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# numpy.fft.irfftn

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

  Computes the inverse of `rfftn`.

  This function computes the inverse of the N-dimensional discrete Fourier Transform for real input over any number of axes in an M-dimensional array by means of the Fast Fourier Transform (FFT). In other words, `irfftn(rfftn(a), a.shape) == a` to within numerical accuracy. (The `a.shape` is necessary like `len(a)` is for `irfft`, and for the same reason.)

  The input should be ordered in the same way as is returned by `rfftn`, i.e. as for `irfft` for the final transformation axis, and as for `ifftn` along all the other axes.

  Parameters:
    a array_like

      Input array.

    s sequence of ints, optional

      Shape (length of each transformed axis) of the output (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.). s is also the number of input points used along this axis, except for the last axis, where `s[-1]//2+1` points of the input are used. Along any axis, if the shape indicated by s 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. Except for the last axis which is taken to be `2*(m-1)` where `m` is the length of the input along that axis.

      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 inverse FFT. If not given, the last len(s) axes are used, or all axes if s is also not specified. Repeated indices in axes means that the inverse transform over that axis is performed multiple times.

      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.

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

    out ndarray, optional

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

      Added in version 2.0.0.

  Returns:
    out ndarray

      The truncated or zero-padded input, transformed along the axes indicated by axes, or by a combination of s or a, as explained in the parameters section above. The length of each transformed axis is as given by the corresponding element of s, or the length of the input in every axis except for the last one if s is not given. In the final transformed axis the length of the output when s is not given is `2*(m-1)` where `m` is the length of the final transformed axis of the input. To get an odd number of output points in the final axis, s must be specified.

  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

  `rfftn`

    The forward n-dimensional FFT of real input, of which `ifftn` is the inverse.

  `fft`

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

  `irfft`

    The inverse of the one-dimensional FFT of real input.

  `irfft2`

    The inverse of the two-dimensional FFT of real input.

  Notes

  See `fft` for definitions and conventions used.

  See `rfft` for definitions and conventions used for real input.

  The correct interpretation of the hermitian input depends on the shape of the original data, as given by s. This is because each input shape could correspond to either an odd or even length signal. By default, `irfftn` assumes an even output length which puts the last entry at the Nyquist frequency; aliasing with its symmetric counterpart. When performing the final complex to real transform, the last value is thus treated as purely real. To avoid losing information, the correct shape of the real input must be given.

  Examples

      >>> import numpy as np
      >>> a = np.zeros((3, 2, 2))
      >>> a[0, 0, 0] = 3 * 2 * 2
      >>> np.fft.irfftn(a)
      array([[[1.,  1.],
              [1.,  1.]],
             [[1.,  1.],
              [1.,  1.]],
             [[1.,  1.],
              [1.,  1.]]])