Question

Where does the poles of the transfer function of normalized low pass Butterworth filter exists?

(a) Inside unit circle

(b) Outside unit circle

(c) On unit circle

(d) None of the mentioned

I have been asked this question in my homework.

The above asked question is from Butterworth Filters in chapter Digital Filters Design of Digital Signal Processing

Answer 1

Right answer is (c) On unit circle

To explain: The transfer function of normalized low pass Butterworth filter is given as

HN(s).HN(-s)=\(\frac{1}{1+(\frac{s}{j})^{2N}}\)

The poles of the above equation is obtained by equating the denominator to zero.

=> \(1+(\frac{s}{j})^{2N}\)=0

=> s=(-1)^1/2N.j

=> sk=\(e^{jπ(\frac{2k+1}{2N})} e^{jπ/2}\), k=0,1,2…2N-1

The poles are therefore on a circle with radius unity.