RANDOM SIGNALSRandom VariablesA random variable x(ξ) is a mapping that assigns a real numberx to every outcome ξ from an abstract probability space. Themapping should satisfy the following two c onditions:• the interval {x(ξ) ≤ x} is an event in the abstract probabiltyspace for every x;• Pr[x(ξ) < ∞] = 1 and Pr[x(ξ) = −∞] = 0.Cumulative distribution function (cdf) of a random variablex(ξ):Fx(x) = Pr{x(ξ) ≤ x}.Probability density function (pdf):fx(x) =dFx(x)dx.ThenFx(x) =Zx−∞fx(x) dx.EE 524, Fall 2004, # 7 1Since Fx(∞) = 1, we have normalization condition:Z∞−∞fx(x)dx = 1.Several important properties:0 ≤ Fx(x) ≤ 1, Fx(−∞) = 0, Fx(∞) = 1,fx(x) ≥ 0,Z∞−∞fx(x) dx = 1.Simple interpretation:fx(x) = lim∆→0Pr{x − ∆/2 ≤ x(ξ) ≤ x + ∆/2}∆.EE 524, Fall 2004, # 7 2Expectation of an arbitrary function g(x(ξ)):E {g(x(ξ))} =Z∞−∞g(x)fx(x) dx.Mean:µx= E {x(ξ)} =Z∞−∞xfx(x) dx.Variance of a real random variable x(ξ):var{x} = σ2x= E {(x − E {x})2}= E {x2− 2xE {x} + E {x}2}= E {x2} − (E {x})2= E {x2} − µ2x.Complex random variables: A complex random variablex(ξ) = xR(ξ) + jxI(ξ).Although the definition of t he mean remains unchanged, thedefinition of variance changes for complex x(ξ):var{x} = σ2x= 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} − |µx|2.EE 524, Fall 2004, # 7 3Random VectorsA real-valued vector containing N random variablesx(ξ) =x1(ξ)x2(ξ)...xN(ξ)is called a random N vector or a random vector whendimensionality is unimportant. A real-valued random vector≡ mapping from an abstract probability space to a vector-valued real space RN.A random vector is completely characterized by its jointcumulative distribution function, which is defined byFx(x1, x2, . . . , xN)4= P [{x1(ξ) ≤ x1}∩. . . ∩{xN(ξ) ≤ xN}]and is often written asFx(x) = P [x(ξ) ≤ x].A random vector can also be characterized by its jointEE 524, Fall 2004, # 7 4probability density function (pdf), defined as follows:fx(x) = lim∆x1→ 0∆x2→ 0...∆xN→ 0P [{x1< x1(ξ) ≤ x1+ ∆x1} ∩ . . . ∩ {xN< xN(ξ) ≤ xN+ ∆xN}]∆x1···∆xN=∂∂x1···∂∂xNFx(x).The functionfxi(xi) =Z···Z(N−1)fx(x) dx1···dxi−1dxi+1···dxNis known as marginal pdf and describes individual randomvariables.The cdf of x can be computed from the joint pdf as:Fx(x) =Zx1−∞···ZxN−∞fx(v) dv1dv2···dvN4=Zx−∞fx(v) dv.EE 524, Fall 2004, # 7 5Complex random vectors:x(ξ) = xR(ξ) + jxI(ξ) =xR,1(ξ)xR,2(ξ)...xR,N(ξ)+ jxI,1(ξ)xI,2(ξ)...xI,N(ξ).Complex random vector ≡ mapping from an abstract probabilityspace to a vector-valued complex space CN. The cdf of acomplex-valued random vector x(ξ) is defined as:Fx(x)4= P [x(ξ) ≤ x]4= P [{xR(ξ) ≤ xR} ∩ {xI(ξ) ≤ xI}]and its joint pdf is de fined asfx(x) = lim∆xR,1→ 0∆xI,1→ 0...