Unformatted text preview: 31 Introduction To Random Processes I Thus for, the probability models and random variables We consider have been ﬁtted
and did not change with other variables, such as time. Let X (a) be the value of the
random variable for s E 5 (eg an event 5 in the state space S). This does not allow X
to be different at different times. a To allow for X (sl to be dilierent at different times, we extend the parameter set or
state space to include the time domain. Let X (s7 35) be the value of the random variable
for s E 5. Such a random variable is called a random process or a. stochastic process — Suppose we ﬁx 5 E S which is a value of interest. Then, X(s,t) is a function of
only time and details how X (5,15) and the state 5 Suppose we fix 7: E T which is a time period of interest. Then X (s t) is a function
of only the state and details hoW X (s. t) different states behave for a. ﬁxed time _),__ U.» kl": a, 5 1:1}: ( . Ti" .;
 X(s1t) for s 6 Sit E T for theleiitire State space of the random protess. Depending on the time domain there are different elassiﬁcetions 7’ 5 5
. 5 1‘ .. " a 
5 $55552": 5’ 3 “r If T is discrete, then X(s, t.) is said to be a discrete—time random process
— If T is eontinuous, then X{s1 1:) is said to be a continuous—time random process
“ If S is discrete1 then X15, :3) is said to be a discretestate random process
“7 If S is continuous, then X (s, t) is said to be a continuousstate random process
0 Example: suppose an electromagneticrsensor (EM). passively records and processes
local electromagnetic information which consists of two parts (i) the actual random signal X(t, s) and (ii) noise NOE, s) so that the observed signal RR, 5) is given by Rﬁﬁs) = Xﬁ, s) + NH, .3) o Read Sections 8.1, 8.2 71 ...
View
Full Document
 Fall '07
 Carlton

Click to edit the document details