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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 14 Today: (1) Linear Combinations of R.V.s (2) Decorrelation Transform • HW 6 is due Tue Oct 27. • You must: watch videos prior to Tuesday’s class . Two are up now; one to come tonight. 1 Linear Combinations of R.V.s Consider two random vectors: X = [ X 1 ,...,X m ] T Y = [ Y 1 ,...,Y n ] T Let each r.v. Y i be a linear combination of the random variables in vector X . Specifically, create an n × m matrix A of known real- valued constants: A = A 1 , 1 A 1 , 2 ··· A 1 ,m A 2 , 1 A 2 , 2 ··· A 2 ,m . . . . . . . . . . . . A n, 1 A n, 2 ··· A n,m Then the vector Y is given as the product of A and X : Y = A X (1) We can represent many types of systems as linear combinations. Just for some specific motivation, some examples: • Multiple antenna transceivers, such as 802.11n. The channel gain between each pair of antennas is represented as a matrix A . Then what is received is a linear combination of what is sent. Note that A in this case would be a complex matrix. • Secret key generation. In application assignment 4, you will come up with linear combinations in order to eliminate corre- lation between RSS samples. ECE 5510 Fall 2009 2 • Finance. A mutual fund or index is a linear combination of many different stocks or equities. A i,j is the quantity of stock j contained in mutual fund i . • Finite impulse response (FIR) filters, for example, for audio or image processing. Each value in matrix A would be a filter tap. Matrix A would have special structure: each row has identical values but delayed one column (shifted one element to the right). Let’s study what happens to the mean and covariance when we take a linear transformation. Mean of a Linear Combination The expected value is a linear operator. Thus the constant matrix A can be brought outside of the expected value. μ Y = E Y [ Y ] = E X [ A X ] = AE X [ X ] = Aμ X The result? Just apply the transform A to the vector of means of each component. Covariance of a Linear Combination Use the definition of covariance matrix to come up with the covariance of Y . C Y = E Y bracketleftbig ( Y − μ Y )( Y − μ Y ) T bracketrightbig = E X bracketleftbig ( A X − Aμ X )( A x − Aμ X ) T bracketrightbig Now, we can factor out A from each term inside the expected value. But note that ( CD ) T = D T C T (This is a linear algebra relationship you should know). C Y = E X bracketleftbig A ( X − μ X )( A ( x − μ X )) T bracketrightbig = E X bracketleftbig A ( X − μ X )( x − μ X ) T A T bracketrightbig Because the expected value is a linear operator, we again can bring the A and the A T outside of the expected value....
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This note was uploaded on 09/15/2011 for the course ECE 5510 taught by Professor Chen,r during the Fall '08 term at University of Utah.
- Fall '08