信號與系統(tǒng)英文版課件：Chap10 Z Transform

1、The Z-TransformChapter 10 Lecture 1Signals and Systems Spring 2015Homework 10.2 10.3 10.6 10.7 10.9 10.10 10.11 10.13 10.16 10.17 10.18 10.31 10.47OutlineThe bilateral and unilateral z-transformRegion of convergence for z-transformProperties of the z-transformAnalysis of DT LTI systems using z-trans

2、formBlock diagram of DT LTI systemsDiscrete-Time Fourier TransformDiscrete-time Fourier Transform (DTFT)Inverse discrete-time Fourier TransformOften use notationFrom DTFT to Z-TransformThe discrete-time Fourier transform (DTFT)requires the f n being absolutely summable Absolutely summable is a strin

3、gent constraint that cannot be satisfied by many signals of interest.Z-TransformFor an arbitrary DT signal fn, we define a new signal via properly choosing r, such that the new signal gn is absolutely summable, which ensures the existence of the DTFT of gn. wherelet z = rejz-transformCan we derive f

4、n from its z-transform F(z)?Inverse Z-TransformSince , we have let z = rejdz = jrej d= jz dInverse z-transformZ-TransformZ-Transform (ZT)Inverse Z-Transform Often use notationZ-TransformIntuitively, fn is decompose into a sum of weighted complex exponentials zn-1, the magnitudes of which are Example

5、 Determine the ZT of the following signals (1) Solution: (2) (3) Two distinct signals lead to identical algebraic expression for the z-transforms. The set of values of z for which the expression is valid is very different in the two examples. In specifying the z-transform of a signal, both the algeb

6、raic expression and the range of values of z for which this expression is valid are required.Range of ConvergenceIn general, the range of values of z for which the summation involved in the z-transform converges is referred to as the region of convergence (ROC) of the z-transform.The ROC consists of

7、 the values of z=rej for which the discrete time Fourier transform of fnr-n converges.Range of ConvergenceSince the variable z is a complex number, we display the ROC in the complex plane, generally called the z-plane, associated with this complex variable. The coordinate axes are Rez along the hori

8、zontal axis and Imz along the vertical axis. Zeros and PolesThe roots of equation are called the zeros of F(z), denoted by z1, z2, , zM.The roots of equation are called the poles of F(z), denoted by p1, p2, , pN.In general, the z-transform F(z) is a rational function and can be expressed aspole-zero

9、 plotExampleGiven a DT signal draw the pole-zero plot of F(z).Solution: poles: p1=1 and p2=2zeros: z1=0 and z2=3The Properties of ROCProperty 1: The ROC of F(z) consists of a ring in the z-plane centered about the origin. The ROC of F(z) consists of the values of z=rej for which the DTFT of fnr-n co

10、nverges, i.e. the ROC should make fnr-n absolutely summableROC depend only on r=|z| and not on .Property 2: The ROC does not contain any poles. Since F(z) equals infinity at a pole, the sum in does not converge, and hence the ROC cannot contain any poles.Property 3: If f n is of finite duration, the

11、n the ROC is the entire z-plane, except possibly z=0 and/or z=.When f n has finite duration, sayEx: the summation in must be well defined. Property 4: If the z-transform F(z) of f n is rational, and if f n is right sided, then the ROC is the region in the z-plane outside the outermost pole. Consider

12、 AsEx: Property 5: If the z-transform F(z) of f n is rational, and if f n is left sided, then the ROC is the region in the z-plane inside the innermost nonzero pole. Consider AsEx. z-planeProperty 6: If the z-transform F(z) of f n is rational, and if f n is two sided, then the ROC is a ring in the z

13、-plane centered about the origin or an empty set . Ex. poles of the right-sided signal poles of the left-sided signalz-planeZ-Transform and Fourier TransformDTFT:ZT:If is in the ROC of , i.e. the ROC contains the unit circle, then Otherwise, the FT dose not exist.ExampleGiven the ZT determine the si

14、gnal fn under the following assumptions: (1) fn is left sided, (2) fn is right sided, (3) The FT of fn exists.Solution: The pole-zero plot of F(z)(1) When fn is left sided, the ROC is .(2) When fn is right sided, the ROC is .(3) When the FT of fn exists, the ROC contains the unit circle.ExampleDeter

15、mine the ZT of Solution: If If , the ROC is . F (z) does not exist. The Properties of Z-TransformLinearity Time shiftingScaling in the z-domainTime reversalTime expansionConjugationConvolutionDifferentiation in the z-domainThroughout this part, we assumeLinearityExample: Determine the ZT of . Soluti

