🚀 go-pugleaf

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