NAME

PDL::DSP::Windows - Window functions for signal processing

SYNOPSIS

use PDL;
use PDL::DSP::Windows 'window';

# Get an ndarray with a window's samples with the helper
my $samples = window( 10, tukey => { params => .5 });

# Or construct a window object with the same parameters
my $window = PDL::DSP::Windows->new( 10, tukey => { params => .5 });

# These two are equivalent
$samples = $window->samples;

# The window object gives access to additional methods
print $window->coherent_gain, "\n";

$window->plot; # Requires PDL::Graphics::Simple

DESCRIPTION

This module provides symmetric and periodic (DFT-symmetric) window functions for use in filtering and spectral analysis. It provides a high-level 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($N,$beta)" window function requires PDL::GSLSF::BESSEL. The "dpss($N,$beta)" 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 higher-level filtering interface, such as PDL::DSP::Fir::Simple. The next easiest usage is to return a pdl of real-space 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 (non-object oriented) is described in "'FUNCTIONAL INTERFACE'". Next, the object methods are described in "METHODS". Next the low-level 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 case-insensitive.

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 option periodic => 1
params

ref to array of parameter or parameters for the window-function subroutine. Only some window-function 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). As of 0.101, throws exception if the value for N is undefined or zero.
periodic

If value is true, return a periodic rather than a symmetric window function. Defaults to false, meaning "symmetric".

list_windows

print join ", ", list_windows(), "\n"
print join ", ", list_windows(STR), "\n"

list_windows returns the names all of the available windows. list_windows STR returns only the names of windows matching the regular expression STR.

METHODS

new

my $win = PDL::DSP::Windows->new(ARGS);

Create an instance of a window object. If ARGS are given, the instance is initialized. ARGS are interpreted in exactly the same way as arguments for the "window" subroutine.

For example:

my $win1 = PDL::DSP::Windows->new( 8, 'hann' );
my $win2 = PDL::DSP::Windows->new({ N => 8, name => 'hann' });

init

$win->init(ARGS);

Initialize (or reinitialize) a window object. ARGS are interpreted in exactly the same way as arguments for the "window" subroutine. As of 0.101, throws exception if the value for N is undefined or zero.

For example:

my $win = PDL::DSP::Windows->new( 8, 'hann' );
$win->init( 10, 'hamming' );

samples

$win->samples;

Generate and return a reference to the ndarray of $N samples for the window $win. This is the real-space 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 = PDL::DSP::Windows->new( 8, 'hann' );
print $win->samples, "\n";

modfreqs

$win->modfreqs;

Generate and return a reference to the ndarray 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. Defaults to 1000. If necessary, the ndarray of window samples are padded with zeroes 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 = PDL::DSP::Windows->new( 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 passed as a hash reference will be passed to the "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". Defaults to 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;
$win->plot($pgswin); # can supply e.g. for multi-plotting

Plot the samples. Uses PDL::Graphics::Simple. The default display type is used.

plot_freq

Can be called like this

$win->plot_freq;

Or this

$win->plot_freq({ ordinate => ORDINATE });
$win->plot_freq($pgswin, { ordinate => ORDINATE }); # can supply e.g. for multi-plotting

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. Uses PDL::Graphics::Simple. The default display type is used.

options

coord => COORD

This sets the units of frequency of the co-ordinate axis. COORD must be one of nyquist, for fraction of the nyquist frequency (range -1, 1); sample, for fraction of the sampling frequncy (range -0.5, 0.5); or bin for frequency bin number (range 0, $N - 1). The default value is nyquist.
min_bins => MIN

This sets the minimum number of frequency bins. See "get_modfreqs". Defaults to 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;

Compute and return the scalloping loss of the window.

WINDOW FUNCTIONS

These window-function 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 window-function 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 'Blackman-Harris 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 2π.

