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From: "Andy Robb" <ajrobb@hotmail.com>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Hilbert Transform
Date: Wed, 7 May 2008 10:09:03 +0000 (UTC)
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"David Egger" <eggerd@sbox.tugraz.at> wrote in message
<fvriev$b1i$1@fred.mathworks.com>...
> Hey,
> 
> 
> I unterstand the algorithm hilbert.m in Matlab.
> But can anyone tell me:
> 
> 1)Is this the ideal Hilbert or an approximation?
> 2)Who invented the algorithm?
> 3)do anyone know a book where I can find the alg.?
> 
> Regards!

I spent many years applying Hilbert transforms, often
combining them with re-sampling techniques (Shannon et al).

There are two approaches to the Hilbert transform. Both
synthesise an imaginary component of a complex analytic
waveform from the 'real' signal. The real component should
be unchanged.

From memory, hilbert.m uses an FFT approach, it zeros
frequency components below 0 and double frequency components
between zero and Nyquist. The IFFT then produces a complex
analytic waveform. The problems with this approach are the
same as any FFT technique and can suffer the effects of
truncation.

An alternative approach is to synthesise the imaginary
component directly from the real component using a
time-domain filter. From my dim and distant pass, I think
you can see the shape of an FIR by synthesising a spectrum
with 1i in positive frequencies and -1i in negative
frequencies and zero in all real components (including 0 and
Nyquist). Then take the IFFT.