Fourier Transforms (:mod:`scipy.fft`)
=========================================

.. sectionauthor:: SciPy Developers

.. currentmodule:: scipy.fft

.. contents::


Fourier analysis is a method for expressing a function as a sum of periodic
components, and for recovering the signal 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 [CT65]_. Press et al. [NR07]_ provide an accessible introduction to
Fourier analysis and its applications.


.. _tutorial_FFT:

Fast Fourier transforms
-----------------------

1-D discrete Fourier transforms
___________________________________________

The FFT `y[k]` of length :math:`N` of the length-:math:`N` sequence `x[n]` is
defined as

.. math::

    y[k] = \sum_{n=0}^{N-1} e^{-2 \pi j \frac{k n}{N} } x[n] \, ,

and the inverse transform is defined as follows

.. math::

    x[n] = \frac{1}{N} \sum_{k=0}^{N-1} e^{2 \pi j \frac{k n}{N} } y[k] \, .

These transforms can be calculated by means of :func:`fft` and :func:`ifft`,
respectively, as shown in the following example.

>>> from scipy.fft import fft, ifft
>>> import numpy as np
>>> x = np.array([1.0, 2.0, 1.0, -1.0, 1.5])
>>> y = fft(x)
>>> y
array([ 4.5       +0.j        ,  2.08155948-1.65109876j,
       -1.83155948+1.60822041j, -1.83155948-1.60822041j,
        2.08155948+1.65109876j])
>>> yinv = ifft(y)
>>> yinv
array([ 1.0+0.j,  2.0+0.j,  1.0+0.j, -1.0+0.j,  1.5+0.j])


From the definition of the FFT it can be seen that

.. math::

    y[0] = \sum_{n=0}^{N-1} x[n] \, .

In the example

>>> np.sum(x)
4.5

which corresponds to :math:`y[0]`. For N even, the elements
:math:`y[1]...y[N/2-1]` contain the positive-frequency terms, and the elements
:math:`y[N/2]...y[N-1]` contain the negative-frequency terms, in order of
decreasingly negative frequency. For N odd, the elements
:math:`y[1]...y[(N-1)/2]` contain the positive-frequency terms, and the
elements :math:`y[(N+1)/2]...y[N-1]` contain the negative-frequency terms, in
order of decreasingly negative frequency.

In case the sequence x is real-valued, the values of :math:`y[n]` for positive
frequencies is the conjugate of the values :math:`y[n]` for negative
frequencies (because the spectrum is symmetric). Typically, only the FFT
corresponding to positive frequencies is plotted.

The example plots the FFT of the sum of two sines.

.. plot::
    :alt: "This code generates an X-Y plot showing amplitude on the Y axis vs frequency on the X axis. A single blue trace has an amplitude of zero all the way across with the exception of two peaks. The taller first peak is at 50 Hz with a second peak at 80 Hz."

    >>> from scipy.fft import fft, fftfreq
    >>> import numpy as np
    >>> # Number of sample points
    >>> N = 600
    >>> # sample spacing
    >>> T = 1.0 / 800.0
    >>> x = np.linspace(0.0, N*T, N, endpoint=False)
    >>> y = np.sin(50.0 * 2.0*np.pi*x) + 0.5*np.sin(80.0 * 2.0*np.pi*x)
    >>> yf = fft(y)
    >>> xf = fftfreq(N, T)[:N//2]
    >>> import matplotlib.pyplot as plt
    >>> plt.plot(xf, 2.0/N * np.abs(yf[0:N//2]))
    >>> plt.grid()
    >>> plt.show()


The FFT input signal is inherently truncated. This truncation can be modeled
as multiplication of an infinite signal with a rectangular window function. In
the spectral domain this multiplication becomes convolution of the signal
spectrum with the window function spectrum, being of form :math:`\sin(x)/x`.
This convolution is the cause of an effect called spectral leakage (see
[WPW]_). Windowing the signal with a dedicated window function helps mitigate
spectral leakage. The example below uses a Blackman window from scipy.signal
and shows the effect of windowing (the zero component of the FFT has been
truncated for illustrative purposes).

.. plot::
    :alt: "This code generates an X-Y log-linear plot with amplitude on the Y axis vs frequency on the X axis. The first trace is the FFT with two peaks at 50 and 80 Hz and a noise floor around an amplitude of 1e-2. The second trace is the windowed FFT and has the same two peaks but the noise floor is much lower around an amplitude of 1e-7 due to the window function."

