Fourier decomposition

[David Havelock <david.havelock@xxxxxx>]

A fairly detailed discussion of the problem, and a solution,
can be found in 
"Accurate analysis of multitone signals using a DFT" 
John C. Burgess
J. Acoust. Soc. Am. 116, 389 (2004)

The optimized estimation of phase and magnitude take into
account the effects of windowing, using an optimized window
design.  The paper addresses details of discrete signal
analysis that are often glossed over. 


David I. Havelock
Acoustics and Signal Processing Group 
Institute for Microstructural Sciences (M36)
National Research Council Canada 
-----------

Date:    Sun, 18 Sep 2005 20:15:31 -0500
From:    beaucham <beaucham@xxxxxxxxxxxxxxxxxxxxxx>
Subject: Re: Fourier decomposition

What is meant by "accurately"?. I.e., how do you test and
how close do the results have to be before you'd say the
analysis method is accurate? The method described at

http://ccrma.stanford.edu/~jos/parshl/parshl.html

does a pretty darn good job. We have an implementation
similar to this in 
our SNDAN package which performs very well on harmonic and
inharmonic 
sounds. See

http://ems.music.uiuc.edu/beaucham/software/sndan/

Regards,

Jim Beauchamp
Univ. of Illinois at Urbana-Champaign

On Sat, 17 Sep 2005, Bob Masta wrote:

> On 16 Sep 2005 at 19:20, Fred Herzfeld wrote:
> 
> > Hello List:
> > 
> > I am now about to make public some work on signal
decomposition. As 
> > part
> > of the disclosure I will make the statement:
> > ------------------
> > It is not possible to accurately recover the
coefficients (amplitude and 
> > phase of the individual harmonics) of a function
consisting of harmonic 
> > sinusoidal components, when the Period of the not
necessarily present 
> > fundamental is not known, by using the normal
computational procedure of 
> > either the Fourier Series or the Short term Fourier
Transform.
> > -----------------------
> > 
> > I would appreciate any and all comments.
> > 
> 
> Interesting question.  Pre-multiplication by ordinary
windowing 
> functions seems to do a pretty good job when recovering
the 
> magnitudes, but I've never looked into what this does to
the phase.  I 
> gather from your statement that it screws it up, and that
there is no 
> analogous window function for obtaining phase?
> 
> Thanks for your insights...
> 
> Bob Masta
>