16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde l=x(1)x( =R(11) Thus for a Gaussian process, the autocorrelation function completely defines all the statistical properties of the process since it defines the probability density functions of all order. This means IfR,(,t,)=R(-4), the process is stationary If two processes x(o), y(o)are jointly Gaussian, and are uncorrelated (R,(L, t, )=0), they are independent processes Most important: Gaussian input> linear system> Gaussian output. In this case all the statistical properties of the output are determined by the correlation function of the output- for which we shall require only the correlation functions for the inputs Upcoming lectures will not cover several sections that deal with Narrow band Gaussian processes Fast Fourier transform Pseudorandom binary coded signals These are important topics for your general knowledge Characteristics of Linear Systems Definition of linear system: If l1(D)→>y1(1) l2(1)→>y2() a4(1)+bn2(1)→ayv(1)+by2( Page 5 of 816.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 8 ()( ) (, ) ij i j xx i j M xt xt R t t = = Thus for a Gaussian process, the autocorrelation function completely defines all the statistical properties of the process since it defines the probability density functions of all order. This means: If Ryx i j xx j i (, ) tt R t t = − ( ), the process is stationary. If two processes x( ), ( ) t yt are jointly Gaussian, and are uncorrelated ( ) (, ) 0 R tt xy i j = , they are independent processes. Most important: Gaussian input Æ linear system Æ Gaussian output. In this case all the statistical properties of the output are determined by the correlation function of the output – for which we shall require only the correlation functions for the inputs. Upcoming lectures will not cover several sections that deal with: Narrow band Gaussian processes Fast Fourier Transform Pseudorandom binary coded signals These are important topics for your general knowledge. Characteristics of Linear Systems Definition of linear system: If 1 1 2 2 () () () () ut yt ut yt → → Then 12 12 au t bu t ay t by t () () () () +→+