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* Octave Signal: (signal).          Signal Toolkit for Octave
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Octave Signal Toolkit
*********************

Copyright © The Octave Project Developers

Permission is granted to make and distribute verbatim copies of this
manual provided the copyright notice and this permission notice are
preserved on all copies.

Permission is granted to copy and distribute modified versions of this
manual under the conditions for verbatim copying, provided that the
entire resulting derived work is distributed under the terms of a
permission notice identical to this one.

Permission is granted to copy and distribute translations of this manual
into another language, under the above conditions for modified versions.
* Menu:

* Overview::
* Installing and loading::
* Function Reference::


File: signal.info,  Node: Overview,  Next: Installing and loading,  Prev: Top,  Up: Top

1 Overview
**********

The Signal Toolkit contains signal processing tools, including
filtering, windowing and display functions.


File: signal.info,  Node: Installing and loading,  Next: Function Reference,  Prev: Overview,  Up: Top

2 Installing and loading
************************

The Signal Toolkit must be installed and then loaded to be used.

It can be installed in GNU Octave directly from octave-forge,

2.1 Windows install
===================

If running in Windows, the package may already be installed, to check
run:

     pkg list signal

2.2 Installing
==============

With an internet connection available, the Signal package can be
installed from octave-forge using the following command within GNU
Octave:

     pkg install -forge signal

The latest released version of the toolkit will be downloaded and
installed.

Otherwise, if the package file has already been downloaded it can be
installed using the follwoing command in GNU Octave:

     pkg install signal-1.4.5.tar.gz

2.3 Loading
===========

Regardless of the method of installing the toolkit, in order to use its
functions, the toolkit must be loaded using the pkg load command:

     pkg load signal

The toolkit must be loaded on each GNU Octave session.


File: signal.info,  Node: Function Reference,  Prev: Installing and loading,  Up: Top

3 Function Reference
********************

* Menu:

* Signals::
* Signal Measurement::
* Correlation and Convolution::
* Filtering::
* Filter Analysis::
* Filter Conversion::
* IIR Filter Design::
* FIR Filter Design::
* Transforms::
* Power Spectrum Analysis::
* Window Functions::
* System Identification::
* Sample Rate Change::
* Pulse Metrics::
* Utility::


File: signal.info,  Node: Signals,  Next: Signal Measurement,  Up: Function Reference

3.1 Signals
===========

3.1.1 buffer
------------

 -- Function File: Y = buffer (X, N, P, OPT)
 -- Function File: [Y, Z, OPT] = buffer (...)
     Buffer a signal into a data frame.  The arguments to ‘buffer’ are

     X
          The data to be buffered.

     N
          The number of rows in the produced data buffer.  This is an
          positive integer value and must be supplied.

     P
          An integer less than N that specifies the under- or overlap
          between column in the data frame.  The default value of P is
          0.

     OPT
          In the case of an overlap, OPT can be either a vector of
          length P or the string ’nodelay’.  If OPT is a vector, then
          the first P entries in Y will be filled with these values.  If
          OPT is the string ’nodelay’, then the first value of Y
          corresponds to the first value of X.

          In the can of an underlap, OPT must be an integer between 0
          and ‘-P’.  The represents the initial underlap of the first
          column of Y.

          The default value for OPT the vector ‘zeros (1, P)’ in the
          case of an overlap, or 0 otherwise.

     In the case of a single output argument, Y will be padded with
     zeros to fill the missing values in the data frame.  With two
     output arguments Z is the remaining data that has not been used in
     the current data frame.

     Likewise, the output OPT is the overlap, or underlap that might be
     used for a future call to ‘code’ to allow continuous buffering.

3.1.2 chirp
-----------

 -- Function File: chirp (T)
 -- Function File: chirp (T, F0)
 -- Function File: chirp (T, F0, T1)
 -- Function File: chirp (T, F0, T1, F1)
 -- Function File: chirp (T, F0, T1, F1, SHAPE)
 -- Function File: chirp (T, F0, T1, F1, SHAPE, PHASE)

     Evaluate a chirp signal at time T.  A chirp signal is a frequency
     swept cosine wave.

     T
          vector of times to evaluate the chirp signal
     F0
          frequency at time t=0 [ 0 Hz ]
     T1
          time t1 [ 1 sec ]
     F1
          frequency at time t=t1 [ 100 Hz ]
     SHAPE
          shape of frequency sweep ’linear’ f(t) = (f1-f0)*(t/t1) + f0
          ’quadratic’ f(t) = (f1-f0)*(t/t1)^2 + f0 ’logarithmic’ f(t) =
          (f1/f0)^(t/t1) * f0
     PHASE
          phase shift at t=0

     For example:

           specgram (chirp ([0:0.001:5]));  # default linear chirp of 0-100Hz in 1 sec
           specgram (chirp ([-2:0.001:15], 400, 10, 100, "quadratic"));
           soundsc (chirp ([0:1/8000:5], 200, 2, 500, "logarithmic"), 8000);

     If you want a different sweep shape f(t), use the following:

      y = cos (2 * pi * integral (f(t)) + phase);

3.1.3 cmorwavf
--------------

 -- Function File: [PSI, X] = cmorwavf (LB, UB, N, FB, FC)
     Compute the Complex Morlet wavelet.

3.1.4 diric
-----------

 -- Function File: Y = diric (X,N)
     Compute the dirichlet function.

     See also: sinc, gauspuls, sawtooth.

3.1.5 gauspuls
--------------

 -- Function File: Y = gauspuls (T)
 -- Function File: Y = gauspuls (T, FC)
 -- Function File: Y = gauspuls (T, FC, BW)
     Generate a Gaussian modulated sinusoidal pulse sampled at times T.

     See also: pulstran, rectpuls, tripuls.

3.1.6 gmonopuls
---------------

 -- Function File: Y = gmonopuls (T,FC)
     Return the gaussian monopulse.

3.1.7 mexihat
-------------

 -- Function File: [PSI, X] = mexihat (LB, UB, N)
     Compute the Mexican hat wavelet.

3.1.8 meyeraux
--------------

 -- Function File: Y = meyeraux (X)
     Compute the Meyer wavelet auxiliary function.

3.1.9 morlet
------------

 -- Function File: [PSI, X] = morlet (LB, UB, N)
     Compute the Morlet wavelet.

3.1.10 pulstran
---------------

 -- Function File: Y = pulstran (T, D, FUNC, ...)
 -- Function File: Y = pulstran (T, D, P)
 -- Function File: Y = pulstran (T, D, P, FS)
 -- Function File: Y = pulstran (T, D, P, FS, METHOD)

     Generate the signal y=sum(func(t+d,...))  for each d.  If d is a
     matrix of two columns, the first column is the delay d and the
     second column is the amplitude a, and y=sum(a*func(t+d)) for each
     d,a.  Clearly, func must be a function which accepts a vector of
     times.  Any extra arguments needed for the function must be tagged
     on the end.

     Example:
           fs = 11025;  # arbitrary sample rate
           f0 = 100;    # pulse train sample rate
           w = 0.001;   # pulse width of 1 millisecond
           auplot (pulstran (0:1/fs:0.1, 0:1/f0:0.1, "rectpuls", w), fs);

     If instead of a function name you supply a pulse shape sampled at
     frequency Fs (default 1 Hz), an interpolated version of the pulse
     is added at each delay d.  The interpolation stays within the the
     time range of the delayed pulse.  The interpolation method defaults
     to linear, but it can be any interpolation method accepted by the
     function interp1.

     Example:
           fs = 11025;      # arbitrary sample rate
           f0 = 100;        # pulse train sample rate
           w = boxcar(10);  # pulse width of 1 millisecond at 10 kHz
           auplot (pulstran (0:1/fs:0.1, 0:1/f0:0.1, w, 10000), fs);

3.1.11 rectpuls
---------------

 -- Function File: Y = rectpuls (T)
 -- Function File: Y = rectpuls (T, W)
     Generate a rectangular pulse over the interval [-W/2,W/2), sampled
     at times T.  This is useful with the function ‘pulstran’ for
     generating a series of pulses.

     Example:
           fs = 11025;  # arbitrary sample rate
           f0 = 100;    # pulse train sample rate
           w = 0.3/f0;  # pulse width 3/10th the distance between pulses
           plot (pulstran (0:1/fs:4/f0, 0:1/f0:4/f0, "rectpuls", w));

     See also: gauspuls, pulstran, tripuls.

3.1.12 sawtooth
---------------

 -- Function File: Y = sawtooth (T)
 -- Function File: Y = sawtooth (T, WIDTH)
     Generates a sawtooth wave of period ‘2 * pi’ with limits ‘+1/-1’
     for the elements of T.

     WIDTH is a real number between ‘0’ and ‘1’ which specifies the
     point between ‘0’ and ‘2 * pi’ where the maximum is.  The function
     increases linearly from ‘-1’ to ‘1’ in ‘[0, 2 * pi * WIDTH]’
     interval, and decreases linearly from ‘1’ to ‘-1’ in the interval
     ‘[2 * pi * WIDTH, 2 * pi]’.

     If WIDTH is 0.5, the function generates a standard triangular wave.

     If WIDTH is not specified, it takes a value of 1, which is a
     standard sawtooth function.

3.1.13 shanwavf
---------------

 -- Function File: [PSI, X] = shanwavf (LB, UB, N, FB, FC)
     Compute the Complex Shannon wavelet.

3.1.14 shiftdata
----------------

 -- Function File: [OUT PERM SHIFTS] = shiftdata (IN)
 -- Function File: [OUT PERM SHIFTS] = shiftdata (IN, DIM)
     Shift data IN to permute the dimension DIM to the first column.

     See also: unshiftdata.

3.1.15 sigmoid_train
--------------------

 -- Function File: [Y S] = sigmoid_train (T, RANGES, RC)

     Evaluate a train of sigmoid functions at T.

     The number and duration of each sigmoid is determined from RANGES.
     Each row of RANGES represents a real interval, e.g.  if sigmoid ‘i’
     starts at ‘t=0.1’ and ends at ‘t=0.5’, then ‘RANGES(i,:) = [0.1
     0.5]’.  The input RC is an array that defines the rising and
     falling time constants of each sigmoid.  Its size must equal the
     size of RANGES.

     The individual sigmoids are returned in S.  The combined sigmoid
     train is returned in the vector Y of length equal to T, and such
     that ‘Y = max (S)’.

     Run ‘demo sigmoid_train’ to some examples of the use of this
     function.

3.1.16 specgram
---------------

 -- Function File: specgram (X)
 -- Function File: specgram (X, N)
 -- Function File: specgram (X, N, FS)
 -- Function File: specgram (X, N, FS, WINDOW)
 -- Function File: specgram (X, N, FS, WINDOW, OVERLAP)
 -- Function File: [S, F, T] = specgram (...)

     Generate a spectrogram for the signal X.  The signal is chopped
     into overlapping segments of length N, and each segment is windowed
     and transformed into the frequency domain using the FFT. The
     default segment size is 256.  If FS is given, it specifies the
     sampling rate of the input signal.  The argument WINDOW specifies
     an alternate window to apply rather than the default of ‘hanning
     (N)’.  The argument OVERLAP specifies the number of samples overlap
     between successive segments of the input signal.  The default
     overlap is ‘length (WINDOW)/2’.

     If no output arguments are given, the spectrogram is displayed.
     Otherwise, S is the complex output of the FFT, one row per slice, F
     is the frequency indices corresponding to the rows of S, and T is
     the time indices corresponding to the columns of S.

     Example:
              x = chirp([0:0.001:2],0,2,500);  # freq. sweep from 0-500 over 2 sec.
              Fs=1000;                  # sampled every 0.001 sec so rate is 1 kHz
              step=ceil(20*Fs/1000);    # one spectral slice every 20 ms
              window=ceil(100*Fs/1000); # 100 ms data window
              specgram(x, 2^nextpow2(window), Fs, window, window-step);

              ## Speech spectrogram
              [x, Fs] = auload(file_in_loadpath("sample.wav")); # audio file
              step = fix(5*Fs/1000);     # one spectral slice every 5 ms
              window = fix(40*Fs/1000);  # 40 ms data window
              fftn = 2^nextpow2(window); # next highest power of 2
              [S, f, t] = specgram(x, fftn, Fs, window, window-step);
              S = abs(S(2:fftn*4000/Fs,:)); # magnitude in range 0<f<=4000 Hz.
              S = S/max(S(:));           # normalize magnitude so that max is 0 dB.
              S = max(S, 10^(-40/10));   # clip below -40 dB.
              S = min(S, 10^(-3/10));    # clip above -3 dB.
              imagesc (t, f, log(S));    # display in log scale
              set (gca, "ydir", "normal"); # put the 'y' direction in the correct direction

     The choice of window defines the time-frequency resolution.  In
     speech for example, a wide window shows more harmonic detail while
     a narrow window averages over the harmonic detail and shows more
     formant structure.  The shape of the window is not so critical so
     long as it goes gradually to zero on the ends.

     Step size (which is window length minus overlap) controls the
     horizontal scale of the spectrogram.  Decrease it to stretch, or
     increase it to compress.  Increasing step size will reduce time
     resolution, but decreasing it will not improve it much beyond the
     limits imposed by the window size (you do gain a little bit,
     depending on the shape of your window, as the peak of the window
     slides over peaks in the signal energy).  The range 1-5 msec is
     good for speech.

     FFT length controls the vertical scale.  Selecting an FFT length
     greater than the window length does not add any information to the
     spectrum, but it is a good way to interpolate between frequency
     points which can make for prettier spectrograms.

     After you have generated the spectral slices, there are a number of
     decisions for displaying them.  First the phase information is
     discarded and the energy normalized:

     S = abs(S); S = S/max(S(:));

     Then the dynamic range of the signal is chosen.  Since information
     in speech is well above the noise floor, it makes sense to
     eliminate any dynamic range at the bottom end.  This is done by
     taking the max of the magnitude and some minimum energy such as
     minE=-40dB. Similarly, there is not much information in the very
     top of the range, so clipping to a maximum energy such as maxE=-3dB
     makes sense:

     S = max(S, 10^(minE/10)); S = min(S, 10^(maxE/10));

     The frequency range of the FFT is from 0 to the Nyquist frequency
     of one half the sampling rate.  If the signal of interest is band
     limited, you do not need to display the entire frequency range.  In
     speech for example, most of the signal is below 4 kHz, so there is
     no reason to display up to the Nyquist frequency of 10 kHz for a 20
     kHz sampling rate.  In this case you will want to keep only the
     first 40% of the rows of the returned S and f.  More generally, to
     display the frequency range [minF, maxF], you could use the
     following row index:

     idx = (f >= minF & f <= maxF);

     Then there is the choice of colormap.  A brightness varying
     colormap such as copper or bone gives good shape to the ridges and
     valleys.  A hue varying colormap such as jet or hsv gives an
     indication of the steepness of the slopes.  The final spectrogram
     is displayed in log energy scale and by convention has low
     frequencies on the bottom of the image:

     imagesc(t, f, flipud(log(S(idx,:))));

3.1.17 square
-------------

 -- Function File: S = square (T, DUTY)
 -- Function File: S = square (T)
     Generate a square wave of period 2 pi with limits +1/-1.

     If DUTY is specified, it is the percentage of time the square wave
     is "on".  The square wave is +1 for that portion of the time.

                        on time * 100
         duty cycle = ------------------
                      on time + off time

     See also: cos, sawtooth, sin, tripuls.

3.1.18 tripuls
--------------

 -- Function File: Y = tripuls (T)
 -- Function File: Y = tripuls (T, W)
 -- Function File: Y = tripuls (T, W, SKEW)
     Generate a triangular pulse over the interval [-W/2,W/2), sampled
     at times T.  This is useful with the function ‘pulstran’ for
     generating a series of pulses.

     SKEW is a value between -1 and 1, indicating the relative placement
     of the peak within the width.  -1 indicates that the peak should be
     at -W/2, and 1 indicates that the peak should be at W/2.  The
     default value is 0.

     Example:
           fs = 11025;  # arbitrary sample rate
           f0 = 100;    # pulse train sample rate
           w = 0.3/f0;  # pulse width 3/10th the distance between pulses
           plot (pulstran (0:1/fs:4/f0, 0:1/f0:4/f0, "tripuls", w));

     See also: gauspuls, pulstran, rectpuls.

3.1.19 udecode
--------------

 -- Function File: OUT = udecode (IN, N)
 -- Function File: OUT = udecode (IN, N, V)
 -- Function File: OUT = udecode (IN, N, V, OVERFLOWS)
     Invert the operation of uencode.

     See also: uencode.

3.1.20 uencode
--------------

 -- Function File: OUT = uencode (IN, N)
 -- Function File: OUT = uencode (IN, N, V)
 -- Function File: OUT = uencode (IN, N, V, SIGNED)
     Quantize the entries of the array IN using 2^N quantization levels.

     See also: udecode.

3.1.21 unshiftdata
------------------

 -- Function File: [OUT] = unshiftdata (IN, PERM, SHIFTS)
     Reverse what is done by shiftdata.

     See also: shiftdata.

3.1.22 vco
----------

 -- : y = vco (X, FC, FS)
 -- : y = vco (X, [FMIN, FMAX], FS)
     Creates a signal that oscillates at a frequency determined by input
     X with a sampling frequency FS.

     Inputs:
        • X - input data with a range of -1 to 1.  A value of -1 means
          no output, 0 cuoreesponds to FC, and 1 corresponds to 2*FC.

        • FC - Carrier frequency

        • FS - Sampling rate

        • FMIN, FMAX - Frequency modulation range limits.

     Outputs:
        • Y - output signal


File: signal.info,  Node: Signal Measurement,  Next: Correlation and Convolution,  Prev: Signals,  Up: Function Reference

3.2 Signal Measurement
======================

3.2.1 findpeaks
---------------

 -- Function File: [PKS, LOC, EXTRA] = findpeaks (DATA)
 -- Function File: ... = findpeaks (..., PROPERTY, VALUE)
 -- Function File: ... = findpeaks (..., "DoubleSided")
     Finds peaks on DATA.

     Peaks of a positive array of data are defined as local maxima.  For
     double-sided data, they are maxima of the positive part and minima
     of the negative part.  DATA is expected to be a single column
     vector.

     The function returns the value of DATA at the peaks in PKS.  The
     index indicating their position is returned in LOC.

     The third output argument is a structure with additional
     information:

     "parabol"
          A structure containing the parabola fitted to each returned
          peak.  The structure has two fields, "x" and "pp".  The field
          "pp" contains the coefficients of the 2nd degree polynomial
          and "x" the extrema of the interval where it was fitted.

     "height"
          The estimated height of the returned peaks (in units of DATA).

     "baseline"
          The height at which the roots of the returned peaks were
          calculated (in units of DATA).

     "roots"
          The abscissa values (in index units) at which the parabola
          fitted to each of the returned peaks realizes its width as
          defined below.

     This function accepts property-value pair given in the list below:

     "MinPeakHeight"
          Minimum peak height (non-negative scalar).  Only peaks that
          exceed this value will be returned.  For data taking positive
          and negative values use the option "DoubleSided".  Default
          value ‘eps’.

     "MinPeakDistance"
          Minimum separation between (positive integer).  Peaks
          separated by less than this distance are considered a single
          peak.  This distance is also used to fit a second order
          polynomial to the peaks to estimate their width, therefore it
          acts as a smoothing parameter.  The neighborhood size is equal
          to the value of "MinPeakDistance".  Default value 1.

     "MinPeakWidth"
          Minimum width of peaks (positive integer).  The width of the
          peaks is estimated using a parabola fitted to the neighborhood
          of each peak.  The width is caulculated with the formula
          a * (width - x0)^2 = 1
          where a is the the concavity of the parabola and x0 its
          vertex.  Default value 1.

     "MaxPeakWidth"
          Maximum width of peaks (positive integer).  Default value
          ‘Inf’.

     "DoubleSided"
          Tells the function that data takes positive and negative
          values.  The base-line for the peaks is taken as the mean
          value of the function.  This is equivalent as passing the
          absolute value of the data after removing the mean.

     Run ‘demo findpeaks’ to see some examples.

3.2.2 peak2peak
---------------

 -- Function File: Y = peak2peak (X)
 -- Function File: Y = peak2peak (X, DIM)
     Compute the difference between the maximum and minimum values in
     the vector X.

     If X is a matrix, compute the difference for each column and return
     them in a row vector.

     If the optional argument DIM is given, operate along this
     dimension.

     See also: max, min, peak2rms, rms, rssq.

3.2.3 peak2rms
--------------

 -- Function File: Y = peak2rms (X)
 -- Function File: Y = peak2rms (X, DIM)
     Compute the ratio of the largest absolute value to the
     root-mean-square (RMS) value of the vector X.

     If X is a matrix, compute the peak-magnitude-to-RMS ratio for each
     column and return them in a row vector.

     If the optional argument DIM is given, operate along this
     dimension.

     See also: max, min, peak2peak, rms, rssq.

3.2.4 rms
---------

 -- Function File: Y = rms (X)
 -- Function File: Y = rms (X, DIM)
     Compute the root-mean-square (RMS) of the vector X.

     The root-mean-square is defined as

           rms (X) = SQRT (1/N SUM_i X(i)^2)

     If X is a matrix, compute the root-mean-square for each column and
     return them in a row vector.

     If the optional argument DIM is given, operate along this
     dimension.

     See also: mean, meansq, peak2rms, rssq, sumsq.

3.2.5 rssq
----------

 -- Function File: Y = rssq (X)
 -- Function File: Y = rssq (X, DIM)
     Compute the root-sum-of-squares (RSS) of the vector X.

     The root-sum-of-squares is defined as

           rssq (X) = SQRT (SUM_i X(i)^2)

     If X is a matrix, compute the root-sum-of-squares for each column
     and return them in a row vector.

     If the optional argument DIM is given, operate along this
     dimension.

     See also: mean, meansq, sumsq, rms.


File: signal.info,  Node: Correlation and Convolution,  Next: Filtering,  Prev: Signal Measurement,  Up: Function Reference

3.3 Correlation and Convolution
===============================

3.3.1 cconv
-----------

 -- Function File: C = cconv (A, B, N)
 -- Function File: C = cconv (A, B)
     Compute the modulo-N circular convolution.

     A and B are input vectors and C is the modolo-N convolution of A
     and B.  If N is not provided, its assumed default value is
     ‘length(A) + length(B) - 1’, which provides the same result as a
     linear convolution.

