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Chapter 3

Chapter 3. CT LTI Systems Updated: 9/16/13. A Continuous-Time System. How do we know the output?. X(t). y(t). System . 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)]

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Chapter 3

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  1. Chapter 3 CT LTI Systems Updated: 9/16/13

  2. A Continuous-Time System • How do we know the output? X(t) y(t) System

  3. 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 solution What is the input-output relationship for LTI-CT Systems?

  4. Convolution Integral • An approach (available tool or operation) to describe the input-output relationship for LTI Systems • In a LTI system • d(t) h(t) • Remember h(t) is T[d(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 X(t)=d(t) y(t)=h(t) LTI System X(t) y(t) LTI System: h(t)

  5. Convolution Integral • Remember • So what is the general solution for X(t)=Ad(t-kto) y(t)=Ah(t-kto) LTI System X(t) y(t) LTI System ?

  6. Convolution Integral • Any input can be expressed using the unit impulse function Proof: Sifting Property tot and integrate by dt X(t) y(t) LTI System

  7. Convolution Integral • Given • We obtain Convolution Integral • That is: A system can be characterized using its impulse response: y(t)=x(t)*h(t) X(t) y(t) LTI System X(t) y(t) LTI System: h(t) By definition Do not confuse convolution with multiplication! y(t)=x(t)*h(t)

  8. Convolution Integral X(t) y(t) LTI System: h(t)

  9. Convolution Integral - Properties • Commutative • Associative • Distributive • Thus, using commutative property: Next: We draw the block diagram representation!

  10. Convolution Integral - Properties • Commutative • Associative • Distributive

  11. Simple Example • What if a step unit function is the input of a LTI system? • S(t) is called the Step Response u(t) y(t)=S{u(t)}=s(t) LTI System Step response can be obtained by integrating the impulse response! Impulse response can be obtained by differentiating the step response

  12. Example 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. u(t) y(t) h(t)=e^-at Because t>0 Draw x(t), h(t), h(t-t),etc.  next slide The fact that a>0 is not an issue!

  13. Example 1 – Cont. y(t); for a=3 y(t) * t t U(-(t-t)) U(-(t-t)) t>0 t<0 Remember we are plotting it over t and t is the variable

  14. Example 1 – Cont. Graphical Representation (similar) a=1 In this case! http://www.wolframalpha.com/input/?i=convolution+of+two+functions&lk=4&num=4&lk=4&num=4

  15. Example 1 – Cont. Graphical Representation (similar) Note in our case We have u(t) rather than rectangular function! http://www.jhu.edu/signals/convolve/

  16. 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. x(t) y(t) h(t)=e^-at Note that for t>0; x(t) =0; so the integration can only be valid up to t=0 Draw x(t), h(t), h(t-t),etc.  next slide

  17. Example – Cont. * x(t)= e^at u(-t) h(t)=e^-at u(t) ?

  18. Another Example notes

  19. Properties 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? notes

  20. Example • Is this an stable system? • What about this? notes

  21. Differential-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 is

  22. Is 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 output delay by the same amount? • Is this a linear system? notes notes + e(t) Y(t) Sum Integrator X(t) - a

  23. Example • Is this a time invariant linear system? R L V(t) Ldi(t)/dt + Ri(t)= v(t) a= -R/L b=1/L y(t)=i(t) x(t) = V(t)

  24. Solution of DE • A classical model for the solution of DE is called method of undermined coefficients • yc(t) is called the complementary or natural solution • yp(t) is called the particular or forced solution

  25. Solution of DE Thus, for x(t) =constant  yp(t)=P x(t) =Ce^-7t  yp(t)= Pe^-7t x(t) =2cos(3t)  yp(t)=P1cos(3t)+P2sin(3t)

  26. Example • Solve • Assume x(t) = 2 and y(0) = 4 • What happens if notes yc(t) = Ce^-2t; yp(t) = P; P = 1 y(t) = Ce^-2t + 1 y(0) = 4  C = 3  y(t) = 3e^-2t + 1

  27. Schaum’s Examples • Chapter 2: • 2, 4-6, 8, 10, 11-14, 18, 19, 48, 52, 53,

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