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 ‘b0‘ such that the frequency response H(ω) satisfies the condition |H(π/4)|^2=1/2 and H(0)=1?

(a) 0.36

(b) 0.38

(c) 0.32

(d) 0.46

The question was asked in examination.

The above asked question is from LTI System as Frequency Selective Filters in section Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right option is (d) 0.46

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.

Already we have

b0=(1-p)^2=(1-0.32)^2

=>b0 = 0.46