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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»

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.


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