信号与系统重邮课件第三章

1、 信号的单、双边频谱，利用性质求信号的信号的单、双边频谱，利用性质求信号的 傅氏变换，系统的频域分析，取样定理。傅氏变换，系统的频域分析，取样定理。 利用性质求信号的傅氏变换，系统的频域利用性质求信号的傅氏变换，系统的频域 分析。分析。 1 000 )sincos()( n nn tnbtnaatf 2/ 2/ 0 )( 1 T T dttf T a ), 3 , 2 , 1( cos)( 2 2/ 2/ 0 ntdtntf T a T T n ), 3 , 2 , 1( sin)( 2 2/ 2/ 0 ntdtntf T b T T n T/2 0 nn baa, 0 1 00 )cos()

2、( n nn tnAAtf 22 nnn baA )( n n n a b arctg 00 aA )cos( 0nn tnA 0 A n tjn ne Ftf 0 )( 2/ 2/ 0 )( 1 T T tjn n dtetf T F n F 0000 00 00 000 1 0 1 0 1 0 1 ( )cossin 22 22 nn n jntjntjntjnt nn n jntjnt nnnn n jntjnt nn n f taantbnt eeee aab j ajbajb aee aF eF e F-n=1/2(a-n-jb-n)= 1/2(an+jbn)= Fn* n tjn

3、n eFtf 0 2 2 0 1 T T tjn n dtetf T F 其中复系数其中复系数 00 1 1 nn jntjnt nn nn FeF e 令令 记记F0=a0 又有又有 所以所以 n j nnnn eFjbaF |)( 2 1 nnnn AbaF 2 1 2 1 | 22 nn |F-n| =|Fn| -n = -n 关于关于n偶函数偶函数 关于关于n奇函数奇函数 f(t) t0T -T A /2- /2 T A dttf T a T T 2/ 2/ 0 )( 1 ) 2 sin( 2 cos)( 2 0 2/ 2/ 0 n n A tdtntf T a T T n 0 n b

4、 1 0 0 cos) 2 sin( 2 )( n tn n n A T A tf ) 2 () 2 sin( )( 2 1 )( 1 00 2/ 2/ 0 n Sa T An n A jbadtetf T F nn T T tjn n n tjn n tjn n e n Sa T A eFtf 00 ) 2 ()( 0 x x xSa sin )( 0 x 1 - 2 -2 -3 3 dxxSa)( x x xSa sin )( /4 0 /4 1/2 22 1/2 4 ( )sin 48 2sinsin() 22 T n T bf tntdt T An Atn tdt n 0 0 0 n

5、a a f(t) t02-2 A 1-13 -A 奇函数、奇谐函数奇函数、奇谐函数只含正弦的奇次谐波。只含正弦的奇次谐波。 其它 20.50.5 ( )2 (1)0.51.5 0 Att f tAtt 000 1 22 1 2 ( )(cossin) sin() 8 2 sin 8111 (sinsin3sin5sin7) 92549 nn n n f taantbnt n A n t n A tttt 0 /2 /2 22 11 ( )() 2 4 sin() 22 T jnt nnn T n Ff t edtajb T jAjn b n 0 22 4 ( )sin() 2 jntjn t

6、n nn Ajn f tF ee n 1 00 )cos()( n nn tnAAtf n tjn ne Ftf 0 )( 0 n 0 An 0 A0 A1 A2 A3 2 03 0 0 n 0 n 0 1 2 3 2 03 0 0 n 0 |Fn| 0 F0 |F1| |A2| |A3| 2 03 0 |F-1| |F-2| |F-3| - 0-2 0-3 0 0 n 0 n 02 03 0 - 0-2 0-3 0 1 2 3 -1 -2 -3 2 8111 ( )(sinsin3sin5sin7) 92549 A f ttttt 2 8111 ( )cos()cos(3)(cos5)(co

7、s7) 292252492 A f ttttt f(t) t02-2 A 1-13 -A 2 8111 ( )cos()cos(3)(cos5)(cos7) 292252492 A f ttttt n n 0 0 An n 00 3 5 2 8A 2 2 8 9 A 2 2 8 25 A n n0 Fn n00 0 3535 2 4 9 A 2 4 25 A 2 4 9 A 2 4A 2 4 25 A 22 4 sin() 2 n Ajn F n 22 4 ( )sin() 2 jn t n Ajn f te n 2 4A 2 2 ) 2 ()( 1 0 2/ 2/ 0 n Sa T A dt

8、etf T F T T tjn n f(t) t0T-T A /2- /2 n 311 21 3 3 E S(costcost ) 11 2 (cos) E St 5111 211 (coscos3cos5) 35 E Sttt -0.5-0.4-0.3-0.2-0.100.10.20.30.40.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 111111 2111 (coscos3cos5cos11) 3511 E Stttt -0.5-0.4-0.3-0.2-0.100.10.20.30.40.5 -1 -0.8 -0.6 -0.4 -0.2 0

