1. Z-Transforms and Transfer Functions

2. Outline Signals and Systems Z-Transforms Transfer Functions
How to do Z-Transforms
How to do inverse Z-Transforms
How to infer properties of a signal from its Z-transform
Transfer Functions
How to obtain Transfer Functions
How to infer properties of a system from its Transfer Function

3. Signals The signals we are studying in this course – Discrete Signals
A discrete signal takes value at each non-negative time instance

4. Example of a System Filter raw readings from a noisy temperature
sensor
- Input Signal
smooth temperature
values after filtering
- Output Signal
A (SISO) system takes an input signal, manipulates it
and gives a corresponding output signal.

5. Control System Reference Input Control error Controller Control Input
Target
System
Measured
Output
Transduced
Output
Transducer

6. Common Signals exponential (ak) |a|>1 impulse |a|<1
delayed impulse
sine
step
cosine
ramp
exponentially
modulated
cosine/sine

7. Z-Transform of a Signal
U(z) Z
u(0) · z0
+u(1) · z-1
+u(2) · z-2
+u(3) · z-3
+u(4) · z-4 …
u(k)
Z-1
u(0)
u(1)
u(2)
u(3)
u(4) …

8. Z-Transform – Cont’d Mapping from a discrete signal to a function of z
Many Z-Transforms have this form:
Helps intuitively derive the signal properties
Does it converge?
To which value does it converge?
How fast does it converges to the value?
Rational Function of z

9. Z Transform of Unit Impulse Signal
uimpulse(k)
Uimpulse(z)
Z-1
u(0) = 1
u(1) = 0
u(2) = 0
u(3) = 0
u(4) = 0 …
1 · z0
+0 · z-1
+0 · z-2
+0 · z-3
+0 · z-4 …

10. Delayed Unit Impulse SignalZ
udelay(k)
Udelay(z)
Z-1
u(0) = 0
u(1) = 1
u(2) = 0
u(3) = 0
u(4) = 0 …
0 · z0
+1 · z-1
+0 · z-2
+0 · z-3
+0 · z-4 …

11. Z-Transform of Unit Step Signal
ustep(k)
Ustep(z)
Z-1
u(0) = 1
u(1) = 1
u(2) = 1
u(3) = 1
u(4) = 1 …
1 · z0
+1 · z-1
+1 · z-2
+1 · z-3
+1 · z-4 …

12. Unit Step Signal - continued
A little bit more math …
assuming

13. Z-Transform of Exponential Signal
uexp(k)
Uexp(z)
Z-1
u(0) = 1
u(1) = a
u(2) = a2
u(3) = a3
u(4) = a4 …
1 · z0
+a · z-1
+a2 · z-2
+a3 · z-3
+a4 · z-4 …
Remember this!

14. LTI Systems Linear, Time Invariant (LTI) System
Many systems we analyze or design are or can be approximated by LTI systems
We have a well-established theory for LTI system analysis and design
Example - A simple moving average
y(k)=[u(k-1)+u(k-2)+u(k-3)]/3
3-MA
u(k)
y(k)

15. Control System Reference Input Control error Controller Control Input
Target
System
Measured
Output
Transduced
Output
Transducer

16. What does “Linear” mean exactly?
Scaling
Superposition
3-MA
u(k)
y(k)
3-MA
λu(k)
λy(k)
3-MA
u1(k)
y1(k)
3-MA
u2(k)
y2(k)
3-MA
u1(k)+u2(k)
y1(k)+y2(k)

17. Time Invariance 3-MA 3-MA u(k) y(k) u’(k)=u(k-n) y’(k)=y(k-n) Idiom:
u(k-n) is u(k) delayed by n time units!
y(k+1)=[u(k)+u(k-1)+u(k-2)]/3
y(k+1-n)=[u(k-n)+u(k-1-n)+u(k-2-n)]/3
y’(k+1)=[u’(k)+u’(k-1)+u’(k-2)]/3

18. Reality Check Typically speaking, are computing systems linear? Why?
Consider saturation …
Typically speaking, are computing systems time-invariant? Why?

