Is White Noise Necessarily Zero-Mean?

Hi Maurice,

I still don't understand... for example using your terminology above
lets assume we have a
WN with nonzero mean that is the result of a dc component added to a
zero-mean normal WN:

y= x + c

Where x *is* normal zero-mean WN. Then using the properties you
mentioned above we have (with E[x(t)]=0):

Ry(t1,t2)         =E[y(t1)*y(t2)]

                     =E[ (x(t1)+c)*(x(t2)+c) ]

	     =E[x(t1)*x(t2) + c*x(t1) + cx(t2) + c^2]

	     =E[x(t1)*x(t2)] + c^2

	     =E[x(t1)]E[x(t2)] + c^2 = c^2     for t1!=t2  and using 0 = E[XY]
- E[X]E[Y]

	     = Rx(0) + c^2                         for t1=t2

note that because x is WN zero mean then R(0)=var != 0 then (using
delta as the impulse function):

Ry(t1,t2)= var*delta(t1-t2) + c^2      %note that the process is still
WS stationary

But it's Ppwer spectrum is now

Sy(w)= var + c2*delta(w) !=  constant as a function of w

which is not flat anymore, thus non-white. Unless I am making a mistake
;)
it seems like we are missing something from the definition.

Randy Yates wrote:
> Let Z(t) be a white-noise (stationary) random process.
> Can we conclude that Z(t) has zero mean?

I haven't read comp.dsp for a couple of weeks now, and this thread has
gone
on for a while. I am sure somebody else already have come up with this,

but here are my 2c anyway:

There is no connection between the noise being Gaussian and it being
zero mean. Knowing that the noise is zero mean does not imply it
being Gaussian. Knowing that the noise is Gaussian does not
imply that it is 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.

We can separate the noise as

   Z(t) =3D z(t) + m_z

where z(t) is the zero mean stationary process and m_z is the
man of Z(t). The autocorrelation becomes

E[Z(t)^2] =3D E[(z(t)+m_z)^2] =3D E[=A8z(t)^2] + 2E[z(t)]m_z + m_z^2
   =3D E[z(t)^2] + m_z^2

since by assumption, E[z(t)]=3D=3D0.

So the only effect of adding a DC term to the zero mean
Gaussian process is that the autocorrelation is offset by a
constant value. It is no drama by that, but it does provide a
motivation why the autocovariance may be a preferred quantity
in the analysis of Gaussian processes.

Rune

Rune Allnor wrote:
> Randy Yates wrote:
> 
>>Let Z(t) be a white-noise (stationary) random process.
>>Can we conclude that Z(t) has zero mean?
> 
> 
> I haven't read comp.dsp for a couple of weeks now, and this thread has
> gone
> on for a while. I am sure somebody else already have come up with this,
> 
> but here are my 2c anyway:
> 
> There is no connection between the noise being Gaussian and it being
> zero mean. Knowing that the noise is zero mean does not imply it
> being Gaussian. Knowing that the noise is Gaussian does not
> imply that it is 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.
> 
> 
> We can separate the noise as
> 
>    Z(t) = z(t) + m_z
> 
> where z(t) is the zero mean stationary process and m_z is the
> man of Z(t). The autocorrelation becomes
> 
> E[Z(t)^2] = E[(z(t)+m_z)^2] = E[�z(t)^2] + 2E[z(t)]m_z + m_z^2
>    = E[z(t)^2] + m_z^2
> 
> since by assumption, E[z(t)]==0.
> 
> So the only effect of adding a DC term to the zero mean
> Gaussian process is that the autocorrelation is offset by a
> constant value. It is no drama by that, but it does provide a
> motivation why the autocovariance may be a preferred quantity
> in the analysis of Gaussian processes.

Presumably, an essential property of AWGN is equal power per unit 
bandwidth at all frequencies. In other words, every band (f + delta_f), 
delta_f being constant and f being arbitrary, has the same power. This, 
and the assumption that the power density is not infinite, implies that 
power ->zero as delta_f -> zero. I ask again, what is the bandwidth of 
DC? If zero, as I believe, we have a paradox.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Hi Jerry,

Even if the DC bandwidth is zero we don't have a paradox because
according to the post above:

Ry(t1,t2)= var*delta(t1-t2) + c^2
Sy(w)= var + c2*delta(w) !=  constant as a function of w

Now, from a theoretical point of view, the delta function has zero
bandwidth, but a finite area of one. So the DC component is *not* zero,
and in the case of non-zero mean, higher than the other avegare
spectrum.

