# nlcpy.fft.fft のソースコード

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from nlcpy.fft import _fft

[ドキュメント]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.

--------
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)

[ドキュメント]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 : 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.

--------
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)

[ドキュメント]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.

--------
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)

[ドキュメント]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.

--------
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)

[ドキュメント]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.

--------
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)

[ドキュメント]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.

--------
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)

[ドキュメント]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.

--------
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)

[ドキュメント]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.

--------
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)

[ドキュメント]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.

--------
rfftn : Computes the n-dimensional discrete fourier transform for a real array.

"""
return _fft.rfft2(a, s, axes, norm)

[ドキュメント]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.

--------
irfftn : Computes the inverse of the n-dimensional FFT of a real array.

"""
return _fft.irfft2(a, s, axes, norm)

[ドキュメント]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.

--------
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)

[ドキュメント]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.

--------
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)

[ドキュメント]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.

--------
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)

[ドキュメント]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.

--------
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)

[ドキュメント]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)

[ドキュメント]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)

[ドキュメント]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.

--------
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)

[ドキュメント]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.

--------
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)