Re: does WGN(white Gaussian noise) imple zero mean?

From: Tim Wescott (tim_at_wescottnospamdesign.com)
Date: 12/26/04 

Date: Sun, 26 Dec 2004 09:06:02 -0800

kiki wrote:

> Hi all,
>
> I read through several books but did not get clarification on whether
> WGN(white Gaussian noise process) imply zero mean or not...

OK. Think.

Hmm. The definition of a white noise process is that the PSD is 1
everywhere. OK, I understand that.

If I know the PSD of a function I can find the expected power between
any two frequencies by just integrating the PSD over that interval. OK,
I've read my books, I understand that.

Now, DC means the frequency interval between 0 and 0 (technically
between 0- and 0+). Integrating 1 between 0 and 0 I get -- ZERO! WOW!
>
> Another confusion I have is that the definition of WGN is it has flat power
> spectrum density, let's say S(f)=1, then Rx(t)=delta(t) is its
> autocorrelation function, I don't see how people say the power of this noise
> process is E((x(t))^2)=sigma_x, something like that... the power should be
> infinite, right?

For a truly white PSD the power is infinite (see, I'm not raking you
over the coals for this -- this is actually a bit of a brain twister and
therefore not a blindingly obvious question).

This is actually the mathematical difficulty that led Plank to his
discovery of black-body radiation -- before Plank you had to just
describe thermal noise in a resistor or whatnot as white noise, and
shrug your shoulders when the question of infinite power came up.

So for all practical purposes you should remember that white noise of
any sort is a mathematical fiction, and only it to predict the responses
of physical systems who's bandwidths are much lower than the bandwidth
limitation of the noise you have at hand.
>
> Any clarifications? Thanks a lot!
>
>
> 

-- 
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

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