Question

What is the magnitude squared response of the normalized low pass Butterworth filter?

(a) \(\frac{1}{1+Ω^{-2N}}\)

(b) 1+Ω^-2N

(c) 1+Ω^2N

(d) \(\frac{1}{1+Ω^{2N}}\)

I got this question in an online quiz.

Asked question is from Butterworth Filters topic in chapter Digital Filters Design of Digital Signal Processing

Answer 1

Right option is (d) \(\frac{1}{1+Ω^{2N}}\)

Explanation: We know that the magnitude response of a low pass Butterworth filter of order N is given as

|H(jΩ)|=\(\frac{1}{\sqrt{1+(\frac{Ω}{Ω_C})^{2N}}}\)

For a normalized filter, ΩC =1

=> |H(jΩ)|=\(\frac{1}{\sqrt{1+(Ω)^{2N}}}\) => |H(jΩ)|^2=\(\frac{1}{1+Ω^{2N}}\)

Thus the magnitude squared response of the normalized low pass Butterworth filter of order N is given by the equation,

|H(jΩ)|^2=\(\frac{1}{1+Ω^{2N}}\).