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1、We note in particular that, for r=1 (or ), X(z) reduces to the Fourier transform, that is, ,where is a complex variable. .1. Definition of bilateral z-transform: 10 The Z-Transform 10.1 The Z - TransformThe motivations for and properties of the z-transform closely parallel those of the Laplace trans
2、form. 10 The Z-Transform For convergence of the z-transform, we require that the Fourier transform of xnr -n converge. The range of values of z for which X(z) converges is referred to as the ROC. If the ROC includes the unit circle, then the Fourier transform also converges.Example 1.ROC:For , the D
3、TFT of xn does not converge.For , Unit circle1a 10.2 The ROC for the z- TransformExample 2.a a 1 1Z-planeUnit circle 10 The Z-Transform Example 3.Z-plane1 1（ROC） 10 The Z-Transform As , the DTFT of xn does not converge.Example 4.2 21/21/2Z-plane Property 1: In general, the ROC of consists of a ring
4、in the z-plane centered about the origin. In some case, the inner boundary can extend inward to the origin, in which case the ROC becomes a disc. In other cases, the outer boundary can extend outward to infinity. 10 The Z-Transform Property 2: The ROC does not contain any poles. This is a consequenc
5、e of the fact that at a pole X(z) in infinite and therefore, by definition, does not converge. 10 The Z-Transform Property 3: If xn is of finite duration, then the ROC is the entire z-plane, except possibly z = 0 and/or z = .For N10, the ROC does not include z = 0 and z = . If , the ROC includes z =
6、 . If , the ROC includes z = 0. 10 The Z-Transform Property 4: If xn is a 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. As is right-sided, i.e., If , thenAnd if ，we know thatProof:If ， ，thenProperty 5: If xn is a le
7、ft-sided sequence, and if the circle |z|=r0 is in the ROC, then all finite values of z for which 0|z|r0 will also be in the ROC. 10 The Z-Transform Proof:As is left-sided, i.e., 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
8、 that includes the circle |z|=r0 . Property 7: If the z-transform X(z) of xn is rational, then its ROC is bounded by poles or extends to infinity. 10 The Z-Transform Property 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 z-plane outside the
9、 outermost polei.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 n0), then the ROC also includes z = 0.Poles：( pole of first order )(pole of order N1)Zeros：0 0 10 The Z-Transform Example
10、 10.6otherwisePole-zero cancellation at z = a There are no poles other than at z = 0. For ，the ROC is Example 10.7For ，there is no common ROC, and thus the xn will not have a z-transform.b b1/b1/bZ-plane 10 The Z-Transform Example 10.80 0Zeros：(two order)Poles:Three possible ROCs：1， is right sided，c
11、ausal 10 The Z-Transform As , the DTFT of xn does not converge. 10 The Z-Transform 2， is left sided，anti-causalAs , the DTFT of xn does not converge.3， is two sided. Let 10 The Z-Transform 10.3 The Inverse Z- TransformThe integration is over a 2 interval in , which, in terms of z, corresponds to one
12、 traversal around the circle |z|=r. Example： 10 The Z-Transform (2)The calculation for inverse Laplace transform: 1. Integration of complex function by equation. 2. Compute by Fraction expansion.3. Power-series expansion (Long division & Taylors series expansion) 10 The Z-Transform (rational)(non-ra
