Source code for numpy.fft.fftpack

"""
Discrete Fourier Transforms

Routines in this module:

fft(a, n=None, axis=-1)
ifft(a, n=None, axis=-1)
rfft(a, n=None, axis=-1)
irfft(a, n=None, axis=-1)
hfft(a, n=None, axis=-1)
ihfft(a, n=None, axis=-1)
fftn(a, s=None, axes=None)
ifftn(a, s=None, axes=None)
rfftn(a, s=None, axes=None)
irfftn(a, s=None, axes=None)
fft2(a, s=None, axes=(-2,-1))
ifft2(a, s=None, axes=(-2, -1))
rfft2(a, s=None, axes=(-2,-1))
irfft2(a, s=None, axes=(-2, -1))

i = inverse transform
r = transform of purely real data
h = Hermite transform
n = n-dimensional transform
2 = 2-dimensional transform
(Note: 2D routines are just nD routines with different default
behavior.)

The underlying code for these functions is an f2c-translated and modified
version of the FFTPACK routines.

"""
from __future__ import division, absolute_import, print_function

__all__ = ['fft', 'ifft', 'rfft', 'irfft', 'hfft', 'ihfft', 'rfftn',
'irfftn', 'rfft2', 'irfft2', 'fft2', 'ifft2', 'fftn', 'ifftn']

from numpy.core import (array, asarray, zeros, swapaxes, shape, conjugate,
take, sqrt)
from . import fftpack_lite as fftpack

_fft_cache = {}
_real_fft_cache = {}

def _raw_fft(a, n=None, axis=-1, init_function=fftpack.cffti,
work_function=fftpack.cfftf, fft_cache=_fft_cache):
a = asarray(a)

if n is None:
n = a.shape[axis]

if n < 1:
raise ValueError("Invalid number of FFT data points (%d) specified."
% n)

try:
# Thread-safety note: We rely on list.pop() here to atomically
# retrieve-and-remove a wsave from the cache.  This ensures that no
# other thread can get the same wsave while we're using it.
wsave = fft_cache.setdefault(n, []).pop()
except (IndexError):
wsave = init_function(n)

if a.shape[axis] != n:
s = list(a.shape)
if s[axis] > n:
index = [slice(None)]*len(s)
index[axis] = slice(0, n)
a = a[index]
else:
index = [slice(None)]*len(s)
index[axis] = slice(0, s[axis])
s[axis] = n
z = zeros(s, a.dtype.char)
z[index] = a
a = z

if axis != -1:
a = swapaxes(a, axis, -1)
r = work_function(a, wsave)
if axis != -1:
r = swapaxes(r, axis, -1)

# As soon as we put wsave back into the cache, another thread could pick it
# up and start using it, so we must not do this until after we're
# completely done using it ourselves.
fft_cache[n].append(wsave)

return r

def _unitary(norm):
if norm not in (None, "ortho"):
raise ValueError("Invalid norm value %s, should be None or \"ortho\"."
% norm)
return norm is not None

def fft(a, n=None, axis=-1, norm=None):
"""
Compute the one-dimensional discrete Fourier Transform.

This function computes the one-dimensional *n*-point discrete Fourier
Transform (DFT) with the efficient Fast Fourier Transform (FFT)
algorithm [CT].

Parameters
----------
a : array_like
Input array, can be complex.
n : int, optional
Length of the transformed axis of the output.
If n is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros.  If n is not given,
the length of the input along the axis specified by axis is used.
axis : int, optional
Axis over which to compute the FFT.  If not given, the last axis is
used.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axis
indicated by axis, or the last one if axis is not specified.

Raises
------
IndexError
if axes is larger than the last axis of a.

--------
numpy.fft : for definition of the DFT and conventions used.
ifft : The inverse of fft.
fft2 : The two-dimensional FFT.
fftn : The *n*-dimensional FFT.
rfftn : The *n*-dimensional FFT of real input.
fftfreq : Frequency bins for given FFT parameters.

Notes
-----
FFT (Fast Fourier Transform) refers to a way the discrete Fourier
Transform (DFT) can be calculated efficiently, by using symmetries in the
calculated terms.  The symmetry is highest when n is a power of 2, and
the transform is therefore most efficient for these sizes.

The DFT is defined, with the conventions used in this implementation, in
the documentation for the numpy.fft module.

References
----------
.. [CT] Cooley, James W., and John W. Tukey, 1965, "An algorithm for the
machine calculation of complex Fourier series," *Math. Comput.*
19: 297-301.