9、 0.2 0.4 0.6 0.8 1 2/ 2/ 2 )( 1 T T dttf T P 1 2 2 0 2 2 | n n n n A AFP n n T T Fdttf T P 2 2/ 2/ 2 |)( 1 物理意义：物理意义：周期信号在时域中的平均功率等周期信号在时域中的平均功率等 于频域中的直流分量和各次谐波分量的平均于频域中的直流分量和各次谐波分量的平均 功率之和。功率之和。 0 2 | nFn 0 n 2 4 0 |Fn|2 0 n Fn t f (t) t 2 2 0 n Fn F( ) 2 2 A T 0 2 T fT(t) t A dtetfFtf tj )()()( dt

10、tf| )(| )(tfT )(lim)(tftf T T 2/ 2/ 0 )( 1 T T tjn Tn dtetf T F lim( )( ) j t n T TFf t edtF 记为 F() () ( )( )|( )| ( )cos( )sin j tj Ff t edtFe f ttdtjf ttdt * ()()F jFj |( )|F )( Re( )F Im( )F deFtfF tj )( 2 1 )()( deF edfFTeTF eFtftf tj tj n tjn n T n tjn n T T T )( 2 1 )(/lim lim)(lim)( 0 0 n tjn

11、 ne Ftf 0 )( 0 n n F tjn e 0 deFtf tj )( 2 1 )( )2/()( dF tj e )( F f F FTFF n f nn T 00 lim 2 limlim)( )( F 0 )( n n T F F n T TFF lim)( )( F n F 1 )(t j t 1 )( )( )(2 AA )( )( Ftf aj A atAe at )0( )( 2 ) 2 () 2 ( SaAttA )( )( Ftf j t 2 )sgn( )()( )cos( 000 t )( )( Ftf 0 00 t et f t t () ( ) () 1 0

12、 jt Fft edt j ()( )() 1.1.单边指数信号单边指数信号 f (t) t0 1 22 1 F () arctg ()() 幅频：幅频：相频：相频： )( 0 2 2 )(F 0 1 0 jt Fft edt j ()( )() 0 0 t t t et f te et ( ) 2.2.双边指数信号双边指数信号 0 0 22 112 tjttjt Feedteedt jj () f(t) 0t 22 2 F () 0( ) )(F 0 f(t) 0t 0 0 t t t et f te et ( ) 2 2 0 At ft t () ( ) () 2 2 2 2 22 2 j

13、t A FAedtAASa / / sin() ()sin()() 3.3.矩形脉冲信号矩形脉冲信号 f(t) t 0 2 2 A 4 )( )(F 2 6 E f(t) t 0 2 2 A 10 10 t f tt t () ( )sgn( ) () 0 atat a f tetet ( )lim( )() 0 112 a F ajajj ()lim 2 F( ) 2 2 0 0 () ( ) () 4.4.符号函数符号函数 )(F )( 2 2 Sgn(t) +1 -1 1)()(dtetF tj f t ( ) 1 0 t 1 0 )(F t f t ( ) 0 2F( ) 2 0 5.

14、 5. 冲激函数傅立叶变换对冲激函数傅立叶变换对 1 11 ( )( ) 22 j t FTed 00 ttsin?cos? 00 0 2 jtjtjt FFT eeedt ()() 1 2 ( ) jt ted 6.6.虚指信号的傅立叶变换虚指信号的傅立叶变换 7.7.周期信号的傅立叶变换周期信号的傅立叶变换 n tjn ne Ftf 0 )( 0 2 n n FFn ()() 0 n n F 2 )(tfT t02/T2/T TT A )(tf t02/T2/T A ) 4 ( 2 1 )()( 2 T ATSaFtf ) 2 ( 2 1 ) 4 ( 2 1)( 2 0 2 0 n ASa

15、 Tn ASa T F F n n n n nTT n n ASa nFFtf )() 2 ( )(2)()( 0 2 0 )()( )()( 22 11 Ftf Ftf )()()()( 2121 bFaFtbftaf ?)( t j j tt 1 )( 1 )()(2 )(1)( ? f(t) 1 t0.5-0.5 2 1.5-1.50 )5 . 0()5 . 1(3)( SaSaF )()()( 13 tgtgtf )()( Ftf )( | 1 )( a F a atf ( )f tF 例例: ()ft 的频谱函数。的频谱函数。 ，求，求 解解: Ftf)( 1a j t 1 )()(

