信息与系统公式整理.pdf

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1、( , ) ( , )N N ( , )2( )f t2( )TTE f t dt22lim ( ) ( )TT TE f t dt f t dt2( )Nn NE x n2( )nE x n212( )TTTP f t dt212lim ( )TT TTP f t dt21( )2 1Nn NP x nN21( )2 1limNn NNP x nN002T1 1 2 21 2 1 2( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )f t y taf t ay tf t y t f t y tf t f t y t y t0( ) ( ) ( )( )

2、( ) ( )x fny t y t y ty n y n y n( ) ( )ff t y t0 0( ) ( )x n n y n n( )stf t Ke s j0, 0 ( )f t K0, 0100, 000 0cos sin( )j tK t jK tf t Ke( )jf t Ke0 00Im Im sin( )j t j j ttKe Ke e K00, 0Z( )A t( ) 0 0( ) 0( )A t tA t tA t dt A: ( ) ( ) (0)A t t dt A ( )A t0 0 0( ) ( ) ( ) ( )f t t t f t t t0( ) (

3、) ( ) ( )f t t f t t0 0( ) ( ) ( )f t t t dt f t ( ) ( ) (0)f t t dt f1( ) ( )ba aat b t 0a 0a1( ) ( ) ( ) ( )aa bg t at b dt g t t dt( )Au t00 0( )A ttAu t0t ,110,2( ) ( )tA d Au t ( )( ) Au tdAdt( )Ar t0( )0 0At tAr tt( ) ( )tAu t dt Ar t ( ) ( )AdAu t r tdt( )( )nt( )0( ) ( ) ( 1) ( ):nn nntdf t

4、t dt f tdt( )t 0( ) ( ) ( ) (0):tdf t t dt f t fdt( )t2s A2Vs( )t0t0 0 0( ) ( )( ) ( )t t t tf t f t 0 0( ) ( ) ( )f t t t dt f t 2 21( ) ( ) 01( ) ( ) 0bat b t aa abat b t aa a 1, 0 ( ) ( )a b t t( )t1 21 2( ) ( )( ) ( ) f f t df t f t1 2 1 2( ) ( ) ( ) ( )kx n x n x k x n k1 2 2 1( ) ( ) ( ) ( )f

5、t f t f t f t1 2 2 1( ) ( ) ( ) ( )x n x n x n x n( ) ( ) ( )f t t f t0 0( ) ( ) ( )f t t t f t t( )( ) ( )tf du t f t1( )( ) ( )( 1)!( ) ( )nt tntf t dt dt f tnu t f t ( ) ( ) ( )kx k x n u n ( ) ( ) ( )t f t f t( ) ( )( ) ( ) ( )n nt f t f tu i( )( )u tRi t( )( )RRU tRI t( )( )RRU sI sR1( ) ( )tu

6、t i t dtC( ) 1( )u ti t pC( ) 1( )CCU tI t j C1 1(0 )( ) ( )CC CuCs sU s I s(0 )( ) ( )CC CuI s CsU s C( ) ( )du t L i tdt( )( )u tpLi t( )( )CCU tI tj L1 1(0 )( ) ( )LL LiLs sI s U s( ) ( ) ( ) ( )D p y t N p f t0( ) ( )pD D p( ) ( ) 0( ) ( ) ( ) 0( ) ( ) ( ) ( )( )( ) ( )( ) ( )xff x fffD p y ty t

7、 f t h t tN p y t y t y tN py t tD p D p( )xty0( )y n1 2 n1 1 2 20( )n n nk ky n c c c01 2 n-10t11 2 10)( ) (q nqc n ny n c c1 1 1 1n n nq q k kc c c11q k1 1 1 2 2 2,j ji ij11 1 1 1( ) cos( ) sin( )txy t e k t k t cos( ) sin( )iti i i ie k t k t 0t1 20( ) ( ) ( )j n j ny n c re c re,j jre re( )H p (

