numpy.convolve¶

numpy.convolve(a,v,mode='full')[source]¶

Returns the discrete, linear convolution of two one-dimensional sequences.

The convolution operator is often seen in signal processing, where itmodels the effect of a linear time-invariant system on a signal[1]. Inprobability theory, the sum of two independent random variables isdistributed according to the convolution of their individualdistributions.

Ifv is longer thana, the arrays are swapped before computation.

Parameters:

a:(N,) array_like First one-dimensional input array. v:(M,) array_like Second one-dimensional input array. mode:{‘full’, ‘valid’, ‘same’}, optional ‘full’: By default, mode is ‘full’. This returns the convolutionat each point of overlap, with an output shape of (N+M-1,). Atthe end-points of the convolution, the signals do not overlapcompletely, and boundary effects may be seen. ‘same’: Mode ‘same’ returns output of lengthmax(M,N). Boundaryeffects are still visible. ‘valid’: Mode ‘valid’ returns output of lengthmax(M,N)-min(M,N)+1. The convolution product is only givenfor points where the signals overlap completely. Values outsidethe signal boundary have no effect.

Returns:

out:ndarray Discrete, linear convolution ofa andv.

See also

scipy.signal.fftconvolve

Convolve two arrays using the Fast Fourier Transform.

scipy.linalg.toeplitz

Used to construct the convolution operator.

polymul

Polynomial multiplication. Same output as convolve, but also accepts poly1d objects as input.

Notes

The discrete convolution operation is defined as

(a * v)[n] = \sum_{m = -\infty}^{\infty} a[m] v[n - m]

It can be shown that a convolutionx(t) * y(t) in time/spaceis equivalent to the multiplicationX(f) Y(f) in the Fourierdomain, after appropriate padding (padding is necessary to preventcircular convolution). Since multiplication is more efficient (faster)than convolution, the functionscipy.signal.fftconvolve exploits theFFT to calculate the convolution of large data-sets.

References

[1]	(1,2) Wikipedia, “Convolution”,http://en.wikipedia.org/wiki/Convolution.

Examples

Note how the convolution operator flips the second arraybefore “sliding” the two across one another:

>>>np.convolve([1,2,3],[0,1,0.5])array([ 0. ,  1. ,  2.5,  4. ,  1.5])

Only return the middle values of the convolution.Contains boundary effects, where zeros are takeninto account:

>>>np.convolve([1,2,3],[0,1,0.5],'same')array([ 1. ,  2.5,  4. ])

The two arrays are of the same length, so thereis only one position where they completely overlap:

>>>np.convolve([1,2,3],[0,1,0.5],'valid')array([ 2.5])