Chapter 2 Discrete System Analysis – Discrete Signals

1. Chapter 2Discrete System Analysis – Discrete Signals

2. Continuous-time analog signal Sampling of Continuous-time Signals sampler Output of sampler T How to treat the sampling process mathematically ? For convenience, uniform-rate sampler(1/T) with finite sampling duration (p) is assumed. p 1 time T where p(t) is a carrier signal (unit pulse train)

3. carrier signal p(t) PAM This procedure is called a pulse amplitude modulation (PAM) The unit pulse train is written as By Fourier series or magnitude phase

4. -4s -3s -2s -s 0 s 2s 3s 4s

5. |F(j)| 1 c : Cutoff frequency -s /2 -c 0 c s/2

6. 0 s -s frequency folding 0 -s s

7. Theorem : Shannon’s Sampling Theorem To recover a signal from its sampling, you must sample at least twice the highest frequency in the signal. Remarks: i) A practical difficulty is that real signals do not have Fourier transforms that vanish outside a given frequency band. To avoid the frequency folding (aliasing) problem, it is necessary to filter the analog signal before sampling. Note: Claude Shannon (1917 - 2001)

8. ii) Many controlled systems have low-pass filter characteristics. iii) Sampling rates > 10 ~ 30 times of the BW of the system. iv) For the train of unit impulses -2s -s s 0

9. |X(j)| |X*(j)| |Y(j)| Ideal filterG(j) 1 1/T 1/T -c 0 c -c 0 c -c 0 c reconstruction Remarks : • Impulse response of ideal low-pass filter (non-causality) • Ideal low-pass filter is not realizable in a physical system. •  How to realize it in a physical system ? ZOH or FOH T sampling

10. ii)(Aliasing) It is not possible to reconstruct exactly a continuous-time signal in a practical control system once it is sampled.

11. iii)(Hidden oscillation) If the continuous-time signal involves a frequency component equal to n times the sampling frequency (where n is an integer), then that component may not appear in the sampled signal.

13. Signal Reconstruction How to reconstruct (approximate) the original signal from the sampled signal? -ZOH (zero-order hold) - FOH (first-order hold)

14. ZOH (Zero-order Hold) zero-order hold reconstruction k-1 k k+1 time T

15. phase lag

16. Ideal low-pass filter

17. Remarks: i) The ZOH behaves essentially as a low-pass filter. ii) The accuracy of the ZOH as an extrapolator depends greatly on the sampling frequency, . • iii) In general, the filtering property of the ZOH is used almost • exclusively in practice.

18. k-1 k k+1 T FOH (First-order Hold)

19. When k =0,

20. 2 1 0 T 2T 3T time -1

21. first zero zero first Large lag(delay) in high frequency makes a system unstable

22. Remark: • At low frequencies, the phase lag produced by the ZOH exceeds that of FOH, but as the frequencies become higher, the opposite is true

23. T Z-transform

25. Because R(z) is a power series in z-1 , the theory of power series may be applied to determine the convergence of • the z-transform. ii) The series in z-1 has a radius of convergence  such that the series converges absolutely when | z-1 |< iii) If   0, the sequence {rk} is said to be z-transformable.

26. Remark: unit impulse = Z-transform of Elementary Functions i) Unit pulse function ii) Unit step function

27. iii) Ramp function

28. iv) Polynomial function v) Exponential function

29. vi) Sinusoidal function

30. Remark: Refer Table 2-1 in pp.29-30 (Ogata) Also, refer Appendix B.2 Table in pp. 702-703(Franklin)

31. Correspondence with Continuous Signals z=esT s-plane z-plane

32. ② ③ ① -1 0 1 ⑤ ④ ImZ ImZ 0 1 ReZ ReZ j ③ ② ① 0  ④ ⑤ j -2 1 0 

33. ImZ ImZ 1 ReZ ReZ j 0 0  fixed j 0 

34. Important Properties and Theorems of the z-transform 1. Linearity 2. Time Shifting

36. 3. Convolution 4. Scaling 5. Initial Value Theorem

37. 6. Final Value Theorem

38. Remark: Refer Table 2-2 in p. 38.(Ogata) Also, refer Appendix B.1 Table in p.701(Franklin)

39. Discrete-time domain Continuous-time domain S-plane Z-plane

42. Inverse z-transform • Power Series Method (Direct Division) • ii) Computational Method : • - MATLAB Approach - Difference Equation Approach • iii) Partial Fraction Expansion Method • iv) Inversion Integral Method

43. Example 1) Power Series Method

44. Example 2) Computational Method

46. Example 3) Partial Fraction Expansion Method Remark:

47. Example 4) Inverse Integral Method : where cis a circle with its center at the origin of the z plane such that all poles of F(z)zk-1 are inside it.

48. Case 1) simple pole Case 2) m multiple poles