 OVERVIEW
 NAME
 SYNOPSIS
 DESCRIPTION
 FUNCTIONAL INTERFACE
 METHODS
 WINDOW FUNCTIONS
 Symmetric window functions
 bartlett($N)
 bartlett_hann($N)
 blackman($N)
 blackman_bnh($N)
 blackman_ex($N)
 blackman_gen($N,$alpha)
 blackman_gen3($N,$a0,$a1,$a2)
 blackman_gen4($N,$a0,$a1,$a2,$a3)
 blackman_gen5($N,$a0,$a1,$a2,$a3,$a4)
 blackman_harris($N)
 blackman_harris4($N)
 blackman_nuttall($N)
 bohman($N)
 cauchy($N,$alpha)
 chebyshev($N,$at)
 cos_alpha($N,$alpha)
 cosine($N)
 dpss($N,$beta)
 exponential($N)
 flattop($N)
 gaussian($N,$beta)
 hamming($N)
 hamming_ex($N)
 hamming_gen($N,$a)
 hann($N)
 hann_matlab($N)
 hann_poisson($N,$alpha)
 kaiser($N,$beta)
 lanczos($N)
 nuttall($N)
 nuttall1($N)
 parzen($N)
 parzen_octave($N)
 poisson($N,$alpha)
 rectangular($N)
 triangular($N)
 tukey($N,$alpha)
 welch($N)
 AUXILIARY SUBROUTINES
 REFERENCES
 AUTHOR
 ACKNOWLEDGMENTS
 LICENSE AND COPYRIGHT
OVERVIEW
A module for PDL providing window functions for signal processing. Distributions of this module can be downloaded most easily from CPAN. (Commits tagged with version numbers, may also be downloaded from github.)
NAME
PDL::DSP::Windows  Window functions for signal processing
SYNOPSIS
use PDL;
use PDL::DSP::Windows('window');
my $samples = window( 10, 'tukey', { params => .5 });
use PDL;
use PDL::DSP::Windows;
my $win = new PDL::DSP::Windows(10, 'tukey', { params => .5 });
print $win>coherent_gain , "\n";
$win>plot;
DESCRIPTION
This module provides symmetric and periodic (DFTsymmetric) window functions for use in filtering and spectral analysis. It provides a highlevel access subroutine "window". This functional interface is sufficient for getting the window samples. For analysis and plotting, etc. an object oriented interface is provided. The functional subroutines must be either explicitly exported, or fully qualified. In this document, the word function refers only to the mathematical window functions, while the word subroutine is used to describe code.
Window functions are also known as apodization functions or tapering functions. In this module, each of these functions maps a sequence of $N
integers to values called a samples. (To confuse matters, the word sample also has other meanings when describing window functions.) The functions are often named for authors of journal articles. Be aware that across the literature and software, some functions referred to by several different names, and some names refer to several different functions. As a result, the choice of window names is somewhat arbitrary.
The "kaiser" window function requires PDL::GSLSF::BESSEL. The "dpss" window function requires PDL::LinearAlgebra. But the remaining window functions may be used if these modules are not installed.
The most common and easiest usage of this module is indirect, via some higherlevel filtering interface, such as PDL::DSP::Fir::Simple. The next easiest usage is to return a pdl of realspace samples with the subroutine "window". Finally, for analyzing window functions, object methods, such as "new", "plot", "plot_freq" are provided.
In the following, first the functional interface (nonobject oriented) is described in "FUNCTIONAL INTERFACE". Next, the object methods are described in "METHODS". Next the lowlevel subroutines returning samples for each named window are described in "WINDOW FUNCTIONS". Finally, some support routines that may be of interest are described in "AUXILIARY SUBROUTINES".
FUNCTIONAL INTERFACE
window
$win = window({OPTIONS});
$win = window($N,{OPTIONS});
$win = window($N,$name,{OPTIONS});
$win = window($N,$name,$params,{OPTIONS});
$win = window($N,$name,$params,$periodic);
Returns an $N
point window of type $name
. The arguments may be passed positionally in the order $N,$name,$params,$periodic
, or they may be passed by name in the hash OPTIONS
.