Examples
--------
>>> np.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8))
array([ -3.44505240e-16 +1.14383329e-17j,
8.00000000e+00 -5.71092652e-15j,
2.33482938e-16 +1.22460635e-16j,
1.64863782e-15 +1.77635684e-15j,
9.95839695e-17 +2.33482938e-16j,
0.00000000e+00 +1.66837030e-15j,
1.14383329e-17 +1.22460635e-16j,
-1.64863782e-15 +1.77635684e-15j])

>>> import matplotlib.pyplot as plt
>>> t = np.arange(256)
>>> sp = np.fft.fft(np.sin(t))
>>> freq = np.fft.fftfreq(t.shape[-1])
>>> plt.plot(freq, sp.real, freq, sp.imag)
[<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>]
>>> plt.show()

In this example, real input has an FFT which is Hermitian, i.e., symmetric
in the real part and anti-symmetric in the imaginary part, as described in
the numpy.fft documentation.

"""

a = asarray(a).astype(complex, copy=False)
if n is None:
n = a.shape[axis]
output = _raw_fft(a, n, axis, fftpack.cffti, fftpack.cfftf, _fft_cache)
if _unitary(norm):
output *= 1 / sqrt(n)
return output

def ifft(a, n=None, axis=-1, norm=None):
"""
Compute the one-dimensional inverse discrete Fourier Transform.

This function computes the inverse of the one-dimensional *n*-point
discrete Fourier transform computed by fft.  In other words,
ifft(fft(a)) == a to within numerical accuracy.
For a general description of the algorithm and definitions,
see numpy.fft.

The input should be ordered in the same way as is returned by fft,
i.e.,

* a[0] should contain the zero frequency term,
* a[1:n//2] 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. See numpy.fft for details.

Parameters
----------
a : array_like
Input array, can be complex.
n : int, optional
Length of the transformed axis of the output.
If n is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros.  If n is not given,
the length of the input along the axis specified by axis is used.
axis : int, optional
Axis over which to compute the inverse DFT.  If not given, the last
axis is used.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axis
indicated by axis, or the last one if axis is not specified.

Raises
------
IndexError
If axes is larger than the last axis of a.

--------
numpy.fft : An introduction, with definitions and general explanations.
fft : The one-dimensional (forward) FFT, of which ifft is the inverse
ifft2 : The two-dimensional inverse FFT.
ifftn : The n-dimensional inverse FFT.

Notes
-----
If the input parameter n is larger than the size of the input, the input
is padded by appending zeros at the end.  Even though this is the common
approach, it might lead to surprising results.  If a different padding is
desired, it must be performed before calling ifft.

Examples
--------
>>> np.fft.ifft([0, 4, 0, 0])
array([ 1.+0.j,  0.+1.j, -1.+0.j,  0.-1.j])

Create and plot a band-limited signal with random phases:

>>> import matplotlib.pyplot as plt
>>> t = np.arange(400)
>>> n = np.zeros((400,), dtype=complex)
>>> n[40:60] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20,)))
>>> s = np.fft.ifft(n)
>>> plt.plot(t, s.real, 'b-', t, s.imag, 'r--')
...
>>> plt.legend(('real', 'imaginary'))
...
>>> plt.show()

"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=complex)
if n is None:
n = a.shape[axis]
unitary = _unitary(norm)
output = _raw_fft(a, n, axis, fftpack.cffti, fftpack.cfftb, _fft_cache)
return output * (1 / (sqrt(n) if unitary else n))

def rfft(a, n=None, axis=-1, norm=None):
"""
Compute the one-dimensional discrete Fourier Transform for real input.

This function computes the one-dimensional *n*-point discrete Fourier
Transform (DFT) of a real-valued array by means of an efficient algorithm
called the Fast Fourier Transform (FFT).

Parameters
----------
a : array_like
Input array
n : int, optional
Number of points along transformation axis in the input to use.
If n is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros. If n is not given,
the length of the input along the axis specified by axis is used.
axis : int, optional
Axis over which to compute the FFT. If not given, the last axis is
used.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axis
indicated by axis, or the last one if axis is not specified.
If n is even, the length of the transformed axis is (n/2)+1.
If n is odd, the length is (n+1)/2.

Raises
------
IndexError
If axis is larger than the last axis of a.

--------
numpy.fft : For definition of the DFT and conventions used.
irfft : The inverse of rfft.
fft : The one-dimensional FFT of general (complex) input.
fftn : The *n*-dimensional FFT.
rfftn : The *n*-dimensional FFT of real input.

Notes
-----
When the DFT is computed for purely real input, the output is
Hermitian-symmetric, i.e. the negative frequency terms are just the complex
conjugates of the corresponding positive-frequency terms, and the
negative-frequency terms are therefore redundant.  This function does not
compute the negative frequency terms, and the length of the transformed
axis of the output is therefore n//2 + 1.

When A = rfft(a) and fs is the sampling frequency, A[0] contains
the zero-frequency term 0*fs, which is real due to Hermitian symmetry.

If n is even, A[-1] contains the term representing both positive
and negative Nyquist frequency (+fs/2 and -fs/2), and must also be purely
real. If n is odd, there is no term at fs/2; A[-1] contains
the largest positive frequency (fs/2*(n-1)/n), and is complex in the
general case.

If the input a contains an imaginary part, it is silently discarded.

Examples
--------
>>> np.fft.fft([0, 1, 0, 0])
array([ 1.+0.j,  0.-1.j, -1.+0.j,  0.+1.j])
>>> np.fft.rfft([0, 1, 0, 0])
array([ 1.+0.j,  0.-1.j, -1.+0.j])

Notice how the final element of the fft output is the complex conjugate
of the second element, for real input. For rfft, this symmetry is
exploited to compute only the non-negative frequency terms.

"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=float)
output = _raw_fft(a, n, axis, fftpack.rffti, fftpack.rfftf,
_real_fft_cache)
if _unitary(norm):
output *= 1 / sqrt(a.shape[axis])
return output

def irfft(a, n=None, axis=-1, norm=None):
"""
Compute the inverse of the n-point DFT for real input.

This function computes the inverse of the one-dimensional *n*-point
discrete Fourier Transform of real input computed by rfft.
In other words, irfft(rfft(a), len(a)) == a to within numerical
accuracy. (See Notes below for why len(a) is necessary here.)

The input is expected to be in the form returned by rfft, i.e. the
real zero-frequency term followed by the complex positive frequency terms
in order of increasing frequency.  Since the discrete Fourier Transform of
real input is Hermitian-symmetric, the negative frequency terms are taken
to be the complex conjugates of the corresponding positive frequency terms.

Parameters
----------
a : array_like
The input array.
n : int, optional
Length of the transformed axis of the output.
For n output points, n//2+1 input points are necessary.  If the
input is longer than this, it is cropped.  If it is shorter than this,
it is padded with zeros.  If n is not given, it is determined from
the length of the input along the axis specified by axis.
axis : int, optional
Axis over which to compute the inverse FFT. If not given, the last
axis is used.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : ndarray
The truncated or zero-padded input, transformed along the axis
indicated by axis, or the last one if axis is not specified.
The length of the transformed axis is n, or, if n is not given,
2*(m-1) where m is the length of the transformed axis of the
input. To get an odd number of output points, n must be specified.

Raises
------
IndexError
If axis is larger than the last axis of a.

--------
numpy.fft : For definition of the DFT and conventions used.
rfft : The one-dimensional FFT of real input, of which irfft is inverse.
fft : The one-dimensional FFT.
irfft2 : The inverse of the two-dimensional FFT of real input.
irfftn : The inverse of the *n*-dimensional FFT of real input.

Notes
-----
Returns the real valued n-point inverse discrete Fourier transform
of a, where a contains the non-negative frequency terms of a
Hermitian-symmetric sequence. n is the length of the result, not the
input.

If you specify an n such that a must be zero-padded or truncated, the
extra/removed values will be added/removed at high frequencies. One can
thus resample a series to m points via Fourier interpolation by:
a_resamp = irfft(rfft(a), m).

Examples
--------
>>> np.fft.ifft([1, -1j, -1, 1j])
array([ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j])
>>> np.fft.irfft([1, -1j, -1])
array([ 0.,  1.,  0.,  0.])

Notice how the last term in the input to the ordinary ifft is the
complex conjugate of the second term, and the output has zero imaginary
part everywhere.  When calling irfft, the negative frequencies are not
specified, and the output array is purely real.

"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=complex)
if n is None:
n = (a.shape[axis] - 1) * 2
unitary = _unitary(norm)
output = _raw_fft(a, n, axis, fftpack.rffti, fftpack.rfftb,
_real_fft_cache)
return output * (1 / (sqrt(n) if unitary else n))

def hfft(a, n=None, axis=-1, norm=None):
"""
Compute the FFT of a signal which has Hermitian symmetry (real spectrum).