8、 )h t ( )H E ( )h na ( )a t1( )nap( )au tEE( )nu n1p a( )ate u t2( )EE1( )nn u n1( )np a1( 1)!( )nattne u t22( )EE2 2( ) bbp a ( )mEE1( 1) ( 2)( 1)!( )n mn n n mmu n2 2( ) bp ap a( )n0 01 0( )nnn ( ) ( ) ( )kx n x k n k( )u n0 01 0( )nnu n ( ) ( ) ( 1)n u n u n( )nu n0 0( )1 0nnu nn( )kG n1 0 1( )0k

9、n kG n( ) , ,n jx n z n z re0, , ( )nz r x n r00 0 01, , ( ) cos sinj nr x n e j02N k02N k0sin n00 0( , )( ) ( )ky n x k( ) ( 1) ( )x n x n x n( ) ( ) ( 1)x n x n x n( ) ( ) ( )e of t f t f t1( ) ( ) ( )21( ) ( ) ( )2eof t f t f tf t f t f t( )( ) ( ) ( ) ( )j t j t j t jy t e h t e h d e e h d( )H

10、j e hj( )d) ( )j ty(t e H jf t( )0)0t Ttf (t dtf t( )f t( )01sin( ) ( cos )n nnf t a a n t b n t00000001( )2( )cos2( )sint Ttt Tntt Tnta f t dtTa f t n tdtTb f t n tdtT000000cos sin 0 ,cos cos20sin sin20t Ttt Ttt Ttn t m tdt m nTm nn t m tdtm nTm nn t m tdtm n( )jn tnnf t F e001( )t Tjn tntF f t e d

11、tT000t Tjn t jm ttT n me e dtn mnanb( ) ( )f t f t204( )cos( )Tna f t n t dtT0nb( ) ( )f t f t0na204( )sin( )Tnb f t n t dtT( ) ( )2Tf t f t200 0,2,4,4( )cos( ) 1,3,5,Tnnaf t n t dt nT200 0,2,4,4( )sin( ) 1,3,5,Tnnbf t n t dt nT( ) ( )2Tf t f t200 1,3,5,4( )cos( ) 0,2,4,Tnnaf t n t dt nT200 1,3,5,4(

12、 )sin( ) 0,2,4,Tnnbf t n t dt nT( ) ( )2jn tnE nf t Sa eT21T( )jn tnnF ef t( )( )jn tnnF H jn t ey t1 1 2 ( ) ( )1( )2j tj t je dF e d f e df t2 2 2( )( ) ( )( ) ( )j tj tf d e d f te d f tsgn( )t1( ) ( 0)te u tj2 21 1 2( ) ( )t tje u t e u tj j1( ) ( 0)te u tj2 20 02 2sgn( ) lim ( ) ( ) limt tjt e

13、u t e u tj( ) ( )12j tf t F e dt( ) 2 ( )( ) ( )12j j tf t f tF e d F t e d( ) ( )12j tf t F e d00( )0( ) ( )( )j t j tf t t e dt f e df t t0 0( ) ( )j t j tje ef e d F( ) ( ) ( )nnndf t j Fdt( ) ( )( ) ( )12j t j td df t e e j Fdt dtF d j F d( ) ( )1( )j( ) ( )u t u t( ) ( ) ( )u t u t tu t( ) ( )

14、( )tf d f t u t1( ) ( )u tj1 ( )( ) ( ) ( ) ( ) ( ) (0) ( )tFf d f t u t F Fj j( ) ( )j tF f t e dt( )1( )2j tF e df t( )f t ( )F ( )f t ( )F1 2( ) ( )f t f t1 2( ) ( )F F( ), 0f at a1( )Fa a( )F jt 2 ( )f0( )j tF e0( )j tf t e0( )F( )df tdt( )j F ( )jtf t( )dFd( )tf d( )(0) ( )FFj( )(0) ( )f tf tjt

