Correlation

Cross - Correlation¶

For two continuous functions $a(t)$ and $b(t)$, the correlation is defined with the shift operator $\tau$ as:

$$ a(t) \star b(t) = \int_{-\infty}^{\infty} a(t + \tau) b(\tau) d \tau$$

In the discrete case: $$ a[n] \star b[n] = \sum_{m=-\infty}^{\infty} a[n + m] b[m]$$

Note the similarity to the convolution - only the sign of the shift operator $\tau$ has changed. This flip also means that the cross correlation is not commutative:

$$ a(t) \star b(t) \neq b(t) \star a(t)$$

Cross correlation can give information on the similarity of two signals and is thus frequently used in audio DSP.

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Example: Correlation of Random Signals¶

The below example shows the result of a cross correlation between to different random sequences $b(t)$ and $b(t)$. As with convolution, the length of the signal is the lenght of both input signals combined -1. Although the signals are not identical, the cross correlation function has its maximum in the center.

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Comparison with Convolution¶

The following example shows how correlation and convolution work differently for two input signals. $a(t)$ is a sine wave and $b(t)$ a cosine wave, both with the same amplitude and frequency.

Correlation:

To inspect the commutative properties, both $a \star b$ and '$b \star a$ ar calculated and plotted. The results of both correlations are not equal:

Convolution:

As for the correlation, both $a * b$ and '$b * a$ ar calculated and plotted. The results of both correlations are identical: