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@node Signal Processing
@chapter Signal Processing


This chapter describes the signal processing and fast Fourier
transform functions available in Octave.  Fast Fourier transforms are
computed with the @sc{fftw} or @sc{fftpack} libraries depending on how
Octave is built.
 


@c detrend scripts/signal/detrend.m
@anchor{doc-detrend}
@deftypefn {Function File} {} detrend (@var{x}, @var{p})
If @var{x} is a vector, @code{detrend (@var{x}, @var{p})} removes the
best fit of a polynomial of order @var{p} from the data @var{x}.

If @var{x} is a matrix, @code{detrend (@var{x}, @var{p})} does the same
for each column in @var{x}.

The second argument is optional.  If it is not specified, a value of 1
is assumed.  This corresponds to removing a linear trend.

The order of the polynomial can also be given as a string, in which case
@var{p} must be either @t{"constant"} (corresponds to @code{@var{p}=0}) or
@t{"linear"} (corresponds to @code{@var{p}=1}).
@seealso{@ref{doc-polyfit,,polyfit}}
@end deftypefn


@c fft src/DLD-FUNCTIONS/fft.cc
@anchor{doc-fft}
@deftypefn  {Loadable Function} {} fft (@var{x})
@deftypefnx {Loadable Function} {} fft (@var{x}, @var{n})
@deftypefnx {Loadable Function} {} fft (@var{x}, @var{n}, @var{dim})
Compute the discrete Fourier transform of @var{A} using
a Fast Fourier Transform (FFT) algorithm.

The FFT is calculated along the first non-singleton dimension of the
array.  Thus if @var{x} is a matrix, @code{fft (@var{x})} computes the
FFT for each column of @var{x}.

If called with two arguments, @var{n} is expected to be an integer
specifying the number of elements of @var{x} to use, or an empty
matrix to specify that its value should be ignored.  If @var{n} is
larger than the dimension along which the FFT is calculated, then
@var{x} is resized and padded with zeros.  Otherwise, if @var{n} is
smaller than the dimension along which the FFT is calculated, then
@var{x} is truncated.

If called with three arguments, @var{dim} is an integer specifying the
dimension of the matrix along which the FFT is performed
@seealso{@ref{doc-ifft,,ifft}, @ref{doc-fft2,,fft2}, @ref{doc-fftn,,fftn}, @ref{doc-fftw,,fftw}}
@end deftypefn


Octave uses the @sc{fftw} libraries to perform FFT computations.  When Octave
starts up and initializes the @sc{fftw} libraries, they read a system wide
file (on a Unix system, it is typically @file{/etc/fftw/wisdom}) that
contains information useful to speed up FFT computations.  This
information is called the @emph{wisdom}.  The system-wide file allows
wisdom to be shared between all applications using the @sc{fftw} libraries.

Use the @code{fftw} function to generate and save wisdom.  Using the
utilities provided together with the @sc{fftw} libraries
(@command{fftw-wisdom} on Unix systems), you can even add wisdom
generated by Octave to the system-wide wisdom file.

@c fftw src/DLD-FUNCTIONS/fftw.cc
@anchor{doc-fftw}
@deftypefn  {Loadable Function} {@var{method} =} fftw ('planner')
@deftypefnx {Loadable Function} {} fftw ('planner', @var{method})
@deftypefnx {Loadable Function} {@var{wisdom} =} fftw ('dwisdom')
@deftypefnx {Loadable Function} {} fftw ('dwisdom', @var{wisdom})

Manage @sc{fftw} wisdom data.  Wisdom data can be used to significantly
accelerate the calculation of the FFTs, but implies an initial cost
in its calculation.  When the @sc{fftw} libraries are initialized, they read
a system wide wisdom file (typically in @file{/etc/fftw/wisdom}), allowing
wisdom to be shared between applications other than Octave.  Alternatively,
the @code{fftw} function can be used to import wisdom.  For example,

@example
@var{wisdom} = fftw ('dwisdom')
@end example

@noindent
will save the existing wisdom used by Octave to the string @var{wisdom}.
This string can then be saved to a file and restored using the @code{save}
and @code{load} commands respectively.  This existing wisdom can be
reimported as follows

@example
fftw ('dwisdom', @var{wisdom})
@end example

If @var{wisdom} is an empty matrix, then the wisdom used is cleared.

During the calculation of Fourier transforms further wisdom is generated.
The fashion in which this wisdom is generated is also controlled by
the @code{fftw} function.  There are five different manners in which the
wisdom can be treated:

@table @asis
@item 'estimate'
Specifies that no run-time measurement of the optimal means of
calculating a particular is performed, and a simple heuristic is used
to pick a (probably sub-optimal) plan.  The advantage of this method is
that there is little or no overhead in the generation of the plan, which
is appropriate for a Fourier transform that will be calculated once.

