Mean and Autocorrelation of a Deterministic Function

[OhMyMarkov]

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 ?Any help/clarification is appreciated.

 

[CaptainBlack]

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

 

[OhMyMarkov]

Hello CaptainBlack,

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

 

[CaptainBlack]

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