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Hi all,

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

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?

Any clarifications? Thanks a lot!

Hi all,

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

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,

I read through several books but did not get clarification on whether

WGN(white Gaussian noise process) imply zero mean or not...

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?

Any clarifications? Thanks a lot!

Hi all,

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

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?

Any clarifications? Thanks a lot!

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.

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!

Tim Wescott <tim@wescottnospamdesign.com> writes:


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!


Hey Tim,

You get the same result when integrating between 1- and 1+, or
253,392- and 253,392+, etc., and we know have power at there.

What have you answered, then?

Randy Yates wrote:

Tim Wescott <tim@wescottnospamdesign.com> writes:

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!
Hey Tim,
You get the same result when integrating between 1- and 1+, or
253,392- and 253,392+, etc., and we know have power at there. What
have you answered, then?

Well, _I've_ answered that there's no DC content (which is another way
of saying zero mean), when you take DC to it's mathematical limit
(note that white noise will appear to have DC content if you only
observe it for a finite amount of time, such as the time from the big
bang to right now). _You've_ extended this to show that you can pick
any one, zero-bandwidth, filter and find no energy there.

Hopefully I've also pointed out to Kiki that he has all this
information at his fingertips if he'd just collate, think, and use a
pencil and paper every once in a while.

Tim Wescott <tim@wescottnospamdesign.com> writes:


Randy Yates wrote:


Tim Wescott <tim@wescottnospamdesign.com> writes:


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!

Hey Tim,
You get the same result when integrating between 1- and 1+, or
253,392- and 253,392+, etc., and we know have power at there. What
have you answered, then?

Well, _I've_ answered that there's no DC content (which is another way
of saying zero mean), when you take DC to it's mathematical limit
(note that white noise will appear to have DC content if you only
observe it for a finite amount of time, such as the time from the big
bang to right now). _You've_ extended this to show that you can pick
any one, zero-bandwidth, filter and find no energy there.


No, you've shown that there is no power there. There is indeed energy
there since, for white noise, Sxx(w) at w = w0 is strictly greater
than zero for any value of w0 (including 0), and the units of power
spectral density are [joules] ([watts/Hz] == [joules]). One obtains
power upon integration of the Sxx(w) (no matter how small of an
integration interval is chosen) since \int_{w_0-}^{w_0+} Sxx(w) dw has
units of [joules] * [1/seconds], i.e., power.

Oy -- good point. Geeze these limits-to-infinity things get tricky.

Hopefully I've also pointed out to Kiki that he has all this
information at his fingertips if he'd just collate, think, and use a
pencil and paper every once in a while.


If you confuse me and I've had two classes in it, I can't imagine
what's going on in kiki's mind. It is very possible that my mind
is screwed on wrong - if you think so, show me where my thinking
has gone astray.

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

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?

Randy Yates wrote:

Tim Wescott <tim@wescottnospamdesign.com> writes:


Randy Yates wrote:


Tim Wescott <tim@wescottnospamdesign.com> writes:


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!

Hey Tim,
You get the same result when integrating between 1- and 1+, or
253,392- and 253,392+, etc., and we know have power at there. What
have you answered, then?

Well, _I've_ answered that there's no DC content (which is another way
of saying zero mean), when you take DC to it's mathematical limit
(note that white noise will appear to have DC content if you only
observe it for a finite amount of time, such as the time from the big
bang to right now). _You've_ extended this to show that you can pick
any one, zero-bandwidth, filter and find no energy there.
No, you've shown that there is no power there. There is indeed energy

there since, for white noise, Sxx(w) at w = w0 is strictly greater
than zero for any value of w0 (including 0), and the units of power
spectral density are [joules] ([watts/Hz] == [joules]). One obtains
power upon integration of the Sxx(w) (no matter how small of an
integration interval is chosen) since \int_{w_0-}^{w_0+} Sxx(w) dw has
units of [joules] * [1/seconds], i.e., power.


Oy -- good point. Geeze these limits-to-infinity things get tricky.

There must be energy there because if you integrate a white noise
process the variance of the result goes up with the integration
time. But if you take the average of the white noise process (average
= integral / integration time) then the variance goes _down_ with the
integration time, eventually going to zero as the integration time
goes to infinity.

So; zero mean, infinite energy.

I was confused about this stuff, too, and asking questions didn't
clear it up. What _did_ clear it up (for the most part; see your
comment above) was thinking about it. I had the advantage that I take
long bike rides, and for some reason it really worked for me to ponder
these questions while riding.

This is why I'm trying to get the guy pulled away from Matlab
simulations.

"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.
%% 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 <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

Hi all,

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

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?

Any clarifications? Thanks a lot!



That's because I believe most of the time the PSD for white noise is quoted