     Examples:

           cconv (1:2, 1:4)
              ⇒  1   4   7   10   8

           cconv (1:2, 1:4, 2)
              ⇒  16   14

           cconv (1:2, 1:4, 4)
              ⇒  9   4   7   10

     See also: conv, circshift.

3.3.2 convmtx
-------------

 -- Function File: convmtx (A, N)
     If A is a column vector and X is a column vector of length N, then

     ‘convmtx(A, N) * X’

     gives the convolution of of A and X and is the same as ‘conv(A,
     X)’.  The difference is if many vectors are to be convolved with
     the same vector, then this technique is possibly faster.

     Similarly, if A is a row vector and X is a row vector of length N,
     then

     ‘X * convmtx(A, N)’

     is the same as ‘conv(X, A)’.

See also: conv.

3.3.3 wconv
-----------

 -- Function File: Y = wconv (TYPE, X, F)
 -- Function File: Y = wconv (TYPE, X, F, SHAPE)
     1-D or 2-D convolution.

     *Inputs*
     TYPE
          Type of convolution.
     X
          Signal vector or matrix.
     F
          Coefficients of FIR filter.
     SHAPE
          Shape.

     *Outputs*
     Y
          Convoluted signal.

3.3.4 xcorr
-----------

 -- Function File: [R, LAG] = xcorr ( X )
 -- Function File: ... = xcorr ( X, Y )
 -- Function File: ... = xcorr ( ..., MAXLAG)
 -- Function File: ... = xcorr ( ..., SCALE)
     Estimates the cross-correlation.

     Estimate the cross correlation R_xy(k) of vector arguments X and Y
     or, if Y is omitted, estimate autocorrelation R_xx(k) of vector X,
     for a range of lags k specified by argument "maxlag".  If X is a
     matrix, each column of X is correlated with itself and every other
     column.

     The cross-correlation estimate between vectors "x" and "y" (of
     length N) for lag "k" is given by

                      N
           R_xy(k) = sum x_{i+k} conj(y_i),
                     i=1

     where data not provided (for example x(-1), y(N+1)) is zero.  Note
     the definition of cross-correlation given above.  To compute a
     cross-correlation consistent with the field of statistics, see
     ‘xcov’.

     *ARGUMENTS*
     X
          [non-empty; real or complex; vector or matrix] data

     Y
          [real or complex vector] data

          If X is a matrix (not a vector), Y must be omitted.  Y may be
          omitted if X is a vector; in this case xcorr estimates the
          autocorrelation of X.

     MAXLAG
          [integer scalar] maximum correlation lag If omitted, the
          default value is N-1, where N is the greater of the lengths of
          X and Y or, if X is a matrix, the number of rows in X.

     SCALE
          [character string] specifies the type of scaling applied to
          the correlation vector (or matrix).  is one of:
          ‘none’
               return the unscaled correlation, R,
          ‘biased’
               return the biased average, R/N,
          ‘unbiased’
               return the unbiased average, R(k)/(N-|k|),
          ‘coeff or normalized’
               return the correlation coefficient, R/(rms(x).rms(y)),
               where "k" is the lag, and "N" is the length of X.  If
               omitted, the default value is "none".  If Y is supplied
               but does not have the same length as X, scale must be
               "none".

     *RETURNED VARIABLES*
     R
          array of correlation estimates
     LAG
          row vector of correlation lags [-maxlag:maxlag]

     The array of correlation estimates has one of the following forms:
     (1) Cross-correlation estimate if X and Y are vectors.

     (2) Autocorrelation estimate if is a vector and Y is omitted.

     (3) If X is a matrix, R is an matrix containing the
     cross-correlation estimate of each column with every other column.
     Lag varies with the first index so that R has 2*maxlag+1 rows and
     P^2 columns where P is the number of columns in X.

     If Rij(k) is the correlation between columns i and j of X

     ‘R(k+maxlag+1,P*(i-1)+j) == Rij(k)’

     for lag k in [-maxlag:maxlag], or

     ‘R(:,P*(i-1)+j) == xcorr(X(:,i),X(:,j))’.

     ‘reshape(R(k,:),P,P)’ is the cross-correlation matrix for ‘X(k,:)’.

     See also: xcov.

3.3.5 xcorr2
------------

 -- Function File: xcorr2 (A)
 -- Function File: xcorr2 (A, B)
 -- Function File: xcorr2 (..., SCALE)
     Compute the 2D cross-correlation of matrices A and B.

     If B is not specified, computes autocorrelation of A, i.e., same as
     ‘xcorr (A, A)’.

     The optional argument SCALE, defines the type of scaling applied to
     the cross-correlation matrix.  Possible values are:

     "none" (default)
          No scaling.

     "biased"
          Scales the raw cross-correlation by the maximum number of
          elements of A and B involved in the generation of any element
          of C.

     "unbiased"
          Scales the raw correlation by dividing each element in the
          cross-correlation matrix by the number of products A and B
          used to generate that element.

     "coeff"
          Scales the normalized cross-correlation on the range of [0 1]
          so that a value of 1 corresponds to a correlation coefficient
          of 1.

     See also: conv2, corr2, xcorr.

3.3.6 xcov
----------

 -- Function File: [R, LAG] = xcov ( X )
 -- Function File: ... = xcov ( X, Y )
 -- Function File: ... = xcov ( ..., MAXLAG)
 -- Function File: ... = xcov ( ..., SCALE)
     Compute covariance at various lags
     [=correlation(x-mean(x),y-mean(y))].

     X
          input vector
     Y
          if specified, compute cross-covariance between X and Y,
          otherwise compute autocovariance of X.
     MAXLAG
          is specified, use lag range [-maxlag:maxlag], otherwise use
          range [-n+1:n-1].
     SCALE:
          ‘biased’
               for covariance=raw/N,
          ‘unbiased’
               for covariance=raw/(N-|lag|),
          ‘coeff’
               for covariance=raw/(covariance at lag 0),
          ‘none’
               for covariance=raw
          ‘none’
               is the default.

     Returns the covariance for each lag in the range, plus an optional
     vector of lags.

     See also: xcorr.


File: signal.info,  Node: Filtering,  Next: Filter Analysis,  Prev: Correlation and Convolution,  Up: Function Reference

3.4 Filtering
=============

3.4.1 filtfilt
--------------

 -- Function File: Y = filtfilt (B, A, X)

     Forward and reverse filter the signal.  This corrects for phase
     distortion introduced by a one-pass filter, though it does square
     the magnitude response in the process.  That’s the theory at least.
     In practice the phase correction is not perfect, and magnitude
     response is distorted, particularly in the stop band.

     Example
           [b, a]=butter(3, 0.1);                  # 5 Hz low-pass filter
           t = 0:0.01:1.0;                         # 1 second sample
           x=sin(2*pi*t*2.3)+0.25*randn(size(t));  # 2.3 Hz sinusoid+noise
           y = filtfilt(b,a,x); z = filter(b,a,x); # apply filter
           plot(t,x,';data;',t,y,';filtfilt;',t,z,';filter;')

3.4.2 filtic
------------

 -- Function File: ZF = filtic (B, A, Y)
 -- Function File: ZF = filtic (B, A, Y, X)

     Set initial condition vector for filter function The vector zf has
     the same values that would be obtained from function filter given
     past inputs x and outputs y

     The vectors x and y contain the most recent inputs and outputs
     respectively, with the newest values first:

     x = [x(-1) x(-2) ...  x(-nb)], nb = length(b)-1 y = [y(-1) y(-2)
     ...  y(-na)], na = length(a)-a

     If length(x)<nb then it is zero padded If length(y)<na then it is
     zero padded

     zf = filtic(b, a, y) Initial conditions for filter with
     coefficients a and b and output vector y, assuming input vector x
     is zero

     zf = filtic(b, a, y, x) Initial conditions for filter with
     coefficients a and b input vector x and output vector y

3.4.3 medfilt1
--------------

 -- : Y = medfilt1 (X, N)
 -- : Y = medfilt1 (X, N, [], DIM)
 -- : Y = medfilt1 (..., NAN_FLAG, PADDING)

     Apply a one dimensional median filter with a window size of N to
     the data X, which must be real, double and full.  For N = 2m+1,
     Y(i) is the median of X(i-m:i+m).  For N = 2m, Y(i) is the median
     of X(i-m:i+m-1).

     The calculation is performed over the first non-singleton
     dimension, or over dimension DIM if that is specified as the fourth
     argument.  (The third argument is ignored; Matlab used to use it to
     tune its algorithm.)

     NAN_FLAG may be omitnan or includenan (the default).  If it is
     omitnan then any NaN values are removed from the window before the
     median is taken.  Otherwise, any window containing an NaN returns a
     median of NaN.

     PADDING determines how the partial windows at the start and end of
     X are treated.  It may be truncate or zeropad (the default).  If it
     is truncate then the window for Y(i) is the intersection of the
     window stated above with 1:length(X).  If it is zeropad, then
     partial windows have additional zeros to bring them up to size N.

     See also: filter, medfilt2.

3.4.4 movingrms
---------------

 -- Function File: [RMSX,W] = movingrms (X,W,RC,FS=1)
     Calculate moving RMS value of the signal in X.

     The signal is convoluted against a sigmoid window of width W and
     risetime RC.  The units of these parameters are relative to the
     value of the sampling frequency given in FS (Default value = 1).

     Run ‘demo movingrms’ to see an example.

     See also: sigmoid_train.

3.4.5 sgolayfilt
----------------

 -- Function File: Y = sgolayfilt (X)
 -- Function File: Y = sgolayfilt (X, P)
 -- Function File: Y = sgolayfilt (X, P, N)
 -- Function File: Y = sgolayfilt (X, P, N, M)
 -- Function File: Y = sgolayfilt (X, P, N, M, TS)
 -- Function File: Y = sgolayfilt (X, P, N, M, TS)
 -- Function File: Y = sgolayfilt (X, F)
     Smooth the data in x with a Savitsky-Golay smoothing filter of
     polynomial order p and length n, n odd, n > p.  By default, p=3 and
     n=p+2 or n=p+3 if p is even.

     If F is given as a matrix, it is expected to be a filter as
     computed by ‘sgolay’.

     These filters are particularly good at preserving lineshape while
     removing high frequency squiggles.  Particularly, compare a 5
     sample averager, an order 5 butterworth lowpass filter (cutoff 1/3)
     and sgolayfilt(x, 3, 5), the best cubic estimated from 5 points:

           [b, a] = butter (5, 1/3);
           x = [zeros(1,15), 10*ones(1,10), zeros(1,15)];
           plot (sgolayfilt (x), "r;sgolayfilt;", ...
                 filtfilt (ones (1,5)/5, 1, x), "g;5 sample average;", ...
                 filtfilt (b, a, x), "c;order 5 butterworth;", ...
                 x, "+b;original data;");

     See also: sgolay.

3.4.6 sosfilt
-------------

 -- Loadable Function: Y = sosfilt (SOS, X)
     Second order section IIR filtering of X.  The second order section
     filter is described by the matrix SOS with:

             [ B1 A1 ]
     SOS =   [ ... ],
             [ BN AN ]

     where ‘B1 = [b0 b1 b2]’ and ‘A1 = [1 a1 a2]’ for section 1, etc.
     The b0 entry must be nonzero for each section.


File: signal.info,  Node: Filter Analysis,  Next: Filter Conversion,  Prev: Filtering,  Up: Function Reference

3.5 Filter Analysis
===================

3.5.1 filternorm
----------------

 -- Function File: L = filternorm (B, A)
 -- Function File: L = filternorm (B, A, PNORM)
 -- Function File: L = filternorm (B, A, 2, TOL)

     Compute the 2-norm of a digital filter defined by the numerator
     coefficients, B, and the denominator coefficients, A.  It is also
     possible to compute the infinity-norm by passing inf in the PNORM
     parameter.  PNORM only accepts 2 or inf.

     Example:
           [b, a] = butter (8, 0.5);
           filternorm (b, a)

3.5.2 filtord
-------------

 -- Function File: N = filtord (B, A)
 -- Function File: N = filtord (SOS)

     Returns the filter order N for a filter defined by the numerator
     coefficients, B, and the denominator coefficients, A.  It also
     accepts a filter defined by a matrix of second-order sections, SOS.

     Example:
           [b, a] = butter (8, 0.5);
           filtord (b, a)

3.5.3 freqs
-----------

 -- Function File: H = freqs (B, A, W)
 -- Function File: freqs (B, A, W)

     Compute the s-plane frequency response of the IIR filter B(s)/A(s)
     as H = polyval(B,j*W)./polyval(A,j*W). If called with no output
     argument, a plot of magnitude and phase are displayed.

     Example:
           b = [1 2]; a = [1 1];
           w = linspace (0, 4, 128);
           freqs (b, a, w);

3.5.4 freqs_plot
----------------

 -- Function File: freqs_plot (W, H)
     Plot the amplitude and phase of the vector H.

3.5.5 fwhm
----------

 -- Function File: F = fwhm (Y)
 -- Function File: F = fwhm (X, Y)
 -- Function File: F = fwhm (..., "zero")
 -- Function File: F = fwhm (..., "min")
 -- Function File: F = fwhm (..., "alevel", LEVEL)
 -- Function File: F = fwhm (..., "rlevel", LEVEL)

     Compute peak full-width at half maximum (FWHM) or at another level
     of peak maximum for vector or matrix data Y, optionally sampled as
     y(x).  If Y is a matrix, return FWHM for each column as a row
     vector.

     The default option "zero" computes fwhm at half maximum, i.e.
     0.5*max(y).  The option "min" computes fwhm at the middle curve,
     i.e.  0.5*(min(y)+max(y)).

     The option "rlevel" computes full-width at the given relative level
     of peak profile, i.e.  at rlevel*max(y) or rlevel*(min(y)+max(y)),
     respectively.  For example, ‘fwhm (..., "rlevel", 0.1)’ computes
     full width at 10 % of peak maximum with respect to zero or minimum;
     FWHM is equivalent to ‘fwhm(..., "rlevel", 0.5)’.

     The option "alevel" computes full-width at the given absolute level
     of Y.

     Return 0 if FWHM does not exist (e.g.  monotonous function or the
     function does not cut horizontal line at rlevel*max(y) or
     rlevel*(max(y)+min(y)) or alevel, respectively).

3.5.6 grpdelay
--------------

 -- Function File: [G, W] = grpdelay (B)
 -- Function File: [G, W] = grpdelay (B, A)
 -- Function File: [G, W] = grpdelay (..., N)
 -- Function File: [G, W] = grpdelay (..., N, "whole")
 -- Function File: [G, F] = grpdelay (..., N, FS)
 -- Function File: [G, F] = grpdelay (..., N, "whole", FS)
 -- Function File: [G, W] = grpdelay (..., W)
 -- Function File: [G, F] = grpdelay (..., F, FS)
 -- Function File: grpdelay (...)
     Compute the group delay of a filter.

     [g, w] = grpdelay(b) returns the group delay g of the FIR filter
     with coefficients b.  The response is evaluated at 512 angular
     frequencies between 0 and pi.  w is a vector containing the 512
     frequencies.  The group delay is in units of samples.  It can be
     converted to seconds by multiplying by the sampling period (or
     dividing by the sampling rate fs).

     [g, w] = grpdelay(b,a) returns the group delay of the rational IIR
     filter whose numerator has coefficients b and denominator
     coefficients a.

     [g, w] = grpdelay(b,a,n) returns the group delay evaluated at n
     angular frequencies.  For fastest computation n should factor into
     a small number of small primes.

     [g, w] = grpdelay(b,a,n,’whole’) evaluates the group delay at n
     frequencies between 0 and 2*pi.

     [g, f] = grpdelay(b,a,n,Fs) evaluates the group delay at n
     frequencies between 0 and Fs/2.

     [g, f] = grpdelay(b,a,n,’whole’,Fs) evaluates the group delay at n
     frequencies between 0 and Fs.

     [g, w] = grpdelay(b,a,w) evaluates the group delay at frequencies w
     (radians per sample).

     [g, f] = grpdelay(b,a,f,Fs) evaluates the group delay at
     frequencies f (in Hz).

     grpdelay(...)  plots the group delay vs.  frequency.

     If the denominator of the computation becomes too small, the group
     delay is set to zero.  (The group delay approaches infinity when
     there are poles or zeros very close to the unit circle in the z
     plane.)

     Theory: group delay, g(w) = -d/dw [arg{H(e^jw)}], is the rate of
     change of phase with respect to frequency.  It can be computed as:

                         d/dw H(e^-jw)
                  g(w) = -------------
                           H(e^-jw)

     where

                   H(z) = B(z)/A(z) = sum(b_k z^k)/sum(a_k z^k).

     By the quotient rule,

                              A(z) d/dw B(z) - B(z) d/dw A(z)
                  d/dw H(z) = -------------------------------
                                         A(z) A(z)

     Substituting into the expression above yields:

                          A dB - B dA
                  g(w) =  ----------- = dB/B - dA/A
                              A B

     Note that,

                  d/dw B(e^-jw) = sum(k b_k e^-jwk)
                  d/dw A(e^-jw) = sum(k a_k e^-jwk)

     which is just the FFT of the coefficients multiplied by a ramp.

     As a further optimization when nfft>>length(a), the IIR filter
     (b,a) is converted to the FIR filter conv(b,fliplr(conj(a))).  For
     further details, see
     http://ccrma.stanford.edu/~jos/filters/Numerical_Computation_Group_Delay.html

3.5.7 impz
----------

 -- Function File: [X, T] = impz (B)
 -- Function File: [X, T] = impz (B, A)
 -- Function File: [X, T] = impz (B, A, N)
 -- Function File: [X, T] = impz (B, A, N, FS)
 -- Function File: impz (...)

     Generate impulse-response characteristics of the filter.  The
     filter coefficients correspond to the the z-plane rational function
     with numerator b and denominator a.  If a is not specified, it
     defaults to 1.  If n is not specified, or specified as [], it will
     be chosen such that the signal has a chance to die down to -120dB,
     or to not explode beyond 120dB, or to show five periods if there is
     no significant damping.  If no return arguments are requested, plot
     the results.

     See also: freqz, zplane.

3.5.8 isallpass
---------------

 -- Function File: L = isallpass (B, A)
 -- Function File: L = isallpass (SOS)

     Determine whether a digital filter is allpass.  The filter might be
     defined by the numerator coefficients, B, and the denominator
     coefficients, A, or, alternatively, by a matrix of second-order
     sections, SOS.

     Example:
           a = [1 2 3];
           b = [3 2 1];
           isallpass (b, a)

     Ref [1] Shyu, Jong-Jy, & Pei, Soo-Chang, A new approach to the
     design of complex all-pass IIR digital filters, Signal Processing,
     40(2–3), 207–215, 1994.
     https://doi.org/10.1016/0165-1684(94)90068-x

     Ref [2] Vaidyanathan, P. P. Multirate Systems and Filter Banks.
     1st edition, Pearson College Div, 1992.

3.5.9 ismaxphase
----------------

 -- Function File: L = ismaxphase (B, A)
 -- Function File: L = ismaxphase (SOS)
 -- Function File: L = ismaxphase (..., TOL)

     Determine whether a digital filter is maximum phase (maximum
     energy-delay).  The filter might be defined by the numerator
     coefficients, B, and the denominator coefficients, A, or,
     alternatively, by a matrix of second-order sections, SOS.  A
     tolerance TOL might be given to define when two numbers are close
     enough to be considered equal.

     Example:
           b = [1 2 4 4 2 1];
           zplane (b);
           ismaxphase (b)

     Ref [1] Oppenheim, Alan, and Ronald Schafer.  Discrete-Time Signal
     Processing.  3rd edition, Pearson, 2009.

3.5.10 isminphase
-----------------

 -- Function File: L = isminphase (B, A)
 -- Function File: L = isminphase (SOS)
 -- Function File: L = isminphase (..., TOL)

     Determine whether a digital filter is minimum phase.  The filter
     might be defined by the numerator coefficients, B, and the
     denominator coefficients, A, or, alternatively, by a matrix of
     second-order sections, SOS.  A toleranve TOL might be given to
     define when two numbers are close enough to be considered equal.

     Example:
           a = [1 0.5]; b = [3 1];
           isminphase (b, a)

     Ref [1] Oppenheim, Alan, and Ronald Schafer.  Discrete-Time Signal
     Processing.  3rd edition, Pearson, 2009.

3.5.11 isstable
---------------

 -- Function File: FLAG = isstable (B, A)
     Returns a logical output equal to TRUE, if the filter is stable.
     This can be done with coeffients of the filer B and A.
     Alternatively by using a second order sections matrix (SOS).

     Inputs:
        • B: Numerator coefficients of the filter

        • A: Denominator coeffients of the filter.  Can be an empty
          vector.

     Output:
        • FLAG: Returns a logical output, equal to TRUE if the filter is
          stable.

     Examples:
             b = [1 2 3 4 5 5 1 2];
             a = [4 5 6 7 9 10 4 6];
             flag = isstable (b, a)
             flag = 0

     Using SOS
             [z, p, k] = butter (6, 0.7, 'high');
             sos = zp2sos (z, p, k);
             flag = isstable (sos)
             flag = 1

3.5.12 phasez
-------------

 -- Function File: [PHI, W] = phasez (B, A, N)
 -- Function File: [PHI, W] = phasez (B, A)
 -- Function File: [PHI, W] = phasez (SOS, N)
 -- Function File: [PHI, W] = phasez (SOS)
 -- Function File: [PHI, W] = phasez (..., N, "whole")
 -- Function File: [PHI, W] = phasez (..., N, Fs)
 -- Function File: phasez (...)

     Compute the phase response of digital filter defined either by its
     coefficients (B and A are the numerator and denominator
     coefficients respectively) or by its second-order sections
     representation, given by the matrix SOS.  The output PHI is the
     phase response computed in a vector the vector of frequencies W.

     The phase response is evaluated at N angular frequencies between 0
     and pi.

     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.

     If the third/forth argument, "whole", is given, the response is
     evaluated at N angular frequencies between 0 and 2*pi.  It is
     possible also to pass the value "half", which will lead to the
     default behaviour.

     Example:
           [b, a] = butter (2, [.15,.3]);
           phasez (b, a);

     Ref [1] Oppenheim, Alan, and Ronald Schafer.  Discrete-Time Signal
     Processing.  3rd edition, Pearson, 2009.

     See also: freqz, phasedelay.

