Signal & Linear System Analysis

Transcription

1 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Sigal & Liar Sym Aalyi Sigal Modl ad Claificaio Drmiiic v. Radom Drmiiic igal: complly pcifid fucio of im. Prdicabl, o ucraiy.g., < < ; whr A ad ω ar coa x A coω Radom igal ochaic igal: ak o radom valu a ay giv im ia ad characrizd by pdf probabiliy diy fucio o complly prdicabl, wih ucraiy.g. x dic valu how wh od a im idx Modl: A larg, mayb of wavform ach aociad wih a probabiliy maur.g. pdf characrizig h ou-of-bad radio oi CU EE

2 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Priodic v. Apriodic Priodic igal: A igal x i priodic iff a coa, uch ha x +, h mall uch i calld fudamal priod or imply priod. Apriodic igal: cao fid a fii uch ha x +, Phaor igal & Spcra A pcial priodic fucio ~ j ω + θ jθ j ω A A ~ jθ x roaig phaor ; A phaor ; A, θ ral umbr Why hi complx igal?. Ky par of modulaio hory. Corucio igal for almo ay igal 3. Eay mahmaical aalyi for igal 4. Pha carri im dlay iformaio Mor o Phaor Sigal:. Iformaio i coaid i A ad θ giv a fixd f.. h rlad ral iuoidal fucio: or ω ~ x Aco ω + θ R{ } ~ or x Aco ω ~ + θ + x 3. I vcor form graphically: CU EE

3 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi 4. Frqucy-domai rpraio Sigl-idd SS amp. & phaor v. doubl-idd DS: Li pcra: E.g. fid SS ad DS pcra of x Ai ω + θ A: pu io coi form fir Sigulariy Fucio: oppod o rgular fucio Ui impul fucio δ :. Dfid by + + x δ d δ d δ d ; δ d. I dfi a prci ampl poi of a im or if δ x δ d 3. Baic fucio for liarly corucig a im igal x τ δ τ dτ 4. Propri: Z &, pp. 5-6 do bohr propri 5 ad 6 δ a δ ; δ δ : v fucio a + CU EE 3

4 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi 5. Wha i δ prcily? om of iuiiv way of ralizig i: E.g. δ lim, ε ε, < ε ohrwi or lwhr E.g. π δ limε i ε π ε Ui p fucio u: u δ λ dλ; δ du d Sigal yp claifid by rgy & powr hi claificaio will b dd for h lar aalyi of drmiiic ad radom igal Ergy: Powr: E lim P lim d d Ergy Sigal: iff < E < P Powr Sigal: iff < P < E CU EE 4

5 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi α Exampl- x A u Exampl- x Au Exampl-3 x Aco ω + 3 θ o:. If i priodic, h i i maigl o fid i rgy, w oly d o chck i powr P +. oi i of pri ad i of a powr igal 3. Drmiiic ad apriodic igal ar of rgy igal 4. A ralizabl LI ym ca b rprd by a igal ad moly i a rgy igal 5. Powr maur i uful for igal ad oi aalyi 6. h rgy ad powr claificaio of igal ar muually xcluiv cao b boh a h am im. Bu a igal ca b ihr rgy or powr igal Sigal Spac & Orhogoal Bai Applyig h Sophomor Liar Algbra Bai vcor for vcor pac: ial i DSP & commuicaio hory -dimioal bai vcor: b, b,, b L Dgr of frdom ad idpdc: E.g.: I gomry, ay -D vcor d p x ca b dcompod io com- q po alog wo orhogoal bai vcor, or xpadd by h wo vcor x x + Maig of liar i liar algbra: b xb x + y x + b + y b + x y CU EE 5

6 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Bai fucio for fucio pac: idipabl for gral igal aalyi A gral fucio ca alo b xpadd by a of bai fucio i a approximaio or mor faibly x φ φ Dfi h machig or corrlaio opraio a x φ d φ φ d m m If w dfi orhogoaliy a h E.g. co mω, m φ φm d δ m, o. w. x φ d m m φ m for priodic v ; of φm orhogoal? E.g. aylor xpaio for coi fucio, bai fucio? orhogoal? Rmark:. Uig Frhm calculu ca how ha fucio approximaio xpaio by orhogoal bai fucio i a opimal LSE approximaio. I hr a vry good of orhogoal bai fucio? 3. Cocp ad rlaiohip of pcrum, badwidh ad ifii coiuou bai fucio CU EE 6

