content: """
  Discrete Fourier Transforms
  
  Routines in this module:
  
  fft(a, n=None, axis=-1)
  ifft(a, n=None, axis=-1)
  rfft(a, n=None, axis=-1)
  irfft(a, n=None, axis=-1)
  hfft(a, n=None, axis=-1)
  ihfft(a, n=None, axis=-1)
  fftn(a, s=None, axes=None)
  ifftn(a, s=None, axes=None)
  rfftn(a, s=None, axes=None)
  irfftn(a, s=None, axes=None)
  fft2(a, s=None, axes=(-2,-1))
  ifft2(a, s=None, axes=(-2, -1))
  rfft2(a, s=None, axes=(-2,-1))
  irfft2(a, s=None, axes=(-2, -1))
  
  i = inverse transform
  r = transform of purely real data
  h = Hermite transform
  n = n-dimensional transform
  2 = 2-dimensional transform
  (Note: 2D routines are just nD routines with different default
  behavior.)
  
  The underlying code for these functions is an f2c-translated and modified
  version of the FFTPACK routines.
  
  """
  from __future__ import division, absolute_import, print_function
  
  __all__ = ['fft', 'ifft', 'rfft', 'irfft', 'hfft', 'ihfft', 'rfftn',
             'irfftn', 'rfft2', 'irfft2', 'fft2', 'ifft2', 'fftn', 'ifftn']
  
  from numpy.core import (array, asarray, zeros, swapaxes, shape, conjugate,
                          take, sqrt)
  from . import fftpack_lite as fftpack
  
  _fft_cache = {}
  _real_fft_cache = {}
  
  
  def _raw_fft(a, n=None, axis=-1, init_function=fftpack.cffti,
               work_function=fftpack.cfftf, fft_cache=_fft_cache):
      a = asarray(a)
  
      if n is None:
          n = a.shape[axis]
  
      if n < 1:
          raise ValueError("Invalid number of FFT data points (%d) specified."
                           % n)
  
      try:
          # Thread-safety note: We rely on list.pop() here to atomically
          # retrieve-and-remove a wsave from the cache.  This ensures that no
          # other thread can get the same wsave while we're using it.
          wsave = fft_cache.setdefault(n, []).pop()
      except (IndexError):
          wsave = init_function(n)
  
      if a.shape[axis] != n:
          s = list(a.shape)
          if s[axis] > n:
              index = [slice(None)]*len(s)
              index[axis] = slice(0, n)
              a = a[index]
          else:
              index = [slice(None)]*len(s)
              index[axis] = slice(0, s[axis])
              s[axis] = n
              z = zeros(s, a.dtype.char)
              z[index] = a
              a = z
  
      if axis != -1:
          a = swapaxes(a, axis, -1)
      r = work_function(a, wsave)
      if axis != -1:
          r = swapaxes(r, axis, -1)
  
      # As soon as we put wsave back into the cache, another thread could pick it
      # up and start using it, so we must not do this until after we're
      # completely done using it ourselves.
      fft_cache[n].append(wsave)
  
      return r
  
  
  def _unitary(norm):
      if norm not in (None, "ortho"):
          raise ValueError("Invalid norm value %s, should be None or \"ortho\"."
                           % norm)
      return norm is not None
  
  
  def fft(a, n=None, axis=-1, norm=None):
      """
      Compute the one-dimensional discrete Fourier Transform.
  
      This function computes the one-dimensional *n*-point discrete Fourier
      Transform (DFT) with the efficient Fast Fourier Transform (FFT)
      algorithm [CT].
  
      Parameters
      ----------
      a : array_like
          Input array, can be complex.
      n : int, optional
          Length of the transformed axis of the output.
          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 : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      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.
  
      Raises
      ------
      IndexError
          if `axes` is larger than the last axis of `a`.
  
      See Also
      --------
      numpy.fft : for definition of the DFT and conventions used.
      ifft : The inverse of `fft`.
      fft2 : The two-dimensional FFT.
      fftn : The *n*-dimensional FFT.
      rfftn : The *n*-dimensional FFT of real input.
      fftfreq : Frequency bins for given FFT parameters.
  
