第1章連續(xù)時(shí)間信號(hào)分析

1、x(t)t0t0Asin(t+ )A)2cos()sin()( tAtAtxttt/sin)( Sa 3 2 321t0Sa(t)1sinlim0 ttt2)(0 dttSa dtt)(Sa 0,10,0)(tttu 000,1,0)(ttttttu0u(t)1t0u(t t0)1tt0 1)(0,0)(dtttt )()()()()()(),0()()(00txdtxtxdttttxxdtttx t0(1)(t)()()()(),()0()()(000tttxtttxtxttx )(|1)(),(|1)(00attatattaat tjeteeetttjts sincos t0e t1=00

2、t0Re e jt 1t0Re e st 10t210 x(t)1t420 x(-t/2)1t-1/2-10 x(-2t)1t1/2 10 x(-2t+2)1t420 x(t/2)1t1/2 10 x(2t)1 nntuntunxntuntunxtutuxtutuxntuntunxtutuxtutuxtx )()()()()()()2()()()()()0()()()()2()()()()()0()(x(t)x(0)t0 n 時(shí)時(shí)，有有故故當(dāng)當(dāng)，且且時(shí)時(shí)，由由于于當(dāng)當(dāng)0)()()(0 tntuntudn， dtxntnxntuntunxtxnn)()()()(lim)()()(lim)(00

3、dtxxtxtx)()()()(212110tx1(t)=u(t)(a) 單位階躍信號(hào)單位階躍信號(hào)x2(t)=e- atu(t)t01(b) 單邊指數(shù)信號(hào)單邊指數(shù)信號(hào)x2(-)01(c) 翻褶翻褶y(t)t01/a(f) 卷積值卷積值(e) 相乘并積分相乘并積分x1()x2(t )01t(d) 時(shí)移時(shí)移x2(t )t 01)()()()(1221txtxtxtx )()()()()()(321321txtxtxtxtxtx )()()()()()()(3121321txtxtxtxtxtxtx )()()()()()()()()()()()()()()()()()()(2)(1)()1(212

4、)1(1)1()1(212)1(1)1(211)(1)txtxtytxtxtxtxtyxtxtxtxtytxtxtytxtxtxjiji 推推廣廣為為一一般般形形式式則則，且且有有的的一一階階導(dǎo)導(dǎo)數(shù)數(shù)和和一一次次積積分分分分別別表表示示任任意意信信號(hào)號(hào)若若、)()()()()()(00ttxtttxtxttx tdxtutx )()()()()()(txttx (a) 鋸齒波鋸齒波-T03T02T0 x(t)tT00(b) 半波整流半波整流-T03T02T0 x(t)tT00),()()2()()(000 tnTtxTtxTtxtx均均為為整整數(shù)數(shù)有有理理數(shù)數(shù)2112212211, nnnnT

5、TTnTn 2/2/00| )(|TTdttx，21sin)(2210cos)(2)sincos(2)(2/2/002/2/0010000000 ntdtntxTbntdtntxTatnbtnaatxTTnTTnnnn式式中中，傅傅里里葉葉系系數(shù)數(shù))()(1)()(02/2/0000000 nXdtetxTcenXectxTTtjnnntjnntjnn式式中中，傅傅里里葉葉系系數(shù)數(shù)為相頻特性為相頻特性為幅頻特性，為幅頻特性，其中，其中，)(| )(| )(|)()(00)(000 nnXenXnXtxnj (b) 相頻特性相頻特性0-00-20-302030 (n0)n0(a) 幅頻特性幅頻特

6、性0-00-20-302030|X(n0)|n0周期鋸齒波信號(hào)離散頻譜周期鋸齒波信號(hào)離散頻譜)()()()()()()()()()()()(2221112211210220112211021222111 nXanXatxatxanXanXatxatxanXtxnXtx最最小小公公倍倍數(shù)數(shù)時(shí)時(shí)，則則有有，但但兩兩信信號(hào)號(hào)的的周周期期存存在在當(dāng)當(dāng)時(shí)時(shí)，則則有有當(dāng)當(dāng)若若周周期期信信號(hào)號(hào)，)()()()(00000 nXettxnXtxtjn則則若若周周期期信信號(hào)號(hào)為實(shí)常數(shù)為實(shí)常數(shù)則則若周期信號(hào)若周期信號(hào)anXatxnXtx)()()()(00 )()(| )(| )(|0000nnnXnX 100c

7、os2)(nntnaatx 10sin)(nntnbtx)(20txTtx 0( )2Tx tx t 0)()()()()()()()()()(00)1(00)(000 njnnXdxtxnXjndttxdtxnXjntxnXtxtkkkk 積積分分的的情情況況，即即推推廣廣到到高高階階導(dǎo)導(dǎo)數(shù)數(shù)和和函函數(shù)數(shù)則則若若周周期期信信號(hào)號(hào)t0 x(t)At0 xT(t)AT周期信號(hào)與非周期信號(hào)的關(guān)系：周期信號(hào)與非周期信號(hào)的關(guān)系：)(lim)(txtxTT 000/20/2( )()1()( )jntTnTjntTTx tX neX nx t edtT( )周期信號(hào)展成傅里級(jí)數(shù)，得其中Txt)()(21

