numpy.correlate

numpy.correlate(a, v, mode='valid')

Cross-correlation of two 1-dimensional sequences.

This function computes the correlation as generally defined in signal processing texts [1]:

\[c_k = \sum_n a_{n+k} \cdot \overline{v}_n\]

with a and v sequences being zero-padded where necessary and \(\overline v\) denoting complex conjugation.

Parameters:

a, varray_like

Input sequences.

mode{‘valid’, ‘same’, ‘full’}, optional

Refer to the convolve docstring. Note that the default is ‘valid’, unlike convolve, which uses ‘full’.

Returns:

outndarray

Discrete cross-correlation of a and v.

See also

convolve

Discrete, linear convolution of two one-dimensional sequences.

scipy.signal.correlate

uses FFT which has superior performance on large arrays.

Notes

The definition of correlation above is not unique and sometimes correlation may be defined differently. Another common definition is [1]:

\[c'_k = \sum_n a_{n} \cdot \overline{v_{n+k}}\]

which is related to \(c_k\) by \(c'_k = c_{-k}\).

numpy.correlate may perform slowly in large arrays (i.e. n = 1e5) because it does not use the FFT to compute the convolution; in that case, scipy.signal.correlate might be preferable.

References

[1] (1,2)

Wikipedia, “Cross-correlation”, https://en.wikipedia.org/wiki/Cross-correlation

Examples

>>> np.correlate([1, 2, 3], [0, 1, 0.5])
array([3.5])
>>> np.correlate([1, 2, 3], [0, 1, 0.5], "same")
array([2. ,  3.5,  3. ])
>>> np.correlate([1, 2, 3], [0, 1, 0.5], "full")
array([0.5,  2. ,  3.5,  3. ,  0. ])

Using complex sequences:

>>> np.correlate([1+1j, 2, 3-1j], [0, 1, 0.5j], 'full')
array([ 0.5-0.5j,  1.0+0.j ,  1.5-1.5j,  3.0-1.j ,  0.0+0.j ])

Note that you get the time reversed, complex conjugated result (\(\overline{c_{-k}}\)) when the two input sequences a and v change places:

>>> np.correlate([0, 1, 0.5j], [1+1j, 2, 3-1j], 'full')
array([ 0.0+0.j ,  3.0+1.j ,  1.5+1.5j,  1.0+0.j ,  0.5+0.5j])