Continuous-Time Signals and LTI Systems

Transcription

1 Coninuous-Time Signals and LTI Sysems Chaper 9 A he sar of he course boh coninuous and discree-ime signals were inroduced. In he world of signals and sysems modeling, analysis, and implemenaion, boh discree-ime and coninuous-ime signals are a realiy. We live in an analog world, is ofen said. The follow-on courses o ECE26, Circuis and Sysems I (ECE225) and Circuis and Sysems II (ECE325) focus on coninuous-ime signals and sysems. In paricular circuis based implemenaion of sysems is invesigaed in grea deail. There sill remains a lo o discuss abou coninuous-ime signals and sysems wihou he need o consider a circui implemenaion. This chaper begins ha discussion. Coninuous-Time Signals To begin wih signals will be classified by heir suppor inerval Two-Sided Infinie-Lengh Signals Sinusoids are a primary example of infinie duraion signals, ha are also periodic ECE 26 Signal and Sysems 9

2 x () Acos( ω + φ), < < x () Ae jφ e jω, < < Coninuous-Time Signals (9.) The period for boh he real sinusoid and complex sinusoid signals is T 2π ω The signal may be any periodic signal, say a pulse rain or squarewave A wo-sided exponenial is anoher example x () Ae β, < < (9.2) 4 x () 5 cos 2π x () Pulse Train Period 2s Pulse Widh.5s x () 2e Two-sided exponenial ECE 26 Signals and Sysems 9 2

3 Coninuous-Time Signals One-Sided Signals Anoher class of signals are hose ha exis on a semi-infinie inerval, i.e., are zero for < (suppor [, ) ) The coninuous-ime uni-sep funcion, u (), is useful for describing one-sided signals u (),, oherwise (9.3) When we muliply he previous wo-side signals by he sepfuncion a one-side signal is creaed x () 5 2π cos 2 -- π -- u () x () u () x () 2e 2 u () One-sided exponenial ECE 26 Signals and Sysems 9 3

4 Coninuous-Time Signals The sar ime can easily be changed by leing x () u ( 2), 2, oherwise (9.4) Finie-Duraion Signals Finie duraion signals will have suppor over jus a finie ime inerval, e.g., [4, ) A convenien way of craing such signals is via pulse gaing funcion such as p () u ( 4) u ( ), 4 <, oherwise (9.5) p () x () 5 2π cos 2 -- π -- p () ECE 26 Signals and Sysems 9 4

5 The Uni Impulse The Uni Impulse The opics discussed up o his poin have all followed logically from our previous sudy of discree-ime signals and sysems The uni impulse signal, δ(), however is more difficul o define han he uni impulse sequence, δ[ n] Recall ha δ[ n], n, oherwise The uni impulse signal is defined as and Wha does his mean? (9.6) (9.7) I would seem ha δ() mus have zero widh, ye have area of uniy A es funcion, δ() as Δ δ Δ () δ(), δ () d δ Δ (), can be defined ha in fac becomes , Δ < < Δ 2Δ, oherwise (9.8) ECE 26 Signals and Sysems 9 5

6 The Uni Impulse δ Δ () Δ Δ 2 Δ 2 Δ Δ Δ 2 The claim is ha lim Δ Check (9.6) and (9.7) δ Δ () δ() lim δ Δ (), Δ δ () Δ d (9.9) In ploing a scaled uni-impulse signal, e.g., Aδ(), we plo a verical arrow wih he ampliude acually corresponding o he area Aδ() ( A) ECE 26 Signals and Sysems 9 6

7 The Uni Impulse Sampling Propery of he Impulse A noeworhy propery of δ() is ha Sampling Propery f ()δ ( ) f ( )δ( ) (9.) Discussion Since δ( ) is zero everywhere excep, only he value f ( ) is of ineres Using he es funcion δ Δ () we also noe ha f ()δ Δ () f () ( 2Δ), Δ< < Δ, oherwise (9.) so as Δ he only value of f () ha maers is f( ) f ()δ Δ () f( ) Δ f ()δ Δ () f( )δ Δ () f () f ()δ f( )δ() ( f( ) ) f () Also observe ha f ()δ d f( )δ() d f( ) δ() d f( ) (9.2) ECE 26 Signals and Sysems 9 7

