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Hilbert Transform

Hello List,
Earlier today I asked two questions about the Hilbert Transform. From the response I now realize that I did not make myself very clear. The problem is a real one which I believe the mamalian auditory system has solved. I too have solved the problem but require an exact knowledge of the frequency of the sinusoidal signal of the collected sampled data S(n). ( S(t) = A*Cos(2*pi*f*t - phi) ) I have absolutely no knowldge of A or of phi but I do know that the frequency is absolutely constant. If I have exact knowledge of the frequency f then I can determine the sampling frequency so that DFT[ S(n) ] has no "leakage" and all bins are exactly zero except for a single bin showing the single frequency component (real and imaginary) of S(t) from which I can recover both A and phi. I now calculate IDFT {DFT [S(n)]} and find that this is exactly equal to S(n) (to 14 decimal digits !)

If I take G(t) = A*Sin(2*pi*f*t - phi) )using the same conditions as above then G(n)= IDFT {DFT [G(n)]}.

I can now take the DFT of S(n) modift the phase of the components and then I have IDFT {modified-phase-DFT [S(n)]}= G(n) (exactly and again to 14 decimal digits).

Thus far no problem.

Now if I change the frequency in S(t) and call the new function SS(t) so that the components of DFT(SS(n)) no longer fall into a single bin and then do IDFT{DFT(SS(n))} I also get IDFT{DFT(SS(n))}=SS(n) exactly (again to 14 digits of accuracy). When I form the equivalent GG(t) and GG(n) I get the same results.

What I am looking for is the transformation that I need
(call it "TRANS" ) so that IDFT {TRANS-DFT [S(n)]}= G(n)}

Thats my problem in a nutshell.

Any ideas ?


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