Continuous-Time Chapter Signals and LTI Systems

1 ECE 2610 signal and Systems9 1 Continuous-Time Signals and LTI SystemsAt the start of the course both continuous and discrete - time sig-nals were introduced. In the world of Signals and Systems model-ing, analysis, and implementation, both discrete - time andcontinuous- time Signals are a reality. We live in an analog world,is often said. The follow-on courses to ECE2610, Circuits andSystems I (ECE2205) and Circuits and Systems II (ECE3205)focus on Continuous-Time Signals and Systems . In particular cir-cuits based implementation of Systems is investigated in greatdetail. There still remains a lot to discuss about continuous-timesignals and Systems without the need to consider a circuit imple-mentation. This Chapter begins that Signals To begin with Signals will be classified by their support inter-valTwo-Sided Infinite-Length Signals Sinusoids are a primary example of infinite duration Signals ,that are also periodicChapter9 Continuous-Time SignalsECE 2610 Signals and Systems9 2( ) The period for both the real sinusoid and complex sinusoidsignals is The signal may be any periodic signal , say a pulse train orsquarewave A two-sided exponential is another example( )xt()A 0t +() t <<,cos=xt()Aej ej 0t t <<,=T02 0 =xt()Ae t t <<,= 10 4 224 4 224 4 ()52 t2--- cos=xt()2et2 =xt()

2 Pulse Train=Period = 2sPulse Width = exponentialContinuous- time SignalsECE 2610 Signals and Systems9 3 One-Sided Signals Another class of Signals are those that exist on a semi-infiniteinterval, , are zero for (support ) The Continuous-Time unit-step function, , is useful fordescribing one-sided Signals ( ) When we multiply the previous two-side Signals by the step-function a one-side signal is createdtt0<t[0 ), ut()ut()1,t0 0, otherwise = 11234 4 224xt()52 t2--- 4--- ut()cos=xt()2et2 ut()=xt()ut()=One-sided exponentialtttContinuous- time SignalsECE 2610 Signals and Systems9 4 The start time can easily be changed by letting ( )Finite-Duration Signals Finite duration Signals will have support over just a finitetime interval, , A convenient way of crating such Signals is via pulse gatingfunction such as( )ttt0 xt()ut2 ()1,t2 0, otherwise ==t[4 10), pt()ut4 ()ut10 () 1, 4t10< 0, otherwise ==ttpt()xt()52 t2--- 4--- pt()cos=24681012 4 202424681012 4 224 The Unit ImpulseECE 2610 Signals and Systems9 5 The Unit Impulse The topics discussed up to this point have all followed logi-cally from our previous study of discrete - time Signals andsystems The unit impulse signal , , however is more difficult todefine than the unit impulse sequence, Recall that The unit impulse signal is defined as( )and( ) What does this mean?]]

3 It would seem that must have zero width, yet havearea of unity A test function, , can be defined that in fact becomes as ( ) t() n[] n[]1,n0=0, otherwise = t()0t0 ,= t()td 1= t() t() t() 0 t()12 -------, t <<0,otherwise =The Unit ImpulseECE 2610 Signals and Systems9 6 The claim is that( ) Check ( ) and ( ) In plotting a scaled unit-impulse signal , , , we plot avertical arrow with the amplitude actually corresponding tothe areat t()12 1---------12 2--------- 1 20 1 2 t() 0 lim t()= t() 0 lim0t0 ,= t()td 1=A t()tA t()A()0 The Unit ImpulseECE 2610 Signals and Systems9 7 Sampling Property of the Impulse A noteworthy property of is that( ) Discussion Since is zero everywhere except , only thevalue is of interest Using the test function we also note that( )so as the only value of that matters is Also observe that( ) t()ft() tt0 ()ft0() tt0 ()=Sampling Property tt0 ()tt0=ft0() t()ft() t()ft()2 (), t <<0,otherwise = 0 ft()f0()ttft()ft()ft() t()f0() t() ft() t()f0() t() ft() t()f0()2 ---------f0()()ft() t()td f0() t()td =f0() t()td f0()==The Unit ImpulseECE 2610 Signals and Systems9 8 Integral Form( )Example.

