MATLAB Tutorial - ELEC 360 | Dr. Mohammad Al Bataineh

1️⃣ Getting Started with MATLAB

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Accessing MATLAB at UAEU

🖥️ On Campus

MATLAB is installed on all engineering lab computers. Visit any computer lab in the College of Engineering.

💻 Personal Computer

Download MATLAB through UAEU's campus license. Contact IT support for installation instructions.

🌐 MATLAB Online

Access MATLAB through your web browser at matlab.mathworks.com using your UAEU credentials.

MATLAB Interface Overview

When you open MATLAB, you'll see several key windows:

Command Window: Where you type commands and see results
Workspace: Shows all variables currently in memory
Current Folder: Displays files in your working directory
Editor: For writing and saving scripts (.m files)

2️⃣ MATLAB Basics

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Basic Operations

MATLAB Code

% Basic arithmetic
a = 5;              % Assignment (semicolon suppresses output)
b = 3;
sum = a + b         % Addition (no semicolon shows result)
difference = a - b; % Subtraction
product = a * b;    % Multiplication
quotient = a / b;   % Division
power = a^2;        % Exponentiation

% Useful commands
clear              % Clear all variables
clc                % Clear command window
close all          % Close all figures

Vectors and Arrays

MATLAB Code

% Creating vectors
t = 0:0.1:10;              % Time vector from 0 to 10, step 0.1
t = linspace(0, 10, 100);  % 100 points from 0 to 10
x = [1 2 3 4 5];           % Row vector
y = [1; 2; 3; 4; 5];       % Column vector

% Array operations (element-by-element)
a = [1 2 3];
b = [4 5 6];
c = a .* b;      % Element-by-element multiplication [4 10 18]
d = a .^ 2;      % Element-by-element power [1 4 9]

% Useful array functions
length(x)        % Number of elements
size(x)          % Dimensions
max(x)           % Maximum value
min(x)           % Minimum value
sum(x)           % Sum of elements
mean(x)          % Average value

⚠️ Important: Always use .*, ./, and .^ for element-by-element operations on arrays. This is crucial for signal processing!

Common Mathematical Functions

Function	MATLAB Code	Description
Sine	`sin(x)`	Sine function
Cosine	`cos(x)`	Cosine function
Exponential	`exp(x)`	e^x
Natural Log	`log(x)`	ln(x)
Absolute Value	`abs(x)`	\|x\|
Square Root	`sqrt(x)`	√x

3️⃣ Plotting Signals

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Basic Plotting

MATLAB Code

% Simple plot
t = 0:0.01:2*pi;
y = sin(t);
plot(t, y)
xlabel('Time (s)')
ylabel('Amplitude')
title('Sine Wave')
grid on

% Multiple plots on same figure
t = 0:0.01:2*pi;
y1 = sin(t);
y2 = cos(t);
plot(t, y1, 'b-', t, y2, 'r--')
legend('sin(t)', 'cos(t)')
xlabel('Time (s)')
ylabel('Amplitude')
title('Sine and Cosine')
grid on

Subplot

MATLAB Code

% Create multiple plots in one figure
t = 0:0.01:2*pi;

subplot(2,1,1)
plot(t, sin(t))
title('Sine Wave')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(2,1,2)
plot(t, cos(t))
title('Cosine Wave')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

Plot Customization

Option	Code	Effect
Line Color	`'r'`, `'b'`, `'g'`, `'k'`	Red, Blue, Green, Black
Line Style	`'-'`, `'--'`, `':'`, `'-.'`	Solid, Dashed, Dotted, Dash-dot
Marker	`'o'`, `'*'`, `'+'`, `'x'`	Circle, Star, Plus, Cross
Line Width	`'LineWidth', 2`	Thicker line

💡 Tip: Use figure command to create new figure windows: figure; plot(x,y)

4️⃣ Signal Operations

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Creating Basic Signals

MATLAB Code

% Unit Step Function
t = -5:0.01:5;
u = (t >= 0);        % Unit step at t=0
figure;
plot(t, u, 'LineWidth', 2)
title('Unit Step Function u(t)')
xlabel('Time (s)')
ylabel('u(t)')
grid on
ylim([-0.2 1.2])

% Shifted Unit Step
u_shifted = (t >= 2);  % Step at t=2
figure;
plot(t, u, 'b-', t, u_shifted, 'r--', 'LineWidth', 2)
legend('u(t)', 'u(t-2)')
title('Unit Step and Shifted Step')
xlabel('Time (s)')
grid on

% Rectangular Pulse
t = -5:0.01:5;
rect = ((t >= -1) & (t <= 1));
figure;
plot(t, rect, 'LineWidth', 2)
title('Rectangular Pulse')
xlabel('Time (s)')
ylabel('rect(t)')
grid on
ylim([-0.2 1.2])

