text: #
  # * The source code in this file is based on the soure code of NumPy and CuPy.
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  from nlcpy.fft import _fft
  
  
  def fft(a, n=None, axis=-1, norm=None):
      """Computes 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.
  
      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. If
          *axis* is larger than the last axis of a, *IndexError* occurs.
      norm : {None, "ortho"},optional
          Normalization mode. By default(None), the transforms are unscaled. It *norm* is
          set to "ortho", the return values will be scaled by :math:`1/\\sqrt{n}`.
  
      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.
  
      Note
      ----
      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.
  
      See Also
      --------
      ifft : Computes the one-dimensional inverse discrete fourier transform.
      fft2 : Computes the 2-dimensional discrete fourier transform.
      fftn : Computes the n-dimensional discrete fourier transform.
      rfftn : Computes the n-dimensional discrete fourier transform for a real array.
      fftfreq : Returns the discrete fourier transform sample frequencies.
  
      Examples
      --------
      .. plot::
          :align: center
  
          >>> import nlcpy as vp
          >>> vp.fft.fft(vp.exp(2j * vp.pi * vp.arange(8) / 8))    # doctest: +SKIP
          array([-3.44509285e-16+1.14423775e-17j,  8.00000000e+00-8.52069395e-16j,
                  2.33486982e-16+1.22464680e-16j,  0.00000000e+00+1.22464680e-16j,
                  9.95799250e-17+2.33486982e-16j,  0.00000000e+00+1.17281316e-16j,
                  1.14423775e-17+1.22464680e-16j,  0.00000000e+00+1.22464680e-16j])
  
          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:
  
          >>> import matplotlib.pyplot as plt
          >>> t = vp.arange(256)
          >>> sp = vp.fft.fft(vp.sin(t))
          >>> freq = vp.fft.fftfreq(t.shape[-1])
          >>> _ = plt.plot(freq, sp.real, freq, sp.imag)
          >>> plt.show()
  
      """
      return _fft.fft(a, n, axis, norm)
  
  
  def ifft(a, n=None, axis=-1, norm=None):
      """Computes the one-dimensional inverse discrete fourier transform.
  
      This function computes the inverse of the one-dimensional n-point discrete fourier
      transform computed by :func:`fft`.
      In other words, ``ifft( fft(a) ) == a`` to within numerical accuracy.
      The input should be ordered in the same way as is returned by :func:`fft`, i.e.,
  
      - ``a[0]`` should contain the zero frequency term,
      - ``a[1:n//2]`` should contain the positive-frequency terms,
      - ``a[n//2 + 1:]`` should contain the negative-frequency terms, in increasing order
        starting from the most negative frequency.
  
      For an even number of input points, ``A[n//2]`` represents the sum of the values at
      the positive and negative Nyquist frequencies, as the two are aliased together.
  
      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.
          If *axis* is larger than the last axis of *a*, *IndexError* occurs.
      norm : {None, "ortho"},optional
          Normalization mode. By default(None), the transforms are scaled by :math:`1/n`.
          It *norm* is set to "ortho", the return values will be scaled by
          :math:`1/\\sqrt{n}`.
  
      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.
  
      Note
      ----
      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.
  
      See Also
      --------
      fft : Computes the one-dimensional discrete fourier transform.
      ifft2 : Computes the 2-dimensional inverse discrete fourier transform.
      ifftn : Computes the n-dimensional inverse discrete fourier transform.
  
      Examples
      --------
      .. plot::
          :align: center
  
          >>> import nlcpy as vp
          >>> vp.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 = vp.arange(400)
          >>> n = vp.zeros((400,), dtype=complex)
          >>> n[40:60] = vp.exp(1j*vp.random.uniform(0, 2*vp.pi, (20,)))
          >>> s = vp.fft.ifft(n)
          >>> _ = plt.plot(t, s.real, 'b-', t, s.imag, 'r--')
          >>> _ = plt.legend(('real', 'imaginary'))
          >>> plt.show()
  
      """
      return _fft.ifft(a, n, axis, norm)
  
  
  def fft2(a, s=None, axes=(-2, -1), norm=None):
      """Computes 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. If s and axes have
          different length, or axes not given and ``len(s) != 2``, *ValueError* occurs.
      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. If
          an element of axes is larger than than the number of axes of *a*, *IndexError*
          occurs.
      norm : {None, "ortho"},optional
          Normalization mode. By default(None), the transforms are unscaled. It *norm* is
          set to "ortho", the return values will be scaled by :math:`1/\\sqrt{n}`.
  
