(octave.info)Signal Processing
31 Signal Processing
********************
This chapter describes the signal processing and fast Fourier transform
functions available in Octave. Fast Fourier transforms are computed
with the FFTW or FFTPACK libraries depending on how Octave is built.
-- : fft (X)
-- : fft (X, N)
-- : fft (X, N, DIM)
Compute the discrete Fourier transform of X using a Fast Fourier
Transform (FFT) algorithm.
The FFT is calculated along the first non-singleton dimension of
the array. Thus if X is a matrix, ‘fft (X)’ computes the FFT for
each column of X.
If called with two arguments, N is expected to be an integer
specifying the number of elements of X to use, or an empty matrix
to specify that its value should be ignored. If N is larger than
the dimension along which the FFT is calculated, then X is resized
and padded with zeros. Otherwise, if N is smaller than the
dimension along which the FFT is calculated, then X is truncated.
If called with three arguments, DIM is an integer specifying the
dimension of the matrix along which the FFT is performed.
See also: Note: ifft, Note: fft2, *note fftn:
XREFfftn, Note: fftw.
-- : ifft (X)
-- : ifft (X, N)
-- : ifft (X, N, DIM)
Compute the inverse discrete Fourier transform of X using a Fast
Fourier Transform (FFT) algorithm.
The inverse FFT is calculated along the first non-singleton
dimension of the array. Thus if X is a matrix, ‘fft (X)’ computes
the inverse FFT for each column of X.
If called with two arguments, N is expected to be an integer
specifying the number of elements of X to use, or an empty matrix
to specify that its value should be ignored. If N is larger than
the dimension along which the inverse FFT is calculated, then X is
resized and padded with zeros. Otherwise, if N is smaller than the
dimension along which the inverse FFT is calculated, then X is
truncated.
If called with three arguments, DIM is an integer specifying the
dimension of the matrix along which the inverse FFT is performed.
See also: Note: fft, Note: ifft2, *note ifftn:
XREFifftn, Note: fftw.
-- : fft2 (A)
-- : fft2 (A, M, N)
Compute the two-dimensional discrete Fourier transform of A using a
Fast Fourier Transform (FFT) algorithm.
The optional arguments M and N may be used specify the number of
rows and columns of A to use. If either of these is larger than
the size of A, A is resized and padded with zeros.
If A is a multi-dimensional matrix, each two-dimensional sub-matrix
of A is treated separately.
See also: Note: ifft2, Note: fft, *note fftn:
XREFfftn, Note: fftw.
-- : ifft2 (A)
-- : ifft2 (A, M, N)
Compute the inverse two-dimensional discrete Fourier transform of A
using a Fast Fourier Transform (FFT) algorithm.
The optional arguments M and N may be used specify the number of
rows and columns of A to use. If either of these is larger than
the size of A, A is resized and padded with zeros.
If A is a multi-dimensional matrix, each two-dimensional sub-matrix
of A is treated separately.
See also: Note: fft2, Note: ifft, *note ifftn:
XREFifftn, Note: fftw.
-- : fftn (A)
-- : fftn (A, SIZE)
Compute the N-dimensional discrete Fourier transform of A using a
Fast Fourier Transform (FFT) algorithm.
The optional vector argument SIZE may be used specify the
dimensions of the array to be used. If an element of SIZE is
smaller than the corresponding dimension of A, then the dimension
of A is truncated prior to performing the FFT. Otherwise, if an
element of SIZE is larger than the corresponding dimension then A
is resized and padded with zeros.
See also: Note: ifftn, Note: fft, *note fft2:
XREFfft2, Note: fftw.
-- : ifftn (A)
-- : ifftn (A, SIZE)
Compute the inverse N-dimensional discrete Fourier transform of A
using a Fast Fourier Transform (FFT) algorithm.
The optional vector argument SIZE may be used specify the
dimensions of the array to be used. If an element of SIZE is
smaller than the corresponding dimension of A, then the dimension
of A is truncated prior to performing the inverse FFT. Otherwise,
if an element of SIZE is larger than the corresponding dimension
then A is resized and padded with zeros.