@item 'measure'
In this case a range of algorithms to perform the transform is considered
and the best is selected based on their execution time.

@item 'patient'
Similar to 'measure', but a wider range of algorithms is considered.

@item 'exhaustive'
Like 'measure', but all possible algorithms that may be used to
treat the transform are considered.

@item 'hybrid'
As run-time measurement of the algorithm can be expensive, this is a
compromise where 'measure' is used for transforms up to the size of 8192
and beyond that the 'estimate' method is used.
@end table

The default method is 'estimate'.  The current method can
be queried with

@example
@var{method} = fftw ('planner')
@end example

@noindent
or set by using

@example
fftw ('planner', @var{method})
@end example

Note that calculated wisdom will be lost when restarting Octave.  However,
the wisdom data can be reloaded if it is saved to a file as described
above.  Saved wisdom files should not be used on different platforms since
they will not be efficient and the point of calculating the wisdom is lost.
@seealso{@ref{doc-fft,,fft}, @ref{doc-ifft,,ifft}, @ref{doc-fft2,,fft2}, @ref{doc-ifft2,,ifft2}, @ref{doc-fftn,,fftn}, @ref{doc-ifftn,,ifftn}}
@end deftypefn


@c ifft src/DLD-FUNCTIONS/fft.cc
@anchor{doc-ifft}
@deftypefn  {Loadable Function} {} ifft (@var{x})
@deftypefnx {Loadable Function} {} ifft (@var{x}, @var{n})
@deftypefnx {Loadable Function} {} ifft (@var{x}, @var{n}, @var{dim})
Compute the inverse discrete Fourier transform of @var{A}
using a Fast Fourier Transform (FFT) algorithm.

The inverse FFT is calculated along the first non-singleton dimension
of the array.  Thus if @var{x} is a matrix, @code{fft (@var{x})} computes
the inverse FFT for each column of @var{x}.

If called with two arguments, @var{n} is expected to be an integer
specifying the number of elements of @var{x} to use, or an empty
matrix to specify that its value should be ignored.  If @var{n} is
larger than the dimension along which the inverse FFT is calculated, then
@var{x} is resized and padded with zeros.  Otherwise, if @var{n} is
smaller than the dimension along which the inverse FFT is calculated,
then @var{x} is truncated.

If called with three arguments, @var{dim} is an integer specifying the
dimension of the matrix along which the inverse FFT is performed
@seealso{@ref{doc-fft,,fft}, @ref{doc-ifft2,,ifft2}, @ref{doc-ifftn,,ifftn}, @ref{doc-fftw,,fftw}}
@end deftypefn


@c fft2 src/DLD-FUNCTIONS/fft2.cc
@anchor{doc-fft2}
@deftypefn  {Loadable Function} {} fft2 (@var{A})
@deftypefnx {Loadable Function} {} fft2 (@var{A}, @var{m}, @var{n})
Compute the two-dimensional discrete Fourier transform of @var{A} using
a Fast Fourier Transform (FFT) algorithm.

The optional arguments @var{m} and @var{n} may be used specify the
number of rows and columns of @var{A} to use.  If either of these is
larger than the size of @var{A}, @var{A} is resized and padded with
zeros.

If @var{A} is a multi-dimensional matrix, each two-dimensional sub-matrix
of @var{A} is treated separately.
@seealso {ifft2, fft, fftn, fftw}
@end deftypefn


@c ifft2 src/DLD-FUNCTIONS/fft2.cc
@anchor{doc-ifft2}
@deftypefn  {Loadable Function} {} ifft2 (@var{A})
@deftypefnx {Loadable Function} {} ifft2 (@var{A}, @var{m}, @var{n})
Compute the inverse two-dimensional discrete Fourier transform of @var{A}
using a Fast Fourier Transform (FFT) algorithm.

The optional arguments @var{m} and @var{n} may be used specify the
number of rows and columns of @var{A} to use.  If either of these is
larger than the size of @var{A}, @var{A} is resized and padded with
zeros.

If @var{A} is a multi-dimensional matrix, each two-dimensional sub-matrix
of @var{A} is treated separately
@seealso {fft2, ifft, ifftn, fftw}
@end deftypefn


@c fftn src/DLD-FUNCTIONS/fftn.cc
@anchor{doc-fftn}
@deftypefn  {Loadable Function} {} fftn (@var{A})
@deftypefnx {Loadable Function} {} fftn (@var{A}, @var{size})
Compute the N-dimensional discrete Fourier transform of @var{A} using
a Fast Fourier Transform (FFT) algorithm.

