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from scipy._lib.uarray import generate_multimethod, Dispatchable 

import numpy as np 

 

 

def _x_replacer(args, kwargs, dispatchables): 

""" 

uarray argument replacer to replace the transform input array (``x``) 

""" 

if len(args) > 0: 

return (dispatchables[0],) + args[1:], kwargs 

kw = kwargs.copy() 

kw['x'] = dispatchables[0] 

return args, kw 

 

 

def _dispatch(func): 

""" 

Function annotation that creates a uarray multimethod from the function 

""" 

return generate_multimethod(func, _x_replacer, domain="numpy.scipy.fft") 

 

 

@_dispatch 

def fft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the 1-D discrete Fourier Transform. 

 

This function computes the 1-D *n*-point discrete Fourier 

Transform (DFT) with the efficient Fast Fourier Transform (FFT) 

algorithm [1]_. 

 

Parameters 

---------- 

x : 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. Default is None, meaning no normalization on the 

forward transforms and scaling by ``1/n`` on the `ifft`. 

For ``norm="ortho"``, both directions are scaled by ``1/sqrt(n)``. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See the notes below for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. See below for more 

details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

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 `x`. 

 

See Also 

-------- 

ifft : The inverse of `fft`. 

fft2 : The 2-D FFT. 

fftn : The N-D FFT. 

rfftn : The N-D FFT of real input. 

fftfreq : Frequency bins for given FFT parameters. 

next_fast_len : Size to pad input to for most efficient transforms 

 

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. For poorly factorizable sizes, 

`scipy.fft` uses Bluestein's algorithm [2]_ and so is never worse than 

O(`n` log `n`). Further performance improvements may be seen by zero-padding 

the input using `next_fast_len`. 

 

If ``x`` is a 1d array, then the `fft` is equivalent to :: 

 

y[k] = np.sum(x * np.exp(-2j * np.pi * k * np.arange(n)/n)) 

 

The frequency term ``f=k/n`` is found at ``y[k]``. At ``y[n/2]`` we reach 

the Nyquist frequency and wrap around to the negative-frequency terms. So, 

for an 8-point transform, the frequencies of the result are 

[0, 1, 2, 3, -4, -3, -2, -1]. To rearrange the fft output so that the 

zero-frequency component is centered, like [-4, -3, -2, -1, 0, 1, 2, 3], 

use `fftshift`. 

 

Transforms can be done in single, double, or extended precision (long 

double) floating point. Half precision inputs will be converted to single 

precision and non-floating-point inputs will be converted to double 

precision. 

 

If the data type of ``x`` is real, a "real FFT" algorithm is automatically 

used, which roughly halves the computation time. To increase efficiency 

a little further, use `rfft`, which does the same calculation, but only 

outputs half of the symmetrical spectrum. If the data are both real and 

symmetrical, the `dct` can again double the efficiency, by generating 

half of the spectrum from half of the signal. 

 

When ``overwrite_x=True`` is specified, the memory referenced by ``x`` may 

be used by the implementation in any way. This may include reusing the 

memory for the result, but this is in no way guaranteed. You should not 

rely on the contents of ``x`` after the transform as this may change in 

future without warning. 

 

The ``workers`` argument specifies the maximum number of parallel jobs to 

split the FFT computation into. This will execute independent 1-D 

FFTs within ``x``. So, ``x`` must be at least 2-D and the 

non-transformed axes must be large enough to split into chunks. If ``x`` is 

too small, fewer jobs may be used than requested. 

 

References 

---------- 

.. [1] Cooley, James W., and John W. Tukey, 1965, "An algorithm for the 

machine calculation of complex Fourier series," *Math. Comput.* 

19: 297-301. 

.. [2] Bluestein, L., 1970, "A linear filtering approach to the 

computation of discrete Fourier transform". *IEEE Transactions on 

Audio and Electroacoustics.* 18 (4): 451-455. 

