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.. _c-fftw:
*******************
Discrete Transforms
*******************
The :mod:`cvxopt.fftw` module is an interface to the FFTW library and
contains routines for discrete Fourier, cosine, and sine transforms.
This module is optional, and only installed when the FFTW library is made
available during the CVXOPT installation.
.. seealso::
`FFTW3 code, documentation, copyright and license
<http://www.fftw.org>`_
Discrete Fourier Transform
==========================
.. function:: cvxopt.fftw.dft(X)
Replaces the columns of a dense complex matrix with their discrete
Fourier transforms: if ``X`` has :math:`n` rows,
.. math::
X[k,:] := \sum_{j=0}^{n-1} e^{-2\pi j k \sqrt{-1}/n} X[j,:],
\qquad k=0,\ldots,n-1.
.. function:: cvxopt.fftw.idft(X)
Replaces the columns of a dense complex matrix with their inverse
discrete Fourier transforms: if ``X`` has :math:`n` rows,
.. math::
X[k,:] :=
\frac{1}{n} \sum_{j=0}^{n-1} e^{2\pi j k \sqrt{-1}/n} X[j,:],
\qquad k=0,\ldots,n-1.
The module also includes a discrete *N*-dimensional Fourier transform.
The input matrix is interpreted as an *N*-dimensional matrix stored
in column-major order. The discrete *N*-dimensional Fourier transform
computes the corresponding one-dimensional transform along each dimension.
For example, the two-dimensional transform applies a one-dimensional
transform to all the columns of the matrix, followed by a one-dimensional
transform to all the rows of the matrix.
.. function:: cvxopt.fftw.dftn(X[, dims = X.size])
Replaces a dense complex matrix with its *N*-dimensional discrete
Fourier transform. The dimensions of the *N*-dimensional matrix
are given by the *N*-tuple ``dims``. The two-dimensional transform is
computed as ``dftn(X, X.size)``.
.. function:: cvxopt.fftw.idftn(X[, dims = X.size])
Replaces a dense complex *N*-dimensional matrix with its inverse
*N*-dimensional discrete Fourier transform. The dimensions of the
matrix are given by the tuple ``dims``. The two-dimensional inverse
transform is computed as ``idftn(X, X.size)``.
Discrete Cosine Transform
=========================
.. function:: cvxopt.fftw.dct(X[, type = 2])
Replaces the columns of a dense real matrix with their discrete
cosine transforms. The second argument, an integer between 1 and 4,
denotes the type of transform (DCT-I, DCT-II, DCT-III, DCT-IV).
The DCT-I transform requires that the row dimension of ``X`` is at
least 2. These transforms are defined as follows (for a matrix with
:math:`n` rows).
.. math::
\mbox{DCT-I:} \qquad
X[k,:] & := X[0,:] + (-1)^k X[n-1,:] +
2 \sum_{j=1}^{n-2} X[j,:] \cos(\pi j k /(n-1)),
\qquad k=0,\ldots,n-1.\\
\mbox{DCT-II:} \qquad
X[k,:] & := 2 \sum_{j=0}^{n-1} X[j,:] \cos(\pi(j+1/2)k/n),
\qquad k=0,\ldots,n-1.\\
\mbox{DCT-III:} \qquad
X[k,:] & :=
X[0,:] + 2 \sum_{j=1}^{n-1} X[j,:] \cos(\pi j(k+1/2)/n),
\qquad k=0,\ldots,n-1.\\
\mbox{DCT-IV:} \qquad
X[k,:] & :=
2 \sum_{j=0}^{n-1} X[j,:] \cos(\pi (j+1/2)(k+1/2)/n),
\qquad k=0,\ldots,n-1.
.. function:: cvxopt.fftw.idct(X[, type = 2])
Replaces the columns of a dense real matrix with the inverses
of the discrete cosine transforms defined above.
The module also includes a discrete *N*-dimensional cosine transform.
The input matrix is interpreted as an *N*-dimensional matrix stored
in column-major order. The discrete *N*-dimensional cosine transform
computes the corresponding one-dimensional transform along each dimension.
For example, the two-dimensional transform applies a one-dimensional
transform to all the rows of the matrix, followed by a one-dimensional
transform to all the columns of the matrix.
.. function:: cvxopt.fftw.dctn(X[, dims = X.size, type = 2])
Replaces a dense real matrix with its *N*-dimensional discrete cosine
transform. The dimensions of the *N*-dimensional matrix are given by
the *N*-tuple ``dims``. The two-dimensional transform is computed as
``dctn(X, X.size)``.
.. function:: cvxopt.fftw.idctn(X[, dims = X.size, type = 2])
Replaces a dense real *N*-dimensional matrix with its inverse
*N*-dimensional discrete cosine transform. The dimensions of the
matrix are given by the tuple ``dims``. The two-dimensional inverse
transform is computed as ``idctn(X, X.size)``.
Discrete Sine Transform
=======================
.. function:: cvxopt.fftw.dst(X, dims[, type = 1])
Replaces the columns of a dense real matrix with their discrete
sine transforms. The second argument, an integer between 1 and 4,
denotes the type of transform (DST-I, DST-II, DST-III, DST-IV).
These transforms are defined as follows (for a matrix with :math:`n`
rows).
.. math::
\mbox{DST-I:} \qquad
X[k,:] & :=
2 \sum_{j=0}^{n-1} X[j,:] \sin(\pi(j+1)(k+1)/(n+1)),
\qquad k=0,\ldots,n-1.\\
\mbox{DST-II:} \qquad
X[k,:] & := 2 \sum_{j=0}^{n-1} X[j,:] \sin(\pi(j+1/2)(k+1)/n),
\qquad k=0,\ldots,n-1.\\
\mbox{DST-III:} \qquad
X[k,:] & := (-1)^k X[n-1,:] + 2 \sum_{j=0}^{n-2}
X[j,:] \sin(\pi(j+1)(k+1/2)/n), \qquad k=0,\ldots,n-1. \\
\mbox{DST-IV:} \qquad
X[k,:] & :=
2 \sum_{j=0}^{n-1} X[j,:] \sin(\pi (j+1/2)(k+1/2)/n),
\qquad k=0,\ldots,n-1.
.. function:: cvxopt.fftw.idst(X, dims[, type = 1])
Replaces the columns of a dense real matrix with the inverses of the
discrete sine transforms defined above.
The module also includes a discrete *N*-dimensional sine transform. The
input matrix is interpreted as an *N*-dimensional matrix stored in
column-major order. The discrete *N*-dimensional sine transform computes
the corresponding one-dimensional transform along each dimension. For
example, the two-dimensional transform applies a one-dimensional
transform to all the rows of the matrix, followed by a one-dimensional
transform to all the columns of the matrix.
.. function:: cvxopt.fftw.dstn(X[, dims = X.size, type = 2])
Replaces a dense real matrix with its *N*-dimensional discrete sine
transform. The dimensions of the *N*-dimensional matrix are given by
the *N*-tuple ``dims``. The two-dimensional transform is computed as
``dstn(X, X.size)``.
.. function:: cvxopt.fftw.idstn(X[, dims = X.size, type = 2])
Replaces a dense real *N*-dimensional matrix with its inverse
*N*-dimensional discrete sine transform. The dimensions of the
matrix are given by the tuple ``dims``. The two-dimensional inverse
transform is computed as ``idstn(X, X.size)``.
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