Re: Gaussian vs uniform noise audibility (Julius Smith )


Subject: Re: Gaussian vs uniform noise audibility
From:    Julius Smith  <jos(at)CCRMA.STANFORD.EDU>
Date:    Tue, 27 Jan 2004 09:13:38 -0800

Yes, a "sliding cosine transform" can be used in place of the usual "hopping short-time Fourier transform", and in that case, phase information is contained in the time variation of the sliding transform coefficients. I didn't realize you were doing something like that, so my argument was based on different assumptions. Even the short-time Fourier transform hopping by half its window length each frame can be stripped of all phase information and still be used as the basis of a convincing sound synthesis, at least for smoothly changing sounds. -- jos At 03:08 AM 1/26/2004, Eckard Blumschein wrote: >At 12:06 23.01.2004 -0800, Julius Smith wrote: > >At 11:16 AM 1/23/2004, Eckard Blumschein wrote: > >>First of all, forget the wrong idea that the cochlea performs a complex > >>Fourier transform. > > > >This implies phase is discarded. > >No! Do not consider me a moron. You and largely the rest of the world grew >up with the erroneous believe that there is no equivalent alternative to >complex spectral analysis. Complex calculus is indeed tremendously useful. >No matter whether one prefers magnitude and phase or real and imaginary >part, one always has to consider both constituents except for the case one >of them equals zero. Given, a function of time like 2A cos(omega t) does >not have any imaginary part at all. Entrance into complex plane is payed by >mandatory arbitrary omission of A exp(- i omega t) or A exp(i omega t). >Neither the magnitude A nor the phase omega t can be discarded. >At that point, you will object: Aren't anti-symmetrical functions, i.e. >functions of time with odd symmetry like sinus, also needed in frequency >analysis? > >No again, on condition, causality has been taken into account. In brief: >Future signals cannot be analyzed yet. Even sin(omega t) can be continued >as its mirror into fictive future time like an even function. Of course, >this wouldn't hold for its derivative or antiderivative. However, our topic >is just frequency analysis within cochlea. > > >However, phase information does exist as > >the phase of the basilar membrane vibration,... > >I don't take amiss this fallacy. It has to do with the missing natural >justification for fixing any reference point on the time scale. Our ears >are not synchronized with anything. When Descartes introduced Cartesian >coordinates, he imagined a spatially infinite world. Time is >correspondingly believed to also expand from minus infinite to plus >infinite. However, elapsed time definitely ends at the 'NOW' being the only >clever choice for a natural time scale. Take subsequent snapshots of a >sinusoid at NOW each. Try the same with any cochlear pattern. By chance, >you might observe sin or cos. In other words, so called linear phase is >arbitrary as is time. I don't deny that delay or according phase difference >is reasonable with respect to a second signal or a different reference. >Without such reference, a sinusoidal function cannot be a identified as >sin, cos or something complex in between, and the reference is lacking in >nature. The only natural reference is the NOW, which is steadily on the >move. This causes the trouble of permanently lagging window position in >case of arbitrarily centered complex Fourier transform. > > > >Since basilar membrane filtering is generally > >modeled as linear, any corresponding short-time-Fourier-transform would > >have to be complex to model basilar membrane filtering. Subsequent > >half-wave rectification does not eliminate all phase information, > >An old specialist of power electronics like me cannot retrace how you >imagine rectification of a complex-valued function of time. > >My wife is a teacher for adults. Perhaps she would more heedfully >anticipate what you and many others are feeling rather than thinking. I >will try and elucidate how engineers handle a similar case: Consider an >ideal sinusoidal voltage as a real input into a circuit that may also >contain a first (small) resistor and a reactance in series. Parallel to the >first resistor there are a diod and a much larger second impedance in >series. The voltage across the first resistor is a complex quantity with >respect to the source but pretty independent of the diod. However, >piecewise linear calculation requires to refer to the current through the >diod as a real one. In case of hearing, phase of the stimulus does not >matter since it anyway relates to an arbitrary reference. > >As a rule, recognized experts like you tend to be cautious against >radically uncommon views. Therefore I would like to ask you: Look at >pattern of BM motion (e.g. T. Ren's) or of firing in the auditory nerve. >They do not resemble magnitude, nothing to say about phase. As far as I can >judge, they resemble the pattern of the natural (real-valued) spectrogram. >More in detail: Magnitude cannot account for the different patterns with >rarefaction vs. condensation clicks while positve and negative amplitudes >of the natural spectrogram clearly differ from each other. > >In all, I didn't find any tenable argument in favor of complex cochlear >function. On the other hand, Fourier cosine transform, the natural >spectrogram and joint autocorrelation already resolved a lot of so far >poorly understood questions. > >Incidentally, I recall a textbook denying any difference between time >domain and frequency domain. I do not fully share this opinion. In >particular, I consider it necessary to clearly distinguish between real >world and fictitious complex domain. > >Eckard


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