7 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Fourir Sri & Fourir raform Fourir Sri: x j f π + jπf d Siuoidal Rpraio xˆ If i ral, + j πf j π f [ + ] j, j j xˆ + [ + jπf + jπf+ ] + o: Idx ar from o coπ f + Coi FS rigoomric FS: [ co coπf i iπf ] xˆ + + a coπ f + b iπf whr a co, b i Or, + a coπ f + b iπ f d d xˆ a + a + coπ f b iπf CU EE 7

8 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Propri of Fourir ri + λ j πf d DC coffici: + + jπ f d d avrag valu of AC coffici: + [coπ f jiπf ] d + + coπ f d j iπf d If i ral, h, R[ ] + j Im[ ] j whr + R[ ] coπ f d x f d + Im[ ] iπ Hc, R[ ] R[ ] Im[ ] Im[ ] v fucio ad odd fucio Liariy a k y A+By b k Aa k + Bb k im Rvral a k - a -k im Shifig jk f π a k CU EE 8

9 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi im Scalig α a k Bu h fudamal frqucy chag Muliplicaio a k y b k y l a l b k l Cojugaio ad Cojuga Symmry a k x a -k If i ral a -k a k Parval horm Powr i im domai powr i frqucy domai + Px d o Px o CU EE 9

10 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Exampl: half-rcifid iwav Exampl: CU EE

11 Pricipl of Commuicaio I Fall, Fourir Sri Sigal & Liar Sym Aalyi Fourir raform Good orhogoal bai fucio for a Good orhogoal bai fucio for a priodic fucio: apriodic fucio:. Iuiivly, bai fucio hould. Alrady kow iuoidal fucio ar good choic b alo priodic. Iuiivly, priod of h bai. Siuoidal compo hould fucio hould b qual o h o b i a fudamal & harmoic rlaiohip priod or igr fracio of h arg igal 3. Apriodic igal ar moly fii duraio 3. Fourir foud ha iuoidal fucio ar good ad mooh 4. Coidr h apriodic fucio fucio o xpad a priodic a a pcial ca of priodic fucio wih ifii fucio priod Syhi & aalyi: rcorucio & projcio Giv priodic x wih priod, ω πf f Syhi & aalyi: rcorucio & projcio Giv apriodic x wih priod, i ca b yh- df, ω dω πdf, w ca yhiz i a izd a x jω : Spcra coffici, pcra ampliud rpo + lim dω > f jπf df jdω π ω o yhiz i w mu fir aalyz i By orhogoaliy j f ad fid ou. f π d By orhogoaliy Fourir raform of jω d frqucy rpo of Hc, f jπf df Ivr F jω dω CU EE

12 Pricipl of Commuicaio I Fall, Frqucy compo:. Dcompo a priodic igal io couabl frqucy compo. Ha a fudamal frq. ad may ohr harmoic jω j jω : ampliud : pha of 3. Dicr li pcra Sigal & Liar Sym Aalyi Frqucy compo:. Dcompo a apriodic igal io ucouabl frqucy compo. o fudamal frq. ad coai all poibl frq. j πf j f j πf f f < f < 3. Coiuou pcral diy Powr Spcral Diy: ad by Parval qualiy P + d Ergy Spcral Diy: G f f ad E d f df I ral bai fucio: I ral bai fucio: + jω + B + + jω co ω + A i ω co ω co ω + o ha for ral A your xrci xampl! CU EE

13 Pricipl of Commuicaio I Fall, Wha ha Fourir ri?. Expaio by orhogoal bai fucio ca b how i quival o fidig uig h LSE or MSE co fucio: E{[ ] } E{[ jω ] } Sigal & Liar Sym Aalyi Wha ha Fourir raform?. Expaio by orhogoal bai fucio ca b how i quival o fidig f uig h LSE or MSE co fucio: E{[ ] } E{[ f jπf df ] }. Would E{[ ] } a?. Would E{[ ] } a? 3. hi rquir quar igrabl codiio for h powr igal: io for h rgy 3. hi rquir quar igrabl codi- igal: x d < ad o carily 4. Dirichl codiio: a fii o. of fii dicoiuii b fii o. of fii max & mi. c abolu igrabl x d < Dirichl codiio impli covrgc almo vrywhr, xcp a om dicoiuii. d < 4. Dirichl codiio: a fii o. of fii dicoiuii b fii o. of fii max & mi. c abolu igrabl x d < Dirichl codiio impli covrgc almo vrywhr, xcp a om dicoiuii. Som Gral Symmry Dfiiio a Symmric v:, ral b Cojuga ymmric: x c Ai-ymmric odd:, ral d Cojuga ai-ymmric: x CU EE 3