The optional vector argument @var{size} may be used specify the
dimensions of the array to be used.  If an element of @var{size} is
smaller than the corresponding dimension of @var{A}, then the dimension of
@var{A} is truncated prior to performing the FFT@.  Otherwise, if an element
of @var{size} is larger than the corresponding dimension then @var{A}
is resized and padded with zeros.
@seealso{@ref{doc-ifftn,,ifftn}, @ref{doc-fft,,fft}, @ref{doc-fft2,,fft2}, @ref{doc-fftw,,fftw}}
@end deftypefn


@c ifftn src/DLD-FUNCTIONS/fftn.cc
@anchor{doc-ifftn}
@deftypefn  {Loadable Function} {} ifftn (@var{A})
@deftypefnx {Loadable Function} {} ifftn (@var{A}, @var{size})
Compute the inverse N-dimensional discrete Fourier transform of @var{A}
using a Fast Fourier Transform (FFT) algorithm.

The optional vector argument @var{size} may be used specify the
dimensions of the array to be used.  If an element of @var{size} is
smaller than the corresponding dimension of @var{A}, then the dimension of
@var{A} is truncated prior to performing the inverse FFT@.  Otherwise, if an
element of @var{size} is larger than the corresponding dimension then @var{A}
is resized and padded with zeros.
@seealso{@ref{doc-fftn,,fftn}, @ref{doc-ifft,,ifft}, @ref{doc-ifft2,,ifft2}, @ref{doc-fftw,,fftw}}
@end deftypefn


@c fftconv scripts/signal/fftconv.m
@anchor{doc-fftconv}
@deftypefn  {Function File} {} fftconv (@var{x}, @var{y})
@deftypefnx {Function File} {} fftconv (@var{x}, @var{y}, @var{n})
Convolve two vectors using the FFT for computation.

@code{c = fftconv (@var{x}, @var{y})} returns a vector of length equal to
@code{length (@var{x}) + length (@var{y}) - 1}.
If @var{x} and @var{y} are the coefficient vectors of two polynomials, the
returned value is the coefficient vector of the product polynomial.

The computation uses the FFT by calling the function @code{fftfilt}.  If
the optional argument @var{n} is specified, an N-point FFT is used.
@seealso{@ref{doc-deconv,,deconv}, @ref{doc-conv,,conv}, @ref{doc-conv2,,conv2}}
@end deftypefn


@c fftfilt scripts/signal/fftfilt.m
@anchor{doc-fftfilt}
@deftypefn {Function File} {} fftfilt (@var{b}, @var{x}, @var{n})

With two arguments, @code{fftfilt} filters @var{x} with the FIR filter
@var{b} using the FFT.

Given the optional third argument, @var{n}, @code{fftfilt} uses the
overlap-add method to filter @var{x} with @var{b} using an N-point FFT.

If @var{x} is a matrix, filter each column of the matrix.
@seealso{@ref{doc-filter,,filter}, @ref{doc-filter2,,filter2}}
@end deftypefn


@c filter src/DLD-FUNCTIONS/filter.cc
@anchor{doc-filter}
@deftypefn  {Loadable Function} {y =} filter (@var{b}, @var{a}, @var{x})
@deftypefnx {Loadable Function} {[@var{y}, @var{sf}] =} filter (@var{b}, @var{a}, @var{x}, @var{si})
@deftypefnx {Loadable Function} {[@var{y}, @var{sf}] =} filter (@var{b}, @var{a}, @var{x}, [], @var{dim})
@deftypefnx {Loadable Function} {[@var{y}, @var{sf}] =} filter (@var{b}, @var{a}, @var{x}, @var{si}, @var{dim})
Return the solution to the following linear, time-invariant difference
equation:
@tex
$$
\sum_{k=0}^N a_{k+1} y_{n-k} = \sum_{k=0}^M b_{k+1} x_{n-k}, \qquad
 1 \le n \le P
$$
@end tex
@ifnottex
@c Set example in small font to prevent overfull line

@smallexample
@group
   N                   M
  SUM a(k+1) y(n-k) = SUM b(k+1) x(n-k)      for 1<=n<=length(x)
  k=0                 k=0
@end group
@end smallexample

@end ifnottex

@noindent
where
@ifnottex
 N=length(a)-1 and M=length(b)-1.
@end ifnottex
@tex
 $a \in \Re^{N-1}$, $b \in \Re^{M-1}$, and $x \in \Re^P$.
@end tex
over the first non-singleton dimension of @var{x} or over @var{dim} if
supplied.  An equivalent form of this equation is:
@tex
$$
y_n = -\sum_{k=1}^N c_{k+1} y_{n-k} + \sum_{k=0}^M d_{k+1} x_{n-k}, \qquad
 1 \le n \le P
$$
@end tex
@ifnottex
@c Set example in small font to prevent overfull line

@smallexample
@group
            N                   M
  y(n) = - SUM c(k+1) y(n-k) + SUM d(k+1) x(n-k)  for 1<=n<=length(x)
           k=1                 k=0
@end group
@end smallexample

@end ifnottex

@noindent
where
@ifnottex
 c = a/a(1) and d = b/a(1).
@end ifnottex
@tex
$c = a/a_1$ and $d = b/a_1$.
@end tex

If the fourth argument @var{si} is provided, it is taken as the
initial state of the system and the final state is returned as
@var{sf}.  The state vector is a column vector whose length is
equal to the length of the longest coefficient vector minus one.
If @var{si} is not supplied, the initial state vector is set to all
zeros.