19. Unit Impulse Response 3-MA uimpulse(k) yimpulse(k) Claim:
If we know yimpulse(k), we can obtain y(k) corresponing to ANY input u(k)!
yimpulse(k) contains ALL information about
the input-output relationship of an LTI system.

20. An Example: 3-MA 3MA 3MA uimpulse(k) yimpulse(k) u (k) y (k) ?+
uimpulse(k-1)
9 x
u(k) = +
uimpulse(k-2)
3 x +…

21. An Example: 3-MA 3MA 3MA uimpulse(k) yimpulse(k) u (k) y (k) ?+
yimpulse(k-1)
9 x
y(k) = +
yimpulse(k-2)
3 x +…

22. Convolution y(5)= u(0) · yimpulse(k) + u(1) · yimpulse(k-1)
u(0) x +
yimpulse(k-1)
u(1) x
y(k) = +
yimpulse(k-2)
u(2) x +…

23. Important Theorem = * Z Z-1 Z Z-1 Z Z-1 = · Time Domain u(k) v(k) y(k)
(convolution)
v(k)
y(k) Z
Z-1 Z
Z-1 Z
Z-1 =
U(z)
(multiplication)
V(z)
Y(z)
Z Domain

24. Z-Transform/Inverse Z-Transform
LTI: yimpuse(k)=0.3k-1
u (k)=0.7k
y (k)?
Z-1 = *
(convolution) Z Z
Transfer Function
(multiplication) =

25. Delay the Unit Step Signal
y(k)=u(k-1)
LTI: yimpuse(k)
=udelayed(k)
u (k)
y (k) =
ustep (k) *
(convolution)
udelayed(k)
udstep(k) Z Z
Transfer Function Z
z-1
(multiplication) =

26. Delayed Unit Step Signal – Cont’dZ
udstep(k)
Udstep(z)
Z-1
u(0) = 0
u(1) = 1
u(2) = 1
u(3) = 1
u(4) = 1 …
0 · z0
+1 · z-1
+1 · z-2
+1 · z-3
+1 · z-4 …
Remember this!

27. Transfer Function
Transfer function provides a much more intuitive way to understand the input-output relationship, or system characteristics of an LTI system
Stability
Accuracy
Settling time
Overshoot …

28. Signals and Systems in Computer Systems
Spike, one-time fluctuation in input/output,
or disturbance
Change of reference value
Multiple changes of reference value
Sum of delayed step signals
ustep(k)+8ustep(k-3)-4ustep(k-6)
y(k)=u(k-n)
Input got delayed for n time units

29. n-Delay
y(k)=u(k-n)
Transfer function: z-n

30. Unit Shift and n-Shift
y(k)=u(k+1)
Caveat: u(0) disappears
y(k)=u(k+n)

31. Other properties of Z-Transform
Linearity
Time Domain
Z-Transform
y(k)=au(k)
Y(z)=aU(z)
Scaling
y(k)=u(k)+v(k)
Y(z)=U(z)+V(z)
Superposition

32. sin? cos?

33. From Exponential to Trigonometric?
Z[cos(kθ)]?
Z[sin(kθ)]?
Euler Formula:

34. Z-Transform of sin/cos
Time Domain
Z-Transform

35. Exponentially Modulated sin/cos
A damped oscillating signal – a typical output of a second order system

36. An LTI System – Discrete Integrator
y(k)=y(k-1)+u(k-1)
Y(k)=u(k-1)+u(k-2)+…+u(1)+u(0)
LTI: yimpuse(k)
=udstep(k)
u (k)
y (k) =
ustep(k) *
(convolution)
udstep(k)
uramp(k)
Transfer Function
Z-1 Z Z =
(multiplication)

37. Inverse Z-Transform Z Z-1? Z-1? u(k) U(z)
Table Lookup – if the Z-Transform looks familiar, look it up in the Z-Transform table!
Long Division
Partial Fraction Expansion
Z-1?

