Power Spectral Density Calculation Using FFT in MATLAB

• 📘 Overview
• 🧮 Steps to calculate the PSD of a signal
• 🧮 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.

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

MATLAB Script

% 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(2π * 50)t + sin(2π * 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 output visualization

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