1、1Fourier Series Representationsof Periodic SignalsChapter 32 Chapter 3 Fourier Series tyste th3.2 The Response of LTI Systems to Complex ExponentialsLTI 系统对复指数信号的响应系统对复指数信号的响应1. Continuous-time system thetyst dehts dehesst Defining dtethsHstste stesH Eigenfunction特征函数特征函数Eigenvalue (特征值）特征值） sstee与时
2、间无关与时间无关 steth3与时间无关与时间无关 Chapter 3 Fourier Series nh nynz2. Discrete-time system nhznyn knkzkh kknzkhz nz nzzH Eigenfunction特征函数特征函数Eigenvalue (特征值）特征值） nnznhzHDefining nznh4 Chapter 3 Fourier Series nkkkzanx nkkkkzzHany tskktskkkesHaea kk tx ty tsktskkesHeContinuous-time system nkknkzzHz Discrete-
3、time systemParticularly tjtjejHe Fourier AnalysisnjjnjeeHe Fourier Analysis5 Chapter 3 Fourier SeriesExample 3.1Consider an LTI system : 3tth 3txty tjetx 2 1 dtethsHst S 2tjetx tjjsesH2232tje 37cos34cos7cos4cos 2tttytttx 21212121 7 7 4 4tjtjtjtjeeeetxtjjee41221tjjee72121tjjee41221tjjee72121 32tjetyd
4、tetst3 se3tjjee266 Chapter 3 Fourier Series3.3 Fourier Series Representation（傅立叶级数）（傅立叶级数） of Continuous-time Periodic Signals3.3.1 Linear Combinations （线性组合）（线性组合）of Harmonically Related Complex Exponentials txTtx0002T Fundamental frequency tjkket 0 , 2, 1, 0k tjkkkeatx0 Fourier SerieskaFourier Ser
5、ies Coefficients Spectral Coefficients （频谱系数）（频谱系数）0aConstantComponent 1aFundamentalComponent 2aSecond HarmonicComponent 7 Chapter 3 Fourier SeriesExample 3.2 tjkkkeatx 233 3/1 , 2/14/1 , 13210aaaa tjtjtjtjtjtjeeeeeetx 6 6 4 4 2 23121411 ttttx 6cos324cos2cos211Consider a real periodic signal txtxtjk
6、kktjkkkeaea00 tjkkkea0 real periodickkaa txtjkkkea0 8 Chapter 3 Fourier Series tjkkkeatx0 tjkkea0 tjkkkeaatx0Re210 kjkkeAa 1 tjkjkkeeAatxk0Re210 kkktkAatx 010cos2 kkkjCBa 2 tjkkkkejCBatx0Re210 tkCtkBatxkkk0010sincos2 tjkktjkkkeaeaa0010 9 Chapter 3 Fourier SeriesExample :Consider an LTI system for wh
7、ich the inputand the impulse response determine the output ttx2cos211 tuetht ty tjtjtjeeetx 2 2041 dtetuejHtjt 0 11 jejHtj11 j tjejHty 00 21412141122jejetytjtjdteetjt 0 0tjejH 2241 tjejH 2241 10 Chapter 3 Fourier Series tjkkkeatx0 tjkkkejkHaty00 3.3.2 Determination of Fourier Series Representation t
8、jkket 0 , 2, 1, 0k002 T dtedtttTtnkjTnk00000 nknk 0 T 0010000Ttnkjnkenkj tjkkkeatx0 dtetxtjnT000 dteeadtetxtjntjkTkktjnT0000000 0TandteeatjntjkkkT0000 11 Chapter 3 Fourier Series dtetxTatjnTn00 0 01 tjkkkeatx0 dtetxTatjkTk0001 Synthesis equation综合公式综合公式Analysis equation分析公式分析公式Specially dttxTaT0 0 0
