Module 1-2: LTI Systems. Prof. Ali M. Niknejad
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1 Modu -: LTI Sysms Prof. Ai M. Niknad Dparmn of EECS Univrsiy of Caifornia, Brky
2 EE 5 Fa 6 Prof. A. M. Niknad LTI Dfiniion Sysm is inar sudid horoughy in 6AB: Sysm is im invarian: Thr is no cock or im rfrnc Th ransfr funcion is no a funcion of im I dos no mar hn you appy h inpu. Th ransfr funcion is going o b h sam Univrsiy of Caifornia, Brky
3 EE 5 Fa 6 Prof. A. M. Niknad Linar Sysms Coninuous im inar sysms hav a o in common ih fini dimnsiona inar sysms sudid in 6AB: Linariy: Basis cors à basis funcions: Suprposiion: Marix Rprsnaion à Ingra rprsnaion: 3 Univrsiy of Caifornia, Brky
4 EE 5 Fa 6 Linar Sysms con Prof. A. M. Niknad Eignvcors à ignfuncions Orhonorma basis Eignfuncion xpansion Opraors acing on ignfuncion xpansion 4 Univrsiy of Caifornia, Brky
5 EE 5 Fa 6 LTI Sysms Prof. A. M. Niknad Sinc mos priodic non-priodic signas can b dcomposd ino a summaion ingraion of sinusoids via Fourir Sris Transform, h rspons of a LTI sysm o viruay any inpu is characrizd by h frquncy rspons of h sysm: 5 Univrsiy of Caifornia, Brky
6 EE 5 Fa 6 Examp: Lo Pass Fir LPF Prof. A. M. Niknad Inpu signa: W kno ha: v v s cos s K cos f #" o! s Phas shif Amp shif v v s - i R dv i C d dv v vs - RC d dv vs v d 6 Univrsiy of Caifornia, Brky
7 EE 5 Fa 6 LPF h hard ay con. Prof. A. M. Niknad Pug h knon form of h oupu ino h quaion and s if i can saisfy KL and KCL cos cos x sin x s s y y cos cos f - cos x cos y sin x cos y cos xsin sin f coscosf - sinf - - sin xsin y y sin sinf cosf Sinc sin and cosin ar inary indpndn funcions: a sin a cos IFF a º a º 7 Univrsiy of Caifornia, Brky
8 EE 5 Fa 6 LPF: Soving for rspons Prof. A. M. Niknad Appying inar indpndnc - sinf - cosf anf - cosf - sinf - s Phas Rspons: f - an - cosf - sinf s cosf - anf s 8 Ampiud Rspons: s cosf / s s Univrsiy of Caifornia, Brky
9 EE 5 Fa 6 LPF Magniud Rspons Prof. A. M. Niknad 9 Univrsiy of Caifornia, Brky
10 EE 5 Fa 6 LPF Phas Rspons Prof. A. M. Niknad Univrsiy of Caifornia, Brky
11 EE 5 Fa 6 Prof. A. M. Niknad db: Honor h invnor of h phon Th LPF rspons quicky dcays o zro W can xpand rang by aking h og of h magniud rspons db dcib dci Univrsiy of Caifornia, Brky
12 EE 5 Fa 6 Why? Por! Prof. A. M. Niknad Why muipy og by rahr han? Por is proporiona o voag squard: æ ö æ db og ç og ç ès ø è s ö ø A brakpoin: æ ö / ç è s ødb æ ö / ç -4 db ès ø æ / ç è - -6 db 3dB Obsrv: sop of signa anuaion is db/dcad in frquncy s db ö ø db Univrsiy of Caifornia, Brky
13 EE 5 Fa 6 Why inroduc compx numbrs? Prof. A. M. Niknad Thy acuay mak hings asir On insighfu drivaion of Considr a scond ordr homognous DE y y '' y ìsin x í îcos x Sinc sin and cosin ar inary indpndn, any souion is a inar combinaion of h fundamna souions ix 3 Univrsiy of Caifornia, Brky