% Exponential Signals
t = 0:0.01:5;
figure;
subplot(2,1,1)
x1 = exp(-t);        % Decaying exponential
plot(t, x1, 'LineWidth', 2)
title('Decaying Exponential: e^{-t}')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(2,1,2)
x2 = exp(-2*t);      % Faster decay
plot(t, x2, 'r', 'LineWidth', 2)
title('Faster Decay: e^{-2t}')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

Signal Transformations

MATLAB Code

% Time Shifting (Delay)
t = -5:0.01:5;
x = exp(-abs(t));    % Original signal

% Different time shifts
x_shift1 = exp(-abs(t-1));   % Shifted right by 1
x_shift2 = exp(-abs(t+1));   % Shifted left by 1

figure;
plot(t, x, 'b-', 'LineWidth', 2)
hold on
plot(t, x_shift1, 'r--', 'LineWidth', 2)
plot(t, x_shift2, 'g-.', 'LineWidth', 2)
legend('x(t)', 'x(t-1)', 'x(t+1)')
title('Time Shifting')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

% Time Scaling (Compression and Expansion)
t = -5:0.01:5;
x = exp(-abs(t)) .* sin(2*pi*t);

figure;
subplot(3,1,1)
plot(t, x, 'LineWidth', 2)
title('Original: x(t)')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(3,1,2)
x_compressed = exp(-abs(2*t)) .* sin(2*pi*2*t);
plot(t, x_compressed, 'r', 'LineWidth', 2)
title('Compressed: x(2t)')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(3,1,3)
x_expanded = exp(-abs(0.5*t)) .* sin(2*pi*0.5*t);
plot(t, x_expanded, 'g', 'LineWidth', 2)
title('Expanded: x(t/2)')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

Even and Odd Decomposition

MATLAB Code

% Decompose signal into even and odd parts
% x(t) = x_e(t) + x_o(t)
% x_e(t) = [x(t) + x(-t)]/2  (even part)
% x_o(t) = [x(t) - x(-t)]/2  (odd part)

t = -5:0.01:5;
x = exp(-abs(t)) .* sin(t);  % Original signal

% Create x(-t) by flipping
x_reversed = fliplr(x);

% Even and odd parts
x_even = (x + x_reversed) / 2;
x_odd = (x - x_reversed) / 2;

figure;
subplot(4,1,1)
plot(t, x, 'LineWidth', 2)
title('Original Signal x(t)')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(4,1,2)
plot(t, x_even, 'r', 'LineWidth', 2)
title('Even Part: x_e(t) = [x(t) + x(-t)]/2')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(4,1,3)
plot(t, x_odd, 'g', 'LineWidth', 2)
title('Odd Part: x_o(t) = [x(t) - x(-t)]/2')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(4,1,4)
plot(t, x, 'b-', t, x_even + x_odd, 'r--', 'LineWidth', 2)
legend('x(t)', 'x_e(t) + x_o(t)')
title('Verification: x(t) = x_e(t) + x_o(t)')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

💡 Key Concepts:

Time Shifting: x(t-t₀) delays signal by t₀
Time Scaling: x(at) compresses signal if a>1, expands if 0
Time Reversal: x(-t) flips signal about t=0

5️⃣ System Analysis

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Convolution - Graphical Method

MATLAB Code

% Convolution: y(t) = x(t) * h(t) = ∫x(τ)h(t-τ)dτ

% Define signals
t = -2:0.01:5;
x = (t >= 0 & t <= 1);      % Rectangular pulse
h = exp(-2*t) .* (t >= 0);  % Exponential

% Method 1: Using conv function
dt = 0.01;
y_conv = conv(x, h, 'same') * dt;

% Method 2: Manual convolution (for understanding)
y_manual = zeros(size(t));
for i = 1:length(t)
    tau = t;
    h_shifted = exp(-2*(t(i)-tau)) .* ((t(i)-tau) >= 0);
    y_manual(i) = sum(x .* h_shifted) * dt;
end

% Plot results
figure;
subplot(3,1,1)
plot(t, x, 'LineWidth', 2)
title('Input Signal: x(t)')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(3,1,2)
plot(t, h, 'r', 'LineWidth', 2)
title('Impulse Response: h(t)')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(3,1,3)
plot(t, y_conv, 'b-', t, y_manual, 'r--', 'LineWidth', 2)
legend('Using conv()', 'Manual calculation')
title('Output: y(t) = x(t) * h(t)')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

Differential Equations - Zero-Input and Zero-State Response

MATLAB Code

% Solving: dy/dt + ay = x(t)
% Total response = Zero-Input Response + Zero-State Response

a = 2;  % System parameter
t = 0:0.01:5;
dt = 0.01;

