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*To*: AUDITORY@xxxxxxxxxxxxxxx*Subject*: Re: Gaussian vs uniform noise audibility*From*: Julius Smith <jos@xxxxxxxxxxxxxxxxxx>*Date*: Fri, 23 Jan 2004 23:04:00 -0800*Comments*: To: John Hershey <jhershey@COGSCI.UCSD.EDU>*Delivery-date*: Sat Jan 24 02:19:24 2004*In-reply-to*: <0db901c3e1f2$68870440$d80aef84@PUTANESKA>*References*: <200401231611.i0NGBf9B014296@staff2.cso.uiuc.edu> <6.0.1.1.2.20040123104845.06701ec0@w3k.org> <0db901c3e1f2$68870440$d80aef84@PUTANESKA>*Reply-to*: Julius Smith <jos@xxxxxxxxxxxxxxxxxx>*Sender*: AUDITORY Research in Auditory Perception <AUDITORY@xxxxxxxxxxxxxxx>

With Gaussian random variables, uncorrelated implies independent. At 12:49 PM 1/23/2004, John Hershey wrote:

So, according to the central limit theorem, each frequency component, being a weighted sum of a large number of independent random variables approaches a Gaussian distribution. However the sums are all over the same independent random variables, so in general the sums are not independent. It seems clear, though, that the frequency components are uncorrelated, because the Fourier transform is orthogonal, and they were assumed to be uncorrelated in the time domain. However, unless I'm missing something, if the time domain distributions are not Gaussian, then the frequency components are in general not jointly Gaussian, despite being individually Gaussian and being uncorrelated. Lack of correlation is necessary but not sufficient for independence, so in general there still may be higher-order statistical dependencies between the frequency components. ----- Original Message ----- From: "Julius Smith" <jos@CCRMA.STANFORD.EDU> To: <AUDITORY@LISTS.MCGILL.CA> Sent: Friday, January 23, 2004 11:11 AM Subject: Re: Gaussian vs uniform noise audibility > I am surprised nobody seems to have mentioned the central limit theorem > which shows that the sum of random variables from most any distribution > (including uniform) converges to a Gaussian random variable. As a result, > the Fourier transform of almost any type of stationary random process > yields a set of iid complex Gaussian random variables. On a more practical > level, two spectral samples from a (finite-length) FFT can be regarded as > independent as long as they are separated by at least one "resolution cell" > --- i.e., the "band slices" they represent do not overlap > significantly. For a rectangular window, the width of a resolution cell > can be defined conservatively as twice the sampling rate divided by the > window length. For Hamming and Hann windows, it's double that of the > rectangular window, Blackman three times, and so on. > > In summary, any time a noise process has been heavily filtered, it can be > regarded as approximately Gaussian, by the central limit theorem, and > disjoint spectral regions are statistically independent. > > -- Julius > > Reference: >

_____________________________ Julius O. Smith III <jos@ccrma.stanford.edu> Assoc. Prof. of Music and (by courtesy) Electrical Engineering CCRMA, Stanford University http://www-ccrma.stanford.edu/~jos/

**Follow-Ups**:**Re: Gaussian vs uniform noise audibility***From:*John Hershey

**References**:**Re: Gaussian vs uniform noise audibility***From:*beauchamp james w

**Re: Gaussian vs uniform noise audibility***From:*Julius Smith

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

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