38. Long Division
Sort both nominator and denominator with descending order of z first
u(0)=3, u(1)=5, u(2)=7, u(3)=9, …, guess: u(k)=3ustep(k)+2uramp(k)

39. Partial Fraction Expansion
Many Z-transforms of interest can be expressed as division of polynomials of z
May be trickier:
complex root
duplicate root
where
k>0

40. An Example Z-1 Z-1 Z-1 (z-2)(z-4) U1(z)=c0 u1(k)=c0*uimpulse(k)
u2(k)=c1*2k-1, k>0
Z-1
u2(k)=c2*4k-1, k>0
c0? c1? c2?

41. Get The Constants!
(z-2)(z-4)

42. Partial Fraction Expansion – cont’d
How to get c0 and cj’s ?
define

43. An Example – Complete Solution

44. Solving Difference EquationsZ
Z-1
Transfer Function

45. A Difference Equation Example
Exponentially Weighted Moving Average
y(k)=cy(k-1)+(1-c)u(k-1)

46. LTI: y(k)=0.4y(k-1)+0.6u(k-1)
Solve it!
LTI: y(k)=0.4y(k-1)+0.6u(k-1)
u (k)=0.8k
y (k)? Z Z
Z-1

47. Signal Characteristics from Z-Transform
If U(z) is a rational function, and
Then Y(z) is a rational function, too
Poles are more important – determine key characteristics of y(k)
zeros
poles

48. Why are poles important?
Z domain
poles
Z-1
Time domain
components

49. Various pole values (1)
p=1.1
p=-1.1
p=1
p=-1
p=0.9
p=-0.9

50. Various pole values (2)
p=0.9
p=-0.9
p=0.6
p=-0.6
p=0.3
p=-0.3

51. Conclusion for Real Poles
If and only if all poles’ absolute values are smaller than 1, y(k) converges to 0
The smaller the poles are, the faster the corresponding component in y(k) converges
A negative pole’s corresponding component is oscillating, while a positive pole’s corresponding component is monotonous

52. How fast does it converge?
U(k)=ak, consider u(k)≈0 when the absolute value of u(k) is smaller than or equal to 2% of u(0)’s absolute value
Remember
This!

53. LTI: y(k)=0.4y(k-1)+0.6u(k-1)
Example
LTI: y(k)=0.4y(k-1)+0.6u(k-1)
u (k)=0.8k
y (k)? Z Z
Z-1

54. When There Are Complex Poles …If If
Or in polar coordinates,

55. What If Poles Are Complex
If Y(z)=N(z)/D(z), and coefficients of both D(z) and N(z) are all real numbers, if p is a pole, then p’s complex conjugate must also be a pole
Complex poles appear in pairs
Z-1
Time domain

56. An Example Z-Domain: Complex Poles Time-Domain:
Exponentially Modulated Sin/Cos

57. Poles Everywhere

58. Observations Using poles to characterize a signal
The smaller is |r|, the faster converges the signal
|r| < 1, converge
|r| > 1, does not converge, unbounded
|r|=1?
When the angle increase from 0 to pi, the frequency of oscillation increases
Extremes – 0, does not oscillate, pi, oscillate at the maximum frequency

59. Change Angles Im
-0.9 Re
0.9

60. Changing Absolute ValueIm Re 1

61. Conclusion for Complex Poles
A complex pole appears in pair with its complex conjugate
The Z-1-transform generates a combination of exponentially modulated sin and cos terms
The exponential base is the absolute value of the complex pole
The frequency of the sinusoid is the angle of the complex pole (divided by 2π)

62. Steady-State Analysis
If a signal finally converges, what value does it converge to?
When it does not converge
Any |pj| is greater than 1
Any |r| is greater than or equal to 1
When it does converge
If all |pj|’s and |r|’s are smaller than 1, it converges to 0
If only one pj is 1, then the signal converges to cj
If more than one real pole is 1, the signal does not converge … (e.g. the ramp signal)

63. An Example
converge to 2

64. Final Value Theorem
Enable us to decide whether a system has a steady state error (yss-rss)

65. Final Value Theorem If any pole of (1-z)Y(z) lies out of or ON the
unit circle, y(k) does not converge!

66. What Can We Infer from TF?
Almost everything we want to know
Stability
Steady-State
Transients
Settling time
Overshoot …