    >>> from scipy.fft import fft, fftfreq
    >>> import numpy as np
    >>> # Number of sample points
    >>> N = 600
    >>> # sample spacing
    >>> T = 1.0 / 800.0
    >>> x = np.linspace(0.0, N*T, N, endpoint=False)
    >>> y = np.sin(50.0 * 2.0*np.pi*x) + 0.5*np.sin(80.0 * 2.0*np.pi*x)
    >>> yf = fft(y)
    >>> from scipy.signal.windows import blackman
    >>> w = blackman(N)
    >>> ywf = fft(y*w)
    >>> xf = fftfreq(N, T)[:N//2]
    >>> import matplotlib.pyplot as plt
    >>> plt.semilogy(xf[1:N//2], 2.0/N * np.abs(yf[1:N//2]), '-b')
    >>> plt.semilogy(xf[1:N//2], 2.0/N * np.abs(ywf[1:N//2]), '-r')
    >>> plt.legend(['FFT', 'FFT w. window'])
    >>> plt.grid()
    >>> plt.show()


In case the sequence x is complex-valued, the spectrum is no longer symmetric.
To simplify working with the FFT functions, scipy provides the following two
helper functions.

The function :func:`fftfreq` returns the FFT sample frequency points.

>>> from scipy.fft import fftfreq
>>> freq = fftfreq(8, 0.125)
>>> freq
array([ 0., 1., 2., 3., -4., -3., -2., -1.])

In a similar spirit, the function :func:`fftshift` allows swapping the lower
and upper halves of a vector, so that it becomes suitable for display.

>>> from scipy.fft import fftshift
>>> x = np.arange(8)
>>> fftshift(x)
array([4, 5, 6, 7, 0, 1, 2, 3])

The example below plots the FFT of two complex exponentials; note the
asymmetric spectrum.

.. plot::
    :alt: "This code generates an X-Y plot with amplitude on the Y axis vs frequency on the X axis. The trace is zero-valued across the plot except for two sharp peaks at -80 and 50 Hz. The 50 Hz peak on the right is twice as tall."

    >>> from scipy.fft import fft, fftfreq, fftshift
    >>> import numpy as np
    >>> # number of signal points
    >>> N = 400
    >>> # sample spacing
    >>> T = 1.0 / 800.0
    >>> x = np.linspace(0.0, N*T, N, endpoint=False)
    >>> y = np.exp(50.0 * 1.j * 2.0*np.pi*x) + 0.5*np.exp(-80.0 * 1.j * 2.0*np.pi*x)
    >>> yf = fft(y)
    >>> xf = fftfreq(N, T)
    >>> xf = fftshift(xf)
    >>> yplot = fftshift(yf)
    >>> import matplotlib.pyplot as plt
    >>> plt.plot(xf, 1.0/N * np.abs(yplot))
    >>> plt.grid()
    >>> plt.show()


The function :func:`rfft` calculates the FFT of a real sequence and outputs the
complex FFT coefficients :math:`y[n]` for only half of the frequency range. The
remaining negative frequency components are implied by the Hermitian symmetry of
the FFT for a real input (``y[n] = conj(y[-n])``). In case of N being even:
:math:`[Re(y[0]) + 0j, y[1], ..., Re(y[N/2]) + 0j]`; in case of N being odd
:math:`[Re(y[0]) + 0j, y[1], ..., y[N/2]`. The terms shown explicitly as
:math:`Re(y[k]) + 0j` are restricted to be purely real since, by the hermitian
property, they are their own complex conjugate.

The corresponding function :func:`irfft` calculates the IFFT of the FFT
coefficients with this special ordering.

>>> from scipy.fft import fft, rfft, irfft
>>> x = np.array([1.0, 2.0, 1.0, -1.0, 1.5, 1.0])
>>> fft(x)
array([ 5.5 +0.j        ,  2.25-0.4330127j , -2.75-1.29903811j,
        1.5 +0.j        , -2.75+1.29903811j,  2.25+0.4330127j ])
>>> yr = rfft(x)
>>> yr
array([ 5.5 +0.j        ,  2.25-0.4330127j , -2.75-1.29903811j,
        1.5 +0.j        ])
>>> irfft(yr)
array([ 1. ,  2. ,  1. , -1. ,  1.5,  1. ])
>>> x = np.array([1.0, 2.0, 1.0, -1.0, 1.5])
>>> fft(x)
array([ 4.5       +0.j        ,  2.08155948-1.65109876j,
       -1.83155948+1.60822041j, -1.83155948-1.60822041j,
        2.08155948+1.65109876j])
>>> yr = rfft(x)
>>> yr
array([ 4.5       +0.j        ,  2.08155948-1.65109876j,
        -1.83155948+1.60822041j])

