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*To*: AUDITORY@xxxxxxxxxxxxxxx*Subject*: Re: Gaussian vs uniform noise audibility*From*: Israel Nelken <israel@xxxxxxxxxxxxx>*Date*: Thu, 22 Jan 2004 04:46:25 +0200*Delivery-date*: Wed Jan 21 22:04:41 2004*Reply-to*: Israel Nelken <israel@xxxxxxxxxxxxx>*Sender*: AUDITORY Research in Auditory Perception <AUDITORY@xxxxxxxxxxxxxxx>*User-agent*: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.4.1) Gecko/20031008

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 -- ================================================================== 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@md.huji.ac.il ==================================================================

**Follow-Ups**:**Re: Gaussian vs uniform noise audibility***From:*Eckard Blumschein

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