1、1The Continuous-Time Fourier TransformChapter 42 Chapter 4 Fourier Transform4.1 Representation of Aperiodic Signals :The Continuous-Time Fourier Transform1 tx-T -T/2 T1 0 T1 T/2 T t101sin2TkcTTak 011sin2 kkTcTTa3 Chapter 4 Fourier TransformT1 0 T1 t tx 1 , 0Tttx -T T1 0 T1 T t tx 12 TT kTtxtxk 2 0 o
2、therwisex ttT/x t tjkkeatx0 T 204 Chapter 4 Fourier Transform 01jktkax t edtT Defining dtetxjXtj 0110 jkXTjXTakk tjkkejkXTtx001 00021 tjkkejkXtx 0/2/21TjktkTatTx t ed 0/2/21TjktTxedtTt 012T 5 Chapter 4 Fourier Transform01ktjejX 0 00021 tjkkejkXtxtjkejkX00T tx dejXtj 2100 0kArea 6 Chapter 4 Fourier T
3、ransform jtXjx t edt 1 2jtx tXjed Synthesis equationAnalysis equationFourier Transform Pair1. A linear combination of complex exponentials.factor txjXSpectrum（频谱）（频谱） of 2.Consider a periodic signal txTtxDefining others 0 00Tttttxtx jXtxF7 Chapter 4 Fourier Transform 000 1tTjktktx taedtT 000 1tTjktk
4、taex tdtT 0 -1jktkax t edtT 0k0110 jkXTjXTakkThe Fourier coefficients of are proportional to samples of the Fourier transform of one period of ka tx tx8 Chapter 4 Fourier Transform4.1.2 Convergence of Fourier TransformsDirichlet Conditions: dttx1. be absolutely integrable. tx2. have a finite number
5、of maxima and minimawithin any finite interval . tx3. have a finite number of discontinuities within any interval. Futhermore ,each of these discontinuities must be finite. txLet dejXtxtj 21Defining txtxte dttx2 02dtte9 Chapter 4 Fourier Transform4.1.3 Fourier Transforms of Typical SignalsExample 4.
6、1 0 atuetxat dtetuejXtjat dtetja 001tjaeja jajX1221 ajXajX/arctg jatueFat12/2aa/1aajXaajX2/2/4/4/10 Chapter 4 Fourier TransformExample 4.2 0 aetxtadteejXtjta dteetjat 0 ja 1222 aajX2220 aaaeFtadteetjat 0 ja 111 Chapter 4 Fourier TransformExample 4.3 ttx dtetjXtj 1Ft 10jX 2jX detxtj 2211 21 F 20112 C
7、hapter 4 Fourier TransformT1 0 T1 t1 txExample 4.4 t 0 t 1 11TTtxdtejXtjTT 11 11 1TTtjej 1sin2TjX11sin2TcT 12T01/T1/TjX jeeTjTj11 13 Chapter 4 Fourier Transform-W 0 W 1jX detxtjWW 21WWtjetj 21 tWttx sin 0 1 WWjX Example 4.5WtcWsin tjeetjWtjW 2 14 Chapter 4 Fourier Transform 0 1 WWjX F tWttx sin11sin
8、2TcT t 0 t 1 11TTtxF WtcWtxsin /Wt0W/W/-W 0 W 1jXF WtcWtWsinlim 15 Chapter 4 Fourier TransformExample 1 0 1-0 0 0 1sgnttttt11 tsgn tuetuetatata0limsgn tuetuetatataFlimsgnF0 lim0a jt2sgnF tueata0limtueata0limaj 1aj 116F02 Chapter 4 Fourier Transformt01 tuExample 2 tutx jtuF1 1sgn21ttu jt1sgn21F F2/1E
9、xample 3 tjetx0 0tt 0 tje detj 0221tje0 dtetttj 0F 200Xj 17 Chapter 4 Fourier Transform jtXjx t edt 1 2jtx tXjed Synthesis equationAnalysis equation jXtxF 11. 0Fateu taaj 2222. 0a tFaeaa 3. 1 12FFt 18 Chapter 4 Fourier Transform 0 1 WWjX F tWttx sin11sin2TcT t 0 t 1 11TTtxF F25. sgn tj 16. Fu tj 007
10、. 2jtFe 4.19 Chapter 4 Fourier Transform4.2 The Fourier Transforms for Periodic Signalstjtjeet0021cos0 tjtjeejt0021sin0 0F20 tje0F20 tje00F0cos t0000F0sin jjtj00j20 Chapter 4 Fourier TransformMore generally txTtx tjkkeatx0 02 kajXkkPeriodic square wave kTkak10sin010sin2 kkTkjXkT/2 0 21 Chapter 4 Fou