13、tional)Example： 10 The Z-Transform The power-series expansion method for obtaining the inverse z-transform is particularly useful for non-rational z-transform.Example 10.14 10.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plota a1 1 10 The Z-Transform 10.4.1 First-Order SystemsF
14、or ， as ， monotonically 10 The Z-Transform Magnitude responsePhase responsea a1 1 10 The Z-Transform Magnitude responsePhase responsePoles：Zeros：（second order） 10 The Z-Transform 10.4.2 Second-Order Systems(Under damping)1 1 10 The Z-Transform as ， 10 The Z-Transform Magnitude responsePhase response
15、then 10 The Z-Transform 10.5 Properties of the Z-Transformv The derivation of many results are analogous to those of the corresponding properties for the DTFT. 1. Linearity ：Ifwith ROC containingwith , except for the possible addition or deletion of the or Ifthen 10 The Z-Transform 2. Time Shifting:
16、3. Scaling in the z-Domain:IfthenSpecially, if ,Ifthen 10 The Z-Transform 4. Time Reversal:then the ROC of is0 0Example：If the ROC of is IfIf n is a multiple of kOthersthenProof： 10 The Z-Transform 5. Time Expansion:6. Conjugation：If is real, i.e., , we can conclude that Thus, if has a pole(or zero)
17、 at z=z0, it must also have a pole (or zero) at the complex conjugate point z=z0* Ifthen 10 The Z-Transform For an LTI system, we havethen 10 The Z-Transform 7. The Convolution Property：Ifwith ROC containingProof：8. Differentiation in the z-Domain：Example 10.17 By differentiating, a non-rational z-t
18、ransform X(z) can be converted to a rational expression. Ifthen 10 The Z-Transform Example 10.18 10 The Z-Transform 9. The Initial-Value Theorem：thenIf and , 10 The Z-Transform Proof：Consider the expression for the z- transform , with the constraint AsThus, If and , all the poles of X(z) lie inside
19、the unit circle, except one possible first order pole at z = 1 , then we have Proof： From the above constraints, we know that all the poles of (z-1)X(z) lie inside the unit circle, and there is no any pole lie on the unit circle. Thus, 10 The Z-Transform 10. The Final-Value Theorem： 10 The Z-Transfo
20、rm 10 The Z-Transform Poles and the Final-valueTable 10.2 10 The Z-Transform 10.6 Some Common Z-Transform Pairs10.7 Analysis and Characterization of LTI Systems Using Z-Transforms1. System Function H(z)H(z) is also referred to as the transfer function. If the ROC of H(z) includes the unit circle, th
21、en for z=ej , H(z) is the frequency response of the LTI system. 10 The Z-Transform 10.7.1 CausalityA discrete-time LTI system is causal if and only if the ROC of its system function is the exterior of a circle, including infinity . A discrete-time LTI system with rational system function H(z) is cau
22、sal if and only if : (a) the ROC is the 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 be greater than the order of the denominator. 10 The Z-Transform 10.7.2 StabilityA discrete-time LTI system is stable
23、 if and only if the ROC of its system function H(z) includes the unit circle . A causal LTI system with rational system function H(z) is stable if and only if all the poles of H(z) lie inside the unit circlei.e., they must all have magnitude smaller than 1. 10 The Z-Transform 10.7.3 LTI Systems Char
