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fft.py
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fft.py
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# Copyright (c) 2021 PaddlePaddle Authors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
from typing import Sequence
import numpy as np
import paddle
from . import _C_ops
from .fluid.data_feeder import check_variable_and_dtype
from .fluid.layer_helper import LayerHelper
from .framework import in_dynamic_mode
from .tensor.attribute import is_floating_point, is_integer
from .tensor.creation import _complex_to_real_dtype, _real_to_complex_dtype
__all__ = [
'fft',
'ifft',
'rfft',
'irfft',
'hfft',
'ihfft',
'fft2',
'ifft2',
'rfft2',
'irfft2',
'hfft2',
'ihfft2',
'fftn',
'ifftn',
'rfftn',
'irfftn',
'hfftn',
'ihfftn',
'fftfreq',
'rfftfreq',
'fftshift',
'ifftshift',
]
def _check_normalization(norm):
if norm not in ['forward', 'backward', 'ortho']:
raise ValueError(
"Unexpected norm: {}. Norm should be forward, backward or ortho".format(
norm
)
)
def _check_fft_n(n):
if not isinstance(n, int):
raise ValueError(
f"Invalid FFT argument n({n}), it shoule be an integer."
)
if n <= 0:
raise ValueError(f"Invalid FFT argument n({n}), it should be positive.")
def _check_fft_shape(x, s):
ndim = x.ndim
if not isinstance(s, Sequence):
raise ValueError(
"Invaid FFT argument s({}), it should be a sequence of integers."
)
if len(s) > ndim:
raise ValueError(
"Length of FFT argument s should not be larger than the rank of input. "
"Received s: {}, rank of x: {}".format(s, ndim)
)
for size in s:
if not isinstance(size, int) or size <= 0:
raise ValueError(f"FFT sizes {s} contains invalid value ({size})")
def _check_fft_axis(x, axis):
ndim = x.ndim
if not isinstance(axis, int):
raise ValueError(f"Invalid FFT axis ({axis}), it shoule be an integer.")
if axis < -ndim or axis >= ndim:
raise ValueError(
"Invalid FFT axis ({}), it should be in range [-{}, {})".format(
axis, ndim, ndim
)
)
def _check_fft_axes(x, axes):
ndim = x.ndim
if not isinstance(axes, Sequence):
raise ValueError(
"Invalid FFT axes ({}), it should be a sequence of integers.".format(
axes
)
)
if len(axes) > ndim:
raise ValueError(
"Length of fft axes should not be larger than the rank of input. "
"Received, len of axes: {}, rank of x: {}".format(len(axes), ndim)
)
for axis in axes:
if not isinstance(axis, int) or axis < -ndim or axis >= ndim:
raise ValueError(
"FFT axes {} contains invalid value ({}), it should be in range [-{}, {})".format(
axes, axis, ndim, ndim
)
)
def _resize_fft_input(x, s, axes):
if len(s) != len(axes):
raise ValueError("length of `s` should equals length of `axes`.")
shape = x.shape
ndim = x.ndim
axes_to_pad = []
paddings = []
axes_to_slice = []
slices = []
for i, axis in enumerate(axes):
if shape[axis] < s[i]:
axes_to_pad.append(axis)
paddings.append(s[i] - shape[axis])
elif shape[axis] > s[i]:
axes_to_slice.append(axis)
slices.append((0, s[i]))
if axes_to_slice:
x = paddle.slice(
x,
axes_to_slice,
starts=[item[0] for item in slices],
ends=[item[1] for item in slices],
)
if axes_to_pad:
padding_widths = [0] * (2 * ndim)
for axis, pad in zip(axes_to_pad, paddings):
padding_widths[2 * axis + 1] = pad
x = paddle.nn.functional.pad(x, padding_widths)
return x
def _normalize_axes(x, axes):
ndim = x.ndim
return [item if item >= 0 else (item + ndim) for item in axes]
def _check_at_least_ndim(x, rank):
if x.ndim < rank:
raise ValueError(f"The rank of the input ({x.ndim}) should >= {rank}")
# public APIs 1d
def fft(x, n=None, axis=-1, norm="backward", name=None):
"""
Calculate one-dimensional discrete Fourier transform.
