Question

What is the value of XI(ω) given \(\frac{1}{1-ae^{-jω}}\), |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) \(\frac{-asinω}{1-2acosω+a^2}\)

The question was asked in exam.

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

Answer 1

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

Easiest explanation: 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

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