% Input signal
x = sin(2*pi*t);

% Initial condition
y0 = 1;

% Zero-Input Response (x(t) = 0, y(0) = y0)
y_zi = y0 * exp(-a*t);

% Zero-State Response (y(0) = 0)
h = exp(-a*t);  % Impulse response
y_zs = conv(x, h, 'same') * dt;

% Total Response
y_total = y_zi + y_zs;

% Plot
figure;
subplot(4,1,1)
plot(t, x, 'LineWidth', 2)
title('Input: x(t) = sin(2πt)')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(4,1,2)
plot(t, y_zi, 'r', 'LineWidth', 2)
title('Zero-Input Response')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(4,1,3)
plot(t, y_zs, 'g', 'LineWidth', 2)
title('Zero-State Response')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(4,1,4)
plot(t, y_total, 'b', 'LineWidth', 2)
title('Total Response = ZIR + ZSR')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

Second-Order System Analysis

MATLAB Code

% Second-order system: d²y/dt² + 2ζωn(dy/dt) + ωn²y = ωn²x(t)

wn = 2*pi;  % Natural frequency
zeta_values = [0.1, 0.5, 0.7, 1.0, 2.0];  % Damping ratios

t = 0:0.01:5;
figure;
hold on

for i = 1:length(zeta_values)
    zeta = zeta_values(i);
    num = wn^2;
    den = [1, 2*zeta*wn, wn^2];
    sys = tf(num, den);
    [y, t_step] = step(sys, t);
    plot(t_step, y, 'LineWidth', 2, 'DisplayName', ['\zeta = ' num2str(zeta)])
end

title('Step Response of Second-Order System')
xlabel('Time (s)')
ylabel('Amplitude')
legend('show')
grid on
hold off

✅ Key System Analysis Tools:

conv(x, h) - Convolution of two signals
tf(num, den) - Create transfer function
step(sys, t) - Step response
impulse(sys, t) - Impulse response

6️⃣ Fourier Analysis

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Fourier Series - Trigonometric Form

MATLAB Code

% Fourier Series: x(t) = a0 + Σ[an*cos(nω0t) + bn*sin(nω0t)]

% Square Wave Example
T = 2;           % Period
f0 = 1/T;        % Fundamental frequency
w0 = 2*pi*f0;    % Angular frequency
t = -2*T:0.001:2*T;

% Number of harmonics
N_values = [1, 3, 5, 10, 50];
figure;

for idx = 1:length(N_values)
    N = N_values(idx);
    x = zeros(size(t));
    
    % Fourier series for square wave (odd harmonics only)
    for n = 1:2:N
        bn = 4/(n*pi);
        x = x + bn * sin(n*w0*t);
    end
    
    subplot(3,2,idx)
    plot(t, x, 'LineWidth', 2)
    ylim([-1.5 1.5])
    title(['N = ' num2str(N) ' harmonics'])
    xlabel('Time (s)')
    ylabel('Amplitude')
    grid on
end

subplot(3,2,6)
N = 100;
x = zeros(size(t));
for n = 1:2:N
    bn = 4/(n*pi);
    x = x + bn * sin(n*w0*t);
end
plot(t, x, 'LineWidth', 2)
ylim([-1.5 1.5])
title(['N = ' num2str(N) ' harmonics'])
xlabel('Time (s)')
ylabel('Amplitude')
grid on

fprintf('Notice Gibbs phenomenon at discontinuities!\n');

Fourier Series - Exponential Form

MATLAB Code

% Exponential Fourier Series: x(t) = Σ cn*exp(jnω0t)

% Sawtooth Wave Example
T = 2;
w0 = 2*pi/T;
t = -2*T:0.001:2*T;
N = 20;  % Number of harmonics

% Compute Fourier coefficients
n = -N:N;
cn = zeros(size(n));
for i = 1:length(n)
    if n(i) == 0
        cn(i) = 0;  % DC component
    else
        cn(i) = 1j/(pi*n(i)) * (-1)^n(i);
    end
end

% Reconstruct signal
x = zeros(size(t));
for i = 1:length(n)
    x = x + cn(i) * exp(1j*n(i)*w0*t);
end
x = real(x);

% Plot
figure;
subplot(2,1,1)
plot(t, x, 'LineWidth', 2)
title(['Sawtooth Wave - Exponential Fourier Series (N = ' num2str(N) ')'])
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(2,1,2)
stem(n, abs(cn), 'filled')
title('Magnitude Spectrum |c_n|')
xlabel('Harmonic number n')
ylabel('|c_n|')
grid on

Fast Fourier Transform (FFT)

MATLAB Code

% FFT Example: Multi-component signal analysis

Fs = 1000;        % Sampling frequency (Hz)
T = 1/Fs;         % Sampling period
L = 1024;         % Length of signal
t = (0:L-1)*T;   % Time vector