In terms of the Z Transform, y is the result of passing the discrete-
time signal x through a system characterized by the following rational
system function:
@tex
$$
H(z) = {\displaystyle\sum_{k=0}^M d_{k+1} z^{-k}
        \over 1 + \displaystyle\sum_{k+1}^N c_{k+1} z^{-k}}
$$
@end tex
@ifnottex

@example
@group
             M
            SUM d(k+1) z^(-k)
            k=0
  H(z) = ----------------------
               N
          1 + SUM c(k+1) z^(-k)
              k=1
@end group
@end example

@end ifnottex
@seealso{@ref{doc-filter2,,filter2}, @ref{doc-fftfilt,,fftfilt}, @ref{doc-freqz,,freqz}}
@end deftypefn


@c filter2 scripts/signal/filter2.m
@anchor{doc-filter2}
@deftypefn  {Function File} {@var{y} =} filter2 (@var{b}, @var{x})
@deftypefnx {Function File} {@var{y} =} filter2 (@var{b}, @var{x}, @var{shape})
Apply the 2-D FIR filter @var{b} to @var{x}.  If the argument
@var{shape} is specified, return an array of the desired shape.
Possible values are:

@table @asis
@item 'full'
pad @var{x} with zeros on all sides before filtering.

@item 'same'
unpadded @var{x} (default)

@item 'valid'
trim @var{x} after filtering so edge effects are no included.
@end table

Note this is just a variation on convolution, with the parameters
reversed and @var{b} rotated 180 degrees.
@seealso{@ref{doc-conv2,,conv2}}
@end deftypefn


@c freqz scripts/signal/freqz.m
@anchor{doc-freqz}
@deftypefn {Function File} {[@var{h}, @var{w}] =} freqz (@var{b}, @var{a}, @var{n}, "whole")
Return the complex frequency response @var{h} of the rational IIR filter
whose numerator and denominator coefficients are @var{b} and @var{a},
respectively.  The response is evaluated at @var{n} angular frequencies
between 0 and
@ifnottex
 2*pi.
@end ifnottex
@tex
 $2\pi$.
@end tex

@noindent
The output value @var{w} is a vector of the frequencies.

If the fourth argument is omitted, the response is evaluated at
frequencies between 0 and
@ifnottex
 pi.
@end ifnottex
@tex
 $\pi$.
@end tex

If @var{n} is omitted, a value of 512 is assumed.

If @var{a} is omitted, the denominator is assumed to be 1 (this
corresponds to a simple FIR filter).

For fastest computation, @var{n} should factor into a small number of
small primes.

@deftypefnx {Function File} {@var{h} =} freqz (@var{b}, @var{a}, @var{w})
Evaluate the response at the specific frequencies in the vector @var{w}.
The values for @var{w} are measured in radians.

@deftypefnx {Function File} {[@dots{}] =} freqz (@dots{}, @var{Fs})
Return frequencies in Hz instead of radians assuming a sampling rate
@var{Fs}.  If you are evaluating the response at specific frequencies
@var{w}, those frequencies should be requested in Hz rather than radians.

@deftypefnx {Function File} {} freqz (@dots{})
Plot the pass band, stop band and phase response of @var{h} rather
than returning them.
@end deftypefn


@c freqz_plot scripts/signal/freqz_plot.m
@anchor{doc-freqz_plot}
@deftypefn {Function File} {} freqz_plot (@var{w}, @var{h})
Plot the pass band, stop band and phase response of @var{h}.
@end deftypefn


@c sinc scripts/signal/sinc.m
@anchor{doc-sinc}
@deftypefn {Function File} {} sinc (@var{x})
Return
@tex
$ \sin (\pi x)/(\pi x)$.
@end tex
@ifnottex
 sin(pi*x)/(pi*x).
@end ifnottex
@end deftypefn


@c unwrap scripts/signal/unwrap.m
@anchor{doc-unwrap}
@deftypefn  {Function File} {@var{b} =} unwrap (@var{x})
@deftypefnx {Function File} {@var{b} =} unwrap (@var{x}, @var{tol})
@deftypefnx {Function File} {@var{b} =} unwrap (@var{x}, @var{tol}, @var{dim})

Unwrap radian phases by adding multiples of 2*pi as appropriate to
remove jumps greater than @var{tol}.  @var{tol} defaults to pi.

Unwrap will work along the dimension @var{dim}.  If @var{dim}
is unspecified it defaults to the first non-singleton dimension.
@end deftypefn


@c FIXME -- someone needs to organize these...

