Z-Transform(Z变换).

1、Cha pter 10 The Z-Transform10.1 The Z-Transform (Z 变换)Recall 3.2 yn Hzzn and H (z)When zX(z)hnzIProp erty 1: The ROC of X(z) consists of a ring in thezp laneI centered about the origin.；X(z)的ROC在z平面内以原点为中心的圆环。I Prop erty 2: The ROC does not contain any po les. i ROC内不包括任何极点。X(z)nT ZTxn ej |z| ej X(e

2、jX(rej )nx nrnExa mp le 10.1xn anu nxnz nz- transformX(z)1)FTxnxn(re jX (ejImUnit criclez-pla ne.1严ReIProp erty 3: If xn is of finite duration, then the ROC is the entireIz-p lane, exce pt p ossibz=0 and/orz=.j如果xn是有限长序列，那么ROC就是整个z平面，可能除去z=0I和 /或 z=8。Prop erty 4: If xn is right-sided/sequence, and i

3、f the circle z| = r0 is in the ROC, then all finite values ofz for which | z| r。will also be in the ROC.如果xn是右边序列，而且如果| z| = r0的圆位于ROC内，那么 |z r0的全部有限Z值都一定在ROC内。X(z)n11 azanu nz/1 n(az ),011 ae|az 11，|a|Prop erty 5: If xn is left sided sequence, and if the circle z| = r0 is in the ROC, then all value

4、s ofz for which 0| z| r。will also be in the ROC.如果xn是左边序列，而且如果| z| = r0的圆位于ROC内，那么 满足0| z| r0的全部Z值都一定在 ROC内。Exa mp le 10.2xnX(z)n P a u| z | a when aanu n小 n1z1 unZT1Property 6: If xn is two sided, and if the circle| z| = ro is in the ROC, then the ROC will consist of a ring in the z-planethat inclu

5、des the circle| z| = r。.如果x(t)是双边序列，而且如果I z| = r0的圆位于ROC内,那么该 ROC就一定是由包括|z| = r0的圆环所组成。(a01a z1 n .z) , |a1z| 1Prop erty 7: If the z-transform X(z) of xn is rational, then its ROC is bounded by po les or extends to infinity.如果xn的z变换X(z)是有理的，那么它的ROC是被极点所界定 或延伸到无限远。z a, |z|region of convergencea ROCIm

6、z-planeReROC for Exa mp le 10.1: 10.2X(z) will be rational whenever xn is a linear combination of real or complex expo nentials!只要xn是实指数或复指数的线性组合，X(z)就一定是有理的。Prop erty 8: If the z-transform X(z) of xn is rational, and if xn is right sided, then the ROC is the region in the;-pl ane outside the outmos

7、t po le-i.e., outside the circle of radius equal to the largest magnitude of the po les o1X(z). Furthermore, if xn is causal(i.e., i it is right sided and equal to 0 fom0), then the ROC also include z=s.如果xn的z变换X(z)是有理的，而且若xn是右边序列，那么， ROC就位于z平面内最外层极点的外边；也就是半径等于 X(z 极点中最大模值的圆的外边。而且，若xn是因果序列(即xn； 为n0)

8、, then the ROC also includesz=0.如果xn的z变换X(z)是有理的，而且若xn是左边序列，那么， ROC就位于z平面内最里层的非零极点的里边；也就是半径等 于X(z)中除去z=0的极点中最小模值的圆的里边，并且，向内 延伸到可能包括z=0o特别是，若xn是反因果序列(即xn为 n0等于0的左边序列)，那么，ROC也包括z=0 o501Z-Transform (Z变换收敛域)Exam ple 10.7 Let xn bnunxnbnunZTb叫b nu n111 bz 1，|z|b,xnunu n 142 -31zHc 513 - z63襲111 b 1z,1I1,

9、 |z| b!If b1, there is no commonROC, xn will not have z-transform .If b1 Inx n ZTmZTn x(t)dz dzdzmA causal LTI system with rational system functionH(z) is stable if and only if all of the po les of H (z) lie inside the unit circle -i.e., they must all have magnitude smaller then 1.一个具有有理系统函数H(z)的因果L

10、TI系统，当且仅当H(z) 的全部极点都位于单位圆内时，也即全部极点其模均小于 1 时，该系统就是稳定的。X(z)dzdzdzdzExa mp le 10.18X(z)For Exam pie 10.11az11(1 az) anun ZT -|z|z| aso nanun ZTdz dz11 az1az 1| a9. The Initial- value Theorems初值定理)If xn=0, nThe causal LTI system function with inp uts that are identically zero for t1/2and with the con di

11、ti on of in itial rest, characterizes the system:51131yn 6yn 1 6yn 2 xn -xn 1 3 2Differentiation in the TimeDomainx n1 UZTz1 (z)xx n1 UZTz(z)zx0UZT1 ,x nxn 1(1xnmu nUZTm z(z)x nmu nUZTm z(z)1k m(z) x 11kxkz m1kxkz 0Exam ple 10.27, see it by yourself !Exam ple 10.3010.8 System Function Algebra and Bl

12、ock Diagram Rep rese ntati onsH(z)ynTI systems(1 1z4yn41加n8that is:1,11 1 - z -z482 xnn H1(z)xn 一 rn h2(z)P arallel connectionhn h1n blnH(z)WIynH1(z) H2(z)yn4yn411-yn 28xnxn_J h1 nSeries conn ecti onh2 nynH2(z)or:H(z)xn+聖h2 nIH2(z)hnH(z)hin h2nH1(z)H2(z)11 z14131丄z14Feedback connectionY(z)E(z)Y(z)丫h1 nH1ynH1(z) E(z)X(z) H2(z)Y (z)H1(z) X (z) H2(z)丫(z)H(z)H1(z)丿1也巴See book page 787 Figure 10.2010.9 The Unilateral Z-Transform (单边 Z 变换)xnz n0Homework #8r 1 U