Parameters
----------
a : array_like
The input array.
n : int, optional
Length of the transformed axis of the output.
For n output points, n//2+1 input points are necessary.  If the
input is longer than this, it is cropped.  If it is shorter than this,
it is padded with zeros.  If n is not given, it is determined from
the length of the input along the axis specified by axis.
axis : int, optional
Axis over which to compute the FFT. If not given, the last
axis is used.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : ndarray
The truncated or zero-padded input, transformed along the axis
indicated by axis, or the last one if axis is not specified.
The length of the transformed axis is n, or, if n is not given,
2*(m-1) where m is the length of the transformed axis of the
input. To get an odd number of output points, n must be specified.

Raises
------
IndexError
If axis is larger than the last axis of a.

--------
rfft : Compute the one-dimensional FFT for real input.
ihfft : The inverse of hfft.

Notes
-----
hfft/ihfft are a pair analogous to rfft/irfft, but for the
opposite case: here the signal has Hermitian symmetry in the time domain
and is real in the frequency domain. So here it's hfft for which
you must supply the length of the result if it is to be odd:
ihfft(hfft(a), len(a)) == a, within numerical accuracy.

Examples
--------
>>> signal = np.array([1, 2, 3, 4, 3, 2])
>>> np.fft.fft(signal)
array([ 15.+0.j,  -4.+0.j,   0.+0.j,  -1.-0.j,   0.+0.j,  -4.+0.j])
>>> np.fft.hfft(signal[:4]) # Input first half of signal
array([ 15.,  -4.,   0.,  -1.,   0.,  -4.])
>>> np.fft.hfft(signal, 6)  # Input entire signal and truncate
array([ 15.,  -4.,   0.,  -1.,   0.,  -4.])

>>> signal = np.array([[1, 1.j], [-1.j, 2]])
>>> np.conj(signal.T) - signal   # check Hermitian symmetry
array([[ 0.-0.j,  0.+0.j],
[ 0.+0.j,  0.-0.j]])
>>> freq_spectrum = np.fft.hfft(signal)
>>> freq_spectrum
array([[ 1.,  1.],
[ 2., -2.]])

"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=complex)
if n is None:
n = (a.shape[axis] - 1) * 2
unitary = _unitary(norm)
return irfft(conjugate(a), n, axis) * (sqrt(n) if unitary else n)

def ihfft(a, n=None, axis=-1, norm=None):
"""
Compute the inverse FFT of a signal which has Hermitian symmetry.

Parameters
----------
a : array_like
Input array.
n : int, optional
Length of the inverse FFT.
Number of points along transformation axis in the input to use.
If n is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros. If n is not given,
the length of the input along the axis specified by axis is used.
axis : int, optional
Axis over which to compute the inverse FFT. If not given, the last
axis is used.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axis
indicated by axis, or the last one if axis is not specified.
If n is even, the length of the transformed axis is (n/2)+1.
If n is odd, the length is (n+1)/2.

--------
hfft, irfft

Notes
-----
hfft/ihfft are a pair analogous to rfft/irfft, but for the
opposite case: here the signal has Hermitian symmetry in the time domain
and is real in the frequency domain. So here it's hfft for which
you must supply the length of the result if it is to be odd:
ihfft(hfft(a), len(a)) == a, within numerical accuracy.

Examples
--------
>>> spectrum = np.array([ 15, -4, 0, -1, 0, -4])
>>> np.fft.ifft(spectrum)
array([ 1.+0.j,  2.-0.j,  3.+0.j,  4.+0.j,  3.+0.j,  2.-0.j])
>>> np.fft.ihfft(spectrum)
array([ 1.-0.j,  2.-0.j,  3.-0.j,  4.-0.j])

"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=float)
if n is None:
n = a.shape[axis]
unitary = _unitary(norm)
output = conjugate(rfft(a, n, axis))
return output * (1 / (sqrt(n) if unitary else n))

def _cook_nd_args(a, s=None, axes=None, invreal=0):
if s is None:
shapeless = 1
if axes is None:
s = list(a.shape)
else:
s = take(a.shape, axes)
else:
shapeless = 0
s = list(s)
if axes is None:
axes = list(range(-len(s), 0))
if len(s) != len(axes):
raise ValueError("Shape and axes have different lengths.")
if invreal and shapeless:
s[-1] = (a.shape[axes[-1]] - 1) * 2
return s, axes

def _raw_fftnd(a, s=None, axes=None, function=fft, norm=None):
a = asarray(a)
s, axes = _cook_nd_args(a, s, axes)
itl = list(range(len(axes)))
itl.reverse()
for ii in itl:
a = function(a, n=s[ii], axis=axes[ii], norm=norm)
return a

[docs]def fftn(a, s=None, axes=None, norm=None):
"""
Compute the N-dimensional discrete Fourier Transform.

