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