      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.
  
      Note
      ----
      fft2 is just :func:`fftn` with a different default for axes.
  
      The output, analogously to :func:`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 :func:`fftn` for details and a plotting example.
  
      See Also
      --------
      ifft2 : Computes the 2-dimensional inverse discrete fourier transform.
      fft : Computes the one-dimensional discrete fourier transform.
      fftn : Computes the n-dimensional discrete fourier transform.
      fftshift : Shifts the zero-frequency component to the center of the spectrum.
  
      Examples
      --------
      >>> import nlcpy as vp
      >>> import numpy as np
      >>> a = np.mgrid[:5, :5][0]
      >>> vp.fft.fft2(a)   # doctest: +SKIP
      array([[ 50. +0.0000000000000000e+00j,   0. +0.0000000000000000e+00j, # may vary
                0. +0.0000000000000000e+00j,   0. +0.0000000000000000e+00j,
                0. +0.0000000000000000e+00j],
             [-12.5+1.7204774005889668e+01j,   0. +8.8817841970012523e-16j,
                0. +8.8817841970012523e-16j,   0. +8.8817841970012523e-16j,
                0. +8.8817841970012523e-16j],
             [-12.5+4.0614962029113286e+00j,   0. -2.2204460492503131e-16j,
                0. -2.2204460492503131e-16j,   0. -2.2204460492503131e-16j,
                0. -2.2204460492503131e-16j],
             [-12.5-4.0614962029113286e+00j,   0. +2.2204460492503131e-16j,
                0. +2.2204460492503131e-16j,   0. +2.2204460492503131e-16j,
                0. +2.2204460492503131e-16j],
             [-12.5-1.7204774005889668e+01j,   0. -8.8817841970012523e-16j,
                0. -8.8817841970012523e-16j,   0. -8.8817841970012523e-16j,
                0. -8.8817841970012523e-16j]])
  
      """
      return _fft.fft2(a, s, axes, norm)
  
  
  def ifft2(a, s=None, axes=(-2, -1), norm=None):
      """Computes 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 :func:`ifft`, should be ordered in the same way as is
      returned by :func:`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. If s and axes have different length, or axes not given and ``len(s) !=
          2``, *ValueError* occurs.
      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. If
          an element of axes is larger than than the number of axes of a, *IndexError*
          occurs.
      norm : {None, "ortho"},optional
          Normalization mode. By default(None), the transforms are scaled by :math:`1/n`.
          It *norm* is set to "ortho", the return values will be scaled by
          :math:`1/\\sqrt{n}`.
  
      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.
  
      Note
      ----
      :func:`ifft2` is just :func:`ifftn` with a different default for axes.
  
      Zero-padding, analogously with :func:`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 :func:`ifft2` is called.
  
      See Also
      --------
      fft2 : Computes the 2-dimensional discrete fourier transform.
      ifftn : Computes the n-dimensional inverse discrete fourier transform.
      fft : Computes the one-dimensional discrete fourier transform.
      ifft : Computes the one-dimensional inverse discrete fourier transform.
  
      Examples
      --------
      >>> import nlcpy as vp
      >>> a = 4 * vp.eye(4)
      >>> vp.fft.ifft2(a)    # doctest: +SKIP
      array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],    # may vary
             [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 _fft.ifft2(a, s, axes, norm)
  
  
  def fftn(a, s=None, axes=None, norm=None):
      """Computes 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. If s and axes have different
          length, *ValueError* occurs.
      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. If an element of axes
          is larger than than the number of axes of a, *IndexError* occurs.
      norm : {None, "ortho"},optional
          Normalization mode. By default(None), the transforms are unscaled. It *norm* is
          set to "ortho", the return values will be scaled by :math:`1/\\sqrt{n}`.
  
      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.
  
      Note
      ----
      The output, analogously to :func:`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 Also
      --------
      ifftn : Computes the n-dimensional inverse discrete fourier transform.
      fft : Computes the one-dimensional discrete fourier transform.
      rfftn : Computes the n-dimensional discrete fourier transform for a real array.
      fft2 : Computes the 2-dimensional discrete fourier transform.
      fftshift : Shifts the zero-frequency component to the center of the spectrum.
  
