numpy.correlate#

numpy.correlate(a, v, mode='valid')[source]#

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

>>> import numpy as np
>>> 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])