[Eeglablist] Heisenberg, .....

[Rami Niazy]

Dear All,

May I add that for a number of years now the Hilbert-Huang Transform  
(HHT) has been applied successfully in many fields to localise the  
energy of non-stationary and non-linear signals.  Actually, it was  
invented specifically for such cases.  It is completely adaptive and  
data-driven (no a priori-specified basis).  Its temporal resolution  
is that of the signal (instantaneous) and its frequency resolution is  
only limited by computational constraints.     However, it is purely  
algorithmic and doesn't have a mathematical foundation, yet.  With  
regards to the Heisenberg uncertainty, it applies in FT/wavelets and  
so on largely because of the definition adopted for frequency, f=1/ 
T.  However,  one could define frequency as the rate of change in  
phase, which would allow the calculation of instantaneous frequency   
if the phase function is known.  HHT provides an algorithm called  
empirical mode decomposition (EMD), which decomposes any signal into  
its intrinsic modes of oscillations, in such away that no two  
components have the same instantaneous frequency at the same time.   
You can then use the Hilbert transform or other methods on these  
components to calculate the phase functions, instantaneous frequency   
and power.  Of course the methods has its drawback, but my purpose  
here is just to point out an alternative rather that go into the  
details.

Best,

________________________________________________________
Rami K. Niazy
DPhil Candidate
University of Oxford, Centre for Functional MRI of the Brain (FMRIB)
John Radcliffe Hospital, Headington, Oxford OX3 9DU, United Kingdom
Tel:    +44 (0) 1865 222739   /  Fax:  +44 (0) 1865 222717
e-mail: rami at fmrib.ox.ac.uk  /  URL:  http://www.fmrib.ox.ac.uk/~rami



On 5 Oct 2006, at 10:32, M V Chilukuri wrote:

> Dear Asefa,
>
> S-Transform(Stockwell Transform) is a time-frequency technique with  
> better
> resolution in time and frequency. In fact, it is very useful for  
> non-stationary
> signal analysis, unlike Fourier transform which is applicable to only
> stationary signals. Many practical application involves analysis of  
> non
> stationary signals, and they are often corrupted with noise. In  
> such a scenario
> Generalized S-Transform/Hyperbolic S-Transform/Complex S-Transform  
> are very
> useful in extracting information from the signal. However, these  
> techniques use
> analytical signal to obtain time-frequency plots and analytical  
> signal is
> obtained from Hilbert Transform of real signal. Also, they give better
> resolution than STFT(poor resolution)/Wigner-Ville distribution 
> (suffers from
> cross terms). There may be many techniques coming out in future.
>
> Sincerely,
> M V Chilukuri
>
> "A. Debebe" wrote:
>
>> Dear All,
>>
>> There are excellent discussions on this forum, having
>> listening to  all the technical discussions about
>> Heisenberg uncertainty, etc...., I can not resist
>> participating. When one discuss about Heisenberg
>> uncertainty, one should wonder what it means in
>> relation to event related signal processing, i.e.,
>> separation of the coherent signal from the noise, and
>> locating an event.
>>
>> In signal processing, we are using windows( or certain
>> resolution)to walk us through the signal during
>> processing. A window size can not be less a certain
>> threshold, in general there is a limitation to it.
>> "Heisenberg uncertainty says, one cannot know the
>> exact time-frequency representation of a signal, i.e.,
>> one cannot know what spectral components exist at what
>> instances of times. What one can know are the time
>> intervals in which certain band of frequencies exist,
>> which is a resolution problem( Robi Polikar Tutorial,
>> and a host of papers on signal processing)".
>>
>> Each signal processing algorithms, FT,FFT(STFT),
>> wavelet transform, assume a certain size of the
>> resolution of the window used for processing, FT
>> assumes the the size as never ending, thus not good
>> for time related events, FFT assumes the same size of
>> resolution through out the life of the signal,but
>> event related signals are not uniformly distributed(
>> not in term of statistics), wavelet assumes
>> variability of the window size or resolution based
>> upon the size of a band of signals at that location.
>> Wavelet adapts to the variability of the signal or
>> event related signal.
>>
>> There are also continuous  and discrete signals,
>> morlet and FT are good for processing  continuous
>> signal, continuous signal can  easily be transformed
>> to discrete signal though.
>>
>> Every transformation algorithms are not equal, i.e.,
>> after transformation the coherent signal may not be
>> the same as the original signal specially the
>> location, therefore, one has to take into
>> consideration a large signal or a periodic signal for
>> processing to get the original signal minus noise.
>>
>> Somebody has mentioned s-transformation, my
>> understanding is s-transformation is periodic Fourier
>> transformation, it solved the problem mentioned in the
>> previous paragraph.
>>
>> Regards,
>>
>> Debebe Asefa, PhD
>>
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