EXAMPLES
# Each of the following return a 100 point symmetric hamming window.
$win = window(100);
$win = window(100, 'hamming');
$win = window(100, { name => 'hamming' );
$win = window({ N=> 100, name => 'hamming' );
# Each of the following returns a 100 point symmetric hann window.
$win = window(100, 'hann');
$win = window(100, { name => 'hann' );
# Returns a 100 point periodic hann window.
$win = window(100, 'hann', { periodic => 1 } );
# Returns a 100 point symmetric Kaiser window with alpha=2.
$win = window(100, 'kaiser', { params => 2 });
OPTIONS
The options follow default PDL::Options rules They may be abbreviated, and are caseinsensitive.
 name

(string) name of window function. Default:
hamming
. This selects one of the window functions listed below. Note that the suffix '_per', for periodic, may be ommitted. It is specified with the optionperiodic => 1
 params

ref to array of parameter or parameters for the windowfunction subroutine. Only some windowfunction subroutines take parameters. If the subroutine takes a single parameter, it may be given either as a number, or a list of one number. For example
3
or[3]
.  N

number of points in window function (the same as the order of the filter) No default value.
 periodic

If value is true, return a periodic rather than a symmetric window function. Default: 0 (that is, false. that is, symmetric.)
list_windows
list_windows
list_windows STR
list_windows
prints the names all of the available windows. list_windows STR
prints only the names of windows matching the string STR
.
METHODS
new
my $win = new PDL::DSP::Windows(ARGS);
Create an instance of a Windows object. If ARGS
are given, the instance is initialized. ARGS
are interpreted in exactly the same way as arguments the subroutine "window".
For example:
my $win1 = new PDL::DSP::Windows(8,'hann');
my $win2 = new PDL::DSP::Windows( { N => 8, name => 'hann' } );
init
$win>init(ARGS);
Initialize (or reinitialize) a Windows object. ARGS are interpreted in exactly the same way as arguments the subroutine "window".
For example:
my $win = new PDL::DSP::Windows(8,'hann');
$win>init(10,'hamming');
samples
$win>samples();
Generate and return a reference to the piddle of $N samples for the window $win
. This is the realspace representation of the window.
The samples are stored in the object $win
, but are regenerated every time samples
is invoked. See the method "get_samples" below.
For example:
my $win = new PDL::DSP::Windows(8,'hann');
print $win>samples(), "\n";
modfreqs
$win>modfreqs();
Generate and return a reference to the piddle of the modulus of the fourier transform of the samples for the window $win
.
These values are stored in the object $win
, but are regenerated every time modfreqs
is invoked. See the method "get_modfreqs" below.
options
 min_bins => MIN

This sets the minimum number of frequency bins. Default 1000. If necessary, the piddle of window samples are padded with zeros before the fourier transform is performed.
get
my $windata = $win>get('samples');
Get an attribute (or list of attributes) of the window $win
. If attribute samples
is requested, then the samples are created with the method "samples" if they don't exist.
For example:
my $win = new PDL::DSP::Windows(8,'hann');
print $win>get('samples'), "\n";
get_samples
my $windata = $win>get_samples
Return a reference to the pdl of samples for the Window instance $win
. The samples will be generated with the method "samples" if and only if they have not yet been generated.
get_modfreqs
my $winfreqs = $win>get_modfreqs;
my $winfreqs = $win>get_modfreqs({OPTS});
Return a reference to the pdl of the frequency response (modulus of the DFT) for the Window instance $win
.
Options are passed to the method "modfreqs". The data are created with "modfreqs" if they don't exist. The data are also created even if they already exist if options are supplied. Otherwise the cached data are returned.
options
 min_bins => MIN

This sets the minimum number of frequency bins. See "modfreqs". Default 1000.
get_params
my $params = $win>get_params
Create a new array containing the parameter values for the instance $win
and return a reference to the array. Note that not all window types take parameters.
get_name
print $win>get_name , "\n";
Return a name suitable for printing associated with the window $win. This is something like the name used in the documentation for the particular window function. This is static data and does not depend on the instance.
plot
$win>plot;
Plot the samples. Currently, only PDL::Graphics::Gnuplot is supported. The default display type is used.
plot_freq
Can be called like this
$win>plot_freq;
Or this
$win>plot_freq( {ordinate => ORDINATE });
Plot the frequency response (magnitude of the DFT of the window samples). The response is plotted in dB, and the frequency (by default) as a fraction of the Nyquist frequency. Currently, only PDL::Graphics::Gnuplot is supported. The default display type is used.
options
 coord => COORD

This sets the units of frequency of the coordinate axis.
