《信号与线性系统分析》重要公式汇总

1、学习必备欢迎下载第一章：信号与系统1.1 单位阶跃函数(t)1.2 冲激函数的性质：f (t )(t)f (0)(t)f (t )(t) dtf(0)f (t )'(t )f (0)'(t)f '(0) (t )f (t )' (t)dtf '(0)f (t )(n) (t)dt(1)( n) f (n) (0)(at )1(t)a' (at )1 1'(t )a a(n) (at)1 1 (n )(t )a a (n)1.3 线形系统的性质：齐次性T af ( )af ( )信号与线形系统重要公式单位冲激函数(t)f (t)(tt1)f

2、 (t1)(tt1)f (t)(tt1)dtf (t1)(t t1 )dtf (t1)f (t)'(tt1)f (t1)'(tt1 )f ' (t1 ) (tt1 )f (t)' (tt1 )dtf ' (t1)( n)( t)(n)为偶数(t )n( n)( t)( n)为奇数(t) n可加性T f1 ( )f 2 ( )T f1 ( )T f2 ( )T a f ( )a f ( )aT f ( )a T f ( )11221122零输入响应，零状态响应，全响应yx ( )T x(0),0y f ()T 0 ,f ()y()yx () yf()第二章

3、连续系统的时域分析法全解 =齐次解（自由响应）yh (t ) +特解（强迫响应）y p (t )全响应 =零输入响应yx (t) + 零状态响应y f (t)y(t)yh (t)yp (t ) =yx (t)y f (t)零输入响应是指激励为零，仅由系统的初始状态所引起的响应，用yx (t) 表示。零状态响应是指初始状态为零，仅由激励所引起的响应，用y f (t ) 表示。学习必备欢迎下载ni tnityx (t )C xi ey f ( t )Cifyp ( )tCxi和 C fi 都为待定系数ei 1i 1ni tnitni ty(t)Cei（强迫响应）Cfi e（零状态响应）（自由响应）

4、（零输入响应）yp(t)Cxi eyp(t)i 1i 1i 12.2冲激响应和阶跃响应一个 LTI 系统，当其初始状态为零，输入为单位冲激函数(t) 时所引起的响应，简称为冲激响应。用 h(t ) 表示，即冲激响应为激励为(t) 时的零状态响应。一个 LTI系统，当其初始状态为零、 输入为单位阶跃函数(t ) 时所引起的响应， 称为单位阶跃响应，简称阶跃响应。用g(t) 表示。阶跃响应是(t)时，系统的零状态响应。冲激响应(t ) 与阶跃响应(t ) 的关系：(t)d (t)(t)t(t )dxdtdg (t)同一系统阶跃响应h(t ) 与冲激响应 g(t) 的关系 h(t)g(t )t(t

5、)dxdt2.3卷积积分 f (t )f1 (t )* f 2 (t)f1 () f 2 (t)d零状态响应的另一种方法y f()*h()f tt2.4卷积积分性质f 1( t) *f 2 ( t )f 2 ( t ) *f 1 ( t )f 1( t) *f 2( t )f 3 ( t )f 1 ( t ) *f f 1 ( t ) *f 2( t )*f 3 ( t )f 1 ( t ) *f22( t )f 1 ( t ) *f 3 ( t )( t ) *f 3 ( t )函数与冲激函数的卷积f ( t ) *( t )( t ) * f ( t )f ( t )f ( t ) *(

6、tt 1 )( tt 1 ) * f ( t )f ( tt 1 )( tt 1 ) *( tt 2 )( tt 1t 2 )f ( tt 1 ) *( tt 2 )f ( tt 2 ) *( tt1 )f ( tt 1t 2 )若 f ( t )f 1 ( t ) *f 2 ( t ),则f 1 ( tt1 ) *f 2 ( tt 2 )f 1 ( tt 2 ) *f 2 ( tt1 )f ( tt 1t 2 )卷积的微分与积分学习必备欢迎下载若 f ( t )f 1 ( t ) *f 2 ( t )f 2 ( t ) *f 1 ( t ), 则导数f(1)( t )f 1(1)( t )*

7、f 2(t)f 1 ( t)*f2(1)( t)积分f (1)( t )f 1 (1) ( t ) *f2 ( t )f 1 (t ) *f 2 (1)( t )推论f( t)f 1(1)( t ) *f2( 1)(t )f 1(1)( t)*f 2(1)( t )f ( i ) ( t )f 1 ( j ) ( t ) *f 2 ( ij ) ( t )第三章离散系统的时域分析3 1 全响应 y( k) =零输入响应 yx (k) +零状态响应 y f (k )nknknknky(k)Cy( k )Cy ( k)y(k)CCfy(k )xiiffipiiiipii 1i1i 1i 1差分方程