8、)()()()()(21)(T1 jXdejXtxtxdtetxjXdedtetxtxtjtjtjtjFF 即即為為則則上上式式方方括括號(hào)號(hào)中中的的部部分分時(shí)時(shí)取取極極限限，可可得得對(duì)對(duì)上上式式兩兩邊邊在在傅里葉變換對(duì)傅里葉變換對(duì))()( jXtx0000/2/2/20/21( )( )( )2TjntjntTTTnTjntjntTTnx tx t edt eTx t edt e例題例題 ： 例例1.3.1)()()()()()()()(221122112211 jXajXatxatxajXtxjXtx則則若若，)(2)()()( xjtXjXtx 則則若若，為為非非零零實(shí)實(shí)常常數(shù)數(shù)若若則則a

9、ajXaatxjXtx)(|1)()()( ，)()()()()()()()(21212211 jXjXtxtxjXtxjXtx則則若若，0)()()()(0tjejXttxjXtx 則則若若， 00)()()( jXetxjXtxtj則則若若，)()(21)()()()()()(21212211 jXjXtxtxjXtxjXtx 則則若若， nnnnnndjXdtxjtdjdXtxjtjXjdttxdjXjdttdxjXtx )()()()()()()()()()()()(，頻頻域域微微分分特特性性時(shí)時(shí)域域微微分分特特性性若若則則 ttdjXjXtxjttxjXjXdxtxjXtx )()(

10、)(1)()0()(1)()0()()()()()1()1(頻域積分特性頻域積分特性時(shí)域積分特性時(shí)域積分特性若若則則 )()(21)()()()()(21)()(21)()()()()()()(1)(1)(sXdsesXjtxtxdtetxsXjddsdejXtxedejXjXetxjXdtetxdteetxetxetxjjststtjttjttjtjtttss LLFF 于于是是有有j j時(shí)時(shí)且且，則則j j令令，可可得得上上式式兩兩邊邊同同乘乘以以進(jìn)進(jìn)行行傅傅里里葉葉變變換換，得得對(duì)對(duì)信信號(hào)號(hào)，拉氏變換對(duì)拉氏變換對(duì))()(sXtx 0)()(dtetxsXst jsLTejXsesX)(2

11、)(1)(2sinc，信號(hào)拉普拉斯變信號(hào)拉普拉斯變換的曲面圖在截?fù)Q的曲面圖在截面面Res=0上的上的曲線就是該信號(hào)曲線就是該信號(hào)傅里葉變換的頻傅里葉變換的頻譜譜 dtetxt|)(| ttte22,ttet)()()(sXsYsH )()(sHth L jssHjH)()( dttytxdttxtyRdttxtydttytxRttxyxxyy)()()()()()()()()()()()(* 是是能能量量有有限限信信號(hào)號(hào)，則則若若、)()(* yxxyRR)()( yxxyRR dttxtxdttxtxRxx)()()()()(* )()( xxxxRR dttxtxdttxtxRdttytx

12、dttytxRdttytxdttytxRxxyxxy)()()()()()()()()()()()()()()( dttxtxTRdttxtyTRdttytxTRTTTxxTTTyxTTTxy 2/2/*2/2/*2/2/*)()(1lim)()()(1lim)()()(1lim)( )()()()()()()()()()()()()()()()()(tytxtRtytgdtyxdttytxtRdtgxtgtxtyttxxyxyg 則則相相關(guān)關(guān)與與卷卷積積的的關(guān)關(guān)系系為為令令互互相相關(guān)關(guān)卷卷積積，有有和和對(duì)對(duì)于于實(shí)實(shí)信信號(hào)號(hào)， 、)()()()()()()()()()(* jXjYRjYjXR

13、jYtyjXtxyxxy FFFF則則若若，2)()()()( jXRtytxxx F則則若若，)()()()()()()()( )()()()()()()()(* jXjYRjYjXdtejYtxdtdetytxdedttytxdeRRdttytxRyxtjjjjxyxyxy FF同理可得同理可得)()()()()()()()()()(* jXjYRjYjXRjYtyjXtxyxxy FFFF則則若若，2)()()()( jXRtytxxx F則則若若，*( )( )()xyRx t y tdt*( )( )( )()jjxyxyRRedx t y tdt ed F*( )()( )()jjtx ty teddtx t Yjedt *()()X jYj *( )()()yxRY jXj 同同理理可可得得F)(square)(squaredutytxtx， 或或)(sawtooth)(sawtoothwidthtxtx， 或或)(S