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Signal & Linear system

Signal & Linear system. Chapter 5 D T System Analysis : Z Transform Basil Hamed. Introduction. Z-Transform does for DT systems what the Laplace Transform does for CT systems In this chapter we will: -Define the ZT -See its properties -Use the ZT and its properties to analyze D-T systems.

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Signal & Linear system

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  1. Signal & Linear system Chapter5DT System Analysis : ZTransform Basil Hamed

  2. Introduction Z-Transform does for DT systems what the Laplace Transform does for CT systems In this chapter we will: -Define the ZT -See its properties -Use the ZT and its properties to analyze D-T systems Z-T is used to Solve difference equations with initial conditions Solve zero-state systems using the transfer function Basil Hamed

  3. 5.1 The Z-transform We define X(z),the direct Z-transform of x[n],as Where z is the complex variable. The unilateral z-Transform Basil Hamed

  4. Z-Transform of Elementary Functions: Example 5.2 P 499 find the Z-transform of • U[n] Solution • x[n]=={ =1 Z 1 Basil Hamed

  5. Z-Transform of Elementary Functions: b) x[n]=u[n]={ We have from power series from Book P 48 +……..= Basil Hamed

  6. Z-Transform of Elementary Functions: c) Basil Hamed

  7. Z-Transform of Elementary Functions: d) x(t)= { t nT X[n]= X[z]= Basil Hamed

  8. Z-Transform of Elementary Functions: Example given y Find X(z) & Y(z) Solution Basil Hamed

  9. Region of Convergence Basil Hamed

  10. Region of Convergence Basil Hamed

  11. Z-Transform of Elementary Functions: Y- Let n=-m Y - - As seen in the example above, X(z) & Y(z) are identical, the only different is ROC Basil Hamed

  12. Relationship between ZT & LT Basil Hamed

  13. Relationship between ZT & LT Basil Hamed

  14. ROC Basil Hamed

  15. ROC Example given Find X(z) Solution ROC Basil Hamed

  16. 5.2 Some Properties of The Z-Transform As seen in the Fourier & Laplace transform there are many properties of the Z-transform will be quite useful in system analysis and design. If Then a Basil Hamed

  17. 5.2 Some Properties of The Z-Transform Right Shift of x[n] (delay) Then … Note that if x[n]=0 for n=-1,-2,-3,…, then Z{x[n]}= Basil Hamed

  18. 5.2 Some Properties of The Z-Transform Left Shift in Time (Advanced) : : Example given Find y[n] Basil Hamed

  19. 5.2 Some Properties of The Z-Transform Basil Hamed

  20. 5.2 Some Properties of The Z-Transform Example Given For y[n], n x[n]=u[n], y[1]=1, y[0]=1 Solve the difference equation Solution take inverse z and find y[n] Basil Hamed

  21. 5.2 Some Properties of The Z-Transform Frequency Scaling (Multiplication by ) Then Example given Find Y[z] Solution From Z-Table ( Basil Hamed

  22. 5.2 Some Properties of The Z-Transform Differentiation with Respect to Z Then Example; given y[n]=n[n+1]u[n], find Y[z] Solution y[n]= Z[n u[n]]= And Basil Hamed

  23. 5.2 Some Properties of The Z-Transform Initial Value Theorem Then Example find x(0) Solution Basil Hamed

  24. 5.2 Some Properties of The Z-Transform The initial value theorem is a convenient tool for checking if the Z-transform of a given signal is in error. Using Matlab software we can have x[n]; The initial value is x(0)=1, which agrees with the result we have. Final value Theorem Basil Hamed

  25. 5.2 Some Properties of The Z-Transform As in the continuous-time case, care must be exercised in using the final value thm. For the existence of the limit; all poles of the system must be inside the unit circle. (system must be stable) Example given Find x Solution Example given x[n]=Find x Solution The system is unstable because we have one pole outside the unit circle so the system does not have final value, Basil Hamed