This function uses the efficient fast Fourier transform (FFT) algorithm [1] to
calculate the 1-D * n * point discrete Fourier transform (DFT).
Args:
x (Tensor): The input data. It's a Tensor type. It's a complex.
n (int, optional): The length of the output transform axis. If `n` is less than
the length input, the input will be cropped. If larger, the input is filled
with zeros. If `n` is not given, the input length along the axis specified
by `axis` is used.
axis (int, optional): Axis used to calculate FFT. If not specified, the last axis
is used by default.
norm (str, optional): Indicates which direction to scale the `forward` or `backward` transform
pair and what normalization factor to use. The parameter value must be one
of "forward" or "backward" or "ortho". Default is "backward", meaning no normalization on
the forward transforms and scaling by ``1/n`` on the `ifft`. "forward" instead applies
the ``1/n`` factor on the forward tranform. For ``norm="ortho"``, both directions are
scaled by ``1/sqrt(n)``.
name (str, optional): The default value is None. Normally there is no need for user to set
this property. For more information, please refer to :ref:`api_guide_Name`.
Returns:
complex tensor. The truncated or zero-padded input, transformed along the axis indicated
by `axis`, or the last one if `axis` is not specified.
Examples:
.. code-block:: python
import numpy as np
import paddle
x = np.exp(3j * np.pi * np.arange(7) / 7)
xp = paddle.to_tensor(x)
fft_xp = paddle.fft.fft(xp).numpy()
print(fft_xp)
# [1.+1.25396034e+00j 1.+4.38128627e+00j 1.-4.38128627e+00j
# 1.-1.25396034e+00j 1.-4.81574619e-01j 1.+8.88178420e-16j
# 1.+4.81574619e-01j]
"""
if is_integer(x) or is_floating_point(x):
return fft_r2c(
x, n, axis, norm, forward=True, onesided=False, name=name
)
else:
return fft_c2c(x, n, axis, norm, forward=True, name=name)
def ifft(x, n=None, axis=-1, norm="backward", name=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.
Args:
x (Tensor): The input data. It's a Tensor type. It's a complex.
n (int, optional): The length of the output transform axis. If `n` is less than
the length input, the input will be cropped. If larger, the input is filled
with zeros. If `n` is not given, the input length along the axis specified
by `axis` is used.
axis (int, optional): Axis used to calculate FFT. If not specified, the last axis
is used by default.
norm (str, optional): Indicates which direction to scale the `forward` or `backward` transform
pair and what normalization factor to use. The parameter value must be one
of "forward" or "backward" or "ortho". Default is "backward", meaning no normalization on
the forward transforms and scaling by ``1/n`` on the `ifft`. "forward" instead applies
the ``1/n`` factor on the forward tranform. For ``norm="ortho"``, both directions are
scaled by ``1/sqrt(n)``.
name (str, optional): The default value is None. Normally there is no need for user to set
this property. For more information, please refer to :ref:`api_guide_Name`.
Returns:
complex tensor. The truncated or zero-padded input, transformed along the axis indicated
by `axis`, or the last one if `axis` is not specified.
Examples:
.. code-block:: python
import numpy as np
import paddle
x = np.exp(3j * np.pi * np.arange(7) / 7)
xp = paddle.to_tensor(x)
ifft_xp = paddle.fft.ifft(xp).numpy()
print(ifft_xp)
# [0.14285714+1.79137191e-01j 0.14285714+6.87963741e-02j
# 0.14285714+1.26882631e-16j 0.14285714-6.87963741e-02j
# 0.14285714-1.79137191e-01j 0.14285714-6.25898038e-01j
# 0.14285714+6.25898038e-01j]
"""
if is_integer(x) or is_floating_point(x):
return fft_r2c(
x, n, axis, norm, forward=False, onesided=False, name=name
)
else:
return fft_c2c(x, n, axis, norm, forward=False, name=name)
def rfft(x, n=None, axis=-1, norm="backward", name=None):
"""
The one dimensional FFT for real input.
