Notes

Chapter 3: Data and Signals

Time Domain Signal (Sine Wave)
% Parameters
fs = 1000; % Sampling frequency (Hz)
t = 0:1/fs:1; % Time vector from 0 to 1 second
f = 5; % Frequency of the sine wave (Hz)
% Generate the sine wave
signal = sin(2 * pi * f * t);
% Plot the signal
figure;
plot(t, signal);
title('Sine Wave');
xlabel('Time (s)');
ylabel('Amplitude');
grid on;
Unit Step Signal

% Parameters
t = -1:0.01:1; % Time vector from -1 to 1 second
% Generate the unit step signal
u = t >= 0;
% Plot the unit step signal
figure;
plot(t, u);
title('Unit Step Signal');
xlabel('Time (s)');
ylabel('Amplitude');
grid on;

Unit Impulse Signal
% Parameters
n = -5:5; % Discrete time vector
% Generate the unit impulse signal
delta = (n == 0);
% Plot the unit impulse signal
figure;
stem(n, delta);
title('Unit Impulse Signal');
xlabel('n');
ylabel('Amplitude');
grid on;

Generate and Compare Analog and Digital Signals
% Parameters
fs = 1000; % Sampling frequency for analog (Hz)
t = 0:1/fs:1; % Time vector from 0 to 1 second
f = 5; % Frequency of the sine wave (Hz)
% Generate the analog signal (sine wave)
analog_signal = sin(2 * pi * f * t);
% Sampling the analog signal to create a digital signal
fs_digital = 100; % Sampling frequency for digital (Hz)
t_digital = 0:1/fs_digital:1; % Digital time vector
digital_signal = sin(2 * pi * f * t_digital); % Sampled signal
% Plot both analog and digital signals
figure;
hold on;
plot(t, analog_signal, 'b', 'LineWidth', 1.5); % Analog signal
stem(t_digital, digital_signal, 'r', 'filled', 'LineWidth', 1.5); % Digital signal
title('Comparison of Analog and Digital Signals');
xlabel('Time (s)');
ylabel('Amplitude');
legend('Analog Signal', 'Digital Signal');

grid on;
hold off;

MATLAB Code for Two Signals with Same Amplitude and Phase, Different

Frequencies

% Parameters
fs = 1000; % Sampling frequency (Hz)
t = 0:1/fs:1; % Time vector from 0 to 1 second
A = 1; % Amplitude of the signals
phi = 0; % Phase (in radians)
% Frequencies for the two signals
f1 = 5; % Frequency of the first signal (Hz)
f2 = 10; % Frequency of the second signal (Hz)
% Generate the two signals
signal1 = A * sin(2 * pi * f1 * t + phi); % First signal
signal2 = A * sin(2 * pi * f2 * t + phi); % Second signal
% Plot both signals
figure;
hold on;
plot(t, signal1, 'b', 'LineWidth', 1.5); % First signal in blue
plot(t, signal2, 'r', 'LineWidth', 1.5); % Second signal in red
title('Two Signals with Same Amplitude and Phase, Different Frequencies');
xlabel('Time (s)');
ylabel('Amplitude');
legend('Signal 1 (f=5 Hz)', 'Signal 2 (f=10 Hz)');
grid on;
hold off;

Calculate the wavelength in cm if the frequency = 6 GHz
% Given frequency
frequency_GHz = 6; % Frequency in GHz
frequency_Hz = frequency_GHz * 1e9; % Convert to Hz
% Speed of light
speed_of_light = 3e8; % Speed of light in m/s

% Calculate wavelength in meters
wavelength_m = speed_of_light / frequency_Hz; % Wavelength in meters
% Convert wavelength to centimeters
wavelength_cm = wavelength_m * 100; % Wavelength in centimeters
% Display the result
fprintf('The wavelength for a frequency of %.2f GHz is %.4f cm.n', frequency_GHz,
wavelength_cm);

Explanation
1. Frequency Conversion: The frequency is converted from gigahertz (GHz) to hertz (Hz).
2. Speed of Light: The speed of light is set to 3×1083 times 10^83×108 m/s.
3. Wavelength Calculation: The wavelength is calculated in meters using the formula