@c arch_fit scripts/signal/arch_fit.m
@anchor{doc-arch_fit}
@deftypefn {Function File} {[@var{a}, @var{b}] =} arch_fit (@var{y}, @var{x}, @var{p}, @var{iter}, @var{gamma}, @var{a0}, @var{b0})
Fit an ARCH regression model to the time series @var{y} using the
scoring algorithm in Engle's original ARCH paper.  The model is

@example
@group
y(t) = b(1) * x(t,1) + @dots{} + b(k) * x(t,k) + e(t),
h(t) = a(1) + a(2) * e(t-1)^2 + @dots{} + a(p+1) * e(t-p)^2
@end group
@end example

@noindent
in which @math{e(t)} is @math{N(0, h(t))}, given a time-series vector
@var{y} up to time @math{t-1} and a matrix of (ordinary) regressors
@var{x} up to @math{t}.  The order of the regression of the residual
variance is specified by @var{p}.

If invoked as @code{arch_fit (@var{y}, @var{k}, @var{p})} with a
positive integer @var{k}, fit an ARCH(@var{k}, @var{p}) process,
i.e., do the above with the @math{t}-th row of @var{x} given by

@example
[1, y(t-1), @dots{}, y(t-k)]
@end example

Optionally, one can specify the number of iterations @var{iter}, the
updating factor @var{gamma}, and initial values @math{a0} and
@math{b0} for the scoring algorithm.
@end deftypefn


@c arch_rnd scripts/signal/arch_rnd.m
@anchor{doc-arch_rnd}
@deftypefn {Function File} {} arch_rnd (@var{a}, @var{b}, @var{t})
Simulate an ARCH sequence of length @var{t} with AR
coefficients @var{b} and CH coefficients @var{a}.  I.e., the result
@math{y(t)} follows the model
@c Set example in small font to prevent overfull line

@smallexample
y(t) = b(1) + b(2) * y(t-1) + @dots{} + b(lb) * y(t-lb+1) + e(t),
@end smallexample

@noindent
where @math{e(t)}, given @var{y} up to time @math{t-1}, is
@math{N(0, h(t))}, with
@c Set example in small font to prevent overfull line

@smallexample
h(t) = a(1) + a(2) * e(t-1)^2 + @dots{} + a(la) * e(t-la+1)^2
@end smallexample
@end deftypefn


@c arch_test scripts/signal/arch_test.m
@anchor{doc-arch_test}
@deftypefn {Function File} {[@var{pval}, @var{lm}] =} arch_test (@var{y}, @var{x}, @var{p})
For a linear regression model

@example
y = x * b + e
@end example

@noindent
perform a Lagrange Multiplier (LM) test of the null hypothesis of no
conditional heteroscedascity against the alternative of CH(@var{p}).

I.e., the model is

@example
y(t) = b(1) * x(t,1) + @dots{} + b(k) * x(t,k) + e(t),
@end example

@noindent
given @var{y} up to @math{t-1} and @var{x} up to @math{t},
@math{e}(t) is @math{N(0, h(t))} with

@example
h(t) = v + a(1) * e(t-1)^2 + @dots{} + a(p) * e(t-p)^2,
@end example

@noindent
and the null is @math{a(1)} == @dots{} == @math{a(p)} == 0.

If the second argument is a scalar integer, @math{k}, perform the same
test in a linear autoregression model of order @math{k}, i.e., with

@example
[1, y(t-1), @dots{}, y(t-@var{k})]
@end example

@noindent
as the @math{t}-th row of @var{x}.

Under the null, LM approximately has a chisquare distribution with
@var{p} degrees of freedom and @var{pval} is the @math{p}-value (1
minus the CDF of this distribution at LM) of the test.

If no output argument is given, the @math{p}-value is displayed.
@end deftypefn


@c arma_rnd scripts/signal/arma_rnd.m
@anchor{doc-arma_rnd}
@deftypefn {Function File} {} arma_rnd (@var{a}, @var{b}, @var{v}, @var{t}, @var{n})
Return a simulation of the ARMA model

@example
@group
x(n) = a(1) * x(n-1) + @dots{} + a(k) * x(n-k)
     + e(n) + b(1) * e(n-1) + @dots{} + b(l) * e(n-l)
@end group
@end example

@noindent
in which @var{k} is the length of vector @var{a}, @var{l} is the
length of vector @var{b} and @var{e} is Gaussian white noise with
variance @var{v}.  The function returns a vector of length @var{t}.