16、 11 ( 0.5 )() | 0.5|0.5/( 0.5) 1 ( ) t j j 1)(t 5 . 0)2(t ?)2(t ?)5 . 0( t f(t/2) 0t 2 2(2 )F 0 (2 )ft 0 t 2 4 4 压缩压缩 扩展扩展 1 4 4 1 1 () 22 F 0 扩展扩展压缩压缩 )()( Ftf )(2)( ftF ?A? 1 t ?)(tSa )(2)(2 AAA AtA)( j t 2 )sgn( )sgn(2 2 jt )sgn()sgn( 1 jj t ?A? 1 t ?)(tSa ) 2 ()( SaAtAg )(2) 2 ( Ag t SaA )()()(

17、22 ggtSa ?A? 1 t ?)(tSa 2( ) ( ) t 1 1 1 ()f t ( )F ( )F ( )f t ( )F 2 2 2 2 ( )f t ( )F 2 c 2 c 2 c 2 c t t 1 2 c 1 00 00 )(? )(? e)2cos(? j t 1 )()( )(2 1 )( jt t )( 2 1 2 )( t j t 1 1 )( j te t )(2 1 1 e jt )( )1(2 1 e jt )(? )(? e)2cos(? )2()2(2cos t )2cos(2)2()2( tt )2cos()2()2( 2 1 tt )(? )(?

18、e)2cos(? )()( Ftf 0 )()( 0 tj eFttf 00 0 () () ( )( ) ( ) j tj tj tj FeFee Fe j t 1 )()( 0 0 0 1 ()( ) 1 ( ) j t j t tte j e j ?)( 0 tt ?)( 0 tt 1)(t 0 )( 0 tj ett )()( Ftf?)( 0 tatf 0 )( | 1 )( 0 t a j e a F a tatf )()( Ftf 0 )()( 0 tj eFttf )()( Ftf )()( 0 0 Fetf tj )()( Ftf?cos)( 0 ttf )(tf t 0 c

19、os ttf 0 cos)( )()( 2 1 )()( 2 1 cos)( 00 0 00 FF etfetfttf tjtj )( Y m 00 0 2/A m 0m 00 m 0 )( F m m 0 A ttfty 0 cos)()( ttf 0 cos)( )()( Ftf?)42( 3 tj etf )()( Ftf 4 )()4( j eFtf 2 4 ) 2 ( |2| 1 )42( j eFtf )3(23 ) 2 3 ( 2 1 )42( jtj eFetf )()( Xtx )()( Hth )()()()(*)()( HXYthtxty )(tx )(*)()(thtx

20、ty )( X )()()( HXY ?)(*)( 2 tete tt )2)(1( 1 2 1 1 1 )(*)( 2 jjjj tete tt ? f(t) t 0 A - ) 2 () 2 () 2 ( )( 1 *)()( 2 SaASaSaA tgtAgtf FTFT FT 乘 卷 ( ) T t 0 t T 2T 3T-T-2T-3T (1)(1)(1)(1)(1)(1)(1) ( )f t t 0 2 T 2 T A )(tfT t0 2/T2/T TT A * = )(tfT t0 2/T2/T TT A 2 0 2 0 ( )( )( )()( ) 12 ()() 24 ()

21、 () 2 TTT n n ftf ttFFTt T ATSan T n ASan )()( Ftf )()( Gtg )(*)( 2 1 )()( GFtgtf )()( Ftf?cos)( 0 ttf )()(cos 000 t )()( 2 1 )()(*)( 2 1 cos)( 00 00 0 FF F ttf )( )( Fj dt tdf )()( )( Fj dt tfd n n n )()( Ftf ?)(t ?)( )( t n 1 1 )()()( j jtt nn jt)()( )( )()( Ftf?)( 0 tatf )()( Ftf 0 )()( 0 tj eFtt

22、f 0 )( | 1 )( 0 t a j e a F a tatf 0 )( | 1 )( 0 t a j e a F a jtatf )()( Ftf )()( Fjtf 0 )()( 0 tj eFjttf 0 )( | 1 )( 0 t a j e a F a j a tatf )()( Ftf j jF Fdftf t )( )()0()()( )1( 00 (0) j t FFf t edtf t dt 1)(t j tt 1 )()()( )1( ?)(t ( )( )( )tft ( ) ( )( )( )(0) ( ) t f tdF j ( )f t ( ) ( )( )( ) t f tdF j ( ) ( )( )( ) ( )() ( ) t f tdFff j ( )( )()() 22 AA tftgtgt ) 2 ()()( 22 jj eeASat ? f(t) t 0 A - t 0 - dt tdf t )( )( /A /A t 0 ? )(tf 1 2 ) 2 ( )( )()0()( 2 SaA j F t dtf )()( 1)()()( tft j dtf t )( )()0()(2 )(1)( j F 1 )(3)( 0 (1) t )()(tft t 0 ? )(tf 1 2 )()()()(