Chapter 3A Continuous-Time SystemLTI SystemsConvolution IntegralSlide 5Slide 6Slide 7Slide 8Convolution Integral - PropertiesSlide 10Simple ExampleExample 1Example 1 – Cont.Example 1 – Cont. Graphical Representation (similar)Slide 15Example 2Example – Cont.Another ExampleProperties of CT LTI SystemsExampleDifferential-Equations ModelsIs the First-Order DE Linear?ExampleSolution of DESlide 25Slide 26Schaum’s ExamplesChapter 3CT LTI SystemsUpdated: 9/16/13A Continuous-Time System•How do we know the output?System X(t) y(t)LTI Systems•Time Invariant –X(t) y(t) & x(t-to)  y(t-to) •Linearity–a1x1(t)+ a2x2(t) a1y1(t)+ a2y2(t)–a1y1(t)+ a2y2(t)= T[a1x1(t)+a2x2(t)]•Meet the description of many physical systems•They can be modeled systematically –Non-LTI systems typically have no general mathematical procedure to obtain solutionWhat is the input-output relationship for LTI-CT Systems?•An approach (available tool or operation) to describe the input-output relationship for LTI Systems•In a LTI system t) h(t) –Remember h(t) is T[t)]–Unit impulse function  the impulse response•It is possible to use h(t) to solve for any input-output relationship•One way to do it is by using the Convolution Integral Convolution IntegralLTI SystemX(t)=(t)y(t)=h(t)LTI System: h(t)X(t) y(t)Convolution Integral•Remember•So what is the general solution for LTI SystemX(t)=A(t-kto) y(t)=Ah(t-kto)LTI SystemX(t) y(t)?Convolution Integral•Any input can be expressed using the unit impulse function)()1)(()()()()()()()()()()()()()()()(txtxdttxdtxdttxtttxtttxtdttttttooooodtxtx )()()( LTI SystemX(t) y(t)Sifting PropertyProof:toand integrate by dConvolution Integral•Given•We obtain Convolution Integral•That is: A system can be characterized using its impulse response: y(t)=x(t)*h(t) LTI SystemX(t) y(t) dthxtydtTxtyLinearitydtxTtytTthtTthtxTty)()()()()()(:)()()()}({)()}({)()}({)(LTI System: h(t)X(t) y(t)Do not confuse convolution with multiplication!y(t)=x(t)*h(t)By definitionConvolution IntegralLTI System: h(t)X(t) y(t)Convolution Integral - Properties•Commutative•Associative •Distributive •Thus, using commutative property: dtxhdthxtx )()()()()( )](*)([)](*)([)]()([*)()](*)([*)()(*)](*)([)(*)()(*)(21212121thtxthtxththtxththtxththtxtxththtxNext: We draw the block diagram representation!Convolution Integral - Properties)](*)([)](*)([)]()([*)()](*)([*)()(*)](*)([)(*)()(*)(21212121thtxthtxththtxththtxththtxtxththtx•Commutative•Associative •DistributiveSimple Example•What if a step unit function is the input of a LTI system?•S(t) is called the Step Response LTI Systemu(t) y(t)=S{u(t)}=s(t)dttsdttythNotedhdtuhdthutstytuthtxthtstytuStyt/)(/)()(:)()()()()()()()()()()()()()}({)(Step response can be obtained by integrating the impulse response!Impulse response can be obtained by differentiating the step responseExample 1•Consider a CT-LTI system. Assume the impulse response of the system is h(t)=e^(-at) for all a>0 and t>0 and input x(t)=u(t). Find the output. h(t)=e^-at u(t) y(t)  )()1(1)1(1)()()()()()()()()()(0tueaeadedtuuedtuhtytuthtxthtyatattaaDraw x(), h(), h(t-),etc.  next slide Because t>0The fact that a>0 is not an issue!Example 1 – Cont. y(t)t>0t<0Remember we are plotting it over and t is the variableU(-(-t))U(-(-t))*  )()1(1)1(1)()()()()()()()()()(0tueaeadedtuuedtuhtytuthtxthtyatattaaty(t); for a=3tExample 1 – Cont. Graphical Representation (similar)http://www.wolframalpha.com/input/?i=convolution+of+two+functions&lk=4&num=4&lk=4&num=4http://www.wolframalpha.com/input/?i=convolution+of+two+functions&lk=4&num=4&lk=4&num=4a=1In this case!Example 1 – Cont. Graphical Representation (similar)http://www.jhu.edu/signals/convolve/http://www.jhu.edu/signals/convolve/Note in our caseWe have u(t) rather than rectangular function!Note in our caseWe have u(t) rather than rectangular function!Example 2•Consider a CT-LTI system. Assume the impulse response of the system is h(t)=e^-at for all a>0 and t>0 and input x(t)= e^at u(-t). Find the output. h(t)=e^-at x(t) y(t)            021)(21]01[21021210)()()()()()()()()()(0222)(aeatyeaaedeeteaeaedeetdtueuedthxtytuthtxthtytaatataatatatattaattaaDraw x(), h(), h(t-),etc.  next slide Note that for t>0; x(t) =0; so the integration can only be valid up to t=0Example – Cont.021)()()()()()(aeatytuthtxthtyta*h(t)=e^-at u(t)x(t)= e^at u(-t)dtxhtytuthtxthty)()()()()()()()(?Another ExamplenotesProperties of CT LTI Systems•When is a CT LTI system memory-less (static)•When does a CT LTI system have an inverse system (invertible)?•When is a CT LTI system considered to be causal? Assuming the input is causal:•When is a CT LTI system considered to be Stable? )()()()( tKxtytKth )()(*)( tththidtxhdthxtytt)()()()()(00notesdtthty )()(Example•Is this an stable system?•What about this? )()(3tuetht)()(3tuethtnotes3/1)()()(033dtedttuedtthtyttDifferential-Equations Models •This is a linear first order differential equation with constant coefficients (assuming a and b are constants)•The general nth order linear DE with constant equations isIs the First-Order DE Linear? •Consider•Does a1x1(t)+ a2x2(t) a1y1(t)+ a2y2(t)?•Is it time-invariant? Does input delay results in an