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

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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 ( , A ,P ) denote a probability space consisting of a sample space , a - algebra of events A , and a probability measure P that satisfies the Kolmogorov axioms. For example, 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 ( , A ,P ) by X T : T R n . Restated, X T ( ,t ) R n . (b) For a given value of = , X T ( ,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 = T = R , A be the Borel algebra (smallest - 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 e- x 2 2 2 . Define X T ( ,t ) = Y ( ) cos( t ) . The sample function for = is a cosine wave cos( t ) ,t T , multiplied by the scalar amplitude Y ( ) . For t = t , we have a random variable Y ( ) = 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 = ( , A ,P ) that is the mathematical model for what we might observe ( sample space ), 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...

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