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

Subject: Re: Fourier decomposition
From:    "Richard F. Lyon"  <DickLyon(at)ACM.ORG>
Date:    Sat, 17 Sep 2005 14:06:47 -0700

At 9:51 AM +0100 9/17/05, Mark Every wrote:
>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.

I think Fred's point, which I claim is self-evident, is that all such 
methods have inherent accuracy limits.     I would go further and 
argue that Fourier techniques make the problem harder, not easier, 
than a direct time-domain or autocorrelation technique.

>So, knowing the period of the signal is not a prerequisite to 
>accurately finding the parameters of the harmonics of that signal. 
>...

And my point is that "knowing the period" is not a prequisite, but 
that finding the period is equivalent to solving the stated problem. 
That is, the period is found if the harmonics are found, and the 
harmonics are trivial to find if the period is found.  So the problem 
can be reduced to the problem of identifying the period of a signal 
that is known to be periodic but of unknown period.

The solution to this problem is impossible if you are given less than 
a period of the signal, and has a unique canonical solution otherwise 
(that is, a unique shortest period; longer periods are always also 
possible), assuming you aren't given data with noise or aliasing.

Dick