14 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Symmry Propri of ad I Fourir Fucio For ral priodic,, or,. For ral apriodic, f f or f f, f f Exampl: Fourir raform of Sigular Fucio δ i o a rgy igal hc do aify Dirichl codiio. Howvr, i F ca b obaid by formal dfiiio. F F δ, δ f, F j πf jπf Aδ A, F A Aδ f f, Exampl: δ Fourir raform of Priodic Sigal Priodic igal ar o rgy igal do aify Dirichl codiio Giv a priodic x, x jω f δ f f Exampl- co π f Exampl- δ CU EE 4

15 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Propri of Fourir raform p.3 of O.W.Y CU EE 5

16 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Fourir raform Pair p.5 of O.W.Y CU EE 6

17 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Rlaiohip Bw F of a Apriodic Sigal & I Priodic Exio L F of a apriodic pul igal p b I { p } P f, W ca gra a priodic igal x by duplicaig p a vry irval, h From covoluio horm, x [ δ ] p p f I{[ f δ f f δ P f ]} P f f P f δ f f CU EE 7

18 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Poio um formula: By akig ivr F of abov q. I { f } f P f I { δ f f p } I { f P f f P f δ f jπf f } jπf p f P f --> Poio um formula Powr Spcral Diy & Corrlaio Ergy Sigal φ τ I { G f } I τ τ [ f f ] I λ λ + τ dλ lim im-avrag auocorrlaio fucio [ f ] I λ λ + τ dλ [ f ] φ E igal rgy im avrag auocorrlaio fucio of powr igal R τ τ x + τ lim x x + τ d, + τ d, if if apriodic powr igal priodic powr igal R S f df S f I{ R τ } Powr pcral diy For priodic powr igal f I{ R τ } δ f f S CU EE 8

19 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Irpraio. φ τ ad R τ maur h imilariy bw h igal a im ad + τ. G f ad S f rpr h igal rgy or powr pr ui frqucy a frq. f. Propri of R τ :. R powr x R τ, τ, max{ R τ } R. R τ i v for ral igal: R τ x τ R τ 3. If do o coai a priodic compo lim R τ τ 4. If i priodic wih priod, h R τ i priodic i τ wih h am priod 5. S f I{ R τ }, f Crocorrlaio of wo powr igal x, y R xy τ y lim τ y + τ y τ d Crocorrlaio of wo rgy igal x, y φ xy τ y τ Rmark: R xy τ R yx τ, φ xy τ φ yx τ d CU EE 9

20 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Sigal & Liar Sym. y H{ }, Η x y. H: a opraor rprig om mchaim ad opraio o ad/or y o produc y; grally i ca b approximad by a diffrial quaio. 3. Mor pcifically, h liar coa-coffici diffrial q. ui for mo applicaio ad mak lif a lo air! Liar & im-ivaria LI Sym Liar ym: Saifi uprpoiio pricipl y H{ α x + α x } H{ αx } + H{ α x } y + y im-ivaria: Dlayd ipu produc a dlayd oupu du o h o-dlayd ipu y H{ } --> H } y { Compl characrizaio of LI ym: h ui impul fucio i ky o h characrizaio. h H{ δ } x λ δ λ dλ y H{ } H{ λ δ λ dλ} λ H{ δ λ} dλ CU EE

21 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi CU EE If I, } { } { y d h x x H h x d h x x H y λ λ λ λ λ λ