      Examples
      --------
      .. plot::
          :align: center
  
          >>> import nlcpy as vp
          >>> import numpy as np
          >>> a = np.mgrid[:3, :3, :3][0]
          >>> vp.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]],
          <BLANKLINE>
                 [[ 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]],
          <BLANKLINE>
                 [[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]]])
          >>> vp.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]],
          <BLANKLINE>
                 [[-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 * vp.pi * vp.arange(200) / 12,
          ...                      2 * vp.pi * vp.arange(200) / 34)
          >>> S = vp.sin(X) + vp.cos(Y) + vp.random.uniform(0, 1, X.shape)
          >>> FS = vp.fft.fftn(S)
          >>> _ = plt.imshow(vp.log(vp.absolute(vp.fft.fftshift(FS))**2))
          >>> plt.show()
  
      """
      return _fft.fftn(a, s, axes, norm)
  
  
  def ifftn(a, s=None, axes=None, norm=None):
      """Computes 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.
  
      The input, analogously to :func:`ifft`, should be ordered in the same way as is
      returned by :func:`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. If s and axes have different length, *ValueError* occurs.
      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. If an element
          of *axes* is larger than than the number of axes of a, *IndexError* occurs.
      norm : {None, "ortho"},optional
          Normalization mode. By default(None), the transforms are scaled by :math:`1/n`.
          It *norm* is set to "ortho", the return values will be scaled by
          :math:`1/\\sqrt{n}`.
  
      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.
  
      Note
      ----
      Zero-padding, analogously with :func:`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.
  
      See Also
      --------
      fftn : Computes the n-dimensional discrete fourier transform.
      ifft : Computes the one-dimensional inverse discrete fourier transform.
      ifft2 : Computes the 2-dimensional inverse discrete fourier transform.
      ifftshift : The inverse of fftshift.
  
      Examples
      --------
      .. plot::
          :align: center
  
          >>> import nlcpy as vp
          >>> a = vp.eye(4)
          >>> vp.fft.ifftn(vp.fft.fftn(a, axes=(0,)), axes=(1,))  # doctest: +SKIP
          array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],     # may vary
                 [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 = vp.zeros((200,200), dtype=complex)
          >>> n[60:80, 20:40] = vp.exp(1j*vp.random.uniform(0, 2*vp.pi, (20, 20)))
          >>> im = vp.fft.ifftn(n).real
          >>> _ = plt.imshow(im)
          >>> plt.show()
  
      """
      return _fft.ifftn(a, s, axes, norm)
  
  
  def rfft(a, n=None, axis=-1, norm=None):
      """Computes the one-dimensional discrete fourier transform for a real array.
  
      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. If
          *axes* is larger than the last axis of *a*, *IndexError* occurs.
      norm : {None, "ortho"},optional
          Normalization mode. By default(None), the transforms are unscaled. It *norm* is
          set to "ortho", the return values will be scaled by :math:`1/\\sqrt{n}`.
  
      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``.
  
      Note
      ----
      When the DFT is computed for purely a real array, 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.
  
      See Also
      --------
      irfft : Computes the inverse of the n-point DFT for a real array.
      fft : Computes the one-dimensional discrete fourier transform.
      fftn : Computes the n-dimensional discrete fourier transform.
      rfftn : Computes the n-dimensional discrete fourier transform for a real array.
  
      Examples
      --------
      >>> import nlcpy as vp
      >>> vp.fft.fft([0, 1, 0, 0])   # doctest: +SKIP
      array([ 1.+0.j,  0.-1.j, -1.+0.j,  0.+1.j])  # may vary
      >>> vp.fft.rfft([0, 1, 0, 0])  # doctest: +SKIP
      array([ 1.+0.j,  0.-1.j, -1.+0.j])   # may vary
  
      Notice how the final element of the :func:`fft` output is the complex conjugate
      of the second element, for a real array.
      For rfft, this symmetry is exploited to compute only the non-negative frequency
      terms.
  
      """
      return _fft.rfft(a, n, axis, norm)
  
  
  def irfft(a, n=None, axis=-1, norm=None):
      """Computes the inverse of the n-point DFT for a real array.
  