The optional parameter @var{n} gives the number of dummy
@var{x}(@var{i}) used for initialization, i.e., a sequence of length
@var{t}+@var{n} is generated and @var{x}(@var{n}+1:@var{t}+@var{n})
is returned.  If @var{n} is omitted, @var{n} = 100 is used.
@end deftypefn


@c autoreg_matrix scripts/signal/autoreg_matrix.m
@anchor{doc-autoreg_matrix}
@deftypefn {Function File} {} autoreg_matrix (@var{y}, @var{k})
Given a time series (vector) @var{y}, return a matrix with ones in the
first column and the first @var{k} lagged values of @var{y} in the
other columns.  I.e., for @var{t} > @var{k}, @code{[1,
@var{y}(@var{t}-1), @dots{}, @var{y}(@var{t}-@var{k})]} is the t-th row
of the result.  The resulting matrix may be used as a regressor matrix
in autoregressions.
@end deftypefn


@c bartlett scripts/signal/bartlett.m
@anchor{doc-bartlett}
@deftypefn {Function File} {} bartlett (@var{m})
Return the filter coefficients of a Bartlett (triangular) window of
length @var{m}.

For a definition of the Bartlett window, see e.g., A. V. Oppenheim &
R. W. Schafer, @cite{Discrete-Time Signal Processing}.
@end deftypefn


@c blackman scripts/signal/blackman.m
@anchor{doc-blackman}
@deftypefn {Function File} {} blackman (@var{m})
Return the filter coefficients of a Blackman window of length @var{m}.

For a definition of the Blackman window, see e.g., A. V. Oppenheim &
R. W. Schafer, @cite{Discrete-Time Signal Processing}.
@end deftypefn


@c diffpara scripts/signal/diffpara.m
@anchor{doc-diffpara}
@deftypefn {Function File} {[@var{d}, @var{dd}] =} diffpara (@var{x}, @var{a}, @var{b})
Return the estimator @var{d} for the differencing parameter of an
integrated time series.

The frequencies from @math{[2*pi*a/t, 2*pi*b/T]} are used for the
estimation.  If @var{b} is omitted, the interval
@math{[2*pi/T, 2*pi*a/T]} is used.  If both @var{b} and @var{a} are
omitted then @math{a = 0.5 * sqrt (T)} and @math{b = 1.5 * sqrt (T)}
is used, where @math{T} is the sample size.  If @var{x} is a matrix,
the differencing parameter of each column is estimated.

The estimators for all frequencies in the intervals
described above is returned in @var{dd}.  The value of @var{d} is
simply the mean of @var{dd}.

Reference: P.J. Brockwell & R.A. Davis. @cite{Time Series:
Theory and Methods}. Springer 1987.
@end deftypefn


@c durbinlevinson scripts/signal/durbinlevinson.m
@anchor{doc-durbinlevinson}
@deftypefn {Function File} {} durbinlevinson (@var{c}, @var{oldphi}, @var{oldv})
Perform one step of the Durbin-Levinson algorithm.

The vector @var{c} specifies the autocovariances @code{[gamma_0, @dots{},
gamma_t]} from lag 0 to @var{t}, @var{oldphi} specifies the
coefficients based on @var{c}(@var{t}-1) and @var{oldv} specifies the
corresponding error.

If @var{oldphi} and @var{oldv} are omitted, all steps from 1 to
@var{t} of the algorithm are performed.
@end deftypefn


@c fftshift scripts/signal/fftshift.m
@anchor{doc-fftshift}
@deftypefn  {Function File} {} fftshift (@var{x})
@deftypefnx {Function File} {} fftshift (@var{x}, @var{dim})
Perform a shift of the vector @var{x}, for use with the @code{fft}
and @code{ifft} functions, in order the move the frequency 0 to the
center of the vector or matrix.

If @var{x} is a vector of @math{N} elements corresponding to @math{N}
time samples spaced of @math{Dt} each, then @code{fftshift (fft
(@var{x}))} corresponds to frequencies

@example
f = ((1:N) - ceil(N/2)) / N / Dt
@end example

If @var{x} is a matrix, the same holds for rows and columns.  If
@var{x} is an array, then the same holds along each dimension.

The optional @var{dim} argument can be used to limit the dimension
along which the permutation occurs.
@end deftypefn


@c ifftshift scripts/signal/ifftshift.m
@anchor{doc-ifftshift}
@deftypefn  {Function File} {} ifftshift (@var{x})
@deftypefnx {Function File} {} ifftshift (@var{x}, @var{dim})
Undo the action of the @code{fftshift} function.  For even length
@var{x}, @code{fftshift} is its own inverse, but odd lengths differ
slightly.
@end deftypefn


@c fractdiff scripts/signal/fractdiff.m
@anchor{doc-fractdiff}
@deftypefn {Function File} {} fractdiff (@var{x}, @var{d})
Compute the fractional differences @math{(1-L)^d x} where @math{L}
denotes the lag-operator and @math{d} is greater than -1.
@end deftypefn


@c hamming scripts/signal/hamming.m
@anchor{doc-hamming}
@deftypefn {Function File} {} hamming (@var{m})
Return the filter coefficients of a Hamming window of length @var{m}.