8 The Uni Impulse Inegral Form Sampling/Sifing Propery f ()δ ( ) d f ( ) (9.3) Example: cos( 2π)δ(.2) The sampling propery of When inegraed we have + u ()δ ( 3) δ() resuls in cos( 2π(.2) )δ(.2) + u( 3)δ( 3) [ cos( 2π)δ(.2) + u ()δ ( 3) ] d cos( 2.4π) + u( 3) cos( 2.4π) + Operaional Mahemaics and he Dela Funcion The impulse funcion is no a funcion in he ordinary sense I is he mos pracical when i appears inside of an inegral From an engineering perspecive a rue impulse signal does no exis We can creae a pulse similar o he es funcion δ Δ () as well as oher es funcions which behave like impulse funcions in he limi The operaional properies of he impulse funcion are very useful in coninuous-ime signals and sysems modeling, as well as in probabiliy and random variables, and in modeling disribuions in elecromagneics ECE 26 Signals and Sysems 9 8

9 The Uni Impulse Derivaive of he Uni Sep A case in poin where he operaional properies are very valuable is when we consider he derivaive of he uni sep funcion From calculus you would say ha he derivaive of he uni sep funcion, u (), does no exis because of he disconinuiy a Consider The area propery of δ() δτ ( ) dτ saes ha (9.4) b a δ() d, a < and b, oherwise (9.5) Invoking he area propery we have δτ ( ) dτ,, oherwise (9.6) which says ha his inegral behaves like he uni sep funcion u () δτ ( ) dτ (9.7) ECE 26 Signals and Sysems 9 9

10 The Uni Impulse From calculus we recognize ha (9.7) implies also ha Similarly, (9.8) (9.9) If we now consider siuaions where a produc exis, i.e., x () f ()u, we can invoke he produc rule for derivaives o obain Example: d ----f()u d The derivaive of x x () δ() δ( ) is d ----u() d d ----u( d ) d ----f() d u () + f () ----u() d d f ()u + f ()δ x () e 4 u () + u ( ) x () d ----x() 4e 4 u () + e 4 δ() + δ ( ) d 4e 4 u () + δ() + δ ( ) () () () dx() d (9.2) ECE 26 Signals and Sysems 9

11 Coninuous-Time Sysems Coninuous-Time Sysems A coninuous-ime sysem operaes on he inpu o produce an oupu y () T{ x() } (9.2) x () T { } y () Basic Sysem Examples Squarer y () [ x ()] 2 Time Delay y () x ( d ) Differeniaor y () dx() d (9.22) (9.23) (9.24) Inegraor y () x( τ) dτ (9.25) In all of he above we can calculae he oupu given he inpu and he definiion of he sysem operaor For linear ime-invarian sysems we are paricularly ineresed in he impulse response, ha is he oupu, y () h (), when x () δ (), for he sysem iniially a res ECE 26 Signals and Sysems 9

12 Linear Time-Invarian Sysems Example: Inegraor Impulse Response Using he definiion y () h () δτ ( ) dτ u () Linear Time-Invarian Sysems In he sudy of discree-ime sysems we learned he imporance of sysems ha are linear and ime-invarian, and how o verify hese properies for a given sysem operaor Time-Invariance A ime invarian sysem obeys he following x ( ) y ( ) for any Boh he squarer and inegraor are ime invarian The sysem y () cos( ω c )x () (9.26) (9.27) is no ime invarian as he gain changes as a funcion of ime ECE 26 Signals and Sysems 9 2