4 The sampling property of results in When integrated we haveOperational Mathematics and the Delta Function The impulse function is not a function in the ordinary sense It is the most practical when it appears inside of an integral From an engineering perspective a true impulse signal doesnot exist We can create a pulse similar to the test function aswell as other test functions which behave like impulsefunctions in the limit The operational properties of the impulse function are veryuseful in Continuous-Time Signals and Systems modeling, aswell as in probability and random variables, and in modelingdistributions in electromagneticsft() tt0 ()td ft0()=Sampling/Sifting Property2 t() ()cosut() t3 ()+ t()2 ()() ()cosu3() t3 ()+2 t() ()cosut() t3 ()+[]td ()cosu3()+ ()cos1+== t()The Unit ImpulseECE 2610 Signals and Systems9 9 Derivative of the Unit Step A case in point where the operational properties are veryvaluable is when we consider the derivative of the unit stepfunction From calculus you would say that the derivative of the unitstep function, , does not exist because of the discontinu-ity at Consider( ) The area property of states that( ) Invoking the area property we have( )which says that this integral behaves like the unit step func-tion( )ut()t0= () d t t() t()tdab 1,a0 and b0 <0, otherwise = () d t 1,t0 0, otherwise =ut() () d t =The Unit ImpulseECE 2610 Signals and Systems9 10 From calculus we recognize that ( ) implies also that( ) Similarly,( )

5 If we now consider situations where a product exits, , , we can invoke the product rule for derivatives toobtain( )Example: The derivative of is t()ddt-----ut()= tt0 ()ddt-----utt0 ()=xt()ft()ut()=ddt-----ft()ut()ddt----- ft() ut()ft()ddt-----ut() +=f t()ut()ft() t()+=xt()e4t ut()ut1 ()+=xt()x t()ddt-----xt()4e4t ut()e4t t() t1 ()++==4e4t ut() t() t1 ()++= ()dt------------xt()(1)(1)Continuous-Tim e SystemsECE 2610 Signals and Systems9 11 Continuous-Time Systems A Continuous-Time system operates on the input to producean output( )Basic system Examples( )( )( )( ) In all of the above we can calculate the output given the inputand the definition of the system operator For linear time -invariant Systems we are particularly inter-ested in the impulse response, that is the output, ,when , for the system initially at restyt()Txt(){}=T {}xt()yt()yt()xt()[]2=Squareryt()xttd ()= time Delayyt()dxt()

6 Dt------------=Differentiatoryt()x () d t =Integratoryt()ht()=xt() t()=Linear time -Invariant SystemsECE 2610 Signals and Systems9 12 Example: Integrator Impulse Response Using the definitionLinear time -Invariant Systems In the study of discrete - time Systems we learned the impor-tance of Systems that are linear and time -invariant, and howto verify these properties for a given system operatorTime-Invariance A time invariant system obeys the following( )for any Both the squarer and integrator are time invariant The system ( )is not time invariant as the gain changes as a function of timeyt()ht() () d t ut()===xtt0 ()ytt0 () t0yt() ct()xt()cos=Linear time -Invariant SystemsECE 2610 Signals and Systems9 13 Linearity A linear system obeys the following( )where the inputs are applied together or applied individuallyand combined via and later The squarer is nonlinear by virtue of the fact thatproduces a cross term which does not exist when the twoinputs are processed separately and then combined The integrator is linear sinceThe Convolution Integral For linear time -invariant (LTI) Systems the convolution inte-gral can be used to obtain the output from the input and thesystem impulse response( ) x1t() x2t()+ y1t() y2t()+ yt() x1t() x2t()+[]2= 2x12t()2 x1t()x2t() 2x2t()++=yt() x1 () x2 ()+[] d t = x1 () d t x2 () d t +=yt()x ()ht () d xt()*ht()

