Re: Fourier decomposition

[Mark Every <mre104@xxxxxxxxxx>]

Hi Fred,

Although the STFT coefficients do not directly give an accurate estimation 
of the sinusoidal/harmonic parameters(frequency, amplitude and phase), 
unless the sinusoid happens to fit an exact integer number of times
into the analysis window, it is possible to improve the estimation of these 
parameters over the values
computed from the STFT maximum frequency bin corresponding to each harmonic. 
For example, in:

(P. Depalle and T. H\'{e}lie, Extraction of Spectral Peak Parameters Using a 
Short-Time Fourier Transform Modeling and No Sidelobe Windows, Proc. IEEE 
ASSP Workshop on Applications of Signal Processing to Audio and Acoustics 
(WASPAA'1997), New Paltz, NY)

, a parametric model of the STFT is used to provide a least-squares solution 
to the sinusoidal parameters. We are assuming that the sinusoid is 
stationary within the analysis window. There are also a number of methods 
for improving the estimation of sinusoidal peak parameters by using 
neighbouring frequency bins to the peak maximum. These are reviewed in:

Sylvain Marchand, Mod\'{e}lisation informatique du son musical (analyse, 
transformation, synth\`{e}se), \'{E}cole doctorale de Math\'{e}matiques et 
d'Informatique, 2000, L'Universit\'{e} Bordeaux 1.

So, knowing the period of the signal is not a prerequisite to accurately 
finding the parameters of the harmonics of
that signal. Although, it is correct to say that the STFT or coefficients of 
the Fourier series do not in general directly give accurate parameter 
estimates.

regards,
Mark



> 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