Calculating cross correlation and auto correlation with python

cross-correlation and auto-correlation are important signal processing techniques used to measure the similarity between two signals or the similarity of a signal with a time-delayed version of itself.

**cross-correlation:**
the cross-correlation of two signals `x` and `y` can be calculated using the formula:

\[ r_{xy}[n] = \sum_{m=-\infty}^{\infty} x[m] \cdot y[m+n] \]

**auto-correlation:**
the auto-correlation of a signal `x` can be calculated as a special case of cross-correlation where both signals are the same:

\[ r_{xx}[n] = \sum_{m=-\infty}^{\infty} x[m] \cdot x[m+n] \]

**code example:**
here is an example code snippet that shows how to calculate cross-correlation and auto-correlation using python:

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