信号与系统 第三章课件

1、Chapter 3 Fourier Series Representation of Periodic Signals第3章周期信号的傅里叶级数表示Main content :1.The Frequency Analysis of PeriodicSiganl(周期信号的频域分析2.The Frequency Analysis of LTI(LTI系统的频域分析3.Properties of Fourier Series(傅立叶级数的性质3.0 Introduction(引言The basis for time domain(chapter21Signal can be represented

2、 as linear combination of shift impulses。2System is LTI。Periodic Singal can be represented as linear combination of complex exponentials.3.1 Historical Perspective (历史的回顾 1、The concept of using trigonometric sums to describe periodic phenomena goes back to Babylonians2、Euler examined the motion of V

3、ibrating string is a linear combination of a few normal mode in 1748.3.1 Historical Perspective (cont 3、Largange criticized the use of trigonometric series to examine vibrating string in 1759.4、Fourier claimed that any periodic signal could be represented by harmonically related sinusoids in1807.som

4、e story about FourierBorn in France in 1768Fourier claimed that any periodicsignal could be represented byharmonically related sinusoids in1807Due to Lagrange s objection his 17681830paper never appearedHis paper appeared in “TheAnalytical Theory of Heat” in 1822Dirichlet provide precise conditionsi

5、n 1829傅里叶的两个最重要的贡献“周期信号都可以表示为成谐波关系的正弦信号的加权和”傅里叶的第一个主要论点“非周期信号都可以用正弦信号的加权积分来表示”傅里叶的第二个主要论点3.2 The Response of LTI Systems to Complex Exponentials(LTI 系统对复指数信号的响应ste nz(h n (h t ste(y t nz(y ncontinuios timediscrete timeUsing Time domain method ,(s t sts sty t eh d eh e d H s e-=(n k nknk k y n zh k z

6、h k zH z z-=-=-=EigenvalueGain is called “Eigenvalue”Eigenfunction in-> Same function out with gain Eigenfunctiondiscrete time(h n (h t ste(stH s enz(nH z Zcontinuious timeEigenfunctionEigenvalue(stH s h t e dt-=(nk H z h n z-=-=The usefulness of decomposition in term of eigenfuction is important

7、 for the analysis of LTI systems . ts kk k k es H a t y =(ts kk k ea t x =(If :nkkk Za n x =(n kkk k ZZ H a n y =(complex exponential signal 、are eigenfuctionof LTI systems、are eignevalue.ste nz (H s (H z Conclusion:How broad a class of signals could berepresented as a linear combination of complex

8、exponentials?qustion Example 1 ( 3.1: a LTI systems y(t=x(t-3 , now the inputx(t=cos(4t+cos(7t, detemin y(t?ss e d e s H 33(-+-=-=y(t= 1/2e -j12e j4t + 1/2e j12e -j4t + 1/2e -j21e j7t + 1/2e j21e -j7t=cos4(t-3+cos7(t-3x(t= 1/2e j4t +1/2e -j4t +1/2e j7t +1/2e -j7tThe set of harmonically related compl

9、ex exponentials0(jk t k t e =0,1,2,k =± ±Each of these signals has a fundamental frequency that is multiple of 0,each is periodic with period 02T =3.3 Fourier Series Representation of Continuous-Time Periodic Signals(连续时间周期信号的傅里叶级数表示3.3.1. Linear Combinations of Harmonically RelatedComplex

10、 exponentialsThus , is also periodic,the form is referred to as the Fourier series representation 这表明用傅里叶级数可以表示连续时间周期信号,即: 连续时间周期信号可以分解成无数多个复指数谐波分量。0(,0,1,2jk t k k x t a e k =-=±±Example 2:0(cos x t t =001122j t j t e e -=+112a ±=Example 3 :00(cos 2cos3x t t t =+00003312j t j t j t j