It seems like the standard definitions of WN *should* require the
process with zero-mean,otherwise it's not white...

Ikaro wrote:
> Hi Jerry,
> 
> Even if the DC bandwidth is zero we don't have a paradox because
> according to the post above:
> 
> Ry(t1,t2)= var*delta(t1-t2) + c^2
> Sy(w)= var + c2*delta(w) !=  constant as a function of w
> 
> Now, from a theoretical point of view, the delta function has zero
> bandwidth, but a finite area of one. So the DC component is *not* zero,
> and in the case of non-zero mean, higher than the other avegare
> spectrum.
> 
> It seems like the standard definitions of WN *should* require the
> process with zero-mean,otherwise it's not white...

There's the paradox.
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

"Ikaro" <ikarosilva@hotmail.com> wrote in message 
news:1140536417.286233.272270@g44g2000cwa.googlegroups.com...
>
> It seems like the standard definitions of WN *should* require the
> process with zero-mean,otherwise it's not white...
>

This thread seems to be converging to agreement with Randy's
assertion that the common definitions of white noise seem not
to necessarily imply that the process has zero mean.  Thus, it
is necessary to explicitly state that the mean of a white noise
process is zero.  Indeed, according to Randy, the definitions
of white noise that have given by Brown and Papoulis are not
complete, and should also include a requirement that the mean
be zero.

So, let us consider a process X(t) with nonzero mean m and
autocorrelation function Rxx(t).  Let us also suppose that
Rxx(t) does not depend on the value of m in any way; in fact,
let's take Rxx(t) = N0 delta(t) so that X(t) is a white noise
process with nonzero mean as per the definition that
everybody seems to agree on.

Let Z(t) = X(t) - m.  Then, I think we are all in agreement that
Z(t) is a zero-mean process.  Is it a white noise process?  Well,
Rzz(t) =  Rxx(t) - m^2 = N0 delta(t) - m^2.  So, it doesn't seem
to fit anybody's definition of a white noise process since ....Bzzzt...
it doesn't satisfy the requirement that Z(t1) and Z(t2) are
uncorrelated if t1 and t2 are distinct time instants (Papoulis), and
since the power spectral density is phizz(f) = N0 - m^2 delta(f)
...Bzzzt... the process Z(t) doesn't satisfy the requirement of
constant spectral density (Brown).  So, Z(t) and X(t) *both*
cannot be called white noise processes.

In fact, getting back to X(t), its autocovariance function Cxx(t)
is the same as Rzz(t) and thus the *autocovariance* function of
the process X(t) does seem to depend on the mean m.  Somehow
this does not seem quite right; covariances should not depend on
the mean(s) since we are subtracting them off.... Perhaps someone
(Randy?) would care to shed further light on this curious result....

Hope this confuses (it certainly is not meant to help; it is
intended to throw fuel on the flames!  :-)  )

Hi,

>This thread seems to be converging to agreement with Randy's
>assertion that the common definitions of white noise seem not
>to necessarily imply that the process has zero mean.

Actually I  (as well as dvsarw) was saying just the opposite. If the
process is WN
than it *must* be zero mean. There can't be a non-zero mean WN process.

>So, let us consider a process X(t) with nonzero mean m and
>autocorrelation function Rxx(t).  Let us also suppose that
>Rxx(t) does not depend on the value of m in any way; in fact,
>let's take Rxx(t) = N0 delta(t)

That's not possible, a constant mean will result in a constant
term (m^2) in the correlation matrix (see several posts above).


-Ikaro

Ikaro wrote:
> Hi Maurice,
>
> I still don't understand... for example using your terminology above
> lets assume we have a
> WN with nonzero mean that is the result of a dc component added to a
> zero-mean normal WN:
>
> y= x + c
>
> Where x *is* normal zero-mean WN. Then using the properties you
> mentioned above we have (with E[x(t)]=0):
>
> Ry(t1,t2)         =E[y(t1)*y(t2)]
>
>                      =E[ (x(t1)+c)*(x(t2)+c) ]
>
> 	     =E[x(t1)*x(t2) + c*x(t1) + cx(t2) + c^2]
>
> 	     =E[x(t1)*x(t2)] + c^2
>
> 	     =E[x(t1)]E[x(t2)] + c^2 = c^2     for t1!=t2  and using 0 = E[XY]
> - E[X]E[Y]
>
> 	     = Rx(0) + c^2                         for t1=t2
>
> note that because x is WN zero mean then R(0)=var != 0 then (using
> delta as the impulse function):
>
> Ry(t1,t2)= var*delta(t1-t2) + c^2      %note that the process is still
> WS stationary
>
> But it's Ppwer spectrum is now
>
> Sy(w)= var + c2*delta(w) !=  constant as a function of w
>
> which is not flat anymore, thus non-white. Unless I am making a mistake
> ;)
> it seems like we are missing something from the definition.