COORD
must be one ofnyquist
, for fraction of the nyquist frequency (range1,1
),sample
, for fraction of the sampling frequncy (range.5,.5
), orbin
for frequency bin number (range0,$N1
). The default value isnyquist
.  min_bins => MIN

This sets the minimum number of frequency bins. See "get_modfreqs". Default 1000.
enbw
$win>enbw;
Compute and return the equivalent noise bandwidth of the window.
coherent_gain
$win>coherent_gain;
Compute and return the coherent gain (the dc gain) of the window. This is just the average of the samples.
process_gain
$win>coherent_gain;
Compute and return the processing gain (the dc gain) of the window. This is just the multiplicative inverse of the enbw
.
scallop_loss
$win>scallop_loss;
**BROKEN**. Compute and return the scalloping loss of the window.
WINDOW FUNCTIONS
These windowfunction subroutines return a pdl of $N samples. For most windows, there are a symmetric and a periodic version. The symmetric versions are functions of $N points, uniformly spaced, and taking values from x_lo through x_hi. Here, a periodic function of $N
points is equivalent to its symmetric counterpart of $N+1
points, with the final point omitted. The name of a periodic windowfunction subroutine is the same as that for the corresponding symmetric function, except it has the suffix _per
. The descriptions below describe the symmetric version of each window.
The term 'BlackmanHarris family' is meant to include the Hamming family and the Blackman family. These are functions of sums of cosines.
Unless otherwise noted, the arguments in the cosines of all symmetric window functions are multiples of $N
numbers uniformly spaced from 0
through 2 pi
.
Symmetric window functions
bartlett($N)
The Bartlett window. (Ref 1). Another name for this window is the fejer window. This window is defined by
1  abs arr,
where the points in arr range from 1 through 1. See also triangular.
bartlett_hann($N)
The BartlettHann window. Another name for this window is the Modified BartlettHann window. This window is defined by
0.62  0.48 * abs arr + 0.38* arr1,
where the points in arr range from 1/2 through 1/2, and arr1 are the cos of points ranging from PI through PI.
blackman($N)
The 'classic' Blackman window. (Ref 1). One of the BlackmanHarris family, with coefficients
a0 = 0.42, a1 = 0.5, a2 = 0.08
blackman_bnh($N)
The BlackmanHarris (bnh) window. An improved version of the 3term BlackmanHarris window given by Nuttall (Ref 2, p. 89). One of the BlackmanHarris family, with coefficients
a0 = 0.4243801, a1 = 0.4973406, a2 = 0.0782793
blackman_ex($N)
The 'exact' Blackman window. (Ref 1). One of the BlackmanHarris family, with coefficients
a0 = 0.426590713671539, a1 = 0.496560619088564, a2 = 0.0768486672398968
blackman_gen($N,$alpha)
The General classic Blackman window. A single parameter family of the 3term Blackman window. This window is defined by
my $cx = arr;
(.5  $alpha) + ($cx * ((.5) + ($cx * ($alpha)))),
where the points in arr are the cos of points ranging from 0 through 2PI.
blackman_gen3($N,$a0,$a1,$a2)
The general form of the Blackman family. One of the BlackmanHarris family, with coefficients
a0 = $a0, a1 = $a1, a2 = $a2
blackman_gen4($N,$a0,$a1,$a2,$a3)
The general 4term BlackmanHarris window. One of the BlackmanHarris family, with coefficients
a0 = $a0, a1 = $a1, a2 = $a2, a3 = $a3
blackman_gen5($N,$a0,$a1,$a2,$a3,$a4)
The general 5term BlackmanHarris window. One of the BlackmanHarris family, with coefficients
a0 = $a0, a1 = $a1, a2 = $a2, a3 = $a3, a4 = $a4
blackman_harris($N)
The BlackmanHarris window. (Ref 1). One of the BlackmanHarris family, with coefficients
a0 = 0.422323, a1 = 0.49755, a2 = 0.07922
Another name for this window is the Minimum three term (sample) BlackmanHarris window.
blackman_harris4($N)
The minimum (sidelobe) four term BlackmanHarris window. (Ref 1). One of the BlackmanHarris family, with coefficients
a0 = 0.35875, a1 = 0.48829, a2 = 0.14128, a3 = 0.01168
Another name for this window is the BlackmanHarris window.
