《第十章ppt课件奥本海姆本信号与系统.ppt》由会员分享，可在线阅读，更多相关《第十章ppt课件奥本海姆本信号与系统.ppt（96页珍藏版）》请在三一办公上搜索。
1、1,The Z-Transform,CH 10,The primary focus of this chapter will be on:The z-Transform and the Region of Convergence for the z-TransformThe Inverse z-TransformGeometric Evaluation of the Fourier Transform from the Pole-Zero PlotProperties of the z-Transform and some Common z-Transform PairsAnalysis an
2、d Characterization of LTI Systems Using z-TransformSystem Function Algebra and Block Diagram RepresentationsThe Unilateral z-Transform,2,10.0 INTRODUCTION,z-transform is the discrete-time counterpart of the Laplace transform, the motivation for and properties of the z-transform closely parallel thos
3、e of the Laplace transform. However, they have some important distinctions that arise from the fundamental differences between continuous-time and discrete-time signals and systems. z-transform expand the application in which Fourier analysis can be used.,3,10.1 The z-Transform,The z-transform of a
4、general discrete-time signal xn is defined as,where z is a complex variable.,We will denote the transform relationship between xn and X(z) as,The defination of the z-transform :,The relationship between xn and X(z),4,The relationship between the z-transform and the discrete-time Fourier transform,Ex
5、pressing the complex variable z in polar form as,is the Fourier transform of xn multiplied by a real exponential So, the z-transform is an extension of the DTFT.,For r = 1, or equivalently, |z| = 1, z-transform equation reduces to the Fourier transform.,5,The z-transform reduces to the Fourier trans
6、form for values of z on the unit circle.,Different from the continuous-time case, the z-transform reduces to the Fourier transform on the contour in the complex z-plane corresponding to a circle with a radius of unity .,The z-transform reduces to the discrete-time Fourier transform,The z-transform r
7、educes to the discrete-time Fourier transform,6,In general, the z-transform of a sequence has associated with it a range of values of z for which X(z) converges, and this range of values is referred to as the region of convergence (ROC).,For convergence of the z-transform, we require that the Fourie
8、r transform of converge. For any specific sequence xn, it is this convergence for some value of r.,If the ROC includes the unit circle, then the Fourier transform also converges.,The region of convergence ( ROC ),depends only on r= |z|, just like the ROC in s-plane only depends on Re(s).,7,For conve
9、rgence of X(z), we require that,Consequently, the region of convergence is the range of values of z for which,Then,Pole-zero plot and region of convergence for Example 10.1 for 0 1,Example 10.1 Consider the signal,8,Now Consider the step signal,9,Example 10.2 Determine the z-transform of,If , this s
10、um converges and,Pole-zero plot and region of convergence for Example 10.2 for 0 1,10,Example 10.3 Consider a signal that is the sum of two real exponentials:,The z-transform is then,11,Example 10.4. Consider the signal:,Generally，the ROC of consists of a ring in the z-plane centered about the origi
11、n.,12,Example 10.5 Consider the signal,The z-transform of this signal is,|z| 1/3,13,if the z-transform is rational, its numerator and denominator polynomial can be factarized.,so, the z-transform is chacterized by all its poles and zeros except a constant factor .,The geometric representation of the
12、 z-transformthe Pole-Zero plot：,14,xn can be only determined by all poles and zeros of X(z) and the ROC of the X(z).,The pole-zero plot, illustrate all poles and zeros of the z-transform in z plane, is the geometric representation of the z-transform .,The pole-zero plot is especially useful for desc
13、ribing and analyzing the properties of the discrete-time LTI system.,15,10.2 The Region of Convergence for the z-Transform,Properties of the ROC for z-transform：,Property 1 The ROC of X(z) consists of a ring in the z-plane centered about the origin.,Property 2 The ROC does not contain any poles.,Pro
14、perty 3 If xn is of finite duration, then the ROC is the entire z-plane, except possibly z = 0 and/or z = .,Convergence is dependent only on and not on .,16,Example 10.6 Consider the unit sample signal n.,with an ROC consisting of the entire z-plane, including z = 0 and z = .,On the other hand,the R
15、OC consists of the entire z-plane, including z = but excluding z = 0.,Similarly,the ROC consists of the entire z-plane, including z = 0 but excluding z = .,17,Property 4 If xn is a right-sided sequence, and if the circle is in the ROC, then all finite values of z for which will also be in the ROC.,1
16、8,5) Property If xn is a left-sided sequence, and if the circle is in the ROC, then all values of z for which will also be in the ROC.,6) Property If xn is two sided, and if the circle is in the ROC, then the ROC will consist of a ring in the z-plane that includes the circle .,19,20,poles：,（order 1）
