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
🎓 GATE SOLUTIONS
📡 WIRELESS & THEORY

Contact Us

Name

Email *

Message *

Popular Posts

OFDM Symbols and Subcarriers Explained

This article explains how OFDM (Orthogonal Frequency Division Multiplexing) symbols and subcarriers work. It covers modulation, mapping symbols to subcarriers, subcarrier frequency spacing, IFFT synthesis, cyclic prefix, and transmission. Step 1: Modulation First, modulate the input bitstream. For example, with 16-QAM , each group of 4 bits maps to one QAM symbol. Suppose we generate a sequence of QAM symbols: s0, s1, s2, s3, s4, s5, …, s63 Step 2: Mapping Symbols to Subcarriers Assume N sub = 8 subcarriers. Each OFDM symbol in the frequency domain contains 8 QAM symbols (one per subcarrier): Mapping (example) OFDM symbol 1 → s0, s1, s2, s3, s4, s5, s6, s7 OFDM symbol 2 → s8, s9, s10, s11, s12, s13, s14, s15 … OFDM sym...
Read more

BER vs SNR for M-ary QAM, M-ary PSK, QPSK, BPSK, ...(MATLAB Code + Simulator)

Bit Error Rate (BER) & SNR Guide Analyze communication system performance with our interactive simulators and MATLAB tools. 📘 Theory 🧮 Simulators 💻 MATLAB Code 📚 Resources BER Definition SNR Formula BER Calculator MATLAB Comparison 📂 Explore M-ary QAM, PSK, and QPSK Topics ▼ 🧮 Constellation Simulator: M-ary QAM 🧮 Constellation Simulator: M-ary PSK 🧮 BER calculation for ASK, FSK, and PSK 🧮 Approaches to BER vs SNR What is Bit Error Rate (BER)? The BER indicates how many corrupted bits are received compared to the total number of bits sent. It is the primary figure of merit f...
Read more

UGC NET Electronic Science Previous Year Question Papers with Solutions

Home / Engineering & Other Exams / UGC NET 2026 PYQ ⬇️ Download Papers and Solutions 📋 Exam Pattern 💡 Preparation Tips ❓ FAQs 📊 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 2025 - Question Paper Download PDF June 2025 - Solved Paper + Explanation ...
Read more

Intel 8086 Transistor Count: Architecture, Specifications, and Comparison with Other Microprocessors

Intel 8086 Transistor Count: Architecture, Specifications, and Comparison with Other Microprocessors Intel 8086 Transistor Count: Complete Guide with Architecture and Processor Comparison The Intel 8086 microprocessor is one of the most important processors in computer history. Released in 1978 , it introduced the x86 architecture that still influences modern CPUs. One of the most frequently asked questions in computer architecture and microprocessor courses is: How many transistors are present in the Intel 8086? The commonly accepted answer is approximately 29,000 transistors . However, reverse-engineering studies have shown that the actual number of physical transistors is closer to 19,618 , while Intel's published figure includes programmable transistor locations used in ROM and PLA structures. Intel 8086 Transistor Count Metric Value Published transistor count ~29,000 Physical transistor count ~19,618 Release year 1978 Word ...
Read more

Choke Input Filter Explained

  Choke Input Filter Choke Input Filter A well-designed choke input filter is a type of power supply filter used to smooth the output of a rectifier (like in DC power supplies). It uses an inductor (choke) as the first component right after the rectifier, followed by a capacitor. Basic Structure Rectifier → Choke (L) → Capacitor (C) → Load What Makes It Well-Designed? Critical Inductance is satisfied: The choke must have enough inductance to keep current flowing continuously. This minimum value is called critical inductance. Low ripple output: A good design significantly reduces AC ripple in the DC output. The choke resists sudden changes in current. Proper load current: Works best when the load current is above a certain minimum level. Too light a load results in poor filter...
Read more

FM Bandwidth and FM Band Explained

FM radio uses the frequency band from 88 MHz to 108 MHz , which is a 20 MHz-wide spectrum . This is the range of carrier frequencies available to stations. 108 MHz − 88 MHz = 20 MHz However, a single FM station occupies only about 200 kHz . This is the bandwidth of the modulated FM signal. 1. Why One FM Station Needs ~200 kHz FM uses frequency modulation . The bandwidth depends on how far the carrier swings. Carson's Rule gives the approximate FM bandwidth: B = 2 ( Δf + f m ) ...
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