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# numpy.fft.rfftn

fft.rfftn(a, s=None, axes=None, norm=None, out=None)

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

  This function computes the N-dimensional discrete Fourier Transform over any number of axes in an M-dimensional real array by means of the Fast Fourier Transform (FFT). By default, all axes are transformed, with the real transform performed over the last axis, while the remaining transforms are complex.

  Parameters:
    a array_like

      Input array, taken to be real.

    s sequence of ints, optional

      Shape (length along each transformed axis) to use from the input. (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.). The final element of s corresponds to n for `rfft(x, n)`, while for the remaining axes, it corresponds to n for `fft(x, n)`. Along any axis, if the given shape is smaller than that of the input, the input is cropped. If it is larger, the input is padded with zeros.

      Changed in version 2.0: If it is `-1`, the whole input is used (no padding/trimming).

      If s is not given, the shape of the input along the axes specified by axes is used.

      Deprecated since version 2.0: If s is not `None`, axes must not be `None` either.

      Deprecated since version 2.0: s must contain only `int` s, not `None` values. `None` values currently mean that the default value for `n` is used in the corresponding 1-D transform, but this behaviour is deprecated.

    axes sequence of ints, optional

      Axes over which to compute the FFT. If not given, the last `len(s)` axes are used, or all axes if s is also not specified.

      Deprecated since version 2.0: If s is specified, the corresponding axes to be transformed must be explicitly specified too.

    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.

      New 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 for all axes (and hence is incompatible with passing in all but the trivial `s`).

      New in version 2.0.0.

  Returns:
    out complex ndarray

      The truncated or zero-padded input, transformed along the axes indicated by axes, or by a combination of s and a, as explained in the parameters section above. The length of the last axis transformed will be `s[-1]//2+1`, while the remaining transformed axes will have lengths according to s, or unchanged from the input.

  Raises:
    ValueError

      If s and axes have different length.

    IndexError

      If an element of axes is larger than than the number of axes of a.

  See also

  `irfftn`

    The inverse of `rfftn`, i.e. the inverse of the n-dimensional FFT of real input.

  `fft`

    The one-dimensional FFT, with definitions and conventions used.

  `rfft`

    The one-dimensional FFT of real input.

  `fftn`

    The n-dimensional FFT.

  `rfft2`

    The two-dimensional FFT of real input.

  Notes

  The transform for real input is performed over the last transformation axis, as by `rfft`, then the transform over the remaining axes is performed as by `fftn`. The order of the output is as for `rfft` for the final transformation axis, and as for `fftn` for the remaining transformation axes.

  See `fft` for details, definitions and conventions used.

  Examples

      >>> import numpy as np
      >>> a = np.ones((2, 2, 2))
      >>> np.fft.rfftn(a)
      array([[[8.+0.j,  0.+0.j], # may vary
              [0.+0.j,  0.+0.j]],
             [[0.+0.j,  0.+0.j],
              [0.+0.j,  0.+0.j]]])

      >>> np.fft.rfftn(a, axes=(2, 0))
      array([[[4.+0.j,  0.+0.j], # may vary
              [4.+0.j,  0.+0.j]],
             [[0.+0.j,  0.+0.j],
              [0.+0.j,  0.+0.j]]])