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"""
Discrete Fourier Transforms - _basic.py
"""
# Created by Pearu Peterson, August,September 2002
__all__ = ['fft','ifft','fftn','ifftn','rfft','irfft',
'fft2','ifft2']
from scipy.fft import _pocketfft
from ._helper import _good_shape
def fft(x, n=None, axis=-1, overwrite_x=False):
"""
Return discrete Fourier transform of real or complex sequence.
The returned complex array contains ``y(0), y(1),..., y(n-1)``, where
``y(j) = (x * exp(-2*pi*sqrt(-1)*j*np.arange(n)/n)).sum()``.
Parameters
----------
x : array_like
Array to Fourier transform.
n : int, optional
Length of the Fourier transform. If ``n < x.shape[axis]``, `x` is
truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
default results in ``n = x.shape[axis]``.
axis : int, optional
Axis along which the fft's are computed; the default is over the
last axis (i.e., ``axis=-1``).
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
Returns
-------
z : complex ndarray
with the elements::
[y(0),y(1),..,y(n/2),y(1-n/2),...,y(-1)] if n is even
[y(0),y(1),..,y((n-1)/2),y(-(n-1)/2),...,y(-1)] if n is odd
where::
y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k* 2*pi/n), j = 0..n-1
See Also
--------
ifft : Inverse FFT
rfft : FFT of a real sequence
Notes
-----
The packing of the result is "standard": If ``A = fft(a, n)``, then
``A[0]`` contains the zero-frequency term, ``A[1:n/2]`` contains the
positive-frequency terms, and ``A[n/2:]`` contains the negative-frequency
terms, in order of decreasingly negative frequency. 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`.
Both single and double precision routines are implemented. Half precision
inputs will be converted to single precision. Non-floating-point inputs
will be converted to double precision. Long-double precision inputs are
not supported.
This function is most efficient when `n` is a power of two, and least
efficient when `n` is prime.
Note that if ``x`` is real-valued, then ``A[j] == A[n-j].conjugate()``.
If ``x`` is real-valued and ``n`` is even, then ``A[n/2]`` is real.
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 is both real and
symmetrical, the `dct` can again double the efficiency by generating
half of the spectrum from half of the signal.
Examples
--------
>>> import numpy as np
>>> from scipy.fftpack import fft, ifft
>>> x = np.arange(5)
>>> np.allclose(fft(ifft(x)), x, atol=1e-15) # within numerical accuracy.
True
"""
return _pocketfft.fft(x, n, axis, None, overwrite_x)
def ifft(x, n=None, axis=-1, overwrite_x=False):
"""
Return discrete inverse Fourier transform of real or complex sequence.
The returned complex array contains ``y(0), y(1),..., y(n-1)``, where
``y(j) = (x * exp(2*pi*sqrt(-1)*j*np.arange(n)/n)).mean()``.
Parameters
----------
x : array_like
Transformed data to invert.
n : int, optional
Length of the inverse Fourier transform. If ``n < x.shape[axis]``,
`x` is truncated. If ``n > x.shape[axis]``, `x` is zero-padded.
The default results in ``n = x.shape[axis]``.
axis : int, optional
Axis along which the ifft's are computed; the default is over the
last axis (i.e., ``axis=-1``).
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
Returns
-------
ifft : ndarray of floats
The inverse discrete Fourier transform.
See Also
--------
fft : Forward FFT
Notes
-----
Both single and double precision routines are implemented. Half precision
inputs will be converted to single precision. Non-floating-point inputs
will be converted to double precision. Long-double precision inputs are
not supported.
This function is most efficient when `n` is a power of two, and least
efficient when `n` is prime.
If the data type of `x` is real, a "real IFFT" algorithm is automatically
used, which roughly halves the computation time.
Examples
--------
>>> from scipy.fftpack import fft, ifft
>>> import numpy as np
>>> x = np.arange(5)
>>> np.allclose(ifft(fft(x)), x, atol=1e-15) # within numerical accuracy.
