2. 3.1 The Z-Transform
• Counterpart of the Laplace transform for discrete-time signals
• Generalization of the Fourier Transform
Fourier Transform does not exist for all signals
• Definition:
• Compare to DTFT definition:
• z is a complex variable that can be represented as z=r ej
• Substituting z=ej will reduce the z-transform to DTFT
Chapter 3: The Z-Transform 1
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(
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:
اندازه
:
فاز
تبدیل
z
طرفهیک
تبدیل
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3.1 The Z-Transform
4. The z-transform and the DTFT
• Convenient to describe on the complex z-plane
• If we plot z=ej for =0 to 2 we get the unit circle
Chapter 3: The Z-Transform 3
Re
Im
Unit Circle
r=1
0
2 0 2
j
e
X
5. Convergence of the z-Transform
• DTFT does not always converge
Example: x[n] = anu[n] for |a|>1 does not have a DTFT
• Complex variable z can be written as r ej so the z-
transform
convert to the DTFT of x[n] multiplied with exponential
sequence r –n
• For certain choices of r the sum
maybe made finite
Chapter 3: The Z-Transform 4
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6. Region of Convergence (ROC)
• ROC: The set of values of z for which the z-transform converges
• The region of convergence is made of circles
Chapter 3: The Z-Transform 5
Re
Im
• Example: z-transform converges for
values of 0.5<r<2
ROC is shown on the left
In this example the ROC includes the unit circle,
so DTFT exists
7. • Example:
Doesn't converge for any r.
DTFT exists.
It has finite energy.
DTFT converges in a mean square sense.
• Example:
Doesn't converge for any r.
It doesn’t have even finite energy.
But we define a useful DTFT with
impulse function.
n
n
x o
cos
sin c n
x n
n
Region of Convergence (ROC)
8. Example 1: Right-Sided Exponential Sequence
• For Convergence we require
• Hence the ROC is defined as
• Inside the ROC series converges to
Chapter 3: The Z-Transform 7
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a 1
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• Region outside the circle of
radius a is the ROC
• Right-sided sequence ROCs
extend outside a circle
9. (
ارچپگدنباله
)
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Example 2: Left-Sided Exponential Sequence
10. Example 3: Two-Sided Exponential Sequence
Chapter 3: The Z-Transform 9
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11. Example 4: Finite Length Sequence
Chapter 3: The Z-Transform 10
otherwise
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Pole-zero plot
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13. SEQUENCE TRANSFORM ROC
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if
or
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ALL
Some common Z-transform pairs
14.
1
:
cos
2
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cos
1
cos
:
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1
:
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Some common Z-transform pairs
15.
0
:
1
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:
cos
2
1
sin
sin
1
2
2
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0
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ROC
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a
otherwise
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:
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2
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sin
sin
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Some common Z-transform pairs
16. 3.2 Properties of The ROC of Z-Transform
• The ROC is a ring or disk centered at the origin
• DTFT exists if and only if the ROC includes the unit circle
• The ROC cannot contain any poles
• The ROC for finite-length sequence is the entire z-plane
except possibly z=0 and z=
• The ROC for a right-handed sequence extends outward from the
outermost pole possibly including z=
• The ROC for a left-handed sequence extends inward from the
innermost pole possibly including z=0
• The ROC of a two-sided sequence is a ring bounded by poles
• The ROC must be a connected region
• A z-transform does not uniquely determine a sequence without
specifying the ROC
Chapter 3: The Z-Transform 15
17. Stability, Causality, and the ROC
• Consider a system with impulse response h[n]
• The z-transform H(z) and the pole-zero plot shown below
• Without any other information h[n] is not uniquely determined
|z|>2 or |z|<½ or ½<|z|<2
• If system stable ROC must include unit-circle: ½<|z|<2
• If system is causal must be right sided: |z|>2
Chapter 3: The Z-Transform 16
18. 3.4 Z-Transform Properties: Linearity
• Notation
• Linearity
– Note that the ROC of combined sequence may be larger than either ROC
– This would happen if some pole/zero cancellation occurs
– Example:
•Both sequences are right-sided
•Both sequences have a pole z=a
•Both have a ROC defined as |z|>|a|
•In the combined sequence the pole at z=a cancels with a zero at z=a
•The combined ROC is the entire z plane except z=0
Chapter 3: The Z-Transform 17
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19. Z-Transform Properties: Time Shifting
• Here no is an integer
– If positive the sequence is shifted right
– If negative the sequence is shifted left
• The ROC can change
– The new term may add or remove poles at z=0 or z=
• Example
Chapter 3: The Z-Transform 18
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20. Z-Transform Properties: Multiplication by
Exponential
• ROC is scaled by |zo|
• All pole/zero locations are scaled
• If zo is a positive real number: z-plane shrinks or expands
• If zo is a complex number with unit magnitude it rotates
• Example: We know the z-transform pair
• Let’s find the z-transform of
Chapter 3: The Z-Transform 19
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21. Z-Transform Properties: Differentiation
• Example: We want the inverse z-transform of
• Let’s differentiate to obtain rational expression
• Making use of z-transform properties and ROC
Chapter 3: The Z-Transform 20
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22. Z-Transform Properties: Conjugation
Chapter 3: The Z-Transform 21
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23. Z-Transform Properties: Time Reversal
• ROC is inverted
• Example:
• Time reversed version of
Chapter 3: The Z-Transform 22
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24. Z-Transform Properties: Convolution
• Convolution in time domain is multiplication in z-domain
• Example: Let’s calculate the convolution of
• Multiplications of z-transforms is
• ROC: if |a|<1 ROC is |z|>1 if |a|>1 ROC is |z|>|a|
• Partial fractional expansion of Y(z)
Chapter 3: The Z-Transform 23
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ROC
assume
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26. 3.3 The Inverse Z-Transform
• Formal inverse z-transform is based on a Cauchy integral
• Less formal ways sufficient most of the time
– Inspection method
– Partial fraction expansion
– Power series expansion
• Inspection Method
Make use of known z-transform pairs such as
Example: The inverse z-transform of
Chapter 3: The Z-Transform 25
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27. Inverse Z-Transform by Partial Fraction
Expansion
• Assume that a given z-transform can be expressed as
• Apply partial fractional expansion
• First term exist only if M>N
– Br is obtained by long division
• Second term represents all first order poles
• Third term represents an order s pole
– There will be a similar term for every high-order pole
• Each term can be inverse transformed by inspection
Chapter 3: The Z-Transform 26
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X
28. Inverse Z-Transform by Partial Fraction
Expansion
• Coefficients are given as
• Easier to understand with examples
Chapter 3: The Z-Transform 27
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29. Example 5: 2nd Order Z-Transform
Chapter 3: The Z-Transform 28
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30. Example 5 Continued
• ROC extends to infinity
– Indicates right sided sequence
Chapter 3: The Z-Transform 29
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31. Example 6
• Long division to obtain Bo
Chapter 3: The Z-Transform 30
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32. Example 5 Continued
• ROC extends to infinity
– Indicates right-sided sequence
Chapter 3: The Z-Transform 31
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33. Inverse Z-Transform by Power Series
Expansion
• The z-transform is power series
• In expanded form
• Z-transforms of this form can generally be inversed easily
• Especially useful for finite-length series
Chapter 3: The Z-Transform 32
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Example 6