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    Convolution
    Correlation

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CONVOLUTION

When two time series (a, b) are convolved together, the result is a new time series (c).

Convolution may be applied in either the time or frequency domains:

The result of the convolution is a new time series of length equal to the sum of N and M decremented, where N and M are the lengths of sequences a and b respectfully.

Convolution may be thought of in terms of taking one time series, and reflecting in the vertical ('folding'), then shifting in steps whilst multiplying corresponding values and summing.

Cyclic convolution is different from ordinary convolution because the shorter timeseries is padded with zeroes to make it the same length as the second sequence.  This results in a different result, although, the central values should be equal in either case, the ends of the new time series should be different, reflecting a kind of "filter warm-up".

Ordinary Convolution

Cyclic Convolution

Convolution with a delta pulse returns the original sequence shifted in time.

Filters are applied to a timeseries via convolution.  Filtering involves convolution of a sequence with a second (usually shorter) sequence used to enhance certain features, and supress others.

The process of convolution may be classed as associative, commutative and distributive, i.e.:

The process opposite to convolution is known as deconvolution.

CORRELATION

Cross-correlation is defined as:

This is a very similar operation to convolution.  However, correlation may be used to investigate the similarities between two time series and returns the lag time between them.

Autocorrelation is equivalent to cross-correlation, but with the two time series equal:

The correlation coefficient is a measure of the similarity between two sequences:

The mathematical procedure for calculating the cross-correlation is very similar to that of convolution. 

Cross-Correlation