电子科大信号系统英文版课件chaptera z-transform-b

,10 The Z-Transform,10.1 The z-Transform,10. The Z-Transform,hn,zn,yn,Definition of z-transform for general xn:,(1) Definition of z-transform,10 The Z-Transform,(2) Region of convergence (ROC),ROC: Range of r for X(z) to convergeRepresentation: A. Inequality B. Graphics,Example 10.1 10.2 10.3 10.4,10 The Z-Transform,10.2 The Region of Convergence for the z-Transform,Property1: The ROC of X(z) consists of a ring in the z-plane centered about the origin.,z-plane,Re,Im,10 The Z-Transform,Property2: The ROC does not contain any poles.Property3: If xn is of finite duration, then the ROC is the entire z-plane, except possibly z=0 and/or z= .Property4: 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.,Example 10.5,10 The Z-Transform,z-plane,10 The Z-Transform,Property5: If xn is a left-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,z-plane,10 The Z-Transform,Property6: If xn is two sided, and if the circle |z|=r0 is in the ROC, then the ROC will consists of a ring in the z-plane that includes the circle |z|=r0.,z-plane,Re,Im,Example 10.7,10 The Z-Transform,Property7: If the z-transform X(z) of xn is rational, then its ROC is bounded by poles or extends to infinity.Property8: 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.,10 The Z-Transform,Property9: 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.,Example 10.8,10 The Z-Transform,10.3 The Inverse z-Transform,The way for inverse z-transform:(1) Partial fraction expansion(2) Integration for complex function(3) Power series,Example 10.9 10.10 10.11 10.12 10.13 10.14,Frequency response(Fourier transform): (1) Analytic form: (2) Geometric method:,10 The Z-Transform,10.4 Geometric Evaluation of the Fourier Transform From the Pole-zero plot,10.4.1 First-order systems,The impulse response :,z-Transform:,10 The Z-Transform,10 The Z-Transform,10.5 Properties of the z-Transform,10.5.1 Linearity,10 The Z-Transform,10.5.2 Time shifting,10 The Z-Transform,10.5.3 Scaling in the z-domain,10 The Z-Transform,10.5.4 Time Reversal,10 The Z-Transform,10.5.5 Time Expansion,10 The Z-Transform,10.5.6 Conjugation,10 The Z-Transform,10.5.7 The Convolution Property,Example 10.15 10.16,10 The Z-Transform,10.5.8 Differentiation in the z-Domain,Example 10.17 10.18,10 The Z-Transform,10.5.9 The Initial-Value Theorem,Example 10.19,10 The Z-Transform,10.5.10 Summary of Properties,Table 10.1,10.6 Some Common z-Transform Pairs,Table 10.2,10 The Z-Transform,10.7 Analysis and Characterization of LTI Systems Using z-Transform,Y(z)=X(z)H(z),hnor H(z),xn,X(z),Y(z),yn,So, system function: H(z)=Y(z)/X(z),10 The Z-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.,z-plane,ROC including infinity(z=),10 The Z-Transform,A discrete-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.,Example 10.20 10.21,10 The Z-Transform,10.7.2 Stability,An LTI system is stable if and only if the ROC of its system function H(z) includes the unit circle, |z|=1.,z-plane,z-plane,10 The Z-Transform,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 smaller than 1.,z-plane,Example 10.23 10.24,10 The Z-Transform,10.7.3 LTI Systems Characterized by Linear Constant-Coefficient Difference Equations,Linear constant-coefficient difference equation:,Taking z-transforms of both sides:,So that the system function:,Example 10.25,10 The Z-Transform,10.7.4 Examples Relating System Behavior to the System Function,Example 10.26 10.27,10 The Z-Transform,10.8 System Function Algebra and Block Diagram Representation,10.8.1 System Functions for Interconnections of LTI Systems,10 The Z-Transform,10.8.2 Block Diagram Representations for Causal LTI Systems Described by Difference Equations and Rational System Function,Example 10.28 10.29 10.30 10.31,10 The Z-Transform,Example 10.28,10 The Z-Transform,Example 10.29,10 The Z-Transform,10 The Z-Transform,Example 10.30,System function:,Difference equation:,Block-Diagram in three forms:(a) Direct form:(b) Cascade form:(c) Parallel form:,10 The Z-Transform,10 The Z-Transform,10 The Z-Transform,Example 10.31,System function:,10 The Z-Transform,10.9 The Unilateral z-Transform,Example 10.32 10.33 10.34,Definition:,or,10.9.1 Examples of Unilateral z-Transfo