Skip to main content

Menu

Home 📘 Wireless Communication (Main Page) Constellation Diagrams BER vs SNR GMSK MSK Channel Impulse Response Modulation Schemes MIMO Technology OFDM MATLAB Projects Contact
Login Try App

Power Spectral Density Calculation Using FFT in MATLAB


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 first Fourier transform (FFT) of a signal
  2. Then, calculate the Fourier magnitude of the signal
  3. The power spectrum is the square of the Fourier magnitude
  4. To calculate power spectrum density (PSD), divide the power spectrum by the total number of samples and the frequency resolution. {Frequency resolution = (sampling frequency / total number of samples)}

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:
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

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

% MINIMAL CHANGE 1: 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)
% MINIMAL CHANGE 2: Swapped plot arguments and corrected labels
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 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 Welch method output plot
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

People are good at skipping over material they already know!

View Related Topics to







Contact Us

Name

Email *

Message *

Popular Posts

BER vs SNR for M-ary QAM, M-ary PSK, QPSK, BPSK, ...

📘 Overview of BER and SNR 🧮 Online Simulator for BER calculation of m-ary QAM and m-ary PSK 🧮 MATLAB Code for BER calculation of M-ary QAM, M-ary PSK, QPSK, BPSK, ... 📚 Further Reading 📂 View Other Topics on M-ary QAM, M-ary PSK, QPSK ... 🧮 Online Simulator for Constellation Diagram of m-ary QAM 🧮 Online Simulator for Constellation Diagram of m-ary PSK 🧮 MATLAB Code for BER calculation of ASK, FSK, and PSK 🧮 MATLAB Code for BER calculation of Alamouti Scheme 🧮 Different approaches to calculate BER vs SNR What is Bit Error Rate (BER)? The abbreviation BER stands for Bit Error Rate, which indicates how many corrupted bits are received (after the demodulation process) compared to the total number of bits sent in a communication process. BER = (number of bits received in error) / (total number of tran...
Read more

Calculation of SNR from FFT bins in MATLAB

📘 Overview 🧮 MATLAB Code for Estimation of SNR from FFT bins of a Noisy Signal 🧮 MATLAB Code for Estimation of Signal-to-Noise Ratio from Power Spectral Density Using FFT and Kaiser Window Periodogram from real signal data 📚 Further Reading   Here, you can find the SNR of a received signal from periodogram / FFT bins using the Kaiser operator. The beta (β) parameter characterizes the Kaiser window, which controls the trade-off between the main lobe width and the side lobe level in the frequency domain. For that you should know the sampling rate of the signal.  The Kaiser window is a type of window function commonly used in signal processing, particularly for designing finite impulse response (FIR) filters and performing spectral analysis. It is a general-purpose window that allows for control over the trade-off between the main lobe width (frequency resolution) and side lobe levels (suppression of spectral leakage). The Kaiser window is defined...
Read more

Constellation Diagrams of ASK, PSK, and FSK

📘 Overview of Energy per Bit (Eb / N0) 🧮 Online Simulator for constellation diagrams of ASK, FSK, and PSK 🧮 Theory behind Constellation Diagrams of ASK, FSK, and PSK 🧮 MATLAB Codes for Constellation Diagrams of ASK, FSK, and PSK 📚 Further Reading 📂 Other Topics on Constellation Diagrams of ASK, PSK, and FSK ... 🧮 Simulator for constellation diagrams of m-ary PSK 🧮 Simulator for constellation diagrams of m-ary QAM BASK (Binary ASK) Modulation: Transmits one of two signals: 0 or -√Eb, where Eb​ is the energy per bit. These signals represent binary 0 and 1.    BFSK (Binary FSK) Modulation: Transmits one of two signals: +√Eb​ ( On the y-axis, the phase shift of 90 degrees with respect to the x-axis, which is also termed phase offset ) or √Eb (on x-axis), where Eb​ is the energy per bit. These signals represent binary 0 and 1.  BPSK (Binary PSK) Modulation: Transmits one of two signals...
Read more

MATLAB code for BER vs SNR for M-QAM, M-PSK, QPSk, BPSK, ...

