Signal & Systems. Forward

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1 Sigal & Sysems Hsi-chia Lu hps://ceiba.u.edu.w/941s_ad_s_vlsi 1 Forward Sigal Sysem Respose Ipu volage, curre Circui Oupu volage, curre Depressio of acceleraor padel Auomobile Auomobile speed 2

2 Sigal: characerizaio, ehaceme, oise suppressio examples: image resoraio, speech ehaceme Sysem: characerizaio, syhesis examples: chemical processig plas Aalysis ool: Fourier aalysis, Fourier series, Fourier rasform Sigal ype: coiuous: volage i a circui, emperaure, discree: closig sock marke average, 3 Quaizaio of coiuous sigals io discree sigals. DSP digial sigal process is possible due o advace i compuer ad digial sigal processig power. Audio CD, MP3, Video VCD, DVD, DVB, Coiuous ad discree formulaios are preseed is parallel i his book. They are similar bu o ideical. 4

3 Wha is a sigal? 5 6

4 Oher Sigal Types: phoos 7 Oher Sigal Types 8

5 Coiuous Time vs. Discree Time 9 Sigal Eergy & Power For a resisor wih volage v ad curre of i, he power is wih oal eergy of ad power of Our defiiio is Eergy: 10

6 Sigal Eergy & Power co. For, he eergy is Power is 11 Class of sigals E P Examples Firs Fiie Zero Fiie duraio sigals Secod Ifiie Fiie X[]=4 Third Ifiie Ifiie X= 12

7 Trasformaios of Sigal Time shif 13 Trasformaios of Sigal Time shif 14

8 Trasformaios of Sigal Reflecio 15 Trasformaios of Sigal Reflecio 16

9 Trasformaios of Sigal Scalig 17 Trasformaios of Sigal Examples 18

10 Periodic Sigals 19 Periodic Sigals 20

11 Eve ad Odd Sigals 21 Eve ad Odd Decomposiio 22

12 Real Expoeial Sigals x = Ce a 23 Periodic Siusoidal Sigals Imagiary Expoeial if 24

13 Relaioship bewee Frequecy ad period 25 26

14 Periodic Siusoidal Sigals co. Euler s relaio We obai Eergy per period 27 Harmoically Relaed Complex Expoeials All complex expoeials wih period of T 0 or Defie Ses of wih period of 28

15 Geeral Complex Expoeial Sigals x = Ce a 29 Discree-Time Complex Expoeial Complex expoeial sigal or sequece x[] = Ca x[] = Ce β a = e β Real complex expoeial: C ad a are real Siusoidal Sigal or 30

16 31 32

17 Geeral Complex Expoeial Sigals x = Ce a 33 Periodiciy of Discree-Time Complex Expoeials Coiuous ime Rae of oscillaio icreases wih ω 0 Periodic for ay value of ω 0 Discree-ime Meaigful i For ω 0 = π, he highes oscillaio is 34

18 35 Periodiciy of Complex Expoeials co. For a period of N we eed or Fudameal period 36

19 37 Harmoically Relaed Periodic Expoeials A commo period of N Frequecies are muliple of 2π/N N disic periodic expoeials φ 0 [] = 1, φ 1 [] = e j2π/n, φ 2 [] = e j4π/n,, φ Ν 1 [] = e j2πν 1/N because 38

20 39 Ui Impulse Fucio = = ] [ δ 40 Ui Sep Fucio < = ] [ u 1] [ ] [ ] [ = u u δ

21 41 Ui Sep Fucio co. + = = m m u ] [ ] [ δ 42 Ui Sep Fucio co. = = = = 0 0 ] [ ] [ ] [ k k k k u δ δ m o k = -m Some properies: ] [ ] [ ] [ ] [ ] [ [0] ] [ ] [ x x x x = = δ δ δ δ

22 Coiuous-Time Ui Sep Fucio u = 0 1 < 0 > 0 Discoiuous a = 0 43 Ui Impulse Fucio δ = du d δ = lim δ 0 du δ = d 44

23 45 = d u τ τ δ Relaio bewee ui sep ad impulse fucios 46 = 0 σ σ τ δ d u Relaio bewee ui sep ad impulse fucios = = 0 σ σ δ τ τ δ d d u

24 47 Scaled Ui Impulse Fucio = d k ku τ τ δ 48 Some Properies of Impulse Fucios x x x x = = δ δ δ δ 0 x x δ δ

25 49 Coiuous- ad discree-ime sysems x y x[ ] y[ ] 50

26 Iercoecio of Sysems Series cascade parallel Series-parallel 51 Feedback Coecios 52

27 Example of Feedback Sysems 53 Sysems wihou Memory Memoryless, depeds o ipu a he same ime Ideiy sysem 54

28 Sysems wih Memory Examples The sysem mus remember or sore somehig 55 Iveribiliy ad Iverse Sysems 56

29 Noiverible Sysems Examples y[ ] = y = 0 x 2 y = x 57 Causaliy A sysem is causal if he oupu a ay ime depeds o values of he ipu a prese ad pas imes. Causal sysem is oaicipaive Examples of causal sysem Examples of ocausal sysem y[ ] = x[ ] x[ + 1] y = x

30 Sysem Sabiliy Sable Sysems Small perurbaio does o give large divergece. Examples of usable sysems Ivered pedulum Populaio growh?? Bak accou wih ieres?? Accumulaor Examples of sable sysems Normal pedulum Populaio growh wih limied resources. RC circuis Pracical iegraor 59 Time Ivariace Sysems The characerisic of he sysem are fixed over ime. Mahemaically, x y x 0 y 0 x[ ] y[ ] x[ 0] y[ 0] Example: Time-ivariace Time-variace y = si[ x ] y [ ] = x[ ] 60

31 61 Liear sysem if he Sysem Lieariy x y x y x + x2 y1 + y2 1 ax1 ay1 Called addiive ad homogeeiy properies Combied codiios: Superposiio properies ax 1 + bx2 ay1 + by2 ax ] + bx [ ] ay [ ] + by [ ] 1[

32 Examples Liear sysems y = x y[ ] = 2x[ ] + 3 Noliear sysems y = x 2 y [ ] = Re{ x[ ] } 63 Icremeally Liear Sysem Homework #1, due: Oc. 12 O: 1.21, 22, 37,54 B: 1.3a,b, 1.7a,b,c 64