      Notes
      -----
      FFT (Fast Fourier Transform) refers to a way the discrete Fourier
      Transform (DFT) can be calculated efficiently, by using symmetries in the
      calculated terms.  The symmetry is highest when `n` is a power of 2, and
      the transform is therefore most efficient for these sizes.
  
      The DFT is defined, with the conventions used in this implementation, in
      the documentation for the `numpy.fft` module.
  
      References
      ----------
      .. [CT] Cooley, James W., and John W. Tukey, 1965, "An algorithm for the
              machine calculation of complex Fourier series," *Math. Comput.*
              19: 297-301.
  
      Examples
      --------
      >>> np.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8))
      array([ -3.44505240e-16 +1.14383329e-17j,
               8.00000000e+00 -5.71092652e-15j,
               2.33482938e-16 +1.22460635e-16j,
               1.64863782e-15 +1.77635684e-15j,
               9.95839695e-17 +2.33482938e-16j,
               0.00000000e+00 +1.66837030e-15j,
               1.14383329e-17 +1.22460635e-16j,
               -1.64863782e-15 +1.77635684e-15j])
  
      >>> import matplotlib.pyplot as plt
      >>> t = np.arange(256)
      >>> sp = np.fft.fft(np.sin(t))
      >>> freq = np.fft.fftfreq(t.shape[-1])
      >>> plt.plot(freq, sp.real, freq, sp.imag)
      [<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>]
      >>> plt.show()
  
      In this example, real input has an FFT which is Hermitian, i.e., symmetric
      in the real part and anti-symmetric in the imaginary part, as described in
      the `numpy.fft` documentation.
  
      """
  
      a = asarray(a).astype(complex)
      if n is None:
          n = a.shape[axis]
      output = _raw_fft(a, n, axis, fftpack.cffti, fftpack.cfftf, _fft_cache)
      if _unitary(norm):
          output *= 1 / sqrt(n)
      return output
  
  
  def ifft(a, n=None, axis=-1, norm=None):
      """
      Compute the one-dimensional inverse discrete Fourier Transform.
  
      This function computes the inverse of the one-dimensional *n*-point
      discrete Fourier transform computed by `fft`.  In other words,
      ``ifft(fft(a)) == a`` to within numerical accuracy.
      For a general description of the algorithm and definitions,
      see `numpy.fft`.
  
      The input should be ordered in the same way as is returned by `fft`,
      i.e., ``a[0]`` should contain the zero frequency term,
      ``a[1:n/2+1]`` should contain the positive-frequency terms, and
      ``a[n/2+1:]`` should contain the negative-frequency terms, in order of
      decreasingly negative frequency.  See `numpy.fft` for details.
  
      Parameters
      ----------
      a : array_like
          Input array, can be complex.
      n : int, optional
          Length of the transformed axis of the output.
          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.
          See notes about padding issues.
      axis : int, optional
          Axis over which to compute the inverse DFT.  If not given, the last
          axis is used.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      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.
  
      Raises
      ------
      IndexError
          If `axes` is larger than the last axis of `a`.
  
      See Also
      --------
      numpy.fft : An introduction, with definitions and general explanations.
      fft : The one-dimensional (forward) FFT, of which `ifft` is the inverse
      ifft2 : The two-dimensional inverse FFT.
      ifftn : The n-dimensional inverse FFT.
  
      Notes
      -----
      If the input parameter `n` is larger than the size of the input, the input
      is padded by appending zeros at the end.  Even though this is the common
      approach, it might lead to surprising results.  If a different padding is
      desired, it must be performed before calling `ifft`.
  