 

Examples 

-------- 

>>> import scipy.fft 

>>> scipy.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8)) 

array([-2.33486982e-16+1.14423775e-17j, 8.00000000e+00-1.25557246e-15j, 

2.33486982e-16+2.33486982e-16j, 0.00000000e+00+1.22464680e-16j, 

-1.14423775e-17+2.33486982e-16j, 0.00000000e+00+5.20784380e-16j, 

1.14423775e-17+1.14423775e-17j, 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: 

 

>>> from scipy.fft import fft, fftfreq, fftshift 

>>> import matplotlib.pyplot as plt 

>>> t = np.arange(256) 

>>> sp = fftshift(fft(np.sin(t))) 

>>> freq = fftshift(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() 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def ifft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the 1-D inverse discrete Fourier Transform. 

 

This function computes the inverse of the 1-D *n*-point 

discrete Fourier transform computed by `fft`. In other words, 

``ifft(fft(x)) == x`` to within numerical accuracy. 

 

The input should be ordered in the same way as is returned by `fft`, 

i.e., 

 

* ``x[0]`` should contain the zero frequency term, 

* ``x[1:n//2]`` should contain the positive-frequency terms, 

* ``x[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, ``x[n//2]`` represents the sum of 

the values at the positive and negative Nyquist frequencies, as the two 

are aliased together. See `fft` for details. 

 

Parameters 

---------- 

x : array_like 

Input array, can be complex. 

n : int, optional 

Length of the transformed axis of the output. 

If `n` is smaller than the length of the input, the input is cropped. 

If it is larger, the input is padded with zeros. If `n` is not given, 

the length of the input along the axis specified by `axis` is used. 

See notes about padding issues. 

axis : int, optional 

Axis over which to compute the inverse DFT. If not given, the last 

axis is used. 

norm : {None, "ortho"}, optional 

Normalization mode (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

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 `x`. 

 

See Also 

-------- 

fft : The 1-D (forward) FFT, of which `ifft` is the inverse. 

ifft2 : The 2-D inverse FFT. 

ifftn : The N-D 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`. 

 

If ``x`` is a 1-D array, then the `ifft` is equivalent to :: 

 

y[k] = np.sum(x * np.exp(2j * np.pi * k * np.arange(n)/n)) / len(x) 

 

As with `fft`, `ifft` has support for all floating point types and is 

optimized for real input. 

 

Examples 

-------- 

>>> import scipy.fft 

>>> scipy.fft.ifft([0, 4, 0, 0]) 

array([ 1.+0.j, 0.+1.j, -1.+0.j, 0.-1.j]) # may vary 

 

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 = scipy.fft.ifft(n) 

>>> plt.plot(t, s.real, 'b-', t, s.imag, 'r--') 

[<matplotlib.lines.Line2D object at ...>, <matplotlib.lines.Line2D object at ...>] 

>>> plt.legend(('real', 'imaginary')) 

<matplotlib.legend.Legend object at ...> 

>>> plt.show() 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def rfft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the 1-D discrete Fourier Transform for real input. 

 

This function computes the 1-D *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 `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

Returns 

------- 

out : complex ndarray 

The truncated or zero-padded input, transformed along the axis 

indicated by `axis`, or the last one if `axis` is not specified. 

If `n` is even, the length of the transformed axis is ``(n/2)+1``. 

If `n` is odd, the length is ``(n+1)/2``. 

 

Raises 

------ 

IndexError 

If `axis` is larger than the last axis of `a`. 

 

See Also 

-------- 

irfft : The inverse of `rfft`. 

fft : The 1-D FFT of general (complex) input. 

fftn : The N-D FFT. 

rfft2 : The 2-D FFT of real input. 

rfftn : The N-D 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 ``X = rfft(x)`` and fs is the sampling frequency, ``X[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 

-------- 

>>> import scipy.fft 

>>> scipy.fft.fft([0, 1, 0, 0]) 

array([ 1.+0.j, 0.-1.j, -1.+0.j, 0.+1.j]) # may vary 

>>> scipy.fft.rfft([0, 1, 0, 0]) 

array([ 1.+0.j, 0.-1.j, -1.+0.j]) # may vary 

 

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. 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def irfft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Computes the inverse of `rfft`. 

