Introduction
z-Transform
Zeros and Poles
Region of Convergence
z-Transform Properties Content

The z-Transform :

The z-Transform Introduction

Why z-Transform? :

A generalization of Fourier transform
Why generalize it?
FT does not converge on all sequence
Notation good for analysis
Bring the power of complex variable theory deal with the discrete-time signals and systems Why z-Transform?

The z-Transform :

The z-Transform z-Transform

Definition :

The z-transform of sequence x(n) is defined by Definition Let z = ej. Fourier Transform

z-Plane :

z-Plane Fourier Transform is to evaluate z-transform on a unit circle.

The z-Transform :

The z-Transform Zeros and Poles

Definition :

Give a sequence, the set of values of z for which the z-transform converges, i.e., |X(z)|<, is called the region of convergence. Definition ROC is centered on origin and consists of a set of rings.

Stable Systems :

A stable system requires that its Fourier transform is uniformly convergent. Stable Systems Fact: Fourier transform is to evaluate z-transform on a unit circle.
A stable system requires the ROC of z-transform to include the unit circle.

Example: A right sided Sequence :

Example: A right sided Sequence For convergence of X(z), we require that

Example: A right sided Sequence ROC for x(n)=anu(n) :

Example: A right sided Sequence ROC for x(n)=anu(n) Which one is stable?

Example: A left sided Sequence :

Example: A left sided Sequence For convergence of X(z), we require that

Example: A left sided Sequence ROC for x(n)=anu( n1) :

Example: A left sided Sequence ROC for x(n)=anu( n1) Which one is stable?

The z-Transform :

The z-Transform Region of Convergence

Represent z-transform as a Rational Function :

Represent z-transform as a Rational Function where P(z) and Q(z) are polynomials in z. Zeros: The values of z’s such that X(z) = 0 Poles: The values of z’s such that X(z) =

Example: A right sided Sequence :

Example: A right sided Sequence ROC is bounded by the pole and is the exterior of a circle.

Example: A left sided Sequence :

Example: A left sided Sequence ROC is bounded by the pole and is the interior of a circle.

Properties of ROC :

A ring or disk in the z-plane centered at the origin.
The Fourier Transform of x(n) is converge absolutely iff the ROC includes the unit circle.
The ROC cannot include any poles
Finite Duration Sequences: The ROC is the entire z-plane except possibly z=0 or z=.
Right sided sequences: The ROC extends outward from the outermost finite pole in X(z) to z=.
Left sided sequences: The ROC extends inward from the innermost nonzero pole in X(z) to z=0. Properties of ROC

More on z-Transform :

More on z-Transform Consider the z-transform
with the pole pattern: Case 1: A right sided Sequence.

More on Rational z-Transform :

More on Rational z-Transform Consider the rational z-transform
with the pole pattern: Case 2: A left sided Sequence.

More on Rational z-Transform :

More on Rational z-Transform Consider the rational z-transform
with the pole pattern: Case 3: A two sided Sequence.

More on Rational z-Transform :

More on Rational z-Transform Consider the rational z-transform
with the pole pattern: Case 4: Another two sided Sequence.

The z-Transform :

The z-Transform Important
z-Transform Pairs

Z-Transform Pairs :

Z-Transform Pairs

The z-Transform :

The z-Transform z-Transform Theorems and Properties

Linearity :

Linearity Overlay of
the above two
ROC’s

Shift :

Shift

Multiplication by an Exponential Sequence :

Multiplication by an Exponential Sequence

Differentiation of X(z) :

Differentiation of X(z)

Reversal :

Reversal

Convolution of Sequences :

Convolution of Sequences

Convolution of Sequences :

33 Convolution of Sequences

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