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"""
Discrete Fourier Transforms - basic.py
"""
# Created by Pearu Peterson, August,September 2002
from __future__ import division, print_function, absolute_import
__all__ = ['fft','ifft','fftn','ifftn','rfft','irfft',
'fft2','ifft2']
from numpy import zeros, swapaxes
import numpy
from . import _fftpack
import atexit
atexit.register(_fftpack.destroy_zfft_cache)
atexit.register(_fftpack.destroy_zfftnd_cache)
atexit.register(_fftpack.destroy_drfft_cache)
atexit.register(_fftpack.destroy_cfft_cache)
atexit.register(_fftpack.destroy_cfftnd_cache)
atexit.register(_fftpack.destroy_rfft_cache)
del atexit
def istype(arr, typeclass):
return issubclass(arr.dtype.type, typeclass)
def _datacopied(arr, original):
"""
Strict check for `arr` not sharing any data with `original`,
under the assumption that arr = asarray(original)
"""
if arr is original:
return False
if not isinstance(original, numpy.ndarray) and hasattr(original, '__array__'):
return False
return arr.base is None
# XXX: single precision FFTs partially disabled due to accuracy issues
# for large prime-sized inputs.
#
# See http://permalink.gmane.org/gmane.comp.python.scientific.devel/13834
# ("fftpack test failures for 0.8.0b1", Ralf Gommers, 17 Jun 2010,
# @ scipy-dev)
#
# These should be re-enabled once the problems are resolved
def _is_safe_size(n):
"""
Is the size of FFT such that FFTPACK can handle it in single precision
with sufficient accuracy?
Composite numbers of 2, 3, and 5 are accepted, as FFTPACK has those
"""
n = int(n)
for c in (2, 3, 5):
while n % c == 0:
n /= c
return (n <= 1)
def _fake_crfft(x, n, *a, **kw):
if _is_safe_size(n):
return _fftpack.crfft(x, n, *a, **kw)
else:
return _fftpack.zrfft(x, n, *a, **kw).astype(numpy.complex64)
def _fake_cfft(x, n, *a, **kw):
if _is_safe_size(n):
return _fftpack.cfft(x, n, *a, **kw)
else:
return _fftpack.zfft(x, n, *a, **kw).astype(numpy.complex64)
def _fake_rfft(x, n, *a, **kw):
if _is_safe_size(n):
return _fftpack.rfft(x, n, *a, **kw)
else:
return _fftpack.drfft(x, n, *a, **kw).astype(numpy.float32)
def _fake_cfftnd(x, shape, *a, **kw):
if numpy.all(list(map(_is_safe_size, shape))):
return _fftpack.cfftnd(x, shape, *a, **kw)
else:
return _fftpack.zfftnd(x, shape, *a, **kw).astype(numpy.complex64)
_DTYPE_TO_FFT = {
# numpy.dtype(numpy.float32): _fftpack.crfft,
numpy.dtype(numpy.float32): _fake_crfft,
numpy.dtype(numpy.float64): _fftpack.zrfft,
# numpy.dtype(numpy.complex64): _fftpack.cfft,
numpy.dtype(numpy.complex64): _fake_cfft,
numpy.dtype(numpy.complex128): _fftpack.zfft,
}
_DTYPE_TO_RFFT = {
# numpy.dtype(numpy.float32): _fftpack.rfft,
numpy.dtype(numpy.float32): _fake_rfft,
numpy.dtype(numpy.float64): _fftpack.drfft,
}
_DTYPE_TO_FFTN = {
# numpy.dtype(numpy.complex64): _fftpack.cfftnd,
numpy.dtype(numpy.complex64): _fake_cfftnd,
numpy.dtype(numpy.complex128): _fftpack.zfftnd,
# numpy.dtype(numpy.float32): _fftpack.cfftnd,
numpy.dtype(numpy.float32): _fake_cfftnd,
numpy.dtype(numpy.float64): _fftpack.zfftnd,
}
def _asfarray(x):
"""Like numpy asfarray, except that it does not modify x dtype if x is
already an array with a float dtype, and do not cast complex types to
real."""
if hasattr(x, "dtype") and x.dtype.char in numpy.typecodes["AllFloat"]:
return x
else:
# We cannot use asfarray directly because it converts sequences of
# complex to sequence of real
ret = numpy.asarray(x)
if not ret.dtype.char in numpy.typecodes["AllFloat"]:
return numpy.asfarray(x)
return ret
def _fix_shape(x, n, axis):
""" Internal auxiliary function for _raw_fft, _raw_fftnd."""
