Bài giảng xử lý tín hiệu số z transform ngô quốc cường

Xử lý tín hiệu số
Z - transform

Ngô Quốc Cường

Ngô Quốc Cường
sites.google.com/a/hcmute.edu.vn/ngoquoccuong


Z - transform
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Z- transform
Properties of Z-transform
Inversion of Z- transform
Analysis of LTI systems in Z domain

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4.1. Z - transform
• Given a discrete-time signal x(n), its z-transform is defined as
the following series:

where z is a complex variable.
• Writing explicitly a few of the terms:

• Z-transform is an infinite power series, it exists only for those

values of z for this series converges.
• The region of convergence (ROC) of X(z) is the set of all
values of z for which X(z) attains a finite value.
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4.1. Z - transform
• Example: Determines the z-transform of the following finite
duration signals

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4.1. Z - transform
• Solution

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4.1. Z - transform
• Example

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4.1. Z - transform
Recall that

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4.1. Z - transform
• Example

• Solution

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4.1. Z - transform
• Example

• We have (l = -n),

• Using the formula (when A<1)

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4.1. Z - transform
• We have identical closed-form expressions for the z
transform

• A closed-form expressions for the z transform does not
uniquely specify the signal in time domain.
• The ambiguity can be resolved if the ROC is specified.
• Z – transform = closed-form expressions + ROC

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4.1. Z - transform
• Example

• Solution

– The first power series converges if |z| > |a|
– The second power series converges if |z| < |b|
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• Case 1

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• Case 2

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• Characteristics families of signals with their corresponding
ROC

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4.1. Z - transform
• The z-transform of the impulse response h(n) is called the
transfer function of a digital filter:

• Determine the transfer function H(z) of the two causal filters

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4.2. Properties of Z-transform

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• Example

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• Example

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Exercise

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