17、,（order N-1）,zeros：,At z=a, the zero cancel the pole. Consequently, there are no poles other than at the origin.,Example 10.7 Conside the finite duration xn,other,21,Example 10.8 Consider a two sided sequence,For b 1, there is no common ROC, and thus the sequence will not have a z-transform.,For b 1
18、, the ROCs overlap, and thus the z-transform for the composite sequence is,22,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 outermost pole i.e., outside the circle of radius equal to the largest magnitude of the
19、poles of X(z). Furthermore, if xn is causal, then the ROC also includes 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 i.e., inside the circle of radius equal to the smallest magnitude o
20、f 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 anticausal, then the ROC also includes z = 0.,Property 7 If the z-transform X(z) of xn is rational, then its ROC is bounded by poles or extends to infinity.,23,Example 10.9 Consi
21、der all of the possible ROCs that can be connected with the function,zeros：,（order 2）,poles：,24,一.The expression of the inverse z-transform：,10.3. The Inverse z-Ttansform,Multiplying both sides by , we obtain,Changing the variable of integration from to z with r fixed and varying over a interval:,Th
22、us, the basic inverse z-transform equation is:,25,The formal evaluation of the integral for a general X(z) requires the use of contour integration in the complex plane.,The symbol denotes integration around a counterclockwise circular contour centered at the origin and with radius r.,There are two a
23、lternative procedures for obtaining a sequence from its z-transform: one is partial-fraction expansion, the other is power-series expansion.,二. The procedures for obtaining a sequence from its z-transform：,26,The partial-fraction expansion,For rathional z-transform X(z), determine all poles of it. T
24、hen, it can be expanded by the method of partial-fraction, and we obtain expression of the z-transform as a linear combination of simpler terms:,Now specify the ROC associated with each term.Finally, determine the inverse z-transform of each of these indivial terms based on the ROC associated with e
25、ach.So that the inverse transform of X(z) equals the sun of the inverse transforms of the individual terms in the equation.,27,Example 10.10 Consider the z-transform,There are two poles, one at z=1/3 and one at z=1/4. Performing the partial-fraction expansion, we obtain,28,Power-series expansion,Thi
26、s procedure is motivated by the observation that the definition of the z-transform can be interpreted as a power series involving both positive and negative powers of z. The coefficients in this power series are, in fact, the sequence values xn.,29,Example 10.11 Consider the z-transform,From the pow
27、er-series definition of the z-transform, we can determine the inverse transform of X(z) by inspection:,That is,Some useful ZT pairs:,30,Example 10.10 Consider the z-transform,Then performing long division:,From the ROC, we can conclude that the corresponding sequence xn is right-sided, so that we ar
28、range the numerator polynomial and the denominator polynomial with a order of the power of z decreasing (or a order of the power of increasing).,31,If the ROC is |z|, before performing long division, we arrange the numerator polynomial and the denominator polynomial with a order of the power of z in
29、creasing (or a order of the power of decreasing).,Then performing long division:,32,For two-sided sequence,performing the long division to the two terms respectively.,33,the Residue Theorem,is a pole outside the contour C 。,，,is a pole inside the contour C。,，,Contour integration,34,Example 10.11 Con
30、sider the z-transform,From the definition of the inverse z-transform,This contour integration can be evaluated through using the Residue Theorem, thus,Since the ROC is |z|1, so the corresponding sequence is right sided.,35,For , has only two first-order poles:,For , has three first-order poles:,Then
31、,For , has two first-order poles:and a second-order pole:,36,Then,Consequently,37,In the discrete-time case, the Fourier transform can be evaluated geometrically by considering the pole and zero vectors in the z-plane. Since in this case the rational function is to be evaluated on the contour |z| =
32、1, we consider the vectors from the poles and zeros to the unit circle.,Consider a first-order causal discrete-time system with a impulse response:,Its z-transform is,10.4. Geometric Evaluation of The Fourier Transform From The Pole-Zero Plot,38,For |a| 1, the ROC includes the unit circle, and conse
33、quently, the Fourier transform of hn converges and is equal to H(z) for .The frequency response for the first-order system is,the pole-zero plot for H(z), including the vectors from the pole (at z = a) and zero (at z = 0) to the unit circle.,39,Magnitude of the frequency response for a = 0.95 and a
