Principles of Communications Lecture 1: Signals and Systems. Chih-Wei Liu 劉志尉 National Chiao Tung University

Transcription

1 Priciples of Commuicaios Lecure : Sigals ad Sysems Chih-Wei Liu 劉志尉 Naioal Chiao ug Uiversiy cwliu@wis.ee.cu.edu.w

2 Oulies Sigal Models & Classificaios Sigal Space & Orhogoal Basis Fourier Series &rasform Sigals & Liear Sysems Correlaio & Power Specral Desiy Samplig heory DF & FF Commu.-Lec

3 Sigal Models ad Classificaios Physical world Models Mah descripio Wha is a sigal? Usually we hik of oe-dimesioal sigals waveforms; ca our schemes be eeded o higher dimesios? How abou represeig somehig ucerai, say, a oise? Radom variables/processes mahemaical models for radom sigals Commu.-Lec cwliu@wis.ee.cu.edu.w 3

4 Deermiisic vs. Radom Deermiisic sigals: Compleely specified fucios of ime. Predicable, o uceraiy, e.g., A cos ω wih A ad ω are fied. Radom sigals sochasic sigals: ake o radom values a ay give ime isa ad are characerized by pdf probabiliy desiy fucio. No compleely predicable, wih uceraiy, e.g. dice value a he -h oss. Commu.-Lec cwliu@wis.ee.cu.edu.w 4

5 Commu.-Lec 5

6 Periodic vs. Aperiodic Sigals Periodic sigals: A sigal is periodic iff if ad oly if here eiss a cosa such ha +, he smalles is called fudameal period or simply period. Aperiodic sigals: Cao fid a fiie such ha +, Commu.-Lec cwliu@wis.ee.cu.edu.w 6

7 Phasor Sigals Phasor: A comple siusoidal fucio: % ~ j ω + θ jθ j ω Ae Ae e j Ae θ : roaio phasor; : phasor hmhag/ee, Commu.-Lec NCU Sep 9 cwliu@wis.ee.cu.edu.w 7 7

8 Why Phasor? Why comple umber? Easy mahemaical aalysis ep fucio wo degree of freedoms a a sigle frequecy I,Q modulaio Iformaio is coaied i A ad θ. he relaed real siusoidal fucio Re % Acos ω + θ. by projecio Epress i erms of roaig phasors: A % % * cos ω + θ +. mahemaical represeaio Commu.-Lec cwliu@wis.ee.cu.edu.w 8

9 Frequecy Domai Represeaios Sigle-sided: Comple epoeial % Ae e ω j j θ. Double-sided: Real-value siusoidal Acos ω + θ. hmhag/ee, Commu.-Lec NCU Sep 9 cwliu@wis.ee.cu.edu.w 9 9

10 Sigulariy Fucios Ui impulse fucio: δ -- No a ordiary fucio. I is a geeralized fucio, defied by is associaed operaio jus a operaor Defied by δ d Noe: should be coiuous, I acs like a samplig device workig oly o oe poi + δ d δ d Commu.-Lec cwliu@wis.ee.cu.edu.w

11 Impulse Fucio δ is approimaed by a arrow pulse wih ui area a b δ lim, ε ε, π δ limε si ε π ε < ε oherwise or elsewhere Commu.-Lec cwliu@wis.ee.cu.edu.w

12 Properies of Impulse Fucio Use δ oly by is properies Z&, pp.~ Shifig: δ d ime-scalig: δ a δ a Symmery: δ δ Ui sep fucio: u δ λ dλ; or δ du d Commu.-Lec cwliu@wis.ee.cu.edu.w

13 Eergy ad Power Sigals Eergy: E lim d Power: P lim d Eergy sigals: iff Power sigals: iff < E < P < P < E Noe: If is periodic, we oly eed o check calculae is power i oe period. Ofe, i has ifiie eergy. Commu.-Lec cwliu@wis.ee.cu.edu.w 3

14 Eamples α Ae u Au Acos ω + 3 θ Commu.-Lec cwliu@wis.ee.cu.edu.w 4

15 Cocludig Remarks Periodic sigals ad radom sigals are ofe power sigals. Deermiisic ad aperiodic sigals are ofe eergy sigals. he eergy ad power classificaios of sigals are muually eclusive cao be boh a he same ime. Bu a sigal ca be eiher eergy or power sigal. E.g. 4 4 u Commu.-Lec cwliu@wis.ee.cu.edu.w 5

