Question

Using MATLAB technology

1. (20 points) Design an analog Butterworth LPF with a,--1 dB at ar = 20π rad/s, a,-40 dB at w 100n rad/s. (a) Determine the order and the cutoff frequency of the filter. (b) Find the transfer function of the filter. (c) Plot the frequency response of the filter. (d) Measure the transition band which is given by A2 (a.)-w (e) Increase the order by 2 for the same cutoff frequency, measure the transition band, compare with the previously obtained. (f) Design the a Butterworth HPF with same specifications as the LPF designed in step(1

Answer 1

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****************** Matlab Code ************

% For data sampled at 1000 Hz, design a lowpass filter with no more than 1 dB of ripple in a passband from 0 to 10 Hz,
% and at least 40 dB of attenuation in the stopband. Find the filter order and cutoff frequency.

Wp = 10/500;    % 20pi corresponds to 10Hz
Ws = 50/500;    % 100pi corresponds to 50Hz
clc
[n,Wn] = buttord(Wp,Ws,1,40)
%Specify the filter in terms of second-order sections and plot the frequency response.
[z,p,k] = butter(n,Wn);
sys = zpk(z,p,k,1/500)
sos = zp2sos(z,p,k);
freqz(sos,512,1000)
title(sprintf('n = %d Butterworth Lowpass Filter',n))
%%%%% high pass filter
[z1,p1,k1] = butter(n,Wn,'high');
sys1 = zpk(z1,p1,k1,1/500)
sos1 = zp2sos(z1,p1,k1);
figure;
freqz(sos1,512,1000)
title(sprintf('n = %d Butterworth Highpass Filter',n))

************** End of Code ***************

Output:

Order = 4, Cutoff = 0.0319 rad/s

sys =

                5.5214e-06 (z+1)^4
----------------------------------------------
(z^2 - 1.822z + 0.831) (z^2 - 1.917z + 0.9263)

Sample time: 0.002 seconds
Discrete-time zero/pole/gain model.

Transition band = 45Hz

High pass:

sys1 =

                 0.87736 (z-1)^4
----------------------------------------------
(z^2 - 1.822z + 0.831) (z^2 - 1.917z + 0.9263)

Sample time: 0.002 seconds
Discrete-time zero/pole/gain model.

*****************************************************************
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