Correlations

References:

[1] Brad Osgood, Lecture Notes for EE263. (http://web.stanford.edu/class/archive/ee/ee263/ee263.1082/notes/ee263coursereader.pdf)

Correlation(in signal processing) described as a measure of how well the values of g, when shifted by x, correlate with the values of f. It depends on x; some shifts of g may correlate better with f than other shifts. The closer the match between f(y) and g(x + y) (as y varies) the larger the integral and the larger the cross-correlation.

Signal correlation:

Convolution:

Convolution algebraic properties:

Convolution and correlation interconnetion:

Auto-correlation:

For example, ideal white noise has uncorrelated values, thus one of possible definition of white noise.

I.e. a signal for which auto-correlation equal to zero.

Normalized cross-correlation for signal w  and it's shifted part by "x" w':

 

Interestingly but normalized cross-correlation looks like cosine of angle between two normalized vectors.

Also another interpretation that due to this equality

We see that this last "cross-correlation" is the same as linear correlation also known as "correlation from probability theory" and also known as "Pirson's coefficient of correletaion"