This function computes the one dimensional *n*-point discrete Fourier
Transform (DFT) of a real-valued tensor by means of an efficient algorithm
called the Fast Fourier Transform (FFT).
When the DFT is computed for purely real input, the output is
Hermitian-symmetric. This function does not compute the negative frequency
terms, and the length of the transformed axis of the output is therefore
``n//2 + 1``.
Args:
x(Tensor) : Real-valued input tensor
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. Default value
is last axis.
norm(str, optional) : Normalization mode, indicates which direction of
the forward/backward pair of transforms is scaled and with what
normalization factor. Include {"backward", "ortho", "forward"},
default value is "backward".
- "backward": The factor of forward direction and backward direction are ``1`` and ``1/n`` respectively;
- "forward": The factor of forward direction and backward direction are ``1/n`` and ``1`` respectively;
- "ortho": The factor of forward direction and backword direction are both ``1/sqrt(n)``.
Where ``n`` is the multiplication of each element in ``s`` .
name(str, optional): The default value is None. Normally there is no
need for user to set this property. For more information, please
refer to :ref:`api_guide_Name` .
Returns:
out(Tensor) : complex tensor
Examples:
.. code-block:: python
import paddle
x = paddle.to_tensor([0.0, 1.0, 0.0, 0.0])
print(paddle.fft.rfft(x))
# Tensor(shape=[3], dtype=complex64, place=CUDAPlace(0), stop_gradient=True,
# [ (1+0j), -1j , (-1+0j)])
"""
return fft_r2c(x, n, axis, norm, forward=True, onesided=True, name=name)
def irfft(x, n=None, axis=-1, norm="backward", name=None):
"""
Computes the inverse of `rfft`.
This function calculates the inverse of the one-dimensional *n* point discrete
Fourier transform of the actual input calculated by "rfft". In other words,
``irfft(rfft(a),len(a)) == a`` is within the numerical accuracy range.
The input shall be in the form of "rfft", i.e. the actual zero frequency term,
followed by the complex positive frequency term, in the order of increasing frequency.
Because the discrete Fourier transform of the actual input is Hermite symmetric,
the negative frequency term is regarded as the complex conjugate term of the corresponding
positive frequency term.
Args:
x (Tensor): The input data. It's a Tensor type. It's a complex.
n (int, optional): The length of the output transform axis. For `n` output
points, ``n//2 + 1``input points are necessary. If the length of the input tensor is greater
than `n`, it will be cropped, if it is shorter than this, fill in zero. If `n` is not given,
it is considered to be ``2 * (k-1)``, where ``k`` is the length of the input axis specified
along the ` axis'.
axis (int, optional): Axis used to calculate FFT. If not specified, the last axis
is used by default.
norm (str, optional): Indicates which direction to scale the `forward` or `backward` transform
pair and what normalization factor to use. The parameter value must be one
of "forward" or "backward" or "ortho". Default is "backward".
name (str, optional): The default value is None. Normally there is no need for user to set
this property. For more information, please refer to :ref:`api_guide_Name` .
Returns:
Real tensor. Truncated or zero fill input for the transformation along the axis indicated by
`axis`, or the last input if `axis` is not specified. The length of the conversion axis
is `n`, or ``2 * k-2``, if `k` is None, where `k` is the length of the input conversion axis.
If the output is an odd number, you need to specify the value of 'n', such as ``2 * k-1``
in some cases.