11、rier TransformTak1TkTjXk 22T2 -0 0 0 20jX00jXPeriodic impulses train kTttxk 22 Chapter 4 Fourier Transform4.3 Properties of the Fourier Transforms 4.3.1 Linearity jYtyjXtxFF jbYjaXtbytaxF-2 -1 0 1 2 t 11 tx-2 0 2 t 1 tx2 tx12-1 0 1 t txtxtx21 2sin2F1tx 2sin2F2tx 2sin24sinFtx23 Chapter 4 Fourier Tran
12、sform4.3.2 Time Shifting jXtxF0 F0tjejXttx dejXttxttj0210 deejXttxtjtj 002100 tjXjtjejXejX Example 4.90 1 2 3 4 t 2/31 tx24 Chapter 4 Fourier Transform4.3.3 Conjugation and Conjugate Symmetry jXtxF dtetxjXtj dtetxjXtj jXjXtxtxLet jXjjXjXImRe jX txtx jXjXImIm jXjXReRe jX Re jXj Im25 Chapter 4 Fourier
13、 TransformLet jXjejXjX jX txtx jXjX jXjX dtetxjXtj dtetxjXtj tLet dexjXj txtx jXjX txtxIf jX jXje26 jXjX Chapter 4 Fourier Transformreal even txreal even jX txtx and txtxtxtx jXjXjXreal odd txPurely imaginary odd jX txOdtxEvtx jXjjXImReF jXtxEvReF jXjtxOdImF27 jXjdttxdnFnn Chapter 4 Fourier Transfor
14、mExample tuetueeatatta tueEveatta 2 0 1ajatueFat 1Re2 jaeFta0 222aaa 4.3.4 Differentiation and Integration1. Differentiation jXjdttdxF28 Chapter 4 Fourier Transform jXjdttxd2F22 dttdx-2 0 2 t1-1 22dttxd-2 0 2 t (1)(-2)(1)2222 jeejXjj2022 txtExample 2F2jet 2F2jet 22Ft 22sin4 Example jtuF129 Chapter 4
15、 Fourier Transform2. Integration jXjXdxFt10 4.3.5 Time and Frequency Scaling ajXaatx 1F dteatxatxtj F adex aja 0 adexaja 0ajXa 1 jXtxFLet 1a30jX1jX2 Chapter 4 Fourier Transform1 tP2t0F 2/sin21jX2/sin c1 tPt202F sin22jX csin22231 Chapter 4 Fourier TransformExample jXtxF?26Ftx 6F6jejXtx 6F6jejXtx 3F2/
16、2126jejXtxMore generallyajbeajXabatx/F/1 32 Chapter 4 Fourier Transform jXtxF xjtX2F dejXtxtj 21tx 2交换交换 , t dtejtXxtj 2 4.3.6 Duality（对偶性）（对偶性）Example 1Ft 21F dejXtj 233 Chapter 4 Fourier Transform t 0 t 1 11TTtx 1sin2TFttT1sin2F x2 0 1 WWjX F tWttx sinExample 4.13 ?12F2 jXttx222 aaeFta2 12 Fte 221
17、2 etF 0 2 11TT 34 Chapter 4 Fourier TransformExample 4.14 ?1F jXttx jt2sgnFjt2 sgn1Fjt4.3.7 Differentiation in Frequency Domain djdXtjtxF jXtxF21t F21t sgn2F sgn2Fj35 Chapter 4 Fourier TransformExample 1 21 F 2Fjt jtF2 jjtF22 222jtFMore generally nnFnjt2 jt2sgnF 2F2sgn jtjt2F2 t36 Chapter 4 Fourier
18、TransformExample 2-2 -1 0 1 2 t jX1-2 -1 0 1 2 t 11djdX111122djXd-2 -1 0 1 2 t jXtxF?112222 djXd00F0cos tttcos12cos1 txjt2 ttttx2coscos12 37 Chapter 4 Fourier TransformExample 3 0 1 ajatueFat tujteat 0 12ajatuteFat tuejtat2 0 1213F2ajatuetat 0 1! 11Fajatuetnnatn 2 jajF3F2 jaj38 Chapter 4 Fourier Tra
19、nsform4.3.8 Frequency Shifting 0F0 jXetxtj jXtxFjX000jX0jX txtje00jX tjetx0tje0jX tx39 Chapter 4 Fourier Transform4.3.9 Parsevals Relation djXdttx2221 dttx2 dttx 交换积分次序交换积分次序 21 djX djXjX212 jXEnergy-density spectrum（能量谱密度）（能量谱密度） dttxtx dejXtj21 dtetxtj 40 Chapter 4 Fourier TransformSolution of Inf
20、inite Differentiation dtetxjXtj dttxX00 dejXtxtj21 djXx210 djXdttx2221 djXjdttdxt21041 Chapter 4 Fourier Transform dttxE2 0tdttdxDExample 4.14 Evaluate the following time-domain expressions:-1 -0.5 0 0.5 1 2/jXFigure (a)jj-1 0 1 jXFigure (b) 8521 a2 djXE 121 b2 djXE djXj21D aeven odd0 djXj21D b 2111
