# scipy.fftpack.fftn

scipy.fftpack.fftn(x, shape=None, axes=None, overwrite_x=False)

  Return multidimensional discrete Fourier transform.

  The returned array contains:

      y[j_1,..,j_d] = sum[k_1=0..n_1-1, ..., k_d=0..n_d-1]
         x[k_1,..,k_d] * prod[i=1..d] exp(-sqrt(-1)*2*pi/n_i * j_i * k_i)

  where d = len(x.shape) and n = x.shape.

  Parameters:
    x array_like

      The (N-D) array to transform.

    shape int or array_like of ints or None, optional

      The shape of the result. If both shape and axes (see below) are None, shape is `x.shape`; if shape is None but axes is not None, then shape is `numpy.take(x.shape, axes, axis=0)`. If `shape[i] > x.shape[i]`, the ith dimension is padded with zeros. If `shape[i] < x.shape[i]`, the ith dimension is truncated to length `shape[i]`. If any element of shape is -1, the size of the corresponding dimension of x is used.

    axes int or array_like of ints or None, optional

      The axes of x (y if shape is not None) along which the transform is applied. The default is over all axes.

    overwrite_x bool, optional

      If True, the contents of x can be destroyed. Default is False.

  Returns:
    y complex-valued N-D NumPy array

      The (N-D) DFT of the input array.

  See also

  `ifftn`

  Notes

  If `x` is real-valued, then `y[..., j_i, ...] == y[..., n_i-j_i, ...].conjugate()`.

  Both single and double precision routines are implemented. Half precision inputs will be converted to single precision. Non-floating-point inputs will be converted to double precision. Long-double precision inputs are not supported.

  Examples

      >>> import numpy as np
      >>> from scipy.fftpack import fftn, ifftn
      >>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16))
      >>> np.allclose(y, fftn(ifftn(y)))
      True