λ = C/f

4. Conversion to Centimeters: The wavelength in meters is then converted to centimeters
by multiplying by 100.
5. Output: The result is printed in the command window.
MATLAB code of the period of a signal is 100 ms. What is its frequency in
kilohertz?
% Given period in milliseconds
period_ms = 100; % Period in milliseconds
period_s = period_ms / 1000; % Convert to seconds
% Calculate frequency in Hz
frequency_Hz = 1 / period_s; % Frequency in Hz
% Convert frequency to kilohertz
frequency_kHz = frequency_Hz / 1000; % Frequency in kHz
% Display the result
fprintf('The frequency of a signal with a period of %.2f ms is %.2f kHz.n', period_ms,
frequency_kHz);

MATLAB Code for Three Sine Waves with Same Amplitude and Frequency,

Different Phases

% Parameters
fs = 1000; % Sampling frequency (Hz)
t = 0:1/fs:1; % Time vector from 0 to 1 second
A = 1; % Amplitude of the signals
f = 5; % Frequency of the sine waves (Hz)
% Different phases (in radians)
phase1 = 0; % Phase of the first sine wave
phase2 = pi/2; % Phase of the second sine wave (90 degrees)
phase3 = pi; % Phase of the third sine wave (180 degrees)
% Generate the three sine waves
signal1 = A * sin(2 * pi * f * t + phase1); % First sine wave
signal2 = A * sin(2 * pi * f * t + phase2); % Second sine wave
signal3 = A * sin(2 * pi * f * t + phase3); % Third sine wave
% Plot the three sine waves
figure;
hold on;
plot(t, signal1, 'b', 'LineWidth', 1.5); % First sine wave in blue
plot(t, signal2, 'r', 'LineWidth', 1.5); % Second sine wave in red
plot(t, signal3, 'g', 'LineWidth', 1.5); % Third sine wave in green
title('Three Sine Waves with Same Amplitude and Frequency, Different Phases');
xlabel('Time (s)');
ylabel('Amplitude');
legend('Phase = 0', 'Phase = pi/2', 'Phase = pi');
grid on;
hold off;

Explanation
1. Parameters: The sampling frequency (fs), time vector (t), amplitude (A), and frequency (f) are
defined.
2. Phase Definitions: Three different phases are defined for the sine waves: 000, π2frac{pi}{2}2π
(90 degrees), and πpiπ (180 degrees).
3. Signal Generation: Three sine waves are generated using the sine function with the specified
phases.
4. Plotting: The sine waves are plotted on the same graph for visual comparison, with appropriate
legends and labels.

MATLAB code that generates a constant signal (0 Hz sine wave) and plots both

its time-domain and frequency-domain representations.

% Parameters
fs = 1000; % Sampling frequency (Hz)
t = 0:1/fs:1; % Time vector from 0 to 1 second
A = 1; % Amplitude of the signal
% Generate the 0 Hz signal (constant value)
signal = A * ones(size(t)); % A constant signal
% Calculate the FFT of the signal
N = length(signal);
Y = fft(signal);
f_freq = (0:N-1) * (fs/N); % Frequency vector
% Plotting
figure;
% Time Domain Plot
subplot(2, 1, 1);
plot(t, signal, 'k', 'LineWidth', 1.5);
title('Time Domain of 0 Hz Signal');
xlabel('Time (s)');
ylabel('Amplitude');
grid on;
% Frequency Domain Plot
subplot(2, 1, 2);
plot(f_freq(1:N/2), abs(Y(1:N/2))/N);
title('Frequency Domain of 0 Hz Signal');
xlabel('Frequency (Hz)');
ylabel('|Y(f)|');
xlim([0 fs/2]); % Limit x-axis to half the sampling frequency
grid on;