34、= 0.5,Phase of the frequency response for a = 0.95 and a = 0.5,the magnitude of the frequency response will be maximum at =0 and will decreases monotonically as increases from 0 to .,40,Magnitude of the frequency response for a = -0.95 and a = -0.5,Phase of the frequency response for a = -0.95 and a
35、 = -0.5,41,1) Linearity,Note: ROC is at least the intersection of R1 and R2, which could be empty, also can be larger than the intersection. For sequence with rational z-transform, if the poles of aX1(z)+bX2(z) consist of all of the poles of X1(z) and X2(z) (if there is no pole-zero cancellation), t
36、hen the ROC will be exactly to the overlap of the individual ROC. If the linear combinition is such that some zeros are introduced that cancel poles, then ROC may be larger.,If,and,then,10.5. Properties of The z-Transform,42,Except for the possible addition or deletion of the origin or infinity.,If,
37、2) Time Shifting,then,43,Example 10.12 Consider the signal,From Example 7.1, we know,So that,In fact,In procedure (), one pole at z = 0 was introduced, and it canceled a zero at the same location.,44,3) Scaling in the z-Domain,If,then,General case: when , the pole and zero locations are rotated by a
38、nd scaled in magnitude by a factor of .,Special case: when ,45,4) Time Reversal,5) Time Expansion,Consequence: if z0 is in the ROC for xn, then 1/ z0 is in the ROC for xn.,If,46,6) Conjugation,7) The Convolution Property,Consequence: if xn is real,Thus, if X(z) has a pole (or zero) at z = z0, it mus
39、t also have a pole (or zero) at the complex conjugate point z = z0*.,then,47,the ROC of X1(z)X2(z) may be larger if pole-zero cancellation occurs in the product.Proof:,48,Since,From the convolution property,If the ROC of X(z) is R, then the ROC of W(z) must includes at least the interconnection of R
40、 with |z| 1.,49,8) Differentiation in the z-Domain,Example 14.,50,Example 10.15 (p773) Determine the inverse z-transform for,From Example 10.1,and hence,51,9) The Initial- and Final-Value Theorems,Initial-value theorem :,Final -value theorem :,If xn is a causal sequence, i.e., xn = 0, for n 0, then,
41、52,终值定理 ：,若 是因果信号，且 ， 除了在 可以有一阶极点外，其它极点均在单位圆内，则,证明：,在单位圆上无极点,除了在 可以有 一阶极点外，其它极点均在单位圆内，,53,这其实表明：如果 有终值存在，则其终值等于 在 处的留数。,54,Z平面上极点位置与信号模式的关系示意图,55,10.6 Some Common Z-Transform Pairs,Table 10.1, the properties of the z-transform.,Table 10.2, a number of useful z-transform pairs.,56,10.7. Analysis and
42、Characterization of LTI Systems Using z Transform,System function,suppose,From Convolution Property,Y(z) = H(z) X(z),For ,H(z) is the frequency response of the LTI system.,57,10.7.1 Relating Causality to the System function,For a causal LTI system, the impulse response is zero for n0 and thus is rig
43、ht sided.,The ROC associated with the system function for a causal system is the exterior of a circle in the z-plane. Because the power series does not include any positive powers of z, the ROC include infinity.,If H(z) is rational, for the system to be causal, the ROC must be outside the outermost
44、pole and infinity must be in the ROC. Equivalently, the numerator of H(z) has degree no larger than the denominator when both are expressed as polynomials.,58,A 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 discret
45、e-time LTI system with rational system function H(z) is causal 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.,Summarizing,
46、we have the follow principles:,59,ROC of hns ZT contains the unit circle,10.7.2 Relating Stability to the System function,The stability of a discrete-time LTI system is equivalent to its impulse response being absolutely summable, in which case the Fourier transform of the impulse response converges
47、. SinceSummarizing:,An LTI system is stable if and only if the ROC of its system function H(z) includes the unit circle, |z| = 1.,A 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 sma
48、ller than 1.,60,For an LTI system which is described by a N-order linear constant-coefficient difference equation of the form,10.7.3 LTI System Characterized by Linear Constant-Coefficient Difference Equations,Then taking z-transforms of both sides of the above equation and using the linearity and t
49、ime-shifting properties, we obtain:,61,So its system function (transfer function) is:,Thus, the system function for a system specified by a difference equation is always rational.,的ROC需要通过其它条件确定，如：,1.系统的因果性或稳定性。 2.系统是否具有零初始条件等。,62,LTI系统的Z变换分析法：,1） 由 求得 及其 。 2） 由系统的描述求得 及其 。,分析步骤：,3） 由 得出 并确定它 的ROC包括
50、 。4） 对 做反变换得到 。,63,We can conclude that the system is not causal, because the numerator of H(z) is of higher order than the denominator.,Example 10.16(p777) Consider a system with system function whose algebraic expression is,In fact, since,Even we dont know the ROC for this system function. However
