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

From: Randy Yates (yates_at_ieee.org)
Date: 12/26/04 

Date: Sun, 26 Dec 2004 14:19:17 GMT

"kiki" <lunaliu3@yahoo.com> writes:

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

Hi Kiki,

Now you've got me wondering. On one hand, I've heard the term "zero-mean
additive white Gaussian noise" many times, but on the other hand, "white"
implies a flat PSD, which in term implies that there is some power at DC.
So I can't answer your question.

> 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,

Rxx(t) is defined to be

  Rxx(tau)= E[x(t)*x(t-tau)]

for a real random process x(t). Then, by definition,

  E[x^2(t)] = E[x(t) * x(t-0)]
            = Rxx(0)
            = undefined (infinity)

when Rxx(t) = delta(t). Thus you're contradicting yourself somewhat.

A truly white-noise process does have infinite power (hence the Dirac
delta function in the autocorrelation), but most transistors I know
of burn out after a few gigawatts, so we usually speak of a band-limited
white noise process, i.e., a process which has a PSD Sxx(w) = c, |w| < a,
and in which case the power is finite and Rxx(0) = a*c/pi. 

-- 
%  Randy Yates                  % "She's sweet on Wagner-I think she'd die for Beethoven.
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