      This function computes the inverse of the one-dimensional *n*-point discrete fourier
      transform of a real array computed by :func:`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 :func:`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 a real array 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
          taken to be ``2*(m-1)`` where ``m`` is 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. If
          *axes* is larger than the last axis of *a*, *IndexError* occurs.
      norm : {None, "ortho"},optional
          Normalization mode. By default(None), the transforms are scaled by :math:`1/n`.
          It *norm* is set to "ortho", the return values will be scaled by
          :math:`1/\\sqrt{n}`.
  
      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. 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.
  
      Note
      ----
      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 )``.
  
      The correct interpretation of the hermitian input depends on the length of the
      original data, as given by *n*. This is because each input shape could correspond to
      either an odd or even length signal. By default, irfft assumes an even output length
      which puts the last entry at the Nyquist frequency; aliasing with its symmetric
      counterpart. By Hermitian symmetry, the value is thus treated as purely real. To
      avoid losing information, the correct length of the real array must be given.
  
      See Also
      --------
      rfft : Computes the one-dimensional discrete fourier transform for a real array.
      fft : Computes the one-dimensional discrete fourier transform.
      irfft2 : Computes the 2-dimensional inverse FFT of a real array.
      irfftn : Computes the inverse of the n-dimensional FFT of a real array.
  
      Examples
      --------
      >>> import nlcpy as vp
      >>> vp.fft.ifft([1, -1j, -1, 1j])  # doctest: +SKIP
      array([0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j])  # may vary
      >>> vp.fft.irfft([1, -1j, -1])
      array([0., 1., 0., 0.])
  
      Notice how the last term in the input to the ordinary :func:`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.
  
      """
      return _fft.irfft(a, n, axis, norm)
  
  
  def rfft2(a, s=None, axes=(-2, -1), norm=None):
      """Computes 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
          Normalization mode. By default(None), the transforms are unscaled. It *norm* is
          set to "ortho", the return values will be scaled by :math:`1/\\sqrt{n}`.
  
      Returns
      -------
      out : ndarray
          The result of the real 2-D FFT.
  
      Note
      ----
      This is really just :func:`rfftn` with different default behavior. For more details
      see :func:`rfftn`.
  
      See Also
      --------
      rfftn : Computes the n-dimensional discrete fourier transform for a real array.
  
      """
      return _fft.rfft2(a, s, axes, norm)
  
  
  def irfft2(a, s=None, axes=(-2, -1), norm=None):
      """Computes the 2-dimensional inverse FFT of a real array.
  
      Parameters
      ----------
      a : arrayi_like
          The input array
      s : sequence of ints, optional
          Shape of the real output to 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
          Normalization mode. By default(None), the transforms are scaled by :math:`1/n`.
          It *norm* is set to "ortho", the return values will be scaled by
          :math:`1/\\sqrt{n}`.
  
      Returns
      -------
      out : ndarray
          The result of the inverse real 2-D FFT.
  
      Note
      ----
  
      This is really :func:`irfftn` with different defaults. For more details see
      :func:`irfftn`.
  
      See Also
      --------
      irfftn : Computes the inverse of the n-dimensional FFT of a real array.
  
      """
      return _fft.irfft2(a, s, axes, norm)
  
  
  def rfftn(a, s=None, axes=None, norm=None):
      """Computes the n-dimensional discrete fourier transform for a real array.
  
      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 int, 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.
          If *s* and *axes* have different length, *ValueError* occurs.
      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. If an element of *axes* is larger
          than than the number of axes of *a*, *IndexError* occurs.
      norm : {None, "ortho"},optional
          Normalization mode. By default(None), the transforms are unscaled. It *norm* is
          set to "ortho", the return values will be scaled by :math:`1/\\sqrt{n}`.
  
      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.
  
      Note
      ----
      The transform for a real array is performed over the last transformation axis, as by
      :func:`rfft`, then the transform over the remaining axes is performed as by
      :func:`fftn`. The order of the output is as for :func:`rfft` for the final
      transformation axis, and as for :func:`fftn` for the remaining transformation axes.
      See :func:`fft` for details, definitions and conventions used.
  
      See Also
      --------
      irfftn : Computes the inverse of the n-dimensional FFT of a real array.
      fft : Computes the one-dimensional discrete fourier transform.
      rfft : Computes the one-dimensional discrete fourier transform for a real array.
      fftn : Computes the n-dimensional discrete fourier transform.
      rfft2 : Computes the 2-dimensional FFT of a real array.
  
