SystemProper,es

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ElectronicSystem

RecallthisfromPhysics?

8/29/08 Comp665–SystemProper5es

v(t) i

s(t)

R L

C

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vR = R i

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vL = L didt

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i = C ds(t)dt

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v (t) = Ri + L didt

+ s(t)

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v (t) = RC ds(t)dt

+ LC d 2s(t)dt 2 + s(t)

ComponentProper5es

MechanicalSystem

•  Lawsofmo5on

•  Observa5on:Differentphysicalsystemsleadtosimilarmathema5calmodels.

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M d 2y (t)dt 2 = x(t) −K y(t) −D dy (t)

dt

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x(t) = M d 2y (t)dt 2 + K y(t) + D dy (t)

dt

ThermalSystem

•  CoolingFininsteadystate

•  Observa5ons–  Space,ratherthan5me,istheindependentvariable–  Somesystemsaremorenaturallyspecifiedintermsof“boundarycondi5ons”than“ini5alcondi5ons”

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y(t)

y(t)

y(t)

x(t)

y0

t t0

t = distance along fin x(t) = air temperature along fin’s length y(t) = fin temperature along fin’s length

€

d 2y (t)dt 2 = k[y(t) − x(t)]

t1

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y(t0) = y0

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dydt

(t1) = 0

FinancialSystem

•  Fluctua5onsinthepriceofzero‐couponbondst = 0     Time of purchase at price y0  t = T     Time of maturity at value yT  y(t) =   Values of bond at ;me t  x(t) = Influence of external factors on fluctua;ons in bond price 

•  Observa5onsEven“5me”systemscanhaveboundarycondi5ons

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€

d 2y (t)

dt 2= f y(t), dy (t)

dt,x1(t),x2(t),...,xN (t),y( )

y(0) = y0 y(T ) = yT

DiscreteSystem

•  Considerthesystem y[i]=x[i+1]–2x[i]+x[i‐1]

• Whathappensifitisappliedtothesequence? x=[0,0,0,0,10,10,10,0,0,0,0]

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10 10

SameSystemAppliedtoanImage

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Generalizing a 1-D result to 2-D. What differences do you see?

ANaturalImage

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Another

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Observa,ons

•  Arichclassofsystems(butbynomeansallsystemsofinteresttous)aredescribedbydifferen5alanddifferenceequa5ons.

•  Suchequa5ons,alone,donotcompletelydescribetheinput‐outputbehaviorofasystem:weneedauxiliaryinforma5on(ini5alcondi5ons,boundarycondi5ons).

•  Insomecasesthesystemofinteresthas5measthenaturalindependentvariableandiscausal.However,thatisnotalwaysthecase.

•  Seeminglydifferentphysicalsystemsmayhaveverysimilarmathema5caldescrip5ons.

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Causality

•  Asystemiscausaliftheoutputdoesnotan5cipatefuturevaluesoftheinput,i.e.,iftheoutputatany5medependsonlyonvaluesoftheinputuptothat5me.

•  All5medependentphysicalsystemsarecausal,because5meonlymovesforward.Effectoccursadercause.

•  Causalitydoesnotapplytospa5alsignals.(Wecanmovebothledandright,upanddown.)

•  Causalitydoesnotapplytorecordedsignals,e.g.tapedsportsgamesvs.livebroadcast.

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ShiBInvariance

•  Mathema5cally,Asystemthatmapsx →  y is shi@ invariant (SI) if for any input x[n] and any domain shi@ k, 

       if S(x[n]) → y[n] then S(x[n+k]) → y[n+k] 

•  Informally, a system is shi@‐invariant if its behavior does not depend on its absolute posi;on within its domain 

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Graphically

•  If

•  then

8/29/08 Comp665–Introduc5on&Signals 13Baraniuk, R. Shift Invariant Systems, Connexions Web site. http://cnx.org/content/m12042/1.1/

ShiBInvariantorNot?

•  xisaninputsignalandyisanoutputsignal

•  y(t)=x(t+1)+x(t‐1)•  y(t)=x(t)+x2(t)•  y(t)=x(t)+x(t2)•  y[i]=Ax[i+1]+Bx[i]+Cx[i‐1]•  y[i]=ix[i]•  y[i]=sin(x[i])–sin(x[i+1])•  y[i]=x[i]sin(i)

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ADeduc,on

•  IftheinputtoanSIsystemisperiodic  (itrepeats),thentheoutputisperiodic

•  Why?Proof:– x(t)=x(t+T) Defini5onofperiodicty– y(t)=f(x(t))y(t+T)=f(x(t+T)) Defini5onofSI

– Sincex(t)=x(t+T),theny(t)=y(t+T)

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Linearity

•  Wewillspendsome5meexploringaspecialclassofsystemscalledlinear

•  Notallsystemsarelinear,butmanyare

•  Mostnon‐linearsystemscanbetreated“locallylinear”withininfinitesimalregionsoftheirdomain

•  Wehavepowerfultoolsforsimplifyingandanalyzinglinearsystems

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Defini,onofLinearity

•  Asystemislinearifsuperposi;on holds•  Forarbitraryscalarsaandb,superposi5onstates:If S(x1[n])  y1[n]  and S(x2[n])  y2[n] then S(a x1[n] + b x2[n])  a y1[n] + b y2[n] 

•  Asystemcanbeshidinvariantbutnotlinear,likewiseasystemcanbelinearbutnotSI

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SummingLotsofSignals

•  Superposi5onadnausea

•  Inlinearsystems,a0inputalwaysyieldsa0output

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S ai xi [n]i =0

N

∑

= ai yi [n]

i =0

N

∑

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S 0( ) ≡S 0xi [n]i =0

N

∑

= 0yi [n] = 0

i =0

N

∑

BeautyofLinearity

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If we know the response of a system to some particular input, and we are able to decompose a new input into a linear combination of our “known” input, then We can exactly predict the output.

TheUl,mateCanonicalInput

•  Theimpulse(con5nuous)orunit(discrete)input

•  Ifweknewtheoutputresponseofalinearsystemtooneofthese,thenwecouldfinditsresponsetoanyinput

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Unit impulse, infinite height with unit area

Ex. Width = a, height = 1/a, as a ∞ (Dirac delta)

Unit input, height = 1

(Kronecker delta)

t=0 n=0

ImpulsesasaSystemBasis

•  Wecanconsideranyinputasalinearcombina5onofshidedandscaledimpulses

•  Forshi@‐invariant linear systems thiswillallowustoexactlypredicttheoutputbymerelysummingtogetheriden5callyscaledversionsoftheknownimpulseresponse

•  Impulsesareourfirst,andthemostnaturalbasisforexploringsignals,butnottheonlyone!

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