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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 13 Today: Random Vectors: (1) Covariance, (2) Linear Transfor- mation; (3) Decorrelation Application Assignment 3 due at 5pm today. HW 6 due on Thursday at 5pm. Discussion Item? 1 Random Vectors 1.1 Covariance of a R.V. Defn: Covariance Matrix The covariance matrix of an n-length random vector X is an n n matrix C X with ( i,j )th element equal to Cov( X i ,X j ). In vector notation, C X = E X bracketleftbig ( X X )( X X ) T bracketrightbig Example: For X = [ X 1 X 2 X 3 ] T C X = Var X 1 [ X 1 ] Cov ( X 1 ,X 2 ) Cov ( X 1 ,X 3 ) Cov( X 2 ,X 1 ) Var X 2 [ X 2 ] Cov ( X 2 ,X 3 ) Cov( X 3 ,X 1 ) Cov ( X 3 ,X 2 ) Var X 3 [ X 3 ] You can see that for two r.v.s, well have just the first two rows and two columns of C X this is what we put on the board when we first talked about covariance as a matrix. Note for n = 1, C X = 2 X 1 . 1.2 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- ECE 5510 Fall 2008 2 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. 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). Lets 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. Y = E Y [ Y ] = E X [ A X ] = AE X [ X ] = A X The result? Just apply the transform A to the vector of means of each component....
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- Fall '08