Lect 2

# Lect 2 - ECE4110 Lectures 1&3 Sep Fall 09 Cornell...

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Unformatted text preview: ECE4110 Lectures: 1&3 Sep Fall 09 Cornell University T.L. Fine A second definition of a random process is perhaps more constructive in that it brings out a random process as a randomly selected function of its index parameter (e.g., time, space). Definition 1 (Probability Space) . Let (Ω , A ,P ) denote a probability space consisting of a sample space Ω , a σ- algebra of events A , and a probability measure P that satisfies the Kolmogorov axioms. For example, Ω T might be the set of all R n-valued functions of argument t ∈ T . Definition 2 (Index Set) . Let T = { t } denote a set that will be referred to as an index set or time set for a random process. The index set may include spatial and other variables and need not be restricted to physical time. Definition 3 (Random/Stochastic Process II) . (a) A random process X T is a (possibly vector-valued) function defined on a given index set T and probability space (Ω , A ,P ) by X T : Ω × T → R n . Restated, X T ( ω,t ) ∈ R n . (b) For a given value of ω = ω ∈ Ω , X T ( ω ,t ) is a particular function of “time” t and it is called a sample function . (c) For a given value of t = t ∈ T , X t ( · ) = X T ( t , · ) is a random variable defined on the probability space and taking values in R n . Often, n = 1 . Example 1. Let Ω = T = R , A be the Borel algebra (smallest σ- algebra containing the finite intervals). Let the random variable Y be described by the N (0 , 1) (standard) normal probability law with proba- bility density function (pdf) f Y ( x ) = 1 √ 2 π e- x 2 2 σ 2 . Define X T ( ω,t ) = Y ( ω ) cos( t ) . The sample function for ω = ω is a cosine wave cos( t ) ,t ∈ T , multiplied by the scalar amplitude Y ( ω ) . For t = t , we have a random variable Y ( ω ) = Y that is multiplied by a scalar to yield cos( t ) Y . 1 2 1. Basic Probability Review • All references are to the class text. • Section 1.3 provides examples of random phenomena of interest to ECE and the reason why we study probability theory and its specialization to the theory of random processes. • Section 1.4 discusses the concepts of a random experiment E = (Ω , A ,P ) that is the mathematical model for what we might observe ( sample space Ω), what we are interested in ( events modeled as sets that lie in the event algebra A ), and what we know about what we might observe (their probability P ). • If you need to review sets and notation, see Appendix 1 to Chapter 1. • The mathematical theory of probability begins in Section 3.5. In 4110 we will not dwell upon the event algebra A and just assume that events that we are interested in are also events for which their probability is defined....
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Lect 2 - ECE4110 Lectures 1&3 Sep Fall 09 Cornell...

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