Notice that the :func:`rfft` of odd and even length signals are of the same shape.
By default, :func:`irfft` assumes the output signal should be of even length. And
so, for odd signals, it will give the wrong result:

>>> irfft(yr)
array([ 1.70788987,  2.40843925, -0.37366961,  0.75734049])

To recover the original odd-length signal, we **must** pass the output shape by
the `n` parameter.

>>> irfft(yr, n=len(x))
array([ 1. ,  2. ,  1. , -1. ,  1.5])



2- and N-D discrete Fourier transforms
_________________________________________________

The functions :func:`fft2` and :func:`ifft2` provide 2-D FFT and
IFFT, respectively. Similarly, :func:`fftn` and :func:`ifftn` provide
N-D FFT, and IFFT, respectively.

For real-input signals, similarly to :func:`rfft`, we have the functions
:func:`rfft2` and :func:`irfft2` for 2-D real transforms;
:func:`rfftn` and :func:`irfftn` for N-D real transforms.

The example below demonstrates a 2-D IFFT and plots the resulting
(2-D) time-domain signals.

.. plot::
    :alt: "This code generates six heatmaps arranged in a 2x3 grid. The top row shows mostly blank canvases with the exception of two tiny red peaks on each image. The bottom row shows the real-part of the inverse FFT of each image above it. The first column has two dots arranged horizontally in the top image and in the bottom image a smooth grayscale plot of 5 black vertical stripes representing the 2-D time domain signal. The second column has two dots arranged vertically in the top image and in the bottom image a smooth grayscale plot of 5 horizontal black stripes representing the 2-D time domain signal. In the last column the top image has two dots diagonally located; the corresponding image below has perhaps 20 black stripes at a 60 degree angle."

    >>> from scipy.fft import ifftn
    >>> import matplotlib.pyplot as plt
    >>> import matplotlib.cm as cm
    >>> import numpy as np
    >>> N = 30
    >>> f, ((ax1, ax2, ax3), (ax4, ax5, ax6)) = plt.subplots(2, 3, sharex='col', sharey='row')
    >>> xf = np.zeros((N,N))
    >>> xf[0, 5] = 1
    >>> xf[0, N-5] = 1
    >>> Z = ifftn(xf)
    >>> ax1.imshow(xf, cmap=cm.Reds)
    >>> ax4.imshow(np.real(Z), cmap=cm.gray)
    >>> xf = np.zeros((N, N))
    >>> xf[5, 0] = 1
    >>> xf[N-5, 0] = 1
    >>> Z = ifftn(xf)
    >>> ax2.imshow(xf, cmap=cm.Reds)
    >>> ax5.imshow(np.real(Z), cmap=cm.gray)
    >>> xf = np.zeros((N, N))
    >>> xf[5, 10] = 1
    >>> xf[N-5, N-10] = 1
    >>> Z = ifftn(xf)
    >>> ax3.imshow(xf, cmap=cm.Reds)
    >>> ax6.imshow(np.real(Z), cmap=cm.gray)
    >>> plt.show()


Discrete Cosine Transforms
--------------------------

SciPy provides a DCT with the function :func:`dct` and a corresponding IDCT
with the function :func:`idct`. There are 8 types of the DCT [WPC]_, [Mak]_;
however, only the first 4 types are implemented in scipy. "The" DCT generally
refers to DCT type 2, and "the" Inverse DCT generally refers to DCT type 3. In
addition, the DCT coefficients can be normalized differently (for most types,
scipy provides ``None`` and ``ortho``). Two parameters of the dct/idct
function calls allow setting the DCT type and coefficient normalization.

For a single dimension array x, dct(x, norm='ortho') is equal to
MATLAB dct(x).


Type I DCT
__________

SciPy uses the following definition of the unnormalized DCT-I
(``norm=None``):

.. math::

    y[k] = x_0 + (-1)^k x_{N-1} + 2\sum_{n=1}^{N-2} x[n]
    \cos\left(\frac{\pi nk}{N-1}\right),
    \qquad 0 \le k < N.

