信号与系统课件第三章7

1、120 nf (t) A Acosnt n n1AnnA0A1A2nn23423400指数形式2t2 jntAn T tf (t)edt11jn tf (t ) 2 A nen A A e jn nn三角函数形式yxAT 2An2 AT2 04 n An 2ATAn2 AT20n4n2 4 0双边频谱单边幅度频谱A 2 A Sa n nT2A A e jn nnAsin(n / 2) jntf (t) e T nn /2 f)n1( () jjFj An limT 0f (t) 1 F ( j) cost () d0F ( j ) lim TAnT 2 And)F ( j ) 幅度频谱( 偶）

2、)e (） 相位频谱（奇）反变换：f ( t ) 1 F ( j )e j t d 2 变换：F ( j ) f ( t ) e j t dt (t) 101 2 ( ) 0e t 2 2 2et (t) 1 jAG (t) A Sa( )23.7 周期信号的变换1 2 ( ) e jct 2 ( c ) (sict) j ( c ) ( c )ncos(ct) ( c ) ( c ) e jct 2 ( c ) FTfT (t) FT ( j) An ( n)nA2T 2f (t ).e jn t dt nT T 2f (t) 1 A e jnt T2nnn n ) 12A.e jnt(t

3、) Tn2Tt T2 e jct 2 ( )cF ( j) FTT (t) ( n)n (t) 1e jnt TT n周期冲激序列(t)F ( j )0(1)1t00T (t)1TFSt2 0TFTF ( j )()220Af (t)FT22022周期重复A1 A2f (t)A nTFS2tTTFTF ( j)A n F( j) ASa 2 ( n)nA 2 F ( j) 2A Sa(n )nT0 nT2A0 t周期矩形脉冲的FS和FTF ( j) A Sa 0 2 3.8变换的基本性质a f1 (t ) b f2 (t ) a F1 ( j ) b F2 ( j ) 1线性特性：jt0f (

4、t t0 ) F ( j )e 2 延时特性：FT f (t t ) )e j t dx证明:f (t t00 x t t0FT f (t t ) f (x)e j ( xt0 ) dx0 e j t0f (x)e j xdx e jt0 F ( j) j tF ( j) FT f (t) f (t)edt 2 延时特性：f (t) F ( j)jt0f (t t0 ) F ( j )e一个信号延时，幅度频谱不变，只是相频增加一个线性因子。三、频移（调制）特性FT f( t ) ( j F)j 0 t F ( j j 0 )则： FT f ( t )eFT f (t)e j0t t j tj

5、F( j j )f (t)eedt00 F(jw)F(jw-w0)f (t )e j0t0例2：已知直流 FT1=2(),求复指数函数FTej 0tFT1= 2()FTej 0 t= 2(- 0) ej 0t的频谱是在= 0 强度为2的冲激函数例3：f(t)=cos(0t) 的变换F ( j )f(t)=cos(0t)1/2(ej 0t+ e-j 0t) F(j )=(+ 0) +(- 0)( )( ) 000f (t) F ( j)112j t e j0t )cos t (e000FT f (t) cos t 1 F ( j j) F ( j j)00021212-00FT f ( t )

6、cos 0 t 频谱搬移技术FT f (t)f (t) cos0t调幅信号都可看成乘积信号矩形调幅三角调幅四、尺度变换特性若则FT f (at) 1 F( j )a 0aaFT f (at ) 1 F ( j )a 0aaa 1FT f (t) F( j)FT f (t) F ( j )FT f ( at ) 1 F ( j )aa例4：求下列函数的变换1j () (t )1) 2 ( ) (t ) (t )2)1t 0t 0s n(t )g 12 (t ) (t )3) 1ja 1FT f (t) F( j)例5： 若 FT f (t ) F ( j ), 求 FT f ( at t0 )

7、延时尺度变换分析：f (t ) f (tt0 )f (att0 )时域中的扩展等于频域中的压缩时域中的压缩等于频域中的扩展2F (2 j )f(t/2)210t0F ( j )122扩展20 44t /4 /40压缩1f (2t)五、时域微分特性FT f (t) F( j)若 d nf (t) ( j)F ( j)nFT 则dtn 1 2FF ( j )f (t) F ( j )e1jtdtjdf (t)dt 1 2 1 2de)dF ( jdtjF ( j)e jt d F 1 jF ( j)FT df (t) jF ( j)dt例6：利用微分性质求三角脉冲的变换Ef (t) 22t0f (

8、t) 1 A n1A cosnt T0nn2F ( j ) FT T (t ) ( n )n F T f T( t ) F T( j ) A n ( n ) n 反变换：f ( t ) 1 F ( j )e j t d 2 变换：F ( j ) f ( t ) e j t dt 2t2An f (t)e jntdt Tt11jn tfT (t ) 2 A nen A A e jn nnf (t)A2tTTA220An2 ATn2 A4 0F ( j) n F( j) A Sa 2 ( n)nAsin(n / 2)f (t) e jntT nn /2 f)n1A 2A Sa(n ) 2 F (