blackman_nuttall($N)
The BlackmanNuttall window. One of the BlackmanHarris family, with coefficients
a0 = 0.3635819, a1 = 0.4891775, a2 = 0.1365995, a3 = 0.0106411
bohman($N)
The Bohman window. (Ref 1). This window is defined by
my $x = abs(arr);
(1$x)*cos(PI*$x) +(1/PI)*sin(PI*$x),
where the points in arr range from 1 through 1.
cauchy($N,$alpha)
The Cauchy window. (Ref 1). Other names for this window are: Abel, Poisson. This window is defined by
1 / (1 + (arr * $alpha)**2),
where the points in arr range from 1 through 1.
chebyshev($N,$at)
The Chebyshev window. The frequency response of this window has $at
dB of attenuation in the stopband. Another name for this window is the DolphChebyshev window. No periodic version of this window is defined. This routine gives the same result as the routine chebwin in Octave 3.6.2.
cos_alpha($N,$alpha)
The Cos_alpha window. (Ref 1). Another name for this window is the Powerofcosine window. This window is defined by
arr**$alpha ,
where the points in arr are the sin of points ranging from 0 through PI.
cosine($N)
The Cosine window. Another name for this window is the sine window. This window is defined by
arr,
where the points in arr are the sin of points ranging from 0 through PI.
dpss($N,$beta)
The Digital Prolate Spheroidal Sequence (DPSS) window. The parameter $beta
is the halfwidth of the mainlobe, measured in frequency bins. This window maximizes the power in the mainlobe for given $N
and $beta
. Another name for this window is the sleppian window.
exponential($N)
The Exponential window. This window is defined by
2 ** (1  abs arr)  1,
where the points in arr range from 1 through 1.
flattop($N)
The flat top window. One of the BlackmanHarris family, with coefficients
a0 = 0.21557895, a1 = 0.41663158, a2 = 0.277263158, a3 = 0.083578947, a4 = 0.006947368
gaussian($N,$beta)
The Gaussian window. (Ref 1). Another name for this window is the Weierstrass window. This window is defined by
exp (0.5 * ($beta * arr )**2),
where the points in arr range from 1 through 1.
hamming($N)
The Hamming window. (Ref 1). One of the BlackmanHarris family, with coefficients
a0 = 0.54, a1 = 0.46
hamming_ex($N)
The 'exact' Hamming window. (Ref 1). One of the BlackmanHarris family, with coefficients
a0 = 0.53836, a1 = 0.46164
hamming_gen($N,$a)
The general Hamming window. (Ref 1). One of the BlackmanHarris family, with coefficients
a0 = $a, a1 = (1$a)
hann($N)
The Hann window. (Ref 1). One of the BlackmanHarris family, with coefficients
a0 = 0.5, a1 = 0.5
Another name for this window is the hanning window. See also hann_matlab.
hann_matlab($N)
The Hann (matlab) window. Equivalent to the Hann window of N+2 points, with the endpoints (which are both zero) removed. No periodic version of this window is defined. This window is defined by
0.5  0.5 * arr,
where the points in arr are the cosine of points ranging from 2PI/($N+1) through 2PI*$N/($N+1). This routine gives the same result as the routine hanning in Matlab. See also hann.
hann_poisson($N,$alpha)
The HannPoisson window. (Ref 1). This window is defined by
0.5 * (1 + arr1) * exp ($alpha * abs arr),
where the points in arr range from 1 through 1, and arr1 are the cos of points ranging from PI through PI.
kaiser($N,$beta)
The Kaiser window. (Ref 1). The parameter $beta
is the approximate halfwidth of the mainlobe, measured in frequency bins. Another name for this window is the KaiserBessel window. This window is defined by
barf "kaiser: PDL::GSLSF not installed" unless $HAVE_BESSEL;
$beta *= PI;
my @n = PDL::GSLSF::BESSEL::gsl_sf_bessel_In ($beta * sqrt(1  arr **2),0);
my @d = PDL::GSLSF::BESSEL::gsl_sf_bessel_In($beta,0);
(shift @n)/(shift @d),
where the points in arr range from 1 through 1.
lanczos($N)
The Lanczos window. Another name for this window is the sinc window. This window is defined by
my $x = PI * arr;
my $res = sin($x)/$x;
my $mid;
$mid = int($N/2), $res>slice($mid) .= 1 if $N % 2;
$res;,
where the points in arr range from 1 through 1.