3.5.13 zplane
-------------

 -- Function File: zplane (Z, P)
 -- Function File: zplane (B, A)
     Plot the poles and zeros on a complex plane.  If the arguments are
     column vectors Z and P, the complex zeros Z and poles P are
     displayed.  If the arguments are row vectors B and A, the zeros and
     poles of the transfer function represented by these filter
     coefficients are displayed.

     If Z and P are matrices, the columns are distinct sets of zeros and
     poles and are displayed together in distinct colors.

     Note that due to the nature of the ‘roots’ function, poles and
     zeros may be displayed as occurring around a circle rather than at
     a single point.

     The transfer function is

                  B(z)   b0 + b1 z^(-1) + b2 z^(-2) + ... + bM z^(-M)
           H(z) = ---- = --------------------------------------------
                  A(z)   a0 + a1 z^(-1) + a2 z^(-2) + ... + aN z^(-N)

                         b0          (z - z1) (z - z2) ... (z - zM)
                       = -- z^(-M+N) ------------------------------
                         a0          (z - p1) (z - p2) ... (z - pN)

     If called with only one argument, the poles P defaults to an empty
     vector, and the denominator coefficient vector A defaults to 1.


File: signal.info,  Node: Filter Conversion,  Next: IIR Filter Design,  Prev: Filter Analysis,  Up: Function Reference

3.6 Filter Conversion
=====================

3.6.1 residued
--------------

 -- Function File: [R, P, F, M] = residued (B, A)
     Compute the partial fraction expansion (PFE) of filter H(z) =
     B(z)/A(z).  In the usual PFE function ‘residuez’, the IIR part
     (poles P and residues R) is driven _in parallel_ with the FIR part
     (F).  In this variant, the IIR part is driven by the _output_ of
     the FIR part.  This structure can be more accurate in signal
     modeling applications.

     INPUTS: B and A are vectors specifying the digital filter H(z) =
     B(z)/A(z).  See ‘help filter’ for documentation of the B and A
     filter coefficients.

     RETURNED:
        • R = column vector containing the filter-pole residues
        • P = column vector containing the filter poles
        • F = row vector containing the FIR part, if any
        • M = column vector of pole multiplicities

     EXAMPLES:
           See test residued verbose to see a number of examples.

     For the theory of operation, see
     ‘http://ccrma.stanford.edu/~jos/filters/residued.html’

     See also: residue, residued.

3.6.2 residuez
--------------

 -- Function File: [R, P, F, M] = residuez (B, A)
     Compute the partial fraction expansion of filter H(z) = B(z)/A(z).

     INPUTS: B and A are vectors specifying the digital filter H(z) =
     B(z)/A(z).  See ‘help filter’ for documentation of the B and A
     filter coefficients.

     RETURNED:
        • R = column vector containing the filter-pole residues
        • P = column vector containing the filter poles
        • F = row vector containing the FIR part, if any
        • M = column vector of pole multiplicities

     EXAMPLES:
           See test residuez verbose to see a number of examples.

     For the theory of operation, see
     ‘http://ccrma.stanford.edu/~jos/filters/residuez.html’

     See also: residue, residued.

3.6.3 sos2ss
------------

 -- Function File: [A, B, C, D] = sos2ss (SOS)
     Convert series second-order sections to state-space.

     See also: sos2ss, ss2tf.

3.6.4 sos2tf
------------

 -- Function File: [B, A] = sos2tf (SOS)
 -- Function File: [B, A] = sos2tf (SOS, G)
     Convert series second-order sections to transfer function.

     INPUTS:

        • SOS = matrix of series second-order sections, one per row:
               SOS = [B1.' A1.'; ...; BN.' AN.']
          where ‘B1.' = [b0 b1 b2] and A1.' = [a0 a1 a2]’ for section 1,
          etc.

          a0 is usually equal to 1 because all 2nd order transfer
          functions can be scaled so that a0 = 1.  However, this is not
          mandatory for this implementation, which supports all kinds of
          transfer functions, including first order transfer functions.
          See ‘filter’ for documentation of the second-order direct-form
          filter coefficients Bi and Ai.

        • G is an overall gain factor that effectively scales the output
          B vector (or any one of the input Bi vectors).  If not given
          the gain is assumed to be 1.

     RETURNED: B and A are vectors specifying the analog or digital
     filter H(s) = B(s)/A(s) or H(z) = B(z)/A(z).  See ‘filter’ for
     further details.

     See also: tf2sos, zp2sos, sos2pz, zp2tf, tf2zp.

3.6.5 sos2zp
------------

 -- Function File: [Z, P, K] = sos2zp (SOS)
 -- Function File: [Z, P, K] = sos2zp (SOS, G)
     Convert series second-order sections to zeros, poles, and gains
     (pole residues).

     INPUTS:

        • SOS = matrix of series second-order sections, one per row:
               SOS = [B1.' A1.'; ...; BN.' AN.']
          where ‘B1.' = [b0 b1 b2] and A1.' = [a0 a1 a2]’ for section 1,
          etc.

          a0 is usually equal to 1 because all 2nd order transfer
          functions can be scaled so that a0 = 1.  However, this is not
          mandatory for this implementation, which supports all kinds of
          transfer functions, including first order transfer functions.
          See ‘filter’ for documentation of the second-order direct-form
          filter coefficients Bi and Ai.

        • G is an overall gain factor that effectively scales any one of
          the input Bi vectors.  If not given the gain is assumed to be
          1.

     RETURNED:
        • Z = column-vector containing all zeros (roots of B(z))
        • P = column-vector containing all poles (roots of A(z))
        • K = overall gain = B(Inf)

     EXAMPLE:
           [z, p, k] = sos2zp ([1 0 1, 1 0 -0.81; 1 0 0, 1 0 0.49])
             ⇒ z =
                0 + 1i
                0 - 1i
                0 + 0i
                0 + 0i
             ⇒ p =
               -0.9000 + 0i
                0.9000 + 0i
                0 + 0.7000i
                0 - 0.7000i
             ⇒ k =  1

     See also: zp2sos, sos2tf, tf2sos, zp2tf, tf2zp.

3.6.6 ss2tf
-----------

 -- Function File: [NUM, DEN] = ss2tf (A, B, C, D)
     Conversion from state-space to transfer function representation.
     The state space system:
                 .
                 x = Ax + Bu
                 y = Cx + Du

     is converted to a transfer function:
                           num(s)
                     G(s)=-------
                           den(s)

3.6.7 ss2zp
-----------

 -- Function File: [Z, P, K] = ss2zp (A, B, C, D)
     Converts a state space representation to a set of poles and zeros;
     K is a gain associated with the zeros.

3.6.8 tf2sos
------------

 -- Function File: [SOS, G] = tf2sos (B, A)
 -- Function File: SOS = tf2sos (B, A)
     Convert direct-form filter coefficients to series second-order
     sections.

     INPUTS:

     B and A are vectors specifying the digital filter H(z) = B(z)/A(z).
     See ‘filter’ for documentation of the B and A filter coefficients.

     RETURNED:
        • SOS = matrix of series second-order sections, one per row:
               SOS = [B1.' A1.'; ...; BN.' AN.']
          where ‘B1.' = [b0 b1 b2] and A1.' = [1 a1 a2]’ for section 1,
          etc.  The b0 entry must be nonzero for each section (zeros at
          infinity not supported).
        • G is an overall gain factor that effectively scales any one of
          the Bi vectors.

     If called with only one output argument, the overall filter gain is
     applied to the first second-order section in the matrix SOS.

     EXAMPLE:
           B = [1 0 0 0 0 1];
           A = [1 0 0 0 0 .9];
           [sos, g] = tf2sos (B, A)

           sos =

              1.00000   0.61803   1.00000   1.00000   0.60515   0.95873
              1.00000  -1.61803   1.00000   1.00000  -1.58430   0.95873
              1.00000   1.00000  -0.00000   1.00000   0.97915  -0.00000

           g = 1

     See also: sos2tf, zp2sos, sos2pz, zp2tf, tf2zp.

3.6.9 tf2ss
-----------

 -- Function File: [A, B, C, D] = tf2ss (NUM, DEN)
     Conversion from transfer function to state-space.  The state space
     system:
                 .
                 x = Ax + Bu
                 y = Cx + Du
     is obtained from a transfer function:
                           num(s)
                     G(s)=-------
                           den(s)

     The state space system matrices obtained from this function will be
     in observable companion form as Wolovich’s Observable Structure
     Theorem is used.

3.6.10 tf2zp
------------

 -- Function File: [Z, P, K] = tf2zp (NUM, DEN)
     Convert transfer functions to poles-and-zero representations.

     Returns the zeros and poles of the system defined by NUM/DEN.  K is
     a gain associated with the system zeros.

3.6.11 zp2sos
-------------

 -- Function File: [SOS, G] = zp2sos (Z)
 -- Function File: [SOS, G] = zp2sos (Z, P)
 -- Function File: [SOS, G] = zp2sos (Z, P, K)
 -- Function File: SOS = zp2sos (...)
     Convert filter poles and zeros to second-order sections.

     INPUTS:
        • Z = column-vector containing the filter zeros
        • P = column-vector containing the filter poles
        • K = overall filter gain factor.  If not given the gain is
          assumed to be 1.

     RETURNED:
        • SOS = matrix of series second-order sections, one per row:
               SOS = [B1.' A1.'; ...; BN.' AN.']
          where ‘B1.' = [b0 b1 b2] and A1.' = [a0 a1 a2]’ for section 1,
          etc.  See ‘filter’ for documentation of the second-order
          direct-form filter coefficients Bi and %Ai, i=1:N.

        • G is the overall gain factor that effectively scales any one
          of the Bi vectors.

     If called with only one output argument, the overall filter gain is
     applied to the first second-order section in the matrix SOS.

     EXAMPLE:
             [z, p, k] = tf2zp ([1 0 0 0 0 1], [1 0 0 0 0 .9]);
             [sos, g] = zp2sos (z, p, k)

           sos =
              1.0000    0.6180    1.0000    1.0000    0.6051    0.9587
              1.0000   -1.6180    1.0000    1.0000   -1.5843    0.9587
              1.0000    1.0000         0    1.0000    0.9791         0

           g =
               1

     See also: sos2zp, sos2tf, tf2sos, zp2tf, tf2zp.

3.6.12 zp2ss
------------

 -- Function File: [A, B, C, D] = zp2ss (Z, P, K)
     Conversion from zero / pole to state space.

     *Inputs*
     Z
     P
          Vectors of (possibly) complex poles and zeros of a transfer
          function.  Complex values must come in conjugate pairs (i.e.,
          x+jy in Z means that x-jy is also in Z).
     K
          Real scalar (leading coefficient).

     *Outputs*
     A
     B
     C
     D
          The state space system, in the form:
                     .
                     x = Ax + Bu
                     y = Cx + Du

3.6.13 zp2tf
------------

 -- Function File: [NUM, DEN] = zp2tf (Z, P, K)
     Converts zeros / poles to a transfer function.

     *Inputs*
     Z
     P
          Vectors of (possibly complex) poles and zeros of a transfer
          function.  Complex values must appear in conjugate pairs.
     K
          Real scalar (leading coefficient).


File: signal.info,  Node: IIR Filter Design,  Next: FIR Filter Design,  Prev: Filter Conversion,  Up: Function Reference

3.7 IIR Filter Design
=====================

3.7.1 besselap
--------------

 -- Function File: [ZERO, POLE, GAIN] = besselap (N)
     Return bessel analog filter prototype.

     References:

     http://en.wikipedia.org/wiki/Bessel_polynomials

3.7.2 besself
-------------

 -- Function File: [B, A] = besself (N, W)
 -- Function File: [B, A] = besself (N, W, "high")
 -- Function File: [Z, P, G] = besself (...)
 -- Function File: [A, B, C, D] = besself (...)
 -- Function File: [...] = besself (..., "z")
     Generate a Bessel filter.  Default is a Laplace space (s) filter.

     [b,a] = besself(n, Wc) low pass filter with cutoff pi*Wc radians

     [b,a] = besself(n, Wc, ’high’) high pass filter with cutoff pi*Wc
     radians

     [z,p,g] = besself(...)  return filter as zero-pole-gain rather than
     coefficients of the numerator and denominator polynomials.

     [...]  = besself(...,’z’) return a discrete space (Z) filter, W
     must be less than 1.

     [a,b,c,d] = besself(...)  return state-space matrices

     References:

     Proakis & Manolakis (1992).  Digital Signal Processing.  New York:
     Macmillan Publishing Company.

3.7.3 bilinear
--------------

 -- Function File: [ZB, ZA] = bilinear (SB, SA, T)
 -- Function File: [ZB, ZA] = bilinear (SZ, SP, SG, T)
 -- Function File: [ZZ, ZP, ZG] = bilinear (...)
     Transform a s-plane filter specification into a z-plane
     specification.  Filters can be specified in either zero-pole-gain
     or transfer function form.  The input form does not have to match
     the output form.  1/T is the sampling frequency represented in the
     z plane.

     Note: this differs from the bilinear function in the signal
     processing toolbox, which uses 1/T rather than T.

     Theory: Given a piecewise flat filter design, you can transform it
     from the s-plane to the z-plane while maintaining the band edges by
     means of the bilinear transform.  This maps the left hand side of
     the s-plane into the interior of the unit circle.  The mapping is
     highly non-linear, so you must design your filter with band edges
     in the s-plane positioned at 2/T tan(w*T/2) so that they will be
     positioned at w after the bilinear transform is complete.

     The following table summarizes the transformation:

           +---------------+-----------------------+----------------------+
           | Transform     | Zero at x             | Pole at x            |
           |    H(S)       |   H(S) = S-x          |    H(S)=1/(S-x)      |
           +---------------+-----------------------+----------------------+
           |       2 z-1   | zero: (2+xT)/(2-xT)   | zero: -1             |
           |  S -> - ---   | pole: -1              | pole: (2+xT)/(2-xT)  |
           |       T z+1   | gain: (2-xT)/T        | gain: (2-xT)/T       |
           +---------------+-----------------------+----------------------+

     With tedious algebra, you can derive the above formulae yourself by
     substituting the transform for S into H(S)=S-x for a zero at x or
     H(S)=1/(S-x) for a pole at x, and converting the result into the
     form:

              H(Z)=g prod(Z-Xi)/prod(Z-Xj)

     Please note that a pole and a zero at the same place exactly
     cancel.  This is significant since the bilinear transform creates
     numerous extra poles and zeros, most of which cancel.  Those which
     do not cancel have a "fill-in" effect, extending the shorter of the
     sets to have the same number of as the longer of the sets of poles
     and zeros (or at least split the difference in the case of the band
     pass filter).  There may be other opportunistic cancellations but I
     will not check for them.

     Also note that any pole on the unit circle or beyond will result in
     an unstable filter.  Because of cancellation, this will only happen
     if the number of poles is smaller than the number of zeros.  The
     analytic design methods all yield more poles than zeros, so this
     will not be a problem.

     References:

     Proakis & Manolakis (1992).  Digital Signal Processing.  New York:
     Macmillan Publishing Company.

3.7.4 buttap
------------

 -- Function File: [Z, P, G] = buttap (N)
     Design lowpass analog Butterworth filter.

     This function exists for MATLAB compatibility only, and is
     equivalent to ‘butter (N, 1, "s")’.

     See also: butter.

3.7.5 butter
------------

 -- Function File: [B, A] = butter (N, WC)
 -- Function File: [B, A] = butter (N, WC, FILTER_TYPE)
 -- Function File: [Z, P, G] = butter (...)
 -- Function File: [A, B, C, D] = butter (...)
 -- Function File: [...] = butter (..., "s")
     Generate a Butterworth filter.  Default is a discrete space (Z)
     filter.

     The cutoff frequency, WC should be specified in radians for analog
     filters.  For digital filters, it must be a value between zero and
     one.  For bandpass filters, WC is a two-element vector with ‘w(1) <
     w(2)’.

     The filter type must be one of "low", "high", "bandpass", or
     "stop".  The default is "low" if WC is a scalar and "bandpass" if
     WC is a two-element vector.

     If the final input argument is "s" design an analog Laplace space
     filter.

     Low pass filter with cutoff ‘pi*Wc’ radians:

           [b, a] = butter (n, Wc)

     High pass filter with cutoff ‘pi*Wc’ radians:

           [b, a] = butter (n, Wc, "high")

     Band pass filter with edges ‘pi*Wl’ and ‘pi*Wh’ radians:

           [b, a] = butter (n, [Wl, Wh])

     Band reject filter with edges ‘pi*Wl’ and ‘pi*Wh’ radians:

           [b, a] = butter (n, [Wl, Wh], "stop")

     Return filter as zero-pole-gain rather than coefficients of the
     numerator and denominator polynomials:

           [z, p, g] = butter (...)

     Return a Laplace space filter, WC can be larger than 1:

           [...] = butter (..., "s")

     Return state-space matrices:

           [a, b, c, d] = butter (...)

     References:

     Proakis & Manolakis (1992).  Digital Signal Processing.  New York:
     Macmillan Publishing Company.

3.7.6 buttord
-------------

 -- Function File: N = buttord (WP, WS, RP, RS)
 -- Function File: N = buttord ([WP1, WP2], [WS1, WS2], RP, RS)
 -- Function File: N = buttord ([WP1, WP2], [WS1, WS2], RP, RS, "s")
 -- Function File: [N, WC_P] = buttord (...)
 -- Function File: [N, WC_P, WC_S] = buttord (...)
     Compute the minimum filter order of a Butterworth filter with the
     desired response characteristics.  The filter frequency band edges
     are specified by the passband frequency WP and stopband frequency
     WS.  Frequencies are normalized to the Nyquist frequency in the
     range [0,1].  RP is the allowable passband ripple measured in
     decibels, and RS is the minimum attenuation in the stop band, also
     in decibels.

     The output arguments N and WC_P (or N and WC_N) can be given as
     inputs to ‘butter’.  Using WC_P makes the filter characteristic
     touch at least one pass band corner and using WC_S makes the
     characteristic touch at least one stop band corner.

     If WP and WS are scalars, then WP is the passband cutoff frequency
     and WS is the stopband edge frequency.  If WS is greater than WP,
     the filter is a low-pass filter.  If WP is greater than WS, the
     filter is a high-pass filter.

     If WP and WS are vectors of length 2, then WP defines the passband
     interval and WS defines the stopband interval.  If WP is contained
     within WS (WS1 < WP1 < WP2 < WS2), the filter is a band-pass
     filter.  If WS is contained within WP (WP1 < WS1 < WS2 < WP2), the
     filter is a band-stop or band-reject filter.

     If the optional argument ‘"s"’ is given, the minimum order for an
     analog elliptic filter is computed.  All frequencies WP and WS are
     specified in radians per second.

     Theory: For Low pass filters, |H(W)|^2 = 1/[1+(W/Wc)^(2N)] =
     10^(-R/10).  With some algebra, you can solve simultaneously for Wc
     and N given Ws,Rs and Wp,Rp.  Rounding N to the next greater
     integer, one can recalculate the allowable range for Wc (filter
     caracteristic touching the pass band edge or the stop band edge).

     For other types of filter, before making the above calculation, the
     requirements must be transformed to LP requirements.  After
     calculation, Wc must be transformed back to original filter type.

     See also: butter, cheb1ord, cheb2ord, ellipord.

3.7.7 cheb
----------

 -- Function File: cheb (N, X)
     Returns the value of the nth-order Chebyshev polynomial calculated
     at the point x.  The Chebyshev polynomials are defined by the
     equations:

                     / cos(n acos(x),    |x| <= 1
             Tn(x) = |
                     \ cosh(n acosh(x),  |x| > 1

     If x is a vector, the output is a vector of the same size, where
     each element is calculated as y(i) = Tn(x(i)).

3.7.8 cheb1ap
-------------

 -- Function File: [Z, P, G] = cheb1ap (N, RP)
     Design lowpass analog Chebyshev type I filter.

     This function exists for MATLAB compatibility only, and is
     equivalent to ‘cheby1 (N, RP, 1, "s")’.

     See also: cheby1.

3.7.9 cheb1ord
--------------

 -- Function File: N = cheb1ord (WP, WS, RP, RS)
 -- Function File: N = cheb1ord ([WP1, WP2], [WS1, WS2], RP, RS)
 -- Function File: N = cheb1ord ([WP1, WP2], [WS1, WS2], RP, RS, "s")
 -- Function File: [N, WC] = cheb1ord (...)
 -- Function File: [N, WC_P, WC_S] = cheb1ord (...)
     Compute the minimum filter order of a Chebyshev type I filter with
     the desired response characteristics.  The filter frequency band
     edges are specified by the passband frequency WP and stopband
     frequency WS.  Frequencies are normalized to the Nyquist frequency
     in the range [0,1].  RP is the allowable passband ripple measured
     in decibels, and RS is the minimum attenuation in the stop band,
     also in decibels.

     The output arguments N and WC_P (or N and WC_S) can be given as
     inputs to ‘cheby1’.  Using WC_P makes the filter characteristic
     touch at least one pass band corner and using WC_S makes the
     characteristic touch at least one stop band corner.

     If WP and WS are scalars, then WP is the passband cutoff frequency
     and WS is the stopband edge frequency.  If WS is greater than WP,
     the filter is a low-pass filter.  If WP is greater than WS, the
     filter is a high-pass filter.

     If WP and WS are vectors of length 2, then WP defines the passband
     interval and WS defines the stopband interval.  If WP is contained
     within WS (WS1 < WP1 < WP2 < WS2), the filter is a band-pass
     filter.  If WS is contained within WP (WP1 < WS1 < WS2 < WP2), the
     filter is a band-stop or band-reject filter.

     If the optional argument ‘"s"’ is given, the minimum order for an
     analog elliptic filter is computed.  All frequencies WP and WS are
     specified in radians per second.

     See also: buttord, cheby1, cheb2ord, ellipord.

3.7.10 cheb2ap
--------------

 -- Function File: [Z, P, G] = cheb2ap (N, RS)
     Design lowpass analog Chebyshev type II filter.

     This function exists for MATLAB compatibility only, and is
     equivalent to ‘cheby2 (N, RS, 1, "s")’.

     Demo
           demo cheb2ap

     See also: cheby2.