See also: Note: fftn, Note: ifft, *note ifft2:
XREFifft2, Note: fftw.
Octave uses the FFTW libraries to perform FFT computations. When
Octave starts up and initializes the FFTW libraries, they read a system
wide file (on a Unix system, it is typically ‘/etc/fftw/wisdom’) that
contains information useful to speed up FFT computations. This
information is called the _wisdom_. The system-wide file allows wisdom
to be shared between all applications using the FFTW libraries.
Use the ‘fftw’ function to generate and save wisdom. Using the
utilities provided together with the FFTW libraries (‘fftw-wisdom’ on
Unix systems), you can even add wisdom generated by Octave to the
system-wide wisdom file.
-- : METHOD = fftw ("planner")
-- : fftw ("planner", METHOD)
-- : WISDOM = fftw ("dwisdom")
-- : fftw ("dwisdom", WISDOM)
-- : fftw ("threads", NTHREADS)
-- : NTHREADS = fftw ("threads")
Manage 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 FFTW libraries are initialized, they read a system wide wisdom
file (typically in ‘/etc/fftw/wisdom’), allowing wisdom to be
shared between applications other than Octave. Alternatively, the
‘fftw’ function can be used to import wisdom. For example,
WISDOM = fftw ("dwisdom")
will save the existing wisdom used by Octave to the string WISDOM.
This string can then be saved to a file and restored using the
‘save’ and ‘load’ commands respectively. This existing wisdom can
be re-imported as follows
fftw ("dwisdom", WISDOM)
If WISDOM is an empty string, 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 ‘fftw’ function. There are five different
manners in which the wisdom can be treated:
"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.
"measure"
In this case a range of algorithms to perform the transform is
considered and the best is selected based on their execution
time.
"patient"
Similar to "measure", but a wider range of algorithms is
considered.
"exhaustive"
Like "measure", but all possible algorithms that may be used
to treat the transform are considered.
"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.
The default method is "estimate". The current method can be
queried with
METHOD = fftw ("planner")
or set by using
fftw ("planner", METHOD)
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.
The number of threads used for computing the plans and executing
the transforms can be set with
fftw ("threads", NTHREADS)
Note that octave must be compiled with multi-threaded FFTW support
for this feature. The number of processors available to the
current process is used per default.
See also: Note: fft, Note: ifft, *note fft2:
XREFfft2, Note: ifft2, Note: fftn, Note:
ifftn.
-- : fftconv (X, Y)
-- : fftconv (X, Y, N)
Convolve two vectors using the FFT for computation.
‘c = fftconv (X, Y)’ returns a vector of length equal to ‘length
(X) + length (Y) - 1’. If X and 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 ‘fftfilt’. If
the optional argument N is specified, an N-point FFT is used.
See also: Note: deconv, Note: conv, Note:
conv2.
-- : fftfilt (B, X)
-- : fftfilt (B, X, N)
Filter X with the FIR filter B using the FFT.
If X is a matrix, filter each column of the matrix.
Given the optional third argument, N, ‘fftfilt’ uses the
overlap-add method to filter X with B using an N-point FFT. The
FFT size must be an even power of 2 and must be greater than or
equal to the length of B. If the specified N does not meet these
criteria, it is automatically adjusted to the nearest value that
does.
See also: Note: filter, Note: filter2.
-- : Y = filter (B, A, X)
-- : [Y, SF] = filter (B, A, X, SI)
-- : [Y, SF] = filter (B, A, X, [], DIM)
-- : [Y, SF] = filter (B, A, X, SI, DIM)
Apply a 1-D digital filter to the data X.
‘filter’ returns the solution to the following linear,
time-invariant difference equation:
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
where N=length(a)-1 and M=length(b)-1. The result is calculated
over the first non-singleton dimension of X or over DIM if
supplied.
An equivalent form of the equation is:
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
where c = a/a(1) and d = b/a(1).
If the fourth argument SI is provided, it is taken as the initial
state of the system and the final state is returned as SF. The
state vector is a column vector whose length is equal to the length
of the longest coefficient vector minus one. If 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:
M
SUM d(k+1) z^(-k)
k=0
H(z) = ---------------------
N
1 + SUM c(k+1) z^(-k)
k=1
See also: Note: filter2, Note: fftfilt,
Note: freqz.