  26. Stability of DT Systems Basil Hamed

  27. 5.2 Some Properties of The Z-Transform Convolution Y(z)= X(z)H(z) Example: given h[n]={1,2,0,-1,1} and x[n]={1,3,-1,-2} Find y[n] Solution y[n]= x[n] * h[n] Y(z)=X(z)H(z) H Y[n]={1,5,5,-5,-6,4,1,-2} Basil Hamed

  28. 5.2 Some Properties of The Z-Transform Example: given Find the T. F of the System Basil Hamed

  29. 5.2 Some Properties of The Z-Transform Solution: Basil Hamed

  30. The Inverse of Z-Transform There are many methods for finding the inverse of Z-transform; Three methods will be discussed in this class. • Direct Division Method (Power Series Method) • Inversion by Partial fraction Expansion • Inversion Integral Method Basil Hamed

  31. The Inverse of Z-Transform 1. Direct Division Method (Power Series Method): The power series can be obtained by arranging the numerator and denominator of X(z) in descending power of Z then divide. Example determine the inverse Z- transform : Solution Z-0.1 ZZ-0.1 0.1 X(z)= Basil Hamed

  32. The Inverse of Z-Transform Example find x[n] Solution X(0)=1, x(1)=1/4, x(2)=13/16,……. In this example, it is not easy to determine the general expression for x[n]. As seen, the direct division method may be carried out by hand calculations if only the first several terms of the sequence are desired. In general the method does not yield a closed form for x[n]. Basil Hamed

  33. The Inverse of Z-Transform 2. Inversion by Partial-fraction Expansion T.F has to be rational function, to obtain the inverse Z transform. The use of partial fractions here is almost exactly the same as for Laplace transforms……the only difference is that you first divide by z beforeperforming the partial fraction expansion…then after expanding you multiply by z to get the final expansion Example find x[n] Basil Hamed

  34. The Inverse of Z-Transform Solution: Using same method used in Laplace transform To find A,B,C,D A=1, B=5/2, C=-9, D=9 Basil Hamed

  35. The Inverse of Z-Transform Example 5.3 P 501 given Find the inverse Z-Transform. Solution: From Table 5-1 (12-b) Basil Hamed

  36. The Inverse of Z-Transform 0.5r=1.6 r=3.2, =-2.246 rad =3+j4=5 Example find y[n] Basil Hamed

  37. The Inverse of Z-Transform 3. Inversion integral Method: If the function X(z) has a simple pole at Z=a then the residue is evaluated as Basil Hamed

  38. The Inverse of Z-Transform For a pole of order m at Z=a the residue is calculated using the following expression: ExampleFind x[n] for Solution: The only method to solve above function is by integral method. has multiple poles at Z= 1 Basil Hamed

  39. The Inverse of Z-Transform Example Obtain the inverse Z transform of Solution: Basil Hamed

  40. The Inverse of Z-Transform has a triple pole at Z=1 at triple pole Z=1] Example Obtain the inverse Z transform of Basil Hamed

  41. The Inverse of Z-Transform By Partial Fraction: By Inversion Integral Method: , has double poles at Z=1 at double poles at Z=1] Basil Hamed

  42. Basil Hamed

  43. Transfer Function Basil Hamed

  44. Transfer Function Basil Hamed

  45. Transfer Function • Zero Input Response Zero State Response Basil Hamed

  46. ZT For Difference Eqs. Given a difference equation that models a D-T system we may want to solve it: -with IC’s -with IC’s of zero Note…the ideas here are very much like what we did with the Laplace Transform for CT systems. We’ll consider the ZT/Difference Eq. approach first… Basil Hamed

  47. Solving a First-order Difference Equation using the ZT Basil Hamed

  48. Solving a First-order Difference Equation using the ZT Basil Hamed

  49. First Order System w/ Step Input Basil Hamed

  50. Solving a Second-order Difference Equation using the ZT Basil Hamed

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