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scipy.signal.

# correlate

scipy.signal.correlate(in1, in2, mode='full', method='auto')

  Cross-correlate two N-dimensional arrays.

  Cross-correlate in1 and in2, with the output size determined by the mode argument.

  Parameters:
    in1 array_like

      First input.

    in2 array_like

      Second input. Should have the same number of dimensions as in1.

    mode str {‘full’, ‘valid’, ‘same’}, optional

      A string indicating the size of the output:

      `full`

        The output is the full discrete linear cross-correlation of the inputs. (Default)

      `valid`

        The output consists only of those elements that do not rely on the zero-padding. In ‘valid’ mode, either in1 or in2 must be at least as large as the other in every dimension.

      `same`

        The output is the same size as in1, centered with respect to the ‘full’ output.

    method str {‘auto’, ‘direct’, ‘fft’}, optional

      A string indicating which method to use to calculate the correlation.

      `direct`

        The correlation is determined directly from sums, the definition of correlation.

      `fft`

        The Fast Fourier Transform is used to perform the correlation more quickly (only available for numerical arrays.)

      `auto`

        Automatically chooses direct or Fourier method based on an estimate of which is faster (default). See `convolve` Notes for more detail.

        Added in version 0.19.0.

  Returns:
    correlate array

      An N-dimensional array containing a subset of the discrete linear cross-correlation of in1 with in2.

  See also

  `choose_conv_method`

    contains more documentation on method.

  `correlation_lags`

    calculates the lag / displacement indices array for 1D cross-correlation.

  Notes

  The correlation z of two d-dimensional arrays x and y is defined as:

      z[...,k,...] = sum[..., i_l, ...] x[..., i_l,...] * conj(y[..., i_l - k,...])

  This way, if `x` and `y` are 1-D arrays and `z = correlate(x, y, 'full')` then

  \[z[k] = \sum_{l=0}^{N-1} x_l \, y_{l-k}^{*}\]

  for \(k = -(M-1), \dots, (N-1)\), where \(N\) is the length of `x`, \(M\) is the length of `y`, and \(y_m = 0\) when \(m\) is outside the valid range \([0, M-1]\). The size of \(z\) is \(N + M - 1\) and \(y^*\) denotes the complex conjugate of \(y\).

  `method='fft'` only works for numerical arrays as it relies on `fftconvolve`. In certain cases (i.e., arrays of objects or when rounding integers can lose precision), `method='direct'` is always used.

  When using `mode='same'` with even-length inputs, the outputs of `correlate` and `correlate2d` differ: There is a 1-index offset between them.

  Examples

  Implement a matched filter using cross-correlation, to recover a signal that has passed through a noisy channel.

      >>> import numpy as np
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
      >>> rng = np.random.default_rng()

      >>> sig = np.repeat([0., 1., 1., 0., 1., 0., 0., 1.], 128)
      >>> sig_noise = sig + rng.standard_normal(len(sig))
      >>> corr = signal.correlate(sig_noise, np.ones(128), mode='same') / 128

      >>> clock = np.arange(64, len(sig), 128)
      >>> fig, (ax_orig, ax_noise, ax_corr) = plt.subplots(3, 1, sharex=True)
      >>> ax_orig.plot(sig)
      >>> ax_orig.plot(clock, sig[clock], 'ro')
      >>> ax_orig.set_title('Original signal')
      >>> ax_noise.plot(sig_noise)
      >>> ax_noise.set_title('Signal with noise')
      >>> ax_corr.plot(corr)
      >>> ax_corr.plot(clock, corr[clock], 'ro')
      >>> ax_corr.axhline(0.5, ls=':')
      >>> ax_corr.set_title('Cross-correlated with rectangular pulse')
      >>> ax_orig.margins(0, 0.1)
      >>> fig.tight_layout()
      >>> plt.show()

  Compute the cross-correlation of a noisy signal with the original signal.

      >>> x = np.arange(128) / 128
      >>> sig = np.sin(2 * np.pi * x)
      >>> sig_noise = sig + rng.standard_normal(len(sig))
      >>> corr = signal.correlate(sig_noise, sig)
      >>> lags = signal.correlation_lags(len(sig), len(sig_noise))
      >>> corr /= np.max(corr)

      >>> fig, (ax_orig, ax_noise, ax_corr) = plt.subplots(3, 1, figsize=(4.8, 4.8))
      >>> ax_orig.plot(sig)
      >>> ax_orig.set_title('Original signal')
      >>> ax_orig.set_xlabel('Sample Number')
      >>> ax_noise.plot(sig_noise)
      >>> ax_noise.set_title('Signal with noise')
      >>> ax_noise.set_xlabel('Sample Number')
      >>> ax_corr.plot(lags, corr)
      >>> ax_corr.set_title('Cross-correlated signal')
      >>> ax_corr.set_xlabel('Lag')
      >>> ax_orig.margins(0, 0.1)
      >>> ax_noise.margins(0, 0.1)
      >>> ax_corr.margins(0, 0.1)
      >>> fig.tight_layout()
      >>> plt.show()