3.7.11 cheb2ord
---------------

 -- Function File: N = cheb2ord (WP, WS, RP, RS)
 -- Function File: N = cheb2ord ([WP1, WP2], [WS1, WS2], RP, RS)
 -- Function File: N = cheb2ord ([WP1, WP2], [WS1, WS2], RP, RS, "s")
 -- Function File: [N, WC_S] = cheb2ord (...)
 -- Function File: [N, WC_S, WC_P] = cheb2ord (...)
     Compute the minimum filter order of a Chebyshev type II filter with
     the desired response characteristics.  The filter frequency band
     edges are specified by the passband frequency WP and stopband
     frequency WS.  Frequencies are normalized to the Nyquist frequency
     in the range [0,1].  RP is the allowable passband ripple measured
     in decibels, and RS is the minimum attenuation in the stop band,
     also in decibels.

     The output arguments N and WC_P (or N and WC_S) can be given as
     inputs to ‘cheby2’.  Using WC_P makes the filter characteristic
     touch at least one pass band corner and using WC_S makes the
     characteristic touch at least one stop band corner.

     If WP and WS are scalars, then WP is the passband cutoff frequency
     and WS is the stopband edge frequency.  If WS is greater than WP,
     the filter is a low-pass filter.  If WP is greater than WS, the
     filter is a high-pass filter.

     If WP and WS are vectors of length 2, then WP defines the passband
     interval and WS defines the stopband interval.  If WP is contained
     within WS (WS1 < WP1 < WP2 < WS2), the filter is a band-pass
     filter.  If WS is contained within WP (WP1 < WS1 < WS2 < WP2), the
     filter is a band-stop or band-reject filter.

     If the optional argument ‘"s"’ is given, the minimum order for an
     analog elliptic filter is computed.  All frequencies WP and WS are
     specified in radians per second.

     See also: buttord, cheb1ord, cheby2, ellipord.

3.7.12 cheby1
-------------

 -- Function File: [B, A] = cheby1 (N, RP, W)
 -- Function File: [B, A] = cheby1 (N, RP, W, "high")
 -- Function File: [B, A] = cheby1 (N, RP, [WL, WH])
 -- Function File: [B, A] = cheby1 (N, RP, [WL, WH], "stop")
 -- Function File: [Z, P, G] = cheby1 (...)
 -- Function File: [A, B, C, D] = cheby1 (...)
 -- Function File: [...] = cheby1 (..., "s")
     Generate a Chebyshev type I filter with RP dB of passband ripple.

     [b, a] = cheby1(n, Rp, Wc) low pass filter with cutoff pi*Wc
     radians

     [b, a] = cheby1(n, Rp, Wc, ’high’) high pass filter with cutoff
     pi*Wc radians

     [b, a] = cheby1(n, Rp, [Wl, Wh]) band pass filter with edges pi*Wl
     and pi*Wh radians

     [b, a] = cheby1(n, Rp, [Wl, Wh], ’stop’) band reject filter with
     edges pi*Wl and pi*Wh radians

     [z, p, g] = cheby1(...)  return filter as zero-pole-gain rather
     than coefficients of the numerator and denominator polynomials.

     [...]  = cheby1(...,’s’) return a Laplace space filter, W can be
     larger than 1.

     [a,b,c,d] = cheby1(...)  return state-space matrices

     References:

     Parks & Burrus (1987).  Digital Filter Design.  New York: John
     Wiley & Sons, Inc.

3.7.13 cheby2
-------------

 -- Function File: [B, A] = cheby2 (N, RS, WC)
 -- Function File: [B, A] = cheby2 (N, RS, WC, "high")
 -- Function File: [B, A] = cheby2 (N, RS, [WL, WH])
 -- Function File: [B, A] = cheby2 (N, RS, [WL, WH], "stop")
 -- Function File: [Z, P, G] = cheby2 (...)
 -- Function File: [A, B, C, D] = cheby2 (...)
 -- Function File: [...] = cheby2 (..., "s")
     Generate a Chebyshev type II filter with RS dB of stopband
     attenuation.

     [b, a] = cheby2(n, Rs, Wc) low pass filter with cutoff pi*Wc
     radians

     [b, a] = cheby2(n, Rs, Wc, ’high’) high pass filter with cutoff
     pi*Wc radians

     [b, a] = cheby2(n, Rs, [Wl, Wh]) band pass filter with edges pi*Wl
     and pi*Wh radians

     [b, a] = cheby2(n, Rs, [Wl, Wh], ’stop’) band reject filter with
     edges pi*Wl and pi*Wh radians

     [z, p, g] = cheby2(...)  return filter as zero-pole-gain rather
     than coefficients of the numerator and denominator polynomials.

     [...]  = cheby2(...,’s’) return a Laplace space filter, W can be
     larger than 1.

     [a,b,c,d] = cheby2(...)  return state-space matrices

     References:

     Parks & Burrus (1987).  Digital Filter Design.  New York: John
     Wiley & Sons, Inc.

3.7.14 ellip
------------

 -- Function File: [B, A] = ellip (N, RP, RS, WP)
 -- Function File: [B, A] = ellip (N, RP, RS, WP, "high")
 -- Function File: [B, A] = ellip (N, RP, RS, [WL, WH])
 -- Function File: [B, A] = ellip (N, RP, RS, [WL, WH], "stop")
 -- Function File: [Z, P, G] = ellip (...)
 -- Function File: [A, B, C, D] = ellip (...)
 -- Function File: [...] = ellip (..., "s")

     Generate an elliptic or Cauer filter with RP dB of passband ripple
     and RS dB of stopband attenuation.

     [b,a] = ellip(n, Rp, Rs, Wp) low pass filter with order n, cutoff
     pi*Wp radians, Rp decibels of ripple in the passband and a stopband
     Rs decibels down.

     [b,a] = ellip(n, Rp, Rs, Wp, ’high’) high pass filter with cutoff
     pi*Wp...

     [b,a] = ellip(n, Rp, Rs, [Wl, Wh]) band pass filter with band pass
     edges pi*Wl and pi*Wh ...

     [b,a] = ellip(n, Rp, Rs, [Wl, Wh], ’stop’) band reject filter with
     edges pi*Wl and pi*Wh, ...

     [z,p,g] = ellip(...)  return filter as zero-pole-gain.

     [...]  = ellip(...,’s’) return a Laplace space filter, W can be
     larger than 1.

     [a,b,c,d] = ellip(...)  return state-space matrices

     References:

     - Oppenheim, Alan V., Discrete Time Signal Processing, Hardcover,
     1999.  - Parente Ribeiro, E., Notas de aula da disciplina TE498 -
     Processamento Digital de Sinais, UFPR, 2001/2002.

3.7.15 ellipap
--------------

 -- Function File: [Z, P, G] = ellipap (N, RP, RS)
     Design lowpass analog elliptic filter.

     This function exists for MATLAB compatibility only, and is
     equivalent to ‘ellip (N, RP, RS, 1, "s")’.

     See also: ellip.

3.7.16 ellipord
---------------

 -- Function File: N = ellipord (WP, WS, RP, RS)
 -- Function File: N = ellipord ([WP1, WP2], [WS1, WS2], RP, RS)
 -- Function File: N = ellipord ([WP1, WP2], [WS1, WS2], RP, RS, "s")
 -- Function File: [N, WC] = ellipord (...)
     Compute the minimum filter order of an elliptic filter with the
     desired response characteristics.  The filter frequency band edges
     are specified by the passband frequency WP and stopband frequency
     WS.  Frequencies are normalized to the Nyquist frequency in the
     range [0,1].  RP is the allowable passband ripple measured in
     decibels, and RS is the minimum attenuation in the stop band, also
     in decibels.  The output arguments N and WC can be given as inputs
     to ‘ellip’.

     If WP and WS are scalars, then WP is the passband cutoff frequency
     and WS is the stopband edge frequency.  If WS is greater than WP,
     the filter is a low-pass filter.  If WP is greater than WS, the
     filter is a high-pass filter.

     If WP and WS are vectors of length 2, then WP defines the passband
     interval and WS defines the stopband interval.  If WP is contained
     within WS (WS1 < WP1 < WP2 < WS2), the filter is a band-pass
     filter.  If WS is contained within WP (WP1 < WS1 < WS2 < WP2), the
     filter is a band-stop or band-reject filter.

     If the optional argument ‘"s"’ is given, the minimum order for an
     analog elliptic filter is computed.  All frequencies WP and WS are
     specified in radians per second.

     Reference: Lamar, Marcus Vinicius, ‘Notas de aula da disciplina TE
     456 - Circuitos Analogicos II’, UFPR, 2001/2002.

     See also: buttord, cheb1ord, cheb2ord, ellip.

3.7.17 firpm
------------

 -- Loadable Function: B = firpm (N, F, A)
 -- Loadable Function: B = firpm (N, F, @RESPFN)
 -- Loadable Function: B = firpm (N, F, {@RESPFN, ...})
 -- Loadable Function: B = firpm (..., W)
 -- Loadable Function: B = firpm (..., CLASS)
 -- Loadable Function: B = firpm (..., {ACCURACY, ...})
 -- Loadable Function: [B, MINIMAX] = firpm (...)
 -- Loadable Function: [B, MINIMAX, RES] = firpm (...)

     Designs a linear-phase FIR filter according to given specifications
     and the ‘minimax’ criterion.  The method (per McClellan et al.(1))
     uses successive approximation to minimize the maximum weighted
     error between the desired and actual frequency response of the
     filter.  Such filters are variably described as being ‘minimax’,
     ‘equiripple’, or ‘optimal (in the Chebyshev sense)’.

     Arguments
     =========

     ...
          Where shown as the first argument to ‘firpm’, indicates that
          any previously-indicated list of arguments may substitute for
          the ellipsis.

     N
          A positive integer giving the filter order.

     F
          A vector of real-numbers, increasing in the range [0,1],
          giving the frequencies of the left and right edges of each
          band for which a specific amplitude response is desired: [l1
          r1 l2 r2 ...].  1 represents the Nyquist-frequency.
          Transition-bands are defined implicitly as the regions between
          or outside the given bands.

     A
          A vector of real-numbers giving the desired amplitude
          response.  An amplitude value is given either for each band
          edge: [a(l1) a(r1) a(l2) a(r2) ...], or for each band: [a1 a2
          ...].  In the former case, in-band amplitude is determined by
          linear interpolation between the given band-edge values.  1
          represents unity-gain, 0 represents infinite attenuation, and
          −1 represents a phase change of pi radians.

          Note that amplitude response is necessarily zero at F=0 for
          type III and IV filters, and at F=1 for type II and III
          filters.

     @RESPFN
          A handle to a ‘response function’ that supplies the desired
          amplitude response and error-weighting.  This, unlike A above,
          allows the response to be arbitrary (subject to the note
          above).  firpm invokes the response function according to the
          following syntax:

               AG = respFn (N,F,G,W, ...)
               [AG WG] = respFn (N,F,G,W, ...)
               SYMMETRY = respFn ("defaults", {N,F,G,W, ...})

          where:
             • N and F are as given to firpm.

             • W is as given to firpm, or ones if not given.

             • AG and WG are the desired amplitude and weighting
               functions evaluated at each frequency in vector G (which
               are frequencies within the non-transition bands of F).
               Returning AG alone gives uniform weighting.

             • SYMMETRY is either "even" or "odd"; this provides an
               alternative to using the CLASS values "symmetric" and
               "antisymmetric".

             • Per the ellipses shown here and above, when @RESPFN is
               given contained in a cell-array, any additionally
               contained values are appended to the RESPFN invocation
               argument-list.

     W
          When used in conjunction with A, W is a vector of positive
          real-numbers giving error-weightings to be applied at each
          given band-edge [w(l1) w(r1) w(l2) w(r2) ...], or for each
          band [w1 w2 ...].  In the former case, in-band weighting is
          determined by linear interpolation between the given band-edge
          values.  A higher relative error weighting yields a lower
          relative error.

          When used in conjunction with @RESPFN, W is a vector
          (constrained as above) that is passed through to RESPFN.

     CLASS
          A string, which may be abbreviated, giving the filter-class:

             • "symmetric" (the default) for type I or II filters,

             • "antisymmetric" (or "hilbert") for standard type III or
               IV filters,

             • "differentiator" for type III or IV filters with inverted
               phase and with error-weighting (further to W) of 2/f
               applied in the pass-band(s).

     ACCURACY, ...
          Up to three properties contained within a cell-array:
          ACCURACY, PERSISTENCE, ROBUSTNESS, that respectively control
          how close the computed filter will be to the ideal minimax
          solution, the number of computation iterations over which the
          required accuracy will be sought, and the precision of certain
          internal processing.  Each can each be set to a small positive
          number (typically ≤3), to increase the relevant item; this may
          increase computation time, but the need to do so should be
          rare.  A value of 0 can be used to leave an item unchanged.

          Alternatively, setting ACCURACY ≥16 emulates MATLAB’s LGRID
          argument.

     Results
     =======

     If a problem occurs during the computation, a diagnostic message
     will normally be displayed.  If this happens, adjusting ACCURACY,
     PERSISTENCE, or ROBUSTNESS may provide the solution.  Some filters
     however, may not be realizable due to machine-precision
     limitations.  If a filter can be computed, returned values are as
     follows:

     B
          A length N+1 row-vector containing the computed filter
          coefficients.

     MINIMAX
          The absolute value of the minimized, maximum weighted error,
          or this number negated if the required accuracy could not be
          achieved.

     RES
          A structure of data relating to the filter computation and a
          partial response-analysis of the resultant filter; fields are
          vectors:
               ‘fgrid’   Analysis frequencies per F.
               ‘des’     Desired amplitude response.
               ‘wt’      Error weighting.
               ‘H’       Complex frequency response.
               ‘error’   Desired minus actual amplitude response.
               ‘iextr’   Indices of local peaks in ‘error’.
               ‘fextr’   Frequencies of local peaks in ‘error’.

     Using RES is not recommended because it can be slow to compute and,
     since the analysis excludes transition-bands, any ‘anomalies’(2)
     therein are not easy to discern.  In general, ‘freqz’ suffices to
     check that the response of the computed filter is satisfactory.

     Examples
     ========

          # Low-pass with frequencies in Hz:
          Fs = 96000; Fn = Fs/2; # Sampling & Nyquist frequencies.
          b = firpm (50, [0 20000 28000 48000] / Fn, [1 0]);

          # Type IV high-pass:
          b = firpm (31, [0 0.5 0.7 1], [0 1], "antisym");

          # Inverse-sinc (arbitrary response):
          b = firpm (20, [0 0.5 0.9 1], @(n,f,g) ...
          deal ((g<=f(2))./sinc (g), (g>=f(3))*9+1));

          # Band-pass with filter-response check:
          freqz (firpm (40, [0 3 4 6 8 10]/10, [0 1 0]))

     Further examples can be found in the ‘firpm’ and ‘firpmord’
     demonstration scripts.

     Compatibility
     =============

     Given invalid filter specifications, Octave emits an error and does
     not produce a filter; MATLAB in such circumstances may still
     produce filter coefficients.

     Unlike with MATLAB, with Octave MINIMAX can be negative; for
     compatibility, take the absolute value.

     See also: firpmord.

3.7.18 firpmord
---------------

 -- Function File: [N, FOUT, A, W] = firpmord (F, A, D)
 -- Function File: [N, FOUT, A, W] = firpmord (F, A, D, FS)
 -- Function File: C = firpmord (F, A, D, "cell")
 -- Function File: C = firpmord (F, A, D, FS, "cell")

     Estimate the filter-order needed for ‘firpm’ to design a type-I or
     type-II linear-phase FIR filter according to the given
     specifications.

     Arguments
     =========

     F
          A vector of real-numbers, increasing in the range (0, FS/2),
          giving the frequencies of the left and right edges of each
          band for which a specific amplitude response is desired
          (omitting 0 and FS/2, which are implied): [r1 l2 r2 ...].
          Transition-bands are defined implicitly as the regions between
          the given bands.

     A
          A vector of real-numbers giving the ideal amplitude response.
          An amplitude value is given for each band specified by F: [a1
          a2 ...].  1 represents unity-gain, 0 represents infinite
          attenuation, and −1 represents a phase change of pi radians.

     D
          A vector of positive real-numbers giving the maximum allowable
          linear deviation from the amplitudes given in A, thus
          constraining the actual amplitude response (where defined by
          F) to be within A +/− D.  Note that, though related, D does
          not equate to ‘firpm’’s W argument.

     FS

          The sampling-frequency, which defaults to 2.

     Usage
     =====

     The function returns the estimated filter-order, together with the
     other design specification values, in one of two forms suitable for
     use with ‘firpm’.  By default, multiple return values are used;
     alternatively, by giving "cell" (or "c") as the last argument to
     ‘firpmord’, the returned values are contained within a cell-array
     that can, if desired, be passed directly to ‘firpm’.

     The following examples illustrate the use of both mechanisms, as
     well as aspects of ‘firpmord’ usage in general:

           # Low-pass; frequencies in kHz:
           [n f a w] = firpmord ([2.5 3], [1 0], [0.01 db2mag(-60)], 8);
           b = firpm (n, f, a, w);

           # Band-pass:
           c = firpmord ([3 4 8 9], [0 1 0], [1e-3 1e-2 1e-3], 20, "cell");
           b = firpm (c{:});

           # High-pass:
           b = firpm (firpmord ([6.4 8]/16, [0 1], [1e-4 0.01], "c"){:});

     In cases where elements of D follow a repeating pattern (e.g. all
     the elements are equal, or elements corresponding to pass-bands are
     equal and elements corresponding to stop-bands are equal), only as
     many elements as are needed to establish the pattern need be given.

     For example, the following ‘firpmord’ invocation pairs are
     equivalent:

           # Low-pass:
           firpmord ([0.4 0.5], [0 1], [db2mag(-72) db2mag(-72)]);
           firpmord ([0.4 0.5], [0 1], [db2mag(-72)]);

           # Multi-band-pass:
           ds = db2mag(-80); dp = 0.01;
           firpmord ([1 2 3 4 5 6 7 8]/10, [0 1 0 1 0], [ds dp ds dp ds]);
           firpmord ([1 2 3 4 5 6 7 8]/10, [0 1 0 1 0], [ds dp]);

     Notes
     =====

     The estimation algorithm used is per Ichige et al.(3)  Accuracy
     tends to decrease as the number of bands increases.  Even with two
     bands (i.e. high-pass or low-pass), the algorithm may under- or
     over-estimate.  See the ‘firpmord’ demonstrations for some
     examples.

     In order to precisely determine the minimum order needed for a
     particular design, ‘firpmord’ could be used to seed an algorithm
     iterating invocations of ‘firpm’ (as exemplified in demonstration
     number five).

     Related documentation
     =====================

     See also: firpm, kaiserord.

3.7.19 impinvar
---------------

 -- Function File: [B_OUT, A_OUT] = impinvar (B, A, FS, TOL)
 -- Function File: [B_OUT, A_OUT] = impinvar (B, A, FS)
 -- Function File: [B_OUT, A_OUT] = impinvar (B, A)
     Converts analog filter with coefficients B and A to digital,
     conserving impulse response.

     If FS is not specified, or is an empty vector, it defaults to 1Hz.

     If TOL is not specified, it defaults to 0.0001 (0.1%) This function
     does the inverse of impinvar so that the following example should
     restore the original values of A and B.

     ‘invimpinvar’ implements the reverse of this function.
           [b, a] = impinvar (b, a);
           [b, a] = invimpinvar (b, a);

     Reference: Thomas J. Cavicchi (1996) “Impulse invariance and
     multiple-order poles”.  IEEE transactions on signal processing, Vol
     44 (9): 2344–2347

     See also: bilinear, invimpinvar.

3.7.20 invimpinvar
------------------

 -- Function File: [B_OUT, A_OUT] = invimpinvar (B, A, FS, TOL)
 -- Function File: [B_OUT, A_OUT] = invimpinvar (B, A, FS)
 -- Function File: [B_OUT, A_OUT] = invimpinvar (B, A)
     Converts digital filter with coefficients B and A to analog,
     conserving impulse response.

     This function does the inverse of impinvar so that the following
     example should restore the original values of A and B.
           [b, a] = impinvar (b, a);
           [b, a] = invimpinvar (b, a);

     If FS is not specified, or is an empty vector, it defaults to 1Hz.

     If TOL is not specified, it defaults to 0.0001 (0.1%)

     Reference: Thomas J. Cavicchi (1996) “Impulse invariance and
     multiple-order poles”.  IEEE transactions on signal processing, Vol
     40 (9): 2344–2347

     See also: bilinear, impinvar.

3.7.21 ncauer
-------------

 -- Function File: [Z, P, G] = cauer(RP, RS, N)
     Analog prototype for Cauer filter.

     Rp
          Passband ripple
     Rs
          Stopband ripple
     n
          Desired order
     z
          complex vector of zeros for the model.
     p
          complex vector of poles for the model.
     g
          gain value.

     References:

     - Serra, Celso Penteado, Teoria e Projeto de Filtros, Campinas:
     CARTGRAF, 1983.

     - Lamar, Marcus Vinicius, Notas de aula da disciplina TE 456 -
     Circuitos Analogicos II, UFPR, 2001/2002.

3.7.22 pei_tseng_notch
----------------------

 -- Function File: [B, A] = pei_tseng_notch (FREQUENCIES, BANDWIDTHS)
     Return coefficients for an IIR notch-filter with one or more filter
     frequencies and according (very narrow) bandwidths to be used with
     ‘filter’ or ‘filtfilt’.  The filter construction is based on an
     allpass which performs a reversal of phase at the filter
     frequencies.  Thus, the mean of the phase-distorted and the
     original signal has the respective frequencies removed.  See the
     demo for an illustration.

     Original source: Pei, Soo-Chang, and Chien-Cheng Tseng "IIR
     Multiple Notch Filter Design Based on Allpass Filter" 1996 IEEE
     Tencon doi: 10.1109/TENCON.1996.608814)

3.7.23 sftrans
--------------

 -- Function File: [SZ, SP, SG] = sftrans (SZ, SP, SG, W, STOP)

     Transform band edges of a generic lowpass filter (cutoff at W=1)
     represented in splane zero-pole-gain form.  W is the edge of the
     target filter (or edges if band pass or band stop).  Stop is true
     for high pass and band stop filters or false for low pass and band
     pass filters.  Filter edges are specified in radians, from 0 to pi
     (the nyquist frequency).