Examples:
.. code-block:: python
import paddle
x = paddle.to_tensor([1, -1j, -1])
irfft_x = paddle.fft.irfft(x)
print(irfft_x)
# Tensor(shape=[4], dtype=float32, place=Place(cpu), stop_gradient=True,
# [0., 1., 0., 0.])
"""
return fft_c2r(x, n, axis, norm, forward=False, name=name)
def hfft(x, n=None, axis=-1, norm="backward", name=None):
"""
Compute the FFT of a signal that has Hermitian symmetry, a real
spectrum.
Args:
x (Tensor): The input data. It's a Tensor type. It's a complex.
n (int, optional): The length of the output transform axis. For `n` output
points, ``n//2 + 1`` input points are necessary. If the length of the input tensor is greater
than `n`, it will be cropped, if it is shorter than this, fill in zero. If `n` is not given,
it is considered to be ``2 * (k-1)``, where ``k`` is the length of the input axis specified
along the ` axis'.
axis (int,optional): Axis used to calculate FFT. If not specified, the last axis
is used by default.
norm (str, optional): Indicates which direction to scale the `forward` or `backward` transform
pair and what normalization factor to use. The parameter value must be one
of "forward" or "backward" or "ortho". Default is "backward".
name (str, optional): The default value is None. Normally there is no need for user to set
this property. For more information, please refer to :ref:`api_guide_Name` .
Returns:
Real tensor. Truncated or zero fill input for the transformation along the axis indicated by
`axis`, or the last input if `axis` is not specified. The length of the conversion axis
is `n`, or ``2 * k-2``, if `k` is None, where `k` is the length of the input conversion axis.
If the output is an odd number, you need to specify the value of 'n', such as ``2 * k-1`` in
some cases.
Examples:
.. code-block:: python
import paddle
x = paddle.to_tensor([1, -1j, -1])
hfft_x = paddle.fft.hfft(x)
print(hfft_x)
# Tensor(shape=[4], dtype=float32, place=Place(cpu), stop_gradient=True,
# [0., 0., 0., 4.])
"""
return fft_c2r(x, n, axis, norm, forward=True, name=name)
def ihfft(x, n=None, axis=-1, norm="backward", name=None):
"""
The inverse FFT of a signal that has Hermitian symmetry.
This function computes the one dimensional *n*-point inverse FFT of a signal
that has Hermitian symmetry by means of an efficient algorithm called
the Fast Fourier Transform (FFT).
When the DFT is computed for purely real input, the output is
Hermitian-symmetric. This function does not compute the negative frequency
terms, and the length of the transformed axis of the output is therefore
``n//2 + 1``.
Args:
x(Tensor): Input tensor.
n(int, optional): 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(str, optional) : Normalization mode, indicates which direction of
the forward/backward pair of transforms is scaled and with what
normalization factor. Include {"backward", "ortho", "forward"},
default value is "backward".
name(str, optional): The default value is None. Normally there is no
need for user to set this property. For more information, please
refer to :ref:`api_guide_Name` .
Returns:
out(Tensor) : complex tensor.
Examples:
.. code-block:: python
import paddle
spectrum = paddle.to_tensor([10.0, -5.0, 0.0, -1.0, 0.0, -5.0])
print(paddle.fft.ifft(spectrum))
# Tensor(shape=[6], dtype=complex64, place=CUDAPlace(0), stop_gradient=True,
# [(-0.1666666716337204+0j), (1-1.9868215517249155e-08j), (2.3333334922790527-1.9868215517249155e-08j), (3.5+0j), (2.3333334922790527+1.9868215517249155e-08j), (1+1.9868215517249155e-08j)])
print(paddle.fft.ihfft(spectrum))
# Tensor(shape = [4], dtype = complex64, place = CUDAPlace(0), stop_gradient = True,
# [(-0.1666666716337204+0j), (1-1.9868215517249155e-08j), (2.3333334922790527-1.9868215517249155e-08j), (3.5+0j)])
"""
return fft_r2c(x, n, axis, norm, forward=False, onesided=True, name=name)
# public APIs nd
def fftn(x, s=None, axes=None, norm="backward", name=None):
"""
Compute the N-D discrete Fourier Transform.
