Experiment 5: Fourier Transform of Continuous-Time Signals
% Fourier Transform using Numerical Integration
clc; clear; close all;
% Time specifications
Fs = 1000; % Sampling frequency
dt = 1/Fs; % Time step
t = -10:dt:10; % Time vector (from -1s to 1s)
% Define signal x(t)
x = exp(-abs(t)); % Example signal: e^(-|t|)
% Frequency specifications
f = -100:1:100; % Frequency range
X = zeros(size(f)); % Initialize Fourier Transform
% Numerical integration for each frequency
for k = 1:length(f)
X(k) = trapz(t, x .* exp(-1j*2*pi*f(k)*t));
end
% Plot time-domain signal
figure;
subplot(2,1,1);
plot(t, x, 'b','LineWidth',1.5);
xlabel('Time (s)');
ylabel('x(t)');
title('Time Domain Signal');
grid on;
% Plot frequency spectrum
subplot(2,1,2);
plot(f, abs(X), 'r','LineWidth',1.5);
xlabel('Frequency (Hz)');
ylabel('|X(f)|');
title('Fourier Transform (Numerical Integration)');
grid on;
Experiment 6: Laplace transform and system response.
clc; clear; close all;
%% ==============================
% PART 1: Numerical Laplace Transform
% ==============================
% Define the signal x(t) = e^(-2t)
t_values = 0:0.01:5; % Time: 0 to 5 seconds
x_t = exp(-2 * t_values); % Signal definition
s_values = 0.1:0.1:10; % Range of s-values
X_s = zeros(size(s_values));
% Numerical Laplace Transform using trapezoidal rule
for i = 1:length(s_values)
current_s = s_values(i);
integrand = x_t .* exp(-current_s * t_values);
X_s(i) = trapz(t_values, integrand);
end
% Plot Numerical Laplace Transform
figure;
plot(s_values, abs(X_s), 'b', 'LineWidth', 1.5);
title('Numerical Laplace Transform |X(s)|');
xlabel('s');
ylabel('|X(s)|');
grid on;
%% ==============================
% PART 2: System Response
% ==============================
tau = 1; % Time constant
% Input: Unit Step
u_t = ones(size(t_values));
% Output: Step Response y(t) = 1 - exp(-t/tau)
y_t = 1 - exp(-t_values / tau);
% Plot system response
figure;
plot(t_values, u_t, 'b--', 'LineWidth', 1.5, 'DisplayName', 'Input u(t)');
hold on;
plot(t_values, y_t, 'r-', 'LineWidth', 1.5, 'DisplayName', 'Output y(t)');
title('First-Order System Step Response');
xlabel('Time (s)');
ylabel('Amplitude');
legend('show');
grid on;
% Fix axis limits
xlim([0 5]); % Time axis from 0 to 5 sec
ylim([0 2]); % Amplitude axis from 0 to 2
Experiment 7: sampling theorem and aliasing.
% Sampling Theorem Demonstration with Two Sampling Rates (0 to 2 sec)
clc;
clear;
close all;
% Original continuous-time signal
fsignal = 5; % Signal frequency (Hz)
t = 0:0.001:2; % Time vector from 0 to 2 sec
x = sin(2*pi*fsignal*t); % Original signal
% --- First Sampling (higher fs, Nyquist satisfied) ---
fs1 = 12; % Sampling frequency 1
Ts1 = 1/fs1;
n1 = 0:Ts1:2;
x_n1 = sin(2*pi*fsignal*n1);
% --- Second Sampling (lower fs, aliasing likely) ---
fs2 = 8; % Sampling frequency 2
Ts2 = 1/fs2;
n2 = 0:Ts2:2;
x_n2 = sin(2*pi*fsignal*n2);
% Plot
figure;
% Original signal
plot(t, x, 'k--', 'LineWidth', 1); hold on;
% First sampling (Nyquist satisfied)
stem(n1, x_n1, 'b', 'filled', 'LineWidth', 1.2);
% Second sampling (Aliasing case)
stem(n2, x_n2, 'r', 'filled', 'LineWidth', 1.2);
% Formatting
title('Sampling Theorem Demonstration with Two Sampling Rates');
xlabel('Time (s)');
ylabel('Amplitude');
legend('Original Signal','Sampled Signal (fs=12 Hz)','Sampled Signal (fs=8 Hz)');
axis([0 2 -1.2 1.2]); % Extended axis from 0 to 2 sec
grid on;
Experiment 8 :Z transform and discrete time system analysis.