s = list(x.shape)
if s[axis] > n:
index = [slice(None)]*len(s)
index[axis] = slice(0,n)
x = x[index]
return x, False
else:
index = [slice(None)]*len(s)
index[axis] = slice(0,s[axis])
s[axis] = n
z = zeros(s,x.dtype.char)
z[index] = x
return z, True
def _raw_fft(x, n, axis, direction, overwrite_x, work_function):
""" Internal auxiliary function for fft, ifft, rfft, irfft."""
if n is None:
n = x.shape[axis]
elif n != x.shape[axis]:
x, copy_made = _fix_shape(x,n,axis)
overwrite_x = overwrite_x or copy_made
if n < 1:
raise ValueError("Invalid number of FFT data points "
"(%d) specified." % n)
if axis == -1 or axis == len(x.shape)-1:
r = work_function(x,n,direction,overwrite_x=overwrite_x)
else:
x = swapaxes(x, axis, -1)
r = work_function(x,n,direction,overwrite_x=overwrite_x)
r = swapaxes(r, axis, -1)
return r
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
Note that ``y(-j) = y(n-j).conjugate()``.
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`.
For `n` even, ``A[n/2]`` contains the sum of the positive and
negative-frequency terms. For `n` even and `x` real, ``A[n/2]`` will
always be real.
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 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
--------
>>> from scipy.fftpack import fft, ifft
>>> x = np.arange(5)
>>> np.allclose(fft(ifft(x)), x, atol=1e-15) # within numerical accuracy.
True
"""
tmp = _asfarray(x)
try:
work_function = _DTYPE_TO_FFT[tmp.dtype]
except KeyError:
raise ValueError("type %s is not supported" % tmp.dtype)
if not (istype(tmp, numpy.complex64) or istype(tmp, numpy.complex128)):
overwrite_x = 1
overwrite_x = overwrite_x or _datacopied(tmp, x)
if n is None:
n = tmp.shape[axis]
elif n != tmp.shape[axis]:
tmp, copy_made = _fix_shape(tmp,n,axis)
overwrite_x = overwrite_x or copy_made
if n < 1:
raise ValueError("Invalid number of FFT data points "
"(%d) specified." % n)
if axis == -1 or axis == len(tmp.shape) - 1:
return work_function(tmp,n,1,0,overwrite_x)
tmp = swapaxes(tmp, axis, -1)
tmp = work_function(tmp,n,1,0,overwrite_x)
return swapaxes(tmp, axis, -1)
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
-----
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.
"""
tmp = _asfarray(x)
try:
work_function = _DTYPE_TO_FFT[tmp.dtype]
except KeyError:
raise ValueError("type %s is not supported" % tmp.dtype)
if not (istype(tmp, numpy.complex64) or istype(tmp, numpy.complex128)):
overwrite_x = 1
overwrite_x = overwrite_x or _datacopied(tmp, x)
if n is None:
n = tmp.shape[axis]
elif n != tmp.shape[axis]:
tmp, copy_made = _fix_shape(tmp,n,axis)
overwrite_x = overwrite_x or copy_made
if n < 1:
raise ValueError("Invalid number of FFT data points "
"(%d) specified." % n)
if axis == -1 or axis == len(tmp.shape) - 1:
return work_function(tmp,n,-1,1,overwrite_x)
tmp = swapaxes(tmp, axis, -1)
tmp = work_function(tmp,n,-1,1,overwrite_x)
return swapaxes(tmp, axis, -1)
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
Note that ``y(-j) == y(n-j).conjugate()``.
See Also
--------
fft, irfft, scipy.fftpack.basic
Notes
-----
Within numerical accuracy, ``y == rfft(irfft(y))``.
Examples
--------
>>> 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.])
"""
tmp = _asfarray(x)
if not numpy.isrealobj(tmp):
raise TypeError("1st argument must be real sequence")
try:
work_function = _DTYPE_TO_RFFT[tmp.dtype]
except KeyError:
raise ValueError("type %s is not supported" % tmp.dtype)
overwrite_x = overwrite_x or _datacopied(tmp, x)
return _raw_fft(tmp,n,axis,1,overwrite_x,work_function)
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
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`.
