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Introduction to Communications Principles

by: Sarina Wintheiser

Introduction to Communications Principles ECE 303

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


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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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