🧮 MATLAB Code for BPSK, M-ary PSK, and M-ary QAM Together 🧮 MATLAB Code for M-ary QAM 🧮 MATLAB Code for M-ary PSK 📚 Further Reading MATLAB Script for BER vs. SNR for M-QAM, M-PSK, QPSK, BPSK % Written by Salim Wireless clc; clear; close all; num_symbols = 1e5; snr_db = -20:2:20; psk_orders = [2, 4, 8, 16, 32]; qam_orders = [4, 16, 64, 256]; ber_psk_results = zeros(length(psk_orders), length(snr_db)); ber_qam_results = zeros(length(qam_orders), length(snr_db)); for i = 1:length(psk_orders) psk_order = psk_orders(i); for j = 1:length(snr_db) data_symbols = randi([0, psk_order-1], 1, num_symbols); modulated_signal = pskmod(data_symbols, psk_order, pi/psk_order); received_signal = awgn(modulated_signal, snr_db(j), 'measured'); demodulated_symbols = pskdemod(received_signal, psk_order, pi/psk_order); ber_psk_results(i, j) = sum(data_symbols ~= demodulated_symbols) / num_symbols; end end for i...
Read more

Online Channel Impulse Response Simulator

  Fundamental Theory of Channel Impulse Response The fundamental theory behind the channel impulse response in wireless communication often involves complex exponential components such as: \( h(t) = \sum_{i=1}^{L} a_i \cdot \delta(t - \tau_i) \cdot e^{j\theta_i} \) Where: \( a_i \) is the amplitude of the \( i^{th} \) path \( \tau_i \) is the delay of the \( i^{th} \) path \( \theta_i \) is the phase shift (often due to Doppler effect, reflection, etc.) \( e^{j\theta_i} \) introduces a phase rotation (complex exponential) The convolution \( x(t) * h(t) \) gives the received signal So, instead of representing the channel with only real-valued amplitudes, each path can be more accurately modeled using a complex gain : \( h[n] = a_i \cdot e^{j\theta_i} \) 1. Simple Channel Impulse Response Simulator  (Here you can input only a unit impulse signal) Input Signal (Unit Impu...
Read more

Comparisons among ASK, PSK, and FSK | And the definitions of each

📘 Comparisons among ASK, FSK, and PSK 🧮 Online Simulator for calculating Bandwidth of ASK, FSK, and PSK 🧮 MATLAB Code for BER vs. SNR Analysis of ASK, FSK, and PSK 📚 Further Reading 📂 View Other Topics on Comparisons among ASK, PSK, and FSK ... 🧮 Comparisons of Noise Sensitivity, Bandwidth, Complexity, etc. 🧮 MATLAB Code for Constellation Diagrams of ASK, FSK, and PSK 🧮 Online Simulator for ASK, FSK, and PSK Generation 🧮 Online Simulator for ASK, FSK, and PSK Constellation 🧮 Some Questions and Answers Modulation ASK, FSK & PSK Constellation MATLAB Simulink MATLAB Code Comparisons among ASK, PSK, and FSK    Comparisons among ASK, PSK, and FSK Comparison among ASK, FSK, and PSK Parameters ASK FSK PSK Variable Characteristics Amplitude Frequency ...
Read more

Coherence Bandwidth and Coherence Time

🧮 Coherence Bandwidth 🧮 Coherence Time 🧮 MATLAB Code s 📚 Further Reading Coherence Bandwidth Coherence bandwidth is a concept in wireless communication and signal processing that relates to the frequency range over which a wireless channel remains approximately constant in terms of its characteristics. Coherence bandwidth is inversely related to the delay spread time (e.g., RMS delay spread). The coherence bandwidth is related to the delay spread of the channel, which is a measure of the time it takes for signals to traverse the channel due to multipath. The two are related by the following approximation: Coherence Bandwidth ≈ 1/(delay spread time) Or, Coherence Bandwidth ≈ 1/(root-mean-square delay spread time) (Coherence bandwidth in Hertz) For instance, if the root-mean-square delay spread is 500 ns (i.e., {1/(2*10^6)} seconds), the coherence bandwidth is approximately 2 MHz (1 / 500e-9) in ...
Read more