% Create signal with multiple frequencies
f1 = 50;   % Hz
f2 = 120;  % Hz
f3 = 200;  % Hz

x = 1.0*sin(2*pi*f1*t) + 0.5*sin(2*pi*f2*t) + 0.3*sin(2*pi*f3*t);

% Add noise
x = x + 0.2*randn(size(t));

% Compute FFT
Y = fft(x);
L_half = floor(L/2) + 1;

% Single-sided spectrum
P2 = abs(Y/L);
P1 = P2(1:L_half);
P1(2:end-1) = 2*P1(2:end-1);
f = Fs*(0:L_half-1)/L;

% Plot results
figure;
subplot(2,1,1)
plot(t(1:200), x(1:200), 'LineWidth', 1.5)
title('Time Domain Signal (first 200 samples)')
xlabel('Time (s)')
ylabel('Amplitude')
grid on

subplot(2,1,2)
plot(f, P1, 'LineWidth', 2)
title('Single-Sided Magnitude Spectrum')
xlabel('Frequency (Hz)')
ylabel('|P1(f)|')
xlim([0 Fs/2])
grid on
hold on
plot([f1 f2 f3], [1.0 0.5 0.3], 'ro', 'MarkerSize', 10)
legend('Spectrum', 'Expected peaks')

% Peak detection
[peaks, locs] = findpeaks(P1, 'MinPeakHeight', 0.2);
detected_freqs = f(locs);
fprintf('Detected frequencies: ');
fprintf('%.1f Hz ', detected_freqs);
fprintf('\n');

Fourier Transform Properties

MATLAB Code

% Demonstrate Time Shifting Property

Fs = 1000;
t = linspace(-2, 2, 4*Fs);
dt = t(2) - t(1);

% Original signal
x = exp(-t.^2);

% Time shifted signal
tau = 0.5;
x_shifted = exp(-(t-tau).^2);

% Compute FFTs
w = 2*pi*(-Fs/2:Fs/(length(t)):Fs/2-Fs/(length(t)));
X = fftshift(fft(x)) * dt;
X_shifted = fftshift(fft(x_shifted)) * dt;

% Plot
figure;
subplot(2,2,1)
plot(t, x, 'b-', t, x_shifted, 'r--', 'LineWidth', 2)
legend('x(t)', 'x(t-\tau)')
title('Time Shifting')
xlabel('Time (s)')
grid on

subplot(2,2,2)
plot(w/(2*pi), abs(X), 'b-', w/(2*pi), abs(X_shifted), 'r--', 'LineWidth', 2)
legend('X(f)', 'X(f)e^{-j2πfτ}')
title('Magnitude Spectrum')
xlabel('Frequency (Hz)')
xlim([-50 50])
grid on

⚠️ Important FFT Considerations:

Sampling: Use Fs > 2*f_max (Nyquist criterion)
Length: Power of 2 for efficiency (1024, 2048, 4096)
Resolution: Frequency resolution = Fs/N
Aliasing: Frequencies above Fs/2 will fold back

7️⃣ Course-Specific Examples

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Example 1: RC Circuit Frequency Response

Problem: Analyze an RC low-pass filter with R = 1kΩ and C = 1µF

MATLAB Code

% RC Circuit: Low-pass filter
% Transfer function: H(jω) = 1/(1 + jωRC)

R = 1000;     % Resistance (Ohms)
C = 1e-6;     % Capacitance (Farads)
fc = 1/(2*pi*R*C);  % Cutoff frequency

fprintf('Cutoff frequency fc = %.2f Hz\n', fc);

% Frequency range
f = logspace(0, 5, 1000);  % 1 Hz to 100 kHz
w = 2*pi*f;

% Frequency response
H = 1 ./ (1 + 1i*w*R*C);
mag = abs(H);
mag_dB = 20*log10(mag);
phase = angle(H) * 180/pi;

% Plot Bode diagram
figure;
subplot(2,1,1)
semilogx(f, mag_dB, 'LineWidth', 2)
grid on
xlabel('Frequency (Hz)')
ylabel('Magnitude (dB)')
title('RC Low-Pass Filter Bode Plot')
hold on
plot([fc fc], [-40 0], 'r--', 'LineWidth', 1.5)
plot([fc], [-3], 'ro', 'MarkerSize', 10, 'MarkerFaceColor', 'r')
legend('|H(f)|', 'Cutoff Frequency', '-3dB point')
xlim([f(1) f(end)])

subplot(2,1,2)
semilogx(f, phase, 'LineWidth', 2)
grid on
xlabel('Frequency (Hz)')
ylabel('Phase (degrees)')
title('Phase Response')
hold on
plot([fc fc], [-90 0], 'r--', 'LineWidth', 1.5)
plot([fc], [-45], 'ro', 'MarkerSize', 10, 'MarkerFaceColor', 'r')
xlim([f(1) f(end)])