Note that the DCT-I is only supported for input size > 1.

Type II DCT
___________

SciPy uses the following definition of the unnormalized DCT-II
(``norm=None``):

.. math::

    y[k] = 2 \sum_{n=0}^{N-1} x[n] \cos \left({\pi(2n+1)k \over 2N} \right)
    \qquad 0 \le k < N.

In case of the normalized DCT (``norm='ortho'``), the DCT coefficients
:math:`y[k]` are multiplied by a scaling factor `f`:

.. math::

    f = \begin{cases} \sqrt{1/(4N)}, & \text{if $k = 0$} \\    \sqrt{1/(2N)},
    & \text{otherwise} \end{cases} \, .

In this case, the DCT "base functions" :math:`\phi_k[n] = 2 f \cos
\left({\pi(2n+1)k \over 2N} \right)` become orthonormal:

.. math::

   \sum_{n=0}^{N-1} \phi_k[n] \phi_l[n] = \delta_{lk}.


Type III DCT
____________

SciPy uses the following definition of the unnormalized DCT-III
(``norm=None``):

.. math::

    y[k] = x_0 + 2 \sum_{n=1}^{N-1} x[n] \cos\left({\pi n(2k+1) \over 2N}\right)
    \qquad 0 \le k < N,

or, for ``norm='ortho'``:

.. math::

    y[k] = {x_0\over\sqrt{N}} + {2\over\sqrt{N}} \sum_{n=1}^{N-1} x[n]
    \cos\left({\pi n(2k+1) \over 2N}\right) \qquad 0 \le k < N.


Type IV DCT
___________

SciPy uses the following definition of the unnormalized DCT-IV
(``norm=None``):

.. math::

    y[k] = 2 \sum_{n=0}^{N-1} x[n] \cos\left({\pi (2n+1)(2k+1) \over 4N}\right)
    \qquad 0 \le k < N,

or, for ``norm='ortho'``:

.. math::

    y[k] = \sqrt{2\over N}\sum_{n=0}^{N-1} x[n] \cos\left({\pi (2n+1)(2k+1) \over 4N}\right)
    \qquad 0 \le k < N


DCT and IDCT
____________


The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up to a
factor of `2N`. The orthonormalized DCT-III is exactly the inverse of the
orthonormalized DCT- II. The function :func:`idct` performs the mappings between
the DCT and IDCT types, as well as the correct normalization.

The following example shows the relation between DCT and IDCT for different
types and normalizations.

>>> from scipy.fft import dct, idct
>>> x = np.array([1.0, 2.0, 1.0, -1.0, 1.5])

The DCT-II and DCT-III are each other's inverses, so for an orthonormal transform
we return back to the original signal.

>>> dct(dct(x, type=2, norm='ortho'), type=3, norm='ortho')
array([ 1. ,  2. ,  1. , -1. ,  1.5])

Doing the same under default normalization, however, we pick up an extra scaling
factor of :math:`2N=10` since the forward transform is unnormalized.

>>> dct(dct(x, type=2), type=3)
array([ 10.,  20.,  10., -10.,  15.])

For this reason, we should use the function `idct` using the same type for both,
giving a correctly normalized result.

>>> # Normalized inverse: no scaling factor
>>> idct(dct(x, type=2), type=2)
array([ 1. ,  2. ,  1. , -1. ,  1.5])

Analogous results can be seen for the DCT-I, which is its own inverse up to a
factor of :math:`2(N-1)`.

>>> dct(dct(x, type=1, norm='ortho'), type=1, norm='ortho')
array([ 1. ,  2. ,  1. , -1. ,  1.5])
>>> # Unnormalized round-trip via DCT-I: scaling factor 2*(N-1) = 8
>>> dct(dct(x, type=1), type=1)
array([ 8. ,  16.,  8. , -8. ,  12.])
>>> # Normalized inverse: no scaling factor
>>> idct(dct(x, type=1), type=1)
array([ 1. ,  2. ,  1. , -1. ,  1.5])

And for the DCT-IV, which is also its own inverse up to a factor of :math:`2N`.

