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Pg 71 - 31 Introduction To Random Processes I Thus for the...

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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 discrete-state random process “7 If S is continuous, then X (s, t) is said to be a continuous-state 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 ...
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