     Theory: Given a low pass filter represented by poles and zeros in
     the splane, you can convert it to a low pass, high pass, band pass
     or band stop by transforming each of the poles and zeros
     individually.  The following table summarizes the transformation:

           Transform         Zero at x                  Pole at x
           ----------------  -------------------------  ------------------------
           Low Pass          zero: Fc x/C               pole: Fc x/C
           S -> C S/Fc       gain: C/Fc                 gain: Fc/C
           ----------------  -------------------------  ------------------------
           High Pass         zero: Fc C/x               pole: Fc C/x
           S -> C Fc/S       pole: 0                    zero: 0
                             gain: -x                   gain: -1/x
           ----------------  -------------------------  ------------------------
           Band Pass         zero: b +- sqrt(b^2-FhFl)  pole: b +- sqrt(b^2-FhFl)
                  S^2+FhFl   pole: 0                    zero: 0
           S -> C --------   gain: C/(Fh-Fl)            gain: (Fh-Fl)/C
                  S(Fh-Fl)   b=x/C (Fh-Fl)/2            b=x/C (Fh-Fl)/2
           ----------------  -------------------------  ------------------------
           Band Stop         zero: b +- sqrt(b^2-FhFl)  pole: b +- sqrt(b^2-FhFl)
                  S(Fh-Fl)   pole: +-sqrt(-FhFl)        zero: +-sqrt(-FhFl)
           S -> C --------   gain: -x                   gain: -1/x
                  S^2+FhFl   b=C/x (Fh-Fl)/2            b=C/x (Fh-Fl)/2
           ----------------  -------------------------  ------------------------
           Bilinear          zero: (2+xT)/(2-xT)        pole: (2+xT)/(2-xT)
                2 z-1        pole: -1                   zero: -1
           S -> - ---        gain: (2-xT)/T             gain: (2-xT)/T
                T z+1
           ----------------  -------------------------  ------------------------

     where C is the cutoff frequency of the initial lowpass filter, Fc
     is the edge of the target low/high pass filter and [Fl,Fh] are the
     edges of the target band pass/stop filter.  With abundant tedious
     algebra, you can derive the above formulae yourself by substituting
     the transform for S into H(S)=S-x for a zero at x or H(S)=1/(S-x)
     for a pole at x, and converting the result into the form:

              H(S)=g prod(S-Xi)/prod(S-Xj)

     The transforms are from the references.  The actual pole-zero-gain
     changes I derived myself.

     Please note that a pole and a zero at the same place exactly
     cancel.  This is significant for High Pass, Band Pass and Band Stop
     filters which create numerous extra poles and zeros, most of which
     cancel.  Those which do not cancel have a "fill-in" effect,
     extending the shorter of the sets to have the same number of as the
     longer of the sets of poles and zeros (or at least split the
     difference in the case of the band pass filter).  There may be
     other opportunistic cancellations but I will not check for them.

     Also note that any pole on the unit circle or beyond will result in
     an unstable filter.  Because of cancellation, this will only happen
     if the number of poles is smaller than the number of zeros and the
     filter is high pass or band pass.  The analytic design methods all
     yield more poles than zeros, so this will not be a problem.

     References:

     Proakis & Manolakis (1992).  Digital Signal Processing.  New York:
     Macmillan Publishing Company.

   ---------- Footnotes ----------

   (1) J. H. McClellan, T. W. Parks and L. R. Rabiner, ‘A Computer
Program for Designing Optimum FIR Linear Phase Digital Filters’, IEEE
Trans. Audio Electroacoust., vol. AU-21, 1973, pp. 506–525.

   (2) Tapio Saramäki, ‘Finite impulse response filter design’, Chapter
4 in ‘Handbook for Digital Signal Processing’, edited by S. K. Mitra and
J. F. Kaiser, John Wiley and Sons, New York, 1993, pp. 155–277.
(<http://www.cs.tut.fi/~ts/Mitra_Kaiser.pdf>)

   (3) K. Ichige, M. Iwaki, algorithm and R. Ishii, ‘Accurate Estimation
of Minimum Filter Length for Optimum FIR Digital Filters’, IEEE
Transactions on Circuits and Systems, Vol. 47, No. 10, 2000, pp.
1008–1017


File: signal.info,  Node: FIR Filter Design,  Next: Transforms,  Prev: IIR Filter Design,  Up: Function Reference

3.8 FIR Filter Design
=====================

3.8.1 cl2bp
-----------

 -- Loadable Function: H = cl2bp (M, W1, W2, UP, LO)
 -- Loadable Function: H = cl2bp (M, W1, W2, UP, LO, GRIDSIZE)

     Constrained L2 bandpass FIR filter design.  This is a fast
     implementation of the algorithm cited below.  Compared to “remez”,
     it offers implicit specification of transition bands, a higher
     likelihood of convergence, and an error criterion combining
     features of both L2 and Chebyshev approaches.

     Inputs:

     M
          degree of cosine polynomial, i.e.  the number of output
          coefficients will be M*2+1
     W1
     W2
          bandpass filter cutoffs in the range 0 <= W1 < W2 <= pi, where
          pi is the Nyquist frequency
     UP
          vector of 3 upper bounds for [stopband1, passband, stopband2]
     LO
          vector of 3 lower bounds for [stopband1, passband, stopband2]
     GRIDSIZE
          search grid size; larger values may improve accuracy, but
          greatly increase calculation time.

     Output:

     A vector of M*2+1 FIR coefficients, or an empty value if the solver
     failed to converge.

     Example:
          h = cl2bp(30, 0.3*pi, 0.6*pi, [0.02, 1.02, 0.02], [-0.02, 0.98, -0.02], 2^11);

     Original Paper: I. W. Selesnick, M. Lang, and C. S. Burrus.  A
     modified algorithm for constrained least square design of multiband
     FIR filters without specified transition bands.  IEEE Trans.  on
     Signal Processing, 46(2):497-501, February 1998.

See also: remez.

3.8.2 fir1
----------

 -- Function File: B = fir1 (N, W)
 -- Function File: B = fir1 (N, W, TYPE)
 -- Function File: B = fir1 (N, W, TYPE, WINDOW)
 -- Function File: B = fir1 (N, W, TYPE, WINDOW, NOSCALE)

     Produce an order N FIR filter with the given frequency cutoff W,
     returning the N+1 filter coefficients in B.  If W is a scalar, it
     specifies the frequency cutoff for a lowpass or highpass filter.
     If W is a two-element vector, the two values specify the edges of a
     bandpass or bandstop filter.  If W is an N-element vector, each
     value specifies a band edge of a multiband pass/stop filter.

     The filter TYPE can be specified with one of the following strings:
     "low", "high", "stop", "pass", "bandpass", "DC-0", or "DC-1".  The
     default is "low" is W is a scalar, "pass" if W is a pair, or "DC-0"
     if W is a vector with more than 2 elements.

     An optional shaping WINDOW can be given as a vector with length
     N+1.  If not specified, a Hamming window of length N+1 is used.

     With the option "noscale", the filter coefficients are not
     normalized.  The default is to normalize the filter such that the
     magnitude response of the center of the first passband is 1.

     To apply the filter, use the return vector B with the ‘filter’
     function, for example ‘y = filter (b, 1, x)’.

     Examples:
           freqz (fir1 (40, 0.3));
           freqz (fir1 (15, [0.2, 0.5], "stop"));  # note the zero-crossing at 0.1
           freqz (fir1 (15, [0.2, 0.5], "stop", "noscale"));

     See also: filter, fir2.

3.8.3 fir2
----------

 -- Function File: B = fir2 (N, F, M)
 -- Function File: B = fir2 (N, F, M, GRID_N)
 -- Function File: B = fir2 (N, F, M, GRID_N, RAMP_N)
 -- Function File: B = fir2 (N, F, M, GRID_N, RAMP_N, WINDOW)

     Produce an order N FIR filter with arbitrary frequency response M
     over frequency bands F, returning the N+1 filter coefficients in B.
     The vector F specifies the frequency band edges of the filter
     response and M specifies the magnitude response at each frequency.

     The vector F must be nondecreasing over the range [0,1], and the
     first and last elements must be 0 and 1, respectively.  A
     discontinuous jump in the frequency response can be specified by
     duplicating a band edge in F with different values in M.

     The resolution over which the frequency response is evaluated can
     be controlled with the GRID_N argument.  The default is 512 or the
     next larger power of 2 greater than the filter length.

     The band transition width for discontinuities can be controlled
     with the RAMP_N argument.  The default is GRID_N/25.  Larger values
     will result in wider band transitions but better stopband
     rejection.

     An optional shaping WINDOW can be given as a vector with length
     N+1.  If not specified, a Hamming window of length N+1 is used.

     To apply the filter, use the return vector B with the ‘filter’
     function, for example ‘y = filter (b, 1, x)’.

     Example:
           f = [0, 0.3, 0.3, 0.6, 0.6, 1]; m = [0, 0, 1, 1/2, 0, 0];
           [h, w] = freqz (fir2 (100, f, m));
           plot (f, m, ";target response;", w/pi, abs (h), ";filter response;");

     See also: filter, fir1.

3.8.4 firls
-----------

 -- Function File: B = firls (N, F, A)
 -- Function File: B = firls (N, F, A, W)

     FIR filter design using least squares method.  Returns a length N+1
     linear phase filter such that the integral of the weighted mean
     squared error in the specified bands is minimized.

     The vector F specifies the frequencies of the band edges,
     normalized so that half the sample frequency is equal to 1.  Each
     band is specified by two frequencies, to the vector must have an
     even length.

     The vector A specifies the amplitude of the desired response at
     each band edge.

     The optional argument W is a weighting function that contains one
     value for each band that weights the mean squared error in that
     band.

     A must be the same length as F, and W must be half the length of F.
     N must be even.  If given an odd value, ‘firls’ increments it by 1.

     The least squares optimization algorithm for computing FIR filter
     coefficients is derived in detail in:

     I. Selesnick, "Linear-Phase FIR Filter Design by Least Squares,"
     http://cnx.org/content/m10577

3.8.5 kaiserord
---------------

 -- Function File: [N, WN, BETA, FTYPE] = kaiserord (F, M, DEV)
 -- Function File: [...] = kaiserord (F, M, DEV, FS)

     Return the parameters needed to produce a filter of the desired
     specification from a Kaiser window.  The vector F contains pairs of
     frequency band edges in the range [0,1].  The vector M specifies
     the magnitude response for each band.  The values of M must be zero
     for all stop bands and must have the same magnitude for all pass
     bands.  The deviation of the filter DEV can be specified as a
     scalar or a vector of the same length as M.  The optional sampling
     rate FS can be used to indicate that F is in Hz in the range
     [0,FS/2].

     The returned value N is the required order of the filter (the
     length of the filter minus 1).  The vector WN contains the band
     edges of the filter suitable for passing to ‘fir1’.  The value BETA
     is the parameter of the Kaiser window of length N+1 to shape the
     filter.  The string FTYPE contains the type of filter to specify to
     ‘fir1’.

     The Kaiser window parameters n and beta are computed from the
     relation between ripple (A=-20*log10(dev)) and transition width (dw
     in radians) discovered empirically by Kaiser:

                     / 0.1102(A-8.7)                        A > 50
              beta = | 0.5842(A-21)^0.4 + 0.07886(A-21)     21 <= A <= 50
                     \ 0.0                                  A < 21

              n = (A-8)/(2.285 dw)

     Example:
           [n, w, beta, ftype] = kaiserord ([1000, 1200], [1, 0], [0.05, 0.05], 11025);
           b = fir1 (n, w, kaiser (n+1, beta), ftype, "noscale");
           freqz (b, 1, [], 11025);

     See also: fir1, kaiser.

3.8.6 qp_kaiser
---------------

 -- Function File: qp_kaiser (NB, AT)
 -- Function File: qp_kaiser (NB, AT, LINEAR)

     Computes a finite impulse response (FIR) filter for use with a
     quasi-perfect reconstruction polyphase-network filter bank.  This
     version utilizes a Kaiser window to shape the frequency response of
     the designed filter.  Tha number nb of bands and the desired
     attenuation at in the stop-band are given as parameters.

     The Kaiser window is multiplied by the ideal impulse response
     h(n)=a.sinc(a.n) and converted to its minimum-phase version by
     means of a Hilbert transform.

     By using a third non-null argument, the minimum-phase calculation
     is omitted at all.

3.8.7 remez
-----------

 -- Loadable Function: B = remez (N, F, A)
 -- Loadable Function: B = remez (N, F, A, W)
 -- Loadable Function: B = remez (N, F, A, W, FTYPE)
 -- Loadable Function: B = remez (N, F, A, W, FTYPE, GRIDDENSITY)
     Parks-McClellan optimal FIR filter design.
     N
          gives the filter order, where the generated filter length taps
          is n+1
     F
          gives frequency at the band edges [b1 e1 b2 e2 b3 e3 ...]
     A
          gives amplitude at the band edges [a(b1) a(e1) a(b2) a(e2)
          ...]
     W
          gives weighting applied to each band
     FTYPE
          is "bandpass", "hilbert" or "differentiator"
     GRIDDENSITY
          determines how accurately the filter will be constructed.  The
          minimum value is 16, but higher numbers are slower to compute.

     Frequency is in the range (0, 1), with 1 being the Nyquist
     frequency.

3.8.8 sgolay
------------

 -- Function File: F = sgolay (P, N)
 -- Function File: F = sgolay (P, N, M)
 -- Function File: F = sgolay (P, N, M, TS)
     Computes the filter coefficients for all Savitzsky-Golay smoothing
     filters of order p for length n (odd).  m can be used in order to
     get directly the mth derivative.  In this case, ts is a scaling
     factor.

     The early rows of F smooth based on future values and later rows
     smooth based on past values, with the middle row using half future
     and half past.  In particular, you can use row i to estimate x(k)
     based on the i-1 preceding values and the n-i following values of x
     values as y(k) = F(i,:) * x(k-i+1:k+n-i).

     Normally, you would apply the first (n-1)/2 rows to the first k
     points of the vector, the last k rows to the last k points of the
     vector and middle row to the remainder, but for example if you were
     running on a realtime system where you wanted to smooth based on
     the all the data collected up to the current time, with a lag of
     five samples, you could apply just the filter on row n-5 to your
     window of length n each time you added a new sample.

     Reference: Numerical recipes in C. p 650

     See also: sgolayfilt.


File: signal.info,  Node: Transforms,  Next: Power Spectrum Analysis,  Prev: FIR Filter Design,  Up: Function Reference

3.9 Transforms
==============

3.9.1 bitrevorder
-----------------

 -- Function File: Y = bitrevorder (X)
 -- Function File: [Y I] = bitrevorder (X)
     Reorder the elements of the vector X in bit-reversed order.
     Equivalent to calling ‘digitrevorder (X, 2)’.

     See also: digitrevorder, fft, ifft.

3.9.2 cceps
-----------

 -- Function File: cceps (X)
 -- Function File: cceps (X, CORRECT)
     Return the complex cepstrum of the vector X.  If the optional
     argument CORRECT has the value 1, a correction method is applied.
     The default is not to do this.

3.9.3 cplxreal
--------------

 -- Function File: [ZC, ZR] = cplxreal (Z)
 -- Function File: [ZC, ZR] = cplxreal (Z, TOL)
 -- Function File: [ZC, ZR] = cplxreal (Z, TOL, DIM)
     Sort the numbers Z into complex-conjugate-valued and real-valued
     elements.  The positive imaginary complex numbers of each complex
     conjugate pair are returned in ZC and the real numbers are returned
     in ZR.

     TOL is a weighting factor in the range [0, 1) which determines the
     tolerance of the matching.  The default value is ‘100 * eps’ and
     the resulting tolerance for a given complex pair is ‘TOL * abs
     (Z(i)))’.

     By default the complex pairs are sorted along the first
     non-singleton dimension of Z.  If DIM is specified, then the
     complex pairs are sorted along this dimension.

     Signal an error if some complex numbers could not be paired.
     Signal an error if all complex numbers are not exact conjugates (to
     within TOL).  Note that there is no defined order for pairs with
     identical real parts but differing imaginary parts.

     See also: cplxpair.

3.9.4 czt
---------

 -- Function File: czt (X)
 -- Function File: czt (X, M)
 -- Function File: czt (X, M, W)
 -- Function File: czt (X, M, W, A)
     Chirp z-transform.  Compute the frequency response starting at a
     and stepping by w for m steps.  a is a point in the complex plane,
     and w is the ratio between points in each step (i.e., radius
     increases exponentially, and angle increases linearly).

     To evaluate the frequency response for the range f1 to f2 in a
     signal with sampling frequency Fs, use the following:

           m = 32;                               ## number of points desired
           w = exp(-j*2*pi*(f2-f1)/((m-1)*Fs));  ## freq. step of f2-f1/m
           a = exp(j*2*pi*f1/Fs);                ## starting at frequency f1
           y = czt(x, m, w, a);

     If you don’t specify them, then the parameters default to a Fourier
     transform: m=length(x), w=exp(-j*2*pi/m), a=1

     If x is a matrix, the transform will be performed column-by-column.

3.9.5 dct
---------

 -- Function File: dct (X)
 -- Function File: dct (X, N)
     Compute the discrete cosine transform of X.  If N is given, then X
     is padded or trimmed to length N before computing the transform.
     If X is a matrix, compute the transform along the columns of the
     the matrix.  The transform is faster if X is real-valued and has
     even length.

     The discrete cosine transform X can be defined as follows:

                         N-1
             X[k] = w(k) sum x[n] cos (pi (2n+1) k / 2N ),  k = 0, ..., N-1
                         n=0

     with w(0) = sqrt(1/N) and w(k) = sqrt(2/N), k = 1, ..., N-1.  There
     are other definitions with different scaling of X[k], but this form
     is common in image processing.

     See also: idct, dct2, idct2, dctmtx.

3.9.6 dct2
----------

 -- Function File: dct2 (X)
 -- Function File: dct2 (X, M, N)
 -- Function File: dct2 (X, [M, N])
     Compute the 2-D discrete cosine transform of matrix X.  If M and N
     are specified, the input is padded or trimmed to the desired size.

     See also: dct, idct, idct2.

3.9.7 dctmtx
------------

 -- Function File: dctmtx (N)
     Return the DCT transformation matrix of size N-by-N.

     If A is an N-by-N matrix, then the following are true:

               T*A    == dct(A),  T'*A   == idct(A)
               T*A*T' == dct2(A), T'*A*T == idct2(A)

     A DCT transformation matrix is useful for doing things like jpeg
     image compression, in which an 8x8 DCT matrix is applied to
     non-overlapping blocks throughout an image and only a subblock on
     the top left of each block is kept.  During restoration, the
     remainder of the block is filled with zeros and the inverse
     transform is applied to the block.

     See also: dct, idct, dct2, idct2.

3.9.8 dftmtx
------------

 -- Function File: D = dftmtx (N)
     Compute the N-by-N Fourier transformation matrix.  This is the
     matrix D such that the Fourier transform of a column vector of
     length N is given by ‘dftmtx(N) * X’ and the inverse Fourier
     transform is given by ‘inv(dftmtx(N)) * X’.

     In general this is less efficient than calling the ‘fft’ and ‘ifft’
     functions directly.

     See also: fft, ifft.

3.9.9 digitrevorder
-------------------

 -- Function File: Y = digitrevorder (X, R)
 -- Function File: [Y, I] = digitrevorder (X, R)
     Reorder the elements of the vector X in digit-reversed order.  The
     elements of X are converted to radix R and reversed.  The reordered
     indices of the elements of X are returned in I.

     See also: bitrevorder, fft, ifft.

3.9.10 dst
----------

 -- Function File: Y = dst (X)
 -- Function File: Y = dst (X, N)
     Computes the type I discrete sine transform of X.  If N is given,
     then X is padded or trimmed to length N before computing the
     transform.  If X is a matrix, compute the transform along the
     columns of the the matrix.

     The discrete sine transform X of x can be defined as follows:

             N
      X[k] = sum x[n] sin (pi n k / (N+1) ),  k = 1, ..., N
             n=1

     See also: idst.

3.9.11 dwt
----------

 -- Function File: [U, V] = dwt (X, WNAME)
 -- Function File: [U, V] = dwt (X, HP, GP)
 -- Function File: [U, V] = dwt (X, HP, GP, ...)
     Discrete wavelet transform (1D).

     *Inputs*
     X
          Signal vector.
     WNAME
          Wavelet name.
     HP
          Coefficients of low-pass decomposition FIR filter.
     GP
          Coefficients of high-pass decomposition FIR filter.

     *Outputs*
     U
          Signal vector of average, approximation.
     V
          Signal vector of difference, detail.

3.9.12 fht
----------

 -- Function File: M = fht (D)
 -- Function File: M = fht (D, N)
 -- Function File: M = fht (D, N, DIM)
     Calculate the Fast Hartley Transform of real input D.  If D is a
     matrix, the Hartley transform is calculated along the columns by
     default.  The options N and DIM are similar to the options of FFT
     function.

     The forward and inverse Hartley transforms are the same (except for
     a scale factor of 1/N for the inverse Hartley transform), but
     implemented using different functions.

     The definition of the forward hartley transform for vector d, m[K]
     = \sum_{i=0}^{N-1} d[i]*(cos[K*2*pi*i/N] + sin[K*2*pi*i/N]), for 0
     <= K < N. m[K] = \sum_{i=0}^{N-1} d[i]*CAS[K*i], for 0 <= K < N.

           fht(1:4)

     See also: ifht, fft.

3.9.13 fwht
-----------

 -- Function File: fwht (X)
 -- Function File: fwht (X, N)
 -- Function File: fwht (X, N, ORDER)
     Compute the Walsh-Hadamard transform of X using the Fast
     Walsh-Hadamard Transform (FWHT) algorithm.  If the input is a
     matrix, the FWHT is calculated along the columns of X.

     The number of elements of X must be a power of 2; if not, the input
     will be extended and filled with zeros.  If a second argument is
     given, the input is truncated or extended to have length N.

     The third argument specifies the ORDER in which the returned
     Walsh-Hadamard transform coefficients should be arranged.  The
     ORDER may be any of the following strings:

     "sequency"
          The coefficients are returned in sequency order.  This is the
          default if ORDER is not given.

     "hadamard"
          The coefficients are returned in Hadamard order.

     "dyadic"
          The coefficients are returned in Gray code order.

     See also: ifwht.

3.9.14 hilbert
--------------

 -- Function File: H = hilbert (F, N, DIM)
     Analytic extension of real valued signal.

     ‘H = hilbert (F)’ computes the extension of the real valued signal
     F to an analytic signal.  If F is a matrix, the transformation is
     applied to each column.  For N-D arrays, the transformation is
     applied to the first non-singleton dimension.

