信號(hào)與系統(tǒng)課件第三章7

1、120 nf (t) A Acosnt n n1AnnA0A1A2nn23423400指數(shù)形式2t2 jntAn T tf (t)edt11jn tf (t ) 2 A nen A A e jn nn三角函數(shù)形式y(tǒng)xAT 2An2 AT2 04 n An 2ATAn2 AT20n4n2 4 0雙邊頻譜單邊幅度頻譜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 ) 幅度頻譜( 偶）

2、)e (） 相位頻譜（奇）反變換：f ( t ) 1 F ( j )e j t d 2 變換：F ( j ) f ( t ) e j t dt (t) 101 2 ( ) 0e t 2 2 2et (t) 1 jAG (t) A Sa( )23.7 周期信號(hào)的變換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周期沖激序列(t)F ( j )0(1)1t00T (t)1TFSt2 0TFTF ( j )()220Af (t)FT22022周期重復(fù)A1 A2f (t)A nTFS2tTTFTF ( j)A n F( j) ASa 2 ( n)nA 2 F ( j) 2A Sa(n )nT0 nT2A0 t周期矩形脈沖的FS和FTF ( j) A Sa 0 2 3.8變換的基本性質(zhì)a f1 (t ) b f2 (t ) a F1 ( j ) b F2 ( j ) 1線性特性：jt0f (

4、t t0 ) F ( j )e 2 延時(shí)特性：FT f (t t ) )e j t dx證明: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 延時(shí)特性：f (t) F ( j)jt0f (t t0 ) F ( j )e一個(gè)信號(hào)延時(shí)，幅度頻譜不變，只是相頻增加一個(gè)線性因子。三、頻移（調(diào)制）特性FT f( t ) ( j F)j 0 t F ( j j 0 )則： 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(),求復(fù)指數(shù)函數(shù)FTej 0tFT1= 2()FTej 0 t= 2(- 0) ej 0t的頻譜是在= 0 強(qiáng)度為2的沖激函數(shù)例3：f(t)=cos(0t) 的變換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 頻譜搬移技術(shù)FT f (t)f (t) cos0t調(diào)幅信號(hào)都可看成乘積信號(hào)矩形調(diào)幅三角調(diào)幅四、尺度變換特性若則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：求下列函數(shù)的變換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、延時(shí)尺度變換分析：f (t ) f (tt0 )f (att0 )時(shí)域中的擴(kuò)展等于頻域中的壓縮時(shí)域中的壓縮等于頻域中的擴(kuò)展2F (2 j )f(t/2)210t0F ( j )122擴(kuò)展20 44t /4 /40壓縮1f (2t)五、時(shí)域微分特性FT f (t) F( j)若 d nf (t) ( j)F ( j)nFT 則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：利用微分性質(zhì)求三角脈沖的變換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 反變換：f ( t ) 1 F ( j )e j t d 2 變換：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、時(shí)域積分特性FT f (t ) F ( j )若則F (0) f (t)dtFTtf ( )d F ( j) ( )F (0)jFT (t ) () 1 e jj例7：

10、用FT積分特性求階躍的變換 ( ) 1例8：求下面信號(hào)的變換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 )時(shí)域積分公式 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 、頻域微分 ddtf (t ) F ( j ) jF ( j )djd例11：f (t) F ( j), 求(1 t) f (1變換 j tF ( j) FT f (t) f (t)edt1f (t) 2jtF ( j)ed若已知?jiǎng)t若f(t)為實(shí)偶函數(shù)F(jw)為實(shí)偶函數(shù)=F(w)若F(jt)為實(shí)偶函數(shù)=F(t)2f(w

12、)為實(shí)偶函數(shù)f ( t ) F ( j )F( jt) 2 f ()八、對(duì)稱（互易）性質(zhì)F ( ) (t)11F ( )f(t)2 ( )1直流和沖激函數(shù)的頻譜的對(duì)稱性是一例子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 時(shí)域卷積2 頻域卷積f (t). f (t) 1 F ( j) F ( j)12212九、 卷積定理 f1(t)* f2(t) F

13、1( j).F2( j)1 時(shí)域卷積2 頻域卷積例15：證明積分性質(zhì)f (t). f (t) 1 F ( j) F ( j)122121000012121 F ( j j) F ( j j)200卷積FTcos0tFT f (t)調(diào)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 性質(zhì)函數(shù)單邊拉氏 線性 Kf (t ) KF ( j ) KF ( s)尺度f(wàn)(at)1 F ( j )aa1 F ( s )aa延時(shí)f (t t0)F ( j )e jt0F ( j )est0時(shí)域微分 f (t)jF ( j )sF( s) f (0 )時(shí)域積分tf (t )F ( j ) F(0) ( )jF( s) G(0)ss移頻f (t )e j0tF ( j( 0 )F ( s 0 )頻域微分 tf(t) d F ( j )dj d F (s) ds卷積f1(t) f2 (t)F1( j )F2 ( j )

15、F1 (s)F2 ( s)對(duì)稱F(jt) 2f(-w)變換為F(j)，求f1(t)=f (2t-1) 的1.已知f(t)的變換。利用變換性質(zhì)解題n e j 2 nF ( j ) 2、信號(hào)的頻譜為求信號(hào)f(t)并畫出其時(shí)域波形及頻譜圖。3-9瓦爾定理與能量頻譜 1：周期信號(hào)的功率及有效值 2：非周期信號(hào)的能量一、周期性信號(hào)的功率譜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設(shè)：R1nn220n1 1 A0 n 0 (22)AnPn22n1功率譜只與幅度的平方有關(guān)，與相位無(wú)關(guān)1 Parseval定理：周期信號(hào)的功率等于該信號(hào)在完備正交函數(shù)集中分解后各個(gè)子信號(hào)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、非周期信號(hào)（能量信號(hào)）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