minimum_phase.m

clc;clear;close all;

%% Introduction
% *Title:* Minimum Phase and Non-Minimum Phase Systems
%
% *Author:* Siavash Emami
%
% *Date:* June 13th, 2022
%
% *Description:* This script shows how minimum and non-minimum phase systems
% work and what are the differences in them. Bode plot, step response and
% harmonic response of two example systems are shown in the following code.
%
% For more information please refer to <https://t.me/takane_iust Takane Magazine> 

%% Bode plots
% two systems are
%
% $$G_{min}(s) = \frac{s+2}{s^2+3s+2} G_{max}(s) = \frac{-s+2}{s^2+3s+2}$$
%
% both systems have same magnitude, but minimum phase system has a phase
% shift of 90 degrees in frequency domain, but maximum phase system has 270
% degrees phase shift. this is shown in figure 1
%
min_phase = tf([1 2],[1 3 2]);
max_phase = tf([-1 2],[1 3 2]);
figure(1)
bode(min_phase)
hold on
bode(max_phase)
legend('min phase', 'max phase')

%% Step Responses
% A non-minimum phase system undershoots in response to a step input and
% then follows the error, in contrast with minimum phase system which
% directly responds to error. 
% 
figure(2)
step(min_phase)
hold on
step(max_phase)
legend('min phase', 'max phase')

%% Harmonic Response
% As seen in the first section, the phase shift to each system is
% different, for example when a harmonic input with frequency of 10Hz is
% applied to the systems, phase of minimum phase system will be
% approximately -90 degrees and for maximum phase system will be +90
% degrees. it can be seen in the plots below that every peak of minimum and
% maximum phase response is located on a zero of input signal, and they
% have 180 degrees difference with each other.
% 
% 
% 
% 
tau = 0.1;
Tf = 10;
Ts = 0.00005;
[u,t] = gensig("sine",tau,Tf,Ts);
y_min = lsim(min_phase,u,t,Ts);
y_max = lsim(max_phase,u,t,Ts);

figure(3)
subplot(3,1,1)
plot(t(190001:end),u(190001:end))
title("Input Signal")
xlabel("Time (s)")
grid on

subplot(3,1,2)
plot(t(190001:end), y_min(190001:end))
title("Minimum Phase System Response")
xlabel("Time (s)")
grid on

subplot(3,1,3)
plot(t(190001:end), y_max(190001:end))
title("Maximum Phase System Response")
xlabel("Time (s)")
grid on