      Examples
      --------
      >>> import nlcpy as vp
      >>> a = vp.ones((2, 2, 2))
      >>> vp.fft.rfftn(a)       # doctest: +SKIP
      array([[[8.+0.j, 0.+0.j],     # may vary
              [0.+0.j, 0.+0.j]],
      <BLANKLINE>
             [[0.+0.j, 0.+0.j],
              [0.+0.j, 0.+0.j]]])
      >>> vp.fft.rfftn(a, axes=(2, 0))   # doctest: +SKIP
      array([[[4.+0.j, 0.+0.j],     # may vary
              [4.+0.j, 0.+0.j]],
      <BLANKLINE>
             [[0.+0.j, 0.+0.j],
              [0.+0.j, 0.+0.j]]])
  
      """
      return _fft.rfftn(a, s, axes, norm)
  
  
  def irfftn(a, s=None, axes=None, norm=None):
      """Computes the inverse of the n-dimensional FFT of a real array.
  
      This function computes the inverse of the n-dimensional discrete fourier transform
      for a real array 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
      :func:`irfft`, and for the same reason.)
  
      The input should be ordered in the same way as is returned by :func:`rfftn`, i.e.
      as for :func:`irfft` for the final transformation axis, and as for
      :func:`ifftn` along all the other axes.
  
      Parameters
      ----------
      a : array_like
          Input array.
      s : sequence of int, 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. Except for the last axis which is taken to be ``2*(m-1)`` where ``m`` is
          the length of the input along that axis. If *s* and *axes* have different length,
          *ValueError* occurs.
      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. If
          an element of *axes* is larger than than the number of axes of *a*, *IndexError*
          occurs.
      norm : {None, "ortho"},optional
          Normalization mode. By default(None), the transforms are scaled by :math:`1/n`.
          It *norm* is set to "ortho", the return values will be scaled by
          :math:`1/\\sqrt{n}`.
  
      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.
  
      Note
      ----
      See :func:`fft` for definitions and conventions used.
  
      See :func:`rfft` for definitions and conventions used for a real array.
  
      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 array **must** be given.
  
      See Also
      --------
      rfftn : Computes the n-dimensional discrete fourier transform for a real array.
      fft : Computes the one-dimensional discrete fourier transform.
      irfft : Computes the inverse of the n-point DFT for a real array.
      irfft2 : Computes the 2-dimensional inverse FFT of a real array.
  
      Examples
      --------
      >>> import nlcpy as vp
      >>> a = vp.zeros((3, 2, 2))
      >>> a[0, 0, 0] = 3 * 2 * 2
      >>> vp.fft.irfftn(a)
      array([[[1., 1.],
              [1., 1.]],
      <BLANKLINE>
             [[1., 1.],
              [1., 1.]],
      <BLANKLINE>
             [[1., 1.],
              [1., 1.]]])
  
  
      """
      return _fft.irfftn(a, s, axes, norm)
  
  
  def hfft(a, n=None, axis=-1, norm=None):
      """Computes the FFT of a signal that has Hermitian symmetry, i.e., a 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
          taken to be ``2*(m-1)`` where ``m`` is 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. If
          *axis* is larger than the last axis of *a*, *IndexError* occurs.
      norm : {None, "ortho"},optional
          Normalization mode. By default(None), the transforms are unscaled. It *norm* is
          set to "ortho", the return values will be scaled by :math:`1/\\sqrt{n}`.
  
      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 - 2`` where m is the length of the
          transformed axis of the input. To get an odd number of output points, *n* must be
          specified, for instance as ``2*m - 1`` in the typical case,
  
      Note
      ----
      :func:`hfft`/:func:`ihfft` are a pair analogous to :func:`rfft`/:func:`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 :func:`hfft` for which you must
      supply the length of the result if it is to be odd.
  
      - even: ``ihfft( hfft(a, 2*len(a) - 2) ) == a``, within roundoff error,
      - odd: ``ihfft( hfft(a, 2*len(a) - 1) ) == a``, within roundoff error.
  
      The correct interpretation of the hermitian input depends on the length of the
      original data, as given by `n`. This is because each input shape could correspond to
      either an odd or even length signal. By default, hfft assumes an even output length
      which puts the last entry at the Nyquist frequency; aliasing with its symmetric
      counterpart. By Hermitian symmetry, the value is thus treated as purely real. To
      avoid losing information, the shape of the full signal **must** be given.
  
