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Suppose I have a finite sampled set of data of length 2^n of a function
s(t) = cos(2*pi*f-phi) where f and phi are constant with time and s(t)
is sampled at time intervals so that s(t) falls exactly into the second
FFT bin. If I perform the FFT, modify the output by 90 degrees and
perform an inverse FFT, the output of the FFT will be exactly the
Hilbert Transform of the original series s(t) namely HT[s(t)].
Now I repeat the same process except that I change the frequency (still
constant) to 7*f/8 and call the time series S1(t). The output of the FFT
will now contain many non zero bins. In theory I should still be able to
modify the FFT output and do the inverse FFT to get the Hilbert
Transform of S1(t).
(1) Can I really compute HT[S1(t)] correctly ?
(2) If I can, how should the output of the FFT of S1(t) be modified
Any thoughts on this will be very much appreciated.
Fred Herzfeld, MIT '54
78 Glynn Marsh Drive #59