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 "whitenoise" 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