8、的经典解全解 y(k) =齐次解 yh (k ) +特解 y p ( k)ny(k) yh (k ) yp (k )Ci iky p ( k)i1不同特征根所对应的齐次解特征根特解 yh (k )单实根Ckr 重实根Cr1k r 1 kCr 2 k r2kC1kkC0k一对共轭复根pk C cos(k )D sin(k) 或1,2a jbpe jAp k cos(k), Ae jC jDr 重共轭复根Ar1r r 1 pkcos(kr1)Ar 2r r2 pkcos(k r 1)A0 Pk cos( k0 )不同激励所对应的特解激励f (k)特解y p (k)k mpmk mpm 1k m 1

9、p1kp0 所有特征根均不为1k r pm k mpm 1km 1p1kp0 有r为1 的特征根ampak当 a 等于特征时学习必备欢迎下载p1kakp0 ak 当 a 是特征单根时pr k r akpr 1kr 1akp1ka kp0 ak 当 a 是 r 重特征根时。cos(k)P cos(k)Q sin(k )sin(k )A cos(k), AejP jQ 当所有特征根均不等于 e j3 2 单位序列和单位序列响应当 LTI 离散系统的激励为单位序列( k) 时，系统的零状态响应称为单位序列响应，用 h(k )表示。当 LTI 离散系统的激励为单位阶跃序列(k ) 时， 系统的零状态响

10、应，称为单位阶跃响应，简称阶跃响应，用 g( k) 表示。g (k )kh (i )h (k j )单位序列响应与阶跃响应的关系ij0h (k )g (k )g ( k1)g( t )t) dh (连续系统冲激响应与阶跃响应的关系dg ( t )h( t )dt几种数列的求和公式序号公式说明1k1ak1k 0j 0a j1a, a1(k1),a12k2ak1ak21k1,k2可为正或负整数，但k2 k1a j1a,a1j k1k2k11,a13a j1a1j 01a4ajak1a1k1可为正或负整数1aj k1学习必备欢迎下载5kjk (k 1)k02j06k2(k1k2 )( k2k1 1)

11、k1,k2可为正或负整数，但k2 k1j2jk17kj 2（ 2k+1）k0k( k 1)j 063 3 卷积和f (k )f1 (k )* f 2 (k)f1(i ) f2 (ki)i卷积和的性质f1 (k )*f 2 ( k )f 2 ( k )*f1 ( k )f1 (k )*f 2 ( k )f 3 ( k )f1 ( k )*f 2 ( k )f 1 ( k )*f 3 (k ) f1 ( k )*f 2 ( k )*f 3 ( k )f1 ( k )*f 2 ( k )*f 3 ( k )任一序列f (k ) 与单位序列的卷积f ( k )*( k )( ki )* f ( i )

12、f (k )i( k k1 )*(k k2 )( kk1k 2 )f ( k )*(tt1 )if (i )*(kik1 )f (kk1 )f ( kk1 )*(kk2 )f (k )*( k k1 )*( k k 2 )f (k )*( k k1 k2 ) f ( k k1 k 2 )若 f (k )f1 (k )*f 2 (k ), 则f1 (k )* f2 (kk1 )f1 ( kk1 )* f 2 (k )f (k k1 )f1 (kk1 )* f2 ( kk2 )f1 (kk 2 )* f 2 ( kk1 )f (kk1 k2 )b k1a k 1h( k ) a k ( k )*

13、b k ( k )b a( k1) b k( k ), ab( k ), ab第四章傅里叶变换和系统的频域分析4.1 信号分解为正交函数4.2傅里叶级数a0an cos(n t )bn sin( n t)f (t)2 n 1n 1学习必备欢迎下载a01T2T2T f (t )dt22 ，2T其中 an ,bn 为傅里叶系数，an2Tf (t)cos( nt) dt ,n0,1,2,TT2bn2Tf (t)sin( nt)dt , n0,1,2,T2T2f (t)nA0An cos(n tn )A0a0Anan2bn2 , n 1,2,3,2n 1bna0A0anAn cosn , n1,2,a

14、 r c t a n ( )anbnAn sinn , n1,2,4 3 傅里叶级数的指数形式f (t)1Anej n e jn t令 1 Anej nFn e j nFnf (t )Fn ejn t2 n2nF1A ejn1 A cosjA sin1 (ajb)n2n2nnnn2nn1Tej nt d,tF2Tf( t)n 0 ,1,2 ,nT24 4 傅立叶变换和逆变换TF n T2T f ( t )ejnt dtF ( j )lim FnTf (t )ej dt2Tf (t )F n Te jnt1f (t)1f ( j )e j t dnT2在 f(t)是实函数时：(1) 若 f(t)