>>> dct(dct(x, type=4, norm='ortho'), type=4, norm='ortho')
array([ 1. ,  2. ,  1. , -1. ,  1.5])
>>> # Unnormalized round-trip via DCT-IV: scaling factor 2*N = 10
>>> dct(dct(x, type=4), type=4)
array([ 10.,  20.,  10., -10.,  15.])
>>> # Normalized inverse: no scaling factor
>>> idct(dct(x, type=4), type=4)
array([ 1. ,  2. ,  1. , -1. ,  1.5])

Example
_______

The DCT exhibits the "energy compaction property", meaning that for many
signals only the first few DCT coefficients have significant magnitude.
Zeroing out the other coefficients leads to a small reconstruction error, a
fact which is exploited in lossy signal compression (e.g. JPEG compression).

The example below shows a signal x and two reconstructions (:math:`x_{20}` and
:math:`x_{15}`) from the signal's DCT coefficients. The signal :math:`x_{20}`
is reconstructed from the first 20 DCT coefficients, :math:`x_{15}` is
reconstructed from the first 15 DCT coefficients. It can be seen that the
relative error of using 20 coefficients is still very small (~0.1%), but
provides a five-fold compression rate.


.. plot::
    :alt: "This code generates an X-Y plot showing amplitude on the Y axis and time on the X axis. The first blue trace is the original signal and starts at amplitude 1 and oscillates down to 0 amplitude over the duration of the plot resembling a frequency chirp. The second red trace is the x_20 reconstruction using the DCT and closely follows the original signal in the high amplitude region but it is unclear to the right side of the plot. The third green trace is the x_15 reconstruction using the DCT and is less precise than the x_20 reconstruction but still similar to x."

    >>> from scipy.fft import dct, idct
    >>> import matplotlib.pyplot as plt
    >>> N = 100
    >>> t = np.linspace(0,20,N, endpoint=False)
    >>> x = np.exp(-t/3)*np.cos(2*t)
    >>> y = dct(x, norm='ortho')
    >>> window = np.zeros(N)
    >>> window[:20] = 1
    >>> yr = idct(y*window, norm='ortho')
    >>> sum(abs(x-yr)**2) / sum(abs(x)**2)
    0.0009872817275276098
    >>> plt.plot(t, x, '-bx')
    >>> plt.plot(t, yr, 'ro')
    >>> window = np.zeros(N)
    >>> window[:15] = 1
    >>> yr = idct(y*window, norm='ortho')
    >>> sum(abs(x-yr)**2) / sum(abs(x)**2)
    0.06196643004256714
    >>> plt.plot(t, yr, 'g+')
    >>> plt.legend(['x', '$x_{20}$', '$x_{15}$'])
    >>> plt.grid()
    >>> plt.show()

Discrete Sine Transforms
------------------------

SciPy provides a DST [Mak]_ with the function :func:`dst` and a corresponding IDST
with the function :func:`idst`.

There are, theoretically, 8 types of the DST for different combinations of
even/odd boundary conditions and boundary offsets [WPS]_, only the first 4
types are implemented in scipy.

Type I DST
__________

DST-I assumes the input is odd around n=-1 and n=N. SciPy uses the following
definition of the unnormalized DST-I (``norm=None``):

.. math::

    y[k] = 2\sum_{n=0}^{N-1} x[n]  \sin\left( \pi {(n+1) (k+1)}\over{N+1}
    \right), \qquad 0 \le k < N.

Note also that the DST-I is only supported for input size > 1. The
(unnormalized) DST-I is its own inverse, up to a factor of `2(N+1)`.

Type II DST
___________

DST-II assumes the input is odd around n=-1/2 and even around n=N. SciPy uses
the following definition of the unnormalized DST-II (``norm=None``):

.. math::

    y[k] = 2 \sum_{n=0}^{N-1} x[n]  \sin\left( {\pi (n+1/2)(k+1)} \over N
    \right), \qquad 0 \le k < N.

Type III DST
____________

DST-III assumes the input is odd around n=-1 and even around n=N-1. SciPy uses
the following definition of the unnormalized DST-III (``norm=None``):

.. math::

    y[k] = (-1)^k x[N-1] + 2 \sum_{n=0}^{N-2} x[n] \sin \left( {\pi
    (n+1)(k+1/2)} \over N \right), \qquad 0 \le k < N.

Type IV DST
___________

SciPy uses the following definition of the unnormalized DST-IV
(``norm=None``):

.. math::

    y[k] = 2 \sum_{n=0}^{N-1} x[n] \sin\left({\pi (2n+1)(2k+1) \over 4N}\right)
    \qquad 0 \le k < N,

or, for ``norm='ortho'``:

.. math::

    y[k] = \sqrt{2\over N}\sum_{n=0}^{N-1} x[n] \sin\left({\pi (2n+1)(2k+1) \over 4N}\right)
    \qquad 0 \le k < N,


DST and IDST
____________


The following example shows the relation between DST and IDST for
different types and normalizations.

