1、Chapter 4The Continuous-Time Fourier Transform Chapter 4 Fourier Transform jtXjx t edt factor Synthesis equationAnalysis equationFourier Transform Pair tx jXSpectrum（频谱）（频谱）of jXtxF4.1 Representation of Aperiodic Signals:The Continuous-Time Fourier Transform 1 2jtx tXjed 0110 jkXTjXTakkThe Fourier c
2、oefficients of are proportional to samples of the Fourier transform of one period of ka tx txConsider a periodic signal txTtxDefining others 0 00Tttttxtx Chapter 4 Fourier Transform Chapter 4 Fourier Transform4.1.3 Fourier Transforms of Typical Signals 11.Fateu taj 2222.0a tFaeaa 1 0 WXjW F sinWtx t
3、t 112sinTcT 11 1 t 3.0 t Tx tTF 4.1 12FFt Chapter 4 Fourier Transform F25.sgn tj 16.Fu tj 007.2jtFe 4.2 The Fourier Transforms for Periodic Signalstjtjeet0021cos0 tjtjeejt0021sin0 00F0cos t0000F0sin jjtj00j Chapter 4 Fourier TransformMore generally TtxTtx/2 0 tjkkeatx0 02 kajXkkPeriodic square wave
4、kTkak10sin010sin2 kkTkjXk Chapter 4 Fourier TransformTak1TkTjXk 22T2 -0 0 0 20jX00jXPeriodic impulses train kTttxk Chapter 4 Fourier Transform Chapter 4 Fourier Transform4.3 Properties of the Fourier Transforms 4.3.1 Linearity FF x tXjy tYj Fax tby taXjbYj Chapter 4 Fourier Transform4.3.2 Time Shift
5、ing Fx tXj 0 F0jtx ttXje Example 4.90 1 2 3 4 t 2/31 tx Chapter 4 Fourier Transform-2 -1 0 1 2 t 11 txExample4.3.3 Conjugation and Conjugate Symmetry F1.xtXj x txtXjXj Chapter 4 Fourier Transform 2.xtx tXjXjreal even txreal even jXreal odd txPurely imaginary odd jX x tEv x tOd x tReImFXjjXj FReEv x
6、tXj FImOd x tjXj Chapter 4 Fourier Transform nnFnd x tjXjdt 4.3.4 Differentiation and Integration1.Differentiation Fdx tjXjdt Chapter 4 Fourier Transform2.Integration 10tFxdXXjj 22F2d x tjXjdt dttdx-2 0 2 t1-1 22dttxd-2 0 2 t(1)(-2)(1)2222jjeeXj 2022 txtExample F22jte F22jte F22t 224sin Chapter 4 Fo
7、urier Transform4.3.5 Time and Frequency ScalingF1jx atXaa FxtXj Let 1a Chapter 4 Fourier TransformExample Fx tXj F62?xt F66jxtXje F66jxtXje F3162/22jxtXje Fx tXj F2Xjtx 12jtx tXjed 2jtxtXjed 交换交换 ,t 2jtxXjt edt 4.3.6 Duality（对偶性）（对偶性）Example 1Ft 122F Chapter 4 Fourier Transform12sinT 11 1 t 0 t Tx t
8、TF12sin T ttF11 2 2 0 TxT 0 1 WWjX F tWttx sinExample 4.13?12F2 jXttx222a tFaea 221tFe 2221Fet Chapter 4 Fourier TransformExample 4.14?1F jXttx F2sgn tj F22 sgnjt F1sgnjt 4.3.7 Differentiation in Frequency Domain FdXjjtx td Fx tXj F221sgnjt F21t Chapter 4 Fourier TransformExample 1 21 F 2Fjt jtF2 jj
9、tF22 222jtFMore generally nnFnjt2 jt2sgnF 2F2sgn jtjt2F2 t Chapter 4 Fourier 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 Chapter 4 Fourier TransformExample 3 0 1 ajatueFat 2 jajtujteFat 0 12ajatuteFat 3F22
