numpy.convolve¶

numpy.
convolve
(a, v, mode='full')[source]¶ Returns the discrete, linear convolution of two onedimensional sequences.
The convolution operator is often seen in signal processing, where it models the effect of a linear timeinvariant system on a signal [1]. In probability theory, the sum of two independent random variables is distributed according to the convolution of their individual distributions.
If v is longer than a, the arrays are swapped before computation.
Parameters:  a : (N,) array_like
First onedimensional input array.
 v : (M,) array_like
Second onedimensional input array.
 mode : {‘full’, ‘valid’, ‘same’}, optional
 ‘full’:
By default, mode is ‘full’. This returns the convolution at each point of overlap, with an output shape of (N+M1,). At the endpoints of the convolution, the signals do not overlap completely, and boundary effects may be seen.
 ‘same’:
Mode ‘same’ returns output of length
max(M, N)
. Boundary effects are still visible. ‘valid’:
Mode ‘valid’ returns output of length
max(M, N)  min(M, N) + 1
. The convolution product is only given for points where the signals overlap completely. Values outside the signal boundary have no effect.
Returns:  out : ndarray
Discrete, linear convolution of a and v.
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 convolution x(t) * y(t) in time/space is equivalent to the multiplication X(f) Y(f) in the Fourier domain, after appropriate padding (padding is necessary to prevent circular convolution). Since multiplication is more efficient (faster) than convolution, the function
scipy.signal.fftconvolve
exploits the FFT to calculate the convolution of large datasets.References
[1] (1, 2) Wikipedia, “Convolution”, http://en.wikipedia.org/wiki/Convolution. Examples
Note how the convolution operator flips the second array before “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 taken into 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 there is only one position where they completely overlap:
>>> np.convolve([1,2,3],[0,1,0.5], 'valid') array([ 2.5])