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

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 Calculation 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 figur...
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

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

How to Mount Google Drive in Google Colab

How to Mount Google Drive in Google Colab Google Colab provides temporary storage during a session. Any files stored in the /content directory will be deleted when the runtime disconnects. To store datasets, trained models, and results permanently, it is recommended to mount your Google Drive in Colab. Mounting Google Drive allows your notebook to access files directly from your Drive and save outputs there so they remain available even after the Colab session ends. Step 1: Import the Drive Module First import the Google Colab drive module. from google.colab import drive Step 2: Mount Google Drive Run the following command to mount your Google Drive. from google.colab import drive drive.mount('/content/drive') After running the command: A link will appear in the output. Click the link and log in to your Google account. Copy the authentication code provided. Paste the code back into the notebook. Or, a Google authentication page will a...
Read more

Constellation Diagrams of ASK, PSK, and FSK (with MATLAB Code + Simulator)

Constellation Diagrams: ASK, FSK, and PSK Comprehensive guide to signal space representation, including interactive simulators and MATLAB implementations. 📘 Overview 🧮 Simulator ⚖️ Theory 📈 Q-function 📚 Resources BASK Modulation Transmits one of two signals: 0 or $\sqrt{E_b}$, representing binary 0 and 1. Simple but sensitive to noise. BFSK Modulation Transmits one of two signals: $\sqrt{E_b}$ on the Y-axis or $\sqrt{E_b}$ on the X-axis. These are orthogonal signals. BPSK Modulation Transmits $+\sqrt{E_b}$ or $-\sqrt{E_b}$ (antipodal signaling). Most efficient binary scheme. ...
Read more

Wiener Filter in MATLAB

  MATLAB Code  % Wiener Filter Based on Wiener-Hopf Equation % This script demonstrates how to apply the Wiener filter to recover % a reference signal from a noisy signal using the Wiener-Hopf equation. % The filter minimizes the mean squared error between the noisy signal and the reference signal. clear; close all; clc; % Signal Parameters fs = 4000; % Sampling frequency (Hz) T = 1; % Total recording time (seconds) L = T * fs; % Signal length (samples) tt = (0:L-1) / fs; % Time vector ff = (0:L-1) * fs / L; % Frequency vector % Generate Reference Signal (a sinusoid) y = sin(2 * pi * 120 * tt); % Reference sinusoidal signal y = y(:); % Ensure column vector % Create Noisy Signal by Adding Gaussian Noise x = 0.50 * randn(L, 1) + y; % Noisy signal x = x(:); % Ensure column vector % Define Filter Order (Number of Coefficients) N = 200; % Apply Wiener Filter using custom function [xest, b, MSE] = wienerFilt(x, y, N); % Plot Results figure; subplot(411); plot(tt, x, 'k'), hold on, p...
Read more

Overmodulation & Distortion in AM

Overmodulation in AM and How It Causes Distortion 1. AM Signal Equation s(t) = A c [1 + μ m(t)] cos(2Ï€ f c t) A c = carrier amplitude m(t) = normalized modulating signal (|m(t)| ≤ 1) μ = modulation index 2. Modulation Index μ = A m / A c - Normal AM: 0 < μ ≤ 1 → no distortion - Overmodulation: μ > 1 → distortion occurs 3. Envelope and Overmodulation A(t) = A c [1 + μ m(t)] - For undistorted AM: 1 + μ m(t) ≥ 0 at all times - If μ > 1: 1 + μ m(t) < 0 at negative peaks → carrier flips Example: Let m(t) = cos(2Ï€ f m t), A c = 1 V, μ = 1.2 Minimum envelope: A min = A c [1 - 1.2] = -0.2 V Negative amplitude → envelope crosses zero → 180° phase flip 4. Mathematical Consequence -A c cos(θ) = A c cos(θ + Ï€) This phase reversal is what causes distortion in the demodulated signal. 5. Instantaneous AM Signal s...
Read more

MUSIC Algorithm Explained (with MATLAB + Simulator)

Practical Implementation of the MUSIC Algorithm The focus is on how the algorithm works computationally , not just theory, and it explains the denominator (a H E n E n H a) mathematically and intuitively. 1. Introduction The MUSIC (Multiple Signal Classification) algorithm is a high-resolution method used in signal processing and array processing to estimate the Direction of Arrival (DOA) of signals received by a sensor array. Unlike classical beamforming methods, MUSIC uses eigenvector decomposition of the covariance matrix to separate the signal subspace and noise subspace , allowing it to achieve much higher angular resolution. In practical implementations, MUSIC works by: Simulating or collecting array signals Computing the covariance matrix Performing eigenvalue decomposition Separating signal and noise subspaces Scanning possible angles using a steering vector Constructing a pseudo-spectrum where peaks indicate signal directions 2. Signal Mod...
Read more