EE313 Linear Systems and Signals Fall 2005 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Z-transforms

Z-transforms For discrete-time systems, z-transforms play the same role of Laplace transforms do in continuous-time systems As with the Laplace transform, we compute forward and inverse z-transforms by use of transforms pairs and properties Bilateral Forward z-transformBilateral Inverse z-transform

Region of Convergence Region of the complex z- plane for which forward z-transform converges Im{z} Re{z} Entire plane Im{z} Re{z} Complement of a disk Im{z} Re{z} Disk Im{z} Re{z} Intersection of a disk and complement of a disk Four possibilities (z=0 is a special case and may or may not be included)

Z-transform Pairs h[n] =  [n] Region of convergence: entire z-plane h[n] =  [n-1] Region of convergence: entire z-plane h[n-1]  z -1 H[z] h[n] = a n u[n] Region of convergence: |z| > |a| which is the complement of a disk

Stability Rule #1: For a causal sequence, poles are inside the unit circle (applies to z-transform functions that are ratios of two polynomials) Rule #2: More generally, unit circle is included in region of convergence. (In continuous-time, the imaginary axis would be in the region of convergence of the Laplace transform.) –This is stable if |a| < 1 by rule #1. –It is stable if |z| > |a| and |a| < 1 by rule #2.

Inverse z-transform Yuk! Using the definition requires a contour integration in the complex z-plane. Fortunately, we tend to be interested in only a few basic signals (pulse, step, etc.) –Virtually all of the signals we’ll see can be built up from these basic signals. –For these common signals, the z-transform pairs have been tabulated (see Lathi, Table 5.1)

Example Ratio of polynomial z- domain functions Divide through by the highest power of z Factor denominator into first-order factors Use partial fraction decomposition to get first-order terms

Example (con’t) Find B 0 by polynomial division Express in terms of B 0 Solve for A 1 and A 2

Example (con’t) Express X[z] in terms of B 0, A 1, and A 2 Use table to obtain inverse z-transform With the unilateral z-transform, or the bilateral z-transform with region of convergence, the inverse z-transform is unique

Z-transform Properties Linearity [Lathi, Section 5.1] Right shift (delay) [Lathi, Section 5.2]

Z-transform Properties Convolution definition Take z-transform Z-transform definition Interchange summation Substitute r = n - m Z-transform definition

Example