11、 t e e e e -=+112a ±=31a ±=Some alternative form for the Fourier series 0000*(jk t jk t jk t jk t k k k k k k k k x t a e a e a e a e-*-=-=-=-=-=or k k a a*-=*k k a a -=(t x t x *=Suppose x(t is real ,then is expressed in polar form as k j k k a A e =k a 0001(01(k k k j jk t j k t j k t k

12、k k k k k x t A e e a A e A e -+=-=-=+Some alternative form for the Fourier series (CONT0001k k jk t j jk t j k k k a A e e A e e -=+*k kj j k k k k a a A e A e -=Q thus :k k A A -=k k-=-Conclusion: is even ,and k a k is odd0001(k k jk t j jk t j k k k x t a A e e A e e -=+0012cos(k k k a A k t =+tr

13、igonometric functions formis expressed in rectangular form as k k ka B jC =+k a 00101(jk t jk tk k k k k k x t a B jC e B jC e -=-=+0001(jk t jk t k k k k k a B jC eB jC e -=+*k k a a -=Q k k k kB jC B jC -=+thus k k B B -=k kC C -=-Conclusion: the real part of is even ,the imaginary part of is odd

14、k a ka0001(jk t jk t k kk k k x t a B jC e B jC e-=+-00012cos sin k k k a B k t C k t =+-trigonometric functions form(another form3.3.2. Determination of the Fourier SeriesRepresentation of a continuous-time Periodic Signal Assuming periodic signal x(t can be represented with the Fourier series0(,jk

15、 t k k x t a e =-=002T =00(jn t j k n tk k x t e a e -=-=0000(00(T T jn tj k n tk k x t e dt a e dt-=-=000(00000cos(sin(T T T j k n t e dt k n tdt j k n tdt -=-+-00,T =k n k n =0000(T jn t n x t e dt a T -=consequently 00001(T jn t n a x t e dt T -=Notice : the integration can be over any interval o

16、f length T01(jk t k a x t e dt -=01(T a x t dtT =a 0is simply the average value of x(t over one period 10T 0T -t (x t The spectrum of periodic square waveExample4 (3.5 :11|1,(|/20,t T x t T t T <=<<The spectrum of periodic square wave (Cont10011101000002sin 11T jk tjk t T k T T k T a e dt e

17、 T jk T k T -=-=101111010010002sin 222Sa(sinc(T k T T T T k T k T k T T T T =sin Sa(x x x =sin sinc(xx x=Where0-(Sa x 1x 0121-sin (c x 1x1根据可绘出的频谱图。称为占空比k a (x t 12T T10212T T =10214T T =10218T T =不变时1T 0T不变时1T 0T 10212T T =10214T T =10218T T =周期性矩形脉冲信号的频谱特征:1. 离散性2. 谐波性3. 收敛性考查周期和脉冲宽度改变时频谱的变化:0T 12

18、T 1.当不变,改变时,随使占空比减小,谱线间隔变小,幅度下降。但频谱包络的形状不变,包络主瓣内包含的谐波分量数增加。2.当改变,不变时,随使占空比减小,谱线间隔不变,幅度下降。频谱的包络改变,包络主瓣变宽。主瓣内包含的谐波数量也增加。1T 1T 0T 0T 1T 0T信号对称性与频谱的关系:当时,有(x t x t =-0000220020012(cos T T jk t T k a x t e dt x t k tdt T T -=表明:偶信号的是关于的偶函数、实函数。k a k 当时,有(x t x t =-0000220020012(sin T T jk t T k a x t e d

19、t j x t k tdt T T -=-表明:奇信号的是关于的奇函数、虚函数。k a k3.4 Convergence of the FourierSeries(连续时间傅里叶级数的收敛3.4.1 The validity of Fourier Series Assume: a given periodic signal (x t Now : approximating by a linear combination of a finite number of harmonically relatedcomplex exponentials(x t 0(Njk tN k k Nx t a e

