2014-06-18 ·设窄带高斯噪声m(n)是平稳的随机信号,其均值为0,方差为 Rn(r)=Emt)n(t-r)=E{lt()cosobt-n2()sinan 2,则振幅分布为 In(t-t)coso(I-r)-n, (I-r)sin@o(I-r)l) p(m)= EIn(o)n(t-t)]cost cosmo(I-r) Eln, (on(t-r)sin @,t coso(I-r) ·下面讨论同相分量n(0与正交分量n(的统计特性 -En(tn (t-r)]cos@tsin @o(t-r) +EIn (on, (t-r)]sin @osino(t-r) Ent]=EIn(t)cosoot-n, (t)sin@) Re(t, I-r)cos@ot cosmo(t-r) En (nIcosoof-EIn, (t)]sin@ot Run(t, t-r)sin@t coso(t-r) n(是平稳的,均值为0→对任何时刻,E()=0 R.(t, t-r)cos@ sin @(t-r) EIn(D]= EIn 0=0 +R (L, I-r)sin@tsino(I-r) sm(是平稳的→Rn(与无关→取0,则有 R, (T)=R(LI-T)cosopT+Rm (t,t-t)sinor R, (r)=R(r)cosoot+Rm(r)sinor 显然要求Rn(和R(均与无关,则有 and R(r)=R ()cosor-R.(r)sinor 它们对阬有r均成立,则有 (r)=Rn(,1-r) R, (r)=R(r) R(r)=-Rm(r R(r)=R(r)cost+Run(r)sin oor ()=En(),(t-)=Eln,(t-t)n() 同理取2吗,则有 =En, (nn(r+r)=Run(r) R, (T)=R, (,t-T)cosooT-Ru (t,I-r)sinor Rn(-)=-R(r) R,(T)=R,, (r)cos @or-Rnn (r)sinor R4r是r的奇函数 = R(r)=R(, t-T) t-T) 同理得:Ran(0)=0 54 n(0和n,(n)在同一时刻的取值互不相关 n(o=n(r)@t-n ()sino.t →m(1)=n2(1 R, (r)=Rn, (r)cos@t+Rm, (r)sin@t =R(O)=R (O) R,(r)=R, (T)cosogr-Rm(r)sinOr R,(O)=R, (O) En(OJ=En (0]= En(0]=0 n是平稳的→m(1)和m(2高斯分布→n(1)和n2)高斯分布 E2()=En2()]=En2()]=a2 n(0)和n、(也是平稳的 P(n2)= n(0和n(的自相关函数与元关 n0和n,(0)是平稳的 √2za(2σ2 52014-06-18 5           2 2 2 exp 2 1 ( )   n p n  设窄带高斯噪声n(t)是平稳的随机信号，其均值为0，方差为 2，则振幅分布为：  下面讨论同相分量nc(t)与正交分量ns(t)的统计特性 54 25 E n t t E n t t E n t E n t t n t t c s c s 0 0 0 0 [ ( )]cos [ ( )]sin [ ( )] [ ( ) cos ( )sin ]         n(t)是平稳的，均值为0  对任何时刻t，E[n(t)]=0  E[n (t)]  E[n (t)]  0 c s [ ( ) ( )]sin sin ( ) [ ( ) ( )]cos sin ( ) [ ( ) ( )]sin cos ( ) [ ( ) ( )]cos cos ( ) [ ( ) cos ( ) ( )sin ( )]} ( ) [ ( ) ( )] {[ ( ) cos ( )sin ] 0 0 0 0 0 0 0 0 0 0 0 0                                                 E n t n t t t E n t n t t t E n t n t t t E n t n t t t n t t n t t R E n t n t E n t t n t t c s s c c c c s n c s 54 n(t)是平稳的  Rn()与t 无关  取t=0，则有： 26 ( , )sin sin ( ) ( , ) cos sin ( ) ( , )sin cos ( ) ( , ) cos cos ( ) [ ( ) ( )]sin sin ( ) 0 0 0 0 0 0 0 0 0 0                                  R t t t t R t t t t R t t t t R t t t t E n t n t t t s c s s c c n n n n n n s s 显然要求 Rnc ()和Rns ()均与t 无关，则有：        0 0 R ( )  R (t,t  ) cos  R (t,t  )sin c c s n n n n ( ) ( , ) ( ) ( , )         R R t t R R t t c s c s c c n n n n n n        0 0 ( ) ( ) cos ( )sin c c s  Rn  Rn  Rn n 54 27 c c s 同理取t=/20，则有：        0 0 R ( )  R (t,t  ) cos  R (t,t  )sin s s c n n n n ( ) ( , ) ( , ) ( ) ( ) ( ) cos 0 ( )sin 0            s s s c s c s s c n n n n n n n n n n R R t t R t t R R R R        它们对所有 均成立，则有： ( ) ( ) ( ) ( ) c s c s s c Rn  Rn Rn n  Rn n               0 0 0 0 and ( ) ( ) cos ( )sin ( ) ( ) cos ( )sin s s c c c s n n n n n n n n R R R R R R     ( )  [ ( ) (  )] [ (  ) ( )] n n c s s c R E n t n t E n t n t 54 28 [ ( ) ( )] ( ) ( ) [ ( ) ( )] [ ( ) ( )]       s c c s s c n n n n c s s c E n t n t R (  ) ( ) nsnc nsnc  R   R Rnsnc ( )是 的奇函数  (0)  0 nsnc R 同理得： (0)  0 ncns R 当 =0 nc(t)和ns(t)在同一时刻的取值互不相关 ( ) ( ) cos ( )sin (0) (0) ( ) ( ) cos ( )sin (0) (0) 0 0 0 0 s s c s c c s c n n n n n n n n n n n n R R R R R R R R R R                        E[nc (t)]  E[ns (t)]  E[n(t)]  0 54 29 nc(t)和ns(t)的自相关函数与t无关 2 2 2 2  E[nc (t)]  E[ns (t)]  E[n (t)]  nc(t)和ns(t)的方差均为 2 nc(t)和ns(t)是平稳的 n(t)是平稳的  n(t1)和n(t2)高斯分布  nc(t1)和ns(t2)高斯分布 ( ) ( ) ( ) ( ) ( ) ( ) cos ( )sin 2 2 2 3 1 1 0 0 0 0 2 1 n t n t n t n t n t n t t n t t s t c t c s             54 30 nc(t)和ns(t)也是平稳的                      2 2 2 2 2 exp 2 1 ( ) 2 exp 2 1 ( )     s s c c n p n n p n