13 Linear Time-Invarian Sysems Lineariy A linear sysem obeys he following αx () + βx 2 () αy () + βy 2 () (9.28) where he inpus are applied ogeher or applied individually and combined via α and β laer The squarer is nonlinear by virue of he fac ha y () [ αx () + βx 2 () ] 2 α 2 2 x () + 2αβx ()x 2 () + β 2 x 2 () produces a cross erm which does no exis when he wo inpus are processed separaely and hen combined The inegraor is linear since y () [ αx ( τ) + βx 2 ( τ) ] dτ α x ( τ ) dτ + β x 2 ( τ) dτ The Convoluion Inegral For linear ime-invarian (LTI) sysems he convoluion inegral can be used o obain he oupu from he inpu and he sysem impulse response y () x( τ)h ( τ) dτ Convoluion Inegral x ()*h() (9.29) ECE 26 Signals and Sysems 9 3

14 Linear Time-Invarian Sysems The noaion used o denoe convoluion is he same as ha used for discree-ime signals and sysems, i.e., he convoluion sum Evaluaion of he convoluion inegral iself can prove o be very challenging Example: y () x ()*h() u ()*u() Seing up he convoluion inegral we have y () u( τ)u ( τ) dτ u ( τ) u( τ) τ y (), < dτ,, <, or simply y () u() r (), which is known as he uni ramp ECE 26 Signals and Sysems 9 4

15 Impulse Response of Basic LTI Sysems Properies of Convoluion Commuaiviy: Associaiviy: Disribuiviy over Addiion: Ideniy Elemen of Convoluion: Wha is x ()? I urns ou ha proof x ()*h() h ()*x() [ x ()*h () ]*h 2 () x ()*[ h ()*h 2 () ] x ()*[ h ()*h 2 () ] x ()*h () + x ()*h 2 () x ()*h() h () δτ ( )h ( τ) dτ x () δ () δ()*h () δτ ( )h ( ) dτ h () δτ ( ) dτ h () h () (9.3) (9.3) (9.32) (9.33) Impulse Response of Basic LTI Sysems For cerain simple sysems he impulse response can be found by driving he inpu wih δ() and observing he oupu For complex sysems ransform echniques, such as he Laplace ransform, are more appropriae ECE 26 Signals and Sysems 9 5

16 Convoluion of Impulses Inegraor h () x( τ) dτ x( τ) δ( τ) u () (9.34) Ideal delay h () x ( d ) δ( d ) x () δ () Noe ha his means ha x ()*δ( d ) x ( d ) (9.35) (9.36) Convoluion of Impulses Basic Theorem: δ( )*δ( 2 ) δ( ( + 2 )) (9.37) Example: [ δ() 2δ( 3) ]*u() Using he ime shif propery (9.36) δ()*u () 2δ( 3)*u() u () 2u( 3) Evaluaing Convoluion Inegrals Sep and Exponenial Consider x () u ( 2) and h () e 3 u () We wish o find y () x ()*h() ECE 26 Signals and Sysems 9 6

17 y () e 3τ u( τ)u ( τ 2) dτ Evaluaing Convoluion Inegrals (9.38) To evaluae his inegral we firs need o consider how he sep funcions in he inegrand conrol he limis of inegraion u ( τ 2 ) 2 < 2 2 u ( τ 2 ) 2 > e 3τ u( τ) τ For 2 < or < 2 here is no overlap in he produc ha comprises he inegrand, so y () For 2 > or > 2 here is overlap for τ [, 2), so here 2 τ y () e 3τ d e 3τ [ e 3( 2) ]u ( 2) 3 (9.39) y () ECE 26 Signals and Sysems 9 7

18 Evaluaing Convoluion Inegrals Noe: The use of he exponenial impulse response in examples is significan because i occurs frequenly in pracice, e.g., an RC lowpass filer circui R x () C y () h () e RC RC u () Example: Find Suppose ha x () e a u () and h () e b u () y () x ()*h() y () e aτ u( τ)e b ( τ) u ( τ) dτ e aτ e b ( τ) dτ e b e ( a b)τ dτ a 2 and b 3 by evaluaing he convoluion inegral e b e ( a b)τ e b a b ( a b) a b e ( a b ) [ ]u () a [ b e b e a ]u ()a, b ECE 26 Signals and Sysems 9 8