>>> from scipy.fft import dst, idst
>>> x = np.array([1.0, 2.0, 1.0, -1.0, 1.5])

The DST-II and DST-III are each other's inverses, so for an orthonormal transform
we return back to the original signal.

>>> dst(dst(x, type=2, norm='ortho'), type=3, norm='ortho')
array([ 1. ,  2. ,  1. , -1. ,  1.5])

Doing the same under default normalization, however, we pick up an extra scaling
factor of :math:`2N=10` since the forward transform is unnormalized.

>>> dst(dst(x, type=2), type=3)
array([ 10.,  20.,  10., -10.,  15.])

For this reason, we should use the function `idst` using the same type for both,
giving a correctly normalized result.

>>> idst(dst(x, type=2), type=2)
array([ 1. ,  2. ,  1. , -1. ,  1.5])

Analogous results can be seen for the DST-I, which is its own inverse up to a
factor of :math:`2(N-1)`.

>>> dst(dst(x, type=1, norm='ortho'), type=1, norm='ortho')
array([ 1. ,  2. ,  1. , -1. ,  1.5])
>>>  # scaling factor 2*(N+1) = 12
>>> dst(dst(x, type=1), type=1)
array([ 12.,  24.,  12., -12.,  18.])
>>>  # no scaling factor
>>> idst(dst(x, type=1), type=1)
array([ 1. ,  2. ,  1. , -1. ,  1.5])

And for the DST-IV, which is also its own inverse up to a factor of :math:`2N`.

>>> dst(dst(x, type=4, norm='ortho'), type=4, norm='ortho')
array([ 1. ,  2. ,  1. , -1. ,  1.5])
>>>  # scaling factor 2*N = 10
>>> dst(dst(x, type=4), type=4)
array([ 10.,  20.,  10., -10.,  15.])
>>>  # no scaling factor
>>> idst(dst(x, type=4), type=4)
array([ 1. ,  2. ,  1. , -1. ,  1.5])


Fast Hankel Transform
---------------------

SciPy provides the functions ``fht`` and ``ifht`` to perform the Fast
Hankel Transform (FHT) and its inverse (IFHT) on logarithmically-spaced input
arrays.

The FHT is the discretised version of the continuous Hankel transform defined
by [Ham00]_

.. math::

    A(k) = \int_{0}^{\infty} \! a(r) \, J_{\mu}(kr) \, k \, dr \;,

with :math:`J_{\mu}` the Bessel function of order :math:`\mu`. Under a change
of variables :math:`r \to \log r`, :math:`k \to \log k`, this becomes

.. math::

    A(e^{\log k})
    = \int_{0}^{\infty} \! a(e^{\log r}) \, J_{\mu}(e^{\log k + \log r})
                                        \, e^{\log k + \log r} \, d{\log r}

which is a convolution in logarithmic space. The FHT algorithm uses the FFT
to perform this convolution on discrete input data.

Care must be taken to minimise numerical ringing due to the circular nature
of FFT convolution. To ensure that the low-ringing condition [Ham00]_ holds,
the output array can be slightly shifted by an offset computed using the
``fhtoffset`` function.


References
----------

.. [CT65] Cooley, James W., and John W. Tukey, 1965, "An algorithm for the
        machine calculation of complex Fourier series," *Math. Comput.*
        19: 297-301.

.. [NR07] 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.

.. [Mak] J. Makhoul, 1980, 'A Fast Cosine Transform in One and Two Dimensions',
       `IEEE Transactions on acoustics, speech and signal processing`
       vol. 28(1), pp. 27-34, :doi:`10.1109/TASSP.1980.1163351`

.. [Ham00] A. J. S. Hamilton, 2000, "Uncorrelated modes of the non-linear power
       spectrum", *MNRAS*, 312, 257. :doi:`10.1046/j.1365-8711.2000.03071.x`

.. [WPW] https://en.wikipedia.org/wiki/Window_function

.. [WPC] https://en.wikipedia.org/wiki/Discrete_cosine_transform

.. [WPS] https://en.wikipedia.org/wiki/Discrete_sine_transform