For a definition of the Hamming window, see e.g., A. V. Oppenheim &
R. W. Schafer, @cite{Discrete-Time Signal Processing}.
@end deftypefn


@c hanning scripts/signal/hanning.m
@anchor{doc-hanning}
@deftypefn {Function File} {} hanning (@var{m})
Return the filter coefficients of a Hanning window of length @var{m}.

For a definition of this window type, see e.g., A. V. Oppenheim &
R. W. Schafer, @cite{Discrete-Time Signal Processing}.
@end deftypefn


@c hurst scripts/signal/hurst.m
@anchor{doc-hurst}
@deftypefn {Function File} {} hurst (@var{x})
Estimate the Hurst parameter of sample @var{x} via the rescaled range
statistic.  If @var{x} is a matrix, the parameter is estimated for
every single column.
@end deftypefn


@c pchip scripts/polynomial/pchip.m
@anchor{doc-pchip}
@deftypefn  {Function File} {@var{pp} =} pchip (@var{x}, @var{y})
@deftypefnx {Function File} {@var{yi} =} pchip (@var{x}, @var{y}, @var{xi})

Piecewise Cubic Hermite interpolating polynomial.  Called with two
arguments, the piecewise polynomial @var{pp} is returned, that may
later be used with @code{ppval} to evaluate the polynomial at
specific points.

The variable @var{x} must be a strictly monotonic vector (either
increasing or decreasing).  While @var{y} can be either a vector or
array.  In the case where @var{y} is a vector, it must have a length
of @var{n}.  If @var{y} is an array, then the size of @var{y} must
have the form
@tex
$$[s_1, s_2, \cdots, s_k, n]$$
@end tex
@ifnottex
@code{[@var{s1}, @var{s2}, @dots{}, @var{sk}, @var{n}]}
@end ifnottex
The array is then reshaped internally to a matrix where the leading
dimension is given by
@tex
$$s_1 s_2 \cdots s_k$$
@end tex
@ifnottex
@code{@var{s1} * @var{s2} * @dots{} * @var{sk}}
@end ifnottex
and each row in this matrix is then treated separately.  Note that this
is exactly the opposite treatment than @code{interp1} and is done
for compatibility.

Called with a third input argument, @code{pchip} evaluates the
piecewise polynomial at the points @var{xi}.  There is an equivalence
between @code{ppval (pchip (@var{x}, @var{y}), @var{xi})} and
@code{pchip (@var{x}, @var{y}, @var{xi})}.

@seealso{@ref{doc-spline,,spline}, @ref{doc-ppval,,ppval}, @ref{doc-mkpp,,mkpp}, @ref{doc-unmkpp,,unmkpp}}
@end deftypefn


@c periodogram scripts/signal/periodogram.m
@anchor{doc-periodogram}
@deftypefn {Function File} {[Pxx, @var{w}] =} periodogram (@var{x})
For a data matrix @var{x} from a sample of size @var{n}, return the
periodogram.  The angular frequency is returned in @var{w}.

[Pxx,w] = periodogram (@var{x}).

[Pxx,w] = periodogram (@var{x},win).

[Pxx,w] = periodogram (@var{x},win,nfft).

[Pxx,f] = periodogram (@var{x},win,nfft,Fs).

[Pxx,f] = periodogram (@var{x},win,nfft,Fs,"range").

@itemize
@item x: data; if real-valued a one-sided spectrum is estimated,
if complex-valued or range indicates "@nospell{twosided}", the full
spectrum is estimated.

@item win: weight data with window, x.*win is used for further computation,
if window is empty, a rectangular window is used.

@item nfft: number of frequency bins, default max(256, 2.^ceil(log2(length(x)))).

@item Fs: sampling rate, default 1.

@item range: "@nospell{onesided}" computes spectrum from [0..nfft/2+1].
"@nospell{twosided}" computes spectrum from [0..nfft-1].  These strings
can appear at any position in the list input arguments after window.

@item Pxx: one-, or two-sided power spectrum.

@item w: angular frequency [0..2*pi) (two-sided) or [0..pi] one-sided.

@item f: frequency [0..Fs) (two-sided) or [0..Fs/2] one-sided.
@end itemize
@end deftypefn


@c rectangle_lw scripts/signal/private/rectangle_lw.m
@anchor{doc-rectangle_lw}
@deftypefn {Function File} {} rectangle_lw (@var{n}, @var{b})
Rectangular lag window.  Subfunction used for spectral density
estimation.
@end deftypefn


@c rectangle_sw scripts/signal/private/rectangle_sw.m
@anchor{doc-rectangle_sw}
@deftypefn {Function File} {} rectangle_sw (@var{n}, @var{b})
Rectangular spectral window.  Subfunction used for spectral density
estimation.
@end deftypefn


@c sinetone scripts/signal/sinetone.m
@anchor{doc-sinetone}
@deftypefn {Function File} {} sinetone (@var{freq}, @var{rate}, @var{sec}, @var{ampl})
Return a sinetone of frequency @var{freq} with length of @var{sec}
seconds at sampling rate @var{rate} and with amplitude @var{ampl}.
The arguments @var{freq} and @var{ampl} may be vectors of common size.

