Hilbert Transform

[Fred Herzfeld <herzfeld@xxxxxxxxxxxx>]

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


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