How to Compute Auto-Correlation in MATLAB Without Using Built-In Functions

Introduction: Auto-correlation is a fundamental concept in signal processing, used to determine the presence of a repeating pattern within a signal. This article provides step-by-step guidance on how to compute the auto-correlation of a signal in MATLAB without relying on any built-in functions. This method enhances understanding of the underlying mathematics and offers flexibility for specific applications.

Define the Signal

The first step involves defining the signal data in a vector. This is the input signal that you want to analyze.

Compute the Mean

The mean of the signal is a crucial parameter in the auto-correlation formula. This value should be calculated before proceeding to the next step. The built-in mean function can be used, but for the sake of this exercise, we will manually compute the mean.

Implement the Auto-Correlation Formula

Auto-correlation, denoted as (R_k), is defined as:

[begin{equation}R_k sum_{n0}^{N-k-1} (x[n] - mu)(x[n-k] - mu)end{equation}]

where:

(N) is the length of the signal (k) is the lag (mu) is the mean of the signal

The following MATLAB code implements this definition:

```matlab function R auto_correlation(x) % x: input signal vector N length(x); % Length of the signal mu sum(x)/N; % Mean of the signal R zeros(1, N); % Initialize the auto-correlation vector % Loop through each lag for k 0:N-1 sum 0; % Initialize sum for this lag % Compute the sum for the current lag for n 1:N-k sum sum (x(n) - mu) * (x(n k) - mu); end R(k 1) sum; % Store the result for this lag end % Normalize the auto-correlation by the number of samples R R / N - 1; end ```

Explanation of the Code:

Function Definition: The function auto_correlation takes a vector x as input. Mean Calculation: The mean of the signal is calculated using sum(x)/N. Loop through Lags: A loop runs through each possible lag from 0 to N-1. Inner Loop for Summation: For each lag, an inner loop calculates the sum according to the auto-correlation formula. Normalization: The result is normalized by the number of samples N - 1.

Usage:

```matlab signal [1 2 3 4 5]; % Example signal R auto_correlation(signal); % Compute and store the auto-correlation disp(R); % Display the auto-correlation results ```

This will compute and display the auto-correlation of the given signal. Adjust the signal vector as needed for your specific data.

Alternative Approach via Inner Product

An alternative method to compute auto-correlation involves taking a vector, making a copy of it shifted by (N) samples, multiplying the two of them element-wise, and adding the results. This operation essentially calculates the inner product at each lag.

Example:

To get the auto-correlation at lag (N):

Copy the signal vector. Shift the copied vector by (N) samples. Multiply the original vector with the shifted vector element-wise. Add the results to get the auto-correlation at that lag. Repeat for as many lags as needed to sweep out the entire autocorrelation function. Consider zero-padding for regions outside the finite range of the signal to ensure accurate results.

This method is efficient for specific applications, though it may be less intuitive than the direct implementation described above.

Conclusion

By following the steps outlined in this article, you can effectively compute the auto-correlation of a signal in MATLAB without using built-in functions. This approach not only enhances understanding but also provides flexibility for various applications.