     ‘real (H)’ contains the original signal F.  ‘imag (H)’ contains the
     Hilbert transform of F.

     ‘hilbert (F, N)’ does the same using a length N Hilbert transform.
     The result will also have length N.

     ‘hilbert (F, [], DIM)’ or ‘hilbert (F, N, DIM)’ does the same along
     dimension DIM.

3.9.15 idct
-----------

 -- Function File: Y = idct (X)
 -- Function File: Y = idct (X, N)
     Compute the inverse discrete cosine transform of X.  If N is given,
     then X is padded or trimmed to length N before computing the
     transform.  If X is a matrix, compute the transform along the
     columns of the the matrix.  The transform is faster if X is
     real-valued and even length.

     The inverse discrete cosine transform X can be defined as follows:

                    N-1
             x[n] = sum w(k) X[k] cos (pi (2n+1) k / 2N ),  n = 0, ..., N-1
                    k=0

     with w(0) = sqrt(1/N) and w(k) = sqrt(2/N), k = 1, ..., N-1

     See also: dct, dct2, idct2, dctmtx.

3.9.16 idct2
------------

 -- Function File: Y = idct2 (X)
 -- Function File: Y = idct2 (X, M, N)
 -- Function File: Y = idct2 (X, [M, N])
     Compute the inverse 2-D discrete cosine transform of matrix X.  If
     M and N are specified, the input is either padded or truncated to
     have M rows and N columns.

3.9.17 idst
-----------

 -- Function File: Y = idst (X)
 -- Function File: Y = idst (X, N)
     Computes the inverse type I discrete sine transform of Y.  If N is
     given, then Y is padded or trimmed to length N before computing the
     transform.  If Y is a matrix, compute the transform along the
     columns of the the matrix.

     See also: dst.

3.9.18 ifht
-----------

 -- Function File: M = ifht (D, N, DIM)
     Calculate the inverse Fast Hartley Transform of real input D.  If D
     is a matrix, the inverse Hartley transform is calculated along the
     columns by default.  The options N and DIM are similar to the
     options of FFT function.

     The forward and inverse Hartley transforms are the same (except for
     a scale factor of 1/N for the inverse hartley transform), but
     implemented using different functions.

     The definition of the forward hartley transform for vector d, m[K]
     = 1/N \sum_{i=0}^{N-1} d[i]*(cos[K*2*pi*i/N] + sin[K*2*pi*i/N]),
     for 0 <= K < N. m[K] = 1/N \sum_{i=0}^{N-1} d[i]*CAS[K*i], for 0 <=
     K < N.

           ifht(1:4)

     See also: fht, fft.

3.9.19 ifwht
------------

 -- Function File: ifwht (X)
 -- Function File: ifwht (X, N)
 -- Function File: ifwht (X, N, ORDER)
     Compute the inverse Walsh-Hadamard transform of X using the Fast
     Walsh-Hadamard Transform (FWHT) algorithm.  If the input is a
     matrix, the inverse FWHT is calculated along the columns of X.

     The number of elements of X must be a power of 2; if not, the input
     will be extended and filled with zeros.  If a second argument is
     given, the input is truncated or extended to have length N.

     The third argument specifies the ORDER in which the returned
     inverse Walsh-Hadamard transform coefficients should be arranged.
     The ORDER may be any of the following strings:

     "sequency"
          The coefficients are returned in sequency order.  This is the
          default if ORDER is not given.

     "hadamard"
          The coefficients are returned in Hadamard order.

     "dyadic"
          The coefficients are returned in Gray code order.

     See also: fwht.

3.9.20 rceps
------------

 -- Function File: [Y, YM] = rceps (X)
     Return the cepstrum of the signal X.

     If X is a matrix, return the cepstrum of each column.

     If called with two output arguments, the minimum phase
     reconstruction of the signal X is returned in YM.

     For example:

           f0 = 70; Fs = 10000;            # 100 Hz fundamental, 10kHz sampling rate
           a = poly (0.985 * exp (1i * pi * [0.1, -0.1, 0.3, -0.3])); # two formants
           s = 0.005 * randn (1024, 1);    # Noise excitation signal
           s(1:Fs/f0:length(s)) = 1;       # Impulse glottal wave
           x = filter (1, a, s);           # Speech signal
           [y, ym] = rceps (x .* hanning (1024));

     Reference: ‘Programs for Digital Signal Processing’, IEEE Press,
     John Wiley & Sons, New York, 1979.


File: signal.info,  Node: Power Spectrum Analysis,  Next: Window Functions,  Prev: Transforms,  Up: Function Reference

3.10 Power Spectrum Analysis
============================

3.10.1 __power
--------------

 -- Function File: [P, W] = __power (B, A)
 -- Function File: [...] = __power (B, A, NFFT)
 -- Function File: [...] = __power (B, A, NFFT, FS)
 -- Function File: [...] = __power (B, A, NFFT, FS, RANGE)
 -- Function File: [...] = __power (B, A, NFFT, FS, RANGE, UNITS)
 -- Function File: __power (...)

     Plot the power spectrum of the given ARMA model.

     b, a: filter coefficients (b=numerator, a=denominator) nfft is
     number of points at which to sample the power spectrum Fs is the
     sampling frequency of x range is ’half’ (default) or ’whole’ units
     is ’squared’ or ’db’ (default) range and units may be specified any
     time after the filter, in either order

     Returns P, the magnitude vector, and w, the frequencies at which it
     is sampled.  If there are no return values requested, then plot the
     power spectrum and don’t return anything.

3.10.2 ar_psd
-------------

 -- Function File: ar_psd (A, V)
 -- Function File: ar_psd (A, V, FREQ)
 -- Function File: ar_psd (A, V, FREQ, FS)
 -- Function File: ar_psd (..., RANGE)
 -- Function File: ar_psd (..., METHOD)
 -- Function File: ar_psd (..., PLOTTYPE)
 -- Function File: [PSD, F_OUT] = ar_psd (...)

     Calculate the power spectrum of the autoregressive model

                                  M
           x(n) = sqrt(v).e(n) + SUM a(k).x(n-k)
                                 k=1

     where x(n) is the output of the model and e(n) is white noise.
     This function is intended for use with ‘[a, v, k] = arburg (x,
     poles, criterion)’ which use the Burg (1968) method to calculate a
     "maximum entropy" autoregressive model of X.

     If the FREQ argument is a vector (of frequencies) the spectrum is
     calculated using the polynomial method and the METHOD argument is
     ignored.  For scalar FREQ, an integer power of 2, or METHOD =
     "FFT", causes the spectrum to be calculated by FFT. Otherwise, the
     spectrum is calculated as a polynomial.  It may be computationally
     more efficient to use the FFT method if length of the model is not
     much smaller than the number of frequency values.  The spectrum is
     scaled so that spectral energy (area under spectrum) is the same as
     the time-domain energy (mean square of the signal).

     ARGUMENTS: All but the first two arguments are optional and may be
     empty.
        • A list of M=(order+1) autoregressive model coefficients.  The
          first element of "ar_coeffs" is the zero-lag coefficient,
          which always has a value of 1.
        • V square of the moving-average coefficient of the AR model.
        • FREQ frequencies at which power spectral density is
          calculated, or a scalar indicating the number of uniformly
          distributed frequency values at which spectral density is
          calculated.  (default = 256)
        • FS sampling frequency (Hertz) (default=1)

     CONTROL-STRING ARGUMENTS – each of these arguments is a character
     string.  Control-string arguments can be in any order after the
     other arguments.

     Range:

     ’half’, ’onesided’ : frequency range of the spectrum is from zero
     up to but not including sample_f/2.  Power from negative
     frequencies is added to the positive side of the spectrum.
     ’whole’, ’twosided’ : frequency range of the spectrum is
     -sample_f/2 to sample_f/2, with negative frequencies stored in
     "wrap around" order after the positive frequencies; e.g.
     frequencies for a 10-point ’twosided’ spectrum are 0 0.1 0.2 0.3
     0.4 0.5 -0.4 -0.3 -0.2 -0.1 ’shift’, ’centerdc’ : same as ’whole’
     but with the first half of the spectrum swapped with second half to
     put the zero-frequency value in the middle.  (See "help fftshift".
     If "freq" is vector, ’shift’ is ignored.  If model coefficients
     "ar_coeffs" are real, the default range is ’half’, otherwise
     default range is ’whole’.

     Method:

     ’fft’: use FFT to calculate power spectrum.  ’poly’: calculate
     power spectrum as a polynomial of 1/z N.B. this argument is ignored
     if the "freq" argument is a vector.  The default is ’poly’ unless
     the "freq" argument is an integer power of 2.

     Plot type:

     ’plot’, ’semilogx’, ’semilogy’, ’loglog’, ’squared’ or ’db’:
     specifies the type of plot.  The default is ’plot’, which means
     linear-linear axes.  ’squared’ is the same as ’plot’.  ’dB’ plots
     "10*log10(psd)".  This argument is ignored and a spectrum is not
     plotted if the caller requires a returned value.

     RETURNED VALUES: If returned values are not required by the caller,
     the spectrum is plotted and nothing is returned.
        • PSD estimate of power-spectral density
        • F_OUT frequency values

     REFERENCE [1] Equation 2.28 from Steven M. Kay and Stanley Lawrence
     Marple Jr.: "Spectrum analysis – a modern perspective", Proceedings
     of the IEEE, Vol 69, pp 1380-1419, Nov., 1981

3.10.3 cohere
-------------

 -- Function File: [PXX, FREQ] =
          cohere(X,Y,NFFT,FS,WINDOW,OVERLAP,RANGE,PLOT_TYPE,DETREND)
     Estimate (mean square) coherence of signals "x" and "y".

     Use the Welch (1967) periodogram/FFT method.

     Compatible with Matlab R11 cohere and earlier.

     See "help pwelch" for description of arguments, hints and
     references — especially hint (7) for Matlab R11 defaults.

3.10.4 cpsd
-----------

 -- Function File: [PXX, FREQ] = cpsd (X, Y)
 -- Function File: [...] = cpsd (X, Y, WINDOW)
 -- Function File: [...] = cpsd (X, Y, WINDOW, OVERLAP)
 -- Function File: [...] = cpsd (X, Y, WINDOW, OVERLAP, NFFT)
 -- Function File: [...] = cpsd (X, Y, WINDOW, OVERLAP, NFFT, FS)
 -- Function File: [...] = cpsd (X, Y, WINDOW, OVERLAP, NFFT, FS, RANGE)
 -- Function File: cpsd (...)

     Estimate cross power spectrum of data X and Y by the Welch (1967)
     periodogram/FFT method.

     See also: pwelch.

3.10.5 csd
----------

 -- Function File: [PXX,FREQ] = csd(X, Y, NFFT, FS, WINDOW, OVERLAP,
          RANGE, PLOT_TYPE, DETREND)
     Estimate cross power spectrum of data "x" and "y" by the Welch
     (1967) periodogram/FFT method.

     Compatible with Matlab R11 csd and earlier.

     See "help pwelch" for description of arguments, hints and
     references — especially hint (7) for Matlab R11 defaults.

3.10.6 db2pow
-------------

 -- Function File: db2pow (X)
     Convert decibels (dB) to power.

     The power of X is defined as P = ‘10 ^ (x/10)’.

     If X is a vector, matrix, or N-dimensional array, the power is
     computed over the elements of X.

     Example:

           db2pow ([-10, 0, 10])
          ⇒ 0.1000 1.0000 10.0000

     See also: pow2db.

3.10.7 mscohere
---------------

 -- Function File: [PXX, FREQ] = mscohere (X, Y)
 -- Function File: [...] = mscohere (X, Y, WINDOW)
 -- Function File: [...] = mscohere (X, Y, WINDOW, OVERLAP)
 -- Function File: [...] = mscohere (X, Y, WINDOW, OVERLAP, NFFT)
 -- Function File: [...] = mscohere (X, Y, WINDOW, OVERLAP, NFFT, FS)
 -- Function File: [...] = mscohere (X, Y, WINDOW, OVERLAP, NFFT, FS,
          RANGE)
 -- Function File: mscohere (...)

     Estimate (mean square) coherence of signals X and Y.  Use the Welch
     (1967) periodogram/FFT method.

     See also: pwelch.

3.10.8 pburg
------------

 -- Function File: [PSD,F_OUT] = pburg(X, POLES, FREQ, FS, RANGE,
          METHOD, PLOT_TYPE, CRITERION)
     Calculate Burg maximum-entropy power spectral density.

     The functions "arburg" and "ar_psd" do all the work.  See "help
     arburg" and "help ar_psd" for further details.

     ARGUMENTS:

     All but the first two arguments are optional and may be empty.

     x
          [vector] sampled data
     poles
          [integer scalar] required number of poles of the AR model
     freq
          [real vector] frequencies at which power spectral density is
          calculated.

          [integer scalar] number of uniformly distributed frequency
          values at which spectral density is calculated.  [default=256]
     Fs
          [real scalar] sampling frequency (Hertz) [default=1]

     CONTROL-STRING ARGUMENTS – each of these arguments is a character
     string.

     Control-string arguments can be in any order after the other
     arguments.

     range
          ’half’, ’onesided’ : frequency range of the spectrum is from
          zero up to but not including sample_f/2.  Power from negative
          frequencies is added to the positive side of the spectrum.

          ’whole’, ’twosided’ : frequency range of the spectrum is
          -sample_f/2 to sample_f/2, with negative frequencies stored in
          "wrap around" order after the positive frequencies; e.g.
          frequencies for a 10-point ’twosided’ spectrum are 0 0.1 0.2
          0.3 0.4 0.5 -0.4 -0.3 -0.2 -0.1

          ’shift’, ’centerdc’ : same as ’whole’ but with the first half
          of the spectrum swapped with second half to put the
          zero-frequency value in the middle.  (See "help fftshift".  If
          "freq" is vector, ’shift’ is ignored.  If model coefficients
          "ar_coeffs" are real, the default range is ’half’, otherwise
          default range is ’whole’.

     method
          ’fft’: use FFT to calculate power spectral density.

          ’poly’: calculate spectral density as a polynomial of 1/z N.B.
          this argument is ignored if the "freq" argument is a vector.
          The default is ’poly’ unless the "freq" argument is an integer
          power of 2.

     plot_type
          ’plot’, ’semilogx’, ’semilogy’, ’loglog’, ’squared’ or ’db’:
          specifies the type of plot.  The default is ’plot’, which
          means linear-linear axes.  ’squared’ is the same as ’plot’.
          ’dB’ plots "10*log10(psd)".  This argument is ignored and a
          spectrum is not plotted if the caller requires a returned
          value.

     criterion
          [optional string arg] model-selection criterion.  Limits the
          number of poles so that spurious poles are not added when the
          whitened data has no more information in it (see Kay & Marple,
          1981).  Recognized values are

          ’AKICc’ – approximate corrected Kullback information criterion
          (recommended),

          ’KIC’ – Kullback information criterion

          ’AICc’ – corrected Akaike information criterion

          ’AIC’ – Akaike information criterion

          ’FPE’ – final prediction error" criterion

          The default is to NOT use a model-selection criterion

     RETURNED VALUES:

     If return values are not required by the caller, the spectrum is
     plotted and nothing is returned.

     psd
          [real vector] power-spectral density estimate
     f_out
          [real vector] frequency values

     HINTS

     This function is a wrapper for arburg and ar_psd.

     See "help arburg", "help ar_psd".

3.10.9 pow2db
-------------

 -- Function File: pow2db (X)
     Convert power to decibels (dB).

     The decibel value of X is defined as D = ‘10 * log10 (x)’.

     If X is a vector, matrix, or N-dimensional array, the decibel value
     is computed over the elements of X.

     Examples:

           pow2db ([0, 10, 100])
          ⇒ -Inf 10 20

     See also: db2pow.

3.10.10 pwelch
--------------

 -- Function File: [SPECTRA,FREQ] = pwelch(X, WINDOW, OVERLAP, NFFT, FS,
          RANGE, PLOT_TYPE, DETREND, SLOPPY)
     Estimate power spectral density of data "x" by the Welch (1967)
     periodogram/FFT method.

     All arguments except "x" are optional.

     The data is divided into segments.  If "window" is a vector, each
     segment has the same length as "window" and is multiplied by
     "window" before (optional) zero-padding and calculation of its
     periodogram.  If "window" is a scalar, each segment has a length of
     "window" and a Hamming window is used.

     The spectral density is the mean of the periodograms, scaled so
     that area under the spectrum is the same as the mean square of the
     data.  This equivalence is supposed to be exact, but in practice
     there is a mismatch of up to 0.5% when comparing area under a
     periodogram with the mean square of the data.

     [spectra,freq] = pwelch(x,y,window,overlap,Nfft,Fs,
     range,plot_type,detrend,sloppy,results)

     Two-channel spectrum analyser.  Estimate power spectral density,
     cross- spectral density, transfer function and/or coherence
     functions of time- series input data "x" and output data "y" by the
     Welch (1967) periodogram/FFT method.

     pwelch treats the second argument as "y" if there is a
     control-string argument "cross", "trans", "coher" or "ypower";
     "power" does not force the 2nd argument to be treated as "y".  All
     other arguments are optional.  All spectra are returned in matrix
     "spectra".

     [spectra,Pxx_ci,freq] = pwelch(x,window,overlap,Nfft,Fs,conf,
     range,plot_type,detrend,sloppy)

     [spectra,Pxx_ci,freq] = pwelch(x,y,window,overlap,Nfft,Fs,conf,
     range,plot_type,detrend,sloppy,results)

     Estimates confidence intervals for the spectral density.

     See Hint (7) below for compatibility options.

     Confidence level "conf" is the 6th or 7th numeric argument.  If
     "results" control-string arguments are used, one of them must be
     "power" when the "conf" argument is present; pwelch can estimate
     confidence intervals only for the power spectrum of the "x" data.
     It does not know how to estimate confidence intervals of the
     cross-power spectrum, transfer function or coherence; if you can
     suggest a good method, please send a bug report.

     ARGUMENTS

     All but the first argument are optional and may be empty, except
     that the "results" argument may require the second argument to be
     "y".
     x
          [non-empty vector] system-input time-series data
     y
          [non-empty vector] system-output time-series data
     window
          [real vector] of window-function values between 0 and 1; the
          data segment has the same length as the window.  Default
          window shape is Hamming.

          [integer scalar] length of each data segment.  The default
          value is window=sqrt(length(x)) rounded up to the nearest
          integer power of 2; see ’sloppy’ argument.

     overlap
          [real scalar] segment overlap expressed as a multiple of
          window or segment length.  0 <= overlap < 1, The default is
          overlap=0.5 .

     Nfft
          [integer scalar] Length of FFT. The default is the length of
          the "window" vector or has the same value as the scalar
          "window" argument.  If Nfft is larger than the segment length,
          "seg_len", the data segment is padded with "Nfft-seg_len"
          zeros.  The default is no padding.  Nfft values smaller than
          the length of the data segment (or window) are ignored
          silently.

     Fs
          [real scalar] sampling frequency (Hertz); default=1.0

     conf
          [real scalar] confidence level between 0 and 1.  Confidence
          intervals of the spectral density are estimated from scatter
          in the periodograms and are returned as Pxx_ci.  Pxx_ci(:,1)
          is the lower bound of the confidence interval and Pxx_ci(:,2)
          is the upper bound.  If there are three return values, or conf
          is an empty matrix, confidence intervals are calculated for
          conf=0.95 .  If conf is zero or is not given, confidence
          intervals are not calculated.  Confidence intervals can be
          obtained only for the power spectral density of x; nothing
          else.

     CONTROL-STRING ARGUMENTS – each of these arguments is a character
     string.  Control-string arguments must be after the other arguments
     but can be in any order.

     range
          ’half’, ’onesided’ : frequency range of the spectrum is zero
          up to but not including Fs/2.  Power from negative frequencies
          is added to the positive side of the spectrum, but not at zero
          or Nyquist (Fs/2) frequencies.  This keeps power equal in time
          and spectral domains.  See reference [2].

          ’whole’, ’twosided’ : frequency range of the spectrum is -Fs/2
          to Fs/2, with negative frequencies stored in "wrap around"
          order after the positive frequencies; e.g.  frequencies for a
          10-point ’twosided’ spectrum are 0 0.1 0.2 0.3 0.4 0.5 -0.4
          -0.3 -0.2 -0.1

          ’shift’, ’centerdc’ : same as ’whole’ but with the first half
          of the spectrum swapped with second half to put the
          zero-frequency value in the middle.  (See "help fftshift".

          If data (x and y) are real, the default range is ’half’,
          otherwise default range is ’whole’.

     plot_type
          ’plot’, ’semilogx’, ’semilogy’, ’loglog’, ’squared’ or ’db’:
          specifies the type of plot.  The default is ’plot’, which
          means linear-linear axes.  ’squared’ is the same as ’plot’.
          ’dB’ plots "10*log10(psd)".  This argument is ignored and a
          spectrum is not plotted if the caller requires a returned
          value.

     detrend
          ’no-strip’, ’none’ – do NOT remove mean value from the data

          ’short’, ’mean’ – remove the mean value of each segment from
          each segment of the data.

          ’linear’, – remove linear trend from each segment of the data.

          ’long-mean’ – remove the mean value from the data before
          splitting it into segments.  This is the default.

     sloppy
          ’sloppy’: FFT length is rounded up to the nearest integer
          power of 2 by zero padding.  FFT length is adjusted after
          addition of padding by explicit Nfft argument.  The default is
          to use exactly the FFT and window/ segment lengths specified
          in argument list.

     results
          specifies what results to return (in the order specified and
          as many as desired).

          ’power’ calculate power spectral density of "x"

          ’cross’ calculate cross spectral density of "x" and "y"

          ’trans’ calculate transfer function of a system with input "x"
          and output "y"

          ’coher’ calculate coherence function of "x" and "y"

          ’ypower’ calculate power spectral density of "y"

          The default is ’power’, with argument "y" omitted.

     RETURNED VALUES:

     If return values are not required by the caller, the results are
     plotted and nothing is returned.

     spectra
          [real-or-complex matrix] columns of the matrix contain results
          in the same order as specified by "results" arguments.  Each
          column contains one of the result vectors.

     Pxx_ci
          [real matrix] estimate of confidence interval for power
          spectral density of x.  First column is the lower bound.
          Second column is the upper bound.

     freq
          [real column vector] frequency values

     HINTS
       1. EMPTY ARGS: if you don’t want to use an optional argument you
          can leave it empty by writing its value as [].