% Time domain step response
t = 0:0.0001:0.005;
v_in = ones(size(t));
tau = R*C;
v_out = 1 - exp(-t/tau);

figure;
plot(t*1000, v_in, 'b--', 'LineWidth', 2)
hold on
plot(t*1000, v_out, 'r-', 'LineWidth', 2)
plot([tau tau]*1000, [0 1-exp(-1)], 'k--', 'LineWidth', 1)
plot(tau*1000, 1-exp(-1), 'ko', 'MarkerSize', 10, 'MarkerFaceColor', 'k')
legend('Input', 'Output', 'Time Constant τ', '63.2% point')
title('Step Response of RC Circuit')
xlabel('Time (ms)')
ylabel('Voltage (V)')
grid on

Example 2: AM Modulation and Demodulation

Problem: Demonstrate amplitude modulation with fm = 1 kHz, fc = 10 kHz

MATLAB Code

% Amplitude Modulation (AM)
fm = 1000;           % Message frequency (1 kHz)
fc = 10000;          % Carrier frequency (10 kHz)
fs = 100000;         % Sampling frequency (100 kHz)
t = 0:1/fs:0.01;    % Time vector (10 ms)

% Message signal
m = cos(2*pi*fm*t);

% Carrier signal
c = cos(2*pi*fc*t);

% Modulation index
mu = 0.8;

% AM signal
s_AM = (1 + mu*m) .* c;

% Plot time domain
figure;
subplot(3,2,1)
plot(t*1000, m, 'LineWidth', 1.5)
title('Message Signal m(t)')
xlabel('Time (ms)')
ylabel('Amplitude')
grid on
xlim([0 5])

subplot(3,2,3)
plot(t*1000, c, 'r', 'LineWidth', 1)
title('Carrier Signal c(t)')
xlabel('Time (ms)')
ylabel('Amplitude')
grid on
xlim([0 1])

subplot(3,2,5)
plot(t*1000, s_AM, 'g', 'LineWidth', 1)
title('AM Signal s(t) = [1 + μm(t)]c(t)')
xlabel('Time (ms)')
ylabel('Amplitude')
grid on
xlim([0 5])
hold on
envelope_upper = 1 + mu*m;
envelope_lower = -(1 + mu*m);
plot(t*1000, envelope_upper, 'k--', t*1000, envelope_lower, 'k--', 'LineWidth', 1.5)

% Frequency domain
N = length(s_AM);
f_axis = (-N/2:N/2-1)*(fs/N);

M = fftshift(fft(m))/N;
subplot(3,2,2)
plot(f_axis/1000, abs(M), 'LineWidth', 2)
title('Message Spectrum M(f)')
xlabel('Frequency (kHz)')
ylabel('|M(f)|')
grid on
xlim([-5 5])

C = fftshift(fft(c))/N;
subplot(3,2,4)
plot(f_axis/1000, abs(C), 'r', 'LineWidth', 2)
title('Carrier Spectrum C(f)')
xlabel('Frequency (kHz)')
ylabel('|C(f)|')
grid on
xlim([-15 15])

S = fftshift(fft(s_AM))/N;
subplot(3,2,6)
plot(f_axis/1000, abs(S), 'g', 'LineWidth', 2)
title('AM Signal Spectrum S(f)')
xlabel('Frequency (kHz)')
ylabel('|S(f)|')
grid on
xlim([-15 15])

fprintf('Modulation Index μ = %.2f\n', mu);
fprintf('Bandwidth = 2fm = %.1f Hz\n', 2*fm);

Example 3: First-Order Differential Equation

Problem: Solve dy/dt + 3y = 5u(t) with y(0) = 2

MATLAB Code

% First-order system: dy/dt + ay = bx(t)
% Given: dy/dt + 3y = 5u(t), y(0) = 2

a = 3;
b = 5;
y0 = 2;
t = 0:0.01:3;

% Analytical Solution
y_zi = y0 * exp(-a*t);              % Zero-input response
y_zs = (b/a) * (1 - exp(-a*t));     % Zero-state response
y_total = y_zi + y_zs;

