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=00t0R
2、e 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 时时,有有故故当当,且且时时,由由于于当当0)()()(0 tntuntudn,dtxntnxntuntunxtxnn)()()()(lim)()()(lim)(00 dtxx
3、txtx)()()()(212110tx1(t)=u(t)(a)单位阶跃信号单位阶跃信号x2(t)=e-atu(t)t01(b)单边指数信号单边指数信号x2(-)01(c)翻褶翻褶y(t)t01/a(f)卷积值卷积值(e)相乘并积分相乘并积分x1()x2(t )01t(d)时移时移x2(t )t 01)()()()(1221txtxtxtx )()()()()()(321321txtxtxtxtxtx )()()()()()()(3121321txtxtxtxtxtxtx )()()()()()()()()()()()()()()()()()()(2)(1)()1(212)1(1)1()1(2
4、12)1(1)1(211)(1)txtxtytxtxtxtxtyxtxtxtxtytxtxtytxtxtxjiji 推推广广为为一一般般形形式式则则,且且有有的的一一阶阶导导数数和和一一次次积积分分分分别别表表示示任任意意信信号号若若、)()()()()()(00ttxtttxtxttx tdxtutx )()()()()()(txttx (a)锯齿波锯齿波-T03T02T0 x(t)tT00(b)半波整流半波整流-T03T02T0 x(t)tT00),()()2()()(000 tnTtxTtxTtxtx均均为为整整数数有有理理数数2112212211,nnnnTTTnTn 2/2/00|)
5、(|TTdttx,21sin)(2210cos)(2)sincos(2)(2/2/002/2/0010000000 ntdtntxTbntdtntxTatnbtnaatxTTnTTnnnn式式中中,傅傅里里叶叶系系数数)()(1)()(02/2/0000000 nXdtetxTcenXectxTTtjnnntjnntjnn式式中中,傅傅里里叶叶系系数数为为相相频频特特性性为为幅幅频频特特性性,其其中中,)(|)(|)(|)()(00)(000 nnXenXnXtxnj (b)相频特性相频特性0-00-20-302030(n0)n0(a)幅频特性幅频特性0-00-20-302030|X(n0)|
6、n0周期锯齿波信号离散频谱周期锯齿波信号离散频谱)()()()()()()()()()()()(2221112211210220112211021222111 nXanXatxatxanXanXatxatxanXtxnXtx最最小小公公倍倍数数时时,则则有有,但但两两信信号号的的周周期期存存在在当当时时,则则有有当当若若周周期期信信号号,)()()()(00000 nXettxnXtxtjn则则若若周周期期信信号号为为实实常常数数则则若若周周期期信信号号anXatxnXtx)()()()(00 )()(|)(|)(|0000nnnXnX 100cos2)(nntnaatx 10sin)(nnt
7、nbtx)(20txTtx )(20txTtx 0)()()()()()()()()()(00)1(00)(000 njnnXdxtxnXjndttxdtxnXjntxnXtxtkkkk 积积分分的的情情况况,即即推推广广到到高高阶阶导导数数和和函函数数则则若若周周期期信信号号t0 x(t)At0 xT(t)AT周期信号与非周期信号的关系:周期信号与非周期信号的关系:)(lim)(txtxTT 000/20/2()()1()()jntTnTjntTTx tX neX nx t edtT()周周期期信信展展成成傅傅里里,得得其其中中Txt)()(21)()()()()(21)(T1 jXdejX
8、txtxdtetxjXdedtetxtxtjtjtjtjFF 即即为为则则上上式式方方括括号号中中的的部部分分时时取取极极限限,可可得得对对上上式式两两边边在在傅里叶变换对傅里叶变换对)()(jXtx0000/2/2/20/21()()()2TjntjntTTTnTjntjntTTnx tx t edt eTx t edt e)()()()()()()()(221122112211 jXajXatxatxajXtxjXtx则则若若,)(2)()()(xjtXjXtx 则则若若,为为非非零零实实常常数数若若则则aajXaatxjXtx)(|1)()()(,)()()()()()()()(2121
9、2211 jXjXtxtxjXtxjXtx则则若若,0)()()()(0tjejXttxjXtx 则则若若,00)()()(jXetxjXtxtj则则若若,)()(21)()()()()()(21212211 jXjXtxtxjXtxjXtx 则则若若,nnnnnndjXdtxjtdjdXtxjtjXjdttxdjXjdttdxjXtx )()()()()()()()()()()()(,频域微分特性频域微分特性时域微分特性时域微分特性若若则则 ttdjXjXtxjttxjXjXdxtxjXtx )()()(1)()0()(1)()0()()()()()1()1(频域积分特性频域积分特性时域积分
10、特性时域积分特性若若则则 )()(21)()()()()(21)()(21)()()()()()()(1)(1)(sXdsesXjtxtxdtetxsXjddsdejXtxedejXjXetxjXdtetxdteetxetxetxjjststtjttjttjtjtttss LLFF 于是有于是有j j时时且且,则,则j j令令,可得,可得上式两边同乘以上式两边同乘以进行傅里叶变换,得进行傅里叶变换,得对信号对信号,拉氏变换对拉氏变换对)()(sXtx 0)()(dtetxsXst jsLTejXsesX)(2)(1)(2sinc,信号拉普拉斯变信号拉普拉斯变换的曲面图在截换的曲面图在截面面Re
11、s=0上的上的曲线就是该信号曲线就是该信号傅里叶变换的频傅里叶变换的频谱谱 dtetxt|)(|ttte2)()()(sXsYsH)()(sHth L jssHjH)()(dttytxdttxtyRdttxtydttytxRttxyxxyy)()()()()()()()()()()()(*是是能能量量有有限限信信号号,则则若若、)()(*yxxyRR)()(yxxyRR dttxtxdttxtxRxx)()()()()(*)()(xxxxRR dttxtxdttxtxRdttytxdttytxRdttytxdttytxRxxyxxy)()()()()()()()()()()()()()()(d
12、ttxtxTRdttxtyTRdttytxTRTTTxxTTTyxTTTxy 2/2/*2/2/*2/2/*)()(1lim)()()(1lim)()()(1lim)()()()()()()()()()()()()()()()()()(tytxtRtytgdtyxdttytxtRdtgxtgtxtyttxxyxyg 则则相相关关与与卷卷积积的的关关系系为为令令互互相相关关卷卷积积,有有和和对对于于实实信信号号,、)()()()()()()()()()(*jXjYRjYjXRjYtyjXtxyxxy FFFF则则若若,2)()()()(jXRtytxxx F则则若若,)()()()()()()(
13、)()()()()()()()()(*jXjYRjYjXdtejYtxdtdetytxdedttytxdeRRdttytxRyxtjjjjxyxyxy FF同同理理可可得得)()()()()()()()()()(*jXjYRjYjXRjYtyjXtxyxxy FFFF则则若若,2)()()()(jXRtytxxx F则则若若,*()()()xyRx t y tdt*()()()()jjxyxyRRedx t y tdt ed F*()()()()jj tx 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)(Sa)sin()(sincttttx )2(Sa)(dirictnntx ,)(intintf)(intintfbatftf,或或x(t)A-T/2-/2/2-TtT0T/2)(fourier)(fourierwtxXxX,或或)(ifourier)(ifouriertwXxXx,或或)(laplace)(laplacestxtxsxtxs,或或)(ilaplace)(ilaplacetsxsxtxsxt,或或