21、0d42 Chapter 4 Fourier TransformExample 1 2sin 1 t 1 0 t 1 x tF12sin2d sintdtt Example 221?1dtt 221tFe 2120121dx 211dtt sin?tdtt 01x43 Chapter 4 Fourier TransformExample 3221?1dtt 221tFe 2221221d 022 0tte dtedt122121dtt 2tedt44 ajXaatx 1F Chapter 4 Fourier Transform xjtX2F djdXtjtxF jXtxF45 Chapter
22、4 Fourier Transform 0F0 jXetxtj jXtxF djXdttx2221 dttxX0 djXx21046 Chapter 4 Fourier TransformExample 4 Consider a signal with Fourier Transform Evaluate the following frequency-domain expressions:-1 0 1 2 3 t 1 tx txjX？djX ？0 jX？jX 1 202Xjdx 2 0Xx t dt124132 is real even function. 3 1x t F1jx tXje
23、jX jXje 0 or Xj or Xj 47 Chapter 4 Fourier Transform4.4 The Convolution Property y tx th tYjXjHj Fx tXj Fh tHj jtxYjeddth t 交换积分次序交换积分次序 YjxdjHje jYjxedHj YjXjHj jth tedt 48FT Chapter 4 Fourier Transform kx tx tkT00012jktkXjke 000012jktkXjkHjke 00 x tx t 12j tx tXjed 12j ty tXjHjed th txjHjX thtxty
24、jHjXjY 00y ty t 49 Chapter 4 Fourier TransformLPF filter 1 0 ccH j 01cjHcYjXjHj 0 ccXj Series Systems th1jH1 th2jH2jH th12HjHjHj 12h th tht50 Chapter 4 Fourier TransformExample 4.150tt 0h ttt 0 jtH je jH01jH00 jtYjXje 0y tx ttExample 4.16 h tt Hjj0jHjH2251 Chapter 4 Fourier Transform 1Hjj This is an
25、 unstable system.Example 4.17 tu ty txd h t dtYjXjHj 10YjXXjj 52 Chapter 4 Fourier TransformExample 4.18 1. Ideal LPF filter01cjHcNote : . It is not causal. It is not easy to approximate closely. The impulse response decays with oscillations./ct0c/c/ tcthccsin53 Chapter 4 Fourier Transform2. Causal
26、LPF filter2/1111jH tuetht11jjHNote : . It is causal. It is easy to approximate closely. The impulse response decays without oscillations.t01 th54 Chapter 4 Fourier Transform tuetht tu ty2 sinttthc tu ty101t ty2 ty101tcc2155 Chapter 4 Fourier TransformThe usefulness of the Convolution PropertyExample
27、 4.19 , 0 ;btx teu tb , 0 ;ath teu ta ?y tx th tYjXjHjSolution 1Yjjajb 1 ab 2 ab21Yjja aty tteu tABYjjajb A jbB jajajb11babaYjjajb 1atbty teeu tba56 Chapter 4 Fourier TransformExample 4.2011 1 0 Xj F 1sintx tt 22 1 0 Hj F 2sinth tt YjXjHj00 1 0 0sinty tt 112sin tt 221sin tt 012min, 57 Chapter 4 Four
28、ier TransformThe Fourier Analysis of periodic signals 00jktkky ta Hjke 0jktkkx ta e 02kkXjak YjXjHj 02kka Hjk 002kka Hjkk 00jktkky ta Hjke 58 Chapter 4 Fourier TransformExample Shows in Figure (a) is an LTI system, and are illustrated in Figure (b),(c) . jH2jH12. If the input determine the output tt
29、tx6sin213cos1 tyjH2jH1 ty tx+-1. Determine the frequency response of the equivalent overall system with input and output jH ty txFigure (a)01jH122Figure (b)01jH244Figure (c)59 Chapter 4 Fourier TransformjH-4 -2 0 2 4 1 cos3y tt121HjHjHjSolution 01jH1122 3366111112222j tj tj tj tx teeeejj 11cos3sin62
30、x ttt 00H31Hj60Hj01jH24401jH12260 Chapter 4 Fourier Transform4.5 The Multiplication property (modulation property)（调制特性）（调制特性） r ts t p t12R jS jP j j ts t p t edt 12jtj ts tP jededt 交换积分顺序交换积分顺序1 2P jd S j 1 2S jP jd F1 2s t p tS jP j jts t edt 61 Chapter 4 Fourier Transform r ts t p t12R jS jP j F