      See Also
      --------
      rfft : Computes the one-dimensional discrete fourier transform for a real array.
      ihfft : Computes the inverse FFT of a signal that has Hermitian symmetry.
  
      Examples
      --------
      >>> import nlcpy as vp
      >>> signal = vp.array([1, 2, 3, 4, 3, 2])
      >>> vp.fft.fft(signal)     # doctest: +SKIP
      array([15.+0.j, -4.+0.j,  0.+0.j, -1.+0.j,  0.+0.j, -4.+0.j])    # may vary
      >>> vp.fft.hfft(signal[:4]) # Input first half of signal
      array([15., -4.,  0., -1.,  0., -4.])
      >>> vp.fft.hfft(signal, 6)  # Input entire signal and truncate
      array([15., -4.,  0., -1.,  0., -4.])
  
      >>> signal = vp.array([[1, 1.j], [-1.j, 2]])
      >>> vp.conj(signal.T) - signal   # check Hermitian symmetry
      array([[ 0.-0.j, -0.+0.j],
             [ 0.+0.j,  0.-0.j]])
      >>> freq_spectrum = vp.fft.hfft(signal)
      >>> freq_spectrum
      array([[ 1.,  1.],
             [ 2., -2.]])
  
      """
      return _fft.hfft(a, n, axis, norm)
  
  
  def ihfft(a, n=None, axis=-1, norm=None):
      """Computes the inverse FFT of a signal that has Hermitian symmetry.
  
      Parameters
      ----------
      a : array_like
          Input array.
      n : int, optional
          Length of the inverse FFT, the 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
          Normalization mode. By default(None), the transforms are scaled by :math:`1/n`.
          It *norm* is set to "ortho", the return values will be scaled by
          :math:`1/\\sqrt{n}`.
  
      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. The length of the transformed
          axis is ``n//2 + 1``.
  
      Note
      ----
      :func:`hfft`/:func:`ihfft` are a pair analogous to :func:`rfft`/:func:`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 :func:`hfft` for which you must supply
      the length of the result if it is to be odd:
  
      - even: ``ihfft( hfft(a, 2*len(a) - 2) ) == a``, within roundoff error,
      - odd: ``ihfft( hfft(a, 2*len(a) - 1) ) == a``, within roundoff error.
  
      See Also
      --------
      hfft : Computes the FFT of a signal that has Hermitian symmetry, i.e., a real
          spectrum.
      irfft : Computes the inverse FFT of a signal that has Hermitian symmetry.
  
      Examples
      --------
      >>> import nlcpy as vp
      >>> spectrum = vp.array([ 15, -4, 0, -1, 0, -4])
      >>> vp.fft.ifft(spectrum)     # doctest: +SKIP
      array([1.+0.j, 2.+0.j, 3.+0.j, 4.+0.j, 3.+0.j, 2.+0.j])   # may vary
      >>> vp.fft.ihfft(spectrum)    # doctest: +SKIP
      array([1.+0.j, 2.+0.j, 3.+0.j, 4.+0.j])    # may vary
  
      """
      return _fft.ihfft(a, n, axis, norm)
  
  
  def fftfreq(n, d=1.0):
      """Returns the Discrete fourier transform sample frequencies.
  
      The returned float array *f* contains the frequency bin centers in cycles per unit of
      the sample spacing (with zero at the start). For instance, if the sample spacing is
      in seconds, then the frequency unit is cycles/second. Given a window length *n* and a
      sample spacing *d*::
  
          f = [0, 1, ...,   n/2-1,     -n/2, ..., -1] / (d*n)   if n is even
          f = [0, 1, ..., (n-1)/2, -(n-1)/2, ..., -1] / (d*n)   if n is odd
  
      Parameters
      ----------
      n : int
          Window length.
      d : scalar, optional
          Sample spacing (inverse of the sampling rate). Defaults to 1.
  
      Returns
      -------
      f : ndarray
          Array of length *n* containing the sample frequencies.
  
      Examples
      --------
      >>> import nlcpy as vp
      >>> signal = vp.array([-2, 8, 6, 4, 1, 0, 3, 5], dtype=float)
      >>> fourier = vp.fft.fft(signal)
      >>> n = signal.size
      >>> timestep = 0.1
      >>> freq = vp.fft.fftfreq(n, d=timestep)
      >>> freq
      array([ 0.  ,  1.25,  2.5 ,  3.75, -5.  , -3.75, -2.5 , -1.25])
  
      """
      return _fft.fftfreq(n, d)
  
  
  def rfftfreq(n, d=1.0):
      """Returns the Discrete fourier transform sample frequencies (for usage with
      :func:`rfft`, :func:`irfft`).
  
