Re: Gaussian vs uniform noise audibility (Israel Nelken )

Subject: Re: Gaussian vs uniform noise audibility
From:    Israel Nelken  <israel(at)MD.HUJI.AC.IL>
Date:    Thu, 22 Jan 2004 04:46:25 +0200

John Hershey's mail indicates the mathematical tools necessary to prove
my claims. These can be found I think in the Papoulis book, for example.
The white noise case is also treated as an example in Percival and Walden.
        More generally, in order to define a random process, you need to define
all of its n-dimensional distributions. This means that it is not enough
to specify that the individual samples are uniform (or gaussian) - you
must specify also the joint distributions of any n samples. Thus, a
gaussian process is not a process whose samples are gaussian
distributed, but a process for which any n samples are jointly gaussian.
This is a very strong statement. It immediately implies that the
spectral components are independent - given John's argument, these are
uncorrelated. Furthermore, they are gaussian, being linear combinations
of jointly-gaussian variables. Therefore they are independent. The
reverse is also true, except for the independence (this will happen only
for a white spectrum).
        Intuitively, the correlations that are necessary in order to reproduce
a non-gaussian amplitude distribution are easy to imagine. For example,
in a heavy-tailed process, you will have very occasionally very large
samples. The phases of the Fourier components must be such that many of
them have maxima at these times, in order to build up these large
samples. Thus, the phases cannot be independent.
        Regarding however the original question of audibility of gaussian vs.
uniform - I must retract my original claim. Some years ago we played
with such white processes that have different amplitude distributions,
and I seem to have badly remembered the result. Gaussian is clearly
distinguishable from heavier-tailed distributions, but not from uniform.
        Eli

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Israel Nelken
Dept. of Neurobiology
The Alexander Silberman Institute of Life Sciences
Edmond Safra Campus, Givat Ram    | Tel: Int-972-2-6584229
Hebrew University                 | Fax: Int-972-2-6586077
Jerusalem 91904, ISRAEL           | Email: israel(at)md.huji.ac.il
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