mirror of
https://github.com/PiBrewing/craftbeerpi4.git
synced 2024-11-14 11:08:15 +01:00
199 lines
7.3 KiB
Python
199 lines
7.3 KiB
Python
"""
|
|
Discrete Fourier Transform (:mod:`numpy.fft`)
|
|
=============================================
|
|
|
|
.. currentmodule:: numpy.fft
|
|
|
|
Standard FFTs
|
|
-------------
|
|
|
|
.. autosummary::
|
|
:toctree: generated/
|
|
|
|
fft Discrete Fourier transform.
|
|
ifft Inverse discrete Fourier transform.
|
|
fft2 Discrete Fourier transform in two dimensions.
|
|
ifft2 Inverse discrete Fourier transform in two dimensions.
|
|
fftn Discrete Fourier transform in N-dimensions.
|
|
ifftn Inverse discrete Fourier transform in N dimensions.
|
|
|
|
Real FFTs
|
|
---------
|
|
|
|
.. autosummary::
|
|
:toctree: generated/
|
|
|
|
rfft Real discrete Fourier transform.
|
|
irfft Inverse real discrete Fourier transform.
|
|
rfft2 Real discrete Fourier transform in two dimensions.
|
|
irfft2 Inverse real discrete Fourier transform in two dimensions.
|
|
rfftn Real discrete Fourier transform in N dimensions.
|
|
irfftn Inverse real discrete Fourier transform in N dimensions.
|
|
|
|
Hermitian FFTs
|
|
--------------
|
|
|
|
.. autosummary::
|
|
:toctree: generated/
|
|
|
|
hfft Hermitian discrete Fourier transform.
|
|
ihfft Inverse Hermitian discrete Fourier transform.
|
|
|
|
Helper routines
|
|
---------------
|
|
|
|
.. autosummary::
|
|
:toctree: generated/
|
|
|
|
fftfreq Discrete Fourier Transform sample frequencies.
|
|
rfftfreq DFT sample frequencies (for usage with rfft, irfft).
|
|
fftshift Shift zero-frequency component to center of spectrum.
|
|
ifftshift Inverse of fftshift.
|
|
|
|
|
|
Background information
|
|
----------------------
|
|
|
|
Fourier analysis is fundamentally a method for expressing a function as a
|
|
sum of periodic components, and for recovering the function from those
|
|
components. When both the function and its Fourier transform are
|
|
replaced with discretized counterparts, it is called the discrete Fourier
|
|
transform (DFT). The DFT has become a mainstay of numerical computing in
|
|
part because of a very fast algorithm for computing it, called the Fast
|
|
Fourier Transform (FFT), which was known to Gauss (1805) and was brought
|
|
to light in its current form by Cooley and Tukey [CT]_. Press et al. [NR]_
|
|
provide an accessible introduction to Fourier analysis and its
|
|
applications.
|
|
|
|
Because the discrete Fourier transform separates its input into
|
|
components that contribute at discrete frequencies, it has a great number
|
|
of applications in digital signal processing, e.g., for filtering, and in
|
|
this context the discretized input to the transform is customarily
|
|
referred to as a *signal*, which exists in the *time domain*. The output
|
|
is called a *spectrum* or *transform* and exists in the *frequency
|
|
domain*.
|
|
|
|
Implementation details
|
|
----------------------
|
|
|
|
There are many ways to define the DFT, varying in the sign of the
|
|
exponent, normalization, etc. In this implementation, the DFT is defined
|
|
as
|
|
|
|
.. math::
|
|
A_k = \\sum_{m=0}^{n-1} a_m \\exp\\left\\{-2\\pi i{mk \\over n}\\right\\}
|
|
\\qquad k = 0,\\ldots,n-1.
|
|
|
|
The DFT is in general defined for complex inputs and outputs, and a
|
|
single-frequency component at linear frequency :math:`f` is
|
|
represented by a complex exponential
|
|
:math:`a_m = \\exp\\{2\\pi i\\,f m\\Delta t\\}`, where :math:`\\Delta t`
|
|
is the sampling interval.
|
|
|
|
The values in the result follow so-called "standard" order: If ``A =
|
|
fft(a, n)``, then ``A[0]`` contains the zero-frequency term (the sum of
|
|
the signal), which is always purely real for real inputs. Then ``A[1:n/2]``
|
|
contains the positive-frequency terms, and ``A[n/2+1:]`` contains the
|
|
negative-frequency terms, in order of decreasingly negative frequency.
|
|
For an even number of input points, ``A[n/2]`` represents both positive and
|
|
negative Nyquist frequency, and is also purely real for real input. For
|
|
an odd number of input points, ``A[(n-1)/2]`` contains the largest positive
|
|
frequency, while ``A[(n+1)/2]`` contains the largest negative frequency.
