Question

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

(a) \(\frac{1}{1+(s/j)^{-2N}}\)

(b) \(1+(\frac{s}{j})^{-2N}\)

(c) \(1+(\frac{s}{j})^{2N}\)

(d) \(\frac{1}{1+(\frac{s}{j})^{2N}}\)

I have been asked this question by my college professor while I was bunking the class.

My enquiry is from Characteristics of Commonly Used Analog Filters topic in portion Digital Filters Design of Digital Signal Processing

Answer 1

Correct choice is (d) \(\frac{1}{1+(\frac{s}{j})^{2N}}\)

For explanation I would say: We know that the magnitude squared frequency response of a normalized low pass Butterworth filter is given as

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

Replacing jΩ by ‘s’ and hence Ω by s/j in the above equation, we get

HN(s).HN(-s)=\(\frac{1}{1+(\frac{s}{j})^{2N}}\) which is called the transfer function.