This function calculates the n-D discrete Fourier transform on any number of axes
in the M-D array by fast Fourier transform (FFT).
Args:
x (Tensor): The input data. It's a Tensor type. It's a 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 used to calculate FFT. If not given, the last ``len(s)``
axes are used, or all axes if `s` is also not specified.
norm (str, optional): Indicates which direction to scale the `forward` or `backward` transform
pair and what normalization factor to use. The parameter value must be one
of "forward" or "backward" or "ortho". Default is "backward", meaning no normalization on
the forward transforms and scaling by ``1/n`` on the `ifft`. "forward" instead applies
the ``1/n`` factor on the forward tranform. For ``norm="ortho"``, both directions are
scaled by ``1/sqrt(n)``.
name (str, optional): The default value is None. Normally there is no need for user to set
this property. For more information, please refer to :ref:`api_guide_Name`.
Returns:
complex tensor. 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.
Examples:
.. code-block:: python
import paddle
arr = paddle.arange(4, dtype="float64")
x = paddle.meshgrid(arr, arr, arr)[1]
fftn_xp = paddle.fft.fftn(x, axes=(1, 2))
print(fftn_xp)
# Tensor(shape=[4, 4, 4], dtype=complex128, place=Place(gpu:0), stop_gradient=True,
# [[[(24+0j), 0j , 0j , -0j ],
# [(-8+8j), 0j , 0j , -0j ],
# [(-8+0j), 0j , 0j , -0j ],
# [(-8-8j), 0j , 0j , -0j ]],
# [[(24+0j), 0j , 0j , -0j ],
# [(-8+8j), 0j , 0j , -0j ],
# [(-8+0j), 0j , 0j , -0j ],
# [(-8-8j), 0j , 0j , -0j ]],
# [[(24+0j), 0j , 0j , -0j ],
# [(-8+8j), 0j , 0j , -0j ],
# [(-8+0j), 0j , 0j , -0j ],
# [(-8-8j), 0j , 0j , -0j ]],
# [[(24+0j), 0j , 0j , -0j ],
# [(-8+8j), 0j , 0j , -0j ],
# [(-8+0j), 0j , 0j , -0j ],
# [(-8-8j), 0j , 0j , -0j ]]])
"""
if is_integer(x) or is_floating_point(x):
return fftn_r2c(
x, s, axes, norm, forward=True, onesided=False, name=name
)
else:
return fftn_c2c(x, s, axes, norm, forward=True, name=name)
def ifftn(x, s=None, axes=None, norm="backward", name=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.
Args:
x (Tensor): The input data. It's a Tensor type. It's a 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 used to calculate FFT. If not given, the last ``len(s)``
axes are used, or all axes if `s` is also not specified.
norm (str, optional): Indicates which direction to scale the `forward` or `backward` transform
pair and what normalization factor to use. The parameter value must be one
of "forward" or "backward" or "ortho". Default is "backward", meaning no normalization on
the forward transforms and scaling by ``1/n`` on the `ifft`. "forward" instead applies
the ``1/n`` factor on the forward tranform. For ``norm="ortho"``, both directions are
scaled by ``1/sqrt(n)``.
name (str, optional): The default value is None. Normally there is no need for user to set
this property. For more information, please refer to :ref:`api_guide_Name`.
Returns:
complex tensor. 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.
