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from functools import update_wrapper, lru_cache
import inspect
from ._pocketfft import helper as _helper
import numpy as np
from scipy._lib._array_api import array_namespace
def next_fast_len(target, real=False):
"""Find the next fast size of input data to ``fft``, for zero-padding, etc.
SciPy's FFT algorithms gain their speed by a recursive divide and conquer
strategy. This relies on efficient functions for small prime factors of the
input length. Thus, the transforms are fastest when using composites of the
prime factors handled by the fft implementation. If there are efficient
functions for all radices <= `n`, then the result will be a number `x`
>= ``target`` with only prime factors < `n`. (Also known as `n`-smooth
numbers)
Parameters
----------
target : int
Length to start searching from. Must be a positive integer.
real : bool, optional
True if the FFT involves real input or output (e.g., `rfft` or `hfft`
but not `fft`). Defaults to False.
Returns
-------
out : int
The smallest fast length greater than or equal to ``target``.
Notes
-----
The result of this function may change in future as performance
considerations change, for example, if new prime factors are added.
Calling `fft` or `ifft` with real input data performs an ``'R2C'``
transform internally.
Examples
--------
On a particular machine, an FFT of prime length takes 11.4 ms:
>>> from scipy import fft
>>> import numpy as np
>>> rng = np.random.default_rng()
>>> min_len = 93059 # prime length is worst case for speed
>>> a = rng.standard_normal(min_len)
>>> b = fft.fft(a)
Zero-padding to the next regular length reduces computation time to
1.6 ms, a speedup of 7.3 times:
>>> fft.next_fast_len(min_len, real=True)
93312
>>> b = fft.fft(a, 93312)
Rounding up to the next power of 2 is not optimal, taking 3.0 ms to
compute; 1.9 times longer than the size given by ``next_fast_len``:
>>> b = fft.fft(a, 131072)
"""
pass
# Directly wrap the c-function good_size but take the docstring etc., from the
# next_fast_len function above
_sig = inspect.signature(next_fast_len)
next_fast_len = update_wrapper(lru_cache(_helper.good_size), next_fast_len)
next_fast_len.__wrapped__ = _helper.good_size
next_fast_len.__signature__ = _sig
def prev_fast_len(target, real=False):
"""Find the previous fast size of input data to ``fft``.
Useful for discarding a minimal number of samples before FFT.
SciPy's FFT algorithms gain their speed by a recursive divide and conquer
strategy. This relies on efficient functions for small prime factors of the
input length. Thus, the transforms are fastest when using composites of the
prime factors handled by the fft implementation. If there are efficient
functions for all radices <= `n`, then the result will be a number `x`
<= ``target`` with only prime factors <= `n`. (Also known as `n`-smooth
numbers)
Parameters
----------
target : int
Maximum length to search until. Must be a positive integer.
real : bool, optional
True if the FFT involves real input or output (e.g., `rfft` or `hfft`
but not `fft`). Defaults to False.
Returns
-------
out : int
The largest fast length less than or equal to ``target``.
Notes
-----
The result of this function may change in future as performance
considerations change, for example, if new prime factors are added.
Calling `fft` or `ifft` with real input data performs an ``'R2C'``
transform internally.
In the current implementation, prev_fast_len assumes radices of
2,3,5,7,11 for complex FFT and 2,3,5 for real FFT.
Examples
--------
On a particular machine, an FFT of prime length takes 16.2 ms:
>>> from scipy import fft
>>> import numpy as np
>>> rng = np.random.default_rng()
>>> max_len = 93059 # prime length is worst case for speed
>>> a = rng.standard_normal(max_len)
>>> b = fft.fft(a)
Performing FFT on the maximum fast length less than max_len
reduces the computation time to 1.5 ms, a speedup of 10.5 times:
>>> fft.prev_fast_len(max_len, real=True)
92160
>>> c = fft.fft(a[:92160]) # discard last 899 samples
"""
pass
# Directly wrap the c-function prev_good_size but take the docstring etc.,
# from the prev_fast_len function above
_sig_prev_fast_len = inspect.signature(prev_fast_len)
prev_fast_len = update_wrapper(lru_cache()(_helper.prev_good_size), prev_fast_len)
prev_fast_len.__wrapped__ = _helper.prev_good_size
prev_fast_len.__signature__ = _sig_prev_fast_len
def _init_nd_shape_and_axes(x, shape, axes):
"""Handle shape and axes arguments for N-D transforms.