True
"""
return _pocketfft.ifft(x, n, axis, None, overwrite_x)
def rfft(x, n=None, axis=-1, overwrite_x=False):
"""
Discrete Fourier transform of a real sequence.
Parameters
----------
x : array_like, real-valued
The data to transform.
n : int, optional
Defines the length of the Fourier transform. If `n` is not specified
(the default) then ``n = x.shape[axis]``. If ``n < x.shape[axis]``,
`x` is truncated, if ``n > x.shape[axis]``, `x` is zero-padded.
axis : int, optional
The axis along which the transform is applied. The default is the
last axis.
overwrite_x : bool, optional
If set to true, the contents of `x` can be overwritten. Default is
False.
Returns
-------
z : real ndarray
The returned real array contains::
[y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2))] if n is even
[y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2)),Im(y(n/2))] if n is odd
where::
y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k*2*pi/n)
j = 0..n-1
See Also
--------
fft, irfft, scipy.fft.rfft
Notes
-----
Within numerical accuracy, ``y == rfft(irfft(y))``.
Both single and double precision routines are implemented. Half precision
inputs will be converted to single precision. Non-floating-point inputs
will be converted to double precision. Long-double precision inputs are
not supported.
To get an output with a complex datatype, consider using the newer
function `scipy.fft.rfft`.
Examples
--------
>>> from scipy.fftpack import fft, rfft
>>> a = [9, -9, 1, 3]
>>> fft(a)
array([ 4. +0.j, 8.+12.j, 16. +0.j, 8.-12.j])
>>> rfft(a)
array([ 4., 8., 12., 16.])
"""
return _pocketfft.rfft_fftpack(x, n, axis, None, overwrite_x)
def irfft(x, n=None, axis=-1, overwrite_x=False):
"""
Return inverse discrete Fourier transform of real sequence x.
The contents of `x` are interpreted as the output of the `rfft`
function.
Parameters
----------
x : array_like
Transformed data to invert.
n : int, optional
Length of the inverse Fourier transform.
If n < x.shape[axis], x is truncated.
If n > x.shape[axis], x is zero-padded.
The default results in n = x.shape[axis].
axis : int, optional
Axis along which the ifft's are computed; the default is over
the last axis (i.e., axis=-1).
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
Returns
-------
irfft : ndarray of floats
The inverse discrete Fourier transform.
See Also
--------
rfft, ifft, scipy.fft.irfft
Notes
-----
The returned real array contains::
[y(0),y(1),...,y(n-1)]
where for n is even::
y(j) = 1/n (sum[k=1..n/2-1] (x[2*k-1]+sqrt(-1)*x[2*k])
* exp(sqrt(-1)*j*k* 2*pi/n)
+ c.c. + x[0] + (-1)**(j) x[n-1])
and for n is odd::
y(j) = 1/n (sum[k=1..(n-1)/2] (x[2*k-1]+sqrt(-1)*x[2*k])
* exp(sqrt(-1)*j*k* 2*pi/n)
+ c.c. + x[0])
c.c. denotes complex conjugate of preceding expression.
For details on input parameters, see `rfft`.
To process (conjugate-symmetric) frequency-domain data with a complex
datatype, consider using the newer function `scipy.fft.irfft`.
Examples
--------
>>> from scipy.fftpack import rfft, irfft
>>> a = [1.0, 2.0, 3.0, 4.0, 5.0]
>>> irfft(a)
array([ 2.6 , -3.16405192, 1.24398433, -1.14955713, 1.46962473])
>>> irfft(rfft(a))
array([1., 2., 3., 4., 5.])
"""
return _pocketfft.irfft_fftpack(x, n, axis, None, overwrite_x)
def fftn(x, shape=None, axes=None, overwrite_x=False):
"""
Return multidimensional discrete Fourier transform.
