第10讲-随机过程：维纳辛钦、希氏变换、高斯白噪声PPT

1、1( )( )21imTTTx tx t dtT1( ) ()( ) ()21imTTTx t x tx t x tdtT1( )( )( )2limTXXTTE X tx tx t dtT)()()(21)()()(limXTTTXRdttxtxTRtXtXE,( , )( )( )X YXYfx yfx fy, 0Cov X Y ( ), ()( )()0 x ty tx t y tdt,0X YdttxdttxETTT22)()(lim221( )()2Ex tdtX jdTTTdttxTP2)(21lim22111( )( )222limlimTTTTTTPx t dtXdTT2221

2、11 ( )( )22211( )22limlimlimTTTTTTTTE PEx t dtEXdTTEXdT2)(21)(limTTXXETGdtetxXtjTT)()(2121()*1212()*12121( )( )( )21( )( )2limlimTTjttXTTTTTTTjttTTTGEx t x t edt dtTE X t Xtedt dtT )()()()(211221ttRttRtXtXEXXTT21tt TTTTttjXTXdtdtettRTG21)(1212)(21)(limTTjXTXdeRTG22)()21 ()(limT(1)2TTTjXXdeRG22)()(TT

3、jXXdeGR22)(21)()sin()(0ttX0其他 020 21)(f),(21ttRX12( , )XGt t000( )( )1cos ()()22jXXjGReded 1( ,)( )()cos()2XRt tE X t X t000022( )1111112j tXj tj tj tj tGeedededeejjjjjj eRX)(1( )fdt1( )( ) ( )ff tH f tdt11( ) ( )gHg tdt1( )( )*f tf tt1( )gdt( )f t1sgn( )jt ( 90 ) 0()sgn (90 ) 0jH jjj sgn( )x1 0sgn(

4、 )1 0 xxx th Ftf Ftf sgnj 1f tf th tf tt j ( 90 ) 0jsgnj (90 ) 0 FFFF 1 ( )( )Hf tf t ( )( )H f tf t 22( )( )ft dtft dt( )f t( ) ( )0f t f t dt( )f t( )( )( )z tf tjf t( ) ( )ef tR z t*1( ) ( )( )2f tz tz t( )( )f tF( )( )z tZ2 ( ) 0( )0 0FZ0011( )2 ( )( )2j tj tz tFedFed0 0( )( )2 ( ) ()2 ( ) 0F z

5、tF z tFuF1( )z t2( )z t12( )( )0z tz t12( )( )0z tz t( )cosf tt( )f t)()()()(00 FtfsgnFjF )()()(00jFttfsin)( 000cossinjtZ tf tj f ttjte解：仍利用频域函数来求解。由频域搬移性可知： 00( )( )cos,ff tm ttf t且，其中求的低通信号，为截至频率为若fMtm)()()()(21)()(00MMFtf ffffjMjMjFF0000002/ )(2/ )(sgn于是 这不是一个常用的函数，所以无法通过查表而只能来做它的付氏反变换。 deFtftj2/

6、1deMjdeMjtjtjffff0000002221带入上式令00,vu 2/12/12)()(00dvevMdueuMjtftvjtujffff则ttm0sin)()(200eetmjtjtj tft0cos t0sin t0sin t0cos t0je t0jje ttm0je ttm0jej tf( )( )( )( ) ()( )()Y tX th tXh tdh u X tu du ( )( )()( ) ()( )( )(0)XXXE Y tEh u X tu duh u E X tu duh u m dumh u dum H1212121212( , ) ( ) ( )( )(

7、)( )()( ) ( )()()()() ( ) ( )() ( ) ( )( )YXYR t tE Y t Y tEh u X tu duh v X tv dvEh u h v X tu X tv dudvE X tu X tv h u h v dudvRuv h u h v dudvR 1212121221( , )( ) ( )( )() ( )( )() ( )() ( )() ( )()( )( )XYXXXXYRt tE X t Y tE X tX tu h u duE X t X tu h u duRtut h u duRu h u duRuhR ( )( )( )( )( )

8、( )XYXYXXGF RF RF hGH( )( )() ( ) ( )( )( )()jYYjXjXGRedRuv h u h v dudv edh uh vRuv ed dudv 2*( )( )( )( )( )( )( )( )( )j uj vjYXXXGh u eh v eRed dvduHHGHGuvuv( )X t12121221( , )( )( )( )() ( )() ( )() ( )( )( )( )XXXXXXRt tE X t X tE X t X tu h u duRtut h u duRu h u duRhR( )( )( )( ) ( )( )()XXXX

9、XGF Rh tF RF h tGH0023( ) ,21.38 10Joules/KelvinnBeBNGNk Tk 0( )( )2nNR 02121()()2nNR tttt0( ) ( )TXn tt dt( ) t2200( )2TXNt dt110( )( )TXn tt dt210( )( )TXn tt dt1( ) t2( ) t012120()( )( )2TNE X Xtt dt1( ) t2( ) t120( )( )0Ttt dt1X2X0H, ( )2, 20, HnHHNGf222001( )( )21(2)2sin2222HHHHHjnnfjfnfjfjfHHHRGedGf edfNfeeN f