信號與系統(tǒng) 第三章課件_圖文

1、Chapter 3 Fourier Series Representation of Periodic Signals第3章周期信號的傅里葉級數(shù)表示Main content :1.The Frequency Analysis of PeriodicSiganl(周期信號的頻域分析2.The Frequency Analysis of LTI(LTI系統(tǒng)的頻域分析3.Properties of Fourier Series(傅立葉級數(shù)的性質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 (歷史的回顧 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傅里葉的兩個最重要的貢獻“周期信號都可以表示為成諧波關系的正弦信號的加權和”傅里葉的第一個主要論點“非周期信號都可以用正弦信號的加權積分來表示”傅里葉的第二個主要論點3.2 The Response of LTI Systems to Complex Exponentials(LTI 系統(tǒng)對復指數(shù)信號的響應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(連續(xù)時間周期信號的傅里葉級數(shù)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 這表明用傅里葉級數(shù)可以表示連續(xù)時間周期信號,即: 連續(xù)時間周期信號可以分解成無數(shù)多個復指數(shù)諧波分量。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根據可繪出的頻譜圖。稱為占空比k a (x t 12T T10212T T =10214T T =10218T T =不變時1T 0T不變時1T 0T 10212T T =10214T T =10218T T =周期性矩形脈沖信號的頻譜特征:1. 離散性2. 諧波性3. 收斂性考查周期和脈沖寬度改變時頻譜的變化:0T 12

18、T 1.當不變,改變時,隨使占空比減小,譜線間隔變小,幅度下降。但頻譜包絡的形狀不變,包絡主瓣內包含的諧波分量數(shù)增加。2.當改變,不變時,隨使占空比減小,譜線間隔不變,幅度下降。頻譜的包絡改變,包絡主瓣變寬。主瓣內包含的諧波數(shù)量也增加。1T 1T 0T 0T 1T 0T信號對稱性與頻譜的關系:當時,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:偶信號的是關于的偶函數(shù)、實函數(shù)。k a k 當時,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:奇信號的是關于的奇函數(shù)、虛函數(shù)。k a k3.4 Convergence of the FourierSeries(連續(xù)時間傅里葉級數(shù)的收斂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(連續(xù)時間傅里葉級數(shù)的性質 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,當在變化時,從變化,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(離散時間周期信號的傅里葉級數(shù)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