% System parameters
y_ss = b/a;  % Steady-state
tau = 1/a;   % Time constant

% Plot
figure;
subplot(3,1,1)
plot(t, y_zi, 'LineWidth', 2)
title('Zero-Input Response (Natural Response)')
xlabel('Time (s)')
ylabel('y_{zi}(t)')
grid on
yline(0, 'k--')

subplot(3,1,2)
plot(t, y_zs, 'r', 'LineWidth', 2)
title('Zero-State Response (Forced Response)')
xlabel('Time (s)')
ylabel('y_{zs}(t)')
grid on
yline(y_ss, 'k--', 'LineWidth', 1)
text(1.5, y_ss*1.1, ['Steady State = ' num2str(y_ss, '%.2f')])

subplot(3,1,3)
plot(t, y_total, 'g', 'LineWidth', 2)
hold on
yline(y_ss, 'k--', 'LineWidth', 1)
plot([tau tau], [0 y0*exp(-1) + y_ss*(1-exp(-1))], 'r--', 'LineWidth', 1)
plot(tau, y0*exp(-1) + y_ss*(1-exp(-1)), 'ro', 'MarkerSize', 8, 'MarkerFaceColor', 'r')
title('Total Response y(t) = y_{zi}(t) + y_{zs}(t)')
xlabel('Time (s)')
ylabel('y(t)')
grid on
legend('Total Response', 'Steady State', 'Time Constant τ')

fprintf('Time Constant τ = %.3f s\n', tau);
fprintf('Steady-State y(∞) = %.3f\n', y_ss);
fprintf('95%% settling time ≈ 3τ = %.3f s\n', 3*tau);

Example 4: Ideal Filters

Problem: Design and apply ideal low-pass, high-pass, and band-pass filters

MATLAB Code

% Ideal Filters Example
fs = 1000;
t = 0:1/fs:2;
N = length(t);

% Composite signal
f1 = 5;    % Low frequency
f2 = 50;   % Mid frequency
f3 = 200;  % High frequency

x = sin(2*pi*f1*t) + 0.5*sin(2*pi*f2*t) + 0.3*sin(2*pi*f3*t);
x = x + 0.2*randn(size(t));  % Add noise

% Compute FFT
X = fftshift(fft(x));
f = (-N/2:N/2-1)*(fs/N);

% Design Filters
H_lp = double(abs(f) <= 30);     % Low-pass (fc = 30 Hz)
H_hp = double(abs(f) >= 100);    % High-pass (fc = 100 Hz)
H_bp = double((abs(f) >= 30) & (abs(f) <= 100));  % Band-pass

% Apply filters
Y_lp = X .* H_lp';
Y_hp = X .* H_hp';
Y_bp = X .* H_bp';

y_lp = real(ifft(ifftshift(Y_lp)));
y_hp = real(ifft(ifftshift(Y_hp)));
y_bp = real(ifft(ifftshift(Y_bp)));

% Plot results
figure;
subplot(4,2,1)
plot(t(1:500), x(1:500), 'LineWidth', 1)
title('Original Signal')
xlabel('Time (s)')
grid on

subplot(4,2,2)
plot(f, abs(X)/N, 'LineWidth', 1.5)
title('Original Spectrum')
xlabel('Frequency (Hz)')
xlim([-fs/2 fs/2])
grid on

subplot(4,2,3)
plot(t(1:500), y_lp(1:500), 'r', 'LineWidth', 1)
title('Low-Pass Filtered (f < 30 Hz)')
xlabel('Time (s)')
grid on

subplot(4,2,4)
plot(f, H_lp, 'r', 'LineWidth', 2)
hold on
plot(f, abs(Y_lp)/N, 'b', 'LineWidth', 1.5)
title('LPF Response')
xlabel('Frequency (Hz)')
xlim([-fs/2 fs/2])
legend('Filter', 'Output')
grid on

subplot(4,2,5)
plot(t(1:500), y_hp(1:500), 'g', 'LineWidth', 1)
title('High-Pass Filtered (f > 100 Hz)')
xlabel('Time (s)')
grid on

subplot(4,2,6)
plot(f, H_hp, 'g', 'LineWidth', 2)
hold on
plot(f, abs(Y_hp)/N, 'b', 'LineWidth', 1.5)
title('HPF Response')
xlabel('Frequency (Hz)')
xlim([-fs/2 fs/2])
grid on

subplot(4,2,7)
plot(t(1:500), y_bp(1:500), 'm', 'LineWidth', 1)
title('Band-Pass (30 Hz < f < 100 Hz)')
xlabel('Time (s)')
grid on

subplot(4,2,8)
plot(f, H_bp, 'm', 'LineWidth', 2)
hold on
plot(f, abs(Y_bp)/N, 'b', 'LineWidth', 1.5)
title('BPF Response')
xlabel('Frequency (Hz)')
xlim([-fs/2 fs/2])
grid on

Example 5: Sampling and Aliasing

Problem: Demonstrate Nyquist sampling theorem and aliasing effects

MATLAB Code

% Sampling and Aliasing Demonstration
f_signal = 5;  % Signal frequency (Hz)
t_continuous = 0:0.0001:2;
x_continuous = sin(2*pi*f_signal*t_continuous);