This function computes the *N*-dimensional discrete Fourier Transform over
any number of axes in an *M*-dimensional array by means of the Fast Fourier
Transform (FFT).

Parameters
----------
a : array_like
Input array, can be complex.
s : sequence of ints, optional
Shape (length of each transformed axis) of the output
(s[0] refers to axis 0, s[1] to axis 1, etc.).
This corresponds to n for fft(x, n).
Along any axis, if the given shape is smaller than that of the input,
the input is cropped.  If it is larger, the input is padded with zeros.
if s is not given, the shape of the input along the axes specified
by axes is used.
axes : sequence of ints, optional
Axes over which to compute the FFT.  If not given, the last len(s)
axes are used, or all axes if s is also not specified.
Repeated indices in axes means that the transform over that axis is
performed multiple times.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axes
indicated by axes, or by a combination of s and a,
as explained in the parameters section above.

Raises
------
ValueError
If s and axes have different length.
IndexError
If an element of axes is larger than than the number of axes of a.

--------
numpy.fft : Overall view of discrete Fourier transforms, with definitions
and conventions used.
ifftn : The inverse of fftn, the inverse *n*-dimensional FFT.
fft : The one-dimensional FFT, with definitions and conventions used.
rfftn : The *n*-dimensional FFT of real input.
fft2 : The two-dimensional FFT.
fftshift : Shifts zero-frequency terms to centre of array

Notes
-----
The output, analogously to fft, contains the term for zero frequency in
the low-order corner of all axes, the positive frequency terms in the
first half of all axes, the term for the Nyquist frequency in the middle
of all axes and the negative frequency terms in the second half of all
axes, in order of decreasingly negative frequency.

See numpy.fft for details, definitions and conventions used.

Examples
--------
>>> a = np.mgrid[:3, :3, :3][0]
>>> np.fft.fftn(a, axes=(1, 2))
array([[[  0.+0.j,   0.+0.j,   0.+0.j],
[  0.+0.j,   0.+0.j,   0.+0.j],
[  0.+0.j,   0.+0.j,   0.+0.j]],
[[  9.+0.j,   0.+0.j,   0.+0.j],
[  0.+0.j,   0.+0.j,   0.+0.j],
[  0.+0.j,   0.+0.j,   0.+0.j]],
[[ 18.+0.j,   0.+0.j,   0.+0.j],
[  0.+0.j,   0.+0.j,   0.+0.j],
[  0.+0.j,   0.+0.j,   0.+0.j]]])
>>> np.fft.fftn(a, (2, 2), axes=(0, 1))
array([[[ 2.+0.j,  2.+0.j,  2.+0.j],
[ 0.+0.j,  0.+0.j,  0.+0.j]],
[[-2.+0.j, -2.+0.j, -2.+0.j],
[ 0.+0.j,  0.+0.j,  0.+0.j]]])

>>> import matplotlib.pyplot as plt
>>> [X, Y] = np.meshgrid(2 * np.pi * np.arange(200) / 12,
...                      2 * np.pi * np.arange(200) / 34)
>>> S = np.sin(X) + np.cos(Y) + np.random.uniform(0, 1, X.shape)
>>> FS = np.fft.fftn(S)
>>> plt.imshow(np.log(np.abs(np.fft.fftshift(FS))**2))
<matplotlib.image.AxesImage object at 0x...>
>>> plt.show()

"""

return _raw_fftnd(a, s, axes, fft, norm)

[docs]def ifftn(a, s=None, axes=None, norm=None):
"""
Compute the N-dimensional inverse discrete Fourier Transform.

This function computes the inverse of the N-dimensional discrete
Fourier Transform over any number of axes in an M-dimensional array by
means of the Fast Fourier Transform (FFT).  In other words,
ifftn(fftn(a)) == a to within numerical accuracy.
For a description of the definitions and conventions used, see numpy.fft.

The input, analogously to ifft, should be ordered in the same way as is
returned by fftn, i.e. it should have the term for zero frequency
in all axes in the low-order corner, the positive frequency terms in the
first half of all axes, the term for the Nyquist frequency in the middle
of all axes and the negative frequency terms in the second half of all
axes, in order of decreasingly negative frequency.

