Subject: Re: Gaussian vs uniform noise audibility From: Julius Smith <jos(at)CCRMA.STANFORD.EDU> Date: Fri, 23 Jan 2004 23:04:00 -0800With 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(at)CCRMA.STANFORD.EDU> >To: <AUDITORY(at)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(at)ccrma.stanford.edu> Assoc. Prof. of Music and (by courtesy) Electrical Engineering CCRMA, Stanford University http://www-ccrma.stanford.edu/~jos/