Wide Sense Stationary Signal (WSS)

 

Main Properties

1. The mean and autocorrelation do not change over time.
2. A wide-sense stationary (WSS) process has a constant mean, constant variance, and an autocorrelation function that depends only on the time difference (lag), not the absolute time.

For a WSS input to an LTI system, you are expected to study the output's statistical properties (such as mean, variance, and autocorrelation). You will find that the output signal is also a WSS signal. If your input signal has zero mean and unit variance, then the LTI output will have the same nature as the input signal, but:

1. The mean of the output is scaled by the DC gain of the LTI system.
2. The variance of the output is scaled by the total power gain of the system.

MATLAB Code

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


% Generate a wide-sense stationary (WSS) signal with 0 mean and unit variance
N = 1000; % Length of the signal
X = randn(1, N); % WSS signal


% Define the time indices t1 and t2
t1 = 0; % Time index 1
t2 = 100; % Time index 2


% Initialize autocorrelation value
Rx_val = 0;


% Loop to compute the sum for autocorrelation at (t1, t2)
for n = 1:N
% Ensure indices (n + t1) and (n + t2) are within bounds
if (n + t1 <= N) && (n + t2 <= N)
Rx_val = Rx_val + X(n + t1) * X(n + t2);
else
break; % Stop if indices go out of bounds
end
end


% Normalize by the length of the signal
Rx_val = Rx_val / N;


% Define the time indices t1 and t2
t3 = 100; % Time index 1
t4 = 200; % Time index 2


% Initialize autocorrelation value
Rx_val1 = 0;


% Loop to compute the sum for autocorrelation at (t1, t2)
for n = 1:N
% Ensure indices (n + t1) and (n + t2) are within bounds
if (n + t3 <= N) && (n + t4 <= N)
Rx_val1 = Rx_val1 + X(n + t3) * X(n + t4);
else
break; % Stop if indices go out of bounds
end
end


% Normalize by the length of the signal
Rx_val1 = Rx_val1 / N;
% Display the result
disp(['R_X(', num2str(t2), ') = ', num2str(Rx_val)]);
disp(['R_X(', num2str(t3), ', ', num2str(t4), ') = ', num2str(Rx_val)]);

Output

R_X( 100) = 0.039786

R_X(100, 200) = 0.039786

Copy the MATLAB Code above from here

%The code is developed by SalimWireless.com clc; clear; close all; % Generate a wide-sense stationary (WSS) signal with 0 mean and unit variance N = 1000; % Length of the signal X = randn(1, N); % WSS signal % Define the time indices t1 and t2 t1 = 0; % Time index 1 t2 = 100; % Time index 2 % Initialize autocorrelation value Rx_val = 0; % Loop to compute the sum for autocorrelation at (t1, t2) for n = 1:N % Ensure indices (n + t1) and (n + t2) are within bounds if (n + t1 <= N) && (n + t2 <= N) Rx_val = Rx_val + X(n + t1) * X(n + t2); else break; % Stop if indices go out of bounds end end % Normalize by the length of the signal Rx_val = Rx_val / N; % Define the time indices t1 and t2 t3 = 100; % Time index 1 t4 = 200; % Time index 2 % Initialize autocorrelation value Rx_val1 = 0; % Loop to compute the sum for autocorrelation at (t1, t2) for n = 1:N % Ensure indices (n + t1) and (n + t2) are within bounds if (n + t3 <= N) && (n + t4 <= N) Rx_val1 = Rx_val1 + X(n + t3) * X(n + t4); else break; % Stop if indices go out of bounds end end % Normalize by the length of the signal Rx_val1 = Rx_val1 / N; % Display the result disp(['R_X(', num2str(t2), ') = ', num2str(Rx_val)]); disp(['R_X(', num2str(t3), ', ', num2str(t4), ') = ', num2str(Rx_val)]); web('https://www.google.com/search?q=salimwireless.com+wide%20sense%20stationary', '-browser');

Further Reading

1. Auto-correlation and periodicity of a signal