MHB Mean and Autocorrelation of a Deterministic Function

AI Thread Summary
The discussion centers on the definitions of mean and autocorrelation in the context of deterministic functions versus random processes. The mean of a process X(t) is defined as E[X(t)], which differs from the time average, particularly for deterministic functions. The autocorrelation function for a stochastic process is given by R_X(k) = E[X(t)X(t+k)], but the definition for deterministic functions requires careful consideration of how values outside the defined set are treated. The participants emphasize the distinction between local averages and global means, as well as the need for clarity in defining expectations for deterministic signals. Overall, the conversation seeks to clarify these concepts in relation to deterministic functions and their statistical properties.
OhMyMarkov
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Hello everyone!

I have a couple of questions related to random processes:

(1) Isn't the mean of a process $X(t)$ defined as $E[X(t)]$ which, for example, if $X(t)$ belongs of a countable and finite set, would defined as the some of the elements of this set weighted by their corresponding probability and divided by the cardinality of the set. For example:

$X(t) \in \{\sin(2\pi t), \sin(2\pi t + 2\pi/3), 8\sin(2\pi t -2\pi /3)\}$ each with probability 1/3, then the mean of $X(t)$ would be $(7/3) \cdot \sin(2*\pi t)$

This is how the mean is defined, and it is different than the "time" average of $X(t)$ whatever that is supposed to mean for a random process (I know what it means for a deterministic function).

(2) I've know before that the autocorrelation function of a stochastic process $X(t)$ that is stationary in the wide sense is $R_X (k) = E[X(t)X(t+k)]$. But what if the function is deterministic, how would the autocorrelation be defined?

I'm considering this example:

$X(t) = \sin(2\pi t)$ for $0<t<\pi /2$ with probability 1. Then, $R_X (k) = \sin(2\pi t)\cdot \sin(2\pi t + 2\pi k)$ which is not maximum at $k=0$ for an arbitrary time instant. Am I missing something here :confused:?Any help/clarification is appreciated.
 
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The expectation of a function is the integral of the function over a space with respect to a measure. In the case of a deterministic signal on \(0<t<\pi/2\):

\[E( f)=\int_{0}^{\pi/2} f(t) \frac{2}{\pi}dt\]

Which is of course the expectation of the RV \( f( T)\) where \( T\sim U(0,\pi/2)\).

With a correlation you need to be careful about how functional values for points outside the set on which the function is defined are handled (usually they are taken as zero )

CB
 
Hello CaptainBlack,

Would you mind giving me a Yes/No answer to questions (1) and (2)? I appreciate it.
 
OhMyMarkov said:
Hello CaptainBlack,

Would you mind giving me a Yes/No answer to questions (1) and (2)? I appreciate it.

1. You need to distinguish between the average at a point and the global mean that is between: \( E( X(t)) \) and \( E(X) \), where the first is still a function of \( t\) and the second is not.

2. Since the auto-correlation is an expectation I have already indicated how it is defined for a deterministic function.

CB
 

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