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Summary

Description Spectra of Kaiser windows for different parametric values. Note that the downward spikes in the side lobes should actually spike all they way to -infinity, since at these points the amplitude goes to zero. The fact that the minima are finite is an artifact of the finite plotting resolution.
Date 2007-09-19, revised 2019-03-21 by Bob K
Source Own work
Author RetoGalli
Permission
(Reusing this file)
I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

SVG development
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This vector image was created with GNU Octave.
Octave/gnuplot source
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click to expand

This graphic was created by the following Octave script:

pkg load signal
graphics_toolkit gnuplot

N  = 2^17;
n  = 0:N-1;
P  = 15;        % Maximum bin index drawn
dr = 100;       % dynamic range of plot
M  = 32;        % Fourier transform size as multiple of window length
k  = ([1:M*N]-1-M*N/2)/M;
k2 = [-P : 1/M : P];

% Uncomment warning() if a text() call includes ("fontname", "Symbol")
% warning("off", "Octave:missing-glyph");

h = figure;
hold on
box on
set(gca,'FontSize',10)

beta=4; alpha = beta/pi
w = besseli(0,beta*sqrt(1-(2*n/(N-1) -1).^2))/besseli(0,beta);

H = abs(fft([w zeros(1,(M-1)*N)]));
H = fftshift(H);
H = H/max(H);
H = 20*log10(H);
H = max(-dr,H);
H2 = interp1 (k, H, k2);

plot(k2, H2, "color", "blue", "linewidth", 2)
xlim([-P P])
ylim([-dr 6])
set(gca,"YTick", [0 : -10 : -dr])
grid("on")
ylabel("decibels")
xlabel("DFT bins")

%text(4.26, -36, '\pi\alpha=4; \alpha=1.27', "color", "blue", "fontsize", 12)
%But let's do it the instructive way:
str = ['\pi\alpha=' num2str(beta,'%1i') '; \alpha=' num2str(beta/pi,'%4.2f')];
text(4.26, -36, str, "color", "blue", "fontsize", 12)

beta=8; alpha = beta/pi
w = besseli(0,beta*sqrt(1-(2*n/(N-1) -1).^2))/besseli(0,beta);

H = abs(fft([w zeros(1,(M-1)*N)]));
H = fftshift(H);
H = H/max(H);
H = 20*log10(H);
H = max(-dr,H);
H2 = interp1 (k, H, k2);

plot(k2, H2, "color", "red", "linewidth", 2)

%text(2.5, -19.5,  '\pi\alpha=8; \alpha=2.55', "color", "red", "fontsize", 12)
%But let's do it the less "manual" way:
str = ['\pi\alpha=' num2str(beta,'%1i') '; \alpha=' num2str(beta/pi,'%4.2f')];
text(2.5, -19.5, str, "color", "red", "fontsize", 12)

title("Fourier transforms of two Kaiser windows")

% The following print() converts plain-text Greek characters in text() strings into Symbol font.
% Therefore it isn't necessary to include ("fontname", "Symbol") in the text() calls above,
% and doing so causes warnings, some of which can be suppressed by warning().
print(h,"-dsvg","-color",'C:\Users\BobK\Kaiser-Window-Spectra.svg')

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Date/TimeThumbnailDimensionsUserComment
current21:25, 22 March 2019Thumbnail for version as of 21:25, 22 March 2019700 × 525 (111 KB)Bob KUse definition of α from Window function article.
19:46, 19 September 2007Thumbnail for version as of 19:46, 19 September 2007560 × 420 (143 KB)RetoGalli{{Information |Description=Spectra of Kaiser windows for α of 4 and 8. Note that the downward spikes in the side lobes should actually spike all they way to -infinity, since at these points the amplitude goes to zero. The fact that the minima are finite

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