2. Z-transform and theorem

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2. Z-transform and theorem

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2. Z-transform and theorem

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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?

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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.

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2.1 Z-transform

• Solution
• Example 1 Find the Z-transform of unit step
  function.

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2.1 Z-transform

• Apply the definition of Z-transform, we have

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2.1 Z-transform

• Another method

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2.1 Z-transform

• Example 2 Find the Z-transform of a exponential
  decay.
• Solution

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2.1 Z-transform

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

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2.1 Z-transform

• Example 3 Find the Z-transform of a cosine
  function.
• Solution As

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2.1 Z-transform

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2.1 Z-transform
Exercise 2 Find the Z-transform for decayed
cosine function
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2.1 Z-transform
Example 4 Find the Z-transform for Solution
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2.1 Z-transform
Exercise 3 Find the Z-transform for
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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(sa))
• etc.

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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)
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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).
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2.2 Z-transform theorems
If f(t)0 for tlt0 has the Z-transform F(z),
then Proving By Z-transform definition, we
have
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2.2 Z-transform theorems
Defining mk-n, we have Since f(mT)0 for mlt0,
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.
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2.2 Z-transform theorems
Final value theoremSuppose that f(t), where
f(t)0 for tlt0, 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.
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2.3 Z-transform examples

• Example 1 Assume that f(k)0 for klt0, find the
  Z-transform of f(k)9k(2k-1)-2k3, k0,1,2.
• 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.

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2.3 Z-transform examples

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2.3 Z-transform examples

• Example 2 Obtain the Z-transform of the curve
  x(t) shown below.

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2.3 Z-transform examples

• Solution From the figure, we have
• K 0 1 2 3 4 5 6
• f(k) 0 0 0 1/3 2/3 1 1
• Apply the definition of Z-transform, we have

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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.
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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.

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2.4 Inverse Z-transform

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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).
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2.4 Inverse Z-transform

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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)1z-1z-2z-3
• f(kT)Z-1F(z)1, for k0, 1, 2,

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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

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2.5 Inverse Z-transform examples

• Multiply (1-z-1) to both side and let z-11, we
  have

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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

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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.

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2.5 Inverse Z-transform examples

• 2) Direct division method
• Example 1 F(z)1/(1z-1), find f(kT).

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2.5 Inverse Z-transform examples

• Finally, we obtain F(z)1-z-1z-2-z-3
• K 0 1 2 3
• F(kT) 1 -1 1 -1
• Example 2 Given ,
• Find f(kT).
• Solution Dividing the numerator by the
  denominator, we obtain

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2.5 Inverse Z-transform examples
Finally, we obtain F(z)1 4z-1 7z-2
10z-3 K 0 1 2 3 F(kT) 1 4 7
10 Exercise 5 , Find
f(kT). Ans. k 0 1 2 3 4 5 f(kT) 0 0.3679 0.84
63 1 1 1
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2.5 Inverse Z-transform examples
3) Computational method using Matlab Example
Given find f(kT). Solution num1 2 0
den1 2 1 Say we want the value of f(kT) for
k0 to 30 u1 zeros(1,30) Ffilter(num, den,
u) 1 4 7 10 13 16 19 22
25 28 31
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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)-8.3333(0.5)k8.333(0.8)k-2k(0.
8)k-1 x(k)00.50.050.6151.2035-1.6257-1.87
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Reading

• Study book
• Module 2 The Z-transform and theorems
• Textbook
• Chapter 2 The Z-transform (pp23-50)

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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.

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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 2816 Krad/s.
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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 ?s10 Krad/s, for i 0, we
have the figure in bold line. For i1, we have
the figure in bold-dot line.
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Tutorial
For I-1, ?2, we have
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Tutorial

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

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Tutorial
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Tutorial
Exercise 2 Find the Z-transform for a decayed
cosine function Solution 1
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Tutorial
Solution 2
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Tutorial
Exercise 3 Find the Z-transform for Solution
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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
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Tutorial
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Tutorial
Exercise 5 Given Find f(kT) using
direct-division method. Solution
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Tutorial
Continuous
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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.
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Tutorial

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

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Tutorial

• Solution 2 Expand F(z) into a polynomial form
• Num0 0.5 1 0
• Den1 2.1 1.44 0.32
• U1 zeros(1,40)
• Ffilter(Num, den,U)
• 0 0.5 0.05 -0.615 -1.2035