14 EE 5 Fa 6 Insigh ino Compx Exponnia Prof. A. M. Niknad ix Bu no ha is aso a souion! Tha mans: ix a sin x a cos x To find h consans of prop, ak drivaiv of his quaion: ix i -a sin x a cos x No sov for h consans using boh quaions: 4 æ sin x ç ècos x cos x öæ a sin ç - xøèa æ a A ç èa ö b ø ö ø æ ç èi ix ix ö ø d A - ¹ Univrsiy of Caifornia, Brky
15 EE 5 Fa 6 Prof. A. M. Niknad Compx Exponnia Iz y z x y θ z φ φ z z z z m φ φ x Rz 5 Univrsiy of Caifornia, Brky
16 EE 5 Fa 6 Prof. A. M. Niknad Th Roaing Compx Exponnia So h compx xponnia is nohing bu a poin racing ou a uni circ on h compx pan: ix cos x isin x i i -i -i 6 Univrsiy of Caifornia, Brky
17 EE 5 Fa 6 Magic: Turn Diff Eq ino Agbraic Eq Prof. A. M. Niknad Ingraion and diffrniaion ar rivia ih compx numbrs: d d i i i Any ODE is no rivia agbraic manipuaions in fac, sho ha you don vn nd o dircy driv h ODE by using phasors Th ky is o obsrv ha h currn/voag raion for any mn can b drivd for compx xponnia xciaion ò i d i i 7 Univrsiy of Caifornia, Brky
18 EE 5 Fa 6 Compx Exponnia is Porfu Prof. A. M. Niknad To find sady sa rspons can xci h sysm ih a compx xponnia i LTI Sysm i f H H Mag Rspons Phas Rspons A any frquncy, h sysm rspons is characrizd by a sing compx numbr H: 8 This is no surprising sinc a sinusoid is a sum of compx xponnias and bcaus of inariy! sin H f! H i - i -i cos From his prspciv, h compx xponnia is vn mor fundamna i -i Univrsiy of Caifornia, Brky
19 EE 5 Fa 6 LPF Examp: Th sof ay Prof. A. M. Niknad L s xci h sysm ih a compx xp: v v v s s o dv v d s f us o avoid confusion ra compx 9 s s Easy!!! s Univrsiy of Caifornia, Brky
20 EE 5 Fa 6 Magniud and Phas Rspons Prof. A. M. Niknad Th sysm is characrizd by h compx funcion H s Th magniud and phas rspons mach our prvious cacuaion: H s ü! H - an - ü Univrsiy of Caifornia, Brky
21 EE 5 Fa 6 Why did i ork? Prof. A. M. Niknad Again, h sysm is inar: y L x x L x L x To find h rspons o a sinusoid, can find h rspons o i and - i and sum h rsus: i LTI Sysm i f H H i - i -i LTI Sysm H LTI Sysm H H - H i i - f H - -i Univrsiy of Caifornia, Brky
22 EE 5 Fa 6 Prof. A. M. Niknad con. Sinc h inpu is ra, h oupu has o b ra: Tha mans h scond rm is h conuga of h firs: Thrfor h oupu is: Univrsiy of Caifornia, Brky i i H H y - - oddfuncion vn funcion f H H H H!! cos f f f - H H y i i ü
23 EE 5 Fa 6 3 Prof. A. M. Niknad Proof for Linar Sysms For an arbirary inar circui L,C,R,M, and dpndn sourcs, dcompos i ino inar subopraors, ik muipicaion by consans, im drivaivs, or ingras: For a compx xponnia inpu x his simpifis o: Univrsiy of Caifornia, Brky òòò òò ò!! x x x x d d b x d d b ax x y L!! òò ò c c d d b d d b a y L!! c c b b a y ø ö ç è æ!! c c b b a Hx y
24 EE 5 Fa 6 4 Prof. A. M. Niknad Proof con. Noic ha h oupu is aso a compx xp ims a compx numbr: Th ampiud of h oupu is h magniud of h compx numbr and h phas of h oupu is h phas of h compx numbr Univrsiy of Caifornia, Brky ø ö ç è æ!! c c b b a Hx y cos ] R[ H H y H y c c b b a Hx y H! " "! ø ö ç è æ