The returned array contains::
y[j_1,..,j_d] = sum[k_1=0..n_1-1, ..., k_d=0..n_d-1]
x[k_1,..,k_d] * prod[i=1..d] exp(-sqrt(-1)*2*pi/n_i * j_i * k_i)
where d = len(x.shape) and n = x.shape.
Parameters
----------
x : array_like
The (N-D) array to transform.
shape : int or array_like of ints or None, optional
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[i] > x.shape[i]``, the ith dimension is padded with zeros.
If ``shape[i] < x.shape[i]``, the ith dimension is truncated to
length ``shape[i]``.
If any element of `shape` is -1, the size of the corresponding
dimension of `x` is used.
axes : int or array_like of ints or None, optional
The axes of `x` (`y` if `shape` is not None) along which the
transform is applied.
The default is over all axes.
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed. Default is False.
Returns
-------
y : complex-valued N-D NumPy array
The (N-D) DFT of the input array.
See Also
--------
ifftn
Notes
-----
If ``x`` is real-valued, then
``y[..., j_i, ...] == y[..., n_i-j_i, ...].conjugate()``.
Both single and double precision routines are implemented. Half precision
inputs will be converted to single precision. Non-floating-point inputs
will be converted to double precision. Long-double precision inputs are
not supported.
Examples
--------
>>> import numpy as np
>>> from scipy.fftpack import fftn, ifftn
>>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16))
>>> np.allclose(y, fftn(ifftn(y)))
True
"""
shape = _good_shape(x, shape, axes)
return _pocketfft.fftn(x, shape, axes, None, overwrite_x)
def ifftn(x, shape=None, axes=None, overwrite_x=False):
"""
Return inverse multidimensional discrete Fourier transform.
The sequence can be of an arbitrary type.
The returned array contains::
y[j_1,..,j_d] = 1/p * sum[k_1=0..n_1-1, ..., k_d=0..n_d-1]
x[k_1,..,k_d] * prod[i=1..d] exp(sqrt(-1)*2*pi/n_i * j_i * k_i)
where ``d = len(x.shape)``, ``n = x.shape``, and ``p = prod[i=1..d] n_i``.
For description of parameters see `fftn`.
See Also
--------
fftn : for detailed information.
Examples
--------
>>> from scipy.fftpack import fftn, ifftn
>>> import numpy as np
>>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16))
>>> np.allclose(y, ifftn(fftn(y)))
True
"""
shape = _good_shape(x, shape, axes)
return _pocketfft.ifftn(x, shape, axes, None, overwrite_x)
def fft2(x, shape=None, axes=(-2,-1), overwrite_x=False):
"""
2-D discrete Fourier transform.
Return the 2-D discrete Fourier transform of the 2-D argument
`x`.
See Also
--------
fftn : for detailed information.
Examples
--------
>>> import numpy as np
>>> from scipy.fftpack import fft2, ifft2
>>> y = np.mgrid[:5, :5][0]
>>> y
array([[0, 0, 0, 0, 0],
[1, 1, 1, 1, 1],
[2, 2, 2, 2, 2],
[3, 3, 3, 3, 3],
[4, 4, 4, 4, 4]])
>>> np.allclose(y, ifft2(fft2(y)))
True
"""
return fftn(x,shape,axes,overwrite_x)
def ifft2(x, shape=None, axes=(-2,-1), overwrite_x=False):
"""
2-D discrete inverse Fourier transform of real or complex sequence.
Return inverse 2-D discrete Fourier transform of
arbitrary type sequence x.
See `ifft` for more information.
See Also
--------
fft2, ifft
Examples
--------
>>> import numpy as np
>>> from scipy.fftpack import fft2, ifft2
>>> y = np.mgrid[:5, :5][0]
>>> y
array([[0, 0, 0, 0, 0],
[1, 1, 1, 1, 1],
[2, 2, 2, 2, 2],
[3, 3, 3, 3, 3],
[4, 4, 4, 4, 4]])
>>> np.allclose(y, fft2(ifft2(y)))
True
"""
return ifftn(x,shape,axes,overwrite_x)
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