Hi Carlos,
Carlos Moreno wrote:
>
> Hi Randy,
>
> Randy Yates wrote:
>
>>
>> Carlos,
>>
>> Although I haven't read all of the dozens of
>> articles in this thread, would the following
>> describe the "question in your head"? ...:
>>
>> Let X(t) be a zero-mean, IID, continuous-time process
>> with variance of sigma^2. Now we know that since this
>> function is IID, it has a white PSD and therefore its
>> autocorrelation function at lag 0, R_XX(0), should be
>> b*delta(tau), where delta(tau) is the usual Dirac delta
>> function and b is some constant.
>>
>> However, we also know that the autocorrelation function R_XX(tau)
>> is defined to be E[X(t)*X(t+tau)], and therefore that R_XX(0)
>> is E[X^2] (where we have dropped the dependence on t since the
>> process is IID). Therefore in this sense R_XX(0) = E[X^2] = sigma^2,
>> which is not infinite.
>>
>> Is this the contradiction you speak of?
>
>
> Maybe. As a matter of fact, this may be at the heart of
> what I perceive as a contradiction. What surprises me is
> that you show the two "contradictory" derivations, but
> don't mention why the contradiction, or if it is indeed
> a contradiction (what I mean is that it might be an
> *apparent* contradiction).
It is indeed a contradiction to me, and I cannot resolve
the contradiction. It is one I have had for years.
> In fact, now that you "equate" the results of those two
> different approaches, I guess one problem (maybe part of
> the same problem?) is that I'm not sure I understand the
> justification of defining the PSD and Rxx as a Fourier
> transform pair.
The justification of the definition? Are you asking how
Wiener and Khinchine proved the theorem that the PSD
is the Fourier transform of the autocorrelation function?
That I cannot say - it is an interesting question, though.
> I mean, I remember from my Signals and
> Systems course, understanding the intuitive interpretation
> of it: the more high-frequency contents, the less correlated
> close samples would be -- in particular, for white noise,
> with strong frequency contents going to infinity, samples
> arbitrarily close are still uncorrelated. That made (and
> still makes) perfect sense to me.
>
> Fine. So, why the contradiction? (or the "apparent"
> contradiction of results?).
Because Rxx(tau) != F^(-1)[Sxx(w)], and they should be.
> In fact, how would Rxx(0) be related to the average
> electrical power of the signal? (the average "watts"
> that a voltage signal would dissipate on a 1-ohm resistor).
> I seem to see clearly how the E{x^2} is related to the
> average power of a voltage signal, but not so sure that
> I understand the link with Rxx.
Rxx(0) = E[X^2(t)], by definition. Remember, Rxx(tau) is
defined to be E[X(t)*X(t+tau)], so when tau = 0, this
yields E[X^2(t)].
>> PS: I'm a little put off that you haven't responded to my post on
>> the units of PSD. Did you see it?
>
>
> Yes!! And I apologize for letting it go unnoticed!!
> The thing is that I read it together with the other
> messages that finally made me decide that I was starting
> to sound like a troll, and thus decided to witdraw...
> My sincere apologies!
Acknowledged. Thanks, Carlos.
> Ironically, that was *the* message that best seemed to
> address my initial point (I mean, it was the one message
> that did not give me that feeling of "this guy didn't
> understand what I was asking" -- not that I'm saying
> that the others didn't understand; but almost all the
> other messages (the initial ones, at least) gave me, at
> some extent, the impression that they were going off a
> tangent).
>
> When you showed me that the units of the PSD are indeed
> watts per herz, that seemed like a very precise attempt
> at convincing me of what the PSD is... (still, after
> reading that message, I could not seem ro reconcile
> ideas that seemed contradictory, and then again, that
> contributed to my decision that "maybe I should go
> re-study these concepts before I continue to bother
> these guys" :-))
>
> So, I'm hoping that you won't be too mad at me and will
> be willing to elaborate a bit on this contradiction of
> Rxx(0) being infinite while E{x^2} being finite.
Carlos, I am indeed not mad with you. Thanks for asking
these questions - it helps remind me and others of defintions
and concepts that tend to fade out unless you revisit them
often.
Regarding the contradiction, I hope my responses to you previously
in this post have clarified.
PS: Go to sci.math, where I posed this very question (in fact,
I had cut and paste it here in my last message to you). As of
a few minutes ago, I hadn't yet gotten any responses.
--
% Randy Yates % "...the answer lies within your soul
%% Fuquay-Varina, NC % 'cause no one knows which side
%%% 919-577-9882 % the coin will fall."
%%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO
http://home.earthlink.net/~yatescr