Examples:
.. code-block:: python
import paddle
x = paddle.eye(3)
ifftn_x = paddle.fft.ifftn(x, axes=(1,))
print(ifftn_x)
# Tensor(shape=[3, 3], dtype=complex64, place=Place(cpu), stop_gradient=True,
# [[ (0.3333333432674408+0j) ,
# (0.3333333432674408-0j) ,
# (0.3333333432674408+0j) ],
# [ (0.3333333432674408+0j) ,
# (-0.1666666716337204+0.28867512941360474j),
# (-0.1666666716337204-0.28867512941360474j)],
# [ (0.3333333432674408+0j) ,
# (-0.1666666716337204-0.28867512941360474j),
# (-0.1666666716337204+0.28867512941360474j)]])
"""
if is_integer(x) or is_floating_point(x):
return fftn_r2c(
x, s, axes, norm, forward=False, onesided=False, name=name
)
else:
return fftn_c2c(x, s, axes, norm, forward=False, name=name)
def rfftn(x, s=None, axes=None, norm="backward", name=None):
"""
The N dimensional FFT for real input.
This function computes the N-dimensional discrete Fourier Transform over
any number of axes in an M-dimensional real array by means of the Fast
Fourier Transform (FFT). By default, all axes are transformed, with the
real transform performed over the last axis, while the remaining
transforms are complex.
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.
Args:
x(Tensor) : Input tensor, taken to be real.
s(Sequence[int], optional) : Shape to use from the exec fft. 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[int], 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(str, optional) : Normalization mode, indicates which direction of
the forward/backward pair of transforms is scaled and with what
normalization factor. Include {"backward", "ortho", "forward"},
default value is "backward". The details of
three operations are shown below:
- "backward": The factor of forward direction and backward direction are ``1``
and ``1/n`` respectively;
- "forward": The factor of forward direction and backward direction are ``1/n``
and ``1`` respectively;
- "ortho": The factor of forward direction and backword direction are both ``1/sqrt(n)``.
Where ``n`` is the multiplication of each element in ``s`` .
name(str, optional): The default value is None. Normally there is no
need for user to set this property. For more information, please
refer to :ref:`api_guide_Name` .
Returns:
out(Tensor), complex tensor
Examples:
.. code-block:: python
import paddle
# default, all axis will be used to exec fft
x = paddle.ones((2, 3, 4))
print(paddle.fft.rfftn(x))
# Tensor(shape=[2, 3, 3], dtype=complex64, place=CUDAPlace(0), stop_gradient=True,
# [[[(24+0j), 0j , 0j ],
# [0j , 0j , 0j ],
# [0j , 0j , 0j ]],
#
# [[0j , 0j , 0j ],
# [0j , 0j , 0j ],
# [0j , 0j , 0j ]]])
# use axes(2, 0)
print(paddle.fft.rfftn(x, axes=(2, 0)))
# Tensor(shape=[2, 3, 3], dtype=complex64, place=CUDAPlace(0), stop_gradient=True,
# [[[(8+0j), 0j , 0j ],
# [(8+0j), 0j , 0j ],
# [(8+0j), 0j , 0j ]],
#
# [[0j , 0j , 0j ],
# [0j , 0j , 0j ],
# [0j , 0j , 0j ]]])
"""
return fftn_r2c(x, s, axes, norm, forward=True, onesided=True, name=name)
def irfftn(x, s=None, axes=None, norm="backward", name=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 ``x.shape`` is necessary like ``len(x)`` 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.
Args:
x (Tensor): The input data. It's a Tensor type.
s (sequence of ints, optional): The length of the output transform axis.
(``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*(k-1)``
where ``k`` 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 (str): Indicates which direction to scale the `forward` or `backward` transform
pair and what normalization factor to use. The parameter value must be one
of "forward" or "backward" or "ortho". Default is "backward". The details of
three operations are shown below:
- "backward": The factor of forward direction and backward direction are ``1`` and ``1/n`` respectively;
- "forward": The factor of forward direction and backward direction are ``1/n`` and ``1`` respectively;
- "ortho": The factor of forward direction and backword direction are both ``1/sqrt(n)``.
Where ``n`` is the multiplication of each element in ``s`` .
name (str, optional): The default value is None. Normally there is no need for user to set
this property. For more information, please refer to :ref:`api_guide_Name`.
Returns:
Real tensor. 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.