15、( )F d( )* ( )f t h t ( ) ( )F H ( ) ( )f t p t1( )* ( )2F P( )f t*( )f t*( )f t( )F*( )F*( )F( )f t( ) ( )of t Od f t( ) ( )ef t Ev f tIm ( )j FRe ( )F( ) ( ) ( )f t f t u t( ) ( ) ( )F R jI1( ) ( )*R I( ) ( )nf t t nT1 2( )kF kT T0 01 2( )nf t n0( ) ( )kF k22 1( ) ( )2f t dt F d2( )F(0) ( )F f t d

16、t lim ( ) 0tf t1(0) ( )2f F d lim ( ) 0F( ) ( )j tF f t e dt1( ) ( )2j tf t F e d( )f t ( )F ( )f t ( )F( )t 2 ( )( )dtdtjt2 ( )djd( )u t1( )j1 1( )2 2tj t( )u( )tu t21( )djd1, 0sgn( )1, 0ttt2j1, 0t, 0( ), 0jFj0( )t t0j te0j te 02 ( )0cos t02cos t0sin t02sinj t1,( )0,tG tt( )2Sa ( )WSa Wt1,( )0,WFW1

17、 ,( )0,t ttt2( )2Sa2( )2 2W WtSa1 ,( )0,W WFW( ),Re 0ate u t a1a j1jt2 ( ),e u,Re 0ate a2 22aa2 2t, 0e0cos ( ),Re 0ate tu t a2 20( )a ja j0sin ( ),Re 0ate tu t a 02 20( )a j( ),Re 0atte u t a21( )a j21, 0( )jt2 ( )e u1( ),Re 0( 1)!k att eu t ak1( )ka j( ) ( )Tlt t lT2 2( )kkT T2( )te2( )2e ( ) ( )co

18、s02 2u t u t t( ) ( )0 0 2 2 2Sa Sa0jk tkkF e02 ( )kkF k( )f t ( )fy t( ) ( )f dy t kf t t( ) ( )dj tfY ke F( )H( )( )( )df j tYH keF( )H k k( )dt 0dt( )t 0t0t( ) ( ) ( )h t h t u t2( )H d22( )1Hd2( ) ( )0ddcj tc j tceH G e1( )1Hj RC21( )1 ( )cH( ) arctanc2( ) 1 ( )0ddcj tc j tceH G e( )1j RCHj RC2

19、2 2( )1RCHR C1( ) arctan( )RC2( )( ) ( ) 1j RCHLC j RC j21( )2H( ) ( )nf t t nT( ) ( )nf nT t nT( )sf t ( )f t( )f tm mf12sm mTf1( ) ( ) ( )2s TF FT f t FT t1 1( ) ( ) ( )2nF F nT1( ) ( ) ( )2s TF FT f t FT P t1 1( ) ( ) ( ) ( )2 2n nnF Sa n F nT( ) ( )2nnSa F nT( ) ( ) ( ) ( ) ( )cs cnTf t h t f nT

20、 t nT Sa t( ) ( ) ( )ccnTf nT Sa t nT t12( ) ( ) ( )cccTh t FT TG Sa t( )0ccTH( ) ( ) ( ) ( ) ( )snF F F n1 11 2( ) ( ) ( ) ( ) ( )s sn nf t FT F h t FT n f t n0j ne0j ne0 0( )02j n N j nme eN02N m02( )N m0j te0j ne0j te0j ne0202 mN( 0), ,N m00m00002o00002( )o m21 1( )0 021 1( )0 0( ) ( ) ( )( ) ( )

21、1 1( ) ( ) ( )N Njk nknNNDFSn nN Njk nknNNn nDFS X k x n e x n Wx n X kIDFS x n X k e X k WN N3 1 2( ) ( ) ( )x n x n x n X3( )k X1( )k X2( )k( ) ( )DFSx n X k ( ) ( )DFSknNx n m W X k ( )DFS x n lN X ( )k( ) ( )DFSx n X k ( ) ( )DFSqnNW x n X k q13 1 20( ) ( ) ( )Nmx n x m x n m X3( )k X1( )k X2( )

