MATLAB Code
clc;
clear;
close all;
% Configuration parameters
fs = 10000; % Sampling rate (Hz)
t = 0:1/fs:1-1/fs; % Time vector creation
% Signal definition
x = sin(2 * pi * 100 * t) + cos(2 * pi * 1000 * t);
% Calculate the Fourier Transform
y = fft(x);
z = fftshift(y);
% Create frequency vector
ly = length(y);
f = (-ly/2:ly/2-1) / ly * fs;
% Calculate phase while avoiding numerical precision issues
tol = 1e-6; % Tolerance threshold for zeroing small values
z(abs(z) < tol) = 0;
phase = angle(z);
% Plot the original Signal
figure;
subplot(3, 1, 1);
plot(t, x, 'b');
xlabel('Time (s)');
ylabel('|y|');
title('Original Messge Signal');
grid on;
% Plot the magnitude of the Fourier Transform
subplot(3, 1, 2);
stem(f, abs(z), 'b');
xlabel('Frequency (Hz)');
ylabel('|y|');
title('Magnitude of the Fourier Transform');
grid on;
% Plot the phase of the Fourier Transform
subplot(3, 1, 3);
stem(f, phase / pi, 'b');
xlabel('Frequency (Hz)');
ylabel('Phase (radians)');
title('Phase of the Fourier Transform');
grid on;
Output
Copy the MATLAB Code above from here
Another MATLAB Code
clc;
clear;
close all;
% Parameters
fs = 100; % Sampling frequency
t = 0:1/fs:1-1/fs; % Time vector
% Signal definition
x = cos(2*pi*15*t - pi/4) - sin(2*pi*40*t);
% Compute Fourier Transform
y = fft(x);
z = fftshift(y);
% Frequency vector
ly = length(y);
f = (-ly/2:ly/2-1)/ly*fs;
% Compute phase
z(abs(z) < 1e-6) = 0;
phase = angle(z);
% Plot magnitude of the Fourier Transform
figure;
subplot(2, 1, 1);
stem(f, abs(z), 'b');
xlabel('Frequency (Hz)');
ylabel('|y|');
title('Magnitude of Fourier Transform');
grid on;
% Plot phase of the Fourier Transform
subplot(2, 1, 2);
stem(f, phase, 'b');
xlabel('Frequency (Hz)');
ylabel('Phase (radians)');
title('Phase of Fourier Transform');
grid on;
web('https://www.salimwireless.com/search?q=fourier%20transform', '-browser');
Output
Copy the MATLAB Code above from here
Why use fftshift? In MATLAB, the fft function returns the DC component at the beginning of the array. To visualize a standard double-sided spectrum where 0 Hz is in the center, we use fftshift. This is essential for analyzing signal symmetry in wireless communications and signal processing.
Handling Phase Noise: Notice the tol = 1e-6 line. This is a pro-tip! We zero out very small magnitude values before calculating the phase to avoid "random" phase noise caused by floating-point errors.
Real-World Applications of FFT in MATLAB
- Wireless Communications: Used in OFDM (5G/Wi-Fi) to split data across multiple sub-carriers.
- Audio Engineering: Analyzing frequency response for noise cancellation and equalization.
- Medical Imaging: Processing MRI and Ultrasound data using Fast Fourier Transforms.
- Radar Systems: Determining the velocity of objects via Doppler shift analysis.
Interactive Online Simulators
Frequently Asked Questions
Q: Why is my magnitude plot showing peaks at the wrong frequency?
A: Ensure your sampling frequency (fs) is at least twice the highest frequency of your signal (Nyquist Theorem).
Q: How do I increase frequency resolution?
A: Increase the number of samples (N) or the time duration of your signal vector.
Further Reading
- Fourier Transform of Sine or Cosine
- Definition of the Fourier Series
- Continuous and Discrete Time Fourier Transform
- Cooley-Tukey algorithm for Fast Fourier Transform (FFT) in MATLAB
- Fourier Spectral Analysis
- Power Spectral Density Calculation Using FFT in MATLAB
- Autocorrelation and Periodicity of a Signal
Power Spectral Density (PSD)
Analyzing Signal Power in the Frequency Domain
In fading channels, **Power Spectral Density (PSD)** is critical because it describes how the power of your signal (or noise) is distributed across frequencies. Unlike a standard Fourier Magnitude, PSD provides a normalized view of power relative to the sampling rate.
Calculation Steps:
- Compute the FFT of the signal \(x(t)\).
- Calculate the Absolute Magnitude.
- Square the magnitude to get the Power Spectrum.
- Divide by \((fs \cdot N)\) to find the Density.
The Mathematical Model
(Determines frequency range)
(Determines resolution)
\[ \Delta f = \frac{fs}{N} \]
Note: Increasing \(N\) improves frequency resolution, while increasing \(fs\) expands the frequency range (Nyquist limit).
Read More & Access Interactive PSD Simulator