-- : Y = filter2 (B, X)
-- : Y = filter2 (B, X, SHAPE)
Apply the 2-D FIR filter B to X.
If the argument SHAPE is specified, return an array of the desired
shape. Possible values are:
"full"
pad X with zeros on all sides before filtering.
"same"
unpadded X (default)
"valid"
trim X after filtering so edge effects are no included.
Note this is just a variation on convolution, with the parameters
reversed and B rotated 180 degrees.
See also: Note: conv2.
-- : [H, W] = freqz (B, A, N, "whole")
-- : [H, W] = freqz (B)
-- : [H, W] = freqz (B, A)
-- : [H, W] = freqz (B, A, N)
-- : H = freqz (B, A, W)
-- : [H, W] = freqz (..., FS)
-- : freqz (...)
Return the complex frequency response H of the rational IIR filter
whose numerator and denominator coefficients are B and A,
respectively.
The response is evaluated at N angular frequencies between 0 and
2*pi.
The output value W is a vector of the frequencies.
If A is omitted, the denominator is assumed to be 1 (this
corresponds to a simple FIR filter).
If N is omitted, a value of 512 is assumed. For fastest
computation, N should factor into a small number of small primes.
If the fourth argument, "whole", is omitted the response is
evaluated at frequencies between 0 and pi.
‘freqz (B, A, W)’
Evaluate the response at the specific frequencies in the vector W.
The values for W are measured in radians.
‘[...] = freqz (..., FS)’
Return frequencies in Hz instead of radians assuming a sampling
rate FS. If you are evaluating the response at specific
frequencies W, those frequencies should be requested in Hz rather
than radians.
‘freqz (...)’
Plot the magnitude and phase response of H rather than returning
them.
See also: Note: freqz_plot.
-- : freqz_plot (W, H)
-- : freqz_plot (W, H, FREQ_NORM)
Plot the magnitude and phase response of H.
If the optional FREQ_NORM argument is true, the frequency vector W
is in units of normalized radians. If FREQ_NORM is false, or not
given, then W is measured in Hertz.
See also: Note: freqz.
-- : sinc (X)
Compute the sinc function.
Return sin (pi*x) / (pi*x).
-- : B = unwrap (X)
-- : B = unwrap (X, TOL)
-- : B = unwrap (X, TOL, DIM)
Unwrap radian phases by adding or subtracting multiples of 2*pi as
appropriate to remove jumps greater than TOL.
TOL defaults to pi.
Unwrap will work along the dimension DIM. If DIM is unspecified it
defaults to the first non-singleton dimension.
-- : [A, B] = arch_fit (Y, X, P, ITER, GAMMA, A0, B0)
Fit an ARCH regression model to the time series Y using the scoring
algorithm in Engle’s original ARCH paper.
The model is
y(t) = b(1) * x(t,1) + ... + b(k) * x(t,k) + e(t),
h(t) = a(1) + a(2) * e(t-1)^2 + ... + a(p+1) * e(t-p)^2
in which e(t) is N(0, h(t)), given a time-series vector Y up to
time t-1 and a matrix of (ordinary) regressors X up to t. The
order of the regression of the residual variance is specified by P.
If invoked as ‘arch_fit (Y, K, P)’ with a positive integer K, fit
an ARCH(K, P) process, i.e., do the above with the t-th row of X
given by
[1, y(t-1), ..., y(t-k)]
Optionally, one can specify the number of iterations ITER, the
updating factor GAMMA, and initial values a0 and b0 for the scoring
algorithm.
-- : arch_rnd (A, B, T)
Simulate an ARCH sequence of length T with AR coefficients B and CH
coefficients A.
The result y(t) follows the model
y(t) = b(1) + b(2) * y(t-1) + ... + b(lb) * y(t-lb+1) + e(t),
where e(t), given Y up to time t-1, is N(0, h(t)), with
h(t) = a(1) + a(2) * e(t-1)^2 + ... + a(la) * e(t-la+1)^2
-- : [PVAL, LM] = arch_test (Y, X, P)
For a linear regression model
y = x * b + e
perform a Lagrange Multiplier (LM) test of the null hypothesis of
no conditional heteroscedascity against the alternative of CH(P).
