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## Introduction to Communications Principles

by: Sarina Wintheiser

71

0

17

# Introduction to Communications Principles ECE 303

Marketplace > Colorado State University > ELECTRICAL AND COMPUTER ENGINEERING > ECE 303 > Introduction to Communications Principles
Sarina Wintheiser
CSU
GPA 3.8

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COURSE
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## Popular in ELECTRICAL AND COMPUTER ENGINEERING

This 17 page Class Notes was uploaded by Sarina Wintheiser on Tuesday September 22, 2015. The Class Notes belongs to ECE 303 at Colorado State University taught by Azimi-Sadjadi in Fall. Since its upload, it has received 71 views. For similar materials see /class/210275/ece-303-colorado-state-university in ELECTRICAL AND COMPUTER ENGINEERING at Colorado State University.

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Date Created: 09/22/15
Chapter 5 Operations on Multiple rv s Joint Moments Correlation and Covariance These describe the relationships between two or more rv s 1 Joint Moments about the Origin The expected value of product XiYj is the joint moment of Order i j ie my 2 EXin zjiyijYxaydXdy 00 Obviously mo 2 EXi and m0 EYj and m10 EX J uX and m01 EY l7 uY are the coordinates of the center of gravity of joint PDF Department of Electrical amp Computer Engineering 1 0 The 2nd order moment 1n11 EXY is called the correlation ofX and Y and rXY m11ElXYl fXY xaydXdy If X and Y are discrete rV s then rXY mll xm2Yzyn m n In general for discrete rV s we have my EXin 22x3me xWY yn m n Department of Electrical amp Computer Engineering Example 1 X and Y have joint PDF m 1 OSxS6andOSyS4 fXY xay 24 0 else Find m22EXY2 Solution EltXY2 HnyZfXYxydxdy 4 6 2 2 J J x y dxdy64 y0x0 24 Department of Electrical amp Computer Engineering Example 2 Experiment is tossing a fair die n Where two rV s X and Y are generated according to the following table l 2 3 4 5 6 X 2 l 0 l 2 3 4 l 0 l 4 9 Find correlation of X and Y This is a discrete case Thus 6 an EXY meymHX w ym mil 2 DWE 11 39 H Department of Electrical amp Computer Engineering Remarks 0 1 The above definition can be extended to any function gXY oftwo rV s X and Y withjoint PDF ie EgltXYgt HgXyfXYxydXdy 2 If X and Y are statistically independent fXY 952 fXxfY y and hence rXY E XY fX xfyydxdy 2 Tx fX xdx of y fy ydy Thus in this case VXY EXY EXEY X7 Department of Electrical amp Computer Engineering If this property is satis ed X and Y are said to be uncorrelated Note that Statistical Independence 9 Uncorrelated But Uncorrelated 71gt Statistical Independence not always only for Gaussian rV s Bivariate Gaussian PDF Jointly Gaussian PDF Two rV s X and Y are said to be jointly Gaussian if their Joint density function is of the form 1 1 x YY 2pltx Tolty Y y W X e fX Y y Z UXUYwl pz XPi21 P2i O39jzr GXOY 0 If p 0 corresponding to uncorrelated X and Y above equation can be written as f X Y x y f X x fY y Thus uncorrelated Gaussian rV s are also statistically Independent Department of Electrical amp Computer Engineering 3 If rXY0 e g when one or both rV s are zero mean then gm the two rV s are said to be orthogonal 4 For two orthogonal rV s X and Y the mean squared of the sum ZXY is and the mean is EZ EX EY gt Z 2X 5 Obviously rXY EXY EYX rYX ie order is not important Department of Electrical amp Computer Engineering 2 Joint Central Moments EM The joint central moments of order i j of two rV s X and Y are denoted by 5M EltX fe 17V j Jamiequ xydxdy For discrete rV s 5w ZZltXmYgtquotyn 7gtJ39PX xmYyn Note that 10 01 0 Department of Electrical amp Computer Engineering 0 Also 20 EX if a 02 EltY Io2 a The other 2nd order joint central moment which is very important is 11 This is called the Covariance ofX and Y CXY 11EX XXY YH Expanding the term in the bracket gives the following relation between covariance and correlation cXY EXY XY X7Xl7 EXY XEY E 7 97 cXY rXY XY Department of Electrical amp Computer Engineering Remarks WM 1 If X and Y are statistically independent or uncorrelated then rXY XI7IgtCXY 0 2 If either X or Y has zero mean rXY CXY 3 If X and Y are discrete rV s then their covariance is CXY 11EX XY Y 222xm Xyn xmYyn Department of Electrical amp Computer Engineering Normalized Moment and Correlation Coef cient gm The normalizeo 2nd order moment is defin d by i CH m cran 2 E m 0X I UY This is known as correlation coe icz ent ofX and Y It is easy to see that l 0 1 Thus p is a measure of dependence of two rV s with extreme cases that are p 0 if rV s are independent or uncorrelated p 1 if X Y Department of Electrical amp Computer Engineering Example 3 Two rv s X and Y have means 7 l and M 7 2 variances 039 4 and 039 l and a correlation coefficient DXY 04 New random variables W and V are de ned by V X 2Y W X 3Y Find a mean b variance c correlation and d correlation coefficient of V and W Solution a I7 EV E X 2Y 727 3 W EWEX3YX3Y7 Department of Electrical amp Computer Engineering b 75 EkV WJ W E X2Y 2Z TrickAlwayspair EXT2Y2 up rv with its mean EX Y24EY 72 4EX Y 2 2 20X 403 4cXY Bun pm CXY gtCXYO4X2O8 UXO39Y Thus a 44 4x0848 lt Similarly 0 EW W2 E X3Y 3Y2EHX 73YH a 90 6ch 2178 Department of Electrical amp Computer Engineering 00 c rVW EVW E X2YX 3Y EX2 EXY6EY2 a Y2 QY6U 72 ya 2 ch 37 J2 41 08265 2222 lt d cVW rVW VW UVUW UVUW 222 21 z 01298 148178 Department of Electrical amp Computer Engineering Example 4 Statistically independent rV s X and Y have WM moments m10 2 m20 14 mo2 12 and m11 6 Find the central moment 22 Solution m11 rXY EXY mlom01 3mm 3 mm 2 22 ZElX 2YY2J EltX mlogt2ltY mmgt2 EX m102EY m012 since independent 0326 520502 kiXZl Y21L5b l YZ quot 20 m120xm02 mdl 14 412 930 Department of Electrical amp Computer Engineering Example 5 Given the joint PDF WM nyYxy 046x 0t6y 2 036x 06y 2 016x a6y a 026x 16y 1 Find 0c which minimizes the correlation of X and Y Find this minimum Are X and Y 01thogona1 Solution 00 rXY EXY I I xy fXYltxygtdxdy 04jx5xadx jy5y 2dyo3 jx5x adx Iy y 2dy 01jx5x adx jy5y ady02jx5x 1dx Iy y ldy Now using the propeIty I f x6 x x0 dx f x0 we get 700 Department of Electrical amp Computer Engineering

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