7 ==Convolution IntegralLinear time -Invariant SystemsECE 2610 Signals and Systems9 14 The notation used to denote convolution is the same as thatused for discrete - time Signals and Systems , , the convolu-tion sum Evaluation of the convolution integral itself can prove to bevery challengingExample: Setting up the convolution integral we haveor simply,which is known as the unit rampyt()xt()*ht()ut()*ut()==yt()u ()ut () d = u ()ut ()t0t1yt()0,t0< ,d0t t0 =0,t0<t,t0 =yt()tut()rt() =Impulse Response of Basic LTI SystemsECE 2610 Signals and Systems9 15 Properties of Convolution Commutativity:( ) Associativity:( ) Distributivity over Addition:( ) Identity Element of Convolution:( )What is ? It turns out that proofImpulse Response of Basic LTI Systems For certain simple Systems the impulse response can befound by driving the input with and observing the output For complex Systems transform techniques, such as theLaplace transform, are more appropriate xt()*ht()ht()*xt()=xt()*h1t()[]*h2t()xt( )*h1t()*h2t()[]=xt()*h1t()*h2t()[]xt()*h 1t()xt()*h2t()+=xt()*ht()ht()=xt()xt() t()= t()*ht() ht()= ()ht () d ()ht0 () d =ht() () d ht()== t()Convolution of ImpulsesECE 2610 Signals and Systems9 16 Integrator( )Ideal delay( ) Note that this means that( )Convolution of Impulses Basic Theorem:( )Example.

8 Using the time shift property ( )Evaluating Convolution IntegralsStep and Exponential Consider and We wish to find ht()x () d t x () ()=ut()==ht()xttd ()xt() t()= ttd ()==xt()* ttd ()xttd ()= tt1 ()* tt2 () tt1t2+() ()= t()2 t3 () []*ut() t()*ut()2 t3 ()*ut() ut()2ut3 () =xt()ut2 ()=ht()e3t ut()=yt()xt()*ht()=Evaluating Convolution IntegralsECE 2610 Signals and Systems9 17( ) To evaluate this integral we first need to consider how thestep functions in the integrand control the limits of integra-tion For or there is no overlap in the product thatcomprises the integrand, so For or there is overlap for , sohere( )yt()e3 u ()ut 2 () d = e3 u ()ut 2 ()t20< t2 0t2 1ut 2 ()t20> t20< t2<yt()0=t20> t2> [0t2) , yt()e3 d0t2 =e3 3 ----------0t2 =13---1e3t2 () []ut2 ()=t2013---yt()Evaluating Convolution IntegralsECE 2610 Signals and Systems9 18 Note: The use of the exponential impulse response in exam-ples is significant because it occurs frequently in practice, , an RC lowpass filter circuitExample.]

9 And Find by evaluating the convolution integral Suppose that and yt()xt()RCht()1RC--------etRC-------- ut()=xt()eat ut()=ht()ebt ut()=yt()xt()*ht()=yt()ea u ()ebt () ut () d =ea ebt () d0t =ebt eab () d0t =ebt ab ------------eab () ab () --------------------0t ebt ab ------------1eab ()t []ut()==1ab ------------ebt eat []ut()ab ,=a2=b3=Evaluating Convolution IntegralsECE 2610 Signals and Systems9 19 Square-Pulse Input Consider a pulse input of the form( )where is the pulse width and The output is( ) From the step response analysis we know that,( )so ()a2b3 ,=txt()ut()utT () =Txt()T0t1ht()eat ut()=yt()ut()*ht()utT ()*ht() =ut()*ht()1a---1eat []ut()=Properties of LTI SystemsECE 2610 Signals and Systems9 20( ) Plot the results for and Properties of LTI SystemsCascade and Parallel Connections We have studied cascade and parallel system earlier For a cascade of two LTI Systems having impulse responses and respectively, the impulse response of the cas-cade is the convolution of the impulse responses( )yt()1a---1aat []ut()1a---1aatT () []utT () =T5=a1= ()a1T,5==th1t()h2t()hcascadet()h1t()*h2t ()=h1t()h2t()ht()h1t()*h2t()=xt()yt()xt( )yt()CascadeProperties of LTI SystemsECE 2610 Signals and Systems9 21 For two Systems connected in parallel, the impulse responseis the sum of the impulse responses( )

10 Differentiation and Integration of Convolution Since the integrator and differentiator are both LTI systemoperations, when used in combination with another syst