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Introduction to Electrophysiology III.. Signals and Systems IV.. Signal Sampling and Reconstruction VI... The z – Transform 2• ROC region of convergence: the set of values of z for whic

Nguyễn Công Phương

PHYSIOLOGICAL SIGNAL PROCESSING

The z – Transform

I Introduction

II Introduction to Electrophysiology

III Signals and Systems

IV Fourier Analysis

V Signal Sampling and Reconstruction

VI The z – Transform

The z – Transform

1 The z – Transform

2 The Inverse z – Transform

3 Properties of the z – Transform

The z – Transform (2)

• ROC (region of convergence):

the set of values of z for which

1 1

1

az− < → > z a ROC: |z| > a

Re

ROC

a

The z – Transform (9)

Ex 6

0 0

, [ ]

,

n n

The z – Transform (13)

• ROC

– There is no pole inside a ROC

– The ROC is a connected region

– For finite duration sequences, the ROC is the

entire z – plane, sometimes except for z = 0 and

z = ∞

• The z – transform

– We need both X(z) and its ROC

– X(z) is not defined outside the ROC

The z – Transform (14)

The z – Transform

1 The z – Transform

2 The Inverse z – Transform

3 Properties of the z – Transform

The Inverse z – Transform (1)

1

1 2

N N

The Inverse z – Transform (2)

The Inverse z – Transform (3)

Ex 1

1

1 1

n

u n z

u n z

u n z

The Inverse z – Transform (4)

The z – Transform

1 The z – Transform

2 The Inverse z – Transform

3 Properties of the z – Transform

Properties of the z – Transform