22 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Covoluio form hold iff LI. Dualiy of igal & ym h y λ h λ dλ h λ λ dλ Covoluio horm: I{ y } Y f I{ h λ λ dλ} H f f Ky applicaio: grally H f f i air ha x h. BIBO abiliy: A ym i BIBO if oupu i boudd, giv ay boudd ipu max{ y } max{ h λ dλ < mai lm of h λ λ dλ } max{ } Dirichl codiio h λ dλ < Caualiy:. A ym i caual if: curr oupu do o dpd o fuur ipu; or curr ipu do o coribu o h oupu i h pa A ruh i raliy.. All h ym i aur world ar caual. Caualiy grally i o a problm i mo circumac. Howvr, i mar frquly wh w wa o courac h ffc of hoil commuicaio chal, uch ha w d a idal qualizr, which i o-caual. y h λ λ dλ h λ λ dλ h, for < CU EE

23 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Paly-Wir Codiio: If H f df < h, < l Hf + f df < Rmark: Hf cao grow oo fa. Hf cao b xacly zro ovr a fii bad of frqucy If H f df < l Hf df < + f Hf uch ha Hf i caual h, < Eigfucio of LI Sym If Η { g } αg, whr α i a coa w.r. im paramr, h α : igvalu g : ig fucio A, : a arbirary complx umbr y α h λ A h λ A λ λ dλ dλ [ h λ λ dλ] A α L i i jπf, h y [ h λ dλ] A α i jπf λ jπf CU EE 3

24 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Corrlaio fucio rlad by LI ym:. R yx τ h τ Rx τ h λ Rx τ λ dλ. R τ h τ h τ R τ y 3. S f H f S f yx 4. S f H f S f y x x x pf & from dfiiio. 3 & 4 from I{ h τ } H f ad I{ h τ } H f Sym ramiio Diorio & Sym Frqucy Rpo a Sic almo ay ipu ca b rprd by a liar combiaio of orhogoal iuoidal bai fucio j π f, w oly d o ipu A j π f o h ym o characriz h ym propri, ad h igvalu α h jπf d H f carri all h ym iformaio rpodig o A j π f. b I commuicaio hory, ramiio diorio i of primary cocr i high-qualiy ramiio of daa. Hc, h ym rprig ramiio chal i h ky ivigaio arg. hr yp of diorio of a ramiio chal:. Ampliud diorio: liar ym bu h ampliud rpo i o coa.. Pha dlay diorio: liar ym bu h pha hif i o a liar fucio of frqucy. 3. oliar diorio: oliar ym CU EE 4

25 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Group dlay: dθ f g f, θ f H f π df For a liar pha ym, θ f H πf g f, a coa If g f i o a coa, iuoidal ipu of diffr frquci hav diffr dlay. θ f Cf. Pha dlay: p f πf Idal gral filr: CU EE 5

26 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Ralizabl filr approximaig h idal filr:. Burworh filr: impl. Chbyhv filr: mallr maximum dviaio 3. Bl filr: approximaly liar pha i pabad CU EE 6

27 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi im-badwidh Produc ucraiy pricipl I ca b argud ha a arrow im igal ha a wid frqucy badwidh, ad vic vra: duraio badwidh coa. Dfiiio of : qual ara x. Dfiiio of W: 3. Combi ad : d x d f f W f df f df W W CU EE 7

28 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi Samplig hory Idal amplig Idal amplig igal: impul rai a aalog igal δ, : amplig priod Aalog coiuou-im igal: x Sampld coiuou-im igal: x δ x δ wo Ca: δ x k δ δ δ f f S f f [ f δ f kf ] k f o aliaig: f > W, ad f δ f kf f f kf k aliaig: f < W, whr W i h high ozro frqucy compo of f. Afr amplig, h rplica of f ovrlap i frqucy domai. ha i, h highr frqucy compo of f ovrlap wih h lowr frqucy compo of f. f CU EE 8

29 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi yqui Samplig horm: L b a badlimid igal wih f for f W. i.., o compo a frquci grar ha W h i uiquly drmid by i ampl x[ ],, ±, ±, K, if f W. -- Udramplig: f < W -- Ovrdamplig: f > W Rcorucio: Idal rcorucio filr: f H f H Π B jπf, W B f W Y f f y f H H f jπf Alraiv xprio y h BH ic[b ] wo yp of rcorucio rror CU EE 9

30 Pricipl of Commuicaio I Fall, Sigal & Liar Sym Aalyi CU EE 3 k k k k W k k H W k Y y DF & FF DF,,,,,,,, k x x k j k k k j k L L π π Fa covoluio by FF / k k j k j z j W h h H k H z H H j π π ω ω ω ω