Examples:
.. code-block:: python
import paddle
x = paddle.to_tensor([2.+2.j, 2.+2.j, 3.+3.j]).astype(paddle.complex128)
print(x)
irfftn_x = paddle.fft.irfftn(x)
print(irfftn_x)
# Tensor(shape=[3], dtype=complex128, place=Place(cpu), stop_gradient=True,
# [(2+2j), (2+2j), (3+3j)])
# Tensor(shape=[4], dtype=float64, place=Place(cpu), stop_gradient=True,
# [ 2.25000000, -1.25000000, 0.25000000, 0.75000000])
"""
return fftn_c2r(x, s, axes, norm, forward=False, name=name)
def hfftn(x, s=None, axes=None, norm="backward", name=None):
"""
Compute the N-D FFT of Hermitian symmetric complex input, i.e., a
signal with a real spectrum.
This function calculates the n-D discrete Fourier transform of Hermite symmetric
complex input on any axis in M-D array by fast Fourier transform (FFT).
In other words, ``ihfftn(hfftn(x, s)) == x`` is within the numerical accuracy range.
(``s`` here are ``x.shape`` and ``s[-1] = x.shape[- 1] * 2 - 1``. This is necessary
for the same reason that ``irfft`` requires ``x.shape``.)
Args:
x (Tensor): The input data. It's a Tensor type.
s (sequence of ints, optional): The length of the output transform axis.
(``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*(k-1)`` where
``k`` 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 (str, optional): Indicates which direction to scale the `forward` or `backward` transform
pair and what normalization factor to use. The parameter value must be one
of "forward" or "backward" or "ortho". Default is "backward".
name (str, optional): The default value is None. Normally there is no need for user to set
this property. For more information, please refer to :ref:`api_guide_Name`.
Returns:
Real tensor. Truncate or zero fill input, transforming along the axis indicated by axis or
a combination of `s` or `X`.
Examples:
.. code-block:: python
import paddle
x = paddle.to_tensor([(2+2j), (2+2j), (3+3j)])
hfftn_x = paddle.fft.hfftn(x)
print(hfftn_x)
# Tensor(shape=[4], dtype=float32, place=Place(cpu), stop_gradient=True,
# [ 9., 3., 1., -5.])
"""
return fftn_c2r(x, s, axes, norm, forward=True, name=name)
def ihfftn(x, s=None, axes=None, norm="backward", name=None):
"""
The n dimensional inverse FFT of a signal that has Hermitian symmetry.
This function computes the n dimensional inverse FFT over any number of axes
in an M-dimensional of a signal that has Hermitian symmetry by means of an
efficient algorithm called the Fast Fourier Transform (FFT).
Args:
x(Tensor): Input tensor.
s(Sequence[int], 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[int], optional) : Axis over which to compute the inverse FFT. If not
given, the last axis is used.
norm(str, optional) : Normalization mode, indicates which direction of
the forward/backward pair of transforms is scaled and with what
normalization factor. Include {"backward", "ortho", "forward"},
default value is "backward".
name(str, optional): The default value is None. Normally there is no
need for user to set this property. For more information, please
refer to :ref:`api_guide_Name` .
Returns:
out(Tensor) : complex tensor.
Examples:
.. code-block:: python
import paddle
spectrum = paddle.to_tensor([10.0, -5.0, 0.0, -1.0, 0.0, -5.0])
print(paddle.fft.ifft(spectrum))
# Tensor(shape=[6], dtype=complex64, place=CUDAPlace(0), stop_gradient=True,
# [(-0.1666666716337204+0j), (1-1.9868215517249155e-08j), (2.3333334922790527-1.9868215517249155e-08j), (3.5+0j), (2.3333334922790527+1.9868215517249155e-08j), (1+1.9868215517249155e-08j)])
print(paddle.fft.ihfft(spectrum))
# Tensor(shape = [4], dtype = complex64, place = CUDAPlace(0), stop_gradient = True,
# [(-0.1666666716337204+0j), (1-1.9868215517249155e-08j), (2.3333334922790527-1.9868215517249155e-08j), (3.5+0j)])
"""
return fftn_r2c(x, s, axes, norm, forward=False, onesided=True, name=name)
# public APIs 2d
def fft2(x, s=None, axes=(-2, -1), norm="backward", name=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.
