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Time-Invariant Statistical Characterization

Suppose that we assign to each sample point s a function of time in accordance with the rule where IT is the total observation interval. For a fixed sample the graph of the function X(r, sz) versus time t is called a realization or sample function of the random process. To simplify the notation, we denote this sample function as x,(t) = X(f,s).

time invariant system
time invariant system

Stationary Processes events

From this figure, wc note that for a fixed time tk inside the observation interval, the set of numbers constitutes a random variable. Thus we have an indexed ensemble (family) of random variables, which is called a random process. To simplify the notation, the practice is to suppress the s and simply use X(t) ro denote a random process.

Defining Random Process: Functions, Probabilities, and Outcomes

We may now formally define a random process X(Z) as cut ensemble of time functions together with probability rule that assigns a probability to any meaningful event associated ivtth an observation of one of the sample functions of the random process.


Moreover, we may distinguish between a random variable and a random process as follows. For a random variable, the outcome of a random experiment is mapped into a number. For a random process, the outcome of a random experiment is mapped into a wave form that is a function of time.


Statiotuiry Processes

In dealing with random processes encountered in the real world, we often find that the statistical characterization of a process is independent of the time at which observation of the process is initiated. That is, if such a process is divided into a number of time intervals.


The various sections of the process exhibit essentially the same statistical properties. Such a process is said to be stationary. Otherwise, it is said to be nonstationan. Generally speaking, a stationary process arises from a stable physical phenomenon that has evolved into a steady-state mode of behavior, whereas a nonscationary process arises from an unstable phenomenon.


To be more precise, consider a random process X(t) that is initiated at t, Let {X(r), X(tk)} denote the random variables obtained by observing the random process X(t) at times respectively. The joint distribution function of this set of random variables.


Suppose next we shift all the observation times by a fixed amount t, thereby obtaining a new set of random variables X(r, 4- t), X(Z, + T),. . . , X(tk + t). The distribution funcnon of this latter set of random variables. The random process X(/) is said to be stationary the strict sense or strictly stationary if the following condition holds.


All possible choices of observation times. In other words, time invariant system X(t), initiated at time strictly stahonary if the joint distribution of any set of random variables obtained by observing the random process X. ls invariant with respect to the location of the origin.