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lecture12 - ECE 5510 Random Processes Lecture Notes Fall...

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ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 12 Today: (1) Joint r.v. Expectation Review (2) Transformations of Joint r.v.s, Y&G 4.6 (3) Random Vectors (R.V.s), Y&G 5.2 HW 5 due Tue, Oct 20 at 5pm; Appl. Assignment 3 due same day (at midnight). I have OH today 1-3. By the end of the lecture today, we will have covered all of Chapter 4 except for 4.11 (Bivariate Gaussian) (and the Mat- lab section, which we won’t cover). Required: Watch three videos on youtube (16 min total) prior to Tue Oct 20 (Posted on web and WebCT.) We will spend at least 30 min in class starting HW 6 and answering other questions. 0.1 Expectation Review Short “quiz”. Given r.v.s. X 1 and X 2 , what is 1. What is Var [ X 1 + X 2 ]? 2. What is the definition of Cov ( X 1 , X 2 )? 3. What do we call two r.v.s with zero covariance? 4. What is the definition of correlation coefficient? Note: We often define several random variables to be independent, and to have identical distributions (CDF or pdf or pmf). We ab- breviate “i.i.d.” for “independent and identically distributed”. 1 Transformations of Joint r.v.s Random variables are often a function of multiple other random variables. The example the book uses is a good one, of a multiple antenna receiver. How do you choose from the antenna signals? 1. Just choose the best one: This uses the max( X 1 , X 2 ) function. 2. Add them together; X 1 + X 2 . ‘Combining’.
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ECE 5510 Fall 2009 2 Figure 1: A function Y of two random variables Y = g ( X 1 , X 2 ), might be viewed as a 3D map of what value Y takes for any given input coordinate ( X 1 , X 2 ), like this topology map of Black Moun- tain, Utah. Contour lines give Y = y , for many values of y , which is useful to find the pre-image. One pre-image of importance is the coordinates ( X 1 , X 2 ) for which Y y . 3. Add them in some ratio: X 1 1 + X 2 2 . ‘Maximal Ratio Combining’. We may have more than one output: we’ll have Y 1 = aX 1 + bX 2 and Y 2 = cX 1 + dX 2 , where a , b , c , d , are constants. If we choose wisely to match to the losses in the channel, we won’t loose any of the information that is contained in X 1 and X 2 . Ideas that exploit
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