Parameters
----------
a : array_like
Input array, can be complex.
s : sequence of ints, optional
Shape (length of each transformed axis) of the output
(s[0] refers to axis 0, s[1] to axis 1, etc.).
This corresponds to n for ifft(x, n).
Along any axis, if the given shape is smaller than that of the input,
the input is cropped.  If it is larger, the input is padded with zeros.
if s is not given, the shape of the input along the axes specified
by axes is used.  See notes for issue on ifft zero padding.
axes : sequence of ints, optional
Axes over which to compute the IFFT.  If not given, the last len(s)
axes are used, or all axes if s is also not specified.
Repeated indices in axes means that the inverse transform over that
axis is performed multiple times.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axes
indicated by axes, or by a combination of s or a,
as explained in the parameters section above.

Raises
------
ValueError
If s and axes have different length.
IndexError
If an element of axes is larger than than the number of axes of a.

--------
numpy.fft : Overall view of discrete Fourier transforms, with definitions
and conventions used.
fftn : The forward *n*-dimensional FFT, of which ifftn is the inverse.
ifft : The one-dimensional inverse FFT.
ifft2 : The two-dimensional inverse FFT.
ifftshift : Undoes fftshift, shifts zero-frequency terms to beginning
of array.

Notes
-----
See numpy.fft for definitions and conventions used.

Zero-padding, analogously with ifft, is performed by appending zeros to
the input along the specified dimension.  Although this is the common
approach, it might lead to surprising results.  If another form of zero
padding is desired, it must be performed before ifftn is called.

Examples
--------
>>> a = np.eye(4)
>>> np.fft.ifftn(np.fft.fftn(a, axes=(0,)), axes=(1,))
array([[ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],
[ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],
[ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],
[ 0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j]])

Create and plot an image with band-limited frequency content:

>>> import matplotlib.pyplot as plt
>>> n = np.zeros((200,200), dtype=complex)
>>> n[60:80, 20:40] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20, 20)))
>>> im = np.fft.ifftn(n).real
>>> plt.imshow(im)
<matplotlib.image.AxesImage object at 0x...>
>>> plt.show()

"""

return _raw_fftnd(a, s, axes, ifft, norm)

def fft2(a, s=None, axes=(-2, -1), norm=None):
"""
Compute the 2-dimensional discrete Fourier Transform

This function computes the *n*-dimensional discrete Fourier Transform
over any axes in an *M*-dimensional array by means of the
Fast Fourier Transform (FFT).  By default, the transform is computed over
the last two axes of the input array, i.e., a 2-dimensional FFT.

Parameters
----------
a : array_like
Input array, can be complex
s : sequence of ints, optional
Shape (length of each transformed axis) of the output
(s[0] refers to axis 0, s[1] to axis 1, etc.).
This corresponds to n for fft(x, n).
Along each axis, if the given shape is smaller than that of the input,
the input is cropped.  If it is larger, the input is padded with zeros.
if s is not given, the shape of the input along the axes specified
by axes is used.
axes : sequence of ints, optional
Axes over which to compute the FFT.  If not given, the last two
axes are used.  A repeated index in axes means the transform over
that axis is performed multiple times.  A one-element sequence means
that a one-dimensional FFT is performed.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axes
indicated by axes, or the last two axes if axes is not given.

Raises
------
ValueError
If s and axes have different length, or axes not given and
len(s) != 2.
IndexError
If an element of axes is larger than than the number of axes of a.

--------
numpy.fft : Overall view of discrete Fourier transforms, with definitions
and conventions used.
ifft2 : The inverse two-dimensional FFT.
fft : The one-dimensional FFT.
fftn : The *n*-dimensional FFT.
fftshift : Shifts zero-frequency terms to the center of the array.
For two-dimensional input, swaps first and third quadrants, and second

Notes
-----
fft2 is just fftn with a different default for axes.

The output, analogously to fft, contains the term for zero frequency in
the low-order corner of the transformed axes, the positive frequency terms
in the first half of these axes, the term for the Nyquist frequency in the
middle of the axes and the negative frequency terms in the second half of
the axes, in order of decreasingly negative frequency.

See fftn for details and a plotting example, and numpy.fft for
definitions and conventions used.

