Z-transform power point presentation2.pdf

This ia Z-transform lecture ppt

1. ECE 8443 – Pattern RecognitionEE 3512 – Signals: Continuous and Discrete
•Objectives:
Relationship to the Laplace Transform
Relationship to the DTFT
Stability and the ROC
ROC Properties
Transform Properties
•Resources:
MIT 6.003: Lecture 22
Wiki: Z-Transform
CNX: Definition of the Z-Transform
CNX: Properties
RW: Properties
MKim: Applications of the Z-Transform
LECTURE 28: THE Z-TRANSFORM
AND ITS ROC PROPERTIES
Audio:URL:

2. EE 3512: Lecture 28, Slide 2
Definition Based on the Laplace Transform
•The z-Transform is a special case of the Laplace transform and results from
applying the Laplace transform to a discrete-time signal:
•Let us consider how this transformation maps the s-plane into the z-plane:
s = j:
s =  + j:
Recall, if a CT system
is stable, its poles lie
in the left-half plane.
Hence, a DT system is
stable if its poles are
inside the unit circle.
The z-Transform behaves
much like the Laplace
transform and can be
applied to difference equations
to produce frequency and time domain responses.

 




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
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




n
n
n
n
z
ts
nx
t
tnt
st
znxX(z)ettnxdtetxsX ][)(lim)()(
][
0 
circleunit the tomaps axis-1 
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
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eez

3. EE 3512: Lecture 28, Slide 3
ROC and the Relationship to the DTFT
•We can derive the DTFT by setting z = re
j
:
•The ROC is the region for which:
Depends only on r = |z| just like the ROC in the s-plane for the Laplace
transform depended only on Re{}.
If the unit circle is in the ROC, then the DTFT, X(e
j
), exists.
Example: (a right-sided signal)
If :
The ROC is outside a circle of radius a,
and includes the unit circle, which means
its DTFT exists. Note also there is a zero
at z = 0.
  
n
n
nn
n
n
r
rnxrnxrnxX(z)

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az
az
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n
n
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azaz 

or,,1
1
1
1
1



az
X(z)

4. EE 3512: Lecture 28, Slide 4
Stability and the ROC
•For a > 0: az
az
X(z)nuanx
n




for
1
1
][][
1
•If the ROC is
outside the unit
circle, the signal
is unstable.
1for
1
1
][][
1





z
z
X(z)
nunx •If the ROC
includes the unit
circle, the signal
is stable.

5. EE 3512: Lecture 28, Slide 5
Stability and the ROC (Cont.)
•For a < 0: az
az
X(z)nuanx
n




for
1
1
][][
1
•If the ROC is
outside the unit
circle, the signal
is unstable.
1for
1
1
][][
1





z
z
X(z)
nunx •If the ROC
includes the unit
circle, the signal
is stable.

6. EE 3512: Lecture 28, Slide 6
More on ROC
•Example:
If:
The z-Transform is the same, but the region of convergence is different.
  
?
1
)(1
1
]1[)(
signal)sided(left]1[][
1
1
1
1
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

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nuanx
n
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azza 

or,,1
1
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z
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X(z)
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1
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1
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7. EE 3512: Lecture 28, Slide 7
Stability and the ROC
•For: az
az
X(z)nuanx
n




for
1
1
]1[][
1
•If the ROC
includes the unit
circle, the signal
is stable.
1for
1
1
][][
1





z
z
X(z)
nunx •If the ROC
includes the unit
circle, the signal
is unstable.

8. EE 3512: Lecture 28, Slide 8
Properties of the ROC
•The ROC is an annular ring in the z-plane centered
about the origin (which is equivalent to a vertical
strip in the s-plane).
•The ROC does not contain any poles (similar to
the Laplace transform).
•If x[n] is of finite duration, then the ROC is the entire
z-plane except possibly z = 0 and/or z = :
•If x[n] is a right-sided sequence, and if |z| = r
0
is in the ROC, then all finite
values of z for which |z| > r
0 are also in the ROC.
•If x[n] is a left-sided sequence, and if |z| = r
0
is in the ROC, then all finite values
of z for which |z| < r
0
are also in the ROC.

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


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

)Re()()()(][]1[][
)Re()()()(0][]1[][
1)()()(1][][][
][
1
sesXTttxzROCzzXnnx
sesXTttxzROCzzXnnx
sXttxzallROCzXnnx
znxX(z)
sT
sT
n
n



:CT:DT

9. EE 3512: Lecture 28, Slide 9
Properties of the ROC (Cont.)
•If x[n] is a two-sided sequence, and if |z| = r
0 is in the ROC, then the ROC
consists of a ring in the z-plane including |z| = r
0
.
•Example:
right-sided left-sided two-sided
0][  bbnx
n
bz
bzbbz
zX
b
z
zb
nub
bz
bz
nub
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n
nn
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1
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1
1
][
][][][
111
11
1

10. EE 3512: Lecture 28, Slide 10
Properties of the Z-Transform
•Linearity:
Proof:
•Time-shift:
Proof:
What was the analog for CT signals and the Laplace transform?
•Multiplication by n:
Proof:
][][][][
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11. EE 3512: Lecture 28, Slide 11
Summary
•Definition of the z-Transform:
•Explanation of the Region of Convergence and its relationship to the
existence of the DTFT and stability.
•Properties of the z-Transform:
Linearity:
Time-shift:
Multiplication by n:
•Basic transforms (see Table 7.1) in the textbook.





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n
znxX(z) ][
][][][][
2121
zbXzaXnbxnax 
][][
0
0 zXznnx
n

dz
zdX
znnx
][
][ 