       2. FOR BEGINNERS: The profusion of arguments may make pwelch
          difficult to use, and an unskilled user can easily produce a
          meaningless result or can easily mis-interpret the result.
          With real data "x" and sampling frequency "Fs", the easiest
          and best way for a beginner to use pwelch is probably
          "pwelch(x,[],[],[],Fs)".  Use the "window" argument to control
          the length of the spectrum vector.  For real data and integer
          scalar M, "pwelch(x,2*M,[],[],Fs)" gives an M+1 point
          spectrum.  Run "demo pwelch" (octave only).

       3. WINDOWING FUNCTIONS: Without a window function, sharp spectral
          peaks can have strong sidelobes because the FFT of a data in a
          segment is in effect convolved with a rectangular window.  A
          window function which tapers off (gradually) at the ends
          produces much weaker sidelobes in the FFT. Hann (hanning),
          hamming, bartlett, blackman, flattopwin etc are available as
          separate Matlab/sigproc or Octave functions.  The sidelobes of
          the Hann window have a roll-off rate of 60dB/decade of
          frequency.  The first sidelobe of the Hamming window is
          suppressed and is about 12dB lower than the first Hann
          sidelobe, but the roll-off rate is only 20dB/decade.  You can
          inspect the FFT of a Hann window by plotting
          "abs(fft(postpad(hanning(256),4096,0)))".  The default window
          is Hamming.

       4. ZERO PADDING: Zero-padding reduces the frequency step in the
          spectrum, and produces an artificially smoothed spectrum.  For
          example, "Nfft=2*length(window)" gives twice as many frequency
          values, but adjacent PSD (power spectral density) values are
          not independent; adjacent PSD values are independent if
          "Nfft=length(window)", which is the default value of Nfft.

       5. REMOVING MEAN FROM SIGNAL: If the mean is not removed from the
          signal there is a large spectral peak at zero frequency and
          the sidelobes of this peak are likely to swamp the rest of the
          spectrum.  For this reason, the default behavior is to remove
          the mean.  However, the matlab pwelch does not do this.

       6. WARNING ON CONFIDENCE INTERVALS Confidence intervals are
          obtained by measuring the sample variance of the periodograms
          and assuming that the periodograms have a Gaussian probability
          distribution.  This assumption is not accurate.  If, for
          example, the data (x) is Gaussian, the periodogram has a
          Rayleigh distribution.  However, the confidence intervals may
          still be useful.

       7. COMPATIBILITY WITH Matlab R11, R12, etc When used without the
          second data (y) argument, arguments are compatible with the
          pwelch of Matlab R12, R13, R14, 2006a and 2006b except that

          1) overlap is expressed as a multiple of window length —
          effect of overlap scales with window length

          2) default values of length(window), Nfft and Fs are more
          sensible, and

          3) Goertzel algorithm is not available so Nfft cannot be an
          array of frequencies as in Matlab 2006b.

          Pwelch has four persistent Matlab-compatibility levels.
          Calling pwelch with an empty first argument sets the order of
          arguments and defaults specified above in the USAGE and
          ARGUMENTS section of this documentation.
                         prev_compat=pwelch([]);
                         [Pxx,f]=pwelch(x,window,overlap,Nfft,Fs,conf,...);
          Calling pwelch with a single string argument (as described
          below) gives compatibility with Matlab R11 or R12, or the R14
          spectrum.welch defaults.  The returned value is the PREVIOUS
          compatibility string.

          Matlab R11: For compatibility with the Matlab R11 pwelch:
                         prev_compat=pwelch('R11-');
                         [Pxx,f]=pwelch(x,Nfft,Fs,window,overlap,conf,range,units);
                         %% units of overlap are "number of samples"
                         %% defaults: Nfft=min(length(x),256), Fs=2*pi, length(window)=Nfft,
                         %%           window=Hanning, do not detrend,
                         %% N.B.  "Sloppy" is not available.

          Matlab R12: For compatibility with Matlab R12 to 2006a pwelch:
                         prev_compat=pwelch('R12+');
                         [Pxx,f]=pwelch(x,window,overlap,nfft,Fs,...);
                         %% units of overlap are "number of samples"
                         %% defaults: length(window)==length(x)/8, window=Hamming,
                         %%           Nfft=max(256,NextPow2), Fs=2*pi, do not detrend
                         %% NextPow2 is the next power of 2 greater than or equal to the
                         %% window length. "Sloppy", "conf" are not available.  Default
                         %% window length gives very poor amplitude resolution.

          To adopt defaults of the Matlab R14 "spectrum.welch" spectrum
          object associated "psd" method.
                         prev_compat=pwelch('psd');
                         [Pxx,f] = pwelch(x,window,overlap,Nfft,Fs,conf,...);
                         %% overlap is expressed as a percentage of window length,
                         %% defaults: length(window)==64, Nfft=max(256,NextPow2), Fs=2*pi
                         %%           do not detrend
                         %% NextPow2 is the next power of 2 greater than or equal to the
                         %% window length. "Sloppy" is not available.
                         %% Default window length gives coarse frequency resolution.

     REFERENCES

     [1] Peter D. Welch (June 1967): "The use of fast Fourier transform
     for the estimation of power spectra: a method based on time
     averaging over short, modified periodograms."  IEEE Transactions on
     Audio Electroacoustics, Vol AU-15(6), pp 70-73

     [2] William H. Press and Saul A. Teukolsky and William T.
     Vetterling and Brian P. Flannery", "Numerical recipes in C, The art
     of scientific computing", 2nd edition, Cambridge University Press,
     2002 — Section 13.7.

3.10.11 pyulear
---------------

 -- Function File: [psd,f_out] =
          pyulear(x,poles,freq,Fs,range,method,plot_type)
     Calculates a Yule-Walker autoregressive (all-pole) model of the
     data "x" and computes the power spectrum of the model.

     This is a wrapper for functions "aryule" and "ar_psd" which perform
     the argument checking.

     See "help aryule" and "help ar_psd" for further details.

     ARGUMENTS:

     All but the first two arguments are optional and may be empty.
     x
          [vector] sampled data

     poles
          [integer scalar] required number of poles of the AR model

     freq
          [real vector] frequencies at which power spectral density is
          calculated

          [integer scalar] number of uniformly distributed frequency
          values at which spectral density is calculated.  [default=256]

     Fs
          [real scalar] sampling frequency (Hertz) [default=1]

     CONTROL-STRING ARGUMENTS – each of these arguments is a character
     string.

     Control-string arguments can be in any order after the other
     arguments.

     range
          ’half’, ’onesided’ : frequency range of the spectrum is from
          zero up to but not including sample_f/2.  Power from negative
          frequencies is added to the positive side of the spectrum.

          ’whole’, ’twosided’ : frequency range of the spectrum is
          -sample_f/2 to sample_f/2, with negative frequencies stored in
          "wrap around" order after the positive frequencies; e.g.
          frequencies for a 10-point ’twosided’ spectrum are 0 0.1 0.2
          0.3 0.4 0.5 -0.4 -0.3 -0.2 -0.1

          ’shift’, ’centerdc’ : same as ’whole’ but with the first half
          of the spectrum swapped with second half to put the
          zero-frequency value in the middle.  (See "help fftshift".  If
          "freq" is vector, ’shift’ is ignored.

          If model coefficients "ar_coeffs" are real, the default range
          is ’half’, otherwise default range is ’whole’.

     method
          ’fft’: use FFT to calculate power spectrum.

          ’poly’: calculate power spectrum as a polynomial of 1/z N.B.
          this argument is ignored if the "freq" argument is a vector.
          The default is ’poly’ unless the "freq" argument is an integer
          power of 2.

     plot_type
          ’plot’, ’semilogx’, ’semilogy’, ’loglog’, ’squared’ or ’db’:
          specifies the type of plot.  The default is ’plot’, which
          means linear-linear axes.  ’squared’ is the same as ’plot’.
          ’dB’ plots "10*log10(psd)".  This argument is ignored and a
          spectrum is not plotted if the caller requires a returned
          value.

     RETURNED VALUES:

     If return values are not required by the caller, the spectrum is
     plotted and nothing is returned.

     psd
          [real vector] power-spectrum estimate
     f_out
          [real vector] frequency values

     HINTS

     This function is a wrapper for aryule and ar_psd.

     See "help aryule", "help ar_psd".

3.10.12 tfe
-----------

 -- Function File: [Pxx,freq] =
          tfe(x,y,Nfft,Fs,window,overlap,range,plot_type,detrend)
     Estimate transfer function of system with input "x" and output "y".

     Use the Welch (1967) periodogram/FFT method.

     Compatible with Matlab R11 tfe and earlier.

     See "help pwelch" for description of arguments, hints and
     references — especially hint (7) for Matlab R11 defaults.

3.10.13 tfestimate
------------------

 -- Function File: tfestimate (X, Y)
 -- Function File: tfestimate (X, Y, WINDOW)
 -- Function File: tfestimate (X, Y, WINDOW, OVERLAP)
 -- Function File: tfestimate (X, Y, WINDOW, OVERLAP, NFFT)
 -- Function File: tfestimate (X, Y, WINDOW, OVERLAP, NFFT, FS)
 -- Function File: tfestimate (X, Y, WINDOW, OVERLAP, NFFT, FS, RANGE)
 -- Function File: [PXX, FREQ] = tfestimate (...)

     Estimate transfer function of system with input X and output Y.
     Use the Welch (1967) periodogram/FFT method.

     See also: pwelch.


File: signal.info,  Node: Window Functions,  Next: System Identification,  Prev: Power Spectrum Analysis,  Up: Function Reference

3.11 Window Functions
=====================

3.11.1 barthannwin
------------------

 -- Function File: barthannwin (M)
     Return the filter coefficients of a modified Bartlett-Hann window
     of length M.

     See also: rectwin, bartlett.

3.11.2 blackmanharris
---------------------

 -- Function File: blackmanharris (M)
 -- Function File: blackmanharris (M, "periodic")
 -- Function File: blackmanharris (M, "symmetric")
     Return the filter coefficients of a Blackman-Harris 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.

     See also: rectwin, bartlett.

3.11.3 blackmannuttall
----------------------

 -- Function File: blackmannuttall (M)
 -- Function File: blackmannuttall (M, "periodic")
 -- Function File: blackmannuttall (M, "symmetric")
     Return the filter coefficients of a Blackman-Nuttall 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.

     See also: nuttallwin, kaiser.

3.11.4 bohmanwin
----------------

 -- Function File: bohmanwin (M)
     Return the filter coefficients of a Bohman window of length M.

     See also: rectwin, bartlett.

3.11.5 boxcar
-------------

 -- Function File: boxcar (M)
     Return the filter coefficients of a rectangular window of length M.

3.11.6 chebwin
--------------

 -- Function File: chebwin (M)
 -- Function File: chebwin (M, AT)

     Return the filter coefficients of a Dolph-Chebyshev window of
     length M.  The Fourier transform of the window has a stop-band
     attenuation of AT dB. The default attenuation value is 100 dB.

     For the definition of the Chebyshev window, see

     * Peter Lynch, "The Dolph-Chebyshev Window: A Simple Optimal
     Filter", Monthly Weather Review, Vol.  125, pp.  655-660, April
     1997.  (http://www.maths.tcd.ie/~plynch/Publications/Dolph.pdf)

     * C. Dolph, "A current distribution for broadside arrays which
     optimizes the relationship between beam width and side-lobe level",
     Proc.  IEEE, 34, pp.  335-348.

     The window is described in frequency domain by the expression:

                    Cheb(m-1, beta * cos(pi * k/m))
             W(k) = -------------------------------
                           Cheb(m-1, beta)

     with

             beta = cosh(1/(m-1) * acosh(10^(at/20))

     and Cheb(m,x) denoting the m-th order Chebyshev polynomial
     calculated at the point x.

     Note that the denominator in W(k) above is not computed, and after
     the inverse Fourier transform the window is scaled by making its
     maximum value unitary.

     See also: kaiser.

3.11.7 flattopwin
-----------------

 -- Function File: flattopwin (M)
 -- Function File: flattopwin (M, "periodic")
 -- Function File: flattopwin (M, "symmetric")

     Return the filter coefficients of a Flat Top window of length M.
     The Flat Top window is defined by the function f(w):

             f(w) = 1 - 1.93 cos(2 pi w) + 1.29 cos(4 pi w)
                      - 0.388 cos(6 pi w) + 0.0322cos(8 pi w)

     where w = i/(m-1) for i=0:m-1 for a symmetric window, or w = i/m
     for i=0:m-1 for a periodic window.  The default is symmetric.  The
     returned window is normalized to a peak of 1 at w = 0.5.

     This window has low pass-band ripple, but high bandwidth.

     According to [1]:

     The main use for the Flat Top window is for calibration, due to its
     negligible amplitude errors.

     [1] Gade, S; Herlufsen, H; (1987) "Use of weighting functions in
     DFT/FFT analysis (Part I)", Bruel & Kjaer Technical Review No.3.

3.11.8 gaussian
---------------

 -- Function File: gaussian (M)
 -- Function File: gaussian (M, A)

     Return a Gaussian convolution window of length M.  The width of the
     window is inversely proportional to the parameter A.  Use larger A
     for a narrower window.  Use larger M for longer tails.

     w = exp ( -(a*x)^2/2 )

     for x = linspace ( -(m-1)/2, (m-1)/2, m ).

     Width a is measured in frequency units (sample rate/num samples).
     It should be f when multiplying in the time domain, but 1/f when
     multiplying in the frequency domain (for use in convolutions).

3.11.9 gausswin
---------------

 -- Function File: gausswin (M)
 -- Function File: gausswin (M, A)

     Return the filter coefficients of a Gaussian window of length M.
     The width of the window is inversely proportional to the parameter
     A.  Use larger A for a narrow window.  Use larger M for a smoother
     curve.

     w = exp ( -(a*x)^2/2 )

     for x = linspace(-(m-1)/m, (m-1)/m, m)

3.11.10 hann
------------

 -- Function File: hann (M)
 -- Function File: hann (M, "periodic")
 -- Function File: hann (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.

     This function exists for MATLAB compatibility only, and is
     equivalent to ‘hanning (M)’.

     See also: hanning.

3.11.11 kaiser
--------------

 -- Function File: kaiser (M)
 -- Function File: kaiser (M, BETA)

     Return the filter coefficients of a Kaiser window of length M.  The
     Fourier transform of the window has a stop-band attenuation that is
     derived from the parameter BETA.

     For the definition of the Kaiser window, see A. V. Oppenheim & R.
     W. Schafer, "Discrete-Time Signal Processing".

     The continuous version of width m centered about x=0 is:

                   besseli(0, beta * sqrt(1-(2*x/m).^2))
           k(x) =  -------------------------------------,  m/2 <= x <= m/2
                          besseli(0, beta)

     See also: kaiserord.

3.11.12 nuttallwin
------------------

 -- Function File: nuttallwin (M)
 -- Function File: nuttallwin (M, "periodic")
 -- Function File: nuttallwin (M, "symmetric")
     Return the filter coefficients of a Blackman-Harris window defined
     by Nuttall 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.

     See also: blackman, blackmanharris.

3.11.13 parzenwin
-----------------

 -- Function File: parzenwin (M)
     Return the filter coefficients of a Parzen window of length M.

     See also: rectwin, bartlett.

3.11.14 rectwin
---------------

 -- Function File: rectwin (M)
     Return the filter coefficients of a rectangular window of length M.

     See also: boxcar, hamming, hanning.

3.11.15 triang
--------------

 -- Function File: triang (M)

     Return the filter coefficients of a triangular window of length M.
     Unlike the Bartlett window, ‘triang’ does not go to zero at the
     edges of the window.  For odd M, ‘triang (M)’ is equal to ‘bartlett
     (M + 2)’ except for the zeros at the edges of the window.

     See also: bartlett.

3.11.16 tukeywin
----------------

 -- Function File: tukeywin (M)
 -- Function File: tukeywin (M, R)
     Return the filter coefficients of a Tukey window (also known as the
     cosine-tapered window) of length M.  R defines the ratio between
     the constant section and and the cosine section.  It has to be
     between 0 and 1.  The function returns a Hanning window for R equal
     to 1 and a rectangular window for R equal to 0.  The default value
     of R is 1/2.

     For a definition of the Tukey window, see e.g.  Fredric J. Harris,
     "On the Use of Windows for Harmonic Analysis with the Discrete
     Fourier Transform, Proceedings of the IEEE", Vol.  66, No.  1,
     January 1978, Page 67, Equation 38.

     See also: hanning.

3.11.17 ultrwin
---------------

 -- Function File: [W, XMU] = ultrwin (M, MU, BETA)
 -- Function File: [W, XMU] = ultrwin (M, MU, ATT, "att")
 -- Function File: [W, XMU] = ultrwin (M, MU, LATT, "latt")
 -- Function File: W = ultrwin (M, MU, XMU, "xmu")
     Return the coefficients of an Ultraspherical window of length M.
     The parameter MU controls the window’s Fourier transform’s
     side-lobe to side-lobe ratio, and the third given parameter
     controls the transform’s main-lobe width/side-lobe-ratio; normalize
     W such that the central coefficient(s) value is unitary.

     By default, the third parameter is BETA, which sets the main lobe
     width to BETA times that of a rectangular window.  Alternatively,
     giving ATT or LATT sets the ripple ratio at the first or last
     side-lobe respectively, or giving XMU sets the (un-normalized)
     window’s Fourier transform according to its canonical definition:

                   (MU)
           W(k) = C   [ XMU cos(pi k/M) ],  k = 0, 1, ..., M-1,
                   M-1

     where C is the Ultraspherical (a.k.a.  Gegenbauer) polynomial,
     which can be defined using the recurrence relationship:

            (l)    1                  (l)                    (l)
           C (x) = - [ 2x(m + l - 1) C   (x) - (m + 2l - 2) C   (x) ]
            m      m                  m-1                    m-2
     
                                      (l)        (l)
           for m an integer > 1, and C (x) = 1, C (x) = 2lx.
                                      0          1

     For given BETA, ATT, or LATT, the corresponding (determined) value
     of XMU is also returned.

     The Dolph-Chebyshev and Saramaki windows are special cases of the
     Ultraspherical window, with MU set to 0 and 1 respectively.  Note
     that when not giving XMU, stability issues may occur with MU <=
     -1.5.  For further information about the window, see

        • Kabal, P., 2009: Time Windows for Linear Prediction of Speech.
          Technical Report, Dept.  Elec.  & Comp.  Eng., McGill
          University.
        • Bergen, S., Antoniou, A., 2004: Design of Ultraspherical
          Window Functions with Prescribed Spectral Characteristics.
          Proc.  JASP, 13/13, pp.  2053-2065.
        • Streit, R., 1984: A two-parameter family of weights for
          nonrecursive digital filters and antennas.  Trans.  ASSP, 32,
          pp.  108-118.

     See also: chebwin, kaiser.

3.11.18 welchwin
----------------

 -- Function File: welchwin (M)
 -- Function File: welchwin (M, "periodic")
 -- Function File: welchwin (M, "symmetric")
     Return the filter coefficients of a Welch window of length M.  The
     Welch window is given by W(n)=1-(n/N-1)^2, n=[0,1, ...  M-1].  The
     optional argument specifies a "symmetric" window (the default) or a
     "periodic" window.

     A symmetric window has zero at each end and maximum in the middle,
     and the length must be an integer greater than 2.  The variable N
     in the formula above is ‘(M-1)/2’.

     A periodic window wraps around the cyclic interval [0,1, ...  M-1],
     and is intended for use with the DFT. The length must be an integer
     greater than 1.  The variable N in the formula above is ‘M/2’.

     See also: blackman, kaiser.

3.11.19 window
--------------

 -- Function File: W = window (F, M)
 -- Function File: W = window (F, M, OPTS)
     Create an M-point window from the function F.  The function F can
     be for example ‘@blackman’.  Any additional arguments OPT are
     passed to the windowing function.


File: signal.info,  Node: System Identification,  Next: Sample Rate Change,  Prev: Window Functions,  Up: Function Reference

3.12 System Identification
==========================

3.12.1 arburg
-------------

 -- Function File: [A, V, K] = arburg (X, POLES)
 -- Function File: [A, V, K] = arburg (X, POLES, CRITERION)

     Calculate coefficients of an autoregressive (AR) model of complex
     data X using the whitening lattice-filter method of Burg (1968).
     The inverse of the model is a moving-average filter which reduces X
     to white noise.  The power spectrum of the AR model is an estimate
     of the maximum entropy power spectrum of the data.  The function
     ‘ar_psd’ calculates the power spectrum of the AR model.

     ARGUMENTS:
        • X sampled data
        • POLES number of poles in the AR model or limit to the number
          of poles if a valid CRITERION is provided.
        • CRITERION model-selection criterion.  Limits the number of
          poles so that spurious poles are not added when the whitened
          data has no more information in it (see Kay & Marple, 1981).
          Recognized values are ’AKICc’ – approximate corrected Kullback
          information criterion (recommended), ’KIC’ – Kullback
          information criterion ’AICc’ – corrected Akaike information
          criterion ’AIC’ – Akaike information criterion ’FPE’ – final
          prediction error" criterion The default is to NOT use a
          model-selection criterion

     RETURNED VALUES:
        • A list of (P+1) autoregression coefficients; for data input
          x(n) and white noise e(n), the model is

                                      P+1
                x(n) = sqrt(v).e(n) + SUM a(k).x(n-k)
                                      k=1

          V mean square of residual noise from the whitening operation
          of the Burg lattice filter.
        • K reflection coefficients defining the lattice-filter
          embodiment of the model

     HINTS:

     (1) arburg does not remove the mean from the data.  You should
     remove the mean from the data if you want a power spectrum.  A
     non-zero mean can produce large errors in a power-spectrum
     estimate.  See "help detrend".  (2) If you don’t know what the
     value of "poles" should be, choose the largest (reasonable) value
     you could want and use the recommended value, criterion=’AKICc’, so
     that arburg can find it.  E.g.  arburg(x,64,’AKICc’) The AKICc has
     the least bias and best resolution of the available model-selection
     criteria.  (3) Autoregressive and moving-average filters are stored
     as polynomials which, in matlab, are row vectors.