Examples
--------
>>> a = np.mgrid[:5, :5][0]
>>> np.fft.fft2(a)
array([[ 50.0 +0.j        ,   0.0 +0.j        ,   0.0 +0.j        ,
0.0 +0.j        ,   0.0 +0.j        ],
[-12.5+17.20477401j,   0.0 +0.j        ,   0.0 +0.j        ,
0.0 +0.j        ,   0.0 +0.j        ],
[-12.5 +4.0614962j ,   0.0 +0.j        ,   0.0 +0.j        ,
0.0 +0.j        ,   0.0 +0.j        ],
[-12.5 -4.0614962j ,   0.0 +0.j        ,   0.0 +0.j        ,
0.0 +0.j        ,   0.0 +0.j        ],
[-12.5-17.20477401j,   0.0 +0.j        ,   0.0 +0.j        ,
0.0 +0.j        ,   0.0 +0.j        ]])

"""

return _raw_fftnd(a, s, axes, fft, norm)

def ifft2(a, s=None, axes=(-2, -1), norm=None):
"""
Compute the 2-dimensional inverse discrete Fourier Transform.

This function computes the inverse of the 2-dimensional discrete Fourier
Transform over any number of axes in an M-dimensional array by means of
the Fast Fourier Transform (FFT).  In other words, ifft2(fft2(a)) == a
to within numerical accuracy.  By default, the inverse transform is
computed over the last two axes of the input array.

The input, analogously to ifft, should be ordered in the same way as is
returned by fft2, i.e. it should have the term for zero frequency
in the low-order corner of the two axes, the positive frequency terms in
the first half of these axes, the term for the Nyquist frequency in the
middle of the axes and the negative frequency terms in the second half of
both axes, in order of decreasingly negative frequency.

Parameters
----------
a : array_like
Input array, can be complex.
s : sequence of ints, optional
Shape (length of each axis) of the output (s[0] refers to axis 0,
s[1] to axis 1, etc.).  This corresponds to n for ifft(x, n).
Along each axis, if the given shape is smaller than that of the input,
the input is cropped.  If it is larger, the input is padded with zeros.
if s is not given, the shape of the input along the axes specified
by axes is used.  See notes for issue on ifft zero padding.
axes : sequence of ints, optional
Axes over which to compute the FFT.  If not given, the last two
axes are used.  A repeated index in axes means the transform over
that axis is performed multiple times.  A one-element sequence means
that a one-dimensional FFT is performed.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axes
indicated by axes, or the last two axes if axes is not given.

Raises
------
ValueError
If s and axes have different length, or axes not given and
len(s) != 2.
IndexError
If an element of axes is larger than than the number of axes of a.

--------
numpy.fft : Overall view of discrete Fourier transforms, with definitions
and conventions used.
fft2 : The forward 2-dimensional FFT, of which ifft2 is the inverse.
ifftn : The inverse of the *n*-dimensional FFT.
fft : The one-dimensional FFT.
ifft : The one-dimensional inverse FFT.

Notes
-----
ifft2 is just ifftn with a different default for axes.

See ifftn for details and a plotting example, and numpy.fft for
definition and conventions used.

Zero-padding, analogously with ifft, is performed by appending zeros to
the input along the specified dimension.  Although this is the common
approach, it might lead to surprising results.  If another form of zero
padding is desired, it must be performed before ifft2 is called.

Examples
--------
>>> a = 4 * np.eye(4)
>>> np.fft.ifft2(a)
array([[ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],
[ 0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j],
[ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],
[ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j]])

"""

return _raw_fftnd(a, s, axes, ifft, norm)

[docs]def rfftn(a, s=None, axes=None, norm=None):
"""
Compute the N-dimensional discrete Fourier Transform for real input.

This function computes the N-dimensional discrete Fourier Transform over
any number of axes in an M-dimensional real array by means of the Fast
Fourier Transform (FFT).  By default, all axes are transformed, with the
real transform performed over the last axis, while the remaining
transforms are complex.

Parameters
----------
a : array_like
Input array, taken to be real.
s : sequence of ints, optional
Shape (length along each transformed axis) to use from the input.
(s[0] refers to axis 0, s[1] to axis 1, etc.).
The final element of s corresponds to n for rfft(x, n), while
for the remaining axes, it corresponds to n for fft(x, n).
Along any axis, if the given shape is smaller than that of the input,
the input is cropped.  If it is larger, the input is padded with zeros.
if s is not given, the shape of the input along the axes specified
by axes is used.
axes : sequence of ints, optional
Axes over which to compute the FFT.  If not given, the last len(s)
axes are used, or all axes if s is also not specified.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axes
indicated by axes, or by a combination of s and a,
as explained in the parameters section above.
The length of the last axis transformed will be s[-1]//2+1,
while the remaining transformed axes will have lengths according to
s, or unchanged from the input.

Raises
------
ValueError
If s and axes have different length.
IndexError
If an element of axes is larger than than the number of axes of a.

