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*To*: AUDITORY@xxxxxxxxxxxxxxx*Subject*: Re: Gaussian vs uniform noise audibility*From*: Eckard Blumschein <Eckard.Blumschein@xxxxxxxxxxxxxxxxxxxxxxxxxx>*Date*: Fri, 23 Jan 2004 12:07:26 +0100*Comments*: To: Israel Nelken <israel@MD.HUJI.AC.IL>*Delivery-date*: Fri Jan 23 06:27:52 2004*In-reply-to*: <4010AF6C.5090901@md.huji.ac.il>*References*: <06f901c3e146$b07bce90$d80aef84@PUTANESKA> <3.0.5.32.20040122130645.00c52550@dfnserv1.urz.uni-magdeburg.de> <06f901c3e146$b07bce90$d80aef84@PUTANESKA>*Reply-to*: Eckard Blumschein <Eckard.Blumschein@xxxxxxxxxxxxxxxxxxxxxxxxxx>*Sender*: AUDITORY Research in Auditory Perception <AUDITORY@xxxxxxxxxxxxxxx>

At 07:21 23.01.2004 +0200, Israel Nelken wrote: >... whether the ear is a fourier analyser, ... I think >everyone's answer (certainly mine) is strictly speaking no - although >spectrograms are reasonable 0th order approximations to what we hear... Don't consider me a knowing-all if I try to clarify: - Usual (complex) Fourier analysis transforms the real signal into a magnitude/phase representation with respect to an arbitrarily chosen temporal reference, missing within the ear. - Function of the inner ear does definitely not resemble the complex Fourier transform but the slightly different real-valued Fourier cosine transform plus one-way rectification. - The usual spectrogram is a rather imperfect freak inbetween. Nonetheless, I agree, it provides a first but partially misleading picture of what we hear. I argue that e.g. pitch perhaps reflects autocorrelation resulting from joint cochlear and subsequent neural signal processing. >...spectral representation is mathematically 'true' (within the >obvious limits of the application of a mathematical theory to real >life). Oliver Heaviside's trick of creating Hermitian symmetry has proven very clever on the expense of physical adequacy. Integration over time from minus infinite to plus infinite is only reasonable if one replaces the unknown future by mutually compensating mirrors of the past. Corresponding results often exhibit non-causality. >Furthermore, for real-life signals the inverse fourier transform is >equal to the original function almost everywhere, The same is true for the Fourier cosine transform. >so that any operation on the signal that is formulated in terms of >its temporal waveform can be also formulated in terms of the fourier >spectrum (amplitdue AND phase). Of course, complex representartion of a signal requires magnitude as well as phase. (Magnitude is always positive while there are positive and negative amplitudes.) This should once again persuade anybody that the inner ear does not perform a complex Fourier transform. What about "any operation" I agree on condition of just a single snapshot. Hearing is, however, a continuous process where the complex Fourier transform is doomed to hop from window to window in a clumsy manner. >...there's a full equivalence between time and spectral processing. Complex spectral analysis is not only restricted to linearity and to snapshots of band-limited signals. It also demands a return into time domain because transition into complexity was based on neglect of either exp(i omega t) or exp(-i omega t). Those who thoughtlessly claim full equivalence between a function of time and its complex spectrum tend to also high-handed deny the actual equivalences between R and R+ in case of (apparent) symmetry and between complex-valued magnitude/phase representation and the underestimated real-valued time/frequency representation. The latter merely lacks the information about the arbitrary reference point of complex analysis, i.e. a single value that does not belong to the original signal. >Finally, there's the question of the usefullness of the spectral >description of random processes, which is yet a somewhat different >question. I consider the adequate model of cochlea (FCT + rectification) a good precondition for correct conclusions concerning random stimuli while non-linearity of rectification further invalidates the pipe dream of equivalence between the signal and its complex matrices. >The independence of the spectral components of a gaussian >process is a mathematical result, independent of the physiology of >hearing, but it has consequences for hearing. Given, adequacy of the mathematical model is questionable. How trustworthy are the results? >Since we are sensitive to spectral correlations, non-gaussianity can >be detectable by such sensitivity. This doesn't assume anything about >the use of Fourier transforms in the ear! I googled for 'non-gaussianity': 3750 results, mostly cosmic microwave background. Eckard

**References**:**Re: Gaussian vs uniform noise audibility***From:*John Hershey

**Re: Gaussian vs uniform noise audibility***From:*Eckard Blumschein

**Re: Gaussian vs uniform noise audibility***From:*Israel Nelken

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