67. Bounded Signals

68. BIBO Stability Bounded Input Bounded Output Stability
If the Input is bounded, we want the Output is bounded, too
If the Input is unbounded, it’s okay for the Output to be unbounded
For some computing systems, the output is intrinsically bounded (constrained), but limit cycle may happen

69. Limit Cycle Output constrained, But oscillating – Bad!
Imagine CPU utilization
Constantly switching from
1 to 0, 0 to 1, …
Solution: make sure the system works in a linearized operating region

70. Are these BIBO? Unity y(k+1) = 1 P Controller y(k+1) = KP u(k)
Integrator
y(k+1) = y(k) + u(k)
I Controller
y(k+1) = y(k) + KI u(k)
M/M/1/K
y(k+1) = 0.49y(k) u(k)
Mystery
y(k+1) = -1.3y(k) + 2.3u(k)

71. Better Way to Decide BIBO or NOT
Theorem:
A system G(z) is BIBO stable iff all the poles of G(z) are inside the unit circle.
System
Time domain Eq
Transfer Function
Poles
Unity
y(k+1) = 1
G(z) = 1
N/A
P Controller
y(k+1) = KP u(k)
G(z) = KP
Integrator
y(k+1) = y(k) + u(k)
G(z) = 1/(z-1)
z=1
I Controller
y(k+1) = y(k) + KI u(k)
G(z) = KI/(z-1)
M/M/1/K
y(k+1) = 0.49y(k) u(k)
G(z) = 0.033/(z-0.49)
z = 0.49
Mystery
y(k+1) = -1.3y(k) + 2.3u(k)
G(z) = 2.3/(z+1.3)
z = -1.3

72. LTI: y(k)=0.4y(k-1)+0.6u(k-1)
Example
LTI: y(k)=0.4y(k-1)+0.6u(k-1)
u (k)=0.8k
y (k)? Z Z
BIBO? – only one pole at 0.4, so BIBO!

73. Steady State Gain
yss

74. Steady-State Gain – Cont’d
Which value does the output converges to when the input is an unit step signal?
First of all, it has to converge
Final Value Theorem
Unit Step Input

75. More General Cases Z
z=1
Transfer Function

76. LTI: y(k)=0.4y(k-1)+0.6u(k-1)
Example
LTI: y(k)=0.4y(k-1)+0.6u(k-1)
u (k)=1
y (k)? Z Z
Yss? G(1)=1, so yss=1

77. System Orders System Order = Number of Poles
The higher the system order is, the more complex the system behavior is
Some poles are more important than others
Why?
If |pi|<|pj|,|pi/pj|k-1 approaches 0 when k is large (pik-1 converges faster than pjk-1)

78. Overshoot and Setting Time
If not all poles are positive real numbers, overshoot may happen
Easy to figure out when the system is first order
For higher order systems, approximation to first order systems works under certain conditions
Setting time
First order system
Higher order systems

79. How fast does it converge?
U(k)=ak, consider u(k)≈0 when the absolute value of u(k) is smaller than or equal to 2% of u(0)’s absolute value
Remember
This!

80. Examples: Positive Pole
Dominant
Pole: 0.9

81. Examples: Negative Pole
Dominant
Pole: -0.9

82. Dominant Pole
We can approximate a high-order system with a first-order system with the dominant pole of the high-order system
IF the dominant pole DOES exist
Can give a pretty good estimation of settling time
Can give a reasonable estimate of the maximum overshoot
Some high-order systems do not have dominant pole! – for example

83. No Dominant Pole

84. Dominant Pole – Cont’d
If there is a dominant pole, it must be the pole with the maximum magnitude
The largest pole should have at least twice the magnitude of the other poles!
If the dominant pole is real (p’), the high-order system can be approximated by a first-order system

85. Summary Signals/Systems Characterize a signal with Z-transform
An LTI system can be specified by
Difference equation
Unit impulse response
Transfer function
If one is known, how to get the other two?
Characterize a signal with Z-transform
Z-domain (poles) -> Time domain (convergence, etc.)
Characterize a system with Transfer function
BIBO stability
Steady-State Gain
Transients: overshoot, settling time
If there exists a dominant pole