      The returned float array *f* contains the frequency bin centers in cycles per unit of
      the sample spacing (with zero at the start). For instance, if the sample spacing is
      in seconds, then the frequency unit is cycles/second. Given a window length *n* and a
      sample spacing *d*::
  
          f = [0, 1, ...,     n/2-1,     n/2] / (d*n)   if n is even
          f = [0, 1, ..., (n-1)/2-1, (n-1)/2] / (d*n)   if n is odd
  
      Unlike :func:`fftfreq` the Nyquist frequency component is considered to be positive.
  
      Parameters
      ----------
      n : int
          Window length.
      d : scalar, optional
          Sample spacing (inverse of the sampling rate). Defaults to 1.
  
      Returns
      -------
      f : ndarray
          Array of length ``n//2 + 1`` containing the sample frequencies.
  
      Examples
      --------
      >>> import nlcpy as vp
      >>> signal = vp.array([-2, 8, 6, 4, 1, 0, 3, 5, -3, 4], dtype=float)
      >>> fourier = vp.fft.rfft(signal)
      >>> n = signal.size
      >>> sample_rate = 100
      >>> freq = vp.fft.fftfreq(n, d=1./sample_rate)
      >>> freq
      array([  0.,  10.,  20.,  30.,  40., -50., -40., -30., -20., -10.])
      >>> freq = vp.fft.rfftfreq(n, d=1./sample_rate)
      >>> freq
      array([ 0., 10., 20., 30., 40., 50.])
  
      """
      return _fft.rfftfreq(n, d)
  
  
  def fftshift(x, axes=None):
      """Shifts the zero-frequency component to the center of the spectrum.
  
      This function swaps half-spaces for all axes listed (defaults to all). Note that
      ``y[0]`` is the Nyquist component only if ``len(x)`` is even.
  
      Parameters
      ----------
      x : array_like
          Input array.
      axes : int or shape tuple, optional
          Axes over which to shift. Default is None, which shifts all axes.
  
      Returns
      -------
      y : ndarray
          The shifted array.
  
      See Also
      --------
      ifftshift : The inverse of fftshift.
  
      Examples
      --------
      >>> import nlcpy as vp
      >>> freqs = vp.fft.fftfreq(10, 0.1)
      >>> freqs
      array([ 0.,  1.,  2.,  3.,  4., -5., -4., -3., -2., -1.])
      >>> vp.fft.fftshift(freqs)
      array([-5., -4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.])
  
      Shift the zero-frequency component only along the second axis:
  
      >>> freqs = vp.fft.fftfreq(9, d=1./9).reshape(3, 3)
      >>> freqs
      array([[ 0.,  1.,  2.],
             [ 3.,  4., -4.],
             [-3., -2., -1.]])
      >>> vp.fft.fftshift(freqs, axes=(1,))
      array([[-3., -2., -1.],
             [ 0.,  1.,  2.],
             [ 3.,  4., -4.]])
      """
      return _fft.fftshift(x, axes)
  
  
  def ifftshift(x, axes=None):
      """The inverse of fftshift. Although identical for even-length *x*, the functions
      differ by one sample for odd-length *x*.
  
      Parameters
      ----------
      x : array_like
          Input array.
      axes : int or shape tuple, optional
          Axes over which to calculate. Defaults to None, which shifts all axes.
  
      Returns
      -------
      y : ndarray
          The shifted array.
  
      See Also
      --------
      fftshift : Shifts the zero-frequency component to the center of the spectrum.
  
      Examples
      --------
      >>> import nlcpy as vp
      >>> freqs = vp.fft.fftfreq(9, d=1./9).reshape(3, 3)
      >>> freqs
      array([[ 0.,  1.,  2.],
             [ 3.,  4., -4.],
             [-3., -2., -1.]])
      >>> vp.fft.ifftshift(vp.fft.fftshift(freqs))
      array([[ 0.,  1.,  2.],
             [ 3.,  4., -4.],
             [-3., -2., -1.]])
  
      """
      return _fft.ifftshift(x, axes)
  