 

This function computes the inverse of the 1-D *n*-point 

discrete Fourier Transform of real input computed by `rfft`. 

In other words, ``irfft(rfft(x), len(x)) == x`` 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 

---------- 

x : 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 inverse FFT. If not given, the last 

axis is used. 

norm : {None, "ortho"}, optional 

Normalization mode (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

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 `x`. 

 

See Also 

-------- 

rfft : The 1-D FFT of real input, of which `irfft` is inverse. 

fft : The 1-D FFT. 

irfft2 : The inverse of the 2-D FFT of real input. 

irfftn : The inverse of the N-D FFT of real input. 

 

Notes 

----- 

Returns the real valued `n`-point inverse discrete Fourier transform 

of `x`, where `x` 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 default value of `n` assumes an even output length. By the Hermitian 

symmetry, the last imaginary component must be 0 and so is ignored. To 

avoid losing information, the correct length of the real input *must* be 

given. 

 

Examples 

-------- 

>>> import scipy.fft 

>>> scipy.fft.ifft([1, -1j, -1, 1j]) 

array([0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]) # may vary 

>>> scipy.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. 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def hfft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the FFT of a signal that has Hermitian symmetry, i.e., a real 

spectrum. 

 

Parameters 

---------- 

x : 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. 

norm : {None, "ortho"}, optional 

Normalization mode (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See `fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

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, 

 

Raises 

------ 

IndexError 

If `axis` is larger than the last axis of `a`. 

 

See also 

-------- 

rfft : Compute the 1-D FFT for real input. 

ihfft : The inverse of `hfft`. 

hfftn : Compute the N-D FFT of a Hermitian signal. 

 

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. 

* even: ``ihfft(hfft(a, 2*len(a) - 2) == a``, within roundoff error, 

* odd: ``ihfft(hfft(a, 2*len(a) - 1) == a``, within roundoff error. 

 

Examples 

-------- 

>>> from scipy.fft import fft, hfft 

>>> a = 2 * np.pi * np.arange(10) / 10 

>>> signal = np.cos(a) + 3j * np.sin(3 * a) 

>>> fft(signal).round(10) 

array([ -0.+0.j, 5.+0.j, -0.+0.j, 15.-0.j, 0.+0.j, 0.+0.j, 

-0.+0.j, -15.-0.j, 0.+0.j, 5.+0.j]) 

>>> hfft(signal[:6]).round(10) # Input first half of signal 

array([ 0., 5., 0., 15., -0., 0., 0., -15., -0., 5.]) 

>>> hfft(signal, 10) # Input entire signal and truncate 

array([ 0., 5., 0., 15., -0., 0., 0., -15., -0., 5.]) 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def ihfft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the inverse FFT of a signal that has Hermitian symmetry. 

 

Parameters 

---------- 

x : 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 (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See `fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

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``. 

 

See also 

-------- 

hfft, irfft 

 

Notes 

----- 

`hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the 

opposite case: here, the signal has Hermitian symmetry in the time 

domain and is real in the frequency domain. So, here, it's `hfft`, for 

which you must supply the length of the result if it is to be odd: 

* even: ``ihfft(hfft(a, 2*len(a) - 2) == a``, within roundoff error, 

* odd: ``ihfft(hfft(a, 2*len(a) - 1) == a``, within roundoff error. 

 

Examples 

-------- 

>>> from scipy.fft import ifft, ihfft 

>>> spectrum = np.array([ 15, -4, 0, -1, 0, -4]) 

>>> ifft(spectrum) 

array([1.+0.j, 2.+0.j, 3.+0.j, 4.+0.j, 3.+0.j, 2.+0.j]) # may vary 

>>> ihfft(spectrum) 

array([ 1.-0.j, 2.-0.j, 3.-0.j, 4.-0.j]) # may vary 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def fftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the N-D discrete Fourier Transform. 