9、01Average valueExample tjtjejejt002121sin 10 tjtjeet002121cos 20 1 0 21 21 11kajajak-otherwise 0 1k 1/2 ka12 Chapter 3 Fourier SeriesExample 3.5 Periodic square wave defined over one period as 2/ t T 0 t 1 11TTtx1 tx-T -T/2 T1 0 T1 T/2 T t dttxTaTT2/ /2- 01dteTatjkTTk011 - 1 1100TTtjkTjke TkTkak010s
10、in2 Defining xxxcsinsin101sin2TkcTTak 0 whenk1T1TTjktkj00sin2 kTk10sinTT1213 Chapter 3 Fourier Series101sin2TkcTTak 011sin2 kTcT11sin2TcT the envelope(包络包络) of kTa 11sinTTc211sinTTc311sinTTc0sin11TkTcT1固定，固定， 的包络的包络 固定固定kTa11sin2TcT14 Chapter 3 Fourier SeriesTT/20 谱线变密谱线变密Figure 4.2 14 aTT 18 bTT 11
11、6 cTT 15 Chapter 3 Fourier SeriesExample Periodic Impulse Trains (周期冲激串周期冲激串) nTttxn txtT0TT2T2 1 dtetTatjkTTk02/ 2/- 1 21jktTkx teT kaT1 -0 0 0 202 2/T/T, 2, 1, 0 1kTak 212jktTky tHjkeTT 16 Chapter 3 Fourier Series3.4 Convergence（收敛）（收敛） of the Fourier Series1. Approximation(近似性近似性) tjkkNNkNeatx0 t
12、xtxteNNError 1NEN dtetxTaatjkTkk01 2 dtteENTN2Energy ktjkkeatxte0 02dtteTEN最小最小0 N NE dtteteTNN17ka Chapter 3 Fourier Series2. Dirichlet Conditions:Condition 1 dttxT dtetxTatjkTk01 dttxTT1 1T , 1t0 , /1ttx18 Chapter 3 Fourier SeriesCondition 2. In any finite interval , is of bounded variation. tx 1T
13、 , 1t0 , /2sinttx 19 Chapter 3 Fourier SeriesCondition 3. In any finite interval , there are only a finite number of discontinuities.20 Chapter 3 Fourier SeriesGibbs Phenomenon: Figure 3.921 Chapter 3 Fourier Series3.5 Properties of Continuous-Time Fourier Series3.5.1 Linearity kkbtyatxFSFS kkkBbAac
14、tBytAxtzFS3.5.2 Time Shifting katxFS00FS0tjkkeattx dtettxTbtjkTk001 deexTtjkjkTtt0000100tjkkea 22 txtx3.5.3 Conjugation and Conjugate Symmetry（共轭及共轭对称性）（共轭及共轭对称性） kkatxatxFSFS Chapter 3 Fourier Series kkaa txtxkkaaor Particularly 00aa kj akkaa eis real0akkaakj akkaaekj akkaa ekkaa 23 Chapter 3 Fouri
15、er Series3.5.4 Time Reversal katxFSkatxFStjkkkeatx0 tjmmmeatx0 mktx Fourier Series Coefficients of kkaakkaa txtxkkaa txtxreal even txkareal eventjmmmea0 24 Chapter 3 Fourier Serieskkaakkaa txtx0kkaa txtxreal odd txkaPurely imaginary odd3.5.5 Time Scaling tjkkkeatx0 tajkkkeaatx0 0 kak0 akak25 Chapter
16、 3 Fourier Series3.5.6 Multiplication（相乘）（相乘） kkbtyatxFSFS tytx3.5.7 Parsevals Relation（帕兹瓦尔关系式）（帕兹瓦尔关系式） 221kkTadttxT dttxtxTdttxTTT112tjkkkTeaT01 nnkkaaT11 knkn ka aTT2kkaFSkmk mmha bConvolution Sum dteatjnnn0 Ttjntjkdtee00 26kajk0 Chapter 3 Fourier Series 221kkTadttxT2kaAverage Powerof kth harmon
