# does WGN(white Gaussian noise) imple zero mean?

Started by December 25, 2004
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!


"kiki" <lunaliu3@yahoo.com> wrote in message
news:cqlbr7$a4m$1@news.Stanford.EDU...
> Hi all,
>
> I read through several books but did not get clarification on whether
> WGN(white Gaussian noise process) imply zero mean or not...

Zero mean.

>
> 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?

Power cannot be infinite, unless you have identical waveforms for
autocorrelation peak.

>
> Any clarifications? Thanks a lot!
>
>
>


"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.

> 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

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
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

Tim Wescott wrote:

...

> 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.

Man, I wish I had had you for a teacher, back when it mattered!

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;

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.

--
%  Randy Yates                  % "With time with what you've learned,
%% Fuquay-Varina, NC            %  they'll kiss the ground you walk
%%% 919-577-9882                %  upon."
%%%% <yates@ieee.org>           % '21st Century Man', *Time*, ELO

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

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
>
> 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.

> 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.
--
%  Randy Yates                  % "With time with what you've learned,
%% Fuquay-Varina, NC            %  they'll kiss the ground you walk
%%% 919-577-9882                %  upon."
%%%% <yates@ieee.org>           % '21st Century Man', *Time*, ELO

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
>>
>>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.
>
>>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.

it up.  What _did_ clear it up (for the most part; see your comment
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.

--

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

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

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

For an infinity of samples, yes.  But if you have just n samples, the
mean you'll get will be 0 only in the mean, but have a variance of V/n
where V is the variance of a single sample.

> 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?

That's a problem of defining your scale factors in a manner that
yields workable results.  Fourier transforms for random processes are
somewhat special here.

--
David Kastrup, Kriemhildstr. 15, 44793 Bochum