Lecture III: Systems and their properties

1 Lecture III: Systems and their propertiesMaxim RaginskyBME 171: Signals and SystemsDuke UniversitySeptember 3, 2008 Maxim RaginskyLecture III: Systems and their propertiesThis lecturePlan for the Lecture :1 What is a system ?2 system properties :causalitymemorylinearitytime invariance3 linear time-invariant (LTI) systems4 Nonlinear systemsMaxim RaginskyLecture III: Systems and their propertiesWhat is a system ?Recap:a system is any physical device, process or computer algorithmthat transforms input signals into output :electronic circuitsbiological Systems : audiovisual system , cardiovascular system , Systems : the stock market, social networks, processors in scientific or medical equipment or inaudio/video devicesWe will state our definitions for continuous-time Systems . They areessentially the same for discrete-time RaginskyLecture III: Systems and their propertiesCausalityA systemSiscausalif the output at timetdoes not depend on thevalues of the input at any timet > predictor:y(t) =x(t+ 1) noncausal since the output attimetdepends on the input at future timet+ 12 Ideal delay:y(t) =x(t 1) causal since the output at timetdepends only on the input at past timet 13 Moving average (MA) filter:y[n] =x[n 1]+x[n]+x[n+1]3 notcausal, since the output at timendepends in part on the input atfuture timen+ 1 Most physical Systems are causal.

2 However, noncausal Systems are widelyused in signal processing, for example, for smoothing of continuous-timeand discrete-time signals for noise removal or quality RaginskyLecture III: Systems and their propertiesMemoryAcausalsystemSismemoryle ssif the output at timetdepends only onthe input at timet. Otherwise, the system is said to have :depending on whom you ask, it may or may not make sense totalk about memory for noncausal Systems . To avoid confusion, in thisclass we will only talk about memory for causal amplifier:y(t) =Kx(t), whereK >0is the amplifier gain memoryless, since the output at timetdepends only on the input attimet2 Integrator:y(t) = t x( )d has memory, since the output at timetdepends on the input forall < RaginskyLecture III: Systems and their propertiesLinearityA systemSisadditiveif for any two inputsx1(t)andx2(t),S{x1(t) +x2(t)}=S{x1(t)}+S{x2(t)}homogeneousif, for any inputx(t)and any numbera,S{ax(t)}=aS{x(t)}.

3 A system that is both additive and homogeneous is calledlinear. In otherwords,Sis linear if, for any two inputsx1(t)andx2(t)and any twonumbersa1anda2,S{a1x1(t) +a2x2(t)}=a1S{x1(t)}+a2S{x2(t)}Maxim RaginskyLecture III: Systems and their propertiesLinearity: Example 1 Suppose the input and the output are related by the differential equationdy(t)dt=x(t).Additive?Yes:dy1(t) dt=x1(t),dy2(t)dt=x2(t)y(t) =S{x1(t)+x2(t)} dy(t)dt=x1(t)+x2(t) =d(y1(t) +y2(t))dtHomogeneous?Yes:dy(t)dt=x(t)ya( t) =S{ax(t)} dya(t)dt=ax(t) =ady(t)dt=d(ay(t))dtThe system is RaginskyLecture III: Systems and their propertiesLinearity: Example 2 Now consider the following system :y(t) =t2x(t)Additive?Yes:S{x1(t) +x2(t)}=t2(x1(t) +x2(t))=t2x1(t) +t2x2(t)=S{x1(t)}+S{x2(t)}Homogeneous?Ye s:S{ax(t)}=t2ax(t) =at2x(t) =aS{x(t)}The system is RaginskyLecture III: Systems and their propertiesLinearity: Example 3 Consider thesquare-law device:y(t) =x2(t)Additive?

4 No:(x1(t) +x2(t))2=x21(t) + 2x1(t)x2(t) +x22(t)6=x21(t) +x22(t)So,S{x1(t) +x2(t)}6=S{x1(t)}+S{x2(t)}Homogeneous?No :(ax(t))2=a2x2(t)6=ax2(t)unlessa= 1 The system is RaginskyLecture III: Systems and their propertiesLinearity: Example 4 Consider the systemy(t) = 3x(t) + 2 Additive?No:S{x1(t) +x2(t)}= 3(x1(t) +x2(t))+ 2On the other hand,S{x1(t)}+S{x2(t)}= 3(x1(t) + 3x2)+ 4S{x1(t) +x2(t)}6=S{x1(t)}+S{x2(t)}Homogeneous?No :S{ax(t)}= 3ax(t) + 2On the other hand,aS{x(t)}= 3ax(t) + 2a6= 3ax(t) + 2unlessa= 1 This system is not RaginskyLecture III: Systems and their propertiesTime invarianceA systemSistime-invariantif, for any inputx(t)and any fixed timet1,the outputS{x(t t1)}is equal toy(t t1), wherey(t)is the outputdue tox(t), ,y(t) =S{x(t)}. Systems that are not time-invariant are example: Systems described by linear differential equations withconstant coefficients, such as5d2y(t)dt2 3y(t) = dx(t)dt+ 2x(t). linear (RLC) circuits are described in this RaginskyLecture III: Systems and their propertiesTime invariance: Example 1 Consider the systemy(t) = 3x2(t)u(t)We haveS{x(t t1)}= 3x2(t t1)u(t).

5 On the other hand,y(t t1) = 3x2(t t1)u(t t1)6= 3x2(t t1)u(t)unlesst1= 0 This system is RaginskyLecture III: Systems and their propertiesTime invariance: Example 2 Consider the systemy(t) = t0e 2 x( )d We haveS{x(t)}= t0e 2 x( )d .ThenS{x(t t1)}= t0e 2 x( t1)d =e 2t1 t t1 t1e 2 x( )d andy(t t1) = t t10e 2 x( )d .SinceS{x(t t1)}6=y(t t1), this system is RaginskyLecture III: Systems and their propertiesTime invariance: Example 3 Consider the systemd2y(t)dt2= 3x(t)In other words, ify(t) =S{x(t)}, thend2y(t)dt= 3x(t).Letv(t) =x(t t1). So ifz(t) =S{v(t)}, thend2z(t)dt2= 3v(t) = 3x(t t1) =d2(y(t t1)) system is RaginskyLecture III: Systems and their propertiesLinear time-invariant (LTI) systemsWe will focus almost exclusively onlinear time-invariant(LTI) will prove later that any such system has aconvolutionrepresentationy(t) = h(t )x( )d (continuous-time)y[n] = k= h[n k]x[k](discrete-time)wherehis called the impulse response of the important property of LTI Systems is their action oncomplexexponentials: ifSis LTI, thenS{ej t}=c( )ej tfor some complex numberc( ).

6 So LTI Systems can attenuate or amplifyvarious frequency components of the RaginskyLecture III: Systems and their propertiesNonlinear systemsAnideal amplifiery(t) =Kx(t), K >0is (t)y(t)However, real amplifiers havesaturation effects: linear partP-P0xyy(t) =Kx(t)in thelinear range P x(t) P, but the outputsaturatesat some value KPfor|x(t)| RaginskyLecture III: Systems and their propertiesNonlinear systemsThesumming neuronmodel used in artificial neural networks and inmathematical models of biological neural Systems :w1wNw2x1(t)x2(t)xN(t)y(t)..y(t) = (N n=1wnxn(t)),where:x1(t), .. , xN(t)areNinputsw1, .. , wNare the synapticweights ( )is a nonlineartransformationThis is an example of amultiple-input, RaginskyLecture III: Systems and their properties