Skip to main content

Power Spectral Density Calculation Using FFT in MATLAB


📘📘 Overview 🧮🧮 Steps to calculate 💻🧮 MATLAB Codes 📚📚 Further Reading

Power spectral density (PSD) tells us how the power of a signal is distributed across different frequency components, whereas Fourier Magnitude gives you the amplitude (or strength) of each frequency component in the signal.

Steps to calculate the PSD of a signal

  1. Firstly, calculate the fast Fourier transform (FFT) of a signal.
  2. Then, calculate the Fourier magnitude (absolute value) of the signal.
  3. Square the Fourier magnitude to get the power spectrum.
  4. To calculate the Power Spectral Density (PSD), divide the squared magnitude by the product of the sampling frequency (fs) and the total number of samples (N).
    Formula: PSD = |FFT|^2 / (fs * N)

Sampling frequency (fs): The rate at which the continuous-time signal is sampled (in Hz).
Total number of samples (N): The number of samples in the time-domain signal used for the DFT/FFT.

Suppose:
    Sampling frequency = 1000 Hz
    Number of samples = 500
Then the frequency resolution is:
Δf = 1000 / 500 = 2 Hz
This means the FFT result will contain frequency components spaced 2 Hz apart: 0 Hz, 2 Hz, 4 Hz, ..., up to fs.

  • Increasing the number of samples (N) → improves frequency resolution
  • Increasing the sampling frequency (fs) → worsens frequency resolution, but increases the total frequency range analyzed (Nyquist limit)

MATLAB Script

matlab_script.m 

% The code is written by SalimWireless.com
clear
close all
clc

fs = 1000; % sampling frequency
T = 1; % total recording time
L = T .* fs; % signal length
tt = (0:L-1)/fs; % time vector
ff = (0:L-1)*fs/L;
y = sin(2*pi*50 .* tt) + sin(2*pi*80 .* tt); y = y(:); % reference sinusoid

% Allow user to input SNR in dB
snr_db = input('Enter the SNR (in dB): '); % User input for SNR
snr_linear = 10^(snr_db / 10); % Convert SNR from dB to linear scale

% Calculate noise variance based on SNR
signal_power = mean(y.^2); % Calculate signal power
noise_variance = signal_power / snr_linear; % Calculate noise variance

% Multiply by standard deviation (sqrt of variance) for correct noise power
x = sqrt(noise_variance)*randn(L,1) + y; x = x(:); % sinusoid with additive Gaussian noise

% Plot results
figure

% Time-domain plot of the original signal
subplot(311)
plot(tt, y,'r')
title('Original Message signal sin(2Ï€ * 50)t + sin(2Ï€ * 80)t (Time Domain)')
legend('Original signal')
xlabel('Time (s)')
ylabel('Amplitude')

% Manual Power Spectral Density plots
subplot(312)
[psd_y, f_y] = manualPSD(y, fs); % PSD of the original signal
plot(f_y,10*log10(psd_y),'r')
title('Power Spectral Density')
legend('Original signal PSD')
xlabel('Frequency (Hz)')
ylabel('Power/Frequency (dB/Hz)')

% Manual Power Spectral Density plots
subplot(313)
[psd_x, f_x] = manualPSD(x, fs); % PSD of the noisy signal
plot(f_x,10*log10(psd_x),'k')
title('Power Spectral Density')
legend('Noisy signal PSD')
xlabel('Frequency (Hz)')
ylabel('Power/Frequency (dB/Hz)')
web('https://www.salimwireless.com/search?q=psd%20fourier%20transform', '-browser'); 

% Manual Periodogram PSD calculation function
function [psd, f] = manualPSD(signal, fs)
 N = length(signal); % Signal length
 fft_signal = fft(signal); % FFT of the signal
 fft_signal = fft_signal(1:N/2+1); % Take only the positive frequencies
 psd = (1/(fs*N)) * abs(fft_signal).^2; % Compute the power spectral density
 psd(2:end-1) = 2*psd(2:end-1); % Adjust the PSD for the one-sided spectrum
 f = (0:(N/2))*fs/N; % Frequency vector
end

Output

Power Spectral Density Periodogram output plot

Power Spectral Density output visualization

Interactive PSD Simulator

Experiment with signal parameters and visualize the Power Spectral Density in real-time below.

Parameters


Base Signal



Input Signal

Magnitude Plot

Phase Plot



Further Reading

  1. Bartlett Method for Spectral Estimation in MATLAB
  2. Periodogram for Spectral Estimation in MATLAB
  3. Welch Method for Spectral Estimation in MATLAB
  4. Calculation of SNR from FFT bins in MATLAB
  5. Add AWGN Directly to PSD in MATLAB
  6. How to Find the Fourier Transform of Any Signal
  7. Fourier Spectral Analysis
  8. Fourier Transform | Electronics Communication
  9. FFT Magnitude and Phase Spectrum using MATLAB
  10. Spectral Estimation Methods - Periodogram, Correlogram, Welch, Bartlett and Blackman-Tukey Methods
10
rectangular
📊 MATLAB & ONLINE SIMULATIONS (Web-Based)
🎓 GATE & UGC - NET SOLUTIONS
📡 WIRELESS & SIGNAL PROCESSING


