1. 2. Z-transform and theorem
Gc(s)
GP(s)
R(s)
E(s)
M(s)
Y(s)
Controller
Plant
Gc(z)
ZOH
GP(s)
R(z)
E(z)
M(z)
GHP(z)
Computer system
Y(z)
Plant
A/D
D/A

2. 2. Z-transform and theorem
time
f(t)
Time kT
f(kT)
A/D
D/A

3. 2. Z-transform and theorem
How can we represent the sampled data mathematically?
For continuous time system, we have a mathematical tool Laplace transform. It helps us to define the transfer function of a control system, analyse system stability and design a controller. Can we have a similar mathematical tool for discrete time system?

4. 2.1 Z-transform
For a continuous signal f(t), its sampled data can be written as,
Then we can define Z-transform of f(t) as
where z-1 represents one sampling period delay in time.

5. 2.1 Z-transform Solution:
Example 1: Find the Z-transform of unit step function.
f(t) t kT
f(kT)

6. 2.1 Z-transform
Apply the definition of Z-transform, we have

7. 2.1 Z-transform
Another method

8. 2.1 Z-transform
Example 2: Find the Z-transform of a exponential decay.
Solution:
f(t) t

9. 2.1 Z-transform
Exercise 1: Find the Z-transform of a exponential decay f(t)=e-at using other method.
f(t) t

10. 2.1 Z-transform Example 3: Find the Z-transform of a cosine function.
Solution: As

11. 2.1 Z-transform

12. 2.1 Z-transform
Exercise 2: Find the Z-transform for decayed cosine function

13. 2.1 Z-transform
Example 4: Find the Z-transform for
Solution:

14. 2.1 Z-transform
Exercise 3: Find the Z-transform for

15. 2.1 Z-transform
The functions can be given either in time domain as f(t) or in S-domain as F(s). They are equivalent. eg.
A unit step function: 1(t) or 1/s
A ramp function: t or 1/s2
f(t)=1-e-at or a/(s(s+a))
etc.

16. 2.2 Z-transform theorems
Linearity: If f(t) and g(t) are Z-transformable and  and  are scalar, then the linear combination f(t)+g(t) has the Z-transform
Z[f(t)+g(t)]= F(z)+ G(z)

17. 2.2 Z-transform theorems Shifting Theorem:
Given that the Z-transform of f(t) is F(z), find the Z-transform for f(t-nT).
f(t) t
f(t-nT) nT

18. 2.2 Z-transform theorems
If f(t)=0 for t<0 has the Z-transform F(z), then
Proving: By Z-transform definition, we have

19. 2.2 Z-transform theorems Defining m=k-n, we have
Since f(mT)=0 for m<0, we can rewrite the above as
Thus, if a function f(t) is delayed by nT, its Z-transform would be multiplied by z-n. Or, multiplication of a Z-transform by z-n has the effect of moving the function to the right by nT time. This is the so-called Shifting Theorem.

20. 2.2 Z-transform theorems
Final value theorem:Suppose that f(t), where f(t)=0 for t<0, has the Z-transform of F(z), then the final value of f(t) can be given by
There are other theorems for Z-transform. Please read the study book or textbook for more details.

21. Theorem
Name
Definition
Linearity
Multiply by e-at
Multiply by at
Time Shift 1
Time Shift 2
Differentiation
Integration
Final Value
Initial Value

22. f(t) F(z) (t) 1 u(t) t e-at 1 – e-at sint cost e-atsint e-atcost
F(s)
F(z)
(t) 1
u(t) t
e-at
1 – e-at
sint
cost
e-atsint
e-atcost

23. 2.3 Z-transform examples
Example 1: Assume that f(k)=0 for k<0, find the Z-transform of f(k)=9k(2k-1)-2k+3, k=0,1,2….
Solution: Obvious f(k) is a combination of three sub-function 9k(2k-1), 2k and 3. Therefore, first we can apply linearity theorem to f(k). Second, sub-function 9k(2k-1) can be considered as a product of k and 2-12k, then we can apply the theorem of multiply by ak. Finally, we can find the answer by combining these three together.

24. 2.3 Z-transform examples

25. 2.3 Z-transform examples
Example 2: Obtain the Z-transform of the curve x(t) shown below. 1 2 3 4 5 6 7 8 t
f(t)

26. 2.3 Z-transform examples Solution: From the figure, we have
K …
f(k) /3 2/3 1 1…
Apply the definition of Z-transform, we have

27. 2.3 Z-transform examples Example 3: Find the Z-transform of
Solution: Apply partial fraction to make F(s) as a sum of simpler terms.

28. 2.4 Inverse Z-transform
The inverse Z-transform: When F(z), the Z-transform of f(kT) or f(t), is given, the operation that determines the corresponding time sequence f(kT) is called as the Inverse Z-transform. We label inverse Z-transform as Z-1.

