Chapter 3 the Z-transform

,单击此处编辑母版标题样式,单击此处编辑母版文本样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Discrete-Time Signal,processing,Chapter,3,the Z-transform,Zhongguo Liu,Biomedical Engineering,School of Control Science and Engineering, Shandong University,2024/9/17,1,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Chapter 3 The z-Transform,3.0 Introduction,3.1 z-Transform,3.2 Properties of the Region of Convergence for the z-transform,3.3 The inverse z-Transform,3.4 z-Transform Properties,2024/9/17,2,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.0 Introduction,Fourier transform plays a key role in representing and analyzing discrete-time signals and systems.,Continuous: Laplace transform. Discrete: z-transform.,2024/9/17,3,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.0 Introduction,Motivation of z-transform,:,The Fourier transform does not converge for all sequences and it is useful to have a generalization of the Fourier transform.,In analytical problem,s,the z-Transform notation is more convenient than the Fourier transform notation.,2024/9/17,4,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.1 z-Transform,one-side,d, unilateral z-transform,z-Transform: two-side,d, bilateral z-transform,If, z-transform,is,Fourier transform.,2024/9/17,5,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Express the complex variable z in polar form as,Relationship between z-transform and Fourier transform,The Fourier transform of the product of and the exponential sequence,2024/9/17,6,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Complex z plane,2024/9/17,7,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Region of convergence (ROC),The set of value,s,of z for which the z-transform converges is called the,R,egion,O,f,C,onvergence (,abbreviated as,ROC).,2024/9/17,8,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Region of convergence (ROC),does not converge uniformly to the discontinuous function .,2024/9/17,9,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example,For,2024/9/17,10,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Zero and pole,Zero: The value of z for which,The z-transform is most useful when the infinite sum can be expressed in closed form, usually a ratio of polynomials in z (or z,-1,),.,Pole: The value of z for which,2024/9/17,11,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.1: Right-sided exponential sequence,For,2024/9/17,12,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,: zeros,: poles,Gray region: ROC,2024/9/17,13,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.2: Left-sided exponential sequence,For,2024/9/17,14,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,2024/9/17,15,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.3: Sum of two exponential sequences,2024/9/17,16,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.3: Sum of two exponential sequences,2024/9/17,17,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,2024/9/17,18,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.4:,Sum of two exponential,2024/9/17,19,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.5: Two-sided exponential sequence,2024/9/17,20,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,2024/9/17,21,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Finite-length sequence,Example :,2024/9/17,22,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.6:,Finite-length sequence,2024/9/17,23,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,N=16,a,is real,2024/9/17,24,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,z-transform pairs,2024/9/17,25,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,z-transform pairs,2024/9/17,26,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,z-transform pairs,2024/9/17,27,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,z-transform pairs,2024/9/17,28,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,z-transform pairs,2024/9/17,29,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.2 Properties of the ROC for the z-transform,For a given,xn,ROC is dependent only on,.,Property 1: The ROC is a ring or disk in the z-plane centered at the origin.,2024/9/17,30,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.2 Properties of the ROC for the z-transform,Property 2: The Fourier transform of converges absolutely if the ROC of the z-transform of,includes the unit circle.,The z-transform reduces to the Fourier transform when,ie,.,2024/9/17,31,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.2 Properties of the ROC for the z-transform,Property 3: The ROC cannot contain any poles.,is infinite at a pole and therefore does not converge.,2024/9/17,32,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.2 Properties of the ROC for the z-transform,Property 4: If,is a,finite-duration sequence, i.e., a sequence that is zero except in a finite,i,nterval,:,then the ROC is the entire z-plane, except possible or,2024/9/17,33,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.2 Properties of the ROC for the z-transform,Property 5: If is a,right-sided sequence, i.e., a sequence that is zero for, the ROC extends outward from the,outermost,finite pole in,to,(possibly including),2024/9/17,34,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Property 5:,right-sided sequence,2024/9/17,35,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.2 Properties of the ROC for the z-transform,Property 6: If,is a,left-sided sequence, i.e., a sequence that is zero for, the ROC extends inward from the,innermost,nonzero pole in,to,2024/9/17,36,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Property 5:,right-sided sequence,2024/9/17,37,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.2 Properties of the ROC for the z-transform,Property 7: A,two-sided sequence,is an infinite-duration sequence that is neither right-sided nor left-sided.,If is a two-sided sequence, the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole and not containing any poles.,2024/9/17,38,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.2 Properties of the ROC for the z-transform,Property 8: ROC must be a connected region.,for,finite-duration sequence,possible,ROC,:,for,right-sided,sequence,possible,ROC,:,for,left,-,sided,sequence,possible,ROC,:,for,two,-,sided,sequence,2024/9/17,39,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example,:,Different possibilities of the ROC,define different sequences,A system with three poles,2024/9/17,40,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,(b) ROC to a,right-sided,sequence,Different possibilities of the ROC.,(c) ROC to a,left-handed,sequence,2024/9/17,41,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,(,e,) ROC to another,two-sided sequence,Unit-circle,included,(,d,) ROC to a,two-sided sequence,.,2024/9/17,42,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Ex,.,3.7,Stability, Causality, and the ROC,A z-transform does not uniquely determine a sequence without specifying the ROC,Its convenient to specify the ROC implicitly through,time-domain property,of a sequence,stable system,: ROC include unit-circle:,causal system,: right sided:,2024/9/17,43,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,ROC: , the impulse response is,left-sided, system is,non-causal,unstable,since,the ROC does not include unit circle.,Ex,.,3.7,Stability, Causality, and the ROC,2024/9/17,44,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,ROC: , the impulse response is,two-sided, system is,non-causal,.,stable,.,Ex,.,3.7,Stability, Causality, and the ROC,2024/9/17,45,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,ROC:,the impulse response is,right-sided,. system is,causal,but,unstable,.,Ex,.,3.7,Stability, Causality, and the ROC,A,system is,causal,and,stable,if all the poles,are,inside the unit circle,.,2024/9/17,46,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.3,The Inverse Z-Transform,Formal inverse z-transform is based on a Cauchy integral theorem.,Less formal ways are sufficient and preferable:,Inspection method,Partial fraction expansion,Power series expansion,2024/9/17,47,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.3 The inverse z-Transform,3.3,.1,Inspection Method,2024/9/17,48,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.3 The inverse z-Transform,3.3,.1,Inspection Method,2024/9/17,49,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.3 The inverse z-Transform,3.3,.2,Partial Fraction Expansion,2024/9/17,50,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.8Second-Order z-Transform,2024/9/17,51,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.8Second-Order z-Transform,2024/9/17,52,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,2024/9/17,53,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Inverse Z-Transform by Partial Fraction Expansion,if MN, and has,an order s pole,Br is obtained by long division,2024/9/17,54,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.9:Inverse by Partial Fractions,2024/9/17,55,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,2024/9/17,56,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,2024/9/17,57,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,2024/9/17,58,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.3 The inverse z-Transform,3.3,.3,Power Series Expansion,2024/9/17,59,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.10:Finite-Length Sequence,2024/9/17,60,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Ex,.,3.1,1,:,Inverse Transform,by power series expansion,2024/9/17,61,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.12: Power Series Expansion by Long Division,2024/9/17,62,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.13: Power Series Expansion for a Left-sided Sequence,2024/9/17,63,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.4 z-Transform Properties,3.4,.1,Linearity,2024/9/17,64,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example of,Linearity,2024/9/17,65,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.4,.2,Time Shifting,is positive, is shifted right,is negative, is shifted left,is an integer,2024/9/17,66,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Time Shifting: Proof,2024/9/17,67,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.14:Shifted Exponential Sequence,2024/9/17,68,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.4,.3,Multiplication by an Exponential sequence,2024/9/17,69,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.15:Exponential Multiplication,2024/9/17,70,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,2024/9/17,71,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.4,.4,Differentiation of,X(z,),2024/9/17,72,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.16:Inverse of Non-Rational z-Transform,2024/9/17,73,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.17: Second-Order Pole,2024/9/17,74,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.4,.5,Conjugation of a complex Sequence,2024/9/17,75,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.4,. 6,Time Reversal,2024/9/17,76,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.18: Time-Reverse Exponential Sequence,2024/9/17,77,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.4,. 7,Convolution of Sequences,2024/9/17,78,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Ex,.,3.19: Evaluating a Convolution Using the z-transform,2024/9/17,79,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Example 3.19: Evaluating a Convolution Using the z-transform,2024/9/17,80,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,3.4,. 8,Initial Value Theorem,2024/9/17,81,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,2024/9/17,82,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,Chapter 3 HW,3.2, 3.3, 3.4, 3.7, 3.8, 3.9, 3.11, 3.16, 3.20,上一页,下一页,返 回,2024/9/17,82,Zhongguo Liu_Biomedical Engineering_Shandong Univ.,