10、jajtuejtat 0 1213F2ajatuetat F111 0 !natnt eu tanaj Chapter 4 Fourier Transform4.3.8 Frequency Shifting 0F0 jXetxtj jXtxFjX000jX0 Chapter 4 Fourier Transform4.3.9 Parsevals Relation djXdttx2221Solution of Infinite Differentiation dtetxjXtj 0Xx t dt0 dejXtxtj21 102xXjd djXdttx2221 Chapter 4 Fourier T
11、ransformExample 1 sin2 1t 0 1t 1 txF dsin221 dtttsinExample 2?112dtt2 12 Fte 1012212xd dtt211?sindttt 10 x Chapter 4 Fourier TransformExample 3?1122dtt2 12 Ftedtedt22221221 dtedtett20 20121122 dtt Chapter 4 Fourier TransformExample 4 Consider a signal with Fourier Transform Evaluate the following fr
12、equency-domain expressions:-1 0 1 2 3 t 1 tx txjX？djX？0 jX？jX 202 xdjX dttxX0 314221 is real even function.1tx jejXtxF1 jXjejX0 or Xj or Xj Chapter 4 Fourier Transform4.4 The Convolution Property thtxty jHjXjY jXtxF jHthF th tx jH jX thtxty jHjXjY Chapter 4 Fourier TransformThe usefulness of the Con
13、volution PropertyExample 4.19 ;0,btuetxbt ;0,atuethat?thtxty1YjXjHjjajbSolution b a ba 21ajjY tutetyatbjabajabjY 11 tueeabtybtat1 Chapter 4 Fourier TransformExample 4.20 0 1 11 jXF tttx 1sin 0 1 22 jHF ttth 2sin jHjXjY21000,min 0 1 ttty 0sin211 sin tt122 sin tt Chapter 4 Fourier Transform4.5 The Mul
14、tiplication property(modulation property)（调制特性）（调制特性）tptstr jPjSjR21 tjetp0 02 F 00jtFs t eS j Chapter 4 Fourier Transform00 F 21 21cos00F0 jXjXttx ttp0cos Example 4.23 Determine the Fourier transform of the signal 22/sinsinttttx tttttx 2/sinsin01jX11101jX22/12/12/1jX212323210 jXjX21F 2 Chapter 4 Fo
15、urier Transform 12jXjY tx ty01W3W3jH2 tx tr1 tr2W3W31jH1W5W5 ty tr3Wt3cosWt5cosjXW20W21ExampleIn 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)Chapter 4 Fourier Transform-7W -5W -3W 0 3W 5W 7W jR1
16、21-5W -3W 0 3W 5W jR221-8W -6W -2W 0 2W 6W 8W jR341-2W 0 2W jY41 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 -11.Fy tYj 2.YjHjXj jHth1F 3.YjXjHj jXtx1F thtxty jHjXjY1.Stable System Chapter 4 Fourier Transform jXjYjHkkNkkkMkjajb 00 2.Linear
17、Constant-Coefficients Differential Equations kkMkkkkNkkdttxdbdttyda00FT jXjbjYjakkMkkkNk 00A ratio of polynomials in j Chapter 4 Fourier TransformExample 4.25Consider a stable LTI system txdttdxtydttdydttyd234223422 jjjjH312 jjj31 jjjH2121 0 1ajatueFat tuetuethtt32121 Chapter 4 Fourier Transform1j3j
18、Example 4.25Consider the LTI system with input txdttdxtydttdydttyd23422 tuetxt 3122 jjj jHjXjY3112 jjjjY1j3j214141132jjjdjd41 0 1!11Fajatuetnnatn tuetuetutetyttt3414121 Chapter 4 Fourier TransformExample Consider an LTI system txdttxddttxdtydttdydttyd22332223231223 jjjjjH21542 jjjjjH23112 jjjjH tuet
19、uettthtt232 Chapter 4 Fourier TransformHomework: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 TransformProblems for Fourier Analysis Consider an LTI system with unit impulse response sin2sintth tttt 2.If the input is determine the output 313sincoscos622x tttt y t1.Determine the frequency response of the whole system.