|
|
The routine ``np.fft.fftfreq(n)`` returns an array giving the frequencies
|
|
of corresponding elements in the output. The routine
|
|
``np.fft.fftshift(A)`` shifts transforms and their frequencies to put the
|
|
zero-frequency components in the middle, and ``np.fft.ifftshift(A)`` undoes
|
|
that shift.
|
|
|
|
When the input `a` is a time-domain signal and ``A = fft(a)``, ``np.abs(A)``
|
|
is its amplitude spectrum and ``np.abs(A)**2`` is its power spectrum.
|
|
The phase spectrum is obtained by ``np.angle(A)``.
|
|
|
|
The inverse DFT is defined as
|
|
|
|
.. math::
|
|
a_m = \\frac{1}{n}\\sum_{k=0}^{n-1}A_k\\exp\\left\\{2\\pi i{mk\\over n}\\right\\}
|
|
\\qquad m = 0,\\ldots,n-1.
|
|
|
|
It differs from the forward transform by the sign of the exponential
|
|
argument and the default normalization by :math:`1/n`.
|
|
|
|
Type Promotion
|
|
--------------
|
|
|
|
`numpy.fft` promotes ``float32`` and ``complex64`` arrays to ``float64`` and
|
|
``complex128`` arrays respectively. For an FFT implementation that does not
|
|
promote input arrays, see `scipy.fftpack`.
|
|
|
|
Normalization
|
|
-------------
|
|
|
|
The default normalization has the direct transforms unscaled and the inverse
|
|
transforms are scaled by :math:`1/n`. It is possible to obtain unitary
|
|
transforms by setting the keyword argument ``norm`` to ``"ortho"`` (default is
|
|
`None`) so that both direct and inverse transforms will be scaled by
|
|
:math:`1/\\sqrt{n}`.
|
|
|
|
Real and Hermitian transforms
|
|
-----------------------------
|
|
|
|
When the input is purely real, its transform is Hermitian, i.e., the
|
|
component at frequency :math:`f_k` is the complex conjugate of the
|
|
component at frequency :math:`-f_k`, which means that for real
|
|
inputs there is no information in the negative frequency components that
|
|
is not already available from the positive frequency components.
|
|
The family of `rfft` functions is
|
|
designed to operate on real inputs, and exploits this symmetry by
|
|
computing only the positive frequency components, up to and including the
|
|
Nyquist frequency. Thus, ``n`` input points produce ``n/2+1`` complex
|
|
output points. The inverses of this family assumes the same symmetry of
|
|
its input, and for an output of ``n`` points uses ``n/2+1`` input points.
|
|
|
|
Correspondingly, when the spectrum is purely real, the signal is
|
|
Hermitian. The `hfft` family of functions exploits this symmetry by
|
|
using ``n/2+1`` complex points in the input (time) domain for ``n`` real
|
|
points in the frequency domain.
|
|
|
|
In higher dimensions, FFTs are used, e.g., for image analysis and
|
|
filtering. The computational efficiency of the FFT means that it can
|
|
also be a faster way to compute large convolutions, using the property
|
|
that a convolution in the time domain is equivalent to a point-by-point
|
|
multiplication in the frequency domain.
|
|
|
|
Higher dimensions
|
|
-----------------
|
|
|
|
In two dimensions, the DFT is defined as
|
|
|
|
.. math::
|
|
A_{kl} = \\sum_{m=0}^{M-1} \\sum_{n=0}^{N-1}
|
|
a_{mn}\\exp\\left\\{-2\\pi i \\left({mk\\over M}+{nl\\over N}\\right)\\right\\}
|
|
\\qquad k = 0, \\ldots, M-1;\\quad l = 0, \\ldots, N-1,
|
|
|
|
which extends in the obvious way to higher dimensions, and the inverses
|
|
in higher dimensions also extend in the same way.
|
|
|
|
References
|
|
----------
|
|
|
|
.. [CT] Cooley, James W., and John W. Tukey, 1965, "An algorithm for the
|
|
machine calculation of complex Fourier series," *Math. Comput.*
|
|
19: 297-301.
|
|
|
|
.. [NR] Press, W., Teukolsky, S., Vetterline, W.T., and Flannery, B.P.,
|
|
2007, *Numerical Recipes: The Art of Scientific Computing*, ch.
|
|
12-13. Cambridge Univ. Press, Cambridge, UK.
|
|
|
|
Examples
|
|
--------
|
|
|
|
For examples, see the various functions.
|
|
|
|
"""
|
|
|
|
from ._pocketfft import *
|
|
from .helper import *
|
|
|
|
from numpy._pytesttester import PytestTester
|
|
test = PytestTester(__name__)
|
|
del PytestTester
|