31、x th tXjHj Fx tXj Fh tHj 62 Chapter 4 Fourier Transform 0jtp te 02F F01 22s t p tS j 0S j tyjX tx0jte 0jte jX txt0cost0sin tx tyRe tyIm ty实现较困难实现较困难63 Chapter 4 Fourier Transform00F F001 2x t p tXj F00011cos 22x ttXjXj 0cosp tt 20cos y tx tt 011cos2 22y tx tx tt LPF x t64 Chapter 4 Fourier Transform
32、jX0MMAjH txt0cos tx trM2jR0002At0cos tyjY002M202M24A2AMM信道信道 2 0 otherscHj M20 2 McM -jHcc65 Chapter 4 Fourier TransformExample 4.23 Determine the Fourier transform of the signal 2sin sin/2ttx tt sin/2sinttx ttt 01jX11101jX22/12/12/1jX212323210F12 2XjXj 66jXjY tx ty01W3W3jH2 tx tr1 tr2W3W31jH1W5W5 t
33、y tr3Wt3cosWt5cosjXW20W21 Chapter 4 Fourier TransformExampleIn Figure (a) ,a system is shown with input and outputThe input signal has the Fourier transform shown in Figure (b)Determine and sketch . Figure (a)Figure (b)67 Chapter 4 Fourier Transform-7W -5W -3W 0 3W 5W 7W jR121 -5W -3W 0 3W 5W jR221
34、-8W -6W -2W 0 2W 6W 8W jR341-2W 0 2W jY4168 Chapter 4 Fourier Transform4.5.1 Frequency-Selective Filtering with Variable Center Frequency中心频率可调的频率选择性滤波器中心频率可调的频率选择性滤波器cjXjY tytjce txtjce tf01WWjH trjX0jY0cjHWW0WWjR jHjYjRW2c69 Chapter 4 Fourier Transform0W2jFcjX0The frequency response of the overall
35、 system is01jGW2c tytjce txtjce tf01WWjH trjGW2c70 Chapter 4 Fourier Transform4.6 Frequency-domain analysis of LTI systemsLTI系统的频域分析系统的频域分析 th txjHjX tyjY jYty-1F 1. jXjYjH . 2 jHth1F jHjYjX . 3 jXtx1F thtxty jHjXjY1. Stable System71kkNkkkMkjajb 00 Chapter 4 Fourier Transform2. Linear Constant-Coeff
36、icients Differential Equations kkMkkkkNkkdttxdbdttyda00FT jXjbjYjakkMkkkNk 00 jXjYjHA ratio of polynomials in j72 Chapter 4 Fourier Transform3. Partial-Fraction Expansion 01110111asasasabsbsbsbsHnnnnmmmm Strictly proper rational function真分式真分式 sHnm , sHnm , sDsNcscscsHkk01 js 1Ft jtF真分式真分式 nFnjt 01F
37、01cjcjctctctckkkk 730 Chapter 4 Fourier Transform真分式的部分分式展开真分式的部分分式展开 分母具有分母具有n个不同的单根个不同的单根 npspspssNsH21 nkkknnpsApsApsApsAsH12211 nnpsApsApsAsHps221111ps 111pssHpsA kpskksHpsA74 Chapter 4 Fourier TransformExample 4.25Consider a stable LTI system txdttdxtydttdydttyd234223422 jjjjH312 jjj31 jjjH1 j3
38、 j2121 0 1ajatueFat tuetuethtt3212175 Chapter 4 Fourier Transform 分母具有分母具有1个个r阶重根，其余均为单根阶重根，其余均为单根 311N sH sspDs 131211321111N sAAAH sspDsspsp 32311312111111N sspH sAAspAspspDs 13131spAspH s 33112111112N sddspH sAAspspdsdsDs0 13112pssHpsdsdA1sp01sp76 311221131222pssDsNdsdAsHpsdsd Chapter 4 Fourier T
39、ransform1ps0 131221121pssHpsdsdA可用数学归纳法证明：可用数学归纳法证明： sDpssNsHr11 111! 1pssHpsdsdkrArkrkrk77 Chapter 4 Fourier TransformExample 4.25Consider the LTI system with input txdttdxtydttdydttyd23422 tuetxt 2213jjj YjXjHj2131Yjjjj 1j3j121441132jjjdjd14 0 1! 11Fajatuetnnatn tuetuetutetyttt341412178 Chapter 4
40、Fourier TransformExample Consider an LTI system txdttxddttxdtydttdydttyd22332223231223 jjjjjH21542 jjjjjH23112 jjjjH tuetuettthtt232 79Homework: 4.3 4.4 4.10 4.11 4.14 4.15 4.24 4.25 4.32 4.35 4.36 4.37 4.43 Chapter 4 Fourier Transform80 Chapter 4 Fourier Transform ttx 2cos1t ttttx 6cos32 4cos 2cos2t tttx 4cos 2cos2t42jX224F22jX1F42jX324326326F