% Experiment 8: Z-Transform and Discrete-Time System Analysis
clc; clear; close all;
% Define discrete-time sequence x[n]
n = 0:10; % index from 0 to N
x = (0.8).^n; % example sequence x[n] = (0.8)^n
% Z values on the unit circle (z = e^(j*theta))
theta = linspace(0, 2*pi, 500);
z = exp(1j*theta);
% Compute Z-transform: X(z) = sum x[n] * z^(-n)
Xz = zeros(size(z));
for k = 1:length(z)
Xz(k) = sum(x .* (z(k)).^(-n));
end
% --- Plot 1: Discrete sequence x[n] ---
figure;
stem(n, x, 'filled', 'LineWidth', 1.5);
xlabel('n'); ylabel('x[n]');
title('Discrete Sequence x[n]');
grid on;
% --- Plot 2: Magnitude of X(z) on the unit circle ---
figure;
plot(theta, abs(Xz), 'LineWidth', 2);
xlabel('Frequency (rad/sample)');
ylabel('|X(e^{j\omega})|');
title('Magnitude of Z-Transform on Unit Circle');
grid on;
Experiment 9: LTI system analysis using difference equations.
% Experiment 9: LTI System Analysis using Difference Equations
clc; clear; close all;
% Define coefficients of the difference equation
a1 = 0.5; % example coefficient
a2 = 0.25; % example coefficient
b0 = 1; % input coefficient
b1 = 0.5; % input coefficient
% Form coefficient vectors for filter()
b = [b0 b1]; % numerator coefficients (x terms)
a = [1 a1 a2]; % denominator coefficients (y terms, with y[n], y[n-1], y[n-2])
% Input signal (example: unit step)
n = 0:20;
x = ones(size(n)); % unit step signal
% System response using filter()
y = filter(b, a, x);
% --- Plot input and output ---
figure;
subplot(2,1,1);
stem(n, x, 'filled', 'LineWidth', 1.5);
xlabel('n'); ylabel('x[n]');
title('Input Signal (Unit Step)');
ylim([0 1.2]); % y-axis from 0 to 1.2
yticks(0:0.2:1.2); % tick marks at 0.2 increments
grid on;
subplot(2,1,2);
stem(n, y, 'filled', 'LineWidth', 1.5);
xlabel('n'); ylabel('y[n]');
title('Output Signal (System Response)');
ylim([0 1.2]); % y-axis from 0 to 1.2
yticks(0:0.2:1.2); % tick marks at 0.2 increments
grid on;
Experiment 10: Filter design frequency response analysis
% Experiment 10: Filter Design and Frequency Response Analysis
clc; clear; close all;
%% Define impulse response h[n]
h = [1 2 3 4]; % Example impulse response (can be changed as needed)
%% Frequency range (from -pi to pi)
omega = linspace(-pi, pi, 500);
%% Compute DTFT: H(e^j?) = ? h[n] e^(-j?n)
H = zeros(size(omega));
for n = 1:length(h)
H = H + h(n) * exp(-1j * omega * (n-1)); % (n-1) because MATLAB index starts at 1
end
%% Plot Magnitude and Phase Response
figure;
subplot(2,1,1);
plot(omega, abs(H), 'LineWidth', 1.5);
xlabel('\omega (rad/sample)');
ylabel('|H(e^{j\omega})|');
title('Magnitude Response');
grid on;
subplot(2,1,2);
plot(omega, angle(H), 'LineWidth', 1.5);
xlabel('\omega (rad/sample)');
ylabel('?H(e^{j\omega}) [rad]');
title('Phase Response');
grid on;
%% Display Result
disp('Impulse Response h[n] = ');
disp(h);
