Z-Transform(Z变换)

1、Property 1: The ROC of X(z) consists of a ring in the zplane centered about the origin.X(z)的ROC在z平面内以原点为中心的圆环。Chapter 10 The Z-Transform10.1 The Z-Transform（Z变换）Property 2: The ROC does not contain any poles.ROC内不包括任何极点。Property 3: If xn is of finite duration, then the ROC is the entire z-plane, exc

2、ept possibly z=0 and/or z=.如果xn是有限长序列，那么ROC就是整个z平面，可能除去z=0和/或z= 。Property 4: If xn is right-sided/sequence, and if the circle |z|= r0 is in the ROC, then all finite values of z for which |z|> r0 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。Property 5: If xn

3、 is left sided sequence, and if the circle |z|= r0 is in the ROC, then all values of z for which 0<|z|< r0 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。Property 6: If xn is two sided, and if the circle |z|= r0 is in the ROC, then the ROC will consi

4、st of a ring in the z-plane that includes the circle |z|= r0 .如果x(t)是双边序列，而且如果|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。Property 7: If the z-transform X(z) of xn is rational, then its ROC is bounded by poles 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。Property 8: If the z-transf

5、orm X(z) of xn is rational, and if xn is right sided, then the ROC is the region in the z-plane outside the outmost pole-i.e., outside the circle of radius equal to the largest magnitude of the poles of X(z). Furthermore, if xn is causal(i.e., if it is right sided and equal to 0 for n<0), then th

6、e ROC also includes z=.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n<0等于0的右边序列），那么，ROC也包括z= 。Property 9: If the z-transform X(z) of xn is rational, and if xn is left sided, then the ROC is the region in the z-plane inside the innermost nonzero pole

7、-i.e., inside the circle of radius equal to the smallest magnitude of the poles of X(z) other than any at z=0 and extending inward to and possibly including z=0. In particular, if xn is anti-causal(i.e., if it is left sided and equal to 0 for n>0), then the ROC also includes z=0.如果xn的z变换X(z)是有理的，

8、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=0。特别是，若xn 是反因果序列（即xn 为n>0等于0的左边序列），那么，ROC也包括z=0 。10.2 The Region of Convergence for the Z-Transform（Z变换收敛域）10.3 The Inverse Z-Transform（Z反变换）10.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plo

9、t(由零极点图对傅立叶变换进行几何求值)Learn it by yourself!10.5 Properties of the Z-Transform(Z变换的性质)10.7 Analysis and Characterization of LTI Systems Using the Z-Transform(利用Z变换分析和表征LTI系统)A discrete-time LTI system is causal if and only if the ROC of its system function is the exterior of a circle, including infinit

10、y.一个离散时间LTI系统当且仅当它的系统函数的ROC是在某一个圆的外边，且包括无限远点，该系统就是因果的。A discrete-time LTI system with rational system function H(z) is causal if and only if: (a) the ROC is exterior of a circle outside the outermost pole; and (b)with H(z) expressed as a ratio of polynomials in z, the order of the numerator cannot b

11、e greater than the order of the denominator.一个具有有理系统函数H(z)的LTI系统是因果的，当且仅当(a)ROC位于最外层极点外边某一个圆的外面;和(b)若H(z)表示成z的多项式之比，其分子的阶次不能大于分母的阶次。2. Stability(稳定性)An LTI system is stable if and only if the ROC of its system function H(z) includes the unit circle, z=1.一个LTI系统当且仅当它的系统函数H(z)的ROC包括单位圆，z=1时，该系统就是稳定的。A

12、 causal LTI system with rational system function H(z) is stable if and only if all of the poles 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LTI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。10.8 System Function Algebra and Block Diagram Representations（系统函数的代数属性与方框图表示）1.