9、j)nT2T0nAG (t) A Sa( )2FT f (t) F ( j )a f1 (t ) b f2 (t ) a F1 ( j ) b F2 ( j )jt0f (t t0 ) F ( j )eFT f (t )e j0 t F ( j j 0 )FT f (t) cos t 1 F ( j j) F ( j j)0002j )1FT f ( at ) F (aa d nf (t) ( j)F ( j)nFT dtn六、时域积分特性FT f (t ) F ( j )若则F (0) f (t)dtFTtf ( )d F ( j) ( )F (0)jFT (t ) () 1 e jj例7：

10、用FT积分特性求阶跃的变换 ( ) 1例8：求下面信号的变换f(t)1t-2-112FTtf ( )d F ( j) ( )F (0)jf ( )d F( j) F(0) ()例9、求F(jw)tFTjf (t)f(t)f (t)111（1）（1）1ttt121f (t)f(t)f (t)11（1）1（2）ttt10.5110.5-0.5d f ( t )时域积分公式 若：f ( t )g ( t ) d tG(0) g(t)dt f (t)dt f () f ()F( j) G( j) G(0) () 2 f ()()j G( j) f () f () ()jttf (t) f ( )d f

11、 () g( )d f ()FT t g( )d G( j) G(0) () jt f ( )d f (t) f ()F ( j ) G ( j ) f () f () ( )j例10： f(t)f1(t)f2(t)11/21ttt-1/21/21/2-1/21/2-1/2七 、频域微分 ddtf (t ) F ( j ) jF ( j )djd例11：f (t) F ( j), 求(1 t) f (1变换 j tF ( j) FT f (t) f (t)edt1f (t) 2jtF ( j)ed若已知则若f(t)为实偶函数F(jw)为实偶函数=F(w)若F(jt)为实偶函数=F(t)2f(w

12、)为实偶函数f ( t ) F ( j )F( jt) 2 f ()八、对称（互易）性质F ( ) (t)11F ( )f(t)2 ( )1直流和冲激函数的频谱的对称性是一例子f (t)例13：求f(t) c20tc2 2 22cccf (t)F ( j )100tc2 c2 22cc1F ( j )0c例14： (c22求f(t) t 0F ( j )10cf1(t) F1( j)f2(t) F2( j)若 f1(t)* f2(t) F1( j).F2( j)1 时域卷积2 频域卷积f (t). f (t) 1 F ( j) F ( j)12212九、 卷积定理 f1(t)* f2(t) F

13、1( j).F2( j)1 时域卷积2 频域卷积例15：证明积分性质f (t). f (t) 1 F ( j) F ( j)122121000012121 F ( j j) F ( j j)200卷积FTcos0tFT f (t)调制f (t). f (t) 1 F ( j) F ( j)12212FT f ( t ) cos 0 t f (t )e j0t 1 F ( j ) 2 ( ) F ( j( 例16：)2001例17： 已知 F ( j ) , 求 f (t )( a j ) 211 : e) j a (t ) te f (t) ed12 : te at (t ) j()dj aF

14、 ( jw ) G ( jw ) f ( ) f ( ) ( w )jw 性质函数单边拉氏 线性 Kf (t ) KF ( j ) KF ( s)尺度f(at)1 F ( j )aa1 F ( s )aa延时f (t t0)F ( j )e jt0F ( j )est0时域微分 f (t)jF ( j )sF( s) f (0 )时域积分tf (t )F ( j ) F(0) ( )jF( s) G(0)ss移频f (t )e j0tF ( j( 0 )F ( s 0 )频域微分 tf(t) d F ( j )dj d F (s) ds卷积f1(t) f2 (t)F1( j )F2 ( j )

15、F1 (s)F2 ( s)对称F(jt) 2f(-w)变换为F(j)，求f1(t)=f (2t-1) 的1.已知f(t)的变换。利用变换性质解题n e j 2 nF ( j ) 2、信号的频谱为求信号f(t)并画出其时域波形及频谱图。3-9瓦尔定理与能量频谱 1：周期信号的功率及有效值 2：非周期信号的能量一、周期性信号的功率谱af ( t ) c o s n t s i n n t )0( abnn2n 1 A0A cos nt f (t) nn2n1 1 1T ( a0 )2T功率：P (a2 b2 )f 2 (t )dt设：R1nn220n1 1 A0 n 0 (22)AnPn22n1功率谱只与幅度的平方有关，与相位无关1 Parseval定理：周期信号的功率等于该信号在完备正交函数集中分解后各个子信号功率的和。 2 有效值R1P ( A0 )2 12n1 I2.1A2=I2n2 1 n0( A0 )2I P A2Pnn22n1(In0)2n有效值（值）二、非周期信号（能量信号）的能量 1 2jt )eddtf (t )F ( j 1 2f (t )e jt dt d )F ( j1 1 2F ( j ) 2d )F ( )dF ( jj2