Defaults are @var{rate} = 8000, @var{sec} = 1 and @var{ampl} = 64.
@end deftypefn


@c sinewave scripts/signal/sinewave.m
@anchor{doc-sinewave}
@deftypefn {Function File} {} sinewave (@var{m}, @var{n}, @var{d})
Return an @var{m}-element vector with @var{i}-th element given by
@code{sin (2 * pi * (@var{i}+@var{d}-1) / @var{n})}.

The default value for @var{d} is 0 and the default value for @var{n}
is @var{m}.
@end deftypefn


@c spectral_adf scripts/signal/spectral_adf.m
@anchor{doc-spectral_adf}
@deftypefn {Function File} {} spectral_adf (@var{c}, @var{win}, @var{b})
Return the spectral density estimator given a vector of
autocovariances @var{c}, window name @var{win}, and bandwidth,
@var{b}.

The window name, e.g., @code{"triangle"} or @code{"rectangle"} is
used to search for a function called @code{@var{win}_sw}.

If @var{win} is omitted, the triangle window is used.  If @var{b} is
omitted, @code{1 / sqrt (length (@var{x}))} is used.
@end deftypefn


@c spectral_xdf scripts/signal/spectral_xdf.m
@anchor{doc-spectral_xdf}
@deftypefn {Function File} {} spectral_xdf (@var{x}, @var{win}, @var{b})
Return the spectral density estimator given a data vector @var{x},
window name @var{win}, and bandwidth, @var{b}.

The window name, e.g., @code{"triangle"} or @code{"rectangle"} is
used to search for a function called @code{@var{win}_sw}.

If @var{win} is omitted, the triangle window is used.  If @var{b} is
omitted, @code{1 / sqrt (length (@var{x}))} is used.
@end deftypefn


@c spencer scripts/signal/spencer.m
@anchor{doc-spencer}
@deftypefn {Function File} {} spencer (@var{x})
Return Spencer's 15 point moving average of each column of
@var{x}.
@end deftypefn


@c stft scripts/signal/stft.m
@anchor{doc-stft}
@deftypefn {Function File} {[@var{y}, @var{c}] =} stft (@var{x}, @var{win_size}, @var{inc}, @var{num_coef}, @var{win_type})
Compute the short-time Fourier transform of the vector @var{x} with
@var{num_coef} coefficients by applying a window of @var{win_size} data
points and an increment of @var{inc} points.

Before computing the Fourier transform, one of the following windows
is applied:

@table @asis
@item @nospell{hanning}
win_type = 1

@item @nospell{hamming}
win_type = 2

@item rectangle
win_type = 3
@end table

The window names can be passed as strings or by the @var{win_type} number.

If not all arguments are specified, the following defaults are used:
@var{win_size} = 80, @var{inc} = 24, @var{num_coef} = 64, and
@var{win_type} = 1.

@code{@var{y} = stft (@var{x}, @dots{})} returns the absolute values
of the Fourier coefficients according to the @var{num_coef} positive
frequencies.

@code{[@var{y}, @var{c}] = stft (@code{x}, @dots{})} returns the
entire STFT-matrix @var{y} and a 3-element vector @var{c} containing
the window size, increment, and window type, which is needed by the
synthesis function.
@end deftypefn


@c synthesis scripts/signal/synthesis.m
@anchor{doc-synthesis}
@deftypefn {Function File} {} synthesis (@var{y}, @var{c})
Compute a signal from its short-time Fourier transform @var{y} and a
3-element vector @var{c} specifying window size, increment, and
window type.

The values @var{y} and @var{c} can be derived by

@example
[@var{y}, @var{c}] = stft (@var{x} , @dots{})
@end example
@end deftypefn


@c triangle_lw scripts/signal/private/triangle_lw.m
@anchor{doc-triangle_lw}
@deftypefn {Function File} {} triangle_lw (@var{n}, @var{b})
Triangular lag window.  Subfunction used for spectral density
estimation.
@end deftypefn


@c triangle_sw scripts/signal/private/triangle_sw.m
@anchor{doc-triangle_sw}
@deftypefn {Function File} {} triangle_sw (@var{n}, @var{b})
Triangular spectral window.  Subfunction used for spectral density
estimation.
@end deftypefn


@c yulewalker scripts/signal/yulewalker.m
@anchor{doc-yulewalker}
@deftypefn {Function File} {[@var{a}, @var{v}] =} yulewalker (@var{c})
Fit an AR (p)-model with Yule-Walker estimates given a vector @var{c}
of autocovariances @code{[gamma_0, @dots{}, gamma_p]}.

Returns the AR coefficients, @var{a}, and the variance of white
noise, @var{v}.
@end deftypefn

