194 lines
7.2 KiB
Python
194 lines
7.2 KiB
Python
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"""
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Discrete Fourier Transform (:mod:`numpy.fft`)
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=============================================
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.. currentmodule:: numpy.fft
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Standard FFTs
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-------------
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.. autosummary::
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:toctree: generated/
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fft Discrete Fourier transform.
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ifft Inverse discrete Fourier transform.
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fft2 Discrete Fourier transform in two dimensions.
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ifft2 Inverse discrete Fourier transform in two dimensions.
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fftn Discrete Fourier transform in N-dimensions.
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ifftn Inverse discrete Fourier transform in N dimensions.
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Real FFTs
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---------
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.. autosummary::
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:toctree: generated/
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rfft Real discrete Fourier transform.
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irfft Inverse real discrete Fourier transform.
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rfft2 Real discrete Fourier transform in two dimensions.
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irfft2 Inverse real discrete Fourier transform in two dimensions.
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rfftn Real discrete Fourier transform in N dimensions.
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irfftn Inverse real discrete Fourier transform in N dimensions.
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Hermitian FFTs
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--------------
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.. autosummary::
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:toctree: generated/
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hfft Hermitian discrete Fourier transform.
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ihfft Inverse Hermitian discrete Fourier transform.
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Helper routines
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---------------
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.. autosummary::
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:toctree: generated/
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fftfreq Discrete Fourier Transform sample frequencies.
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rfftfreq DFT sample frequencies (for usage with rfft, irfft).
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fftshift Shift zero-frequency component to center of spectrum.
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ifftshift Inverse of fftshift.
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Background information
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----------------------
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Fourier analysis is fundamentally a method for expressing a function as a
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sum of periodic components, and for recovering the function from those
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components. When both the function and its Fourier transform are
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replaced with discretized counterparts, it is called the discrete Fourier
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transform (DFT). The DFT has become a mainstay of numerical computing in
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part because of a very fast algorithm for computing it, called the Fast
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Fourier Transform (FFT), which was known to Gauss (1805) and was brought
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to light in its current form by Cooley and Tukey [CT]_. Press et al. [NR]_
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provide an accessible introduction to Fourier analysis and its
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applications.
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Because the discrete Fourier transform separates its input into
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components that contribute at discrete frequencies, it has a great number
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of applications in digital signal processing, e.g., for filtering, and in
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this context the discretized input to the transform is customarily
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referred to as a *signal*, which exists in the *time domain*. The output
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is called a *spectrum* or *transform* and exists in the *frequency
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domain*.
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Implementation details
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----------------------
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There are many ways to define the DFT, varying in the sign of the
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exponent, normalization, etc. In this implementation, the DFT is defined
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as
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.. math::
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A_k = \\sum_{m=0}^{n-1} a_m \\exp\\left\\{-2\\pi i{mk \\over n}\\right\\}
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\\qquad k = 0,\\ldots,n-1.
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The DFT is in general defined for complex inputs and outputs, and a
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single-frequency component at linear frequency :math:`f` is
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represented by a complex exponential
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:math:`a_m = \\exp\\{2\\pi i\\,f m\\Delta t\\}`, where :math:`\\Delta t`
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is the sampling interval.
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The values in the result follow so-called "standard" order: If ``A =
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fft(a, n)``, then ``A[0]`` contains the zero-frequency term (the sum of
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the signal), which is always purely real for real inputs. Then ``A[1:n/2]``
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contains the positive-frequency terms, and ``A[n/2+1:]`` contains the
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negative-frequency terms, in order of decreasingly negative frequency.
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For an even number of input points, ``A[n/2]`` represents both positive and
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negative Nyquist frequency, and is also purely real for real input. For
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an odd number of input points, ``A[(n-1)/2]`` contains the largest positive
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frequency, while ``A[(n+1)/2]`` contains the largest negative frequency.
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The routine ``np.fft.fftfreq(n)`` returns an array giving the frequencies
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of corresponding elements in the output. The routine
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``np.fft.fftshift(A)`` shifts transforms and their frequencies to put the
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zero-frequency components in the middle, and ``np.fft.ifftshift(A)`` undoes
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that shift.
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When the input `a` is a time-domain signal and ``A = fft(a)``, ``np.abs(A)``
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is its amplitude spectrum and ``np.abs(A)**2`` is its power spectrum.
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The phase spectrum is obtained by ``np.angle(A)``.
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The inverse DFT is defined as
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.. math::
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a_m = \\frac{1}{n}\\sum_{k=0}^{n-1}A_k\\exp\\left\\{2\\pi i{mk\\over n}\\right\\}
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\\qquad m = 0,\\ldots,n-1.
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It differs from the forward transform by the sign of the exponential
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argument and the default normalization by :math:`1/n`.
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Normalization
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-------------
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The default normalization has the direct transforms unscaled and the inverse
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transforms are scaled by :math:`1/n`. It is possible to obtain unitary
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transforms by setting the keyword argument ``norm`` to ``"ortho"`` (default is
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`None`) so that both direct and inverse transforms will be scaled by
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:math:`1/\\sqrt{n}`.
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Real and Hermitian transforms
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-----------------------------
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When the input is purely real, its transform is Hermitian, i.e., the
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component at frequency :math:`f_k` is the complex conjugate of the
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component at frequency :math:`-f_k`, which means that for real
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inputs there is no information in the negative frequency components that
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is not already available from the positive frequency components.
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The family of `rfft` functions is
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designed to operate on real inputs, and exploits this symmetry by
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computing only the positive frequency components, up to and including the
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Nyquist frequency. Thus, ``n`` input points produce ``n/2+1`` complex
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output points. The inverses of this family assumes the same symmetry of
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its input, and for an output of ``n`` points uses ``n/2+1`` input points.
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Correspondingly, when the spectrum is purely real, the signal is
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Hermitian. The `hfft` family of functions exploits this symmetry by
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using ``n/2+1`` complex points in the input (time) domain for ``n`` real
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points in the frequency domain.
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In higher dimensions, FFTs are used, e.g., for image analysis and
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filtering. The computational efficiency of the FFT means that it can
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also be a faster way to compute large convolutions, using the property
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that a convolution in the time domain is equivalent to a point-by-point
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multiplication in the frequency domain.
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Higher dimensions
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-----------------
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In two dimensions, the DFT is defined as
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.. math::
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A_{kl} = \\sum_{m=0}^{M-1} \\sum_{n=0}^{N-1}
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a_{mn}\\exp\\left\\{-2\\pi i \\left({mk\\over M}+{nl\\over N}\\right)\\right\\}
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\\qquad k = 0, \\ldots, M-1;\\quad l = 0, \\ldots, N-1,
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which extends in the obvious way to higher dimensions, and the inverses
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in higher dimensions also extend in the same way.
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References
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----------
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.. [CT] Cooley, James W., and John W. Tukey, 1965, "An algorithm for the
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machine calculation of complex Fourier series," *Math. Comput.*
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19: 297-301.
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.. [NR] Press, W., Teukolsky, S., Vetterline, W.T., and Flannery, B.P.,
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2007, *Numerical Recipes: The Art of Scientific Computing*, ch.
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12-13. Cambridge Univ. Press, Cambridge, UK.
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Examples
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--------
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For examples, see the various functions.
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"""
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from __future__ import division, absolute_import, print_function
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from ._pocketfft import *
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from .helper import *
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from numpy._pytesttester import PytestTester
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test = PytestTester(__name__)
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del PytestTester
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