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1、BEIJING JIAOTONG UNIVERSITYThe Course Group of Signals and Systems,Beijing Jiaotong University.P.R.CHINA.Copyright 2020Signals and Systems z-domain representation for signalsz-transform of typical signalsProperties of unilateral z-transformInversion of unilateral z-transformComplex frequency-domain
2、analysis for D-T S&S z-domain description for LTI systems Transfer function and system propertiesImplementation structure for systemsz-domain analysis for system responsefor D-T signalsfor D-T systems How to obtain the yzik and yzsk of the LTI system？Solve the problem effectively in complex frequenc
3、y domain.Time domain:yzik (homogeneous equation)yzsk=xk*hkFrequency domain:Only obtain the The input signal xk is absolutely summable,and the system is stable.Difference equation of a causal LTI system:yk+3yk-1+2yk-2=xkinput signal xk=uk,and initial conditions y-1=0，y-2=0.5ykHX IDTFT(e)(e)zsjjComple
4、x frequency-domain analysis for D-T S&S How to describe the properties of the LTI system?Time domain:Described by the impulse response hk.Frequency domain:Described by frequency response H(ej),only for stable systems.Complex frequency-domain analysis for D-T S&SSolve the problem effectively in compl
5、ex frequency domain.Difference equation of a causal LTI system:yk+3yk-1+2yk-2=xkinput signal xk=uk,and initial conditions y-1=0，y-2=0.5Complex frequency-domain analysis for signalsz-domain representation for D-T signalsUnilateral z-transform of typical signalsProperties of unilateral z-transformInve
6、rsion of unilateral z-transformz-domain representation for D-T signalsz-transform of discrete time signalsDefinition of unilateral z-transformRegion of convergence of z-transformxk=2kuk Discrete Time Fourier Transform?Multiply xk by exponential signal r-k,DTFT again.-u k rrkkkkkke2DTFT2 0j zrLeetj02
7、-zkkkz-1121If|z|2-rkkk)e2(j0From DTFT to z-transformGeometric series of infinite length in the ratio 2/z-rx k rx k rx kkkkkkk)DTFT e(ejj-x k zX zkk()zrLeetj-1 12j()dckx kX z zzFrom DTFT to z-transformGeneralizing the idea,we can getAccording to IDTFT,we can derive inverse z-transformz-transform of D
8、-T signals-X zx k zkk()-1 12j()dckx kX z zzComplex frequencyz=rejxk can be represented as complex exponential zkX(z)is a function of complex frequency z=rej.z-transform:Inverse z-transform:-xX zk zkk)(-X zx kzzck(j)2 d11X(z)=Z Z xkxk=Z Z -1X(z)Z Z x kX z()z-transform of D-T signals-X zx k zkk()0-1 1
9、2j()dckx kX z zzzrejDefinition of unilateral z-transformMost systems in engineering applications are causal systems.To describe these systems and solve difference equations with initial conditions conveniently,unilateral z-transform is defined.z-transform is the DTFT of xkr-k.Hence,a necessary condi
10、tion for convergence of the z-transform is the absolute summability of xkr-k.Region of convergence of z-transform-xX zk zkk)(-r=|z|x k z|=|x k r|=CRxRegion of convergence of z-transformFor unilateral z-transformThe range of r=|z|for which the z-transform converges is termed the region of convergence
11、(ROC).ROC can be depicted in a complex plane(z-plane)Region of convergence of z-transform)z(mI)z(eRxRCORzRRxx,is a positive real number z-planeThis is a geometric series of finite length in the ratio z-1.The sum converges,provided that|z|0.-othersx kkN0,1,01-X zx k zzkkkkN()001-zzzzNN1111)11(|z|00 0
12、)z(mI)z(eRCORExample 7.1:Determine the unilateral z-transform and its ROC.Solution:-x ku kk(2)-X zx k zzkkkkk ()(2)00|z|22-)z(mI)z(eR2COR-zzkk121(2)101Example 7.2:Determine the unilateral z-transform and its ROC.Solution:This is a geometric series of infinite length in the ratio(-2z-1).The sum conve
13、rges,provided that|-2z-1|2.ROC outside of the circle containing the pole.-x ku ku kkk(2)+(3)-X zx k zkk()0|z|332-)z(Im)z(Re3ROC-zz121 311+11Example 7.2:Determine the unilateral z-transform and its ROC.Solution:Causal signal has ROC outside of the circle containing the pole of largest magnitude.-zzkk
14、kkkk (2)+(3)00|z|2|z|3Unilateral z-transform and the ROC|z|Rx-Xzzx kzkc(j)2 d11-X zx k zkk()0Complex frequency-domain representation for D-T signalsSummerySignal xk can be represented by complex exponential signal zk The ROC is outside of the circle containing the pole of largest magnitude.AcknowledgmentsMaterials used here are accumulated by authors for years with helpfrom colleagues,media or other sources,which,unfortunately,cannotbe noted specifically.We gratefully acknowledge those contributors.z-domain representation for D-T signals