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Rectangular Function Impulse Function Continuous Time Systems

Rectangular Function Impulse Function Continuous Time Systems. 2.4 &2.6. How do you represent a unit rectangular function mathematically?. Fundamental(1). u (t). Fundamental(2). u(t+0.5). u(t-0.5). u(-t-0.5). u(-t+0.5). Block Function (window). 1. rect (t/T)

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Rectangular Function Impulse Function Continuous Time Systems

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  1. Rectangular FunctionImpulse FunctionContinuous Time Systems 2.4 &2.6

  2. How do you represent a unit rectangular function mathematically?

  3. Fundamental(1) u(t)

  4. Fundamental(2) u(t+0.5) u(t-0.5) u(-t-0.5) u(-t+0.5)

  5. Block Function (window) 1 • rect(t/T) • Can be expressed as u(T/2-t)-u(-T/2-t) • Draw u(t+T/2) first; then reverse it! • Can be expressed as u(t+T/2)-u(t-T/2) • Can be expressed as u(t+T/2)u(T/2-t) -T/2 T/2 1 -T/2 T/2 1 -T/2 T/2 -T/2 T/2

  6. Application • The rectangular pulse can be used to extract part of a signal

  7. c03f10 A Simple Cell Phone Charger Circuit (R1 is necessary) Another Application: Signal strength indicator

  8. Mathematical Modeling V1(t) V1(t-To) V1(t-2To) Modify the unit rectangular pulse: Shift to the right by To/4 The period is To/2

  9. Application of Impulse Function The unit impulse function is used to model sampling operation, i.e. the selection of a value of function at a particular time instant using analog to digital converter.

  10. Generation of an Impulse Function Ramp function epsilon approaches 0

  11. Shifted Impulse Function d(t) 0 d(t-to) 0 to

  12. The Impulse Function We use a vertical arrow to represent 1/ε because g(t) Increases dramatically as ε approaches 0.

  13. Another Definition of the Impulse Function

  14. Mathematica Connection

  15. Property f[t] f[t-2]

  16. Property

  17. Property Shifted Unit Step Function Slope is sharp at t=2

  18. Property

  19. Property area: 1/ε ε/2=1/2 1/ε2ε=2 g(at), a>1, e.g. 2 g(2t), 1/ ε to/2+ε/2 to/2 δ(t) 1 1/2

  20. Property area: 1/ε2ε=2 g(at), a<1, e.g. 1/2 g(2t), 1/ ε 2to+2ε 2to δ(t) 2 1 2to

  21. Property d(t)

  22. Example

  23. Continuous-Time Systems • A system is an operation for which cause-and-effect relationship exists • Can be described by block diagrams • Denoted using transformation T[.] • System behavior described by mathematical model X(t) y(t) T [.] (meat grinder)

  24. Inverting Amplifier Vout=-(R1/R2)

  25. Inverting Summer Example Vout=-RF(V1/R1+V2/R2) If RF/R1=1, RF/R2=1 Vout=-(V1+V2)

  26. Multiplier

  27. Parallel Connection

  28. Cascade Connection

  29. Feedback

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