Subject: Re: Fourier decomposition From: Eckard Blumschein <Eckard.Blumschein(at)E-Technik.Uni-Magdeburg.DE> Date: Mon, 19 Sep 2005 09:19:37 +0200Hello Fred, Maybe you did not get aware of my reply to Robert Gjerdingen: errors in harmonic chord identification (last Thu 15 Sep 2005), or you did not consider it serious. I anticipate: The majority of experts is either not able or at least not willing to accept what I found out concerning Fourier decompositon of a contiuously incoming signal: Future events cannot be observed in advance. While this insight is beginning to cause a lot of turmoil in quantum physics, it is self-evident in case of hearing. Hearing is a real-time process. Accordingly, the signal of concern can be completely expressed on a 'half-sided' time scale extending not from minus oo to plus oo but from zero to oo 'elapsed time'. I am suggesting the shorter name 'E-time'. Incidentally, if basics of physics is actually restricted to observables then I do not see any basis for future spacetime, time travel, white holes, still missing evidence for Higgs bosons and susy, superluminal propagation of signals and similar highly questionable stuff. The expression 'half-sided' implies what I consider an illusion: Future world is believed to exists in advance. I argue: Even if this was correct, it would not matter at all. Description in E-time is fully sufficient. Check it out: Complex-valued Fourier transform does not at all contain more information than real-valued Cosine transform except for an arbitrarily chosen zero of time or linear phase, respectively. It is just twice padded with redundant copies of the essential information. Fortunately, natural evolution did not learn this human belief. The ear does not have an arbitrary reference point ot time. I see time the other way round. E-time is the basic physical quantity while ordinary time is just a human abstraction and extrapolation, in short: Ordinary time is a human fabrication. Our notorious trouble due to Fourier transform is selfmade. You wrote: > It is not possible to accurately recover the coefficients (amplitude and > phase of the individual harmonics) of a function consisting of harmonic > sinusoidal components, I would like to clarify first what you are intending to deal with. You obviously refer to a function of ordinary time, not of E-time. Notice, with Cosine transform in IR+ there is no decomposition in terms of magnitude and phase but simply the natural decomposition into a spectral profile with different positive or negative amplitudes as it is to be seen at basilar membrane. Moreover, you seem to tacitly assume a given theoretical function to be analysed. This is not what the ear does. To the inner ear, any theory is unknown. It has to decide itself what frequency content is in the signal. From this point of view, it depends on its own frequency analyis what 'is' or is not within the signal. With this respect, the ear is unique while STFT requires a lot of arbitrary windowing. > when the Period of the not necessarily present > fundamental is not known, by using the normal computational procedure of > either the Fourier Series or the Short term Fourier Transform. In order to better understand how the ear's work works, ask for the natural window of time. It is amazing. > ----------------------- > > I would appreciate any and all comments. Hopefully you will express your appreciation by commenting on my comments. For your convenience, I point you once again to http://iesk.et.uni-magdeburg.de/~blumsche/M277.html Yours sincerely, Eckard