Contact Us

Name

Email *

Message *

Popular Posts

Electromyography (EMG) Explained

  Electromyography (EMG) EMG stands for Electromyography . It is a medical test used to check how well your muscles and the nerves that control them are working. What it does EMG measures the electrical activity in your muscles. When nerves send signals to muscles, they create tiny electrical impulses—EMG records these. Why doctors use it Doctors may recommend EMG if you have symptoms like: Muscle weakness Numbness or tingling Muscle pain or cramping Suspected nerve disorders It helps diagnose conditions such as: Carpal Tunnel Syndrome Amyotrophic Lateral Sclerosis (ALS) Peripheral Neuropathy How it’s done Nerve conduction study (NCS) – small electrical pulses are applied to test nerve signals Needle EMG – a thin needle electrode is inserted into muscles to record activity Does it hurt? You might feel mild discomfort (like a quick pinch or muscle soreness) ...
Read more

MATLAB Code for MUSIC

  MATLAB Code clc; clear; close all ; %% Step 1: Define Parameters M = 8; % Number of array sensors d = 0.5; % Sensor spacing (lambda/2) K = 2; % Number of signals N = 200; % Number of snapshots theta = [-20 30]; % True signal angles (degrees) SNR = 10; % Signal-to-noise ratio (dB) fprintf( 'Step 1: Parameters Initialized\n' ); %% Step 2: Generate Signal Sources t = 1:N; s1 = exp(1j*2*pi*0.05*t); s2 = exp(1j*2*pi*0.1*t); S = [s1; s2]; figure; plot(real(S(1,:))) title( 'Signal 1 (Real Part)' ) xlabel( 'Samples' ) ylabel( 'Amplitude' ) figure; plot(real(S(2,:))) title( 'Signal 2 (Real Part)' ) xlabel( 'Samples' ) ylabel( 'Amplitude' ) fprintf( 'Step 2: Source Signals Generated\n' ); %% Step 3: Construct Steering Matrix A = zeros(M,K); for k = 1:K A(:,k) = exp(-1j*2*pi*d*(0:M-1)'*sin(theta(k)*pi/180)); end fprintf( 'Step 3: Steering Matr...
Read more

Direction of Arrival (DoA) Online Simulator (using MUSIC)

Interactive DOA Simulator X-axis XY angle (deg): 45 XZ angle (deg): 30 Noise: 0.05 Y-axis XY angle (deg): 60 YZ angle (deg): 45 Noise: 0.05 Z-axis XZ angle (deg): 60 YZ angle (deg): 30 Noise: 0.05 Estimated DOA (deg): 0 Simulation Workflow and Mathematical Background This simulator demonstrates Direction of Arrival (DOA) estimation using three-axis sensor signals (X, Y, Z), Maximal Ratio Combining (MRC) , and the MUSIC algorithm . It allows interactive control of signal angles and noise for teaching purposes. 1. Signal Generation A pure sinewave signal of frequency f is projected onto three axes using user-defined angles in different planes: X-axis: θ XY , θ XZ Y-axis: θ XY , θ YZ Z-axis: θ XZ , θ YZ Mathematically, for each time sample t : x(t) = s(t) * cos(θ_xy_x) * cos(θ_xz_x) + n_x(t) y(t) = s(t) * sin(θ_xy_y) * cos(θ_yz_y) + n_y(t) z(t) = s(t) * sin(θ_xz_z) * sin(θ_yz_z) + n_z(t) wh...
Read more

Amplitude Demodulation Simulation

Instructions for Amplitude Modulation (AM) Step 1: Click on 'Generate Message' button to generate input message signal Step 2: Then click on 'Generate Carrier' button to generate carrier signal. The carrier frequency has to be more than the message frequency and You can change frequencies using sliders Step 3: Click on 'Generate Amplitude Modulated Signal' button to generate Amplitude Modulated Signal Step 4: Click the 'Show Frequency Spectrums' button to view the AM spectra. Here, the modulation index is defined as the ratio of the message signal amplitude to the carrier signal amplitude. You can adjust both values. 50 Hz Step 1: Generate Message 500 Hz Step 2: ...
Read more

MIMO Channel Matrix | Rank and Condition Number

MIMO / Massive MIMO MIMO Channel Matrix | Rank and Condition...   The channel matrix in wireless communication is a matrix that describes the impact of the channel on the transmitted signal. The channel matrix can be used to model the effects of the atmospheric or underwater environment on the signal, such as the absorption, reflection or scattering of the signal by surrounding objects. When addressing multi-antenna communication, the term "channel matrix" is used. Let's assume that only one TX and one RX are in communication and there's no surrounding object. Here, in our case, we can apply the proper threshold condition to a received signal and get the original transmitted signal at the RX side. However, in real-world situations, we see signal path blockage, reflections, etc.,  (NLOS paths [↗]) more frequently. The obstruction is typically caused by building walls, etc. Multi-antenna communication was introduced to address this issue. It makes diversity app...
Read more

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

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