      Examples
      --------
      >>> np.fft.ifft([0, 4, 0, 0])
      array([ 1.+0.j,  0.+1.j, -1.+0.j,  0.-1.j])
  
      Create and plot a band-limited signal with random phases:
  
      >>> import matplotlib.pyplot as plt
      >>> t = np.arange(400)
      >>> n = np.zeros((400,), dtype=complex)
      >>> n[40:60] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20,)))
      >>> s = np.fft.ifft(n)
      >>> plt.plot(t, s.real, 'b-', t, s.imag, 'r--')
      [<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>]
      >>> plt.legend(('real', 'imaginary'))
      <matplotlib.legend.Legend object at 0x...>
      >>> plt.show()
  
      """
      # The copy may be required for multithreading.
      a = array(a, copy=True, dtype=complex)
      if n is None:
          n = a.shape[axis]
      unitary = _unitary(norm)
      output = _raw_fft(a, n, axis, fftpack.cffti, fftpack.cfftb, _fft_cache)
      return output * (1 / (sqrt(n) if unitary else n))
  
  
  def rfft(a, n=None, axis=-1, norm=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 : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      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 larger than the last 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
      --------
      >>> np.fft.fft([0, 1, 0, 0])
      array([ 1.+0.j,  0.-1.j, -1.+0.j,  0.+1.j])
      >>> np.fft.rfft([0, 1, 0, 0])
      array([ 1.+0.j,  0.-1.j, -1.+0.j])
  
      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.
  
      """
      # The copy may be required for multithreading.
      a = array(a, copy=True, dtype=float)
      output = _raw_fft(a, n, axis, fftpack.rffti, fftpack.rfftf,
                        _real_fft_cache)
      if _unitary(norm):
          output *= 1 / sqrt(a.shape[axis])
      return output
  
  
  def irfft(a, n=None, axis=-1, norm=None):
      """
      Compute the inverse of the n-point DFT for real input.
  
      This function computes the inverse of the one-dimensional *n*-point
      discrete Fourier Transform of real input computed by `rfft`.
      In other words, ``irfft(rfft(a), len(a)) == a`` to within numerical
      accuracy. (See Notes below for why ``len(a)`` is necessary here.)
  
      The input is expected to be in the form returned by `rfft`, i.e. the
      real zero-frequency term followed by the complex positive frequency terms
      in order of increasing frequency.  Since the discrete Fourier Transform of
      real input is Hermitian-symmetric, the negative frequency terms are taken
      to be the complex conjugates of the corresponding positive frequency terms.
  
      Parameters
      ----------
      a : array_like
          The input array.
      n : int, optional
          Length of the transformed axis of the output.
          For `n` output points, ``n//2+1`` input points are necessary.  If the
          input is longer than this, it is cropped.  If it is shorter than this,
          it is padded with zeros.  If `n` is not given, it is determined from
          the length of the input along the axis specified by `axis`.
      axis : int, optional
          Axis over which to compute the inverse FFT. If not given, the last
          axis is used.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      Returns
      -------
      out : ndarray
          The truncated or zero-padded input, transformed along the axis
          indicated by `axis`, or the last one if `axis` is not specified.
          The length of the transformed axis is `n`, or, if `n` is not given,
          ``2*(m-1)`` where ``m`` is the length of the transformed axis of the
          input. To get an odd number of output points, `n` must be specified.
  
      Raises
      ------
      IndexError
          If `axis` is larger than the last axis of `a`.
  
      See Also
      --------
      numpy.fft : For definition of the DFT and conventions used.
      rfft : The one-dimensional FFT of real input, of which `irfft` is inverse.
      fft : The one-dimensional FFT.
      irfft2 : The inverse of the two-dimensional FFT of real input.
      irfftn : The inverse of the *n*-dimensional FFT of real input.
  
      Notes
      -----
      Returns the real valued `n`-point inverse discrete Fourier transform
      of `a`, where `a` contains the non-negative frequency terms of a
      Hermitian-symmetric sequence. `n` is the length of the result, not the
      input.
  
      If you specify an `n` such that `a` must be zero-padded or truncated, the
      extra/removed values will be added/removed at high frequencies. One can
      thus resample a series to `m` points via Fourier interpolation by:
      ``a_resamp = irfft(rfft(a), m)``.
  
      Examples
      --------
      >>> np.fft.ifft([1, -1j, -1, 1j])
      array([ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j])
      >>> np.fft.irfft([1, -1j, -1])
      array([ 0.,  1.,  0.,  0.])
  
      Notice how the last term in the input to the ordinary `ifft` is the
      complex conjugate of the second term, and the output has zero imaginary
      part everywhere.  When calling `irfft`, the negative frequencies are not
      specified, and the output array is purely real.
  
