# Is White Noise Necessarily Zero-Mean?

Started by February 15, 2006
```Let Z(t) be a white-noise (stationary) random process.
Can we conclude that Z(t) has zero mean?

My thought is: yes. The intuitive reason that comes to
mind (and it may be wrong!) is this: If Z(t) has a
non-zero mean, then there would be some amount of
correlation between samples due to the means. Thus
the autocorrelation would be a delta function.
--
%  Randy Yates                  % "Bird, on the wing,
%% Fuquay-Varina, NC            %   goes floating by
%%% 919-577-9882                %   but there's a teardrop in his eye..."
%%%% <yates@ieee.org>           % 'One Summer Dream', *Face The Music*, ELO
```
```Randy Yates <yates@ieee.org> writes:
> [...]
> Thus the autocorrelation would be a delta function.
V
not
--
%  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
```
```Randy Yates wrote:

> Let Z(t) be a white-noise (stationary) random process.
> Can we conclude that Z(t) has zero mean?
>
> My thought is: yes. The intuitive reason that comes to
> mind (and it may be wrong!) is this: If Z(t) has a
> non-zero mean, then there would be some amount of
> correlation between samples due to the means. Thus
> the autocorrelation would be a delta function.

Strictly speaking white noise would be zero mean.  One often
overqualifies one's statement about white noise because in real systems
there is often some DC riding on an otherwise white process, so if you
want to make sure that people realize it's really zero mean you have to
say so.

--

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

```
```Randy Yates wrote:

> Let Z(t) be a white-noise (stationary) random process.
> Can we conclude that Z(t) has zero mean?
>
> My thought is: yes. The intuitive reason that comes to
> mind (and it may be wrong!) is this: If Z(t) has a
> non-zero mean, then there would be some amount of
> correlation between samples due to the means. Thus
> the autocorrelation would be a delta function.

Yes. Whit noise is necessarily zero-mean. This follows from the definition
of white (i.e. covariance matrix is identity matrix).

Let x be a random vector. Covariance matrix of a random vector is Cx =
E{x*x'}. Mean of the random vector is m = E{x}. Let y be a white random
vector with zero mean so that y = x + m;

E{x*x'} = E{(y+m)*(y'+m')} = E{y*y'}+E{y*m'}+E{m*y'}+E{m*m'} = I + E{y}*m' +
m*E{y'} + m*m' = I + m*m' != I => random vector is not white if the mean is
not zero.

--
Jani Huhtanen
Tampere University of Technology, Pori
```
```Tim Wescott <tim@seemywebsite.com> writes:

> Randy Yates wrote:
>
>> Let Z(t) be a white-noise (stationary) random process. Can we
>> conclude that Z(t) has zero mean?
>> My thought is: yes. The intuitive reason that comes to
>> mind (and it may be wrong!) is this: If Z(t) has a non-zero mean,
>> then there would be some amount of
>> correlation between samples due to the means. Thus
>> the autocorrelation would be a delta function.
>
> Strictly speaking white noise would be zero mean.  One often
> overqualifies one's statement about white noise because in real
> systems there is often some DC riding on an otherwise white process,
> so if you want to make sure that people realize it's really zero mean
> you have to say so.

Thanks Tim! You hit the nail on the head. I was asking because, even
in textbooks like Proakis, you see it qualified, and I was wondering
why.

It would be better if let our yes's be yes, our no's be no, and our
white's be white!
--
%  Randy Yates                  % "So now it's getting late,
%% Fuquay-Varina, NC            %    and those who hesitate
%%% 919-577-9882                %    got no one..."
%%%% <yates@ieee.org>           % 'Waterfall', *Face The Music*, ELO
```
```"Jani Huhtanen" <jani.huhtanen@kolumbus.fi> wrote in message
news:dsvpsq\$l2f\$1@phys-news4.kolumbus.fi...
> Randy Yates wrote:
>
>> Let Z(t) be a white-noise (stationary) random process.
>> Can we conclude that Z(t) has zero mean?
>>
>> My thought is: yes. The intuitive reason that comes to
>> mind (and it may be wrong!) is this: If Z(t) has a
>> non-zero mean, then there would be some amount of
>> correlation between samples due to the means. Thus
>> the autocorrelation would be a delta function.
>
> Yes. Whit noise is necessarily zero-mean. This follows from the definition
> of white (i.e. covariance matrix is identity matrix).
>
> Let x be a random vector. Covariance matrix of a random vector is Cx =
> E{x*x'}. Mean of the random vector is m = E{x}. Let y be a white random
> vector with zero mean so that y = x + m;
>
> E{x*x'} = E{(y+m)*(y'+m')} = E{y*y'}+E{y*m'}+E{m*y'}+E{m*m'} = I + E{y}*m'
> +
> m*E{y'} + m*m' = I + m*m' != I => random vector is not white if the mean
> is
> not zero.
>
>
> --
> Jani Huhtanen
> Tampere University of Technology, Pori

Just be careful to note that a *sample* of such white noise doesn't
necessarily have zero mean.

Start with a white noise generator (with zero mean implied / assured).
Grab a sample over some finite temporal epoch such that the mean *is* zero.
It should be easy enough to do this by generating the noise and computing
the mean from the beginning, for some time, and then continuing to compute
the mean of the entire record until the mean is zero - then stop grabbing
data.
Now, cut the data into two equal halves in time.
There is some large probability that each half will not each have zero mean.
Of course the sum of the means of the two halves will be zero because that's
what you constructed in the grab.

Fred

