Lecture 22: The z-Transform

22 The z-Transform In Lecture 20, we developed the Laplace transform as a generalization of the continuous-time Fourier transform. In this lecture, we introduce the corresponding generalization of the discrete-time Fourier transform. The resulting transform is referred to as the z-transform and is motivated in exactly the same way as was the Laplace transform. For example, the discrete-time Fourier transform developed out of choosing complex exponentials as basic building blocks for signals because they are eigenfunctions of discrete-time LTI systems. A more general class of eigenfunctions consists of signals of the form z, where z is a general complex number. A representation of discretetime signals with these more general exponentials leads to the z-transform. As with the Laplace transform and the continuous-time Fourier transform, a close relationship exists between the z-transform and the discretetime Fourier transform. For z = ejn or, equivalently, for the magnitude of z equal to unity, the z-transform reduces to the Fourier transform. More generally, the z-transform can be viewed as the Fourier transform of an exponentially weighted sequence. Because of this exponential weighting, the z-transform may converge for a given sequence even if the Fourier transform does not. Consequently, the z-transform offers the possibility of transform analysis for a broader class of signals and systems. As with the Laplace transform, the z-transform of a signal has associated with it both an algebraic expression and a range of values of z, referred to as the region of convergence (ROC), for which this expression is valid. Two very different sequences can have z-transforms with identical algebraic expressions such that their z-transforms differ only in the ROC. Consequently, the ROC is an important part of the specification of the z-transform. Our principal interest in this and the following lectures is in signals for which the z-transform is a ratio of polynomials in z or in z 1.Transforms of this type are again conveniently described by the location of the poles (roots of the denominator polynomial) and the zeros (roots of the numerator polynomial) in the complex plane. The complex plane associated with the z-transform is referred to as the z-plane. Of particular significance in the z-plane is the circle of radius 1, concentric with the origin, referred to as the unit circle. 22-1 Signals and Systems 22-2 Since this circle corresponds to the magnitude of z equal to unity, it is the contour in the z-plane on which the z-transform reduces to the Fourier transform. In contrast, for continuous time it is the imaginary axis in the s-plane on which the Laplace transform reduces to the Fourier transform. The pole-zero pattern in the z-plane specifies the algebraic expression for the z-transform. In addition, the ROC must be indicated either implicitly or explicitly. There are a number of properties of the ROC in relation to the poles of the z-transform and in relation to characteristics of the signal in the time domain that often imply the ROC. For example, if the sequence is known to be right-sided, then the ROC must be the portion of the z-plane outside the circle bounded by the outermost pole. This and other properties are discussed in detail in the lecture. Suggested Reading Section 10.0, Introduction, page 629 Section 10.1, The z-Transform, pages 630-635 Section 10.2, The Region of Convergence for the z-Transform, pages 635-643 Section 10.3, The Inverse z-Transform, pages 643-646 The z-Transform 22-3 MARKERBOARD 22.1 bisc rp-e-Ti me. {c L-TT 2i . Tr u es Srm SIjS4cb1S H((3)~ lbv1pwi$c repvvs wteo6 Ne+Am~ 'X[V%) * -7 V, 'X(3) I MARKERBOARD 22.2 (a) E-XCILV to.Jle fal.1 On y 27, x C.'3 sq V%---Jo + L Vr VIM-do I -too xr"l n--QQ r "T ] Ickio e)rtUk%pI.lc t f )(chi V,-Vt I Signals and Systems 22-4 Im Unit circle TRANSPARENCY 22.1 The z-plane and the unit circle. Z = ej& z-p lane Re TRANSPARENCY 22.2 The z-transform and associated pole-zero plot for a right-sided exponential sequence. The z-Transform 22-5 TRANSPARENCY 22.3 The z-transform and associated pole-zero plot for a left-sided exponential sequence. TRANSPARENCY 22.4 Pole-zero plot for a discrete-time underdamped secondorder system illustrating the geometric determination of the Fourier transform from the pole-zero plot. Signals and Systems 22-6 PROPERTIES OF THE REGION OF CONVERGENCE TRANSPARENCY 22.5 Properties of the ROC of the z-transform. . The ROC does not contain poles . The ROC of X(z) consists of a ring in the z-plane centered about the origin . y7x[n][converges <=> ROC includes the unit circle in the z-plane . x [n] finite duration => ROC is entire z-plane with the possible exception of z = 0 or z = oo . Ix TRANSPARENCY [n 22.6 Properties of the ROC for a right-sided sequence. ITITI N1 Sx [n] right-sided and IzI= r. is in ROC => all finite values of z for which iz I> r0 are in ROC . x[n] right-sided and X(z) rational => ROC is outside the outermost pole 1. - - 0 & 0 n The z-Transform 22-7 . [n] left-sided and Iz|= ro is in ROC => all values of z for which 0 < Iz I< r0 will also be in ROC x TRANSPARENCY 22.7 Properties of the ROC of the z-transform for a left-sided sequence and for a two-sided sequence. . x[n] left-sided and X(z) rational => ROC inside the innermost pole . x [n] two-sided and Iz I= r is in ROC => ROC is a ring in the z-plane which includes the circle Iz|= rI TRANSPARENCY 22.8 Transparencies 22.822.10 show ROCs for a specified algebraic expression for the ztransform. Shown here is the ROC if the sequence is rightsided. Signals and Systems 22-8 x(z) z = _ _ _ _ _ (Z Z ) (Z - 2) TRANSPARENCY 22.9 The ROC if the sequence is left-sided. Im Unit circle z-plane X- x(z)= TRANSPARENCY 22.10 The ROC if the sequence is two-sided. (z - Re z ) (zZ 2) Im Unit circle z-plane Re I-- The z-Transform 22-9 MARKERBOARD 22.2 (b)