1、10.7 Analysis and characterization of LTI systems using z-transform,System function:,10.7 analysis and characterization of LTI system usingz-transform,10.7.1 Causality,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.,
2、10.7 analysis and characterization of LTI system usingz-transform,A discrete-time LTI system with a rational system function H(z) is causal if and only if:(a) the ROC is the exterior of a circle outside the outmost pole; and(b) with H(z) expressed as a ratio of polynomials in z, the order of the num
3、erator cannot be greater than the order of the denominator.,The two conditions must be satisfied at the same time.,10.7 analysis and characterization of LTI system usingz-transform,Example 10.20 10.21 Determine whether or not the following systems are causal?,(a),(b),10.7 analysis and characterizati
4、on of LTI system usingz-transform,10.7.2 Stability,An LTI system is stable if and only if the ROCof its system function H(z) includes the unit circle, |z|=1.,(2) An causal system with rational system function H(z) is stable if and only if all of the poles of H(z) lie inside the unite circle i.e. , t
5、hey must all have magnitude smaller than 1.,10.7 analysis and characterization of LTI system usingz-transform,Example 10.23 10.24,Whether or not the systems are stable?,(a),(b),10.7 analysis and characterization of LTI system usingz-transform,习题：10.13 10.16 10.17 10.31,10.7.3 LTI systems characteriz
6、ed by linear constant-coefficient difference equations,linear constant-coefficient difference equations:,10.7 analysis and characterization of LTI system usingz-transform,System function:,10.7 analysis and characterization of LTI system usingz-transform,a. to determine hn:,b. zero-state response(or
7、initial rest condition), According to xn, to determine yn.,10.7 analysis and characterization of LTI system usingz-transform,Example 10.25 Determine hn.,10.7 analysis and characterization of LTI system usingz-transform,10.7.4 examples relating system behavior tothe system function,10.7 analysis and
8、characterization of LTI system usingz-transform,Example 10.26 Suppose an LTI system with input and output:,Determine H(z) and the value of a. Determine properties of the system.,1.,2.,10.7 analysis and characterization of LTI system usingz-transform,10.8 system function algebra and block diagramrepr
9、esentations,10.8 system function algebra and block diagram representation,10.8.1 system functions for interconnections of LTI systems,h1n H1(z),h2n H2(s),xn,yn,(a) Parallel interconnection,10.8 system function algebra and block diagram representation,h1n H1(z),h2n H2(z),xn,yn,(b) Series interconnect
10、ion,10.8 system function algebra and block diagram representation,h1n H1(z),h2n H2(z),xn,yn,(c) Feedback interconnection,+,-,10.8 system function algebra and block diagram representation,10.8.2 block diagram representations for causal LTI systems described by difference equations and rational system
11、 functions,An adder,Multiplication by a coefficient,An delayer,a,z-1,10.8 system function algebra and block diagram representation,10.8 system function algebra and block diagram representation,系统函数H(z)的方框图：1）直接型方框图 2）级联型方框图 3）并联型方框图,(1) Direct form of the block diagram: （直接型方框图）,10.8 system function
12、 algebra and block diagram representation,步骤一： H(z)表示成z-1的的多项式之比，并把分母中的常数项变为1,方法：直接对H(z)进行变形，画出直接型方框图,10.8 system function algebra and block diagram representation,根据梅森公式，画出直接型系统方框图：,Example 10.28 Plot the direct form block diagram of the following system.,1/4,10.8 system function algebra and block
13、diagram representation,xn,yn,Example 10.29 Plot the direct form block diagram of the following system.,1/4,-2,10.8 system function algebra and block diagram representation,xn,yn,Example 10.30 Plot the direct form block diagram of the following system.,Solution:,10.8 system function algebra and block
14、 diagram representation,1/4,1/8,-,10.8 system function algebra and block diagram representation,xn,yn,Example 10.31 Plot the direct form block diagram of the following system.,10.8 system function algebra and block diagram representation,1/4,1/8,-,7/4,1/2,-,-,10.8 system function algebra and block d