% Different sampling frequencies
fs1 = 50;   % Well above Nyquist
fs2 = 12;   % Just above Nyquist
fs3 = 8;    % Below Nyquist - aliasing!

t1 = 0:1/fs1:2;
x1 = sin(2*pi*f_signal*t1);

t2 = 0:1/fs2:2;
x2 = sin(2*pi*f_signal*t2);

t3 = 0:1/fs3:2;
x3 = sin(2*pi*f_signal*t3);

% Plot
figure;
subplot(3,1,1)
plot(t_continuous, x_continuous, 'b-', 'LineWidth', 1)
hold on
stem(t1, x1, 'r', 'LineWidth', 1.5, 'MarkerSize', 6)
title(['Sampling at fs = ' num2str(fs1) ' Hz (Good!)'])
xlabel('Time (s)')
ylabel('Amplitude')
legend('Original', 'Sampled')
grid on
xlim([0 1])

subplot(3,1,2)
plot(t_continuous, x_continuous, 'b-', 'LineWidth', 1)
hold on
stem(t2, x2, 'g', 'LineWidth', 1.5, 'MarkerSize', 6)
title(['Sampling at fs = ' num2str(fs2) ' Hz (Marginal)'])
xlabel('Time (s)')
ylabel('Amplitude')
legend('Original', 'Sampled')
grid on
xlim([0 1])

subplot(3,1,3)
plot(t_continuous, x_continuous, 'b-', 'LineWidth', 1)
hold on
stem(t3, x3, 'm', 'LineWidth', 1.5, 'MarkerSize', 6)
x3_interp = interp1(t3, x3, t_continuous, 'linear');
plot(t_continuous, x3_interp, 'r--', 'LineWidth', 2)
title(['Sampling at fs = ' num2str(fs3) ' Hz (ALIASING!)'])
xlabel('Time (s)')
ylabel('Amplitude')
legend('Original 5Hz', 'Sampled', 'Reconstructed')
grid on
xlim([0 1])

fprintf('Nyquist Theorem: fs > 2*f_max\n');
fprintf('For f_max = 5 Hz: fs > 10 Hz required\n');

8️⃣ Tips & Best Practices

↑ Go Back

Code Organization and Style

📝 Good Commenting

Write clear comments that explain WHY, not just WHAT

Good Example

% Calculate time constant for 95% settling
tau = 1/a;
t_settle = 3*tau;

🏷️ Variable Naming

Use descriptive names

Good Practice

% Good
sampling_frequency = 1000;
signal_amplitude = 2.5;

% OK for standard notation
fs = 1000;
fc = 50;

🧹 Clean Workspace

Start scripts properly

Script Header

% ELEC 360 - Assignment 1
% Student: Your Name (ID)
% Date: October 2025

clear all;
close all;
clc;

📁 File Organization

One main script per assignment
Use functions for repeated operations
Save figures with meaningful names
Keep data files organized

Debugging Strategies

Debugging Techniques

% 1. Display intermediate values
disp('Checking variable x:')
disp(x)

% 2. Check array dimensions
fprintf('Size of x: %d x %d\n', size(x,1), size(x,2))
whos x

% 3. Visualize data quickly
plot(x); title('Quick check'); grid on

% 4. Validate results
expected = 5;
actual = sum(x);
assert(abs(actual - expected) < 0.01, 'Sum mismatch!')

% 5. Check for NaN and Inf
if any(isnan(x))
    warning('Found NaN values')
end

% 6. Use try-catch
try
    result = risky_operation();
catch ME
    fprintf('Error: %s\n', ME.message)
end

Performance Optimization

Efficient MATLAB Code

% BAD: Growing arrays (very slow!)
tic
y = [];
for i = 1:10000
    y = [y, sin(i)];
end
time_bad = toc;

% GOOD: Preallocate arrays
tic
y = zeros(1, 10000);
for i = 1:10000
    y(i) = sin(i);
end
time_good = toc;

% BEST: Vectorization
tic
i = 1:10000;
y = sin(i);
time_best = toc;

fprintf('Without preallocation: %.4f s\n', time_bad);
fprintf('With preallocation: %.4f s\n', time_good);
fprintf('Vectorized: %.4f s (%.1fx faster!)\n', time_best, time_bad/time_best);

Common Mistakes and Solutions

Error	Cause	Solution
Matrix dimensions must agree	Size mismatch	Use `size()` and `transpose()`
Undefined variable	Typo or not defined	Check spelling, define before use
Index exceeds array bounds	Accessing non-existent element	Check with `length()`
Subscript indices must be positive	Using 0 or negative index	MATLAB uses 1-based indexing!
Out of memory	Arrays too large	Process in chunks, clear variables

