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

 

[blue_raver22]

Hello there,

I would like to provide some clarification on the concepts of mean and autocorrelation in relation to deterministic functions.

Firstly, you are correct in your understanding of the mean of a process $X(t)$, which is defined as the expected value $E[X(t)]$ of the process. This means that for a finite set of values, the mean would be calculated by taking the sum of the values multiplied by their corresponding probabilities and divided by the total number of values. However, for a continuous process, the mean would be calculated by integrating over all possible values of $X(t)$.

Regarding your second question, the autocorrelation function is a measure of the correlation between a process and a delayed version of itself. In the case of a deterministic function, the autocorrelation function would still be defined as $R_X (k) = E[X(t)X(t+k)]$, but the result may not be maximum at $k=0$ as the function is not stochastic and does not exhibit random behavior. In your example, $X(t)$ is a deterministic function and therefore, its autocorrelation function would also be a deterministic function, which may not be maximum at $k=0$.

I hope this clarifies your doubts. If you have any further questions, please feel free to ask. Thank you.