Skip to main content

FFT Magnitude and Phase Spectrum using MATLAB


 

MATLAB Code 

% Developed by SalimWireless.Com

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;
web('https://www.salimwireless.com/search?q=fourier%20transform', '-browser');


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.
Return to Fourier Transform Main Page →


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

  1. Fourier Transform of Sine or Cosine
  2. Definition of the Fourier Series
  3. Continuous and Discrete Time Fourier Transform
  4. Cooley-Tukey algorithm for Fast Fourier Transform (FFT) in MATLAB
  5. Fourier Spectral Analysis
  6. Power Spectral Density Calculation Using FFT in MATLAB
  7. 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:

  1. Compute the FFT of the signal \(x(t)\).
  2. Calculate the Absolute Magnitude.
  3. Square the magnitude to get the Power Spectrum.
  4. Divide by \((fs \cdot N)\) to find the Density.

The Mathematical Model

\[ PSD = \frac{|FFT(x)|^2}{fs \cdot N} \]
\(fs\): Sampling Frequency
(Determines frequency range)
\(N\): Total Samples
(Determines resolution)
Frequency Resolution (\(\Delta f\)):
\[ \Delta f = \frac{fs}{N} \]

Note: Increasing \(N\) improves frequency resolution, while increasing \(fs\) expands the frequency range (Nyquist limit).

Read More & Access Interactive PSD Simulator
📊 MATLAB & ONLINE SIMULATIONS (Web-Based)
🎓 GATE & UGC - NET SOLUTIONS
📡 WIRELESS & SIGNAL PROCESSING


Contact Us

Name

Email *

Message *

Popular Posts

MATLAB Code for Constellation Diagram of QAM configurations such as 4, 8, 16, 32, 64, 128, and 256-QAM

📘 Overview of QAM 🧮 4-QAM MATLAB 🧮 16-QAM MATLAB 🚀 Online Simulator 📂 Other Topics on Constellation Diagrams... ▼ 🧮 MATLAB Code for 4-QAM 🧮 MATLAB Code for 16-QAM 🧮 MATLAB Code for m-ary QAM 🧮 Simulator for m-ary PSK 🧮 Simulator for m-ary QAM 🧮 Overview of Energy per Bit (Eb / N0) 🧮 Simulator for ASK, FSK, and PSK Overview of QAM One of the best-performing modulation techniques is QAM [↗] . Here, we modulate the symbols by varying the carrier signal's amplitude and phase in response to the variation in the message signal (or voltage variation). So, we may say that QAM is a combination of phase and amplitude modulation. Additionally, it performs better than ASK or PSK [↗] . In fact, any constellation for any type of modulatio...
Read more

UGC NET Electronic Science Previous Year Question Papers with Solutions

Download Papers and Solutions Exam Pattern Preparation Tips FAQs More Home / Engineering & Other Exams / UGC NET 2026 PYQ 📊 Exam Highlights: Electronic Science (88) Feature Details Junior Research Fellowship (JRF) ₹37,000 + HRA per month Eligibility M.Sc/M.Tech in Electronics (55%) Validity of Certificate JRF (3 Years) | Lectureship (Lifetime) 📥 Download UGC NET Electronics PDFs Complete collection of previous year question papers, answer keys and explanations for Subject Code 88. Start Downloading 📂 View All Question Papers June 2026 - Question Paper Download PDF June 202...
Read more

Flat vs Frequency Selective Online Simulator

Flat vs Frequency Selective Online Simulator Channel Type Without Fading Flat Fading Multipaths Nakagami m SNR(dB) Run Simulation Input Signal Signal After Fading Constellation Diagram BER vs SNR Explore Advanced Flat vs Frequency-Selective Fading Simulator Want to see these equations in action? Visualize it. Launch Simulator Tool Flat Fading and Frequency-Selective Fading Effects of Multipath Components and the Nakagami-m Parameter 1. Flat Fading Flat fading occurs when the signal bandwidth is smaller than the channel's coherence bandwidth, meaning that all frequency components of the transmitted signal experience approximately the same fading. It is typically associated with narrowband signals, where the multipath delay spread is small relative to the symbol duration. ...
Read more