      """
      # The copy may be required for multithreading.
      a = array(a, copy=True, dtype=complex)
      if n is None:
          n = (a.shape[axis] - 1) * 2
      unitary = _unitary(norm)
      output = _raw_fft(a, n, axis, fftpack.rffti, fftpack.rfftb,
                        _real_fft_cache)
      return output * (1 / (sqrt(n) if unitary else n))
  
  
  def hfft(a, n=None, axis=-1, norm=None):
      """
      Compute the FFT of a signal which has Hermitian symmetry (real spectrum).
  
      Parameters
      ----------
      a : array_like
          The input array.
      n : int, optional
          Length of the transformed axis of the output.
          For `n` output points, ``n//2+1`` input points are necessary.  If the
          input is longer than this, it is cropped.  If it is shorter than this,
          it is padded with zeros.  If `n` is not given, it is determined from
          the length of the input along the axis specified by `axis`.
      axis : int, optional
          Axis over which to compute the FFT. If not given, the last
          axis is used.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      Returns
      -------
      out : ndarray
          The truncated or zero-padded input, transformed along the axis
          indicated by `axis`, or the last one if `axis` is not specified.
          The length of the transformed axis is `n`, or, if `n` is not given,
          ``2*(m-1)`` where ``m`` is the length of the transformed axis of the
          input. To get an odd number of output points, `n` must be specified.
  
      Raises
      ------
      IndexError
          If `axis` is larger than the last axis of `a`.
  
      See also
      --------
      rfft : Compute the one-dimensional FFT for real input.
      ihfft : The inverse of `hfft`.
  
      Notes
      -----
      `hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the
      opposite case: here the signal has Hermitian symmetry in the time domain
      and is real in the frequency domain. So here it's `hfft` for which
      you must supply the length of the result if it is to be odd:
      ``ihfft(hfft(a), len(a)) == a``, within numerical accuracy.
  
      Examples
      --------
      >>> signal = np.array([1, 2, 3, 4, 3, 2])
      >>> np.fft.fft(signal)
      array([ 15.+0.j,  -4.+0.j,   0.+0.j,  -1.-0.j,   0.+0.j,  -4.+0.j])
      >>> np.fft.hfft(signal[:4]) # Input first half of signal
      array([ 15.,  -4.,   0.,  -1.,   0.,  -4.])
      >>> np.fft.hfft(signal, 6)  # Input entire signal and truncate
      array([ 15.,  -4.,   0.,  -1.,   0.,  -4.])
  
  
      >>> signal = np.array([[1, 1.j], [-1.j, 2]])
      >>> np.conj(signal.T) - signal   # check Hermitian symmetry
      array([[ 0.-0.j,  0.+0.j],
             [ 0.+0.j,  0.-0.j]])
      >>> freq_spectrum = np.fft.hfft(signal)
      >>> freq_spectrum
      array([[ 1.,  1.],
             [ 2., -2.]])
  
      """
      # The copy may be required for multithreading.
      a = array(a, copy=True, dtype=complex)
      if n is None:
          n = (a.shape[axis] - 1) * 2
      unitary = _unitary(norm)
      return irfft(conjugate(a), n, axis) * (sqrt(n) if unitary else n)
  
  
  def ihfft(a, n=None, axis=-1, norm=None):
      """
      Compute the inverse FFT of a signal which has Hermitian symmetry.
  
      Parameters
      ----------
      a : array_like
          Input array.
      n : int, optional
          Length of the inverse FFT.
          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 inverse FFT. If not given, the last
          axis is used.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      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``.
  
      See also
      --------
      hfft, irfft
  
      Notes
      -----
      `hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the
      opposite case: here the signal has Hermitian symmetry in the time domain
      and is real in the frequency domain. So here it's `hfft` for which
      you must supply the length of the result if it is to be odd:
      ``ihfft(hfft(a), len(a)) == a``, within numerical accuracy.
  
