When order of the filter increases hardware resource optimization becomes the real challenge. Distributed arithmetic (DA) architecture is comparatively more competitive to any of the other architectures in gate efficiency and operating speed. | Chapter The z Transform The two-sided or bilateral z transform of a discrete-time sequence x n is defined by X z f x n z n n co and the one-sided or unilateral z transform is defined by X z x n z n n 0 Some authors for example Rabiner and Gold 1975 use the unqualified term z transform to refer to while others for example Cadzow 1973 use the unqualified term to refer to . In this book z transform refers to the two-sided transform and the one-sided transform is explicitly identified as such. For causal sequences that is x n 0 for n 0 the one-sided and two-sided transforms are equivalent. Some of the material presented in this chapter may seem somewhat abstract but rest assured that the z transform and its properties play a major role in many of the design and realization methods that appear in later chapters. Region of Convergence For some values of z the series in does not converge to a finite value. The portion of the z plane for which the series does converge is called the region of convergence ROC . Whether or not converges depends upon the magnitude of z rather than a specific complex value of z. In other words for a given sequence x n if the series in converges for a value of z zr then the series will converge for all values of z for which z z1 . Conversely if the series diverges for z z2 then the series will diverge for all values of z for which z z2 . Because convergence depends on the magni- 151 152 Chapter Nine Im b Figure Possible configurations of the region of convergence for the z transform. tude of z the region or convergence will always be bounded by circles centered at the origin of the z plane. This is not to say that the region of convergence is always a circle it can be the interior of a circle the exterior of a circle an annulus or the entire z plane as shown in Fig. . Each of these four cases can be loosely viewed as an annulus a circle s interior being an annulus with an inner radius of zero and a finite .
