1、,10.1 The z-Transform,10. The z-Transform,10.1 The z-Transform,LTI,(1) Definition,10.1 The z-Transform,10.1 The z-Transform,10.1 The z-Transform,a. Especially, when z=ej， above equation becomes The Fourier transform of signal xn:,So, the relationship between the Fourier transformand the z-transform
2、is:,(2) The relationship between Z-transform and the Fourier transform of xn,b. On the other hand,10.1 The z-Transform,(2) Region of Convergence ( ROC ),ROC: Range of z for X(z) to converge Representation: A. InequalityB. Region in z-plane,10.1 The z-Transform,Example 10.1 Determine the z-Transform
3、of xn and its ROC.,10.1 The z-Transform,Solution:,10.1 The z-Transform,Example 10.2 Determine the z-Transform of xn and its ROC.,10.1 The z-Transform,Solution:,10.1 The z-Transform,Figure 10.3,10.1 The z-Transform,and,have same Z- transform representation, buttheir ROC is different.,Z,Z,Note: for a
4、signal xn, we must give out thez-transform with its ROC.,10.1 The z-Transform,(3) The pole-zero plot of X(z),X(z) can be represented the ratio of two polynomials, the numerator polynomial;,the denominator polynomial;,10.1 The z-Transform,Definition: The zeros of X(z): the roots of the numerator poly
5、nomial N(z) is called the zeros of X(z).,The poles of X(z): the roots of the denominator polynomial D(z) is called the poles of X(z).,10.1 The z-Transform,The representation of X(z) through its poles and zeros in the z-plane is referred to the pole-zeroplot of X(z).,Definition:,In the z-plane, use “
6、X” to indicate the poles of X(z); and use “O” to indicate the zeros of X(z);,On the other hand,If MN, z, X(z) , X(z) have (M-N) poles at infinity.If MN, z, X(z) 0, X(z) have (N-M) zeros at infinity.,10.1 The z-Transform,Example 10.1 Determine X(z) , its ROC and its pole-zero plot.,10.1 The z-Transfo
7、rm,Figure 10.2,Example,10.1 The z-Transform,Example 10.3 Determine the z-Transform of xn , its ROC and its pole-zero plot.,10.1 The z-Transform,Figure 10.4,10.1 The z-Transform,Example 10.4 Determine the z-Transform of xn , its ROC and its pole-zero plot.,Fugure 10.5,10.2 The Region of Convergence f
8、or the z-Transform,Property 1: The ROC of X(z) consists of a ring inthe z-plane centered the origin.,10.2 The ROC of the z-Transform,10.2 The ROC of the z-Transform,Property 2: the ROC does not contain any poles.Property 3: If xn is of finite duration, then the ROC is the entire z- plane, except pos
9、sibly z=0 and z=,Example:,Solution:,10.2 The ROC of the z-Transform,Example 10.5,10.2 The ROC of the z-Transform,Property 4: If xn is right-side sequence , and ifthe circle |z|=r0 is in the ROC, then allvalues of z for which |z|r0 will also in the ROC.,10.2 The ROC of the z-Transform,Figure 10.7 rig
10、ht-sided sequence xn.,10.2 The ROC of the z-Transform,a,ROC of a right-sided sequence:,Property 5: If xn is left-sided sequence, and if thecircle |z|=r0 is in the ROC, then all values of z for which 0|z|r0 will also be in the ROC.,10.2 The ROC of the z-Transform,10.2 The ROC of the z-Transform,ROC o
11、f left-sided sequence:,Property 6: If xn is two sided, and if the circle |z|=r0 is in the ROC, then the ROC will consist of a ring in the z-plane thatincludes the circle |z|=r0 .,10.2 The ROC of the z-Transform,10.2 The ROC of the z-Transform,ROC of two-sided sequence:,Example 10.6 Determine the z-t
12、ransform of the following signals.,10.2 The ROC of the z-Transform,10.2 The ROC of the z-Transform,Solution:,Zeros of X(z):,N-1 poles of X(z):,pole of X(z):,10.2 The ROC of the z-Transform,Figure 10-9,Example 10.7 Determine the z-transform of the following signals.,10.2 The ROC of the z-Transform,10
13、.2 The ROC of the z-Transform,Property 7: If the z-transform X(z) of xn is rational, then its ROC is bounded by poles or extends to infinity.,10.2 The ROC of the z-Transform,Property 8: If the z-transform X(z) of xn is rational, and if xn is right sided, then the ROCis the region in the z-plane outs
14、ide theoutmost 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 to0 for n0) , then the ROC also includes z=.,10.2 The ROC of the z-Transform,Property 9: If the z-transform X(z) of xn is rat