Professional Plotting

Well-Formatted Plots

% Complete example
t = 0:0.01:10;
x1 = exp(-0.5*t) .* sin(2*pi*t);
x2 = exp(-0.5*t) .* cos(2*pi*t);

figure('Position', [100, 100, 800, 600]);

plot(t, x1, 'b-', 'LineWidth', 2, 'DisplayName', 'x_1(t)')
hold on
plot(t, x2, 'r--', 'LineWidth', 2, 'DisplayName', 'x_2(t)')

xlabel('Time (seconds)', 'FontSize', 12, 'FontWeight', 'bold')
ylabel('Amplitude', 'FontSize', 12, 'FontWeight', 'bold')
title('Damped Oscillations', 'FontSize', 14, 'FontWeight', 'bold')

legend('Location', 'best', 'FontSize', 11)
grid on
grid minor

xlim([0 10])
ylim([-1 1])

saveas(gcf, 'damped_oscillations.png');

MATLAB Shortcuts

Shortcut	Action
`Ctrl + C`	Stop running code
`Ctrl + Enter`	Run current section
`F5`	Run script
`F9`	Run selected code
`↑/↓ arrows`	Command history
`Tab`	Auto-complete
`Ctrl + R`	Comment lines
`Ctrl + T`	Uncomment lines
`Ctrl + I`	Auto-indent

Useful Functions Quick Reference

Signal Generation

sin(), cos(), exp()
square(), sawtooth()
randn(), rand()
linspace(), logspace()

Signal Analysis

fft(), ifft()
conv(), xcorr()
abs(), angle()
findpeaks()

System Analysis

tf(), zpk(), ss()
step(), impulse()
bode(), nyquist()
pole(), zero()

Filter Design

butter(), cheby1()
filter(), filtfilt()
freqz(), grpdelay()
fir1(), fir2()

Getting Help

Finding Documentation

% In Command Window:
help function_name      % Quick help
doc function_name       % Full documentation
lookfor keyword         % Search functions

% Examples:
help fft
doc conv
lookfor convolution

✅ Assignment Submission Checklist

✓ Header with name, ID, assignment number
✓ Code runs without errors from start to finish
✓ All variables properly defined
✓ Comments explain complex operations
✓ Figures have titles, labels, and legends
✓ Results displayed with fprintf()
✓ Code properly indented and readable
✓ Debug code removed
✓ Tested on clean workspace

📚 Additional Resources

MATLAB Documentation: mathworks.com/help/matlab/
Signal Processing Toolbox: mathworks.com/help/signal/
MATLAB Onramp: Free 2-hour interactive tutorial
Course Materials: Check Blackboard for sample codes
YouTube: Dr. Al Bataineh's lecture videos

💡 Pro Tips

Start simple: Test with simple inputs first
Plot often: Visualize at every step
Save incrementally: Don't lose work!
Learn by doing: Type code yourself
Read errors: They tell you what's wrong
Ask early: Don't wait until deadline
Study examples: Learn from working code

Need Help?

Contact Dr. Al Bataineh during office hours (Monday/Wednesday 11:00 AM - 12:00 PM)

Email: mffbataineh@uaeu.ac.ae | Office: F1-1175

← Back to ELEC 360 Course Page

🔒 Protected Content

📑 Table of Contents

1️⃣ Getting Started with MATLAB

Accessing MATLAB at UAEU

🖥️ On Campus

💻 Personal Computer

🌐 MATLAB Online

MATLAB Interface Overview

2️⃣ MATLAB Basics

Basic Operations

Vectors and Arrays

Common Mathematical Functions

3️⃣ Plotting Signals

Basic Plotting

Subplot

Plot Customization

4️⃣ Signal Operations

Creating Basic Signals

Signal Transformations

Even and Odd Decomposition

5️⃣ System Analysis

Convolution - Graphical Method

Differential Equations - Zero-Input and Zero-State Response

Second-Order System Analysis

6️⃣ Fourier Analysis

Fourier Series - Trigonometric Form

Fourier Series - Exponential Form

Fast Fourier Transform (FFT)

Fourier Transform Properties

7️⃣ Course-Specific Examples

Example 1: RC Circuit Frequency Response

Example 2: AM Modulation and Demodulation

Example 3: First-Order Differential Equation

Example 4: Ideal Filters

Example 5: Sampling and Aliasing

8️⃣ Tips & Best Practices

Code Organization and Style

📝 Good Commenting

🏷️ Variable Naming

🧹 Clean Workspace

📁 File Organization

Debugging Strategies

Performance Optimization

Common Mistakes and Solutions

Professional Plotting

MATLAB Shortcuts

Useful Functions Quick Reference

Signal Generation

Signal Analysis

System Analysis

Filter Design

Getting Help

✅ Assignment Submission Checklist

📚 Additional Resources

💡 Pro Tips

Need Help?