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Re: Fourier decomposition

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.