MATLAB Code for Time Domain and Frequency Domain Sine Waves

for f= 8 Hz

% Parameters
fs = 1000; % Sampling frequency (Hz)
t = 0:1/fs:1; % Time vector from 0 to 1 second
A = 1; % Amplitude of the sine wave
% Generate the 8 Hz sine wave
frequency = 8; % Frequency of the sine wave (8 Hz)
signal = A * sin(2 * pi * frequency * t); % Sine wave
% Calculate the FFT of the signal
N = length(signal);
Y = fft(signal);
f_freq = (0:N-1) * (fs/N); % Frequency vector
% Plotting
figure;
% Time Domain Plot
subplot(2, 1, 1);
plot(t, signal, 'k', 'LineWidth', 1.5);
title('Time Domain of 8 Hz Sine Wave');
xlabel('Time (s)');
ylabel('Amplitude');
grid on;
% Frequency Domain Plot
subplot(2, 1, 2);
plot(f_freq(1:N/2), abs(Y(1:N/2))/N);
title('Frequency Domain of 8 Hz Sine Wave');
xlabel('Frequency (Hz)');
ylabel('|Y(f)|');
xlim([0 fs/2]); % Limit x-axis to half the sampling frequency
grid on;

Simple MATLAB Code for Time Domain and Frequency Domain Sine Waves for

f= 16 Hz

% Parameters
fs = 1000; % Sampling frequency (Hz)
t = 0:1/fs:1; % Time vector from 0 to 1 second
A = 1; % Amplitude of the sine wave
% Generate the 16 Hz sine wave
frequency = 16; % Frequency of the sine wave (16 Hz)
signal = A * sin(2 * pi * frequency * t); % Sine wave
% Calculate the FFT of the signal
N = length(signal);
Y = fft(signal);
f_freq = (0:N-1) * (fs/N); % Frequency vector
% Plotting
figure;
% Time Domain Plot
subplot(2, 1, 1);
plot(t, signal, 'k', 'LineWidth', 1.5);
title('Time Domain of 16 Hz Sine Wave');
xlabel('Time (s)');
ylabel('Amplitude');
grid on;
% Frequency Domain Plot
subplot(2, 1, 2);
plot(f_freq(1:N/2), abs(Y(1:N/2))/N);
title('Frequency Domain of 16 Hz Sine Wave');
xlabel('Frequency (Hz)');
ylabel('|Y(f)|');
xlim([0 fs/2]); % Limit x-axis to half the sampling frequency
grid on;

Explanation

1. Parameters:
o fs: The sampling frequency is set to 1000 Hz.
o t: A time vector ranging from 0 to 1 second.
o A: The amplitude of the sine wave.
2. Signal Generation:
o A sine wave with a frequency of 16 Hz is generated using the sine function.
3. FFT Calculation:
o The Fast Fourier Transform (FFT) of the signal is computed to analyze its frequency
components.

4. Plotting:
o Time Domain: The first subplot displays the 16 Hz sine wave over time.
o Frequency Domain: The second subplot shows the frequency spectrum, which should
have a peak at 16 Hz.

MATLAB code of the bandwidth of periodic signals: 100, 200, 300, 400,

500, 600, 700, 800 Hz where BW = Fh – Fl

To compute the bandwidth of a periodic signal composed of frequencies 100 Hz, 200 Hz, 300 Hz, 400 Hz,
500 Hz, 600 Hz, 700 Hz, and 800 Hz using the formula BW=Fh−FlBW = F_h - F_lBW=Fh−Fl (where
FhF_hFh is the highest frequency component and FlF_lFl is the lowest), you can use the following
MATLAB code.
% Parameters
Fs = 2000; % Sampling frequency (2 kHz)
t = 0:1/Fs:1; % Time vector (1 second duration)
% Define frequencies
frequencies = [100, 200, 300, 400, 500, 600, 700, 800]; % Frequencies in Hz
% Create periodic composite signal
periodic_signal = sum(sin(2 * pi * frequencies' * t), 1);
% Calculate the bandwidth
F_l = min(frequencies); % Lowest frequency
F_h = max(frequencies); % Highest frequency
BW = F_h - F_l; % Bandwidth calculation
% Display results
fprintf('Bandwidth of periodic composite signal: %.2f Hzn', BW);