29. 2.4 Inverse Z-transform
Z-transform =
Inverse Z-transform 

30. 2.4 Inverse Z-transform
The inverse Z-transform can yield the corresponding time sequence f(kt) uniquely. However, it says nothing about f(t). There might be numerous f(t) for a given f(kT).
f(t) t T 2T 3T 4T 5T 6T

31. 2.4 Inverse Z-transform
x(kT)
f(t)
Zero-order
Hold
Low-pass
Filter

32. 2.5 Methods for Inverse Z-transform
How can we find the time sequence for a given Z-transform?
Z-transform table
Example 1: F(z)=1/(1-z-1), find f(kT).
F(z)=1+z-1+z-2+z-3+…
f(kT)=Z-1[F(z)]=1, for k=0, 1, 2, …

33. 2.5 Inverse Z-transform examples
Example 2: Given ,
Find f(kT).
Solution: Apply partial-fraction-expansion to simplify F(z), then find the simpler terms from the Z-transform table.
Then we need to determine k1 and k2

34. 2.5 Inverse Z-transform examples
Multiply (1-z-1) to both side and let z-1=1, we have

35. 2.5 Inverse Z-transform examples
Similar as the above, we let multiply (1-e-aTz-1) to both side and let z-1 =eaT, we have
Finally, we have

36. 2.5 Inverse Z-transform examples
Exercise 4: Given the Z-transform
Determine the initial and final values of f(kT), the inverse Z-transform of F(z), in a closed form.
Hint: Partial-fraction-expansion, then use Z-transform table, and finally applying initial & final value theorems of Z-transform.

37. 2.5 Inverse Z-transform examples
2) Direct division method
Example 1: F(z)=1/(1+z-1), find f(kT).

38. 2.5 Inverse Z-transform examples
Finally, we obtain: F(z)=1-z-1+z-2-z-3+…
K = …
F(kT)=
Example 2: Given ,
Find f(kT).
Solution: Dividing the numerator by the denominator, we obtain

40. 2.5 Inverse Z-transform examples
Finally, we obtain: F(z)=1+ 4z-1 + 7z z-3+…
K = …
F(kT)= …
Exercise 5: ,
Find f(kT).
Ans. :k …
f(kT) …

41. 2.5 Inverse Z-transform examples
3) Computational method using Matlab
Example: Given find f(kT).
Solution:
num=[1 2 0]; den=[1 –2 1]
Say we want the value of f(kT) for k=0 to 30
u=[1 zeros(1,30)]; F=filter(num, den, u) …

42. 2.5 Inverse Z-transform examples
Exercise 6: Given the Z-transform
Use 1) the partial-fraction-expansion method and 2) the Matlab to find the inverse Z-transform of F(z).
Answer: x(k)= (0.5)k+8.333(0.8)k-2k(0.8)k-1
x(k)=0;0.5;0.05;–0.615;–1.2035; ; …

43. Reading Study book Module 2: The Z-transform and theorems Textbook
Chapter 2 : The Z-transform (pp23-50)

44. Tutorial
Exercise: The frequency spectrum of a continuous-time signal is shown below.
What is the minimum sampling frequency for this signal to be sampled without aliasing.
If the above process were to be sampled at 10 Krad/s, sketch the resulting spectrum from –20 Krad/s to 20 Krad/s. -8 -4 4 8
 Krad/s
F()

45. Tutorial
Solution: 1) From the spectrum, we can see that the bandwidth of the continuous signal is 8 Krad/s. The Sampling Theorem says that the sampling frequency must be at least twice the highest frequency component of the signal. Therefore, the minimum sampling frequency for this signal is 2*8=16 Krad/s. -8 -4 4 8
 Krad/s
F()

46. Tutorial
2) Spectrum of the sampled signal is formed by shifting up and down the spectrum of the original signal along the frequency axis at i times of sampling frequency. As s=10 Krad/s, for i =0, we have the figure in bold line. For i=1, we have the figure in bold-dot line. 4 8
 Krad/s
F() 12 2 6 14 18 16 10 -4 -8

47. Tutorial For I=-1, 2,… we have -18 F() 4 8  Krad/s 12 2 6 14 18 1610 -2 -4 -6 -8
-14 4 8
 Krad/s 12 2 6 14 18 16 10

48. Tutorial
Exercise 1: Find the Z-transform of a exponential decay f(t)=e-aT using other method.
f(t) t

49. Tutorial

50. Tutorial
Exercise 2: Find the Z-transform for a decayed cosine function
Solution 1:

51. Tutorial
Solution 2:

52. Tutorial
Exercise 3: Find the Z-transform for
Solution:

53. Tutorial Exercise 4: Given the Z-transform
Determine the initial and final values of f(kT), the inverse Z-transform of F(z), in a closed form.
Solution: Apply the initial value theorem and the final value theorem respectively, we have

54. Tutorial

55. Tutorial Solution: Exercise 5: Given
Find f(kT) using direct-division method.
Solution:

56. Tutorial
Continuous

57. Tutorial Exercise 6: Given the Z-transform
Use 1) the partial-fraction-expansion method and 2) the Matlab to find the inverse Z-transform of F(z).
Solution1: To make the expanded terms more recognizable in the Z-transform table, we usually expand F(z)/z into partial fractions.

60. Tutorial Partial fraction for inverse Z-transform
If F(z)/z involve s a multiple pole, eg. P1, then

61. Tutorial Solution 2: Expand F(z) into a polynomial form
Num=[0 0.5 –1 0];
Den=[1 – –0.32];
U==[1 zeros(1,40)];
F=filter(Num, den,U) …