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Bode Plots for Second-Order Butterworth Filters

Filters of this type are nicely viewed in a Bode plot which shows the magnitude frequency response (in dB) versus a log frequency axis. In matlab we can say, for example,

sys = tf(1,[1,sqrt(2),1]);
to see the frequency response of our normalized second-order Butterworth lowpass filter.

Note that our lowpass is easily converted to a bandpass or highpass filter by changing the transfer-function numerator from $ 1$ to $ s$ or $ s^2$ , respectively:

bode(tf([0 0 1],[1,sqrt(2),1])); % lowpass
bode(tf([0 1 0],[1,sqrt(2),1])); % bandpass
bode(tf([1 0 0],[1,sqrt(2),1])); % highpass
bode(tf([1 0 1],[1,sqrt(2),1])); % notch
These frequency responses are shown in Fig.1.

Figure 1: Overlay of normalized second-order Butterworth lowpass, bandpass, highpass, and notch filters.

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``Digital State-Variable Filters'', by Julius O. Smith III.
Copyright © 2021-02-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University