25 EE 5 Fa 6 Phasors Prof. A. M. Niknad Wih our n confidnc in compx numbrs, go fu sam ahad and ork dircy ih hm i can vn drop h im facor sinc i i canc ou of h quaions. Exci sysm ih a phasor: Rspons i aso b phasor: ~ f ~ f For hos ih a Linar Sysm background, r going o ork in h frquncy domain This is h Lapac domain ih s 5 Univrsiy of Caifornia, Brky
26 EE 5 Fa 6 Capacior I- Phasor Raion Prof. A. M. Niknad Find h Phasor raion for currn and voag in a cap: i c dvc C d i c I c ω v c c ω v C _ i c I c ω C d d [ ω c ] d C c d ω ω C c ω I c ω ω C c ω I c ω C c 6 Univrsiy of Caifornia, Brky
27 EE 5 Fa 6 Prof. A. M. Niknad Inducor I- Phasor Raion Find h Phasor raion for currn and voag in an inducor: v L di d i I ω v ω v i ω L d d [I ω ] _ LI d d ω ω LI ω ω ω LI ω ω L I 7 Univrsiy of Caifornia, Brky
28 EE 5 Fa 6 8 Prof. A. M. Niknad Compx Transfr Funcion Exci a sysm ih an inpu voag currn x Dfin h oupu voag y currn o b any nod voag branch currn For a compx xponnia inpu, h ransfr funcion from inpu o oupu: W can ri his in canonica form as a raiona funcion: Univrsiy of Caifornia, Brky ø ö ç è æ º!! c c b b a x y H!! 3 3 d d d n n n H
29 EE 5 Fa 6 Impd h Currns! Prof. A. M. Niknad Suppos ha h inpu is dfind as h currn of a rmina pair por and h oupu is dfind as h voag ino h por: v i Arbirary LTI Circui v i f Th impdanc Z is dfind as h raio of h phasor voag o phasor currn sf ransfr funcion v i Z H I I f -f I I f i v 9 Univrsiy of Caifornia, Brky
30 EE 5 Fa 6 Admi h Currns! Prof. A. M. Niknad Suppos ha h inpu is dfind as h currn of a rmina pair por and h oupu is dfind as h voag ino h por: v i Arbirary LTI Circui v i f f Th admmianc Z is dfind as h raio of h phasor currn o phasor voag sf ransfr funcion I I i v Y H f -f I I i v 3 Univrsiy of Caifornia, Brky
31 EE 5 Fa 6 oag and Currn Gain Prof. A. M. Niknad 3 Th voag currn gain is us h voag currn ransfr funcion from on por o anohr por: v i Gv G i Arbirary LTI Circui I I f -f If G >, h circui has voag currn gain If G <, h circui has oss or anuaion I I f -f i v Univrsiy of Caifornia, Brky
32 EE 5 Fa 6 3 Prof. A. M. Niknad Transimpdanc/admianc Currn/voag gain ar uniss quaniis Somims ar inrsd in h ransfr of voag o currn or vic vrsa Univrsiy of Caifornia, Brky Arbirary LTI Circui v i v i ] [ ] [ S I I K I I J f f f f - - W
33 EE 5 Fa 6 Prof. A. M. Niknad Dirc Cacuaion of H no DEs To dircy cacua h ransfr funcion impdanc, rans-impdanc, c can gnraiz h circui anaysis concp from h ra domain o h phasor domain Wih h concp of impdanc admianc, can no dircy anayz a circui ihou xpiciy riing don any diffrnia quaions Us KL, KCL, msh anaysis, oop anaysis, or nod anaysis hr inducors and capaciors ar rad as compx rsisors 33 Univrsiy of Caifornia, Brky