15、为 t 的偶函数，即f(t)=f(-t)，则 f(t)的频谱函数 F(j )为 的实函数，且为 的偶函数。(2) 若 f(t) 为 t 的奇函数，即 f(-t)=- f(t)，则 f(t)的频谱函数 F(j )为 的虚函数，且为 的奇函数。表 4-1常用傅里叶变换编号f (t)F ( j)1gr (t)Sa()t222 gr( )Sa( )2学习必备欢迎下载345678910111213141516e at (t ), a0te at (t), a0e a t ,a0(t)1(tt1 )cos0 tsin0t(t)Sgn(t )1tT (t )Fn e jn tntn1e at (t), a

16、0(n1)!1aj1(aj)22aa2212 ( )e j t0(0)(0)(0 )(0 )j()1j1,F(0)0jjSgn()()2Fn(n)n1(aj)n4.5 傅里叶变换的性质1 线形 a1 f1 (t )a2 f2 (t)a1 F1 ( j)a2 F2 ( j)2 奇偶性实部虚部F( j)f (t )e j t dtf (t )cos(t) dtjf (t)sin(t)dtR()jX ()F ( j) e j ( )学习必备欢迎下载实部和虚部分别为R()f (t)cos(t) dtX ()f (t )sin(t )dt频谱函数的模和相角分别为F ( j)R()2X ()2()arct

17、an( X () )R()1、若 f(t) 是时间 t的实函数，则频谱函数F ( j)的实部 R() 是角频率的偶函数，虚部 X() 是角频率的奇函数，F ( j) 是的偶函数，() 是的奇函数。、如果f (t )是时间t 的实函数， 并且是偶函数， 则F ( j)R()2f (t )cos( t )dt20频谱函数 F ( j) 等于R() ，它是的实偶函数、如果f (t)是时间 t的实函数，并且是奇函数，则F( j )j X( )j 2f ( t) si n (t d)t30频谱函数 F ( j) 等于 jX () ，它是的虚奇函数。4、 f (t) 的傅里叶变换若 f(t)是时间t的实函

18、数F ( j )R()jX () R()jX ( )F ( j )f ( t )F ( j )F ( j )R()R( ),X( )X ()则有（ 1）F ( j ) , ( )( )F ( j )（2） f ( t )F ( j )F ( j )（3） 如 f (t)f (t), 则 X ( )0, F ( j)R()如 f ( t)f (t)则,R ()0F,j()j X (若 f(t)是时间t的实函数R()R(),X()X ()（1）F ( j )F ( j ) , ( )()（2） f ( t )F ( j )F ( j )3 对称性若 f (t)F ( j), 则 f ( jt )2

19、 F ()学习必备欢迎下载4 尺度变换若 f (t)F ( j), 则对实常数 a(a0), 有 f ( at )1F ( j)aa5 时移特性若 f (t)F ( j), 则 f (t t0 )e j t0 F ( j )实常数 a和b(a0), 有f (atb)1j b)ea F ( jaa6 频移特性若 f (t)F ( j),且0为常数，则 f (t)ej 0 tF j (0 )f (t)cos(0t )1 F j (0 )1 F j (0 )22f (t)sin(0t )10 )10 )F j (jF j (227 卷积定理时域卷积定理若 f1(t)F1 ( j) 则 f1(t)f2

20、 (t )F1( j)F2 ( j)f 2 (t)F2 ( j)频域卷积定理f1(t)F1 ( j)f2 (t)1F1( j) F2 ( j)若F2 ( j则 f1(t)2f 2 (t)其中 f1 (t )f 2 (t)f1 ( ) f 2 (t)d8 时域微分若 f (t)F ( j), 则 f ( n) (t )( j ) n F ( j )时域积分若 f (t)F ( j),则 f ( 1) (t )F (0)()F ( j)j9 频域微分若 f (t)F ( j), 则 ( jt ) n f (t)F n ( j )频域积分若 f (t)F ( j),则 F(0)(0)1f (t )F

21、 (1) ( j)jt学习必备欢迎下载10 能量谱Ef 2 (t )dt1222F ( j ) dF ( j ) df ,取 F ( j )( )2功率谱T222lim 11F ( j )F ( j )F( j )P2T f 2 (t)dtlimdlimdf , 取 lim( )t T T22t TTt TTt TT傅里叶变换的性质4.6周期信号的傅里叶变换一、正、余弦函数的傅里叶变换cos(0t)1 (ej0te j0t )(0 )(0 )2sin(0 t)1 (ej0tej 0 t )j (0 )(0 )2 j学习必备欢迎下载4 7LTI 系统的频域分析1、 虚指数函数 f (t )ej

