1.

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) 0.36I 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»

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.


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