<HTML><BODY style="word-wrap: break-word; -khtml-nbsp-mode: space; -khtml-line-break: after-white-space; ">Dear All,<DIV><BR class="khtml-block-placeholder"></DIV><DIV>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.</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Best,</DIV><DIV><BR><DIV> <SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><DIV>________________________________________________________</DIV><DIV>Rami K. Niazy</DIV><DIV>DPhil Candidate</DIV><DIV>University of Oxford, Centre for Functional MRI of the Brain (FMRIB)</DIV><DIV>John Radcliffe Hospital, Headington, Oxford OX3 9DU, United Kingdom</DIV><DIV>Tel: +44 (0) 1865 222739 / Fax: +44 (0) 1865 222717</DIV><DIV>e-mail: rami@fmrib.ox.ac.uk / URL: <A href="http://www.fmrib.ox.ac.uk/~rami">http://www.fmrib.ox.ac.uk/~rami</A></DIV><DIV><BR class="khtml-block-placeholder"></DIV><BR class="Apple-interchange-newline"></SPAN></SPAN> </DIV><BR><DIV><DIV>On 5 Oct 2006, at 10:32, M V Chilukuri wrote:</DIV><BR class="Apple-interchange-newline"><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Dear Asefa,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">S-Transform(Stockwell Transform) is a time-frequency technique with better</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">resolution in time and frequency. In fact, it is very useful for non-stationary</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">signal analysis, unlike Fourier transform which is applicable to only</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">stationary signals. Many practical application involves analysis of non</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">stationary signals, and they are often corrupted with noise. In such a scenario</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Generalized S-Transform/Hyperbolic S-Transform/Complex S-Transform are very</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">useful in extracting information from the signal. However, these techniques use</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">analytical signal to obtain time-frequency plots and analytical signal is</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">obtained from Hilbert Transform of real signal. Also, they give better</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">resolution than STFT(poor resolution)/Wigner-Ville distribution(suffers from</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">cross terms). There may be many techniques coming out in future.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Sincerely,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">M V Chilukuri</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">"A. Debebe" wrote:</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV> <BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Dear All,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">There are excellent discussions on this forum, having</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">listening to<SPAN class="Apple-converted-space"> </SPAN>all the technical discussions about</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Heisenberg uncertainty, etc...., I can not resist</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">participating. When one discuss about Heisenberg</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">uncertainty, one should wonder what it means in</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">relation to event related signal processing, i.e.,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">separation of the coherent signal from the noise, and</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">locating an event.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">In signal processing, we are using windows( or certain</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">resolution)to walk us through the signal during</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">processing. A window size can not be less a certain</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">threshold, in general there is a limitation to it.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">"Heisenberg uncertainty says, one cannot know the</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">exact time-frequency representation of a signal, i.e.,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">one cannot know what spectral components exist at what</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">instances of times. What one can know are the time</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">intervals in which certain band of frequencies exist,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">which is a resolution problem( Robi Polikar Tutorial,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">and a host of papers on signal processing)".</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Each signal processing algorithms, FT,FFT(STFT),</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">wavelet transform, assume a certain size of the</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">resolution of the window used for processing, FT</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">assumes the the size as never ending, thus not good</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">for time related events, FFT assumes the same size of</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">resolution through out the life of the signal,but</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">event related signals are not uniformly distributed(</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">not in term of statistics), wavelet assumes</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">variability of the window size or resolution based</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">upon the size of a band of signals at that location.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Wavelet adapts to the variability of the signal or</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">event related signal.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">There are also continuous<SPAN class="Apple-converted-space"> </SPAN>and discrete signals,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">morlet and FT are good for processing<SPAN class="Apple-converted-space"> </SPAN>continuous</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">signal, continuous signal can<SPAN class="Apple-converted-space"> </SPAN>easily be transformed</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">to discrete signal though.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Every transformation algorithms are not equal, i.e.,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">after transformation the coherent signal may not be</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">the same as the original signal specially the</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">location, therefore, one has to take into</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">consideration a large signal or a periodic signal for</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">processing to get the original signal minus noise.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Somebody has mentioned s-transformation, my</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">understanding is s-transformation is periodic Fourier</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">transformation, it solved the problem mentioned in the</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">previous paragraph.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Regards,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Debebe Asefa, PhD</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">__________________________________________________</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Do You Yahoo!?</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Tired of spam?<SPAN class="Apple-converted-space"> </SPAN>Yahoo! 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