22、t 作用于 LTI 系统所引起的零状态响应，设冲击响应h(t)yf (t )f (t)h(t)H ( j) ej t2、任意信号输入时的响应Y ( j )H ( j ) F ( j )第五章拉普拉斯变换5 1 在频域分析中，我们以ejt为基本信号，在复频域分析中，我们以est为基本信号sj，由于当0, sj， ej testFb ( s)f (t)e st dt1j称为双边拉普拉斯变换对；f (t)Fb (s)estds2jjFb (s) 称为 f (t ) 的双边拉氏变换（或象函数）；f (t) 称为 Fb (s) 的双边拉氏逆变换（或原函数） 。（单边） 拉普拉斯变换 F (s)0f (t

23、 )e st dtgr (t)1e st，Res(t )1， Res' (t)s ，Re s2ses0 t (t )1， ResRe s0 ss05 2拉普拉斯变换的性质1 线形 a1 f1 (t )a2 f2 (t)a1 F1 (s)a2 F2 (s), Re smax( 1 ,2 )sin t(t)2 ,cos t(t )s2 , Re s0s2s22 尺度变换若 f (t)F (s), Res0则对实常数 a(a0), 有 f (at )1F ( s), Re sa 0aa3 时移特性若 f (t)F (s), Res0且 t00对实常数则 f (tt 0 )(tt0 )e st

24、0 F (s), Re s01sbs若 f (t )F ( s), Re s且 t0对实常数则 f (at b) ( atb)e a其中 a 0,b00F (,Re s00aa学习必备欢迎下载4 复频移特性若 f ( t)F ( s), Re s0且有复常数则 saaj,则 f (t )esa tF ( ssa ), Re sa + 05 时域微分特性若 f (t)F (s), Res0则 f (1) (t)sF( s)f (0)f (2) (t)s2 F (s)sf (0)f (1) (0)f ( n) (t)snF (s)sn 1 f (0) sn2 f (1) (0 )f ( n 1)

25、(0)如果 f (t ) 是因果信号，则由于f (n) (0 ) 0(n0,1,2 ) 有f ( n) (t)sn F (s), Re s06 时域积分定理( n )F ( s)n1( m)Re s0 和 Re s0f(t)(0 )snn m 1 f其收敛域至少是m 1 s相重叠的部分。7 卷积定理时域卷积定理若因果信号 f1 (t)F1 (s),Re s1 则 f1 (t ) f 2 (t ) F1 (s) F2 (s)f 2 (t)F2 (s),Re s2复频域卷积定理F1(t ) F21(t )2 j8s域微分和积分cj2 , 1 c RescF1 ( )F2 ( s)d , Res12

26、j若 f (t) F (s),Re sdF (s)则 ( t) f (t ),( ds0t )nf (t )d nF ( s)f (t)F ( )d ,Re sn,0dsts5.3拉普拉斯逆变换f (t )12j0, t0jF ( s) s st dsj, t0F ( s) msik jkp(t )1 ,' t ( ) s , (n )t ( )sns j(s sp ) q(t)1(t )1, , tn(t)1, ts2sns111e t(t), e 2t (t)s, e nt(t )s12sn学习必备欢迎下载5.4复频域分析1 用拉普拉斯变换求系统的零输入响应和零状态响应y'

27、' (t )s2Y ( s)sy(0 ) y' (0)y' ( t)s Y( s) y( 0 )y(t ) Y(s)f( n)(t)nF ( s)' ''n ( )f中(M ( s)B( s)s代 入 y( t) a y(t)b (y ) t有)t Y( s)F ( s)A( s)A( s)M (s) 为零输入响应的象函数B(s)F (s) 为零状态响应的象函数A(s)A(s)一般题目中有 y' (0 ) 和 y(0) 的值，如果只有 y' (0) 和 y(0) 的值，那么先算出 yzs (t ) 的函数，在根据函数yzs(t )

28、 ， y(0) ， y' (0) 计算 y' (0) 和 y(0) 的值，可得出 yzi (t ) 的函数2 系统函数系统零状态响应的象函数与激励的象函数之比，称为系统函数。用H ( s) 表示。H (s)B( s), H (s) h(t ) ， H (s) 仅与系统的结构， 元件参数有关， 而与激励及初始状态A( s)无关第六章离散系统的z 域分析6 1F ( z)f (k) zkk0 ,1,2 ,称为序列f(k) 的双边z 变换kF ( z)f (k )(k) zk称为序列f(k) 的单边z 变换k0Z 变换简记为：f (k)F (z)常用序列的z 变换：因果序列：a 为正实数ak( k)z, za( a)k(k )z, zazaza学习必备欢迎下载令 a=1，则单位阶跃序列的z 变换：( k)z , z 1zz