24、acterized by LCCDE（Rational function）Example：Consider the LTI systems described by the following difference equations, please determine the system function H(z) and the impulse response hn, then give the block-diagram representation.Solution：FIR 10 The Z-Transform Solution：IIR 10 The Z-Transform 10
25、The Z-Transform 10.8 System Function Algebra and Block Diagram Representations 1. Cascade with ROC containing2. Parallelwith ROC containing9.8.1 System Functions for Interconnections of LTI systems 10 The Z-Transform 3. Feedbackwith ROC containing 10 The Z-Transform 9.8.2 Block Diagram Representatio
26、ns for Causal LTI systems1. Cascade form：(if N is even)D DD DD DD D(if N is even) 10 The Z-Transform 2. Parallel form：DDDD 10 The Z-Transform Parallel formExample 10.30 10.311. Definition：ROC for unilateral z-transform is always the exterior of a circle (containing ). 10 The Z-Transform 10.9 The Uni
27、lateral Z-Transform Therefore, any sequence that is identically zero for n0 has identical bilateral and unilateral z-transforms. (causal signal)The evaluation for the unilateral inverse z-transforms is the same as for bilateral transforms, with the constraint that the ROC for a unilateral transform
28、must always be the exterior of a circle. Example 10.32: 10 The Z-Transform The bilateral z-transform isThe unilateral z-transform isIt can be easily seen thatCausal sequenceExample 10.33: 10 The Z-Transform The bilateral z-transform isThe unilateral z-transform isIt can be easily seen thatNon-causal
29、 10 The Z-Transform 10.9.2 Properties of the Unilateral z-Transform Many properties of unilateral z-transform are the same as their bilateral counterparts and several properties differ in significant ways.1. Time delay：If thenProof 1: 10 The Z-Transform Similarly, we knowProof 2：Similarly, we knowEx
30、ample：then 10 The Z-Transform 10.9.3 Solving Difference Equations Using the Unilateral z-TransformA primary use of the unilateral z-transform is in obtaining the solution of LCCDE with nonzero initial conditions.Forced responseNatural response 10 The Z-Transform 解：解：例例例例1 1：已知已知 试画出其直接型，级联型和并联型的模拟框图
31、。试画出其直接型，级联型和并联型的模拟框图。1 1）直接型）直接型）直接型）直接型解：解：例例例例1 1：已知已知 试画出其直接型，级联型和并联型的模拟框图。试画出其直接型，级联型和并联型的模拟框图。2 2）级联型）级联型）级联型）级联型解：解：例例例例1 1：已知已知 试画出其直接型，级联型和并联型的模拟框图。试画出其直接型，级联型和并联型的模拟框图。3 3）并联型）并联型）并联型）并联型例例例例2 2：已知描述某因果离散已知描述某因果离散LTI系统的差分方程为：系统的差分方程为： 在在z域求解：域求解： (1)系统的零输入响应系统的零输入响应yzik，零状态响应，零状态响应yzsk和完全响
32、应和完全响应y k 。(2)系统的系统函数系统的系统函数H(z)，单位脉冲响应，单位脉冲响应hk，并判断系统是，并判断系统是否稳定。否稳定。(3)若若xk=2 uk- -1 ，重新计算，重新计算(1)(2)。解：解：解：解：对差分方程两边进行对差分方程两边进行z变换得变换得 整理后可得整理后可得解：解：解：解：(1)(1)例例例例2 2：已知描述某因果离散已知描述某因果离散LTI系统的差分方程为：系统的差分方程为： 在在z域求解：域求解： (1)系统的零输入响应系统的零输入响应yzik，零状态响应，零状态响应yzsk和完全响应和完全响应y k 。解：解：解：解：(2)(2)例例例例2 2：已知
33、描述某因果离散已知描述某因果离散LTI系统的差分方程为：系统的差分方程为： 在在z域求解：域求解： (2) 系统函数系统函数H(z)，单位脉冲响应，单位脉冲响应hk，并判断系统是否稳定。，并判断系统是否稳定。根据系统函数的定义，可得根据系统函数的定义，可得 进行进行z反变换即得反变换即得对因果系统，由于其极点为对因果系统，由于其极点为z1=1/2，z2=1/4，均在单位圆内，均在单位圆内，故系统稳定。故系统稳定。解：解：解：解： (3)例例例例2 2：已知描述某因果离散已知描述某因果离散LTI系统的差分方程为：系统的差分方程为： 在在z域求解：域求解： (3)若若xk=2 uk- -1 ，重新
34、计算，重新计算(1)(2)。系统的完全响应也相应地改变为系统的完全响应也相应地改变为 若若xk=2uk- -1，说明系统的输入信号变了，但系统没变，系，说明系统的输入信号变了，但系统没变，系统的初始状态也没变，因此，系统的系统函数，单位脉冲响应统的初始状态也没变，因此，系统的系统函数，单位脉冲响应和稳定性都不变，系统的零输入响应也不变，只有系统的零状和稳定性都不变，系统的零输入响应也不变，只有系统的零状态响应和完全响应会随输入信号发生变化，由线性非时变特性态响应和完全响应会随输入信号发生变化，由线性非时变特性可得可得Problems: 10.2 10.6 10.9 10.10 10.12 10.13 10.1610.17 10.19(b) 10.20 10.21(a)(g) 10.29 10.33 10.38 10.45 10.59 10 The Z-Transform