I.e., the model is
y(t) = b(1) * x(t,1) + ... + b(k) * x(t,k) + e(t),
given Y up to t-1 and X up to t, e(t) is N(0, h(t)) with
h(t) = v + a(1) * e(t-1)^2 + ... + a(p) * e(t-p)^2,
and the null is a(1) == ... == a(p) == 0.
If the second argument is a scalar integer, k, perform the same
test in a linear autoregression model of order k, i.e., with
[1, y(t-1), ..., y(t-K)]
as the t-th row of X.
Under the null, LM approximately has a chisquare distribution with
P degrees of freedom and PVAL is the p-value (1 minus the CDF of
this distribution at LM) of the test.
If no output argument is given, the p-value is displayed.
-- : arma_rnd (A, B, V, T, N)
Return a simulation of the ARMA model.
The ARMA model is defined by
x(n) = a(1) * x(n-1) + ... + a(k) * x(n-k)
+ e(n) + b(1) * e(n-1) + ... + b(l) * e(n-l)
in which K is the length of vector A, L is the length of vector B
and E is Gaussian white noise with variance V. The function
returns a vector of length T.
The optional parameter N gives the number of dummy X(I) used for
initialization, i.e., a sequence of length T+N is generated and
X(N+1:T+N) is returned. If N is omitted, N = 100 is used.
-- : autoreg_matrix (Y, K)
Given a time series (vector) Y, return a matrix with ones in the
first column and the first K lagged values of Y in the other
columns.
In other words, for T > K, ‘[1, Y(T-1), ..., Y(T-K)]’ is the t-th
row of the result.
The resulting matrix may be used as a regressor matrix in
autoregressions.
-- : bartlett (M)
Return the filter coefficients of a Bartlett (triangular) window of
length M.
For a definition of the Bartlett window see, e.g., A.V. Oppenheim &
R. W. Schafer, ‘Discrete-Time Signal Processing’.
-- : blackman (M)
-- : blackman (M, "periodic")
-- : blackman (M, "symmetric")
Return the filter coefficients of a Blackman window of length M.
If the optional argument "periodic" is given, the periodic form of
the window is returned. This is equivalent to the window of length
M+1 with the last coefficient removed. The optional argument
"symmetric" is equivalent to not specifying a second argument.
For a definition of the Blackman window, see, e.g., A.V. Oppenheim
& R. W. Schafer, ‘Discrete-Time Signal Processing’.
-- : detrend (X, P)
If X is a vector, ‘detrend (X, P)’ removes the best fit of a
polynomial of order P from the data X.
If X is a matrix, ‘detrend (X, P)’ does the same for each column in
X.
The second argument P 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 P must be either "constant" (corresponds to ‘P=0’) or "linear"
(corresponds to ‘P=1’).
See also: Note: polyfit.
-- : [D, DD] = diffpara (X, A, B)
Return the estimator D for the differencing parameter of an
integrated time series.
The frequencies from [2*pi*a/t, 2*pi*b/T] are used for the
estimation. If B is omitted, the interval [2*pi/T, 2*pi*a/T] is
used. If both B and A are omitted then a = 0.5 * sqrt (T) and b =
1.5 * sqrt (T) is used, where T is the sample size. If 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 DD.
The value of D is simply the mean of DD.
Reference: P.J. Brockwell & R.A. Davis. ‘Time Series: Theory and
Methods’. Springer 1987.
-- : durbinlevinson (C, OLDPHI, OLDV)
Perform one step of the Durbin-Levinson algorithm.
The vector C specifies the autocovariances ‘[gamma_0, ...,
gamma_t]’ from lag 0 to T, OLDPHI specifies the coefficients based
on C(T-1) and OLDV specifies the corresponding error.
If OLDPHI and OLDV are omitted, all steps from 1 to T of the
algorithm are performed.
-- : fftshift (X)
-- : fftshift (X, DIM)
Perform a shift of the vector X, for use with the ‘fft’ and ‘ifft’
functions, in order to move the frequency 0 to the center of the
vector or matrix.
