The z-Transform Page 1 Chapter 2 Z-transfrom The Z-Transfrom

The z-Transform Page 2 Outline - Introduction - The z-Transform - Inverse z-Transform - z-Transform Properties - The Unilateral z-Transform

The z-Transform Page 3 Introduction In this chapter, we consider a gerneralization of the Fourier transform referred to as the z-transform. We define the z-transform representation of a sequence and study how the properties of a sequence are related to the properties of its z-transform.

The z-Transform Page 4 Generalization

The z-Transform Page 5

The z-Transform Page 6 The z-Transform Definition

The z-Transform Page 7

The z-Transform Page 8 Region of convergence(ROC) ROC is the set of values of z for which the z-transform converges. The z-transform evaluated on the unit circle corresponds to the Fourier transform.

The z-Transform Page 9 The unit circle in the complex z-plane

The z-Transform Page 10 If ROC includes in the unit circle, this implies convergence of the z-transform for |z|=1 or equivalently, the Fourier transform of the sequence converges. Conversely, if the ROC dose not include in the unit circle, the Fourier transform does not conversege absolutly.

The z-Transform Page 11

The z-Transform Page 12

The z-Transform Page 13 The ROC as a ring in z-plane

The z-Transform Page 14 Properties of the Region of Convergence for the z-Transform Property 1 : The ROC is a ring or disk in the z-plane centered at the origin; i.e., 0≤r R < |z| <r L ≤ ∞. Property 2 : The Fourier transform of x[n] converges absolutely if and only if the ROC of the z-transform of x[n] includes the unit circle.

The z-Transform Page 15 Property 3 : The ROC cannot contain any poles. Property 4 : If x[n] is a finite-duration sequence, then the ROC is the entrie z-plane, expect possibly z=0 or z=∞. Property 5 : If x[n] is a right-side sequence, the ROC extends outward from the outermost finite pole in X(z) to z=∞.

The z-Transform Page 16

The z-Transform Page 17 Property 6 : If x[n] is a left-side sequence, the ROC extends inward from the innermost nonzero pole in X(z) to z=∞. Property 7 : A two-sided sequence is an infinte-duration sequence that is neither right sided nor left sided. If x[n] is a two-sided sequence, the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole and consistent with property 3, not contain any poles.

The z-Transform Page 18 Property 5 : The ROC must be a connected region.

The z-Transform Page 19

The z-Transform Page 20 Rational z-transform and ROC for rational z-transform Rational z-transform

The z-Transform Page 21

The z-Transform Page 22 Poles and Zeros

The z-Transform Page 23 Example

The z-Transform Page 24 Example

The z-Transform Page 25

The z-Transform Page 26 Inverse z-Transform 1 Inspection method 2 Partial fraction expansion 3 Power series expansion Methods

The z-Transform Page 27 1 Inspection method Ex.

The z-Transform Page 28

The z-Transform Page 29

The z-Transform Page 30 2 Partial fraction expansion

The z-Transform Page 31

The z-Transform Page 32 Example

The z-Transform Page 33 3 Power series expansion

The z-Transform Page 34 z-Transform Properties Linearity

The z-Transform Page 35 Time Shifting Z The quantity n 0 is an integer. If it is positive, the original sequence x[n] is shifted right, and if n 0 is negative, x[n] is shift left. ROC is unchanged(except for the positive addition or deletion of z=0 or z=∞)

The z-Transform Page 36 Mulitplication by an Exponential Sequence ROC is scaled by |z 0 |

The z-Transform Page 37 Differentiation of X(z) ROC is unchanged(except for the positive addition or deletion of z=0 or z=∞)

The z-Transform Page 38 Conjugation of a Complex Sequence ROC is unchanged.

The z-Transform Page 39 Time Reversal The ROC is inverted.

The z-Transform Page 40 Convolution of Sequence The region of convergence includes the intersection of the regions of convergence of X 1 (z) and X 2 (z). Z

The z-Transform Page 41 Initial Value Theorem If x[n] is zero for n<0, x[n] is the right side then

The z-Transform Page 42

The z-Transform Page 43 The Unilateral z-Transform The bilateral z-transform is define for values of x[n] for - ∞≤n≤∞.

The z-Transform Page 44 The unilateral z-transform differs form the bilateral z- transform in that it incorporates only values of x[n] for n ≥ 0.