Autocorrelation and Periodicity of a Signal

 

Autocorrelation function

Autocorrelation function: For a signal x(t), the autocorrelation is defined as R_xx(τ) = E[x(t)x(t+τ)] for a random process, or R_xx(τ) = ∫ x(t)x(t+τ) dt for an energy signal.

The auto-correlation of a periodic signal preserves the periodicity. For example, we are transmitting a signal x(t) through the wireless medium, and we receive the signal y(t) at the receiver.

y(t) = x(t) + n(t)

where n(t) is the additive white Gaussian noise (AWGN).

You can find that the periodicity of the autocorrelation of y(t) will be the same as the periodicity of x(t).

In other words, we can say that the autocorrelation of the noisy signal is equal to the autocorrelation of the original periodic signal, except at zero lag (τ = 0), where the noise contributes.

To find the spectral density (also known as the power spectral density, or PSD) from the autocorrelation function mathematically, you can use the Wiener–Khinchin theorem. This theorem states that the power spectral density of a wide-sense stationary (WSS) random process is the Fourier transform of its autocorrelation function.

Why WSS is assumed: The WSS assumption ensures that the autocorrelation function depends only on the time difference τ (i.e., R_xx(t₁,t₂) = R_xx(τ)) and not on absolute time. This time-invariance is necessary for the Fourier transform to exist in a consistent way and to define a meaningful power spectral density. Without stationarity, the statistical properties of the signal change with time, and a single PSD cannot fully describe the signal.

 

Wiener-Khinchin Theorem

Given a wide-sense stationary process X(t), let RX(τ) be its autocorrelation function. The power spectral density SX(f) is given by:

\( S_X(f) = \mathcal{F}\{R_X(\tau)\} = \int_{-\infty}^{\infty} R_X(\tau) e^{-j2\pi f \tau} \, d\tau \)

Where F denotes the Fourier transform, j is the imaginary unit, f is the frequency, and τ is the lag.
Steps to Compute PSD from Autocorrelation Function

 

Steps to Compute PSD from Autocorrelation Function

Compute the Autocorrelation Function RX(τ):
The autocorrelation function RX(τ) is defined as:

RX(τ)=E[X(t)X(t+τ)]

For a discrete-time signal x[n], the autocorrelation function RX[k] can be computed as:

RX[k]=∑(n=−∞,∞) x[n]x[n+k]

Apply the Fourier Transform:

To find the PSD, take the Fourier transform of the autocorrelation function RX(τ) (or RX[k] in the discrete case):

For continuous signals:

SX(f)=∫(−∞,∞) RX(τ)exp(−j2πfτ dτ)

For discrete signals:

SX(exp(jω))=∑(k=−∞,∞) RX[k]exp(−jωk)

 

MATLAB Code to find the periodicity from auto-correlation of a periodic signal

% The code is developed by SalimWireless.Com clc; clear; close all; % Improved MATLAB code to accurately detect frequency of a sine wave % Define the sampling parameters Fs = 1000; % Sampling frequency in Hz T = 1/Fs; % Sampling period L = 1000; % Number of samples t = (0:L-1)*T; % Time vector % Generate a sine wave of 50 Hz f_signal = 50; % Frequency of the sine wave in Hz data = sin(2 * pi * f_signal * t); % Calculate autocorrelation using xcorr and normalize autocorr_values = xcorr(data - mean(data), 'coeff'); lags = 1:1000;%(-L+1):(L-1); % Lags vector % Consider only positive lags to find the period positive_lags = lags(lags >= 0); positive_autocorr = autocorr_values(lags >= 0); % Find peaks in the autocorrelation function [peaks, locs] = findpeaks(positive_autocorr); % Ensure the detected period corresponds to a full cycle if ~isempty(locs) % Identify the true period based on distance between peaks possible_period_lags = positive_lags(locs(1:end)); % Choose the first peak that corresponds to the full period peak_lag = possible_period_lags(find(possible_period_lags > Fs/(2*f_signal), 1)); if isempty(peak_lag) error('Could not detect a clear peak for the fundamental period.'); end else error('Could not detect any peaks in autocorrelation function.'); end % Calculate the frequency T_signal = peak_lag * T; % Period of the signal f_detected = (1 / T_signal); % Frequency of the signal % Display the result fprintf('Detected frequency: %.2f Hz\n', f_detected); % Plot the autocorrelation function with peak detection figure; plot(positive_lags * T, positive_autocorr); hold on; plot(positive_lags(locs(2:end)) * T, peaks(2:end), 'ro'); % Highlight detected peaks xlabel('Lag (s)'); ylabel('Autocorrelation'); title('Autocorrelation of the Signal'); grid on; legend('Autocorrelation', 'Detected Peaks'); web('https://www.google.com/search?q=salimwireless.com+autocorrelation%20and%20periodicity', '-browser');  

Output

 

 

 

 

Another MATLAB Code to find the periodicity from autocorrelation of a noisy periodic signal

% The code is written by SalimWireless.Com clc; clear; close all; % Generate a synthetic signal (discrete example) N = 1000; % Number of samples x = randn(1, N); % Gaussian white noise % Compute autocorrelation function [R, lags] = xcorr(x, 'biased'); % Compute the PSD using the Fourier transform of the autocorrelation function R_positive = R(lags >= 0); % Only take the positive lags psd = abs(fft(R_positive)); freq = (0:length(psd)-1)*(fs/length(psd)); % Frequency axis % Plot the results figure; plot(freq, 10*log10(psd)); title('Power Spectral Density'); xlabel('Frequency (Hz)'); ylabel('Power/Frequency (dB/Hz)');  

 

Output