Question

What is the value of XR(ω) given X(ω)=\(\frac{1}{1-ae^{-jω}}\),|a|<1?

(a) \(\frac{asinω}{1-2acosω+a^2}\)

(b) \(\frac{1+acosω}{1-2acosω+a^2}\)

(c) \(\frac{1-acosω}{1-2acosω+a^2}\)

(d) \(\frac{-asinω}{1-2acosω+a^2}\)

I had been asked this question in a job interview.

Enquiry is from Properties of Fourier Transform  for Discrete Time Signals in section Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right choice is (c) \(\frac{1-acosω}{1-2acosω+a^2}\)

The explanation is: Given, X(ω)=\(\frac{1}{1-ae^{-jω}}\), |a|<1

By multiplying both the numerator and denominator of the above equation by the complex conjugate of the denominator, we obtain

X(ω)=\(\frac{1-ae^{jω}}{(1-ae^{-jω})(1-ae^{jω})} = \frac{1-acosω-jasinω}{1-2acosω+a^2}\)

This expression can be subdivided into real and imaginary parts, thus we obtain

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