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   -> Volume 4, Issue 2


Software: ''The Elliptic Sinc Function''
 
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"Soltis James Dr."
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PostPosted: Mon Dec 02, 2002 5:42 pm    
Subject: Software: ''The Elliptic Sinc Function''
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Software: "The Elliptic Sinc Function"

Of promininant importance in communications theory is the
sinc function ,or the Sa(sampling function), or simply
(sin(x)/x).These are,of course all the same. Let us
replace the trig sine function in sinc with the Jacobian
elliptic sine function and study the new sinc. In particular
we study the absolute value of the DFT or FFT of the
new function (sn(x,m)/x) . Here m is the modulus squared
since MATLAB will now be used. In particular the result
is quite interesting . Whereas the standard (sine(x)/x))
provides a typical [ square-wave cum Gibbs response] the
switch from sine(m=0) to arbitrary m<1 shows a multi-level
hierarchy of flat-levels(cum diminished Gibbs) with the
number of levels increasing with m(!!).A family is born.
For example,with m=0.4 there are a few levels and at
m=0.999999999999 there are about 15 for the 1-sided
spectrum.Typically 4096 points are used and only
the simple rectangular windowing is done.
The programs to show this effect can be
obtained by email from soltis@uwindsor.ca
and use Windows-Matlab 4.0 .It should be trivial to
translate to other platforms, although high precision
is needed for m near 1. I used the digits command
from the symbolic toolbox.I can think of applications
in approximation theory and perhaps time-domain-reflectrometry
but I find the scaling nature of the repeated steps
similar to fractals.Since the function sn appears
in the solution of some non-linear DE's ,e.g. the
simple pendulum, this may not be so surprising.
The actual sinc plots are roughly similar to the
standard case but have very slow roll-off near m=1.
Any comments or questions are welcome.
James Soltis, Univ. of Windsor,Windsor,Canada.
All times are GMT + 1 Hour
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