🚀 go-pugleaf

In article <1992Sep30.154718@sdec1.zrz.tu-berlin.de> feiten@sdec1.zrz.tu-berlin.de (Bernhard Feiten) writes:
>
>I'm looking for methods to analyze sounds with DFT.
>Normal window functions have the problem of either having low side lobe level
>or small main lobe bandwidth.

  The "problem of low sidelobe level and small main lobe bandwidth"?
  The lower your sidelobe level, the less spectral leakage problems
  you window will introduce.  The narrower you mainlobe, the better
  your spectral resolution.  The ideal window (an impulse) has no
  mainlobe width and no sidelobes whatsoever.  When creating a
  window, it can be shown mathematically that you will *always*
  trade mainlobe width for reduced sidelobe level, at least in all
  physically realizable windows.

>
>Does anybody knows an algorithm based on DFT (FFT) that optimizes both - 
>for example a main lobe bandwidth similar to a rectangular window and 
>side lobe level similar to a Kaiser-Bessel window ? 

  I refer you to the definitive window reference, 

  "On the Use of Windows for Harmonic Analysis with the Discrete
  Fourier Transform", Frederic J. Harris, Proceedings of the IEEE,
  Vol. 66, No. 1, January, 1978.

  Additionally, for a reference applicable to sidelobe behavior:

  "Some Windows with Very Good Sidelobe Behavior", Albert H.
  Nuttall, IEEE Transactions on A.S.S.P., Vol ASSP-29, No. 1, Feb.
  1981.

  Another good reference for a fascinating window is:

  "A Novel Window for Harmonic Analysis", Fung I. Tseng, Tapan K.
  Sarkar, and Donald D. Weiner, IEEE Transactions on A.S.S.P., Vol
  ASSP-29, No 2, April 1981.

  The latter allows the user the ability to create a very deep notch
  in the sidelobe level at a selectable frequency displacement from
  the main lobe.  Good for harmonic/multitone signals.

/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\

Bob Beauchaine
bobb@vice.ico.tek.com

In article <1992Sep30.154718@sdec1.zrz.tu-berlin.de> feiten@sdec1.zrz.tu-berlin.de (Bernhard Feiten) writes:
>I'm looking for methods to analyze sounds with DFT.
>Normal window functions have the problem of either having low side lobe level
>or small main lobe bandwidth.
>
>Does anybody knows an algorithm based on DFT (FFT) that optimizes both - 
>for example a main lobe bandwidth similar to a rectangular window and 
>side lobe level similar to a Kaiser-Bessel window ? 

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.


--
Stan Reeves
Auburn University, Department of Electrical Engineering, Auburn, AL  36849
INTERNET: sjreeves@eng.auburn.edu

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.

>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.

dak@messua.informatik.rwth-aachen.de (David Kastrup) writes:
>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. 
[..]

The probability density function described above must be positive and sum 
(integrate) to 1: is therefore the normalised square of the (time domain) 
function an appropriate measure (in a mean square error sense) ? 

i.e. 
variance = SUMi { [xi-mean(xi)]^2 * f(xi)^2 } 
where SUMi ( f(xi)^2 ) = 1.

Also , does anybody  know of some good references on this subject ?

James Andrew
Wollongong
g9024557@wampyr.uow.oz.au