      Examples
      --------
      >>> spectrum = np.array([ 15, -4, 0, -1, 0, -4])
      >>> np.fft.ifft(spectrum)
      array([ 1.+0.j,  2.-0.j,  3.+0.j,  4.+0.j,  3.+0.j,  2.-0.j])
      >>> np.fft.ihfft(spectrum)
      array([ 1.-0.j,  2.-0.j,  3.-0.j,  4.-0.j])
  
      """
      # The copy may be required for multithreading.
      a = array(a, copy=True, dtype=float)
      if n is None:
          n = a.shape[axis]
      unitary = _unitary(norm)
      output = conjugate(rfft(a, n, axis))
      return output * (1 / (sqrt(n) if unitary else n))
  
  
  def _cook_nd_args(a, s=None, axes=None, invreal=0):
      if s is None:
          shapeless = 1
          if axes is None:
              s = list(a.shape)
          else:
              s = take(a.shape, axes)
      else:
          shapeless = 0
      s = list(s)
      if axes is None:
          axes = list(range(-len(s), 0))
      if len(s) != len(axes):
          raise ValueError("Shape and axes have different lengths.")
      if invreal and shapeless:
          s[-1] = (a.shape[axes[-1]] - 1) * 2
      return s, axes
  
  
  def _raw_fftnd(a, s=None, axes=None, function=fft, norm=None):
      a = asarray(a)
      s, axes = _cook_nd_args(a, s, axes)
      itl = list(range(len(axes)))
      itl.reverse()
      for ii in itl:
          a = function(a, n=s[ii], axis=axes[ii], norm=norm)
      return a
  
  
  def fftn(a, s=None, axes=None, norm=None):
      """
      Compute the N-dimensional discrete Fourier Transform.
  
      This function computes the *N*-dimensional discrete Fourier Transform over
      any number of axes in an *M*-dimensional array by means of the Fast Fourier
      Transform (FFT).
  
      Parameters
      ----------
      a : array_like
          Input array, can be complex.
      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.).
          This 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.
          if `s` is not given, the shape of the input along the axes specified
          by `axes` is used.
      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.
          Repeated indices in `axes` means that the transform over that axis is
          performed multiple times.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      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.
  
      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
      --------
      numpy.fft : Overall view of discrete Fourier transforms, with definitions
          and conventions used.
      ifftn : The inverse of `fftn`, the inverse *n*-dimensional FFT.
      fft : The one-dimensional FFT, with definitions and conventions used.
      rfftn : The *n*-dimensional FFT of real input.
      fft2 : The two-dimensional FFT.
      fftshift : Shifts zero-frequency terms to centre of array
  
      Notes
      -----
      The output, analogously to `fft`, contains the term for zero frequency in
      the low-order corner of all axes, the positive frequency terms in the
      first half of all axes, the term for the Nyquist frequency in the middle
      of all axes and the negative frequency terms in the second half of all
      axes, in order of decreasingly negative frequency.
  
      See `numpy.fft` for details, definitions and conventions used.
  
      Examples
      --------
      >>> a = np.mgrid[:3, :3, :3][0]
      >>> np.fft.fftn(a, axes=(1, 2))
      array([[[  0.+0.j,   0.+0.j,   0.+0.j],
              [  0.+0.j,   0.+0.j,   0.+0.j],
              [  0.+0.j,   0.+0.j,   0.+0.j]],
             [[  9.+0.j,   0.+0.j,   0.+0.j],
              [  0.+0.j,   0.+0.j,   0.+0.j],
              [  0.+0.j,   0.+0.j,   0.+0.j]],
             [[ 18.+0.j,   0.+0.j,   0.+0.j],
              [  0.+0.j,   0.+0.j,   0.+0.j],
              [  0.+0.j,   0.+0.j,   0.+0.j]]])
      >>> np.fft.fftn(a, (2, 2), axes=(0, 1))
      array([[[ 2.+0.j,  2.+0.j,  2.+0.j],
              [ 0.+0.j,  0.+0.j,  0.+0.j]],
             [[-2.+0.j, -2.+0.j, -2.+0.j],
              [ 0.+0.j,  0.+0.j,  0.+0.j]]])
  