```
```On Wed, 15 Feb 2006 12:41:58 -0800, "Fred Marshall"
<fmarshallx@remove_the_x.acm.org> wrote:
>
>"Jani Huhtanen" <jani.huhtanen@kolumbus.fi> wrote in message
>news:dsvpsq\$l2f\$1@phys-news4.kolumbus.fi...
>> Randy Yates wrote:
>>
>>> Let Z(t) be a white-noise (stationary) random process.
>>> Can we conclude that Z(t) has zero mean?
>>>
>>> My thought is: yes. The intuitive reason that comes to
>>> mind (and it may be wrong!) is this: If Z(t) has a
>>> non-zero mean, then there would be some amount of
>>> correlation between samples due to the means. Thus
>>> the autocorrelation would be a delta function.
>>
>> Yes. Whit noise is necessarily zero-mean. This follows from the definition
>> of white (i.e. covariance matrix is identity matrix).
>>
>> Let x be a random vector. Covariance matrix of a random vector is Cx =
>> E{x*x'}. Mean of the random vector is m = E{x}. Let y be a white random
>> vector with zero mean so that y = x + m;
>>
>> E{x*x'} = E{(y+m)*(y'+m')} = E{y*y'}+E{y*m'}+E{m*y'}+E{m*m'} = I + E{y}*m'
>> +
>> m*E{y'} + m*m' = I + m*m' != I => random vector is not white if the mean
>> is
>> not zero.
>>
>
>Just be careful to note that a *sample* of such white noise doesn't
>necessarily have zero mean.
>

Huh? Mean of a sample? What's the point of speaking of the mean of one
known value?
```
```Just Cocky <just@cocky.com> writes:
> [...]
> Huh? Mean of a sample? What's the point of speaking of the mean of one
> known value?

I think he meant a section of time from a realization of a random process.

I've seen the term "sample waveform" used in Proakis and I don't like it
either.
--
%  Randy Yates                  % "Bird, on the wing,
%% Fuquay-Varina, NC            %   goes floating by
%%% 919-577-9882                %   but there's a teardrop in his eye..."
%%%% <yates@ieee.org>           % 'One Summer Dream', *Face The Music*, ELO
```
```"Randy Yates" <yates@ieee.org> wrote in message
news:m3bqx82t1e.fsf@ieee.org...
> Just Cocky <just@cocky.com> writes:
>> [...]
>> Huh? Mean of a sample? What's the point of speaking of the mean of one
>> known value?
>
> I think he meant a section of time from a realization of a random process.
>
> I've seen the term "sample waveform" used in Proakis and I don't like it
> either.

Right and point taken.  I suppose it leaks over from statistics where a
"sample" is what we'd here call a "sequence" or "vector" or ..... a set of
samples taken over a temporal epoch.

Fred

```
```Jani Huhtanen <jani.huhtanen@kolumbus.fi> writes:

> Randy Yates wrote:
>
>> Let Z(t) be a white-noise (stationary) random process.
>> Can we conclude that Z(t) has zero mean?
>>
>> My thought is: yes. The intuitive reason that comes to
>> mind (and it may be wrong!) is this: If Z(t) has a
>> non-zero mean, then there would be some amount of
>> correlation between samples due to the means. Thus
>> the autocorrelation would be a delta function.
>
> Yes. Whit noise is necessarily zero-mean. This follows from the definition
> of white (i.e. covariance matrix is identity matrix).
>
> Let x be a random vector. Covariance matrix of a random vector is Cx =
> E{x*x'}. Mean of the random vector is m = E{x}. Let y be a white random
> vector with zero mean so that y = x + m;
>
> E{x*x'} = E{(y+m)*(y'+m')} = E{y*y'}+E{y*m'}+E{m*y'}+E{m*m'} = I + E{y}*m' +
> m*E{y'} + m*m' = I + m*m' != I => random vector is not white if the mean is
> not zero.

I constructed essentially the same proof myself (I didn't need and
didn't use vector random variables), but it doesn't feel "right."  It
seems like you should be able to begin with the whiteness property and
conclude the mean is zero.
--
%  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