If X is a vector of N elements corresponding to N time samples
spaced by dt, then ‘fftshift (fft (X))’ corresponds to frequencies
f = [ -(ceil((N-1)/2):-1:1), 0, (1:floor((N-1)/2)) ] * df
where df = 1 / (N * dt).
If X is a matrix, the same holds for rows and columns. If X is an
array, then the same holds along each dimension.
The optional DIM argument can be used to limit the dimension along
which the permutation occurs.
See also: Note: ifftshift.
-- : ifftshift (X)
-- : ifftshift (X, DIM)
Undo the action of the ‘fftshift’ function.
For even length X, ‘fftshift’ is its own inverse, but odd lengths
differ slightly.
See also: Note: fftshift.
-- : fractdiff (X, D)
Compute the fractional differences (1-L)^d x where L denotes the
lag-operator and d is greater than -1.
-- : hamming (M)
-- : hamming (M, "periodic")
-- : hamming (M, "symmetric")
Return the filter coefficients of a Hamming window of length M.
If the optional argument "periodic" is given, the periodic form of
the window is returned. This is equivalent to the window of length
M+1 with the last coefficient removed. The optional argument
"symmetric" is equivalent to not specifying a second argument.
For a definition of the Hamming window see, e.g., A.V. Oppenheim &
R. W. Schafer, ‘Discrete-Time Signal Processing’.
-- : hanning (M)
-- : hanning (M, "periodic")
-- : hanning (M, "symmetric")
Return the filter coefficients of a Hanning window of length M.
If the optional argument "periodic" is given, the periodic form of
the window is returned. This is equivalent to the window of length
M+1 with the last coefficient removed. The optional argument
"symmetric" is equivalent to not specifying a second argument.
For a definition of the Hanning window see, e.g., A.V. Oppenheim &
R. W. Schafer, ‘Discrete-Time Signal Processing’.
-- : hurst (X)
Estimate the Hurst parameter of sample X via the rescaled range
statistic.
If X is a matrix, the parameter is estimated for every column.
-- : PP = pchip (X, Y)
-- : YI = pchip (X, Y, XI)
Return the Piecewise Cubic Hermite Interpolating Polynomial (pchip)
of points X and Y.
If called with two arguments, return the piecewise polynomial PP
that may be used with ‘ppval’ to evaluate the polynomial at
specific points.
When called with a third input argument, ‘pchip’ evaluates the
pchip polynomial at the points XI. The third calling form is
equivalent to ‘ppval (pchip (X, Y), XI)’.
The variable X must be a strictly monotonic vector (either
increasing or decreasing) of length N.
Y can be either a vector or array. If Y is a vector then it must
be the same length N as X. If Y is an array then the size of Y
must have the form ‘[S1, S2, ..., SK, N]’ The array is reshaped
internally to a matrix where the leading dimension is given by ‘S1
* S2 * ... * SK’ and each row of this matrix is then treated
separately. Note that this is exactly opposite to ‘interp1’ but is
done for MATLAB compatibility.
See also: Note: spline, Note: ppval, Note:
mkpp, Note: unmkpp.
-- : [PXX, W] = periodogram (X)
-- : [PXX, W] = periodogram (X, WIN)
-- : [PXX, W] = periodogram (X, WIN, NFFT)
-- : [PXX, F] = periodogram (X, WIN, NFFT, FS)
-- : [PXX, F] = periodogram (..., "RANGE")
-- : periodogram (...)
Return the periodogram (Power Spectral Density) of X.
The possible inputs are:
X
data vector. If X is real-valued a one-sided spectrum is
estimated. If X is complex-valued, or "RANGE" specifies
"twosided", the full spectrum is estimated.
WIN
window weight data. If window is empty or unspecified a
default rectangular window is used. Otherwise, the window is
applied to the signal (‘X .* WIN’) before computing the
periodogram. The window data must be a vector of the same
length as X.
NFFT
number of frequency bins. The default is 256 or the next
higher power of 2 greater than the length of X (‘max (256,
2.^nextpow2 (length (x)))’). If NFFT is greater than the
length of the input then X will be zero-padded to the length
of NFFT.