 

This function computes the N-D discrete Fourier Transform over 

any number of axes in an M-D array by means of the Fast Fourier 

Transform (FFT). 

 

Parameters 

---------- 

x : 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. 

norm : {None, "ortho"}, optional 

Normalization mode (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

Returns 

------- 

out : complex ndarray 

The truncated or zero-padded input, transformed along the axes 

indicated by `axes`, or by a combination of `s` and `x`, 

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 `x`. 

 

See Also 

-------- 

ifftn : The inverse of `fftn`, the inverse N-D FFT. 

fft : The 1-D FFT, with definitions and conventions used. 

rfftn : The N-D FFT of real input. 

fft2 : The 2-D 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. 

 

Examples 

-------- 

>>> import scipy.fft 

>>> x = np.mgrid[:3, :3, :3][0] 

>>> scipy.fft.fftn(x, axes=(1, 2)) 

array([[[ 0.+0.j, 0.+0.j, 0.+0.j], # may vary 

[ 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]]]) 

>>> scipy.fft.fftn(x, (2, 2), axes=(0, 1)) 

array([[[ 2.+0.j, 2.+0.j, 2.+0.j], # may vary 

[ 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 = scipy.fft.fftn(S) 

>>> plt.imshow(np.log(np.abs(scipy.fft.fftshift(FS))**2)) 

<matplotlib.image.AxesImage object at 0x...> 

>>> plt.show() 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def ifftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the N-D inverse discrete Fourier Transform. 

 

This function computes the inverse of the N-D discrete 

Fourier Transform over any number of axes in an M-D array by 

means of the Fast Fourier Transform (FFT). In other words, 

``ifftn(fftn(x)) == x`` to within numerical accuracy. 

 

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 

---------- 

x : 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. 

norm : {None, "ortho"}, optional 

Normalization mode (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

Returns 

------- 

out : complex ndarray 

The truncated or zero-padded input, transformed along the axes 

indicated by `axes`, or by a combination of `s` or `x`, 

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 `x`. 

 

See Also 

-------- 

fftn : The forward N-D FFT, of which `ifftn` is the inverse. 

ifft : The 1-D inverse FFT. 

ifft2 : The 2-D inverse FFT. 

ifftshift : Undoes `fftshift`, shifts zero-frequency terms to beginning 

of array. 

 

Notes 

----- 

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 

-------- 

>>> import scipy.fft 

>>> x = np.eye(4) 

>>> scipy.fft.ifftn(scipy.fft.fftn(x, axes=(0,)), axes=(1,)) 

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 = np.zeros((200,200), dtype=complex) 

>>> n[60:80, 20:40] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20, 20))) 

>>> im = scipy.fft.ifftn(n).real 

>>> plt.imshow(im) 

<matplotlib.image.AxesImage object at 0x...> 

>>> plt.show() 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def fft2(x, s=None, axes=(-2, -1), norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the 2-D discrete Fourier Transform 

 

This function computes the N-D discrete Fourier Transform 

over any axes in an M-D 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 

---------- 

x : 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. 

norm : {None, "ortho"}, optional 

Normalization mode (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

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 `x`. 

 

See Also 

-------- 

ifft2 : The inverse 2-D FFT. 

fft : The 1-D FFT. 

fftn : The N-D FFT. 

fftshift : Shifts zero-frequency terms to the center of the array. 

For 2-D input, swaps first and third quadrants, and second 

and fourth quadrants. 

 

Notes 

----- 

`fft2` is just `fftn` with a different default for `axes`. 

 

The output, analogously to `fft`, contains the term for zero frequency in 

the low-order corner of the transformed axes, the positive frequency terms 

in the first half of these axes, the term for the Nyquist frequency in the 

middle of the axes and the negative frequency terms in the second half of 

the axes, in order of decreasingly negative frequency. 

 

See `fftn` for details and a plotting example, and `fft` for 

definitions and conventions used. 

