Question

What is the equation for magnitude square response of a low pass Butterworth filter?

(a) \(\frac{1}{\sqrt{1+(\frac{Ω}{Ω_C})^{2N}}}\)

(b) \(1+(\frac{Ω}{Ω_C})^{2N}\)

(c) \(\sqrt{1+(\frac{Ω}{Ω_C})^{2N}}\)

(d) None of the mentioned

This question was posed to me in an international level competition.

My question is from Characteristics of Commonly Used Analog Filters topic in section Digital Filters Design of Digital Signal Processing

Answer 1

Right answer is (a) \(\frac{1}{\sqrt{1+(\frac{Ω}{Ω_C})^{2N}}}\)

The explanation is: A Butterworth is characterized by the magnitude frequency response

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

where N is the order of the filter and ΩC is defined as the cutoff frequency.