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LECTURE 28: THE Z-TRANSFORM AND ITS ROC PROPERTIES. Objectives: Relationship to the Laplace Transform Relationship to the DTFT Stability and the ROC ROC Properties Transform Properties - PowerPoint PPT Presentation
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ECE 8443 – Pattern RecognitionEE 3512 – Signals: Continuous and Discrete
• Objectives:Relationship to the Laplace TransformRelationship to the DTFTStability and the ROCROC PropertiesTransform Properties
• Resources:MIT 6.003: Lecture 22Wiki: Z-TransformCNX: Definition of the Z-TransformCNX: PropertiesRW: PropertiesMKim: Applications of the Z-Transform
LECTURE 28: THE Z-TRANSFORMAND ITS ROC PROPERTIES
Audio:URL:
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 systemis stable, its poles lie in the left-half plane.
Hence, a DT system isstable if its poles areinside the unit circle.
The z-Transform behavesmuch like the Laplacetransform and can beapplied to difference equationsto produce frequency and time domain responses.
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EE 3512: Lecture 28, Slide 3
ROC and the Relationship to the DTFT
• We can derive the DTFT by setting z = rej:
• 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(ej), exists.
Example: (a right-sided signal)
If :
The ROC is outside a circle of radius a,and includes the unit circle, which meansits DTFT exists. Note also there is a zeroat z = 0.
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EE 3512: Lecture 28, Slide 4
Stability and the ROC
• For a > 0: azaz
X(z)nuanx n
for1
1][][
1
• If the ROC is outside the unit circle, the signal is unstable.
1for1
1
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1
zz
X(z)
nunx • If the ROC includes the unit circle, the signal is stable.
EE 3512: Lecture 28, Slide 5
Stability and the ROC (Cont.)
• For a < 0: azaz
X(z)nuanx n
for1
1][][
1
• If the ROC is outside the unit circle, the signal is unstable.
1for1
1
][][
1
zz
X(z)
nunx • If the ROC includes the unit circle, the signal is stable.
EE 3512: Lecture 28, Slide 6
More on ROC
• Example:
If:
The z-Transform is the same, but the region of convergence is different.
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EE 3512: Lecture 28, Slide 7
Stability and the ROC
• For: azaz
X(z)nuanx n
for1
1]1[][
1
• If the ROC includes the unit circle, the signal is stable.
1for1
1
][][
1
zz
X(z)
nunx • If the ROC includes the unit circle, the signal is unstable.
EE 3512: Lecture 28, Slide 8
Properties of the ROC
• The ROC is an annular ring in the z-plane centeredabout the origin (which is equivalent to a vertical strip in the s-plane).
• The ROC does not contain any poles (similar tothe Laplace transform).
• If x[n] is of finite duration, then the ROC is the entirez-plane except possibly z = 0 and/or z = :
• If x[n] is a right-sided sequence, and if |z| = r0 is in the ROC, then all finite values of z for which |z| > r0 are also in the ROC.
• If x[n] is a left-sided sequence, and if |z| = r0 is in the ROC, then all finite values of z for which |z| < r0 are also in the ROC.
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EE 3512: Lecture 28, Slide 9
Properties of the ROC (Cont.)
• If x[n] is a two-sided sequence, and if |z| = r0 is in the ROC, then the ROC consists of a ring in the z-plane including |z| = r0.
• Example:
right-sided left-sided two-sided
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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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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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