Re: Fourier decomposition ("Richard F. Lyon" )

Subject: Re: Fourier decomposition
From:    "Richard F. Lyon"  <DickLyon(at)ACM.ORG>
Date:    Fri, 16 Sep 2005 21:40:31 -0700

At 7:20 PM -0400 9/16/05, Fred Herzfeld wrote:
>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.

Fred, I think your result is self-evident.  The STFT (short-time 
Fourier transform) works on a segment of a signal, and fits it with a 
sum of sinusoids that will repeat it with period equal to the 
transform size.  The periods of these components cannot in general be 
related to the period of the original periodic signal if it was 
unknown.  Furthermore, no method will in general be able to extract 
an exact period from a finite record of a signal unless the record is 
at least as long as the period, and the signal is noiseless.  If the 
signal is sampled in time, no finite record is enough to exactly 
determine the period of even a perfectly noiseless periodic signal 
(well, maybe theoretically if it is bandlimited).  If you can't 
accurately recover the period, you can't find the harmonic 
components, but if you can get the period then getting the harmonics 
is easy by a Fourier series.  Any method that finds the components 
implies finding the exact period, so the problem reduces to a period 
detection problem.  Estimation is easy; getting it accurate is hard.

Dick