DSP Practical - Experiment 6

Aim

To design a digital Butterworth low-pass filter with specific specifications using the Impulse Invariance Method.

Problem Statement

Design a digital Butterworth filter with the following specifications:

          0.8 <= |H(e^jw)| <= 1 (Passband)
          |H(e^jw)| <= 0.2 (Stopband)

Use the Impulse Invariance Method to design the filter with the following frequency range:

          0 <= w <= 0.2π (Passband)
          0.6π <= w <= π (Stopband)

Theory

The Butterworth filter is designed to have a flat frequency response in the passband, with no ripples. The Impulse Invariance Method is used to map the continuous-time filter design to a discrete-time filter. The key steps involved in the process are:

Determine the analog low-pass filter specifications based on the given passband and stopband frequencies.
Use the `buttord` function to calculate the filter order and cutoff frequency.
Design the analog Butterworth filter using the `butter` function.
Use the `impinvar` function to apply the Impulse Invariance Method and obtain the digital filter coefficients.
Plot the frequency response of the digital Butterworth filter using `freqz`.

MATLAB Code

% MATLAB Code for Designing Digital Butterworth LPF using Impulse Invariance Method

clc;
clear all;
close all;

% Given Parameters
T = 1;           % Sampling period
Wp = 0.2 * pi;   % Passband frequency (rad/sample)
Ws = 0.6 * pi;   % Stopband frequency (rad/sample)
Op = Wp / T;     % Analog passband frequency
Qs = Ws / T;     % Analog stopband frequency
Sp = 0.8;        % Passband ripple
Ss = 0.2;        % Stopband attenuation

% Calculate the filter order and cutoff frequency using buttord
Ap = -20 * log10(Sp);  % Passband attenuation
As = -20 * log10(Ss);  % Stopband attenuation
[N, CF] = buttord(Op, Qs, Ap, As, 's');  % Analog filter specifications

% Design the analog Butterworth filter using butter
[Bn, An] = butter(N, CF, 's');  % Analog filter coefficients
Hs = tf(Bn, An);  % Analog transfer function

% Apply Impulse Invariance Method to obtain the digital filter coefficients
[B, A] = butter(N, CF, 's');  % Analog filter coefficients
[num, den] = impinvar(B, A, 1/T);  % Digital filter coefficients

% Create the digital transfer function
Hz = tf(num, den, T);  % Digital transfer function

% Frequency response of the digital Butterworth filter
w = 0:0.1963:pi;  % Frequency range (0 to pi)
Hw = freqz(num, den, w);  % Frequency response

% Plot the magnitude response of the digital Butterworth LPF
Hw_mag = abs(Hw);  % Magnitude of the frequency response
plot(w/pi, Hw_mag, 'k');  % Plot magnitude response
title('Magnitude Response of Butterworth Digital LPF Using Impulse Invariance', 'fontweight', 'b');
xlabel('Normalized Frequency (w)', 'fontweight', 'b');
ylabel('Linear Magnitude |H(e^jw)|', 'fontweight', 'b');

Expected Output

The MATLAB code generates a plot of the magnitude response of the designed Butterworth digital low-pass filter. The expected output will display:

The magnitude response of the filter, which will show a smooth, flat response in the passband and a sharp drop in the stopband, characteristic of a Butterworth filter.

This plot helps visualize the performance of the designed filter and validates the design process using the Impulse Invariance Method.