      >>> import matplotlib.pyplot as plt
      >>> [X, Y] = np.meshgrid(2 * np.pi * np.arange(200) / 12,
      ...                      2 * np.pi * np.arange(200) / 34)
      >>> S = np.sin(X) + np.cos(Y) + np.random.uniform(0, 1, X.shape)
      >>> FS = np.fft.fftn(S)
      >>> plt.imshow(np.log(np.abs(np.fft.fftshift(FS))**2))
      <matplotlib.image.AxesImage object at 0x...>
      >>> plt.show()
  
      """
  
      return _raw_fftnd(a, s, axes, fft, norm)
  
  
  def ifftn(a, s=None, axes=None, norm=None):
      """
      Compute the N-dimensional inverse discrete Fourier Transform.
  
      This function computes the inverse of the N-dimensional discrete
      Fourier Transform over any number of axes in an M-dimensional array by
      means of the Fast Fourier Transform (FFT).  In other words,
      ``ifftn(fftn(a)) == a`` to within numerical accuracy.
      For a description of the definitions and conventions used, see `numpy.fft`.
  
      The input, analogously to `ifft`, should be ordered in the same way as is
      returned by `fftn`, i.e. it should have the term for zero frequency
      in all axes in the low-order corner, the positive frequency terms in the
      first half of all axes, the term for the Nyquist frequency in the middle
      of all axes and the negative frequency terms in the second half of all
      axes, in order of decreasingly negative frequency.
  
      Parameters
      ----------
      a : array_like
          Input array, can be complex.
      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.).
          This corresponds to ``n`` for ``ifft(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.
          if `s` is not given, the shape of the input along the axes specified
          by `axes` is used.  See notes for issue on `ifft` zero padding.
      axes : sequence of ints, optional
          Axes over which to compute the IFFT.  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.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      Returns
      -------
      out : complex 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.
  
      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
      --------
      numpy.fft : Overall view of discrete Fourier transforms, with definitions
           and conventions used.
      fftn : The forward *n*-dimensional FFT, of which `ifftn` is the inverse.
      ifft : The one-dimensional inverse FFT.
      ifft2 : The two-dimensional inverse FFT.
      ifftshift : Undoes `fftshift`, shifts zero-frequency terms to beginning
          of array.
  
      Notes
      -----
      See `numpy.fft` for definitions and conventions used.
  
      Zero-padding, analogously with `ifft`, is performed by appending zeros to
      the input along the specified dimension.  Although this is the common
      approach, it might lead to surprising results.  If another form of zero
      padding is desired, it must be performed before `ifftn` is called.
  
      Examples
      --------
      >>> a = np.eye(4)
      >>> np.fft.ifftn(np.fft.fftn(a, axes=(0,)), axes=(1,))
      array([[ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],
             [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],
             [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],
             [ 0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j]])
  
  
      Create and plot an image with band-limited frequency content:
  
      >>> import matplotlib.pyplot as plt
      >>> n = np.zeros((200,200), dtype=complex)
      >>> n[60:80, 20:40] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20, 20)))
      >>> im = np.fft.ifftn(n).real
      >>> plt.imshow(im)
      <matplotlib.image.AxesImage object at 0x...>
      >>> plt.show()
  
      """
  
      return _raw_fftnd(a, s, axes, ifft, norm)
  
  
  def fft2(a, s=None, axes=(-2, -1), norm=None):
      """
      Compute the 2-dimensional discrete Fourier Transform
  
      This function computes the *n*-dimensional discrete Fourier Transform
      over any axes in an *M*-dimensional array by means of the
      Fast Fourier Transform (FFT).  By default, the transform is computed over
      the last two axes of the input array, i.e., a 2-dimensional FFT.
  
      Parameters
      ----------
      a : array_like
          Input array, can be complex
      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.).
          This corresponds to `n` for `fft(x, n)`.
          Along each 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.
          if `s` is not given, the shape of the input along the axes specified
          by `axes` is used.
      axes : sequence of ints, optional
          Axes over which to compute the FFT.  If not given, the last two
          axes are used.  A repeated index in `axes` means the transform over
          that axis is performed multiple times.  A one-element sequence means
          that a one-dimensional FFT is performed.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      Returns
      -------
      out : complex ndarray
          The truncated or zero-padded input, transformed along the axes
          indicated by `axes`, or the last two axes if `axes` is not given.
  
