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Re: Gaussian vs uniform noise audibility

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>
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
> 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
> --- 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