Notes

%% 9.5.1
% Define the numerator and denominator of H(z)
num = [1 0.5]; % Corresponds to z + 0.5
den = [1 -1 3/16]; % Corresponds to z^2 + 0 + 3/16

% Compute the poles and zeros
zeros_H = roots(num);
poles_H = roots(den);

% Plot poles and zeros
figure;
zplane(zeros_H, poles_H);
title('Pole-Zero Plot');
grid on;

% Check stability
stable = all(abs(poles_H) < 1);
%disp(['The system is ', char(stable * 'stable' + ~stable * 'unstable')]);
%% 9.5.2
% Create transfer function
sys = tf(num, den, -1); % Specify -1 for discrete-time

% Impulse response
figure;
impulse(sys);
title('Impulse Response');
grid on;

% Step response
figure;
step(sys);
title('Step Response');
grid on;
%% 9.5.3
% Partial fraction decomposition
[r, p, k] = residue(num, den);
disp('Residue:');
disp(r);
disp('Poles:');
disp(p);
disp('Direct term:');
disp(k);

% Closed-form expression of impulse response
% Impulse response can be derived from r and p

%% 9.5.4
% Frequency response
[H, W] = freqz(num, den, 1024);

% Magnitude response
figure;
subplot(2,1,1);
plot(W/pi, 20*log10(abs(H)));
title('Magnitude Response');
xlabel('Normalized Frequency (times pi rad/sample)');
ylabel('Magnitude (dB)');
grid on;

% Phase response
subplot(2,1,2);
plot(W/pi, unwrap(angle(H)) * (180/pi));
title('Phase Response');
xlabel('Normalized Frequency (times pi rad/sample)');
ylabel('Phase (degrees)');
grid on;




%% 9.5 5


A=[1 -1 3/16];
B=[1 0.5];
n=0:0.01:pi;

x = (n>=0).*(sin(0.3*pi.*n + pi/4) + 2*cos(0.1*pi.*n));
y= filter(B,A,x);
[H2,w2]= freqz(B,A,n);
subplot(2,1,1)
stem (n,y);
title('Y[n]')
subplot(2,1,2)
stem(n,x)
title('X[n]')

%% 9.6
A = [1 1.25 0.75]; %coefficients of y[n]+1.25y[n-1]+0.75y[n-2]
B = [2 1];
n=0:0.01:pi;
[H,w]=freqz(B,A,n);
subplot(2,1,1)
plot(w,abs(H));
title('Magnitude Vs. Frequency')
subplot(2,1,2)
plot(w,angle(H));
title('Phase Vs. Frequency')
h0=abs(freqz(B,A, [0 pi/2 pi]));
h1=angle(freqz(B,A, [0 pi/2 pi]));

%% 9.6.2
% Define coefficients
A = [1 1.25 0.75]; % Coefficients for y[n]
B = [2 1]; % Coefficients for x[n]

% Compute frequency response with 1024 points
[H, w] = freqz(B, A, 1024);

% Find indices for specific frequencies
index_0 = find(w == 0); % H(j0)
index_pi_over_4 = find(w == pi/2); % H(jπ/4)
index_pi = find(w == pi); % H(jπ)

% Display computed values
disp(['H(j0) = ', num2str(H(index_0))]);
disp(['H(jπ/2) = ', num2str(H(index_pi_over_4))]);
disp(['H(jπ) = ', num2str(H(index_pi))])