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# HILBERT TRANSFORM

`Hello list,`

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`

```--
Fred Herzfeld, MIT '54
78 Glynn Marsh Drive #59
Brunswick, Ga.31525
USA```