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# numpy.convolve

`numpy.``convolve`(a, v, mode='full')

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

  The convolution operator is often seen in signal processing, where it models the effect of a linear time-invariant system on a signal [R17]. In probability theory, the sum of two independent random variables is distributed according to the convolution of their individual distributions.

  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 convolution at each point of overlap, with an output shape of (N+M-1,). At the end-points 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.

  Notes

  The discrete convolution operation is defined as

  It can be shown that a convolution in time/space is equivalent to the multiplication 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 data-sets.

  References

  [R17] | (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])

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