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Re: Fourier decomposition
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.