Orthogonal Time Frequency Space (OTFS) (with MATLAB)

In OTFS (Orthogonal Time Frequency Space) modulation — a scheme designed for high-Doppler and time-varying wireless channels — the terms ISFFT and SFFT are key mathematical transformations used to move between different representation domains. Figure: OTFS block diagram 1. ISFFT — Inverse Symplectic Finite Fourier Transform Purpose: Transforms data symbols from the delay-Doppler domain to the time-frequency domain . \[ X[n, m] = \frac{1}{\sqrt{NM}} \sum_{k=0}^{N-1} \sum_{l=0}^{M-1} x[k, l] \, e^{j2\pi \left( \frac{nk}{N} - \frac{ml}{M} \right)} \] Here, \( N \) is the number of Doppler bins (time slots), and \( M \) is the number of delay bins (subcarriers). The ISFFT maps each data symbol from the delay-Doppler grid (where the channel is sparse and easier to equalize) to the time-frequency grid (where standard multicarrier modulation like OFDM can be applied). 2. SFFT — Symplectic Finite Fourier Transform Purpose: Performs the reverse operation ...
Read more

Coherent vs. Non-Coherent Demodulation

  For demodulation In general: 1. Synchronous (coherent) demodulation Receiver generates a local carrier/oscillator . It combines/multiplies the received signal with that carrier. Then a filter extracts the message. Received signal ↓ × Local oscillator ↓ Low-pass filter ↓ Message So yes, carrier multiplication is a common implementation of synchronous demodulation . 2. Asynchronous (non-coherent) demodulation Receiver doesn't need a phase-synchronized local carrier . It uses properties of the received signal itself to recover the information. For ASK , an envelope detector is the classic example: ASK signal ↓ Diode + RC ↓ Envelope ↓ Comparator ↓ Data Modulation Synchronous demodulation Non-coherent/asynchronous demodulation ASK/OOK Local carrier + multiplier Envelope detector AM Product detector Envelope detector FSK Coherent correlator/mixer Frequency discriminator, filters, etc. PSK Local synchronized carrier + multiplier Differential/n...
Read more

Online Simulator for ASK, FSK, and PSK Signal Generation

Interactive Digital Signal Processing (DSP) Tutorial and Simulator for ASK, FSK, and BPSK modulation techniques. Try our new Digital Signal Processing Simulator!   •   Interactive ASK, FSK, and BPSK tools updated for 2025. Start Now Digital Modulation Visualizer: ASK, FSK, & BPSK Simulator Learn and visualize binary modulation techniques (ASK, FSK, BPSK) in real-time with adjustable carrier and sampling parameters. Perfect for DSP students and engineers. 📡 ASK Simulator 📶 FSK Simulator 🎚️ BPSK Simulator 📚 More Topics ASK Modulator FSK Modulator BPSK Modulator Demodulation More Topics 1. ASK (Ampli...
Read more

Random Variable vs. Random Process: Definitions, Examples, and Key Differences

Random Variable vs. Random Process: Definitions, Examples, and Key Differences Random Variable and Random Process In the fields of probability theory and statistics, understanding the distinction between a Random Variable and a Random Process (Stochastic Process) is fundamental. This guide breaks down these concepts with mathematical precision and practical examples. 1. Random Variable (RV) Definition: A Random Variable is a deterministic function that maps each possible outcome of a random experiment to a unique real number. Mathematical Representation A random variable X is defined as a mapping from the sample space S to the set of real numbers ℝ : X : S → ℝ S (Sample Space): The set of all possible outcomes of an experiment. s ∈ S: A specific outcome. X(s): The numerical value asso...
Read more