(ω)| given X(ω)=1/(1-ae^-jω), |a|<1?

(a) \(\frac{1}{\sqrt{1-2acosω+a^2}}\)

(b) \(\frac{1}{\sqrt{1+2acosω+a^2}}\)

(c) \(\frac{1}{1-2acosω+a^2}\)

(d) \(\frac{1}{1+2acosω+a^2}\)

This question was posed to me in a job interview.

Origin of the question is Properties of Fourier Transform  for Discrete Time Signals topic in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Correct answer is (a) \(\frac{1}{\sqrt{1-2acosω+a^2}}\)

To elaborate: For the given X(ω)=1/(1-ae^-jω), |a|<1 we obtain

XI(ω)=(-asinω)/(1-2acosω+a^2) and XR(ω)=(1-acosω)/(1-2acosω+a^2)

We know that |X(ω)|=\(\sqrt{X_R (ω)^2+X_I (ω)^2}\)

Thus on calculating, we obtain

|X(ω)| =  \(\frac{1}{\sqrt{1-2acosω+a^2}}\).