1. Chapter 2 Discrete System Analysis – Discrete Signals

2. Sampling of Continuous-time Signals
sampler
Continuous-time analog signal
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. This procedure is called a pulse amplitude modulation (PAM)
carrier signal p(t)
PAM
The unit pulse train is written as
By Fourier series or
magnitude
phase

4. s
2s
3s
4s
-s
-2s
-3s
-4s

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

6. -s s
frequency folding

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 ( )

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

9. Impulse response of ideal low-pass filter (non-causality)
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
Ideal filter G(j) T c
-c 1
|X(j)|
1/T
|X*(j)|
|Y(j)|
reconstruction
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. FOH (First-order Hold)
k-1 k
k+1 T

19. When k =0,

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

21. Large lag(delay) in high frequency makes a system unstable
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. Z-transform T

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. Z-transform of Elementary Functions
i) Unit pulse function
Remark: unit impulse =
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 (Franklin)

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

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

33. j
ReZ
ImZ 
fixed j
ReZ
ImZ 1 

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. Continuous-time
domain
Discrete-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 c is 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