"""
tmp = _asfarray(x)
if not numpy.isrealobj(tmp):
raise TypeError("1st argument must be real sequence")
try:
work_function = _DTYPE_TO_RFFT[tmp.dtype]
except KeyError:
raise ValueError("type %s is not supported" % tmp.dtype)
overwrite_x = overwrite_x or _datacopied(tmp, x)
return _raw_fft(tmp,n,axis,-1,overwrite_x,work_function)
def _raw_fftnd(x, s, axes, direction, overwrite_x, work_function):
""" Internal auxiliary function for fftnd, ifftnd."""
if s is None:
if axes is None:
s = x.shape
else:
s = numpy.take(x.shape, axes)
s = tuple(s)
if axes is None:
noaxes = True
axes = list(range(-x.ndim, 0))
else:
noaxes = False
if len(axes) != len(s):
raise ValueError("when given, axes and shape arguments "
"have to be of the same length")
for dim in s:
if dim < 1:
raise ValueError("Invalid number of FFT data points "
"(%s) specified." % (s,))
# No need to swap axes, array is in C order
if noaxes:
for i in axes:
x, copy_made = _fix_shape(x, s[i], i)
overwrite_x = overwrite_x or copy_made
return work_function(x,s,direction,overwrite_x=overwrite_x)
# We ordered axes, because the code below to push axes at the end of the
# array assumes axes argument is in ascending order.
id = numpy.argsort(axes)
axes = [axes[i] for i in id]
s = [s[i] for i in id]
# Swap the request axes, last first (i.e. First swap the axis which ends up
# at -1, then at -2, etc...), such as the request axes on which the
# operation is carried become the last ones
for i in range(1, len(axes)+1):
x = numpy.swapaxes(x, axes[-i], -i)
# We can now operate on the axes waxes, the p last axes (p = len(axes)), by
# fixing the shape of the input array to 1 for any axis the fft is not
# carried upon.
waxes = list(range(x.ndim - len(axes), x.ndim))
shape = numpy.ones(x.ndim)
shape[waxes] = s
for i in range(len(waxes)):
x, copy_made = _fix_shape(x, s[i], waxes[i])
overwrite_x = overwrite_x or copy_made
r = work_function(x, shape, direction, overwrite_x=overwrite_x)
# reswap in the reverse order (first axis first, etc...) to get original
# order
for i in range(len(axes), 0, -1):
r = numpy.swapaxes(r, -i, axes[-i])
return r
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.
Note that ``y[..., -j_i, ...] = y[..., n_i-j_i, ...].conjugate()``.
Parameters
----------
x : array_like
The (n-dimensional) array to transform.
shape : tuple of ints, 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 ``scipy.take(x.shape, axes, axis=0)``.
If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros.
If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to
length ``shape[i]``.
axes : array_like of ints, optional
The axes of `x` (`y` if `shape` is not None) along which the
transform is applied.
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed. Default is False.
Returns
-------
y : complex-valued n-dimensional numpy array
The (n-dimensional) DFT of the input array.
See Also
--------
ifftn
Examples
--------
>>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16))
>>> np.allclose(y, fftn(ifftn(y)))
True
"""
return _raw_fftn_dispatch(x, shape, axes, overwrite_x, 1)
def _raw_fftn_dispatch(x, shape, axes, overwrite_x, direction):
tmp = _asfarray(x)
try:
work_function = _DTYPE_TO_FFTN[tmp.dtype]
except KeyError:
raise ValueError("type %s is not supported" % tmp.dtype)
if not (istype(tmp, numpy.complex64) or istype(tmp, numpy.complex128)):
overwrite_x = 1
overwrite_x = overwrite_x or _datacopied(tmp, x)
return _raw_fftnd(tmp,shape,axes,direction,overwrite_x,work_function)
def ifftn(x, shape=None, axes=None, overwrite_x=False):
"""
Return inverse multi-dimensional discrete Fourier transform of
arbitrary type sequence x.
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.
"""
return _raw_fftn_dispatch(x, shape, axes, overwrite_x, -1)
def fft2(x, shape=None, axes=(-2,-1), overwrite_x=False):
"""
2-D discrete Fourier transform.
Return the two-dimensional discrete Fourier transform of the 2-D argument
`x`.
See Also
--------
fftn : for detailed information.
"""
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 two-dimensional discrete Fourier transform of
arbitrary type sequence x.
See `ifft` for more information.
See also
--------
fft2, ifft
"""
return ifftn(x,shape,axes,overwrite_x)
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