Args:
x (Tensor): The input data. It's a Tensor type.
s (sequence of ints, optional): Shape (length of each transformed axis) of the output.
It should be a sequence of 2 integers. 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. Default is None.
axes (sequence of ints, optional): Axes over which to compute the FFT. It should be a
sequence of 2 integers. If not specified, the last two axes are used by default.
norm (str, optional): Indicates which direction to scale the `forward` or `backward` transform
pair and what normalization factor to use. The parameter value must be one
of "forward" or "backward" or "ortho". Default is "backward".
name (str, optional): The default value is None. Normally there is no need for user to set
this property. For more information, please refer to :ref:`api_guide_Name`.
Returns:
Complex tensor. The truncated or zero-padded input, transformed along the axes indicated by `axes`,
or the last two axes if `axes` is not given.
Examples:
.. code-block:: python
import paddle
arr = paddle.arange(2, dtype="float64")
x = paddle.meshgrid(arr, arr)[0]
fft2_xp = paddle.fft.fft2(x)
print(fft2_xp)
# Tensor(shape=[2, 2], dtype=complex128, place=Place(gpu:0), stop_gradient=True,
# [[ (2+0j), 0j ],
# [(-2+0j), 0j ]])
"""
_check_at_least_ndim(x, 2)
if s is not None:
if not isinstance(s, Sequence) or len(s) != 2:
raise ValueError(
"Invalid FFT argument s ({}), it should be a sequence of 2 integers.".format(
s
)
)
if axes is not None:
if not isinstance(axes, Sequence) or len(axes) != 2:
raise ValueError(
"Invalid FFT argument axes ({}), it should be a sequence of 2 integers.".format(
axes
)
)
return fftn(x, s, axes, norm, name)
def ifft2(x, s=None, axes=(-2, -1), norm="backward", name=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.
Args:
x (Tensor): The input data. It's a Tensor type.
s (sequence of ints, optional): Shape (length of each transformed axis) of the output.
It should be a sequence of 2 integers. 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. Default is None.
axes (sequence of ints, optional): Axes over which to compute the FFT. It should be a
sequence of 2 integers. If not specified, the last two axes are used by default.
norm (str, optional): Indicates which direction to scale the `forward` or `backward` transform
pair and what normalization factor to use. The parameter value must be one
of "forward" or "backward" or "ortho". Default is "backward".
name (str, optional): The default value is None. Normally there is no need for user to set
this property. For more information, please refer to :ref:`api_guide_Name`.
Returns:
Complex tensor. The truncated or zero-padded input, transformed along the axes indicated by `axes`,
or the last two axes if `axes` is not given.
Examples:
.. code-block:: python
import paddle
arr = paddle.arange(2, dtype="float64")
x = paddle.meshgrid(arr, arr)[0]
ifft2_xp = paddle.fft.ifft2(x)
print(ifft2_xp)
# Tensor(shape=[2, 2], dtype=complex128, place=Place(gpu:0), stop_gradient=True,
# [[ (0.5+0j), 0j ],
# [(-0.5+0j), 0j ]])
"""
_check_at_least_ndim(x, 2)
if s is not None:
if not isinstance(s, Sequence) or len(s) != 2:
raise ValueError(
"Invalid FFT argument s ({}), it should be a sequence of 2 integers.".format(
s
)
)
if axes is not None:
if not isinstance(axes, Sequence) or len(axes) != 2:
raise ValueError(
"Invalid FFT argument axes ({}), it should be a sequence of 2 integers.".format(
axes
)
)
return ifftn(x, s, axes, norm, name)