 

 

Examples 

-------- 

>>> import scipy.fft 

>>> x = np.mgrid[:5, :5][0] 

>>> scipy.fft.fft2(x) 

array([[ 50. +0.j , 0. +0.j , 0. +0.j , # may vary 

0. +0.j , 0. +0.j ], 

[-12.5+17.20477401j, 0. +0.j , 0. +0.j , 

0. +0.j , 0. +0.j ], 

[-12.5 +4.0614962j , 0. +0.j , 0. +0.j , 

0. +0.j , 0. +0.j ], 

[-12.5 -4.0614962j , 0. +0.j , 0. +0.j , 

0. +0.j , 0. +0.j ], 

[-12.5-17.20477401j, 0. +0.j , 0. +0.j , 

0. +0.j , 0. +0.j ]]) 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def ifft2(x, s=None, axes=(-2, -1), norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the 2-D inverse discrete Fourier Transform. 

 

This function computes the inverse of the 2-D discrete Fourier 

Transform over any number of axes in an M-D array by means of 

the Fast Fourier Transform (FFT). In other words, ``ifft2(fft2(x)) == x`` 

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 

---------- 

x : 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. 

norm : {None, "ortho"}, optional 

Normalization mode (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

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 `x`. 

 

See Also 

-------- 

fft2 : The forward 2-D FFT, of which `ifft2` is the inverse. 

ifftn : The inverse of the N-D FFT. 

fft : The 1-D FFT. 

ifft : The 1-D inverse FFT. 

 

Notes 

----- 

`ifft2` is just `ifftn` with a different default for `axes`. 

 

See `ifftn` for details and a plotting example, and `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 

-------- 

>>> import scipy.fft 

>>> x = 4 * np.eye(4) 

>>> scipy.fft.ifft2(x) 

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 (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def rfftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the N-D discrete Fourier Transform for real input. 

 

This function computes the N-D discrete Fourier Transform over 

any number of axes in an M-D 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 

---------- 

x : 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 `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

Returns 

------- 

out : complex ndarray 

The truncated or zero-padded input, transformed along the axes 

indicated by `axes`, or by a combination of `s` and `x`, 

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 `x`. 

 

See Also 

-------- 

irfftn : The inverse of `rfftn`, i.e., the inverse of the N-D FFT 

of real input. 

fft : The 1-D FFT, with definitions and conventions used. 

rfft : The 1-D FFT of real input. 

fftn : The N-D FFT. 

rfft2 : The 2-D 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 

-------- 

>>> import scipy.fft 

>>> x = np.ones((2, 2, 2)) 

>>> scipy.fft.rfftn(x) 

array([[[8.+0.j, 0.+0.j], # may vary 

[0.+0.j, 0.+0.j]], 

[[0.+0.j, 0.+0.j], 

[0.+0.j, 0.+0.j]]]) 

 

>>> scipy.fft.rfftn(x, axes=(2, 0)) 

array([[[4.+0.j, 0.+0.j], # may vary 

[4.+0.j, 0.+0.j]], 

[[0.+0.j, 0.+0.j], 

[0.+0.j, 0.+0.j]]]) 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def rfft2(x, s=None, axes=(-2, -1), norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the 2-D FFT of a real array. 

 

Parameters 

---------- 

x : 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 `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

Returns 

------- 

out : ndarray 

The result of the real 2-D FFT. 

 

See Also 

-------- 

irfft2 : The inverse of the 2-D FFT of real input. 

rfft : The 1-D FFT of real input. 

rfftn : Compute the N-D discrete Fourier Transform for real 

input. 

 

Notes 

----- 

This is really just `rfftn` with different default behavior. 

For more details see `rfftn`. 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def irfftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Computes the inverse of `rfftn` 

 

This function computes the inverse of the N-D discrete 

Fourier Transform for real input over any number of axes in an 

M-D array by means of the Fast Fourier Transform (FFT). In 

other words, ``irfftn(rfftn(x), x.shape) == x`` 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 

---------- 

x : 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. Except for the last axis which is taken to be 

``2*(m-1)``, where ``m`` is the length of the input along that axis. 

