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Simple Question on Random Variables and Random Processes

Started by Randy Yates February 14, 2006
Randy Yates <yates@ieee.org> writes:

> > I agree. Did I write or state something that required a zero mean? >
Just the auto-correlation expression. -- "And he sees the vision splendid of the sunlit plains extended And at night the wondrous glory of the everlasting stars."
p.kootsookos@remove.ieee.org (Peter K.) writes:

> Randy Yates <yates@ieee.org> writes: > >> >> I agree. Did I write or state something that required a zero mean? >> > > Just the auto-correlation expression.
I'm confused. The definition I've seen of correlation of two WSS random processes W and Z (not necessarily real) is Rwz(tau) = E[W(t)Z'(t+tau)]. That definition holds whether the mean is zero or not, right? What did I miss? -- % Randy Yates % "The dreamer, the unwoken fool - %% Fuquay-Varina, NC % in dreams, no pain will kiss the brow..." %%% 919-577-9882 % %%%% <yates@ieee.org> % 'Eldorado Overture', *Eldorado*, ELO http://home.earthlink.net/~yatescr
Randy Yates <yates@ieee.org> writes:

> p.kootsookos@remove.ieee.org (Peter K.) writes: > > > Randy Yates <yates@ieee.org> writes: > > > >> > >> I agree. Did I write or state something that required a zero mean? > >> > > > > Just the auto-correlation expression. > > I'm confused. The definition I've seen of correlation of two > WSS random processes W and Z (not necessarily real) is > > Rwz(tau) = E[W(t)Z'(t+tau)]. > > That definition holds whether the mean is zero or not, right?
True.
> What did I miss?
You gave the autocorrlation as a constant times a delta function. If the process had been non-zero mean, it would have been something else. I'm too used to seeing autocorrelations of finite duration data, where if there is a DC offset (non-zero mean), then the (unbiassed) autocorrelation estimate comes out as something overlaid on a triangle. In the case you're looking at, you'd get a constant offset (I think). Ciao, Peter K. -- "And he sees the vision splendid of the sunlit plains extended And at night the wondrous glory of the everlasting stars."
Hey Peter - sorry to take so long to respond - I just noticed this.

p.kootsookos@remove.ieee.org (Peter K.) writes:

> Randy Yates <yates@ieee.org> writes: > >> p.kootsookos@remove.ieee.org (Peter K.) writes: >> >> > Randy Yates <yates@ieee.org> writes: >> > >> >> >> >> I agree. Did I write or state something that required a zero mean? >> >> >> > >> > Just the auto-correlation expression. >> >> I'm confused. The definition I've seen of correlation of two >> WSS random processes W and Z (not necessarily real) is >> >> Rwz(tau) = E[W(t)Z'(t+tau)]. >> >> That definition holds whether the mean is zero or not, right? > > True. > >> What did I miss? > > You gave the autocorrlation as a constant times a delta function. If > the process had been non-zero mean, it would have been something else.
Why? As I wrote in the paper on-line, the autocorrelation of a real process X(t) is Rxx(tau) = E[X(t)X(t+tau)]. If X(t) is not uncorrelated, then we don't get to separate this into Rxx(tau) = E[X(t)] * E[X(t+tau)], and so we really can't make any conclusions regarding the mean of that process. We only know (assuming Brown's definition of "white") that E[X(t))X(t+tau)] = 0, and as far as I know, it is possible that this can hold true even when the mean of X(t) is non-zero. -- % Randy Yates % "Though you ride on the wheels of tomorrow, %% Fuquay-Varina, NC % you still wander the fields of your %%% 919-577-9882 % sorrow." %%%% <yates@ieee.org> % '21st Century Man', *Time*, ELO http://home.earthlink.net/~yatescr
Randy Yates <yates@ieee.org> writes:

> Hey Peter - sorry to take so long to respond - I just noticed this.
NO worries. I'm still getting my head around a response to you and Jerry on this thread.
> p.kootsookos@remove.ieee.org (Peter K.) writes: > > > > > You gave the autocorrlation as a constant times a delta function. If > > the process had been non-zero mean, it would have been something else. > > Why? > > As I wrote in the paper on-line, the autocorrelation of a real process > X(t) is > > Rxx(tau) = E[X(t)X(t+tau)]. > > If X(t) is not uncorrelated, then we don't get to separate this > into > > Rxx(tau) = E[X(t)] * E[X(t+tau)], > > and so we really can't make any conclusions regarding the mean of > that process. We only know (assuming Brown's definition of "white") > that E[X(t))X(t+tau)] = 0, and as far as I know, it is possible > that this can hold true even when the mean of X(t) is non-zero.
Not as far as I can tell: X(t) = M + XX(t) where XX(t) is zero mean. E[X(t) X(t+tau)] = E[(M + XX(t))(M + XX(t+tau))] = M^2 + E[M XX(t)] + E[M + XX(t+tau)] + E[XX(t) XX(t+tau)] = M^2 + E[XX(t) + XX(t+tau)] So if we are given: E[X(t) X(t+tau)] = A.delta(t) doesn't that mean M = 0?? Ciao, Peter K. -- "And he sees the vision splendid of the sunlit plains extended And at night the wondrous glory of the everlasting stars."
Hi Peter,

Responses below.

p.kootsookos@remove.ieee.org (Peter K.) writes:

> Randy Yates <yates@ieee.org> writes: > >> Hey Peter - sorry to take so long to respond - I just noticed this. > > NO worries. I'm still getting my head around a response to you and > Jerry on this thread. > >> p.kootsookos@remove.ieee.org (Peter K.) writes: >> >> > >> > You gave the autocorrlation as a constant times a delta function. If >> > the process had been non-zero mean, it would have been something else. >> >> Why? >> >> As I wrote in the paper on-line, the autocorrelation of a real process >> X(t) is >> >> Rxx(tau) = E[X(t)X(t+tau)]. >> >> If X(t) is not uncorrelated, then we don't get to separate this >> into >> >> Rxx(tau) = E[X(t)] * E[X(t+tau)], >> >> and so we really can't make any conclusions regarding the mean of >> that process. We only know (assuming Brown's definition of "white") >> that E[X(t))X(t+tau)] = 0, and as far as I know, it is possible >> that this can hold true even when the mean of X(t) is non-zero. > > Not as far as I can tell: > > X(t) = M + XX(t) where XX(t) is zero mean. > > E[X(t) X(t+tau)] = E[(M + XX(t))(M + XX(t+tau))] > = M^2 + E[M XX(t)] + E[M + XX(t+tau)] + E[XX(t) XX(t+tau)] > = M^2 + E[XX(t) + XX(t+tau)]
Did you mean = M^2 + E[XX(t) * XX(t+tau)]? I'll proceed as if you did.
> So if we are given: > > E[X(t) X(t+tau)] = A.delta(t)
You mean A delta(tau), right? I'll proceed as if you did.
> doesn't that mean M = 0??
It seems to me that it means E[XX(t) * XX(t+tau)] = -M^2 (tau != 0) Why couldn't that happen? In fact, that makes the point of my paper nicely. If we are ONLY given that Rxx(tau) = A delta(tau), then we can't conclude anything about the mean. However, if we, in addition, are given that the process is uncorrelated sample to sample, then E[XX(t) * XX(t+tau)] = E[XX(t)] * E[XX(t+tau)] = M^2 (since XX is stationary), from which we may write M^2 = -M^2 which THEN implies M = 0. -- % 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:

> Hi Peter, > > Responses below. > > Did you mean > > = M^2 + E[XX(t) * XX(t+tau)]? > > I'll proceed as if you did.
Yup.
> > So if we are given: > > > > E[X(t) X(t+tau)] = A.delta(t) > > You mean A delta(tau), right? I'll proceed as if you did.
Yup.
> > doesn't that mean M = 0?? > > It seems to me that it means > > E[XX(t) * XX(t+tau)] = -M^2 (tau != 0) > > Why couldn't that happen?
It could, I suppose, if you we'ren't given that XX(t) was IID (or uncorrelated, as you say below).
> In fact, that makes the point of my paper nicely. If we are ONLY given > that Rxx(tau) = A delta(tau), then we can't conclude anything about > the mean. However, if we, in addition, are given that the process is > uncorrelated sample to sample, then > > E[XX(t) * XX(t+tau)] = E[XX(t)] * E[XX(t+tau)] > = M^2 (since XX is stationary), > > from which we may write > > M^2 = -M^2 > > which THEN implies M = 0.
OK, thanks! Ciao, Peter K. -- "And he sees the vision splendid of the sunlit plains extended And at night the wondrous glory of the everlasting stars."