Question

What is the magnitude of the frequency response of the system described by the difference equation y(n)=ay(n-1)+bx(n), 0<a<1?

(a) \(\frac{|b|}{\sqrt{1+2acosω+a^2}}\)

(b) \(\frac{|b|}{1-2acosω+a^2}\)

(c) \(\frac{|b|}{1+2acosω+a^2}\)

(d) \(\frac{|b|}{\sqrt{1-2acosω+a^2}}\)

I got this question during an interview for a job.

I'd like to ask this question from Frequency Domain Characteristics of LTI System topic in section Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

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

To explain I would say: Given y(n)=ay(n-1)+bx(n)

=>H(ω)=\(\frac{|b|}{1-ae^{-jω}}\)

By calculating the magnitude of the above equation we get

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