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. 

norm : {None, "ortho"}, optional 

Normalization mode (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

Returns 

------- 

out : ndarray 

The truncated or zero-padded input, transformed along the axes 

indicated by `axes`, or by a combination of `s` or `x`, 

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 `x`. 

 

See Also 

-------- 

rfftn : The forward N-D FFT of real input, 

of which `ifftn` is the inverse. 

fft : The 1-D FFT, with definitions and conventions used. 

irfft : The inverse of the 1-D FFT of real input. 

irfft2 : The inverse of the 2-D FFT of real input. 

 

Notes 

----- 

See `fft` for definitions and conventions used. 

 

See `rfft` for definitions and conventions used for real input. 

 

The default value of `s` assumes an even output length in the final 

transformation axis. When performing the final complex to real 

transformation, the Hermitian symmetry requires that the last imaginary 

component along that axis must be 0 and so it is ignored. To avoid losing 

information, the correct length of the real input *must* be given. 

 

Examples 

-------- 

>>> import scipy.fft 

>>> x = np.zeros((3, 2, 2)) 

>>> x[0, 0, 0] = 3 * 2 * 2 

>>> scipy.fft.irfftn(x) 

array([[[1., 1.], 

[1., 1.]], 

[[1., 1.], 

[1., 1.]], 

[[1., 1.], 

[1., 1.]]]) 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def irfft2(x, s=None, axes=(-2, -1), norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Computes the inverse of `rfft2` 

 

Parameters 

---------- 

x : array_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 (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

Returns 

------- 

out : ndarray 

The result of the inverse real 2-D FFT. 

 

See Also 

-------- 

rfft2 : The 2-D FFT of real input. 

irfft : The inverse of the 1-D FFT of real input. 

irfftn : The inverse of the N-D FFT of real input. 

 

Notes 

----- 

This is really `irfftn` with different defaults. 

For more details see `irfftn`. 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def hfftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the N-D FFT of Hermitian symmetric complex input, i.e., a 

signal with a real spectrum. 

 

This function computes the N-D discrete Fourier Transform for a 

Hermitian symmetric complex input over any number of axes in an 

M-D array by means of the Fast Fourier Transform (FFT). In other 

words, ``ihfftn(hfftn(x, s)) == x`` to within numerical accuracy. (``s`` 

here is ``x.shape`` with ``s[-1] = x.shape[-1] * 2 - 1``, this is necessary 

for the same reason ``x.shape`` would be necessary for `irfft`.) 

 

Parameters 

---------- 

x : 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. Except for the last axis which is taken to be 

``2*(m-1)`` where ``m`` is the length of the input along that axis. 

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. 

norm : {None, "ortho"}, optional 

Normalization mode (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

Returns 

------- 

out : ndarray 

The truncated or zero-padded input, transformed along the axes 

indicated by `axes`, or by a combination of `s` or `x`, 

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 `x`. 

 

See Also 

-------- 

ihfftn : The inverse N-D FFT with real spectrum. Inverse of `hfftn`. 

fft : The 1-D FFT, with definitions and conventions used. 

rfft : Forward FFT of real input. 

 

Notes 

----- 

 

For a 1-D signal ``x`` to have a real spectrum, it must satisfy 

the Hermitian property:: 

 

x[i] == np.conj(x[-i]) for all i 

 

This generalizes into higher dimensions by reflecting over each axis in 

turn:: 

 

x[i, j, k, ...] == np.conj(x[-i, -j, -k, ...]) for all i, j, k, ... 

 

This should not be confused with a Hermitian matrix, for which the 

transpose is its own conjugate:: 

 

x[i, j] == np.conj(x[j, i]) for all i, j 

 

 

The default value of `s` assumes an even output length in the final 

transformation axis. When performing the final complex to real 

transformation, the Hermitian symmetry requires that the last imaginary 

component along that axis must be 0 and so it is ignored. To avoid losing 

information, the correct length of the real input *must* be given. 

