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Understanding Convolution: Representation and Properties in DT and CT Systems

Gain insights into convolution representation and properties in DT and CT systems. Explore block diagram representations, computation examples, and analytical explanations. Learn convolution integrals, properties, and system descriptions. Dive deep into convolution with practical Matlab computations.

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Understanding Convolution: Representation and Properties in DT and CT Systems

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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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