FS
sampling rate. The default is 1.
RANGE
range of spectrum. "onesided" computes spectrum from
[0:nfft/2+1]. "twosided" computes spectrum from [0:nfft-1].
The optional second output W are the normalized angular
frequencies. For a one-sided calculation W is in the range [0, pi]
if NFFT is even and [0, pi) if NFFT is odd. Similarly, for a
two-sided calculation W is in the range [0, 2*pi] or [0, 2*pi)
depending on NFFT.
If a sampling frequency is specified, FS, then the output
frequencies F will be in the range [0, FS/2] or [0, FS/2) for
one-sided calculations. For two-sided calculations the range will
be [0, FS).
When called with no outputs the periodogram is immediately plotted
in the current figure window.
See also: Note: fft.
-- : sinetone (FREQ, RATE, SEC, AMPL)
Return a sinetone of frequency FREQ with a length of SEC seconds at
sampling rate RATE and with amplitude AMPL.
The arguments FREQ and AMPL may be vectors of common size.
The defaults are RATE = 8000, SEC = 1, and AMPL = 64.
See also: Note: sinewave.
-- : sinewave (M, N, D)
Return an M-element vector with I-th element given by ‘sin (2 * pi
* (I+D-1) / N)’.
The default value for D is 0 and the default value for N is M.
See also: Note: sinetone.
-- : spectral_adf (C)
-- : spectral_adf (C, WIN)
-- : spectral_adf (C, WIN, B)
Return the spectral density estimator given a vector of
autocovariances C, window name WIN, and bandwidth, B.
The window name, e.g., "triangle" or "rectangle" is used to search
for a function called ‘WIN_lw’.
If WIN is omitted, the triangle window is used.
If B is omitted, ‘1 / sqrt (length (X))’ is used.
See also: Note: spectral_xdf.
-- : spectral_xdf (X)
-- : spectral_xdf (X, WIN)
-- : spectral_xdf (X, WIN, B)
Return the spectral density estimator given a data vector X, window
name WIN, and bandwidth, B.
The window name, e.g., "triangle" or "rectangle" is used to search
for a function called ‘WIN_sw’.
If WIN is omitted, the triangle window is used.
If B is omitted, ‘1 / sqrt (length (X))’ is used.
See also: Note: spectral_adf.
-- : spencer (X)
Return Spencer’s 15 point moving average of each column of X.
-- : Y = stft (X)
-- : Y = stft (X, WIN_SIZE)
-- : Y = stft (X, WIN_SIZE, INC)
-- : Y = stft (X, WIN_SIZE, INC, NUM_COEF)
-- : Y = stft (X, WIN_SIZE, INC, NUM_COEF, WIN_TYPE)
-- : [Y, C] = stft (...)
Compute the short-time Fourier transform of the vector X with
NUM_COEF coefficients by applying a window of WIN_SIZE data points
and an increment of INC points.
Before computing the Fourier transform, one of the following
windows is applied:
"hanning"
win_type = 1
"hamming"
win_type = 2
"rectangle"
win_type = 3
The window names can be passed as strings or by the WIN_TYPE
number.
The following defaults are used for unspecified arguments: WIN_SIZE
= 80, INC = 24, NUM_COEF = 64, and WIN_TYPE = 1.
‘Y = stft (X, ...)’ returns the absolute values of the Fourier
coefficients according to the NUM_COEF positive frequencies.
‘[Y, C] = stft (X, ...)’ returns the entire STFT-matrix Y and a
3-element vector C containing the window size, increment, and
window type, which is needed by the ‘synthesis’ function.
See also: Note: synthesis.
-- : X = synthesis (Y, C)
Compute a signal from its short-time Fourier transform Y and a
3-element vector C specifying window size, increment, and window
type.
The values Y and C can be derived by
[Y, C] = stft (X , ...)
See also: Note: stft.
-- : [A, V] = yulewalker (C)
Fit an AR (p)-model with Yule-Walker estimates given a vector C of
autocovariances ‘[gamma_0, ..., gamma_p]’.
Returns the AR coefficients, A, and the variance of white noise, V.
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