ZTransforms and Transfer Functions

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Z-Transforms and Transfer Functions

• Ting Yan (
  )

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Outline

• Signals and Systems
• Z-Transforms
• 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

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Signals

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

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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.
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Control System
Reference Input
Measured Output
Controller
Target System
Control error
Control Input
Transducer
Transduced Output
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Common Signals
exponential
(ak)
agt1
impulse
alt1
delayed impulse
sine
step
cosine
ramp
exponentially modulated cosine/sine
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Z-Transform of a Signal
u(k)
Z-1
u(0) u(1) u(2) u(3) u(4)
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Z-Transform Contd

• 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
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Z Transform of Unit Impulse Signal
Z
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

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Delayed Unit Impulse Signal
Z
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

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Z-Transform of Unit Step Signal
Z
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

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Unit Step Signal - continued
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Z-Transform of Exponential Signal
Z
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!
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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

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Control System
Reference Input
Measured Output
Controller
Target System
Control error
Control Input
Transducer
Transduced Output
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What does Linear mean exactly?

• Scaling
• Superposition

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Time Invariance
Idiom u(k-n) is u(k) delayed by n time units!
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Reality Check

• Typically speaking, are computing systems linear?
  Why?
• Consider saturation
• Typically speaking, are computing systems
  time-invariant? Why?

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Unit Impulse Response
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.
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An Example 3-MA
uimpulse(k)
6 x

uimpulse(k-1)
9 x
u(k)

uimpulse(k-2)
3 x

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An Example 3-MA
yimpulse(k)
6 x

yimpulse(k-1)
9 x
y(k)

yimpulse(k-2)
3 x

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Convolution

• y(5) u(0) yimpulse(k)
• u(1) yimpulse(k-1)
• u(2) yimpulse(k-2)
• u(3) yimpulse(k-3)
• u(4) yimpulse(k-4)

yimpulse(k)
u(0) x

yimpulse(k-1)
u(1) x
y(k)

yimpulse(k-2)
u(2) x

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Important Theorem
Time Domain

u(k)
(convolution)
v(k)
y(k)

U(z)
V(z)
Y(z)
(multiplication)
Z Domain
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Z-Transform/Inverse Z-Transform
LTI yimpuse(k)0.3k-1
u (k)0.7k
y (k)?

(convolution)
Transfer Function
(multiplication)

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Delay the Unit Step Signal
y(k)u(k-1)
LTI yimpuse(k) udelayed(k)
u (k)
y (k)

ustep (k)
udelayed(k)
(convolution)
udstep(k)
Transfer Function
(multiplication)

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Delayed Unit Step Signal Contd
Z
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!
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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

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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)

Input got delayed for n time units
y(k)u(k-n)
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n-Delay
y(k)u(k-n)
Transfer function z-n
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Unit Shift and n-Shift
y(k)u(k1)
Caveat u(0) disappears
y(k)u(kn)
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Other properties of Z-Transform

• Linearity

Z-Transform
Time Domain
Y(z)aU(z)
y(k)au(k)
Scaling
Y(z)U(z)V(z)
y(k)u(k)v(k)
Superposition
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sin? cos?
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From Exponential to Trigonometric
?
Zcos(k?)? Zsin(k?)?
Euler Formula
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Z-Transform of sin/cos
Z-Transform
Time Domain
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Exponentially Modulated sin/cos
A damped oscillating signal a typical output of
a second order system
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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)
udstep(k)
(convolution)
uramp(k)
Z-1
Transfer Function
(multiplication)
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Inverse Z-Transform
Z
u(k)
U(z)
Z-1?

• Table Lookup if the Z-Transform looks familiar,
  look it up in the Z-Transform table!
• Long Division
• Partial Fraction Expansion

Z-1?
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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)

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Partial Fraction Expansion

• Many Z-transforms of interest can be expressed as
  division of polynomials of z

May be trickier complex root duplicate root
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An Example
(z-2)(z-4)
Z-1
u2(k)c24k-1, kgt0
c0? c1? c2?
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Get The Constants!
(z-2)(z-4)
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Partial Fraction Expansion contd
How to get c0 and cjs ?
define
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An Example Complete Solution
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Solving Difference Equations
Transfer Function
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A Difference Equation Example
Exponentially Weighted Moving Average
y(k)cy(k-1)(1-c)u(k-1)
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Solve it!
LTI y(k)0.4y(k-1)0.6u(k-1)
u (k)0.8k
y (k)?
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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
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Why are poles important?
Z domain
poles
Time domain
components
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Various pole values (1)
p-1.1
p1.1
p-1
p1
p0.9
p-0.9
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Various pole values (2)
p0.9
p-0.9
p0.6
p-0.6
p0.3
p-0.3
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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 poles corresponding component is
  oscillating, while a positive poles
  corresponding component is monotonous

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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!
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Why do we need Z-Transform?

• A signal can be characterized with its
  Z-transform (poles, final value )
• In an LTI system, Z-transform of Y(z) is the
  multiplication of Z-transform of U(z) and the
  transfer function
• The LTI system can be characterized by the
  transfer function, or the Z-transform of the unit
  impulse response

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Transfer Function
Transfer Function
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Example
LTI y(k)0.4y(k-1)0.6u(k-1)
u (k)0.8k
y (k)?
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When There Are Complex Poles
If
If
Or in polar coordinates,
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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 ps complex conjugate must also be a pole
• Complex poles appear in pairs

Time domain
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An Example
Z-Domain Complex Poles
Time-Domain Exponentially Modulated Sin/Cos
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Poles Everywhere
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Observations

• Using poles to characterize a signal
• The smaller is r, the faster converges the
  signal
• r lt 1, converge
• r gt 1, does not converge, unbounded
• r1?
• When the angle increase from 0 to pi, the
  frequency of oscillation increases
• Extremes 0, does not oscillate, pi, oscillate
  at the maximum frequency

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Change Angles
Im
Re
-0.9
0.9
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Changing Absolute Value
Im
Re
1
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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 2p)

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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 pjs and rs 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)

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An Example
converge to 2
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Final Value Theorem

• Enable us to decide whether a system has a steady
  state error (yss-rss)

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Final Value Theorem
If any pole of (1-z)Y(z) lies out of or ON the
unit circle, y(k) does not converge!
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What Can We Infer from TF?

• Almost everything we want to know
• Stability
• Steady-State
• Transients
• Settling time
• Overshoot

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Bounded Signals
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BIBO Stability

• Bounded Input Bounded Output Stability
• If the Input is bounded, we want the Output is
  bounded, too
• If the Input is unbounded, its okay for the
  Output to be unbounded
• For some computing systems, the output is
  intrinsically bounded (constrained), but limit
  cycle may happen

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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
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Are these BIBO?
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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.
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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!
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Steady State Gain
yss
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Steady-State Gain Contd

• 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
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More General Cases
Z
Transfer Function
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Example
LTI y(k)0.4y(k-1)0.6u(k-1)
u (k)1
y (k)?
Z
Z
Yss? G(1)1, so yss1
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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 piltpj,pi/pjk-1 approaches 0 when k is
  large (pik-1 converges faster than pjk-1)

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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

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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!
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Examples Positive Pole
Dominant Pole 0.9
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Examples Negative Pole
Dominant Pole -0.9
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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

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No Dominant Pole
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Dominant Pole Contd

• 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

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Summary

• Signals/Systems
• 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) -gt 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