      Raises
      ------
      ValueError
          If `s` and `axes` have different length, or `axes` not given and
          ``len(s) != 2``.
      IndexError
          If an element of `axes` is larger than than the number of axes of `a`.
  
      See Also
      --------
      numpy.fft : Overall view of discrete Fourier transforms, with definitions
           and conventions used.
      ifft2 : The inverse two-dimensional FFT.
      fft : The one-dimensional FFT.
      fftn : The *n*-dimensional FFT.
      fftshift : Shifts zero-frequency terms to the center of the array.
          For two-dimensional input, swaps first and third quadrants, and second
          and fourth quadrants.
  
      Notes
      -----
      `fft2` is just `fftn` with a different default for `axes`.
  
      The output, analogously to `fft`, contains the term for zero frequency in
      the low-order corner of the transformed axes, the positive frequency terms
      in the first half of these axes, the term for the Nyquist frequency in the
      middle of the axes and the negative frequency terms in the second half of
      the axes, in order of decreasingly negative frequency.
  
      See `fftn` for details and a plotting example, and `numpy.fft` for
      definitions and conventions used.
  
  
      Examples
      --------
      >>> a = np.mgrid[:5, :5][0]
      >>> np.fft.fft2(a)
      array([[ 50.0 +0.j        ,   0.0 +0.j        ,   0.0 +0.j        ,
                0.0 +0.j        ,   0.0 +0.j        ],
             [-12.5+17.20477401j,   0.0 +0.j        ,   0.0 +0.j        ,
                0.0 +0.j        ,   0.0 +0.j        ],
             [-12.5 +4.0614962j ,   0.0 +0.j        ,   0.0 +0.j        ,
                0.0 +0.j        ,   0.0 +0.j        ],
             [-12.5 -4.0614962j ,   0.0 +0.j        ,   0.0 +0.j        ,
                  0.0 +0.j        ,   0.0 +0.j        ],
             [-12.5-17.20477401j,   0.0 +0.j        ,   0.0 +0.j        ,
                0.0 +0.j        ,   0.0 +0.j        ]])
  
      """
  
      return _raw_fftnd(a, s, axes, fft, norm)
  
  
  def ifft2(a, s=None, axes=(-2, -1), norm=None):
      """
      Compute the 2-dimensional inverse discrete Fourier Transform.
  
      This function computes the inverse of the 2-dimensional discrete Fourier
      Transform over any number of axes in an M-dimensional array by means of
      the Fast Fourier Transform (FFT).  In other words, ``ifft2(fft2(a)) == a``
      to within numerical accuracy.  By default, the inverse transform is
      computed over the last two axes of the input array.
  
      The input, analogously to `ifft`, should be ordered in the same way as is
      returned by `fft2`, i.e. it should have the term for zero frequency
      in the low-order corner of the two axes, the positive frequency terms in
      the first half of these axes, the term for the Nyquist frequency in the
      middle of the axes and the negative frequency terms in the second half of
      both axes, in order of decreasingly negative frequency.
  
      Parameters
      ----------
      a : array_like
          Input array, can be complex.
      s : sequence of ints, optional
          Shape (length of each axis) of the output (``s[0]`` refers to axis 0,
          ``s[1]`` to axis 1, etc.).  This corresponds to `n` for ``ifft(x, n)``.
          Along each 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.
          if `s` is not given, the shape of the input along the axes specified
          by `axes` is used.  See notes for issue on `ifft` zero padding.
      axes : sequence of ints, optional
          Axes over which to compute the FFT.  If not given, the last two
          axes are used.  A repeated index in `axes` means the transform over
          that axis is performed multiple times.  A one-element sequence means
          that a one-dimensional FFT is performed.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      Returns
      -------
      out : complex ndarray
          The truncated or zero-padded input, transformed along the axes
          indicated by `axes`, or the last two axes if `axes` is not given.
  
