Question

A two pole low pass filter has a system function H(z)=\(\frac{b_0}{(1-pz^{-1})^2}\), What is the value of ‘p’ such that the frequency response H(ω) satisfies the condition |H(π/4)|^2=1/2 and H(0)=1?

(a) 0.46

(b) 0.38

(c) 0.32

(d) 0.36

I got this question during an interview.

The query is from LTI System as Frequency Selective Filters topic in division Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

The correct option is (c) 0.32

Explanation: Given

H(z)=\(\frac{b_0}{(1-pz^{-1})^2}\) and we know that z=re^jω. Here in this case r=1. So z=e^jω.

Given at ω=0, H(0)=1=>b0=(1-p)^2

Given at ω=π/4, |H(π/4)|^2=1/2

=>\(\frac{(1-p)^2}{(1-pe^{-jπ/4})^2}\) = 1/2

=>\(\frac{(1-p)^4}{((1-p/\sqrt 2)^2+p^2/2)^2}\) = 1/2

=> √2(1-p)^2=1+p^2-√2p

Upon solving the above quadratic equation, we get the value of p as 0.32.