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Chapter 3 Convolution Representation

Chapter 3 Convolution Representation. System. CT Unit-Impulse Response. Consider the CT SISO system: If the input signal is and the system has no energy at , the output is called the impulse response of the system . System.

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Chapter 3 Convolution Representation

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  1. Chapter 3Convolution Representation

  2. System CT Unit-Impulse Response • Consider the CT SISO system: • If the input signal is and the system has no energy at , the output is called the impulse response of the system System

  3. Exploiting Time-Invariance • Let x(t) be an arbitrary input signal with for • Using the sifting property of , we may write • Exploiting time-invariance, it is System

  4. Exploiting Time-Invariance

  5. Exploiting Linearity • Exploiting linearity, it is • If the integrand does not contain an impulse located at , the lower limit of the integral can be taken to be 0,i.e.,

  6. The Convolution Integral • This particular integration is called the convolution integral • Equation is called the convolution representation of the system • Remark: a CT LTI system is completely described by its impulse response h(t)

  7. Block Diagram Representation of CT LTI Systems • Since the impulse response h(t) provides the complete description of a CT LTI system, we write

  8. Example: Analytical Computation of the Convolution Integral • Suppose that where p(t) is the rectangular pulse depicted in figure

  9. Example – Cont’d • In order to compute the convolution integral we have to consider four cases:

  10. Example – Cont’d • Case 1:

  11. Example – Cont’d • Case 2:

  12. Example – Cont’d • Case 3:

  13. Example – Cont’d • Case 4:

  14. Example – Cont’d

  15. Properties of the Convolution Integral • Associativity • Commutativity • Distributivity w.r.t. addition

  16. Properties of the Convolution Integral - Cont’d • Shift property: define • Convolution with the unit impulse • Convolution with the shifted unit impulse then

  17. Properties of the Convolution Integral - Cont’d • Derivative property: if the signal x(t) is differentiable, then it is • If both x(t) and v(t) are differentiable, then it is also

  18. Properties of the Convolution Integral - Cont’d • Integration property: define then

  19. Representation of a CT LTI System in Terms of the Unit-Step Response • Let g(t) be the response of a system with impulse response h(t) when with no initial energy at time , i.e., • Therefore, it is

  20. Representation of a CT LTI System in Terms of the Unit-Step Response – Cont’d • Differentiating both sides • Recalling that it is and or

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