      Raises
      ------
      ValueError
          If `s` and `axes` have different length, or `axes` not given and
          ``len(s) != 2``.
      IndexError
          If an element of `axes` is larger than than the number of axes of `a`.
  
      See Also
      --------
      numpy.fft : Overall view of discrete Fourier transforms, with definitions
           and conventions used.
      fft2 : The forward 2-dimensional FFT, of which `ifft2` is the inverse.
      ifftn : The inverse of the *n*-dimensional FFT.
      fft : The one-dimensional FFT.
      ifft : The one-dimensional inverse FFT.
  
      Notes
      -----
      `ifft2` is just `ifftn` with a different default for `axes`.
  
      See `ifftn` for details and a plotting example, and `numpy.fft` for
      definition and conventions used.
  
      Zero-padding, analogously with `ifft`, is performed by appending zeros to
      the input along the specified dimension.  Although this is the common
      approach, it might lead to surprising results.  If another form of zero
      padding is desired, it must be performed before `ifft2` is called.
  
      Examples
      --------
      >>> a = 4 * np.eye(4)
      >>> np.fft.ifft2(a)
      array([[ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],
             [ 0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j],
             [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],
             [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j]])
  
      """
  
      return _raw_fftnd(a, s, axes, ifft, norm)
  
  
  def rfftn(a, s=None, axes=None, norm=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.
          if `s` is not given, the shape of the input along the axes specified
          by `axes` is used.
      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.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      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
      --------
      >>> a = np.ones((2, 2, 2))
      >>> np.fft.rfftn(a)
      array([[[ 8.+0.j,  0.+0.j],
              [ 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],
              [ 4.+0.j,  0.+0.j]],
             [[ 0.+0.j,  0.+0.j],
              [ 0.+0.j,  0.+0.j]]])
  
      """
      # The copy may be required for multithreading.
      a = array(a, copy=True, dtype=float)
      s, axes = _cook_nd_args(a, s, axes)
      a = rfft(a, s[-1], axes[-1], norm)
      for ii in range(len(axes)-1):
          a = fft(a, s[ii], axes[ii], norm)
      return a
  
  
  def rfft2(a, s=None, axes=(-2, -1), norm=None):
      """
      Compute the 2-dimensional FFT of a real array.
  
      Parameters
      ----------
      a : array
          Input array, taken to be real.
      s : sequence of ints, optional
          Shape of the FFT.
      axes : sequence of ints, optional
          Axes over which to compute the FFT.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      Returns
      -------
      out : ndarray
          The result of the real 2-D FFT.
  
      See Also
      --------
      rfftn : Compute the N-dimensional discrete Fourier Transform for real
              input.
  
      Notes
      -----
      This is really just `rfftn` with different default behavior.
      For more details see `rfftn`.
  
      """
  
      return rfftn(a, s, axes, norm)
  
  
  def irfftn(a, s=None, axes=None, norm=None):
      """
      Compute the inverse of the N-dimensional FFT of real input.
  
      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. If `s` is not given, the shape of the input along the
          axes specified by `axes` is used.
      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.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      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.
  
      Examples
      --------
      >>> 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.]]])
  
      """
      # The copy may be required for multithreading.
      a = array(a, copy=True, dtype=complex)
      s, axes = _cook_nd_args(a, s, axes, invreal=1)
      for ii in range(len(axes)-1):
          a = ifft(a, s[ii], axes[ii], norm)
      a = irfft(a, s[-1], axes[-1], norm)
      return a
  
  
  def irfft2(a, s=None, axes=(-2, -1), norm=None):
      """
      Compute the 2-dimensional inverse FFT of a real array.
  
      Parameters
      ----------
      a : array_like
          The input array
      s : sequence of ints, optional
          Shape of the inverse FFT.
      axes : sequence of ints, optional
          The axes over which to compute the inverse fft.
          Default is the last two axes.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      Returns
      -------
      out : ndarray
          The result of the inverse real 2-D FFT.
  
      See Also
      --------
      irfftn : Compute the inverse of the N-dimensional FFT of real input.
  
      Notes
      -----
      This is really `irfftn` with different defaults.
      For more details see `irfftn`.
  
      """
  
      return irfftn(a, s, axes, norm)
  