z-Transform

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The z-Transform : 

The z-Transform Waqas Ahmed

Content : 

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 = ej. 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( n1) : 

Example: A left sided Sequence ROC for x(n)=anu( n1) 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