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From: "Andy Robb" <ajrobb@hotmail.com>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Hilbert Transform
Date: Wed, 7 May 2008 20:30:21 +0000 (UTC)
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"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.