17、ic Average Power of tx3.5.8 Differential Property katxFS knnajktx0FS kajktx0FS dtetxTbtjkTk01 txdeTtjkT01 dtejktxTetxtjkTTtjk0001 0dteaTtjkkT201 27 Chapter 3 Fourier Series1 tx-4 -2 1 0 1 2 4 tExample 2 40 T21411 -1 0dta dttdx1-4 -2 1 0 1 2 4 t1 1414nntndx tdttn 1214jke 2sin2kjkkajka2 kkak2/sin FSkd
18、x tadt 214jke 28 Chapter 3 Fourier SeriesExample 3.6 tg-2 -1 0 1 2 t21211 tx-4 -2 1 0 1 2 4 t 211 txtg2 40 T kkbtgatxFSFS 02100 ab 2 0 k0 in k/2 k0kkjbske kkak2/sin29 Chapter 3 Fourier Series1 kctxFS -4 -2 0 2 4 t Example 3.72 40 T 0 11Tcx t dtT kbdttdxFS-2 -1 0 1 2 t2121 02 kkbjkc 0k 2/2/kin 0k 0 2
19、22 kjkejksc21420kkbcjk 30 Chapter 3 Fourier Series kddttxdFS22-4 -2 0 2 4 t 11 244 22ntntdttxdnn kjkked1141414122 kkcjkd20 even is k 0k 0 odd is k 0k 2/21 0k 21 2 jkck20 jkdckk31 Chapter 3 Fourier Series3.6 Fourier Series Representation of Discrete-Time Periodic Signals 3.6.1 Linear Combinations of
20、Harmonically Related Complex Exponentials NnxnxN/20 nNjknjkkeen 20; 1, 2 , 1 , 0Nk nnNkk nNjkkNknjkkNkeaeanx 20FourierSerieskaFourier Series Coefficients Spectral Coefficients32 Chapter 3 Fourier SeriesnjkjknjkeeHe000 njkkNkeanx0 3.6.2 Determination of the Fourier Series RepresentationskaUnknown Coe
21、fficients1. kNkax0 NjkkNkeax 21121NNjkkNkeaNx Linearly independent 11NNNNnkxNNa1njkjkkNkeeHa00 33 Chapter 3 Fourier Series2. nNjknjkkeen 20; 1, 2 , 1 , 0Nk Nnrknn nNjrnNjkNnee 22 Nnrknn nNjrnNjkkNkNnnNjrNneeaenx 222 nNjrnNjkNnkNknNjrNneeaenx 222rNamNrk NmNrk-eenNrkjNnNrkj 01122 交换求和交换求和次序次序34 Chapte
22、r 3 Fourier Series nNjrNnrenxNa 21 nNjkNnnjkNnkenxNenxNa 2110 nNjkkNknjkkNkeaeanx 20Synthesis Equation Analysis Equation Discrete-time Fourier Series nNNkjNnNkenxNa 21kakais periodic nnNkk The discrete-time series is a finite series with N terms.35 Chapter 3 Fourier SeriesExample 3.10 nnx0sin NN 212
23、/ 00 nNjnNjejejnx 222121jajarNrN21 2111 22/ 00NMNM nMNjnMNjejejnx 2 22121 jajarNMrNM21 21-6 -1 4 -4 1 6 k 21j 21j5N ka-3 2 7 -2 3 8 k 21j 21j5N , 3M ka36 Chapter 3 Fourier SeriesExample 3.12 Discrete-time periodic square waven1N0N11NNnNjkNNnkeNa 2111mNkeeeNNjkNNjkNNjk 11212211 NjkNjkNjkNNjkNNjkNjkee
24、eeeeN2/122/12111 mNkNN 121 kamNkNN 121N /sin/2/12sin11mkNkNNkN 37 Chapter 3 Fourier Series3.7 Properties of Discrete-time Fourier Series（不要求）（不要求）3.8 Fourier Series and LTI Systems tjkkkeatx0 Linear Combinations of Eigenfunctions Frequency Responseof LTI System dtethjHtj tjkkkejkHaty00 Periodic Sign
25、alContinuous-time LTI System38 Chapter 3 Fourier Series njkkNkeanx0 Linear Combinations of Eigenfunctions Frequency Responseof LTI System njnjenheH Discrete-time LTI System njkjkkNkeeHany00 Periodic Signal39 Chapter 3 Fourier SeriesExample Consider an LTI system with input the unit impulse response