19 Evaluaing Convoluion Inegrals.4 y () a 2, b Square-Pulse Inpu Consider a pulse inpu of he form x () u () u ( T) where T is he pulse widh and h () e a u () x () (9.4) T The oupu is y () u ()*h() u ( T)*h() (9.4) From he sep response analysis we know ha u ()*h() -- [ e a ]u (), (9.42) a so ECE 26 Signals and Sysems 9 9

20 Properies of LTI Sysems y () Plo he resuls for -- a a [ ]u () -- [ a a ( T) ]u ( T) a a T 5 and a (9.43) y ()..8.6 a, T Properies of LTI Sysems Cascade and Parallel Connecions We have sudied cascade and parallel sysem earlier For a cascade of wo LTI sysems having impulse responses h () and h 2 () respecively, he impulse response of he cascade is he convoluion of he impulse responses h cascade () h ()*h 2 () (9.44) Cascade x () h () h 2 () y () x () h () h ()*h 2 () y () ECE 26 Signals and Sysems 9 2

21 Properies of LTI Sysems For wo sysems conneced in parallel, he impulse response is he sum of he impulse responses h parallel () h () + h 2 () (9.45) Parallel h () x () y () h 2 () x () h () h () + h 2 () y () Differeniaion and Inegraion of Convoluion Since he inegraor and differeniaor are boh LTI sysem operaions, when used in combinaion wih anoher sysem having impulse response, h (), we find ha he cascade propery holds Wha his means is ha performing differeniaion or inegraion before a signal eners and LTI sysem, gives he same resul as performing he differeniaion or inegraion afer he signal passes hrough he sysem x () ( ) or d () d h () y () x () h () ( ) or d () d y () ECE 26 Signals and Sysems 9 2

22 Properies of LTI Sysems Example: Sep Response from Knowing he impulse response of a sysem we can find he respond o a sep inpu by jus inegraing he oupu, since un ( ) a he inpu is obained by inegraing δ() Thus we can wrie ha h () e a u () y () u ()*h() h( τ) dτ e aτ a This resul is consisen wih earlier analysis e aτ u( τ) dτ e aτ dτ -- [ e a ]u () a Sabiliy and Causaliy Definiion: A sysem is sable if and only if every bounded inpu produces a bounded oupu. A bounded inpu/oupu is a signal for which x () or y () < for all values of. A heorem which applies o LTI sysems saes ha a sysem (LTI sysem) is sable if and only of Sabiliy for LTI Sysems h () d if and only if holds in eiher direcion < (9.46) ECE 26 Signals and Sysems 9 22

23 Properies of LTI Sysems Example: LTI wih For sabiliy We mus have h () e a u () e a u () d a > for sabiliy e a d Noe ha a resul in h () un ( ), which is an inegraor sysem, hence an inegraor sysem is no sable Definiion: A sysem is causal if and only if he oupu a he presen ime does no depend upon fuure values of he inpu A heorem which applies o LTI sysems is (9.47) This definiion and LTI heorem also holds for discree-ime sysems Example: Simulae an LTI Sysem using Malab lsim() As a final example we consider how we can use MATLAB o simulae LTI sysems The funcion we use is lsim(), which has behavior similar o ha of filer(), which is used for discree-ime sysems e a a Causal for LTI Sysems h () for < --, a > a ECE 26 Signals and Sysems 9 23

24 Properies of LTI Sysems >> -:.:5; % creae a ime axis >> x zeros(size()); % nex 3 lines creae a pulse >> i_pulse find(> & <5); % duraion is 5s >> x(i_pulse) ones(size(i_pulse)); >> subplo(2) >> plo(,x) >> axis([- 5.]); grid >> ylabel('inpu x()') >> subplo(22) >> y lsim(f([],[ ]),x,);% h() e^(-*) u() Warning: Simulaion will sar a he nonzero iniial ime T(). > In li.lsim a >> plo(,y); grid >> ylabel('oupu y()') >> xlabel('time (s)') Inpu x().5 Inpu pulse of duraion 5s 5 5 Oupu y().5 Impulse response e - u() 5 5 Time (s) ECE 26 Signals and Sysems 9 24