     NOTE ON SELECTION CRITERION:

     AIC, AICc, KIC and AKICc are based on information theory.  They
     attempt to balance the complexity (or length) of the model against
     how well the model fits the data.  AIC and KIC are biased estimates
     of the asymmetric and the symmetric Kullback-Leibler divergence
     respectively.  AICc and AKICc attempt to correct the bias.  See
     reference [4].

     REFERENCES:

     [1] John Parker Burg (1968) "A new analysis technique for time
     series data", NATO advanced study Institute on Signal Processing
     with Emphasis on Underwater Acoustics, Enschede, Netherlands, Aug.
     12-23, 1968.

     [2] Steven M. Kay and Stanley Lawrence Marple Jr.: "Spectrum
     analysis – a modern perspective", Proceedings of the IEEE, Vol 69,
     pp 1380-1419, Nov., 1981

     [3] William H. Press and Saul A. Teukolsky and William T.
     Vetterling and Brian P. Flannery "Numerical recipes in C, The art
     of scientific computing", 2nd edition, Cambridge University Press,
     2002 — Section 13.7.

     [4] Abd-Krim Seghouane and Maiza Bekara "A small sample model
     selection criterion based on Kullback’s symmetric divergence", IEEE
     Transactions on Signal Processing, Vol.  52(12), pp 3314-3323, Dec.
     2004

     See also: ar_psd.

3.12.2 aryule
-------------

 -- Function File: A = aryule (X, P)
 -- Function File: [A, V, K] = aryule (X, P)
     Fit an AR (P)-model with Yule-Walker estimates.
     X
          data vector to estimate
     A
          AR coefficients
     V
          variance of white noise
     K
          reflection coefficients for use in lattice filter

     The power spectrum of the resulting filter can be plotted with
     pyulear(x, p), or you can plot it directly with ar_psd(a,v,...).

     See also: pyulear, power, freqz, impz – for observing
     characteristics of the model arburg – for alternative spectral
     estimators

     Example: Use example from arburg, but substitute aryule for arburg.

     Note: Orphanidis ’85 claims lattice filters are more tolerant of
     truncation errors, which is why you might want to use them.
     However, lacking a lattice filter processor, I haven’t tested that
     the lattice filter coefficients are reasonable.

3.12.3 invfreq
--------------

 -- Function File: [B,A] = invfreq(H,F,nB,nA)
 -- : [B,A] = invfreq(H,F,nB,nA,W)
 -- : [B,A] = invfreq(H,F,nB,nA,W,[],[],plane)
 -- : [B,A] = invfreq(H,F,nB,nA,W,iter,tol,plane)
     Fit filter B(z)/A(z) or B(s)/A(s) to complex frequency response at
     frequency points F.

     A and B are real polynomial coefficients of order nA and nB
     respectively.  Optionally, the fit-errors can be weighted vs
     frequency according to the weights W. Also, the transform plane can
     be specified as either ’s’ for continuous time or ’z’ for discrete
     time.  ’z’ is chosen by default.  Eventually, Steiglitz-McBride
     iterations will be specified by iter and tol.

     H: desired complex frequency response It is assumed that A and B
     are real polynomials, hence H is one-sided.

     F: vector of frequency samples in radians

     nA: order of denominator polynomial A

     nB: order of numerator polynomial B

     plane=’z’: F on unit circle (discrete-time spectra, z-plane design)

     plane=’s’: F on jw axis (continuous-time spectra, s-plane design)

     H(k) = spectral samples of filter frequency response at points zk,
     where zk=exp(sqrt(-1)*F(k)) when plane=’z’ (F(k) in [0,.5]) and
     zk=(sqrt(-1)*F(k)) when plane=’s’ (F(k) nonnegative)

     Example:
               [B,A] = butter(12,1/4);
               [H,w] = freqz(B,A,128);
               [Bh,Ah] = invfreq(H,F,4,4);
               Hh = freqz(Bh,Ah);
               disp(sprintf('||frequency response error|| = %f',norm(H-Hh)));

     References:

     J. O. Smith, "Techniques for Digital Filter Design and System
     Identification with Application to the Violin, Ph.D. Dissertation,
     Elec.  Eng.  Dept., Stanford University, June 1983, page 50; or,

     http://ccrma.stanford.edu/~jos/filters/FFT_Based_Equation_Error_Method.html

3.12.4 invfreqs
---------------

 -- Function File: [B,A] = invfreqs(H,F,nB,nA)
 -- : [B,A] = invfreqs(H,F,nB,nA,W)
 -- : [B,A] = invfreqs(H,F,nB,nA,W,iter,tol,'trace')
     Fit filter B(s)/A(s)to the complex frequency response H at
     frequency points F.

     A and B are real polynomial coefficients of order nA and nB.

     Optionally, the fit-errors can be weighted vs frequency according
     to the weights W.

     Note: all the guts are in invfreq.m

     H: desired complex frequency response

     F: frequency (must be same length as H)

     nA: order of the denominator polynomial A

     nB: order of the numerator polynomial B

     W: vector of weights (must be same length as F)

     Example:
                 B = [1/2 1];
                 A = [1 1];
                 w = linspace(0,4,128);
                 H = freqs(B,A,w);
                 [Bh,Ah] = invfreqs(H,w,1,1);
                 Hh = freqs(Bh,Ah,w);
                 plot(w,[abs(H);abs(Hh)])
                 legend('Original','Measured');
                 err = norm(H-Hh);
                 disp(sprintf('L2 norm of frequency response error = %f',err));

3.12.5 invfreqz
---------------

 -- Function File: [B,A] = invfreqz(H,F,nB,nA)
 -- : [B,A] = invfreqz(H,F,nB,nA,W)
 -- : [B,A] = invfreqz(H,F,nB,nA,W,iter,tol,'trace')
     Fit filter B(z)/A(z)to the complex frequency response H at
     frequency points F.

     A and B are real polynomial coefficients of order nA and nB.
     Optionally, the fit-errors can be weighted vs frequency according
     to the weights W.

     Note: all the guts are in invfreq.m

     H: desired complex frequency response

     F: normalized frequency (0 to pi) (must be same length as H)

     nA: order of the denominator polynomial A

     nB: order of the numerator polynomial B

     W: vector of weights (must be same length as F)

     Example:
               [B,A] = butter(4,1/4);
               [H,F] = freqz(B,A);
               [Bh,Ah] = invfreq(H,F,4,4);
               Hh = freqz(Bh,Ah);
               disp(sprintf('||frequency response error|| = %f',norm(H-Hh)));

3.12.6 levinson
---------------

 -- Function File: [A, V, REF] = levinson (ACF)
 -- Function File: [...] = levinson (ACF, P)

     Use the Durbin-Levinson algorithm to solve: toeplitz(acf(1:p)) * x
     = -acf(2:p+1).  The solution [1, x’] is the denominator of an all
     pole filter approximation to the signal x which generated the
     autocorrelation function acf.

     acf is the autocorrelation function for lags 0 to p.  p defaults to
     length(acf)-1.  Returns a=[1, x’] the denominator filter
     coefficients.  v= variance of the white noise = square of the
     numerator constant ref = reflection coefficients = coefficients of
     the lattice implementation of the filter Use freqz(sqrt(v),a) to
     plot the power spectrum.

     REFERENCE [1] Steven M. Kay and Stanley Lawrence Marple Jr.:
     "Spectrum analysis – a modern perspective", Proceedings of the
     IEEE, Vol 69, pp 1380-1419, Nov., 1981

3.12.7 lpc
----------

 -- Function File: A = lpc (X)
 -- Function File: A = lpc (X, P)
 -- Function File: [A, G] = lpc (...)
 -- Function File: [A, G] = lpc (X, P)

     Determines the forward linear predictor by minimizing the
     prediction error in the least squares sense.  Use the
     Durbin-Levinson algorithm to solve the Yule-Walker equations
     obtained by the autocorrelation of the input signal.

     X is a data vector used to estimate the lpc model of P-th order,
     given by the prediction polynomial ‘A = [1 A(2) ... A(P+1)]’.  If P
     is not provided, ‘length(P) - 1’ is used as default.

     X might also be a matrix, in which case each column is regarded as
     a separate signal.  ‘lpc’ will return a model estimate for each
     column of X.

     G is the variance (power) of the prediction error for each signal
     in X.

     See also: aryule,levinson.


File: signal.info,  Node: Sample Rate Change,  Next: Pulse Metrics,  Prev: System Identification,  Up: Function Reference

3.13 Sample Rate Change
=======================

3.13.1 data2fun
---------------

 -- Function File: [FHANDLE, FULLNAME] = data2fun (TI, YI)
 -- Function File: [...] = data2fun (..., PROPERTY, VALUE)
     Create a vectorized function based on data samples using
     interpolation.

     The values given in YI (N-by-k matrix) correspond to evaluations of
     the function y(t) at the points TI (N-by-1 matrix).  The data is
     interpolated and the function handle to the generated interpolant
     is returned.

     The function accepts PROPERTY-VALUE pairs described below.

     ‘file’
          Code is generated and .m file is created.  The VALUE contains
          the name of the function.  The returned function handle is a
          handle to that file.  If VALUE is empty, then a name is
          automatically generated using ‘tempname’ and the file is
          created in the current directory.  VALUE must not have an
          extension, since .m will be appended.  Numerical values used
          in the function are stored in a .mat file with the same name
          as the function.

     ‘interp’
          Type of interpolation.  See ‘interp1’.

     See also: interp1.

3.13.2 decimate
---------------

 -- Function File: Y = decimate (X, Q)
 -- Function File: Y = decimate (X, Q, N)
 -- Function File: Y = decimate (..., "fir")

     Downsample the signal X by a reduction factor of Q.  A lowpass
     antialiasing filter is applied to the signal prior to reducing the
     input sequence.  By default, an order N Chebyshev type I filter is
     used.  If N is not specified, the default is 8.

     If the optional argument ‘"fir"’ is given, an order N FIR filter is
     used, with a default order of 30 if N is not given.

     Note that Q must be an integer for this rate change method.

     Example:
           ## Generate a signal that starts away from zero, is slowly varying
           ## at the start and quickly varying at the end, decimate and plot.
           ## Since it starts away from zero, you will see the boundary
           ## effects of the antialiasing filter clearly.  Next you will see
           ## how it follows the curve nicely in the slowly varying early
           ## part of the signal, but averages the curve in the quickly
           ## varying late part of the signal.
           t = 0:0.01:2;
           x = chirp (t, 2, .5, 10, "quadratic") + sin (2*pi*t*0.4);
           y = decimate (x, 4);
           stem (t(1:121) * 1000, x(1:121), "-g;Original;"); hold on; # original
           stem (t(1:4:121) * 1000, y(1:31), "-r;Decimated;"); hold off; # decimated

3.13.3 downsample
-----------------

 -- Function File: Y = downsample (X, N)
 -- Function File: Y = downsample (X, N, OFFSET)
     Downsample the signal, selecting every Nth element.  If X is a
     matrix, downsample every column.

     For most signals you will want to use ‘decimate’ instead since it
     prefilters the high frequency components of the signal and avoids
     aliasing effects.

     If OFFSET is defined, select every Nth element starting at sample
     OFFSET.

     See also: decimate, interp, resample, upfirdn, upsample.

3.13.4 interp
-------------

 -- Function File: Y = interp (X, Q)
 -- Function File: Y = interp (X, Q, N)
 -- Function File: Y = interp (X, Q, N, WC)

     Upsample the signal x by a factor of q, using an order 2*q*n+1 FIR
     filter.  Note that q must be an integer for this rate change
     method.  n defaults to 4 and Wc defaults to 0.5.

     Example
           # Generate a signal.
           t=0:0.01:2; x=chirp(t,2,.5,10,'quadratic')+sin(2*pi*t*0.4);
           y = interp(x(1:4:length(x)),4,4,1);   # interpolate a sub-sample
           stem(t(1:121)*1000,x(1:121),"-g;Original;"); hold on;
           stem(t(1:121)*1000,y(1:121),"-r;Interpolated;");
           stem(t(1:4:121)*1000,x(1:4:121),"-b;Subsampled;"); hold off;

     See also: decimate, resample.

3.13.5 resample
---------------

 -- Function File: [Y, H] = resample (X, P, Q)
 -- Function File: Y = resample (X, P, Q, H)
     Change the sample rate of X by a factor of P/Q.  This is performed
     using a polyphase algorithm.  The impulse response H of the
     antialiasing filter is either specified or either designed with a
     Kaiser-windowed sinecard.

     Ref [1] J. G. Proakis and D. G. Manolakis, Digital Signal
     Processing: Principles, Algorithms, and Applications, 4th ed.,
     Prentice Hall, 2007.  Chap.  6

     Ref [2] A. V. Oppenheim, R. W. Schafer and J. R. Buck,
     Discrete-time signal processing, Signal processing series,
     Prentice-Hall, 1999

3.13.6 upfirdn
--------------

 -- Loadable Function: Y = upfirdn (X, H, P, Q)
     Upsample, FIR filtering, and downsample.

3.13.7 upsample
---------------

 -- Function File: Y = upsample (X, N)
 -- Function File: Y = upsample (X, N, OFFSET)
     Upsample the signal, inserting N-1 zeros between every element.

     If X is a matrix, upsample every column.

     If OFFSET is specified, control the position of the inserted sample
     in the block of N zeros.

     See also: decimate, downsample, interp, resample, upfirdn.


File: signal.info,  Node: Pulse Metrics,  Next: Utility,  Prev: Sample Rate Change,  Up: Function Reference

3.14 Pulse Metrics
==================

3.14.1 statelevels
------------------

 -- Function File: LEVELS = statelevels (A)
 -- Function File: LEVELS = statelevels (A, NBINS)
 -- Function File: LEVELS = statelevels (A, NBINS, METHOD)
 -- Function File: LEVELS = statelevels (A, NBINS, METHOD, BOUNDS)
 -- Function File: [LEVELS, HISTOGRAMS] = statelevels (...)
 -- Function File: [LEVELS, HISTOGRAMS, BINLEVELS] = statelevels (...)
 -- Function File: statelevels (...)
     Estimate state-level for bilevel waveform A using histogram method

     INPUTS:

     A
          Bylevel waveform
     NBINS
          Number of histogram bins (100 default)
     METHOD
          State-level estimation method ’mode’ (default) or ’mean’.
     BOUNDS
          2 element vector for histogram lower and upper bounds.  Values
          outside of this will be ignored.

     OUTPUTS:

     LEVELS
          Levels of high and low states
     HISTOGRAMS
          Histogram counts
     BINLEVELS
          Histogram bincenters

     If no outputs are provided, the signal and histogram will be
     plotted, and display the levels.


File: signal.info,  Node: Utility,  Prev: Pulse Metrics,  Up: Function Reference

3.15 Utility
============

3.15.1 buffer
-------------

 -- Function File: Y = buffer (X, N, P, OPT)
 -- Function File: [Y, Z, OPT] = buffer (...)
     Buffer a signal into a data frame.  The arguments to ‘buffer’ are

     X
          The data to be buffered.

     N
          The number of rows in the produced data buffer.  This is an
          positive integer value and must be supplied.

     P
          An integer less than N that specifies the under- or overlap
          between column in the data frame.  The default value of P is
          0.

     OPT
          In the case of an overlap, OPT can be either a vector of
          length P or the string ’nodelay’.  If OPT is a vector, then
          the first P entries in Y will be filled with these values.  If
          OPT is the string ’nodelay’, then the first value of Y
          corresponds to the first value of X.

          In the can of an underlap, OPT must be an integer between 0
          and ‘-P’.  The represents the initial underlap of the first
          column of Y.

          The default value for OPT the vector ‘zeros (1, P)’ in the
          case of an overlap, or 0 otherwise.

     In the case of a single output argument, Y will be padded with
     zeros to fill the missing values in the data frame.  With two
     output arguments Z is the remaining data that has not been used in
     the current data frame.

     Likewise, the output OPT is the overlap, or underlap that might be
     used for a future call to ‘code’ to allow continuous buffering.

3.15.2 clustersegment
---------------------

 -- Function File: CLUSTERIDX = clustersegment (UNOS)
     Calculate boundary indexes of clusters of 1’s.

     The function calculates the initial index and end index of the
     sequences of 1’s in the rows of UNOS.  The clusters are sought in
     the rows of the array UNOS.

     The result is returned in a cell array of size 1-by-NP, where NP is
     the number of rows in UNOS.  Each element of the cell has two rows.
     The first row is the initial index of a sequence of 1’s and the
     second row is the end index of that sequence.

     If NP == 1 the output is a matrix with two rows.

     The function works by finding the indexes of jumps between
     consecutive values in the rows of UNOS.

3.15.3 fracshift
----------------

 -- Function File: [Y, H] = fracshift (X, D)
 -- Function File: Y = fracshift (X, D, H)
     Shift the series X by a (possibly fractional) number of samples D.
     The interpolator H is either specified or either designed with a
     Kaiser-windowed sinecard.

     See also: circshift.

3.15.4 marcumq
--------------

 -- Function File: Q = marcumq (A, B)
 -- Function File: Q = marcumq (A, B, M)
 -- Function File: Q = marcumq (A, B, M, TOL)

     Compute the generalized Marcum Q function of order M with
     noncentrality parameter A and argument B.  If the order M is
     omitted it defaults to 1.  An optional relative tolerance TOL may
     be included, the default is ‘eps’.

     If the input arguments are commensurate vectors, this function will
     produce a table of values.

     This function computes Marcum’s Q function using the infinite
     Bessel series, truncated when the relative error is less than the
     specified tolerance.  The accuracy is limited by that of the Bessel
     functions, so reducing the tolerance is probably not useful.

     Reference: Marcum, "Tables of Q Functions", Rand Corporation.

     Reference: R.T. Short, "Computation of Noncentral Chi-squared and
     Rice Random Variables", www.phaselockedsystems.com/publications

3.15.5 primitive
----------------

 -- Function File: F = primitive (F, T, F0)
     Calculate the primitive of a function.

     The function approximates the primitive (indefinite integral) of
     the univariate function handle F with constant of integration F0.
     The output is the primitive evaluated at the points T.  The vector
     T must be ordered and ascending.

     This function is a fancy way of calculating the cumulative sum,

     ‘F = cumsum (f(t).*diff (t)([1 1:end]))’.

     Example:
           f = @(t) sin (2 * pi * 3 * t);
           t = [0; sort(rand (100, 1))];
           F = primitive (f, t, 0);
           t_true = linspace (0, 1, 1e3).';
           F_true = (1 - cos (2 * pi * 3 * t_true)) / (2 * pi * 3);
           plot (t, F, 'o', t_true, F_true);

     See also: quadgk, cumsum.

3.15.6 sampled2continuous
-------------------------

 -- Function File: XT = sampled2continuous (XN, T, T)

     Calculate the x(t) reconstructed from samples x[n] sampled at a
     rate 1/T samples per unit time.

     t is all the instants of time when you need x(t) from x[n]; this
     time is relative to x[0] and not an absolute time.

     This function can be used to calculate sampling rate effects on
     aliasing, actual signal reconstruction from discrete samples.

3.15.7 schtrig
--------------

 -- V =: schtrig (X,LVL,RST=1)
 -- [V,RNG] =: schtrig (...)
     Implements a multisignal Schmitt trigger with levels LVL.

     The triger works along the first dimension of the 2-dimensional
     array X.  It compares each column in X to the levels in LVL, when
     the value is higher ‘max (LVL)’ the output V is high (i.e.  1);
     when the value is below ‘min (LVL)’ the output is low (i.e.  0);
     and when the value is between the two levels the output retains its
     value.

     The threshold levels are passed in the array LVL.  If this is a
     scalar, the thresholds are symmetric around 0, i.e.  ‘[-lvl lvl]’.

     The second output argument stores the ranges in which the output is
     high, so the indexes RNG(1,I):RNG(2,I) point to the i-th segment of
     1s in V.  See ‘clustersegment’ for a detailed explanation.

     The function conserves the state of the trigger across calls
     (persistent variable).  If the reset flag is active, i.e.  ‘RST==
     true’, then the state of the trigger for all signals is set to the
     low state (i.e.  0).

     Example:
           x = [0 0.5 1 1.5 2 1.5 1.5 1.2 1 0 0].';
           y = schtrig (x, [1.3 1.6]);
           disp ([x y]);
             0.0   0
             0.5   0
             1.0   0
             1.5   0
             2.0   1
             1.5   1
             1.5   1
             1.2   0
             1.0   0
             0.0   0
             0.0   0

     Run ‘demo schtrig’ to see further examples.

     See also: clustersegment.

3.15.8 upsamplefill
-------------------

 -- Function File: Y = upsamplefill (X, V)
 -- Function File: Y = upsamplefill (..., COPY)
     Upsamples a vector interleaving given values or copies of the
     vector elements.

     The values in the vector V are placed between the elements of X.

     If the optional argument COPY is TRUE then V should be a scalar and
     each value in X are repeat V times.

     Example:
           upsamplefill (eye (2), 2, true)
          ⇒  1   0
               1   0
               1   0
               0   1
               0   1
               0   1
           upsamplefill (eye (2), [-1 -1 -1])
          ⇒  1   0
              -1  -1
              -1  -1
              -1  -1
               0   1
              -1  -1
              -1  -1
              -1  -1

     See also: upsample.

3.15.9 wkeep
------------

 -- Function File: Y = wkeep (X, L)
 -- Function File: Y = wkeep (X, L, OPT)
     Extract the elements of X of size L from the center, the right or
     the left.

3.15.10 wrev
------------

 -- Function File: Y = wrev (X)
     Reverse the order of the element of the vector X.

     See also: flipud, fliplr.

3.15.11 zerocrossing
--------------------

 -- Function File: X0 = zerocrossing (X, Y)
     Estimates the points at which a given waveform y=y(x) crosses the
     x-axis using linear interpolation.

     See also: fzero, roots.



Tag Table:
Node: Top200
Node: Overview980
Node: Installing and loading1202
Node: Function Reference2312
Node: Signals2764
Node: Signal Measurement18367
Node: Correlation and Convolution23311
Node: Filtering30163
Node: Filter Analysis35283
Node: Filter Conversion47876
Node: IIR Filter Design58088
Ref: IIR Filter Design-Footnote-196181
Ref: IIR Filter Design-Footnote-296385
Ref: IIR Filter Design-Footnote-396652
Node: FIR Filter Design96871
Node: Transforms107654
Node: Power Spectrum Analysis120708
Node: Window Functions151062
Node: System Identification163225
Node: Sample Rate Change174077
Node: Pulse Metrics179382
Node: Utility180620

End Tag Table


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