26、determine the Fourier Series Representation of output nttxnn1 ty tuetht4 tx1-2 -1 0 1 21 0 2T 122ntnttxn jkkea2121k11 21dteejHtjt 40 tjkkkejkHaty kbkb0 k is even41 jkk is odd41 j40 tjkkkeatx0 dtethjHtj 00 jktLTIkky ta HjkeContinuous-time LTI System njkkNkeanx0 njnjenheH Discrete-time LTI System 00 j
27、kjknLTIkkNy na H ee Chapter 3 Fourier Series41 Chapter 3 Fourier Series3.9 Filtering（滤波）（滤波） th tjkkkejkHaty00 tjkkkeatx0 dtethjHtj Filter Frequency-Shaping Filter频率成形滤波器频率成形滤波器Frequency-Selective Filter频率选择性滤波器频率选择性滤波器3.9.1 Frequency-Shaping Filter421. Equalizer（均衡器）（均衡器）43 Chapter 3 Fourier Series
28、2. Differentiator （微分器）（微分器）D tx dttdx tuth1 dtetujHtj 1 sgn2jejjH0jH sgn2jH22 0 1-0 0 0 1sgn 44 Chapter 3 Fourier SeriesFigure 3.2445 Chapter 3 Fourier Series3. Discrete-time system 121nxnxny 121nnnh njnjenheH 2/cos21212/ jjee jjeHeH2/2jH e 222/cosjeH10 njKeKnx0 KeeKHnynjj00 njnenx 1 0 njjeeHny 46
29、Chapter 3 Fourier Series3.9.2 Frequency-Selective Filter 1. Ideal Lowpass Filter (LPF) ccjH 0 1 01cjHcPassband Stopband Stopband Continuous-time systemDiscrete-time system kkeHccj2 02 1 01cjeHc22Passband Stopband Stopband 47 Chapter 3 Fourier Series2. Ideal Highpass Filter (HPF) ccjH 0 1 Continuous-
30、time systemDiscrete-time system 12 012 1 kkeHccj01jeH2201cjHcPassband Stopband Passband 48 Chapter 3 Fourier Series3. Ideal Bandpass Filter (BPF) others 0 1 21 jHContinuous-time system01jH1212Discrete-time systemothers 0 1 21 jeH 21001jeH2249 Chapter 3 Fourier SeriesExample Consider an LTI system wi
31、th input the frequency response of this system is as shown in Figure 1 , determine the output of system kttxk2 tyjH012/3jHFigure 1 (a)2/301jHFigure 1 (b)2/52/32/3 2/5 0 2T21ka tjkktjkkkeeatx 21 tjkkkejkHaty00 50 Chapter 3 Fourier Series tjtjeety 2121 a tty cos21 tjtjeety 2 221 b tty 2cos 3.10 Exampl
32、es of continuous-time filters described by differential equations3.10.1 A simple RC Lowpass Filter tvtvdttdvRCscc tvsR tiC tvc51 Chapter 3 Fourier Series tjctjsejHtvetv tjtjtjeejHejHdtdRC tjtjeejHRCj 1 RCjjH 11211RCjH RCarctgjH 2RC1RC5 . 0RC2RC1RC5 . 0RCjH tvs tvc52 Chapter 3 Fourier Series3.10.2 A simple RC Highpass Filter RtvdttvtvdCrrs RtdvRCdttdvRCtvsrrRCjRCjjG 121RCRCjG RCarctgjG 23.11 Examples of discrete-time filters described by difference equations （不要求（不要求 ） tvsR tiC rvtjG tvs tvr53 Chapter 3 Fourier SeriesHomework: 3.1 3.13 3.15 3.34 3.35 3.43*
