Z-transformsSlide 2Z-transform PairsRegion of ConvergenceApplying Z-transform DefinitionBIBO StabilityInverse z-transformExampleExample (con’t)Slide 10Z-transform PropertiesSlide 12EE313 Linear Systems and Signals Spring 2013Initial conversion of content to PowerPointby Dr. Wade C. SchwartzkopfProf. Brian L. EvansDept. of Electrical and Computer EngineeringThe University of Texas at AustinZ-transforms21 - 2Z-transforms•For discrete-time systems, z-transforms play the same role as 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 nnznhzH )(Bilateral Forward z-transformRndzzzHjnh1 )( 21][Bilateral Inverse z-transform21 - 3Z-transform Pairs•h[n] = [n]Region of convergence: entire z-plane•h[n] = [n-1]Region of convergence: entire z-plane except z = 0h[n-1]  z-1 H(z)   1 )(00nnnnznznzH   111 1 1)(zznznzHnnnn 1 if 11 )(00zazazazaznuazHnnnnnnnn•h[n] = an u[n]Region of convergence: |z| > |a| which is the complement of a disk21 - 4Region of Convergence•Region of the complex z-plane for which forward z-transform convergesIm{z}Re{z}Entire planeIm{z}Re{z}Complement of a diskIm{z}Re{z}DiskIm{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)21 - 5Applying Z-transform Definition•Finite extent signalPolynomial in z-1•Five-tap delay lineImpulse response h[n]Transfer function H(z)h[n]n1 2 3 4 512345   50 )(nnnnznhznhzHh[n] = n ( u[n] – u[n-6] )5432154320)( zzzzzzHn = -1:6;h = n .* ( stepfun(n,0) - stepfun(n-6,0) );stem(n, h);21 - 6 azzanuaZnfor 111BIBO Stability•Rule #1: Poles inside unit circle (causal signals)•Rule #2: Unit circle in region of convergenceAnalogy in continuous-time: imaginary axis would be in region of convergence of Laplace transform•Example:BIBO stable if |a| < 1 by rule #1BIBO stable if |z| > |a| includes unit circle; hence, |a| < 1 by rule #2BIBO means Bounded-Input Bounded-Output21 - 7Inverse z-transform•Definition•Yuk! Using the definition requires a contour integration in the complex z-planeUse Cauchy residue theorem (from complex analysis) ORUse transform tables and transform pairs?•Fortunately, we tend to be interested in only a few basic signals (pulse, step, etc.)Virtually all signals can be built from these basic signals For common signals, z-transform pairs have been tabulatedRndzzzHjnh1 )( 21][21 - 8Example•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 terms212312)(22zzzzzX21212123121)(zzzzzX 1121121121)(zzzzzX121101211)(zAzABzX21 - 9Example (con’t)•Find B0 by polynomial division•Express in terms of B0•Solve for A1 and A21523212123211121212zzzzzzz 1111211512)(zzzzX82112121121921441121112122121111zzzzzAzzzA21 - 10Example (con’t)•Express X(z) in terms of B0, A1, and A2•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 unique111821192)(zzzX       nununnxn 821 9 2 21 - 11Z-transform Properties•Linearity•Right shift (delay)Second property used in solving difference equationsSecond property derived in Appendix N of course reader by decomposing the left-hand side as follows:x[n-m] u[n] = x[n-m] (u[n] – u[n-m]) + x[n-m] u[n-m]   )()(22112211zXazXanxanxa    )( zXzmnumnxm     mllmmzlxzzXznumnx1 )(21 - 12Z-transform Properties                           )()(212121212121212121zFzFzrfzmfzrfmfzmnfmfzmnfmfmnfmfZnfnfZmnfmfnfnfrrmmm rmrm nnnnmmm   •Convolution definition•Take z-transform•Z-transform definition•Interchange summation•Substitute r = n - m•Z-transform