Returns the shape and axes in a standard form, taking into account negative
values and checking for various potential errors.
Parameters
----------
x : array_like
The input array.
shape : int or array_like of ints or None
The shape of the result. If both `shape` and `axes` (see below) are
None, `shape` is ``x.shape``; if `shape` is None but `axes` is
not None, then `shape` is ``numpy.take(x.shape, axes, axis=0)``.
If `shape` is -1, the size of the corresponding dimension of `x` is
used.
axes : int or array_like of ints or None
Axes along which the calculation is computed.
The default is over all axes.
Negative indices are automatically converted to their positive
counterparts.
Returns
-------
shape : tuple
The shape of the result as a tuple of integers.
axes : list
Axes along which the calculation is computed, as a list of integers.
"""
x = np.asarray(x)
return _helper._init_nd_shape_and_axes(x, shape, axes)
def fftfreq(n, d=1.0, *, xp=None, device=None):
"""Return the Discrete Fourier Transform sample frequencies.
The returned float array `f` contains the frequency bin centers in cycles
per unit of the sample spacing (with zero at the start). For instance, if
the sample spacing is in seconds, then the frequency unit is cycles/second.
Given a window length `n` and a sample spacing `d`::
f = [0, 1, ..., n/2-1, -n/2, ..., -1] / (d*n) if n is even
f = [0, 1, ..., (n-1)/2, -(n-1)/2, ..., -1] / (d*n) if n is odd
Parameters
----------
n : int
Window length.
d : scalar, optional
Sample spacing (inverse of the sampling rate). Defaults to 1.
xp : array_namespace, optional
The namespace for the return array. Default is None, where NumPy is used.
device : device, optional
The device for the return array.
Only valid when `xp.fft.fftfreq` implements the device parameter.
Returns
-------
f : ndarray
Array of length `n` containing the sample frequencies.
Examples
--------
>>> import numpy as np
>>> import scipy.fft
>>> signal = np.array([-2, 8, 6, 4, 1, 0, 3, 5], dtype=float)
>>> fourier = scipy.fft.fft(signal)
>>> n = signal.size
>>> timestep = 0.1
>>> freq = scipy.fft.fftfreq(n, d=timestep)
>>> freq
array([ 0. , 1.25, 2.5 , ..., -3.75, -2.5 , -1.25])
"""
xp = np if xp is None else xp
# numpy does not yet support the `device` keyword
# `xp.__name__ != 'numpy'` should be removed when numpy is compatible
if hasattr(xp, 'fft') and xp.__name__ != 'numpy':
return xp.fft.fftfreq(n, d=d, device=device)
if device is not None:
raise ValueError('device parameter is not supported for input array type')
return np.fft.fftfreq(n, d=d)
def rfftfreq(n, d=1.0, *, xp=None, device=None):
"""Return the Discrete Fourier Transform sample frequencies
(for usage with rfft, irfft).
The returned float array `f` contains the frequency bin centers in cycles
per unit of the sample spacing (with zero at the start). For instance, if
the sample spacing is in seconds, then the frequency unit is cycles/second.
Given a window length `n` and a sample spacing `d`::
f = [0, 1, ..., n/2-1, n/2] / (d*n) if n is even
f = [0, 1, ..., (n-1)/2-1, (n-1)/2] / (d*n) if n is odd
Unlike `fftfreq` (but like `scipy.fftpack.rfftfreq`)
the Nyquist frequency component is considered to be positive.