34 EE 5 Fa 6 LPF Examp: Again! Prof. A. M. Niknad Insad of sing up h DE in h im-domain, s do i dircy in h frquncy domain Tra h capacior as an imaginary rsisanc or impdanc: im domain ra circui frquncy domain phasor circui 34 W kno h impdancs: Z R R Z C C Univrsiy of Caifornia, Brky
35 EE 5 Fa 6 LPF oag Dividr Prof. A. M. Niknad Fas ay o sov probm is o say ha h LPF is ray a voag dividr 35 Z C H o C s ZC Z R R RC ü C Univrsiy of Caifornia, Brky
36 EE 5 Fa 6 36 Prof. A. M. Niknad Biggr Examp no probm! Considr a mor compicad xamp: Univrsiy of Caifornia, Brky C C C C C C C s ff C ff s ff C ff C s o Z R Z Z Z R R Z H Z R Z Z R R Z Z Z Z H ff ff Z,
37 EE 5 Fa 6 Scond Ordr Transfr Funcion Prof. A. M. Niknad Sris RLC circui 37 Univrsiy of Caifornia, Brky
38 EE 5 Fa 6 Pos/Zros of Shun RLC Circui Prof. A. M. Niknad 38 Univrsiy of Caifornia, Brky
39 EE 5 Fa 6 Prof. A. M. Niknad Dos i sound br? Appicaion of LPF: Nois Fir Lisn o h fooing sound fi corrupd ih nois Sinc h nois has a fa frquncy spcrum, if LPF h signa shoud g rid of h high-frquncy componns of nois Th fir cuoff frquncy shoud b abov h highs frquncy producd by h human voic ~ 5 khz. A high-pass fir HPF has h opposi ffc, i ampifis h nois and anuas h signa. Tons Tons Nois LPF BPF on s on 39 BPF boh ons Univrsiy of Caifornia, Brky
40 EE 5 Fa 6 4 Prof. A. M. Niknad Buiding Tns: Pos and Zros For mos circuis ha da ih, h ransfr funcion can b shon o b a raiona funcion Th bhavior of h circui can b xracd by finding h roos of h numraor and dnominaor Or anohr form DC gain xpici Univrsiy of Caifornia, Brky!! 3 3 d d d n n n H Õ Õ p z p p z z H i i!! Õ Õ ,, i p i z K p p z z K G G H!!
41 EE 5 Fa 6 Pos and Zros con Prof. A. M. Niknad Th roos of h numraor ar cad h zros sinc a hs frquncis, h ransfr funcion is zro pos Th roos of h dnominaor ar cad h pos, sinc a hs frquncis h ransfr funcion paks ik a po in a n H z p - - z p - -!! 4 Univrsiy of Caifornia, Brky
42 EE 5 Fa 6 Finding h Magniud quicky Prof. A. M. Niknad Th magniud of h rspons can b cacuad quicky by using h propry of h mag opraor: H G G K K z p z p z p z p!!!! 4 Th magniud a DC dpnds on G and h numbr of pos/zros a DC. If K >, gain is zro. If K <, DC gain is infini. Ohris if K, hn gain is simpy G Univrsiy of Caifornia, Brky
43 EE 5 Fa 6 Finding h Phas quicky Prof. A. M. Niknad Th phas can b compud quicky ih h fooing formua: " H " G " G - " - " K p - - K - " -! No h scond rm is simp o cacua for posiiv frquncis: " !! K K Inrpr his as saying ha muipicaion by is quivan o roaion by 9 dgrs z p p p z z p "!! - z 43 Univrsiy of Caifornia, Brky
44 EE 5 Fa 6 Bod Pos Prof. A. M. Niknad Simpy h og-og po of h magniud and phas rspons of a circui impdanc, ransimpdanc, gain, Givs insigh ino h bhavior of a circui as a funcion of frquncy Th og xpands h sca so ha brakpoins in h ransfr funcion ar cary dinad In EECS 4, Bod pos ar usd o compnsa circuis in fdback oops 44 Univrsiy of Caifornia, Brky
45 EE 5 Fa 6 45 Prof. A. M. Niknad Examp: High-Pass Fir Using h voag dividr ru: Univrsiy of Caifornia, Brky H H H H R L R L L R L H