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 Bartlett-Hann window. Another name for this window is the Modified Bartlett-Hann 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 Blackman-Harris family, with coefficients

a0 = 0.42
a1 = 0.5
a2 = 0.08

blackman_bnh($N)

The Blackman-Harris (bnh) window. An improved version of the 3-term Blackman-Harris window given by Nuttall (Ref 2, p. 89). One of the Blackman-Harris family, with coefficients

a0 = 0.4243801
a1 = 0.4973406
a2 = 0.0782793

blackman_ex($N)

The 'exact' Blackman window. (Ref 1). One of the Blackman-Harris 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 3-term 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 Blackman-Harris family, with coefficients

a0 = $a0
a1 = $a1
a2 = $a2

blackman_gen4($N,$a0,$a1,$a2,$a3)

The general 4-term Blackman-Harris window. One of the Blackman-Harris family, with coefficients

a0 = $a0
a1 = $a1
a2 = $a2
a3 = $a3

blackman_gen5($N,$a0,$a1,$a2,$a3,$a4)

The general 5-term Blackman-Harris window. One of the Blackman-Harris family, with coefficients

a0 = $a0
a1 = $a1
a2 = $a2
a3 = $a3
a4 = $a4

blackman_harris($N)

The Blackman-Harris window. (Ref 1). One of the Blackman-Harris family, with coefficients

a0 = 0.422323
a1 = 0.49755
a2 = 0.07922

Another name for this window is the Minimum three term (sample) Blackman-Harris window.

blackman_harris4($N)

The minimum (sidelobe) four term Blackman-Harris window. (Ref 1). One of the Blackman-Harris family, with coefficients

a0 = 0.35875
a1 = 0.48829
a2 = 0.14128
a3 = 0.01168

Another name for this window is the Blackman-Harris window.

blackman_nuttall($N)

The Blackman-Nuttall window. One of the Blackman-Harris 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 stop-band. Another name for this window is the Dolph-Chebyshev 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 Power-of-cosine 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 half-width 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 Blackman-Harris 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 Blackman-Harris family, with coefficients

a0 = 0.54
a1 = 0.46

hamming_ex($N)

The 'exact' Hamming window. (Ref 1). One of the Blackman-Harris family, with coefficients

a0 = 0.53836
a1 = 0.46164

hamming_gen($N,$a)

The general Hamming window. (Ref 1). One of the Blackman-Harris family, with coefficients

a0 = $a
a1 = (1-$a)

hann($N)

The Hann window. (Ref 1). One of the Blackman-Harris 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 Hann-Poisson 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 half-width of the mainlobe, measured in frequency bins. Another name for this window is the Kaiser-Bessel window. This window is defined by

$beta *= PI;
my @n = gsl_sf_bessel_In($beta * sqrt(1 - arr ** 2), 0);
my @d = gsl_sf_bessel_In($beta, 0);
$n[0] / $d[0];

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 Blackman-Harris family, with coefficients

a0 = 0.3635819
a1 = 0.4891775
a2 = 0.1365995
a3 = 0.0106411

nuttall1($N)

The Nuttall (v1) window. A window referred to as the Nuttall window. One of the Blackman-Harris 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, Valle-Poussin. 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 -$N/($N-1) through $N/($N-1). 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 Blackman-Harris 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 Blackman-Harris 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 51-83.
Nuttall, A.H. Some windows with very good sidelobe behavior, IEEE Transactions on Acoustics, Speech, Signal Processing, 1981, vol. ASSP-29, pp. 84-91.

AUTHOR

John Lapeyre, <jlapeyre at cpan.org>

MAINTAINER

José Joaquín Atria, <jjatria 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

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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NAME

SYNOPSIS

DESCRIPTION

FUNCTIONAL INTERFACE

window

EXAMPLES

OPTIONS

list_windows

METHODS

new

init

samples

modfreqs

options

get

get_samples

get_modfreqs

options

get_params

get_name

plot

plot_freq

options

enbw

coherent_gain

process_gain

scallop_loss

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

cos_mult_to_pow

cos_pow_to_mult

chebpoly

REFERENCES

AUTHOR

MAINTAINER

Packages