Parameters
----------
n : int
Window length.
d : scalar, optional
Sample spacing (inverse of the sampling rate). Defaults to 1.
xp : array_namespace, optional
The namespace for the return array. Default is None, where NumPy is used.
device : device, optional
The device for the return array.
Only valid when `xp.fft.rfftfreq` implements the device parameter.
Returns
-------
f : ndarray
Array of length ``n//2 + 1`` containing the sample frequencies.
Examples
--------
>>> import numpy as np
>>> import scipy.fft
>>> signal = np.array([-2, 8, 6, 4, 1, 0, 3, 5, -3, 4], dtype=float)
>>> fourier = scipy.fft.rfft(signal)
>>> n = signal.size
>>> sample_rate = 100
>>> freq = scipy.fft.fftfreq(n, d=1./sample_rate)
>>> freq
array([ 0., 10., 20., ..., -30., -20., -10.])
>>> freq = scipy.fft.rfftfreq(n, d=1./sample_rate)
>>> freq
array([ 0., 10., 20., 30., 40., 50.])
"""
xp = np if xp is None else xp
# numpy does not yet support the `device` keyword
# `xp.__name__ != 'numpy'` should be removed when numpy is compatible
if hasattr(xp, 'fft') and xp.__name__ != 'numpy':
return xp.fft.rfftfreq(n, d=d, device=device)
if device is not None:
raise ValueError('device parameter is not supported for input array type')
return np.fft.rfftfreq(n, d=d)
def fftshift(x, axes=None):
"""Shift the zero-frequency component to the center of the spectrum.
This function swaps half-spaces for all axes listed (defaults to all).
Note that ``y[0]`` is the Nyquist component only if ``len(x)`` is even.
Parameters
----------
x : array_like
Input array.
axes : int or shape tuple, optional
Axes over which to shift. Default is None, which shifts all axes.
Returns
-------
y : ndarray
The shifted array.
See Also
--------
ifftshift : The inverse of `fftshift`.
Examples
--------
>>> import numpy as np
>>> freqs = np.fft.fftfreq(10, 0.1)
>>> freqs
array([ 0., 1., 2., ..., -3., -2., -1.])
>>> np.fft.fftshift(freqs)
array([-5., -4., -3., -2., -1., 0., 1., 2., 3., 4.])
Shift the zero-frequency component only along the second axis:
>>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3)
>>> freqs
array([[ 0., 1., 2.],
[ 3., 4., -4.],
[-3., -2., -1.]])
>>> np.fft.fftshift(freqs, axes=(1,))
array([[ 2., 0., 1.],
[-4., 3., 4.],
[-1., -3., -2.]])
"""
xp = array_namespace(x)
if hasattr(xp, 'fft'):
return xp.fft.fftshift(x, axes=axes)
x = np.asarray(x)
y = np.fft.fftshift(x, axes=axes)
return xp.asarray(y)
def ifftshift(x, axes=None):
"""The inverse of `fftshift`. Although identical for even-length `x`, the
functions differ by one sample for odd-length `x`.
Parameters
----------
x : array_like
Input array.
axes : int or shape tuple, optional
Axes over which to calculate. Defaults to None, which shifts all axes.
Returns
-------
y : ndarray
The shifted array.
See Also
--------
fftshift : Shift zero-frequency component to the center of the spectrum.
Examples
--------
>>> import numpy as np
>>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3)
>>> freqs
array([[ 0., 1., 2.],
[ 3., 4., -4.],
[-3., -2., -1.]])
>>> np.fft.ifftshift(np.fft.fftshift(freqs))
array([[ 0., 1., 2.],
[ 3., 4., -4.],
[-3., -2., -1.]])
"""
xp = array_namespace(x)
if hasattr(xp, 'fft'):
return xp.fft.ifftshift(x, axes=axes)
x = np.asarray(x)
y = np.fft.ifftshift(x, axes=axes)
return xp.asarray(y)
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