15、ional, and if xn is left sided, then the ROC isthe region in the z-plane inside the innermost pole 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 an possibly including z=0. in particular, if xn is anticausal (i.e.,
16、 if it is right sided and equal to 0 for n0) , then the ROC also includes z=0.,10.2 The ROC of the z-Transform,Example 10.8,Consider all of the possible ROCS of X(z).,Figure 10.12,10.2 The ROC of the z-Transform,10.3 The Inverse z-Transform,10.3 The inverse z-Transform,Show:,10.3 The inverse z-Trans
17、form,The calculation for inverse z-transform X(z): (1) Integration of complex function by equation.,(2) using fraction expansion ,10.3 The inverse z-Transform,(3),Long division (Taylors series)长除法（泰勒级数展开法）,Appendix Partial Fraction Expansion,Consider a fraction polynomial:,10.3 The inverse z-Transfo
18、rm,即，X(z)是z的有理分式。,把X(z)表示成z-1的两个多项式之比形式。,10.3 The inverse z-Transform,Discuss two cases of D(z-1)=0, for distinct roots,and same roots.,我们这里对X(z) 以z-1进行部分分式展开。,10.3 The inverse z-Transform,Case 1: Distinct roots:,thus,10.3 The inverse z-Transform,Calculate A1 :,Generally,10.3 The inverse z-Transform
19、,Using the following relationships to obtain xn.,10.3 The inverse z-Transform,10.3 The inverse z-Transform,Example : Compute the inverse z-transform of X(z).,Solution:,10.3 The inverse z-Transform,10.3 The inverse z-Transform,Case 2: Same root:,So,10.3 The inverse z-Transform,For first order poles:,
20、10.3 The inverse z-Transform,Multiply two sides by (1-p1z-1)r :,For r-order poles:,10.3 The inverse z-Transform,So,10.3 The inverse z-Transform,10.3 The inverse z-Transform,using,We can obtain xn.,10.3 The inverse z-Transform,Or using,We can obtain xn.,10.3 The inverse z-Transform,10.3 The inverse z
21、-Transform,Example: Determine the inverse z-transform.,Solution:,10.3 The inverse z-Transform,10.3 The inverse z-Transform,10.3 The inverse z-Transform,Example 10.9 10.10 10.11 Determine the inverse z-transform of X(z).,10.3 The inverse z-Transform,(3).,10.3 The inverse z-Transform,If X(z) is not ra
22、tional , compute xn by the following relationships,Long division (Taylors series)长除法（泰勒级数展开法）,(a),(b),Example 10.12 10.14 Determine the inverse z-transform of X(z).,10.3 The inverse z-Transform,(a),(b),Example 10.13 Determine the inverse z-transform of X(z) by long division.,10.3 The inverse z-Trans
23、form,(a),10.3 The inverse z-Transform,Solution:,(b),10.3 The inverse z-Transform,Solution:,10.5 Properties of the z-Transform,(1) Linearity,10.5 properties of the z-Transform,10.5 properties of the z-Transform,线性性质:线性组合后的收敛域R是线性组合前两个信号的收敛域R1与R2的公共区域.如果在线性组合过程中出现零点与极点相抵消的情况,则收敛域可能会扩大.,10.5 properties
24、 of the z-Transform,Example:,(2) Time shifting,10.5 properties of the z-Transform,10.5 properties of the z-Transform,(3) Scaling in the z-domain,可见： z平面上的尺度展缩，等效于xn乘以指数序列。 当z0为复指数时， z平面上的尺度展缩对应于Z平面上的点沿角度方向进行旋转，沿径向方向伸张或压缩。,(4) Time Reversal,10.5 properties of the z-Transform,(5) Time expansion,10.5 p
25、roperties of the z-Transform,(6) Conjugation,(7) Convolution property,10.5 properties of the z-Transform,Example 10.15 10.16,10.5 properties of the z-Transform,(8) Differentiation in the z-domain,10.5 properties of the z-Transform,Example 10.17 10.18 Find the inverse z- transform of X(z).,(a),(b),10
26、.5 properties of the z-Transform,(9) The initial-value theorem,If xn=0 for n0, then,Table 10.1 properties of z - transform,10.5 properties of the z-Transform,10.6 Some common z-Transform Pairs,Table 10.2,10.6 some common z-transform pairs,Example : Show the following z-transform pairs.,10.6 some com
27、mon z-transform pairs,10.6 some common z-transform pairs,Show:,利用z域微分性质,利用时移性质,10.6 some common z-transform pairs,10.6 some common z-transform pairs,再次利用z域微分性质,10.6 some common z-transform pairs,再利用时移性质,10.6 some common z-transform pairs,继续使用z域微分性质和时移性质，可以得到以下一般的变换关系：,利用这些变换关系，可以对多重极点的X(z)求z的 逆变换。,10.6 some common z-transform pairs,同样，可以得到以下z变换对。,Homework: 10.2 10.3 10.6 10.7 10.9 10.10 10.11 10.12 10.13 10.16 10.17 10.18 10.24 10.31 10.47,
