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# numpy.fft.rfft

fft.rfft(a, n=None, axis=-1, norm=None, out=None)

  Compute the one-dimensional discrete Fourier Transform for real input.

  This function computes the one-dimensional n-point discrete Fourier Transform (DFT) of a real-valued array by means of an efficient algorithm called the Fast Fourier Transform (FFT).

  Parameters:
    a array_like

      Input array

    n int, optional

      Number of points along transformation axis in the input to use. If n is smaller than the length of the input, the input is cropped. If it is larger, the input is padded with zeros. If n is not given, the length of the input along the axis specified by axis is used.

    axis int, optional

      Axis over which to compute the FFT. If not given, the last axis is used.

    norm{“backward”, “ortho”, “forward”}, optional

      Normalization mode (see `numpy.fft`). Default is “backward”. Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor.

      Added in version 1.20.0: The “backward”, “forward” values were added.

    out complex ndarray, optional

      If provided, the result will be placed in this array. It should be of the appropriate shape and dtype.

      Added in version 2.0.0.

  Returns:
    out complex ndarray

      The truncated or zero-padded input, transformed along the axis indicated by axis, or the last one if axis is not specified. If n is even, the length of the transformed axis is `(n/2)+1`. If n is odd, the length is `(n+1)/2`.

  Raises:
    IndexError

      If axis is not a valid axis of a.

  See also

  `numpy.fft`

    For definition of the DFT and conventions used.

  `irfft`

    The inverse of `rfft`.

  `fft`

    The one-dimensional FFT of general (complex) input.

  `fftn`

    The n-dimensional FFT.

  `rfftn`

    The n-dimensional FFT of real input.

  Notes

  When the DFT is computed for purely real input, the output is Hermitian-symmetric, i.e. the negative frequency terms are just the complex conjugates of the corresponding positive-frequency terms, and the negative-frequency terms are therefore redundant. This function does not compute the negative frequency terms, and the length of the transformed axis of the output is therefore `n//2 + 1`.

  When `A = rfft(a)` and fs is the sampling frequency, `A[0]` contains the zero-frequency term 0*fs, which is real due to Hermitian symmetry.

  If n is even, `A[-1]` contains the term representing both positive and negative Nyquist frequency (+fs/2 and -fs/2), and must also be purely real. If n is odd, there is no term at fs/2; `A[-1]` contains the largest positive frequency (fs/2*(n-1)/n), and is complex in the general case.

  If the input a contains an imaginary part, it is silently discarded.

  Examples

      >>> import numpy as np
      >>> np.fft.fft([0, 1, 0, 0])
      array([ 1.+0.j,  0.-1.j, -1.+0.j,  0.+1.j]) # may vary
      >>> np.fft.rfft([0, 1, 0, 0])
      array([ 1.+0.j,  0.-1.j, -1.+0.j]) # may vary

  Notice how the final element of the `fft` output is the complex conjugate of the second element, for real input. For `rfft`, this symmetry is exploited to compute only the non-negative frequency terms.