 

Examples 

-------- 

>>> import scipy.fft 

>>> x = np.ones((3, 2, 2)) 

>>> scipy.fft.hfftn(x) 

array([[[12., 0.], 

[ 0., 0.]], 

[[ 0., 0.], 

[ 0., 0.]], 

[[ 0., 0.], 

[ 0., 0.]]]) 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def hfft2(x, s=None, axes=(-2, -1), norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the 2-D FFT of a Hermitian complex array. 

 

Parameters 

---------- 

x : array 

Input array, taken to be Hermitian complex. 

s : sequence of ints, optional 

Shape of the real output. 

axes : sequence of ints, optional 

Axes over which to compute the FFT. 

norm : {None, "ortho"}, optional 

Normalization mode (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See `fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

Returns 

------- 

out : ndarray 

The real result of the 2-D Hermitian complex real FFT. 

 

See Also 

-------- 

hfftn : Compute the N-D discrete Fourier Transform for Hermitian 

complex input. 

 

Notes 

----- 

This is really just `hfftn` with different default behavior. 

For more details see `hfftn`. 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def ihfftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the N-D inverse discrete Fourier Transform for a real 

spectrum. 

 

This function computes the N-D inverse discrete Fourier Transform 

over any number of axes in an M-D 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 

---------- 

x : 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.). 

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 `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

Returns 

------- 

out : complex ndarray 

The truncated or zero-padded input, transformed along the axes 

indicated by `axes`, or by a combination of `s` and `x`, 

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 `x`. 

 

See Also 

-------- 

hfftn : The forward N-D FFT of Hermitian input. 

hfft : The 1-D FFT of Hermitian input. 

fft : The 1-D FFT, with definitions and conventions used. 

fftn : The N-D FFT. 

hfft2 : The 2-D FFT of Hermitian input. 

 

Notes 

----- 

 

The transform for real input is performed over the last transformation 

axis, as by `ihfft`, then the transform over the remaining axes is 

performed as by `ifftn`. The order of the output is the positive part of 

the Hermitian output signal, in the same format as `rfft`. 

 

Examples 

-------- 

>>> import scipy.fft 

>>> x = np.ones((2, 2, 2)) 

>>> scipy.fft.ihfftn(x) 

array([[[1.+0.j, 0.+0.j], # may vary 

[0.+0.j, 0.+0.j]], 

[[0.+0.j, 0.+0.j], 

[0.+0.j, 0.+0.j]]]) 

>>> scipy.fft.ihfftn(x, axes=(2, 0)) 

array([[[1.+0.j, 0.+0.j], # may vary 

[1.+0.j, 0.+0.j]], 

[[0.+0.j, 0.+0.j], 

[0.+0.j, 0.+0.j]]]) 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def ihfft2(x, s=None, axes=(-2, -1), norm=None, overwrite_x=False, workers=None, *, 

plan=None): 

""" 

Compute the 2-D inverse FFT of a real spectrum. 

 

Parameters 

---------- 

x : array_like 

The input array 

s : sequence of ints, optional 

Shape of the real input 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 (see `fft`). Default is None. 

overwrite_x : bool, optional 

If True, the contents of `x` can be destroyed; the default is False. 

See :func:`fft` for more details. 

workers : int, optional 

Maximum number of workers to use for parallel computation. If negative, 

the value wraps around from ``os.cpu_count()``. 

See :func:`~scipy.fft.fft` for more details. 

plan: object, optional 

This argument is reserved for passing in a precomputed plan provided 

by downstream FFT vendors. It is currently not used in SciPy. 

 

.. versionadded:: 1.5.0 

 

Returns 

------- 

out : ndarray 

The result of the inverse real 2-D FFT. 

 

See Also 

-------- 

ihfftn : Compute the inverse of the N-D FFT of Hermitian input. 

 

Notes 

----- 

This is really `ihfftn` with different defaults. 

For more details see `ihfftn`. 

 

""" 

return (Dispatchable(x, np.ndarray),)