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From: "Andy Robb" <ajrobb@hotmail.com>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Hilbert Transform
Date: Thu, 8 May 2008 20:34:03 +0000 (UTC)
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"David Egger" <eggerd@sbox.tugraz.at> wrote in message
<fvuksv$2q3$1@fred.mathworks.com>...
> "Andy Robb" <ajrobb@hotmail.com> wrote in message
> <fvt3gt$dse$1@fred.mathworks.com>...
> > "David Egger" <eggerd@sbox.tugraz.at> wrote in message
> > <fvs88h$j98$1@fred.mathworks.com>...
> > > "Andy Robb" <ajrobb@hotmail.com> wrote in message
> > > <fvrv3v$8ja$1@fred.mathworks.com>...
> > > > "David Egger" <eggerd@sbox.tugraz.at> wrote in message
> > > > <fvriev$b1i$1@fred.mathworks.com>...
> > > > 
> > > > 
> > > > 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.
> > > ---------------------------------------------------------
> > > Hey,
> > > 
> > > thank you for answering!
> > > Okay, I want to keep the algorithm based on the
manipulation
> > > in the frequency domain you explained.
> > > Do you know who invented this algorithm?
> > > 
> > > You said, it is an approach.So this is not an ideal
filter?
> > > Is an ideal Hilbert filter possible?
> > > 
> > > You also said,the problem with this approach are the
same as
> > > in any fft approach.Could you name some of them?
> > > 
> > > 
> > > Regards!
> > 
> > FFT is just a fast (efficient) form of DFT calculation.
> > 
> > The basic requirement for a DFT to be accurate are:
> > 1. The original signal contains no component at frequencies
> > above half the sample frequency (Nyquist-Shannon*)
> > 2. The signal is either an event completely sampled over the
> > sample period or is periodic in the sample period (the start
> > and 'end' conditions must be the same)
> > 
> > NOTE: the 'end' condition is the next sample AFTER the last
> > sample.
> > 
> > If these conditions are not met then expect some 'ringing'
> > in the synthesised imaginary component at the ends of the
> > sample period. However, if they are met, then the Hilbert
> > transform will also be accurate.
> > 
> > *By convention, Nyquist is often associated with the Nyquist
> > frequency limit. Shannon is associated with reconstruction
> > (interpolation with sinc waveform FIR filters). However,
> > they worked closely together at AT&T Bell Labs.
> 
> 
> Okay, thank you!
> 
> So if the two conditions are satisfied, fft is equal dft?

No FFT is always DFT - just Fast (hence the extra F)

The conditions are for every good DFT.

> 
> Is the hilbert transform using the fft an approach or not?

Yes - if the DFT conditions are met.
> 
> And who invented the fft algorithm for Hilbert?Shannon and
> Nyquist?

Cooley-Tukey invented the modern FFT - Hilbert had already
done the rest.

Nyquist-Shannon are the godfathers of PCM

Wikipedia is good an all these.

> 
> 
> Greetings!!
Greeted - thanks