RANDOM SIGNALS Random Variables A random variable x is a mapping that assigns a real number x to every outcome from an abstract probability space The mapping should satisfy the following two conditions the interval x x is an event in the abstract probabilty space for every x Pr x 1 and Pr x 0 Cumulative distribution function cdf of a random variable x Fx x Pr x x Probability density function pdf fx x Then Z dFx x dx x Fx x fx x dx EE 524 Fall 2004 7 1 Since Fx 1 we have normalization condition Z fx x dx 1 Several important properties 0 Fx x 1 Fx 0 Fx 1 Z fx x 0 fx x dx 1 Simple interpretation Pr x 2 x x 2 0 fx x lim EE 524 Fall 2004 7 2 Expectation of an arbitrary function g x Z E g x g x fx x dx Mean Z x E x xfx x dx Variance of a real random variable x var x x2 E x E x 2 E x2 2xE x E x 2 E x2 E x 2 E x2 2x Complex random variables A complex random variable x xR jxI Although the definition of the mean remains unchanged the definition of variance changes for complex x var x x2 E x E x 2 E x 2 xE x x E x E x 2 E x 2 E x 2 E x 2 x 2 EE 524 Fall 2004 7 3 Random Vectors A real valued vector containing N random variables x1 x2 x xN is called a random N vector or a random vector when dimensionality is unimportant A real valued random vector mapping from an abstract probability space to a vectorvalued real space RN A random vector is completely characterized by its joint cumulative distribution function which is defined by 4 Fx x1 x2 xN P x1 x1 xN xN and is often written as Fx x P x x A random vector can also be characterized by its joint EE 524 Fall 2004 7 4 probability density function pdf defined as follows fx x lim x1 0 x2 0 xN 0 P x1 x1 x1 x1 xN xN xN xN x1 xN Fx x x1 xN The function Z Z fxi xi fx x dx1 dxi 1 dxi 1 dxN N 1 is known as marginal pdf and describes individual random variables The cdf of x can be computed from the joint pdf as Z x1 Z xN EE 524 Fall 2004 7 4 fx v dv1dv2 dvN Fx x Z x fx v dv 5 Complex random vectors xR 1 xI 1 xR 2 xI 2 x xR jxI j xR N xI N Complex random vector mapping from an abstract probability space to a vector valued complex space C N The cdf of a complex valued random vector x is defined as Fx x 4 4 P x x P xR xR xI xI and its joint pdf is defined as fx x lim xR 1 0 xI 1 0 xR N 0 xI N 0 P xR xR xR xR xI xI xI xI x1 xN Fx x xR 1 xI 1 xR N xI N EE 524 Fall 2004 7 6 The cdf of x can be computed from the joint pdf as Z Fx x xR 1 xI N fx v dvR 1dvI 1 dvR N dvI N 4 Z Z xN fx v dv where the single integral in the last expression is used as a compact notation for a multidimensional integral and should not be confused with a complex contour integral Note that Z x e Fx x Z x fx e x de x fx x dx e xTR xTI T where x For two random variables x x y T fx x fx y x y x and y are independent if fx y x y fx x fy y E xy E x E y Expectation of a function g x Z E g x g x fx x dx EE 524 Fall 2004 7 7 For two random variables x x y T Z g x y fx y x y dxdy E g x y Correlation Real correlation Z Z rx y E xy xyfx y x y dxdy Real covariance rx y E x x y y Z Z x x y y fx y x y dxdy Complex correlation rx y E xy Z Z xy fx y x y dxdy Complex covariance rx y E x x y y Z Z x x y y fx y x y dxdy EE 524 Fall 2004 7 8 Covariance Matrix Mean vector x E x Real covariance matrix Rx E x E x x E x T E xxT E x E x T Rx E xxT if E x 0 Complex covariance matrix Rx E x E x x E x H E xxH E x E x H Rx E xxH if E x 0 Observe the following property of complex correlation ri k E xix k E xk x i rk i EE 524 Fall 2004 7 9 Then for E x 0 Rx r1 1 r1 2 r2 1 r2 2 H E xx rN 1 rN 2 r1 1 r1 2 r1 2 r2 2 r1 N r2 N r1 N r2 N rN N r1 N r2 N rN N The covariance matrix is Hermitian It is positive semidefinite because bH Rxb bH E x E x x E x H b z z bH E zz H b E bH z 2 0 Linear Transformation of Random Vectors Linear Transformation y g x Ax Mean Vector y E Ax A x EE 524 Fall 2004 7 10 Covariance Matrix Ry E yy H y H y H E AxxH AH A x H xA H A E xxH x H x A ARxxAH EE 524 Fall 2004 7 11 Gaussian Random Vectors Gaussian random variables fx x fx x x x 2 o exp for real x 2 2 x x 2 n x 2 o 1 x exp for complex x 2 2 x x 1 n Real Gaussian random vectors n o 1 T 1 1 fx x exp x Rx x x x 2 2 N 2 Rx 1 2 Complex Gaussian random vectors n o 1 fx x N exp x x H Rx 1 x x Rx Symbolic notation for real and complex Gaussian random vectors EE 524 Fall 2004 7 x Nr x Rx real x Nc x Rx complex 12 A linear transformation of Gaussian vector is also Gaussian i e if y Ax then y Nr A x ARxAT real y Nc A x ARxAH complex EE 524 Fall 2004 7 13 Complex Gaussian Distribution Consider joint pdf of real and imaginary part of a complex vector x x u jv Assume z uT v T T The 2n variate Gaussian pdf of the real vector z is 1 fz z p 2 2n Rz 1 T 1 exp 2 z z Rz …