From: jpcampb@afterlife.ncsc.mil (Joe Campbell)
Newsgroups: comp.dsp
Subject: Re: Optimizing frequency reolution with DFT
Date: 10 Oct 92 18:49:45 GMT
Organization: The Great Beyond

In article <3061@news.cerf.net> jcbhrb@nic.cerf.net (Jacob Hirbawi) writes:
>In comp.dsp, <dak.718418083@messua>
>David Kastrup , <dak@messua.informatik.rwth-aachen.de> writes:
>
>> sjreeves@eng.auburn.edu (Stan Reeves) writes:
>>
>>  >In order to optimize the tradeoff between main lobe width and side lobe
>>  >height, one has to have a criterion.  Do you want more resolution and more
>>  >variance or do you want less variance and less resolution?  I suppose that
>>  >one could come up with a criterion based on expected mean-square error
>>  >between the estimated and actual power spectrum, but I don't know of any
>>  >neat closed-form expressions for this.
>
>> Well, if you define bandwidth (and timespread) as the variance of a
>> similar-spread probability density function, you get a sort of
>> square mean error. The product of the time width and the bandwidth
>> is limited downwards, it cannot get better than an optimum, that is
>> provided by a gaussian curve which again transforms into a Gaussian
>> in the frequency domain. You cannot get any better, but this is definitely
>> a closed-form expression. So mean square criterions will not get you
>> too far, but will show you the direction.
>
>This is all very true for the measures of optimality you have described.
>There are at least three differnet sets of parameters that we often deal
>with in this kind of work; I think it will be helpful to list them here:
>
>     (1.1) minimize time-domain "variance"
>     (1.2) minimize frequency-domain "variance"
>
>     (2.1) minimize mainlobe width
>     (2.2) minimize sidelobe level
>
>     (3.1) maximize energy within a time interval  
>     (3.2) maximize energy within a freuquency band
>
>In the first case where "variance" of a function is defined by
>
>                       <x^2> - <x>^2  
>
>where < H > is defined as: integral(  H f(x) f^*(x) dx  ).
>Then the optimal functions are in fact as you have noted guassian.
>(please excuse the mixing up of continuous functions with discrete windows)
>
>In the second case, if I remember right, the Tchebychev windows actually
>minimize mainlobe width for a given sidelobe level. There is definitely 
>a closed form solution for these.
>
>In the third case the optimal functions are Prolate Spheroidal functions
>which are approximated by Kaiser-Bessel windows; these windows also have a
>closed form solution.
>
>Jacob Hirbawi
>JcbHrb@CERF.net

I'd like to add my favorite window design criterion:

Lineu C. Barbosa, "A Maximum-Energy-Concentration Spectral Window,"
IBM J. Res. Develop., Vol. 30, No. 3, May 1986, p. 321-325.
An elegant method for designing a time-discrete solution for
realization of a spectral window which is ideal from an energy
concentration viewpoint.  This window is one that concentrates the
maximum amount of energy in a specified bandwidth and hence provides
optimal spectral resolution.  Unlike Kaiser windows, this window is
a discrete-time realization having the same objectives as the
continuous-time prolate spheroidal function; at the expense of not
having a closed form solution.

Joe
-- 
.............................................................................
; Joe Campbell  N3JBC  jpcampb@afterlife.ncsc.mil  74040.305@compuserve.com ;
; My opinions are mine!   Happiness = Reality - Expectations, Click & Clack ;
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Newsgroups: comp.dsp