15、iagram representation,xn,yn,10.8 system function algebra and block diagram representation,习题：10.18,根据方框图，由梅森公式写出系统函数H(z).,2/3,-1/9,8,-6,xn,yn,(2) Cascade form of the block diagram（级联型方框图),Hi(z) is first-order or second-order system. For a real pole, Hi(z) is first-order system. For conjugation compl
16、ex poles, Hi(z) is second-order system.,10.8 system function algebra and block diagram representation,即，把H(z)因式分解为一阶或二阶子系统级联（相乘）的形式。,Example 10.30 Plot the cascade form block diagram of the following system.,Solution:,first-order system,first-order system,10.8 system function algebra and block diagr
17、am representation,Solution:,first-order system,first-order system,10.8 system function algebra and block diagram representation,1/2,-,1/4,10.8 system function algebra and block diagram representation,xn,vn,vn,yn,1/2,-,1/4,10.8 system function algebra and block diagram representation,xn,yn,vn,Example
18、,10.8 system function algebra and block diagram representation,Plot the cascade form block diagram of thefollowing system.,Example,Solution:,first-order system,second-order system,10.8 system function algebra and block diagram representation,1,-,-1,-1/2,10.8 system function algebra and block diagram
19、 representation,xn,yn,vn,vn,1,-,-1,-1/2,10.8 system function algebra and block diagram representation,xn,yn,vn,(3) Parallel form of the block diagram(并联型方框图):,Hi(z) is first-order or second-order system. For a real pole, Hi(z) is first-order system. For conjugation complex poles, Hi(z) is second-ord
20、er system.,10.8 system function algebra and block diagram representation,即，把H(z)部分分式展开为一阶或二阶子系统并联（相加）的形式。,Example 10.30 Plot the parallel form block diagram of the following system.,10.8 system function algebra and block diagram representation,10.8 system function algebra and block diagram represent
21、ation,Solution:,1/2,-,1/4,2/3,1/3,10.8 system function algebra and block diagram representation,xn,yn,10.8 system function algebra and block diagram representation,Example Plot the parallel form block diagram of the following system.,10.8 system function algebra and block diagram representation,Solu
22、tion:,1,-,-1,-1/2,2,-,10.8 system function algebra and block diagram representation,xn,yn,10.9 the unilateral z-transform,Definition:,10.9 The unilateral z-Transform,Note: If xn is causal, its bilateral z-transform is equal toits unilateral z-transform.,10.9 The unilateral z-Transform,为了区分，前面介绍的z变换，
23、称为双边z变换。,bilateral z-transform：,Example 10.32 10.33 Determine the unilateral z-transform for the following signals.,(a),(b),10.9.1 examples of unilateral z-transform,10.9 The unilateral z-Transform,Example 10.34 Determine the inverse unilateral z-transform for the following signals.,10.9 The unilate
24、ral z-Transform,Table 10.3,10.9.2 properties of the unilateral z-transform,10.9 The unilateral z-Transform,单边z变换的性质，与双边z变换的性质主要区别在于时延性质不同，其余性质都相同。,(1) Time delay,10.9 The unilateral z-Transform,证明：,10.9 The unilateral z-Transform,(2) Time advance,证明：,10.9 The unilateral z-Transform,Generally,10.9 Th
25、e unilateral z-Transform,对于m=1,2的情况，有,（3）First difference:,10.9 The unilateral z-Transform,(4) The initial-value theorem,If n0, xn=0,10.9.3 solving difference equations using the unilateral z-transform,Use the unilateral z-transform to solve linear constant-coefficient difference equations with nonz
26、ero initial conditions.,10.9 The unilateral z-Transform,Example 10.37 A system characterized by the difference equation:,With initial conditions:,Consider:,10.9 The unilateral z-Transform,Solution:,10.9 The unilateral z-Transform,Using unilateral z-transform to two sides.,10.9 The unilateral z-Trans
27、form,Zero-input response,Zero-state response,10.9 The unilateral z-Transform,结论： 1）如果给定初始松弛条件，或者求系统的零状态响应，就用双边z变换的时延性质求解线性差分方程。 如例题10.36 2）如果给定系统初始状态不为零，并且求系统的全响应，则用单边z变换的时延性质求解线性差分方程 如例题10.37。,10.9 The unilateral z-Transform,Example : Consider a causal discrete-time system depicted in following fig
28、ure, which,Find the system function H(s). Sketch the pole-zero pattern of H(z) and indicate the ROC of H(z).Determine the unite impulse hn, is this system stable?Compute the output of this system , if the input signal is,10.9 The unilateral z-Transform,10.9 The unilateral z-Transform,Solution:,(a),10.9 The unilateral z-Transform,(b),This system is stable.,10.9 The unilateral z-Transform,(c ),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,
