lecture14 - ECE 5510: Random Processes Lecture Notes Fall...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 8

lecture14 - ECE 5510: Random Processes Lecture Notes Fall...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online