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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 [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 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.

  `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 data-sets.

  References

  [1]

  Wikipedia, “Convolution”, https://en.wikipedia.org/wiki/Convolution

  Examples

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

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