nuttall($N)
The Nuttall window. One of the BlackmanHarris family, with coefficients
a0 = 0.3635819, a1 = 0.4891775, a2 = 0.1365995, a3 = 0.0106411
See also nuttall1.
nuttall1($N)
The Nuttall (v1) window. A window referred to as the Nuttall window. One of the BlackmanHarris family, with coefficients
a0 = 0.355768, a1 = 0.487396, a2 = 0.144232, a3 = 0.012604
This routine gives the same result as the routine nuttallwin in Octave 3.6.2. See also nuttall.
parzen($N)
The Parzen window. (Ref 1). Other names for this window are: Jackson, VallePoussin. This function disagrees with the Octave subroutine parzenwin, but agrees with Ref. 1. See also parzen_octave.
parzen_octave($N)
The Parzen window. No periodic version of this window is defined. This routine gives the same result as the routine parzenwin in Octave 3.6.2. See also parzen.
poisson($N,$alpha)
The Poisson window. (Ref 1). This window is defined by
exp ($alpha * abs arr),
where the points in arr range from 1 through 1.
rectangular($N)
The Rectangular window. (Ref 1). Other names for this window are: dirichlet, boxcar.
triangular($N)
The Triangular window. This window is defined by
1  abs arr,
where the points in arr range from $N/($N1) through $N/($N1). See also bartlett.
tukey($N,$alpha)
The Tukey window. (Ref 1). Another name for this window is the tapered cosine window.
welch($N)
The Welch window. (Ref 1). Other names for this window are: Riez, Bochner, Parzen, parabolic. This window is defined by
1  arr**2,
where the points in arr range from 1 through 1.
AUXILIARY SUBROUTINES
These subroutines are used internally, but are also available for export.
cos_mult_to_pow
Convert BlackmanHarris coefficients. The BH windows are usually defined via coefficients for cosines of integer multiples of an argument. The same windows may be written instead as terms of powers of cosines of the same argument. These may be computed faster as they replace evaluation of cosines with multiplications. This subroutine is used internally to implement the BlackmanHarris family of windows more efficiently.
This subroutine takes between 1 and 7 numeric arguments a0, a1, ...
It converts the coefficients of this
a0  a1 cos(arg) + a2 cos( 2 * arg)  a3 cos( 3 * arg) + ...
To the cofficients of this
c0 + c1 cos(arg) + c2 cos(arg)**2 + c3 cos(arg)**3 + ...
cos_pow_to_mult
This function is the inverse of "cos_mult_to_pow".
This subroutine takes between 1 and 7 numeric arguments c0, c1, ...
It converts the coefficients of this
c0 + c1 cos(arg) + c2 cos(arg)**2 + c3 cos(arg)**3 + ...
To the cofficients of this
a0  a1 cos(arg) + a2 cos( 2 * arg)  a3 cos( 3 * arg) + ...
chebpoly
chebpoly($n,$x)
Returns the value of the $n
th order Chebyshev polynomial at point $x
. $n and $x may be scalar numbers, pdl's, or array refs. However, at least one of $n and $x must be a scalar number.
All mixtures of pdls and scalars could be handled much more easily as a PP routine. But, at this point PDL::DSP::Windows is pure perl/pdl, requiring no C/Fortran compiler.
REFERENCES
Harris, F.J.
On the use of windows for harmonic analysis with the discrete Fourier transform
, Proceedings of the IEEE, 1978, vol 66, pp 5183.Nuttall, A.H.
Some windows with very good sidelobe behavior
, IEEE Transactions on Acoustics, Speech, Signal Processing, 1981, vol. ASSP29, pp. 8491.
AUTHOR
John Lapeyre, <jlapeyre at cpan.org>
ACKNOWLEDGMENTS
For study and comparison, the author used documents or output from: Thomas Cokelaer's spectral analysis software; Julius O Smith III's Spectral Audio Signal Processing web pages; André Carezia's chebwin.m Octave code; Other code in the Octave signal package.
LICENSE AND COPYRIGHT
Copyright 2012 John Lapeyre.
This program is free software; you can redistribute it and/or modify it under the terms of either: the GNU General Public License as published by the Free Software Foundation; or the Artistic License.
See http://dev.perl.org/licenses/ for more information.
This software is neither licensed nor distributed by The MathWorks, Inc., maker and liscensor of MATLAB.
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