∆xR,N→ 0∆xI,N→ 0P [{xR< xR(ξ) ≤ xR+ ∆xR} ∩ {xI< xI(ξ) ≤ xI+ ∆xI}]∆x1···∆xN=∂∂xR,1∂∂xI,1···∂∂xR,N∂∂xI,NFx(x).EE 524, Fall 2004, # 7 6The cdf of x can be computed from the joint pdf as:Fx(x) =ZxR,1−∞···ZxI,N−∞fx(v) dvR,1dvI,1···dvR,NdvI,N4=ZxN−∞fx(v) dv,where the single integral in the last expression is used as acompact notation for a multidimensional integral and shouldnot be confused with a complex contour integral.Note thatFx(x) =Zex−∞fx(ex) dex =Zx−∞fx(x) dx.whereex = [xTR, xTI]T.For two random variables, x = [x, y]T: fx(x) = fx,y(x, y).x and y are independent iffx,y(x, y) = fx(x) · fy(y) =⇒ E {xy} = E {x}E {y}.Expectation of a function g(x(ξ)):E {g(x)} =Z∞−∞g(x)fx(x)dx.EE 524, Fall 2004, # 7 7For two random variables, x = [x, y]T:E {g(x, y)} =Z∞−∞g(x, y)fx,y(x, y) dxdy.Correlation:Real correlation:rx,y= E {xy} =Z∞−∞Z∞−∞xyfx,y(x, y) dxdy.Real covariance:rx,y= E {(x − µx)(y − µy)}=Z∞−∞Z∞−∞(x − µx)(y − µy) 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 − µx)(y − µy)∗}=Z∞−∞Z∞−∞(x − µx)(y − µy)∗fx,y(x, y) dxdy.EE 524, Fall 2004, # 7 8Covariance Matrix:Mean vector:µx= 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 prope rty of complex correlation:ri,k= E {xix∗k} = E {xkx∗i}∗= r∗k,i.EE 524, Fall 2004, # 7 9Then, for E {x} = 0:Rx= E {xxH} =r1,1r1,2··· ··· r1,Nr2,1r2,2··· ··· r2,N...............rN,1rN,2··· ··· rN,N=r1,1r1,2··· ··· r1,Nr∗1,2r2,2··· ··· r2,N...............r∗1,Nr∗2,N··· ··· rN,N.The covariance matrix is Hermitian. It is positive semidefinitebecausebHRxb = bHE {(x − E {x})| {z }z(x − E {x})H}b= bHE {zzH}b = E {|bHz|2} ≥ 0.Linear Transformation of Random VectorsLinear Transformation:y = g(x) = Ax.Mean Vector:µy= E {Ax} = Aµx.EE 524, Fall 2004, # 7 10Covariance Matrix:Ry= E {yyH} − µyµHy= E {AxxHAH} − AµxµHxAH= AE {xxH} − µxµHxAH= ARxxAH.EE 524, Fall 2004, # 7 11Gaussian Random VectorsGaussian random variables:fx(x) =1σx√2πexpn−(x − µx)22σ2xofor real x,fx(x) =1σ2xπexpn−|x − µx|2σ2xofor complex x.Real Gaussian random vectors:fx(x) =1(2π)N/2|Rx|1/2expn−12(x − µx)TR−1x(x − µx)o.Complex Gaussian random vectors:fx(x) =1πN|Rx|expn− (x − µx)HR−1x(x − µx)o.Symbolic notation for real and complex Gaussian randomvectors:x ∼ Nr(µx, Rx), real,x ∼ Nc(µx, Rx), complex.EE 524, Fall 2004, # 7 12A linear transformation of Gaussian vector is also Gaussian, i.e.ify = Axtheny ∼ Nr(Aµx, ARxAT) real,y ∼ Nc(Aµx, ARxAH) complex.EE 524, Fall 2004, # 7 13Complex Gaussian DistributionConsider joint pdf of real and imaginary part of a complexvector xx = u + jv.Assume z = [uT, vT]T. The 2n-variate Gaussian pdf of the(real!) vector z isfz(z) =1p(2π)2n|Rz|exp−12(z − µz)TR−1z(z − µz) ,whereµz=µuµv, Rz=RuuRuvRvuRvv.That isP [z ∈ Ω] =Zz∈ΩpZ(z) dz.EE 524, Fall 2004, # 7 14Complex Gaussian Distribution (cont.)Suppose Rzhappens to have a spe cial structure:Ruu= Rvvand Ruv= −Rvu.(Note that Ruv= RTvuby construction.) Then, we can definea