46 EE 5 Fa 6 HPF Magniud Bod Po Prof. A. M. Niknad Rca ha og of produc is h sum of og H db db db db db db Þ db Incras by db/dcad Equas uniy a brakpoin 4 db db db 46 db. - db Univrsiy of Caifornia, Brky
47 EE 5 Fa 6 HPF Bod Po disscion Prof. A. M. Niknad Th scond rm can b furhr disscd: db db./ / / db - db -4 db -6 db db - db << >> db - db/dc ~ -3dB 47-3dB Univrsiy of Caifornia, Brky
48 EE 5 Fa 6 Composi Po Prof. A. M. Niknad Composi is simpy h sum of ach componn: db db db High frquncy ~ db Gain Lo frquncy anuaion db./ / / - db -4 db db 48 Univrsiy of Caifornia, Brky
49 EE 5 Fa 6 Prof. A. M. Niknad Approxima vrsus Acua Po 49 Approxima curv accura aay from brakpoin A brakpoin hr is a 3 db rror Univrsiy of Caifornia, Brky
50 EE 5 Fa 6 HPF Phas Po Prof. A. M. Niknad Phas can b nauray dcomposd as : 5 p -! H!!! - an Firs rm is simpy a consan phas of 9 dgrs Th scond rm is h arcan funcion Esima arcan funcion: << 45 Acua curv >> Univrsiy of Caifornia, Brky
51 EE 5 Fa 6 s Compx Pan Prof. A. M. Niknad 5 You may s pop aking abou ransfr funcions as a funcion of compx s rahr han frquncy H s z sz s! p sp s! This is a gnraizaion Lapac Domain of frquncy ha you i arn abou ar. For no, us vaua h funcion as foos H s ω z ω z ω! p ω p ω! This is hy you may s pop dfining a funcion ik: H ω Univrsiy of Caifornia, Brky
52 EE 5 Fa 6 Por Fo Prof. A. M. Niknad 5 P av Th insananous por fo ino any mn is h produc of h voag and currn: P i For a priodic xciaion, h avrag por is: ò T In rms of sinusoids hav T P av i v d I cosω ϕ i cosω ϕ v dτ v I cosω cosϕ i sinω sinϕ i cosω cosϕ v sinω sinϕ v dτ T I dτ cos ω cosϕ i cosϕ v sin ω sinϕ i sinϕ v csinω cosω I T cosϕ i cosϕ v sinϕ i sinϕ v I cosϕ i ϕ v Univrsiy of Caifornia, Brky
53 EE 5 Fa 6 Por Fo ih Phasors Prof. A. M. Niknad P av I cos f -f i v p No ha if f f, hn Imporan: Por is a non-inar funcion so can simpy ak h ra par of h produc of h phasors: P ¹ R[ I ] Por Facor i - v P av cos p / I 53 From our prvious cacuaion: I * * P cos fi -fv R[ I ] R[ I ] Univrsiy of Caifornia, Brky
54 EE 5 Fa 6 54 Prof. A. M. Niknad Mor Por o You! In rms of h circui impdanc hav: Chck h rsu for a ra impdanc rsisor Aso, in rms of currn: Univrsiy of Caifornia, Brky ] R[ ] R[ ] R[ ] R[ ] R[ ] R[ * * * * Z Z Z Z Z Z Z Z I P - ] R[ ] R[ ] R[ * * Z I Z I I I P
55 EE 5 Fa 6 Prof. A. M. Niknad Summary 55 Compx xponnias ar ign-funcions of LTI sysms Sady-sa rspons of LCR circuis ar LTI sysms Phasor anaysis aos us o ra a LCR circuis as simp rsisiv circuis by using h concp of impdanc admianc Frquncy rspons aos us o compy characriz a sysm Any inpu can b dcomposd ino ihr a coninuum or discr sum of frquncy componns Th ransfr funcion is usuay pod in h og-og domain Bod po magniud and phas Locaion of pos/zros is ky Univrsiy of Caifornia, Brky
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