16 Sigal Space & Orhogoal Basis Waveform vecor i geomery, liear algebra he cosequece of lieariy: N-dimesioal basis vecors: Degree of freedom ad idepedece: For eample, i geomery, ay -D vecor ca be decomposed io compoes alog wo orhogoal basis vecors, or epaded by hese wo vecors + b, b,, L b b Meaig of liear i liear algebra: b N + y + b + y b + y Commu.-Lec cwliu@wis.ee.cu.edu.w 6

17 A geeral fucio waveform ca also be epaded by a se of basis fucios N, where N ca be. ϕ Defie he ier produc of fucios as * <, y > y d. ad he basis is orhogoal, he, m, o. w * φ φm d δ m. * m φm d. Commu.-Lec cwliu@wis.ee.cu.edu.w 7

18 Basis Fucios Eample: cosie waves Q: How o cosruc a good se of basis fucios? Wha codiios? Wha purposes? Q: Ca ay fucio waveform be represeed by his se of fucios? Q3: How o compue i? Commu.-Lec cwliu@wis.ee.cu.edu.w 8

19 Fourier Series If is periodic wih period Fourier Series: j e Syhesis: πf Aalysis: + e j πf d Noice he iegral bouds Decompose a periodic sigal io couable frequecy compoes si, cos ˆ + [ ] j πf j π f e + e Commu.-Lec cwliu@wis.ee.cu.edu.w 9

20 FS of Real Fucios If is real, is cojugae symmeric. ˆ + [ ] j πf j π f e + e j j e e * ˆ + jπf + j πf+ e + e ˆ + cos πf+ Commu.-Lec cwliu@wis.ee.cu.edu.w

21 Or, use boh cosie ad sie: wih Or rewrie: [ cos cosπf si siπf ] ˆ + + a cosπf + b siπf a a cos b si + cos Re{ } cosπf + b siπf d a ˆ + a + cosπf b siπf d DC or average compoe of + he fudameal harmoic of + he secod harmoic of + Commu.-Lec cwliu@wis.ee.cu.edu.w

22 Commu.-Lec DC ad AC Coefficies DC coefficie AC coefficies If is eve ad real, ha is, he secod erm is zero. Hece is purely real ad eve If is odd ad real, ha is, he firs erm is zero. Hece is purely imagiary ad odd average of d d e f j + + π d f j d f d f j f + si cos ] si [cos π π π π

23 FS Properies Lieariy If a k ad y b k he A+By Aa k + Bb k ime Reversal If a k he - a -k F ime Shifig e ime Scalig F a a Muliplicaio y Cojugaio ad Cojugae Symmery a k ad * a* -k If is real a -k a k * jπf a k a b k bu he fudameal frequecy chages l a b l a k k l Commu.-Lec cwliu@wis.ee.cu.edu.w 3

24 Eample: FS Commu.-Lec 4

25 Eample: ime Scalig Commu.-Lec 5

26 Commu.-Lec 6 Parseval s heorem Power i ime domai Power i frequecy domai d P o + o P

27 Remarks Fourier foud ha he siusoids are good orhoormal basis fucios o epad a periodic fucio he Fourier series is derived from he good orhoormal basis fucios for a periodic fucio, defied over a period ierval, + How abou he aperiodic sigal? We cosider he aperiodic eergy sigal, ha is is iegrable i he ierval -, Noe ha aperiodic sigals are mosly fiie duraio We may ierpre he aperiodc fcio as a special case of periodic fucio wih ifiie period jπf λ jπf lim λ e dλ e jπfλ jπf Commu.-Lec λ e cwliu@wis.ee.cu.edu.w dλ e df 7, <

28 Commu.-Lec 8 Fourier rasform Defiiio he > df e f df e d e e f j f j f j df j df π π λ π π λ λ lim of respose frequecy of Fourier rasform d e f f j π

29 Fourier rasform for Aperiodic Sigals Aperiodic sigals may be viewed as havig periods ha are ifiiely log. Summaio is replaced by iegraio. Decompose a aperioidc sigal io ucouable ifiie frequecy compoes. Similar mahemaical form, ad similar ierpreaio. o be discussed: F of impulses samples Why siusoidal fucios? a eigefucios of liear sysem; b orhogoal & complee basis Commu.-Lec cwliu@wis.ee.cu.edu.w 9

