1The z-TransformECON 397MacroeconometricsCunningham2z-Transform The z-transform is the most general concept for the transformation of discrete-time series.  The Laplace transform is the more general concept for the transformation of continuous time processes.  For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra. The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.3The TransformsThe Laplace transform of a function f(t):∫∞−=0)()( dtetfsFstThe one-sided z-transform of a function x(n):∑∞=−=0)()(nnznxzXThe two-sided z-transform of a function x(n):∑∞−∞=−=nnznxzX )()(4Relationship to Fourier TransformNote that expressing the complex variable z in polar form reveals the relationship to the Fourier transform:∑∑∑∞−∞=−∞−∞=−−−∞−∞======nniinnininniienxXeXrifandernxreXorrenxreXωωωωωωω)()()(,1,)()(,))(()(which is the Fourier transform of x(n).5Region of ConvergenceThe z-transform of x(n) can be viewed as the Fourier transform of x(n) multiplied by an exponential sequence r-n, and the z-transform may converge even when the Fourier transform does not. By redefining convergence, it is possible that the Fourier transform may converge when the z-transform does not. For the Fourier transform to converge, the sequence must have finite energy, or:∞<∑∞−∞=−nnrnx )(6Convergence, continued∑∞−∞=−=nnznxzX )()(The power series for the z-transform is called a Laurent series:The Laurent series, and therefore the z-transform, represents an analytic function at every point inside the region of convergence, and therefore the z-transform and all its derivatives must be continuous functions of z inside the region of convergence.In general, the Laurent series will converge in an annular region of the z-plane.7Some Special FunctionsFirst we introduce the Dirac delta function (or unit sample function):⎩⎨⎧=≠=0,10,0)(nnnδThis allows an arbitrary sequence x(n) or continuous-time function f(t) to be expressed as:∫∑∞∞−∞−∞=−=−=dttxxftfknkxnxk)()()()()()(δδor⎩⎨⎧=≠=0,10,0)(tttδ8Convolution, Unit StepThese are referred to as discrete-time or continuous-time convolution, and are denoted by:)(*)()()(*)()(ttftfnnxnxδδ==We also introduce the unit step function:⎩⎨⎧<≥=⎩⎨⎧<≥=0,00,1)(or0,00,1)(tttunnnuNote also:∑∞−∞==kknu )()(δ9Poles and ZerosWhen X(z) is a rational function, i.e., a ration of polynomials in z, then:1. The roots of the numerator polynomial are referred to as the zeros of X(z), and2. The roots of the denominator polynomial are referred to as the poles of X(z).Note that no poles of X(z) can occur within the region of convergence since the z-transform does not converge at a pole. Furthermore, the region of convergence is bounded by poles.10Example)()( nuanxn=The z-transform is given by:∑∑∞=−∞−∞=−==01)()()(nnnnnazznuazXWhich converges to:azforazzazzX >−=−=−111)(Clearly, X(z) has a zero at z = 0 and a pole at z = a.×aRegion of convergence11Convergence of Finite SequencesSuppose that only a finite number of sequence values are nonzero, so that:∑=−=21)()(nnnnznxzXWhere n1and n2are finite integers. Convergence requires .)(21nnnfornx ≤≤∞<So that finite-length sequences have a region of convergence that is at least 0 < |z| < ∞, and may include either z = 0 or z = ∞.12Inverse z-Transform The inverse z-transform can be derived by using Cauchy’s integral theorem. Start with the z-transform∑∞−∞=−=nnznxzX )()(Multiply both sides by zk-1 and integrate with a contour integral for which the contour of integration encloses the origin and lies entirely within the region of convergence of X(z):transform.-z inverse the is)()(2121)()(21)(211111nxdzzzXidzzinxdzznxidzzzXiCknCknCnknCk===∫∑∫∫∑∫−∞−∞=−+−∞−∞=−+−−ππππ13Properties z-transforms are linear: The transform of a shifted sequence: Multiplication:But multiplication will affect the region of convergence and all the pole-zero locations will be scaled by a factor of a.[])()()()(zbYzaXnbynax+=+Z[])()(00zXznnxn=+Z[])()(1zaZnxan−=Z14Convolution of Sequencesboth. of econvergenc of regions the inside of values for)()()()()()( let)()()()()(Then)()()(zzYzXzWzzmykxzWknmzknykxzknykxzWknykxnwkkmmnknnnkk=⎥⎥⎦⎤⎢⎢⎣⎡=−=−=⎥⎥⎦⎤⎢⎢⎣⎡−=−=−∞−∞=∞−∞=−−∞−∞=∞−∞=−∞−∞=∞−∞=∞−∞=∑∑∑∑∑∑∑15More DefinitionsDefinition. Periodic. A sequence x(n) is periodic with period λ if and only if x(n) = x(n + λ) for all n. Definition. Shift invariant or time-invariant. Consider a sequence y(n) as the result of a transformation T of x(n). Another interpretation is that T is a system that responds to an input or stimulus x(n):y(n) = T[x(n)].The transformation T is said to be shift-invariant or time-invariant if: y(n) = T [x(n)] implies that y(n - k) = T [x(n – k)]For all k. “Shift invariant” is the same thing as “time invariant” when n is time (t).16[]∑∑∑∑∞−∞=∞−∞=∞−∞=∞−∞==−==−=⎥⎥⎦⎤⎢⎢⎣⎡−==−kkkkkknhnxknhkxnyTnhkxnyknTkxknkxTnyknknnh).(*)()()()(then , transform the of invariance time have weIf).()()()()()()()(:Then . at occurringshockorspike""a),( tosystemtheofresponse the be )( LetδδδThis implies that the system can be completely characterized by its impulse response h(n). This obviously hinges on the stationarity of the series.17Definition. Stable System. A system is stable if∞<∑∞−∞=kkh )(Which means that a bounded input will not yield an unbounded output.Definition. Causal System. A causal system is one in which changes in output do not precede changes in input. In other words, [][ ] . for)()( thenfor)()(If021021nnnxTnxTnnnxnx<=≤=Linear, shift-invariant systems are causal iff h(n) = 0 for n < 0.18.)()( that so )()(Let)()()(Then. for )( letis, That .sinusoidal be)( let)()()( Given)(niikkiikkinikkninikkeeHnyekheHekheekhnynenxnxnhkxnyωωωωωωωω====∞<<∞−==∑∑∑∑∞−∞=−∞−∞=−∞−∞=−−∞−∞=Here H(eiω) is called the frequency response of the system whose