--------
irfftn : The inverse of rfftn, i.e. the inverse of the n-dimensional FFT
of real input.
fft : The one-dimensional FFT, with definitions and conventions used.
rfft : The one-dimensional FFT of real input.
fftn : The n-dimensional FFT.
rfft2 : The two-dimensional FFT of real input.

Notes
-----
The transform for real input is performed over the last transformation
axis, as by rfft, then the transform over the remaining axes is
performed as by fftn.  The order of the output is as for rfft for the
final transformation axis, and as for fftn for the remaining
transformation axes.

See fft for details, definitions and conventions used.

Examples
--------
>>> a = np.ones((2, 2, 2))
>>> np.fft.rfftn(a)
array([[[ 8.+0.j,  0.+0.j],
[ 0.+0.j,  0.+0.j]],
[[ 0.+0.j,  0.+0.j],
[ 0.+0.j,  0.+0.j]]])

>>> np.fft.rfftn(a, axes=(2, 0))
array([[[ 4.+0.j,  0.+0.j],
[ 4.+0.j,  0.+0.j]],
[[ 0.+0.j,  0.+0.j],
[ 0.+0.j,  0.+0.j]]])

"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=float)
s, axes = _cook_nd_args(a, s, axes)
a = rfft(a, s[-1], axes[-1], norm)
for ii in range(len(axes)-1):
a = fft(a, s[ii], axes[ii], norm)
return a

def rfft2(a, s=None, axes=(-2, -1), norm=None):
"""
Compute the 2-dimensional FFT of a real array.

Parameters
----------
a : array
Input array, taken to be real.
s : sequence of ints, optional
Shape of the FFT.
axes : sequence of ints, optional
Axes over which to compute the FFT.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : ndarray
The result of the real 2-D FFT.

--------
rfftn : Compute the N-dimensional discrete Fourier Transform for real
input.

Notes
-----
This is really just rfftn with different default behavior.
For more details see rfftn.

"""

return rfftn(a, s, axes, norm)

[docs]def irfftn(a, s=None, axes=None, norm=None):
"""
Compute the inverse of the N-dimensional FFT of real input.

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

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

Parameters
----------
a : array_like
Input array.
s : sequence of ints, optional
Shape (length of each transformed axis) of the output
(s[0] refers to axis 0, s[1] to axis 1, etc.). s is also the
number of input points used along this axis, except for the last axis,
where s[-1]//2+1 points of the input are used.
Along any axis, if the shape indicated by s is smaller than that of
the input, the input is cropped.  If it is larger, the input is padded
with zeros. If s is not given, the shape of the input along the
axes specified by axes is used.
axes : sequence of ints, optional
Axes over which to compute the inverse FFT. If not given, the last
len(s) axes are used, or all axes if s is also not specified.
Repeated indices in axes means that the inverse transform over that
axis is performed multiple times.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

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

Raises
------
ValueError
If s and axes have different length.
IndexError
If an element of axes is larger than than the number of axes of a.

--------
rfftn : The forward n-dimensional FFT of real input,
of which ifftn is the inverse.
fft : The one-dimensional FFT, with definitions and conventions used.
irfft : The inverse of the one-dimensional FFT of real input.
irfft2 : The inverse of the two-dimensional FFT of real input.

Notes
-----
See fft for definitions and conventions used.

See rfft for definitions and conventions used for real input.

Examples
--------
>>> a = np.zeros((3, 2, 2))
>>> a[0, 0, 0] = 3 * 2 * 2
>>> np.fft.irfftn(a)
array([[[ 1.,  1.],
[ 1.,  1.]],
[[ 1.,  1.],
[ 1.,  1.]],
[[ 1.,  1.],
[ 1.,  1.]]])

"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=complex)
s, axes = _cook_nd_args(a, s, axes, invreal=1)
for ii in range(len(axes)-1):
a = ifft(a, s[ii], axes[ii], norm)
a = irfft(a, s[-1], axes[-1], norm)
return a

def irfft2(a, s=None, axes=(-2, -1), norm=None):
"""
Compute the 2-dimensional inverse FFT of a real array.

Parameters
----------
a : array_like
The input array
s : sequence of ints, optional
Shape of the inverse FFT.
axes : sequence of ints, optional
The axes over which to compute the inverse fft.
Default is the last two axes.
norm : {None, "ortho"}, optional
Normalization mode (see numpy.fft). Default is None.

Returns
-------
out : ndarray
The result of the inverse real 2-D FFT.

This is really irfftn with different defaults.
For more details see irfftn.