Well Ikaro, you actually proved my case.  The power spectrum density
describes the variance of a process with respect to frequency.

delta(t - t0) = 0 for t != t0.  This is part of the definition of
delta.

Therefore, delta(w) = 0 for w != 0.  This means

Sy(w)= var + c2*delta(w) = var, for w != 0,     a constant for all w !=
0

Dilip V. Sarwate wrote:
> "Ikaro" <ikarosilva@hotmail.com> wrote in message
> news:1140536417.286233.272270@g44g2000cwa.googlegroups.com...
> >
> > It seems like the standard definitions of WN *should* require the
> > process with zero-mean,otherwise it's not white...
> >
>
> This thread seems to be converging to agreement with Randy's
> assertion that the common definitions of white noise seem not
> to necessarily imply that the process has zero mean.  Thus, it
> is necessary to explicitly state that the mean of a white noise
> process is zero.  Indeed, according to Randy, the definitions
> of white noise that have given by Brown and Papoulis are not
> complete, and should also include a requirement that the mean
> be zero.
>
> So, let us consider a process X(t) with nonzero mean m and
> autocorrelation function Rxx(t).  Let us also suppose that
> Rxx(t) does not depend on the value of m in any way; in fact,
> let's take Rxx(t) = N0 delta(t) so that X(t) is a white noise
> process with nonzero mean as per the definition that
> everybody seems to agree on.
>
> Let Z(t) = X(t) - m.  Then, I think we are all in agreement that
> Z(t) is a zero-mean process.  Is it a white noise process?  Well,
> Rzz(t) =  Rxx(t) - m^2 = N0 delta(t) - m^2.  So, it doesn't seem
> to fit anybody's definition of a white noise process since ....Bzzzt...
> it doesn't satisfy the requirement that Z(t1) and Z(t2) are
> uncorrelated if t1 and t2 are distinct time instants (Papoulis), and
> since the power spectral density is phizz(f) = N0 - m^2 delta(f)
> ...Bzzzt... the process Z(t) doesn't satisfy the requirement of
> constant spectral density (Brown).  So, Z(t) and X(t) *both*
> cannot be called white noise processes.
>
> In fact, getting back to X(t), its autocovariance function Cxx(t)
> is the same as Rzz(t) and thus the *autocovariance* function of
> the process X(t) does seem to depend on the mean m.  Somehow
> this does not seem quite right; covariances should not depend on
> the mean(s) since we are subtracting them off.... Perhaps someone
> (Randy?) would care to shed further light on this curious result....
>
> Hope this confuses (it certainly is not meant to help; it is
> intended to throw fuel on the flames!  :-)  )

Eh... again, I have not read very many posts in this thread so I
may be stating opinions that have already been discused.

Randy asked about the connection between the mean and
and correlation properties of a process. I tried to argue that
the mean has an insignificant (albeit not vanishing) effect
on the autocorrelation and thus can be disregarded, while
you point out a paradox. Jerry and Ikaro discuss various
technical details.

>From the above discussion I tend to adopt the view that "white
noise" baiscally is a white elephant. It is not well-defined
as f -> infinite, since this violates the requirement that the
signal has finite energy, and thus can not be analyzed in
a meaningful way by the Fourier transform. It follows that
we can basically do what we like at DC, since we are
trying to fit a square peg to a round hole anyway.

I think the view have been aired previously on comp.dsp that
"white noise" means "noise that is white within the relevant
bandwidth." I think that seems reasonable. If one agrees
with that, the decision what to do basically depends on
whether DC is "relevant" or not. If it is, use the covariance
instead of the autocorrelation, and all is well. For a textbook
author, either consitently use the autocovariance, or
discuss "autocorrelation of zero-mean signals".

I prefer the former.

Rune