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

From: maTheMatic (mathematic_at_gmail.com)
Date: 12/28/04 

Date: 27 Dec 2004 16:02:55 -0800

Hi, Randy

when T->inf, we got a finite ENERGY, but it doesn't mean the average
Power isn't zero.

E=Lim P*T, it is undetermined by 0*inf in mathematics.
T->inf

actually the power density at DC (or other single frequency point)
which you had noticed it has energy unit is the engergy of the ensemble
average of the time average signal.
apply FT to autocorrelation , and let f =0, we get
PSD(0) = lim T* (1/T * E(int(x(t)))^2, T->inf.

Randy Yates wrote:
> Randy Yates <yates@ieee.org> writes:
>
> > "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.
>
> Kiki I hope you're reading this new post,
>
> I can now partially answer your question. In order to do this, first
> realize that when someone speaks of a zero-mean, white Gaussian noise
> process, they're talking about a random process with an underlying
> distribution, i.e., for each point in time t, the random process x(t)
> has a specific probability density function f(t, s). IT IS THIS
> UNDERLYING PDF THAT IS ZERO-MEAN.
>
> However, that still doesn't completely resolve the issue (at least in
> my mind it doesn't). A random process is "ergodic" if its time-wise
> statistics are the same as its ensemble-wise statistics. So we have a
> dilemma when postulating a zero-mean, white, ergodic random noise
> process because ensemble-wise the mean is zero while time-wise the
> mean is non-zero (since there's non-zero energy at DC). I still don't
> know how to resolve THIS problem!
>
> --Randy
>
>
> >
> > > 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.
> > %% Fuquay-Varina, NC % She love the way Puccini lays
down a tune, and
> > %%% 919-577-9882 % Verdi's always creepin' from her
room."
> > %%%% <yates@ieee.org> % "Rockaria", *A New World Record*,
ELO
> > http://home.earthlink.net/~yatescr
>
> --
> Randy Yates
> Sony Ericsson Mobile Communications
> Research Triangle Park, NC, USA
> randy.yates@sonyericsson.com, 919-472-1124

Relevant Pages

  • Re: does WGN(white Gaussian noise) imple zero mean?
    ... > implies a flat PSD, which in term implies that there is some power at DC. ... Kiki I hope you're reading this new post, ... distribution, i.e., for each point in time t, the random process x ...
    (sci.math)
  • Re: does WGN(white Gaussian noise) imple zero mean?
    ... > implies a flat PSD, which in term implies that there is some power at DC. ... Kiki I hope you're reading this new post, ... distribution, i.e., for each point in time t, the random process x ...
    (sci.stat.math)