22、k3 1 2( ) ( ) ( )x n x n x n X31( )kN11 20( ) ( )NlX l X k l11( )( )0x n n Nx nn N21( ) ( )21( ) ( )j nj nnx n X e dX x n eN( )nx n2( ) 2 ( )knX a k112( )1( )Njk nNkn Na x n eN21( )x n X1( )2( )x n X2( )1 2( ) ( )ax n bx n a X1( ) b X2( )( ) ( )X XRe ( )Im ( )XX( )( )XX( ) ( )x n X00( ) ( )j nx n n

23、e X( ) ( )x n X00( ) ( )j ne x n X1( ) ( ) (0) ( 2 )1njm kx m X X ke1( ) ( 2 )1jku n ke( ) ( )x n X ( ) ( )x n X( )( )( )0knx n kx nkn k( )( ) ( )kx n X k( )( )dXnx n jd2 221( ) ( )2nx n X d2( )X221( )kn N n Nx n aN( ) ( ) ( )y n x n h n ( ) ( ) ( )Y X H( )f t( ) ( ) ( ) ( )t t j t j tFT f t e f t e

24、 e dt f t e dts j( ) ( ) ( )stF s LT f t f t e dt11( ) ( ) ( )2jstjf t LT F s F s e dsj( )F s0( )stf t e dtlim ( ) 0ttf t es,nt tste0sss111 121 211 1 1( ) ( )( ) ( ) ( )pp pKK KF s F ss s s s s s11111 1 112 1 1111 1 11( ) ( )( ) ( )1( ) ( )( 1)!ps sps sips siK s s F sdK s s F sdsdK s s F si dsis p R

25、e ( ) ( ) ( ) ist sti s ps F s e s p F s eis p k111Re ( ) ( ) ( ) ( 1)!ikst sti s pkds F s e s p F s ek ds( )F s( ) ( )tF s f t e dt ( ) ( )12jstjf t F s e dsj( )f t ( )F s ( )f t ( )F s1 2( ) ( )f t f t1 2( ) ( )F s F s1 2,( ), 0f at a1( )sFa ac0( )stF s e0( )s tf t e0( )F s sc c c 0 c( )df tdt( )

26、(0 )sF s f( )tf t ( )dF sds( )tf d1( ) (0 )F s fs s( )f tt1 1( )sF s ds0( 1)(0 ) ( )f f t dt( )* ( )f t h t ( ) ( )F s H s ( ) ( )f t p t1( )* ( )2F s P sj0(0 ) lim ( ) lim ( )t sf f t sF s0( ) lim ( ) lim ( )t sf f t sF s00 01( ) ( ) ( ) ,stacsf at t u at t F ea a0 0 0( )0 0 0( ) ( ) ( )s t t s te

27、f t t u t t e F s s0 0 0 00( ) ( )0 0 0 ( ) ( ) ( )s t t s t t sttLT e f t t u t t e f t t e dt0 0 0 0 0 0 0( ) ( )0 0 00 0 ( ) ( ) ( ) ( ) ( )s t t s x st st s s x stsxLT e f t t u t t e f x e e dt e f x e dt e F s s00( ) ( ) ( )nf t f t nT u t nT1 2 ( 1)( )( ) (0 ) (0 ) (0 )nn n n nnd f ts F s s f s f fdt( )( ) ( )nnnd F st f tds0( ) ( )stF s f t e dt0( ) ( )stF s f t e dt0 0( ) ( ) ( )st stdf t e dt tf t e dt LT tf tds00( ) ( ) ( )t tf x dx f x dx f x dx00 ( ) ( ) ( ) t tLT f x dx LT f x dx LT f x dx0( 1)1 ( ) (0 )LT f x dx fs0 0 01 ( ) ( ) 0 ( )t tstLT f x dx f x dx e dt F ss