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Chapter 3 Convolution Representation

Chapter 3 Convolution Representation. DT Unit-Impulse Response. Consider the DT 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. System.

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Chapter 3 Convolution Representation

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  1. Chapter 3Convolution Representation

  2. DT Unit-Impulse Response • Consider the DT 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 System

  3. Example • Consider the DT system described by • Its impulse response can be found to be

  4. Representing Signals in Terms ofShifted and Scaled Impulses • Let x[n] be an arbitrary input signal to a DT LTI system • Suppose that for • This signal can be represented as

  5. Exploiting Time-Invariance and Linearity

  6. The Convolution Sum • This particular summation is called the convolution sum • Equation is called the convolution representation of the system • Remark: a DT LTI system is completely described by its impulse response h[n]

  7. Block Diagram Representation of DT LTI Systems • Since the impulse response h[n] provides the complete description of a DT LTI system, we write

  8. The Convolution Sum for Noncausal Signals • Suppose that we have two signals x[n] and v[n] that are not zero for negative times (noncausal signals) • Then, their convolution is expressed by the two-sided series

  9. Example: Convolution of Two Rectangular Pulses • Suppose that both x[n] and v[n] are equal to the rectangular pulse p[n] (causal signal) depicted below

  10. The Folded Pulse • The signal is equal to the pulse p[i] folded about the vertical axis

  11. Sliding over

  12. Sliding over - Cont’d

  13. Plot of

  14. Properties of the Convolution Sum • Associativity • Commutativity • Distributivity w.r.t. addition

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

  16. Example: Computing Convolution with Matlab • Consider the DT LTI system • impulse response: • input signal:

  17. Example: Computing Convolution with Matlab – Cont’d

  18. Example: Computing Convolution with Matlab – Cont’d • Suppose we want to compute y[n] for • Matlab code: n=0:40; x=sin(0.2*n); h=sin(0.5*n); y=conv(x,h); stem(n,y(1:length(n)))

  19. Example: Computing Convolution with Matlab – Cont’d

  20. 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

  21. Exploiting Time-Invariance • Let x[n] be an arbitrary input signal with for • Using the sifting property of , we may write • Exploiting time-invariance, it is System

  22. Exploiting Time-Invariance

  23. 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.,

  24. 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)

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

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

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

  28. Example – Cont’d • Case 1:

  29. Example – Cont’d • Case 2:

  30. Example – Cont’d • Case 3:

  31. Example – Cont’d • Case 4:

  32. Example – Cont’d

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

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

  35. 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

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

  37. 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

  38. 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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