20、 =-=3.4.1 The validity of Fourier Series(contApproximation error : (N N e t x t x t =-the criterion : minimize the energy in the error over one period00220011(N N N T T E t e t dt x t x t dt T T =-000*01(N N jk t jk t k k T k N k N x t a e x t a e dt T =-=-=-0001(jk t k T a x t e dtT -=Conclusion (p

21、roblem 3.66:3.4.2 The conditions that periodic signal can be represented by a Fourier Seriestwo problems may be occur :may diverge.may not converge to x(t Two classes of conditions1:x(t have finite energy over a single period.0001(jk tk T a x t e dt T -=0(,jk t k k x t a e =-=02(T x t dt <Two cla

22、sses of conditions(Cont2:The Dirichlet conditionsover any period ,x(t must be absolutely integrable0000011(jk t k T T a x t e dt x t dt T T -=<k a Thus is finiteThe Dirichlet conditions (ContThere are no more than a finite number of maximaand minima during any single period of the signalIn any fi

23、nite interval of time, there are only a finitenumber of discontinuitiesSignal that violate the Dirichlet conditions 3.4.3.Gibbs phenomenonHow the Fourier Series converges for a periodic signal with discontinuities?N=N=3 1N= 7N=19 3.4.3.Gibbs phenomenon(ContAs N increases, the ripple in the partial s

24、umsbecome compressed toward the discontinuity, but for any finite value N, the peak amplitude of the ripples remains constant.3.5 Properties of Continuous-time Fourier Series(连续时间傅里叶级数的性质 These properties are useful for developing conceptual insights into such representations, and can also help to r

25、educe he complexity of the evaluation of the Fourier Series.3.5.1 linearity :,denote two periodic signals with period (F k x t a (Fky t b (x t (y t Tthen (F k kAx t By t Aa Bb +3.5.2 time shifting :000(jk t F k x t t a e -(F k x t a denote a periodic signals with period (x t T then 02T =3.5.3 Time R

26、eversal :(F kx t a -(F k x t a denote a periodic signals with period (x t T then 3.5.4 time Scaling :(F k x t a denote a periodic signals with period (x t T then 0/(jka t F k T a ax at b x at e dtT -=令,当在变化时,从变化,at =t 0/T a 0T 于是有:01(jk k k T b x e d a T -=(F k k x at b a =3.5.5 Multiplication :,den

27、ote two periodic signals with period (Fkx t a (Fky t b (x t (y t Tthen01(jk tFk Tx t y t C x t y t e dtT -=g 001(jl t jk tk l T l C a e y t e dtT -=-=g 0(1(j k l tk l l k lT l l C a y t e dt a b T -=-=-=(Fl k l k kl x t y t a b a b -=-=*3.5.6 Conjugation and Conjugate Symmetry :(Fk x t a denote a pe

28、riodic signals with period (x t Tthen*-*ka t x(If is real signal thenk ka a*-=kka a *-=Some derived consequence:(x t k kk kA A -=-kj k k a A e=3.5.7 Parsevals Relation for Continuous -time periodic signals :+-=k k T a dt t x T 22(1conclusion :the total average power in a period of the periodic signa

29、l equal the sum of the average powers in all of its harmonic components .Example 5(p208:+-=-=k kT t t x (-T1tT(t x 0/2/211(T jk tk T a t edt TT-=01(jk tk x t eT=-=02T=Example 6:periodic square wave(t g 11T -1T +-T.Tt(11'T t x T t x t g -+=Derivate of the periodic square wave(q t g t '=1t1T +

30、1T -1T T -1T T -+(FFkkg t c g t b 'using differential property 0k kb jkc =Using time shifting property0101012sin jk T jk T k k k b a e eja k T -=-=From example 5 1/k a T =02/T=010112sin sin 2k k b k T k T T c jk k T T k T =3.6 Fourier Series Representation of Discrete-Time Periodic Signals(离散时间周期信号的傅里叶级数表示 3.6.1 Linear Combination of Harmonically Related Complex ExponentialsThe set of all discrete-time complex exponential signals are20,1,2,.j kn Nk