30 Commu.-Lec 3 Eergy Specral Desiy For periodic sigal, we have power specral desiy For aperiodic eergy sigal, we have he similar eergy specral desiy df f df f f ddf e f d df e f d d E f j f j * * * * π π f f G By Parseval s heorem

31 Fourier Series : periodic, wih period ω πf Syhesis: : Fourier coefficie specra coefficie Aalysis: + e j e ω jω d f Fourier rasform : aperiodic, Syhesis: Aalysis: f ω π f jω ω e dω π Fourier frequecy jπ f f e df e j πf rasform d respose of of Commu.-Lec cwliu@wis.ee.cu.edu.w 3

32 Frequecy compoes:.has a fudameal freq. ad may harmoics., fudameal, secod harmoic 3, hird harmoic. Discree lie specra jω j jω e e e < < : ampliude :phase Power Specral Desiy: ad by Parseval s equaliy P + d Frequecy compoes:.no fudameal freq. ad coai all possible freq..coiuous specra desiy < Eergy Specral Desiy: ad E f e f j πf j f j πf < f : ampliude f :phase f e G f f d f df e Commu.-Lec cwliu@wis.ee.cu.edu.w 3

33 Codiios of Eisece Does ay periodic fucio have FS? en e ω N e, j [ ] Would N as N? a square iegrable codiio for he power sigal: bu o ecessarily N b Dirichle s codiios: i fiie o. of fiie discoiuiies; ii fiie o. of fiie ma & mi.; iii absolue iegrable: Dirichle s codiio implies covergece almos everywhere, ecep a some discoiuiies. N [ e N ] d as N d < d < e, Does ay aperiodic fucio have F? jπ f [ ] e f e df e, Would as? a square iegrable codiio for he eergy sigal: d < bu o ecessarily b Dirichle s codiios: i fiie o. of fiie discoiuiies; ii fiie o. of fiie ma & mi.; iii absolue iegrable: Dirichle s codiio implies covergece almos everywhere, ecep a some discoiuiies. Commu.-Lec cwliu@wis.ee.cu.edu.w 33 hmhag/ee, NCU Sep 9 33 [ ] as e d d < e,

34 Symmery Properies Real-valued ad Is Fourier Fucio For real periodic, For real aperiodic, * f * f Commu.-Lec cwliu@wis.ee.cu.edu.w 34

35 F of Sigular Fucios δ is o a eergy sigal hece does saisfy Dirichle codiio. However, is F ca be obaied by geeralizaio formal defiiio. I [ δ ] I lim Π lim sic τ τ τ τ δ F, δ F f E: he F of δ? fτ F jπ f jπ f F Aδ Ae, Ae Aδ f f Commu.-Lec cwliu@wis.ee.cu.edu.w 35

36 F of Periodic Sigals Periodic sigals are o eergy sigals do saisfy Dirichle s codiios. Bu we are doig i ayway jusified by advaced mah. Map FS o F: E-: E-: jω e cos πf δ f δ f f A pulse rai! Wha good are hey for? Commu.-Lec cwliu@wis.ee.cu.edu.w 36

37 F of Periodic Sigals Le F of a eergy sigal p be I{ p } P f Aperiodic sigal is geeraed by duplicaig p a every ierval s. he From covoluio heorem, [ δ ]* p p f I{[ δ ]} P f s s s s s s s Commu.-Lec cwliu@wis.ee.cu.edu.w 37 f δ f f P f f P f δ f f s

38 F of Periodic Sigals 3 Sice m ms fsp f s δ f f p ake iverse F of he eq. f p s fsp fs δ f fs I { } I { } j fs s s { δ s} s s fpf I f f fpf e π p s f s P f s e jπf s Poisso sum formula he sample values Pf s of PfI{p} are he Fourier series coefficies of s p-m s Commu.-Lec cwliu@wis.ee.cu.edu.w 38

39 Eamples of F Commu.-Lec cwliu@wis.ee.cu.edu.w 39

40 E of F Periodic Sigals Commu.-Lec cwliu@wis.ee.cu.edu.w 4

41 F Properies Commu.-Lec 4

42 F Pairs Commu.-Lec 4