16、on: The ROC is , so the ZT does not exist. Similarly, the ZT of the following signal does not exist as well. Time ShiftingProof: , except for the possible addition or dele-tion of the origin or infinity(let )Example Determine the ZT of f n. Solution: (method 1)linearityNote that z=1 is not a pole of

17、 F(z).Solution: (method 2)Solution: (method 3)The Z-TransformChapter 10 Lecture 2Signals and Systems Spring Scaling in Z-DomainProof: Scaling in Z-DomainAlso named frequency shifting property. Scaling in Z-DomainletExampleDetermine the ZT of the following signals. (1) (2)Solution: (1) linearity(2) F

18、or and letTime ReversalProof: Exampletime reversal time delay linearity let Time Expansion, k is an integer-11Time ExpansionProof: ExampleConjugationProof: let If f n is real, The ZT F(z) of a real signal f n has real coefficients, so the poles and zeros must be either real or complex conjugate pair

19、s.ExampleGiven that (1) f n is a real signal.(2) F(z) has two poles, one of which is(3) F(z) has two zeros at the origin.(4) The Fourier transform of f n exists.(5) F(1) = 1.Determine F(z) and its ROC.Solution: (4) the ROC contains the unit circle.(5)ConvolutionIf and Proof: (let )Differentiation in

20、 Z-DomainProof: ExampleProve that for ,Proof：differentiation in z-domainlinearitydifferentiation in z-domainDifferencing and Accumulation in TimeProof: Determine the ZT F (z) ofSolution:ExampleExampleGiven that determine the signal . Solution：ROC :The Z-TransformChapter 10 Lecture 3Signals and Syste

21、ms Spring For a right-sided signal fn=0, n n0, we have and we can show that as long as the ROC of contains _ the unit circle.Initial- and Final- Value Theoreminitial-value theoremfinal-value theoremUseful TablesProperties of z-transformpp. 775, Table 10.1Z-transform pairspp. 776, Table 10.2(1) (2)Gi

22、ven thatSolution: (1)Exampledetermine the z-transforms oftime shiftingtime reversal (2)Accumulationdifferentiation in z-domainInverse Z-TransformHow to obtain the original signal using IZT ?By definition, The above integral can be calculated using the residue theorem. Calculation using the above app

23、roach can be very lengthy and complicated. Partial Fraction ExpansionSuppose F(z) can be expressed asAssume M N and denote the poles (can be complex) by p1, p2, pN. Consider two casesall poles are distinct not all poles are distinct LetIf the poles p1, p2, pN are distinctIf some of the poles p1, p2,

24、 pN are identical, say, p1 is a multiple root with multiplicity KFor i = 2, , N-K+1,For the terms related to p1 ExampleSolution:Given computez-planeExampleGiven that determine fn.Solution:Long DivisionClearly, the weighting factor of z-n is fn. This observation can be explored to find fn. Example 1G

25、iven , find fn.Solution: descending order for right-sided signalsExample 2Given , find fn.Solution: ascending order for left-sided signalsExample 3Given , find fn for n1.Solution: The input and output of an DT LTI system can be described by its difference equationDefine DT system functionLTI System

26、Analysis Using the ZTDefine DT system functionThe system function is determined by the system poles and zeros up to an unknown factor K. The pole-zero plot can be used to describe an LTI system.Pole-Zero Plot of DT LTI SystemEquivalent Descriptions of DT LTI SysDifference equationSystem functionImpu

27、lse responsePole-zero plotCausalityFor a causal LTI system, the impulse response is a causal function, i.e. and is thereby right sided. Due to the ROC property, the ROC of a right sided signal is the region in the z-plane outside the outermost pole.A DT LTI system is causal if and only if the ROC of

28、 its system function is the exterior of a circle, including infinity.InvertibilityThe impulse response hinvn of the inverse system of an invertible system hn satisfiesThe system function Hinv(z) of the inverse system is the reciprocal of the invertible system H(z).The Z-TransformChapter 10 Lecture 4

29、Signals and Systems Spring Consider a causal system with distinct poles.Thus, the system impulse response isStabilityThe system hn can be regarded as a parallel interconnection of N subsystems hin.The system is stable when all the subsystems are stable.Subsystem i is stable only when satisfiesFor a

30、causal CT LTI system Generally, for any DT LTI system with a rational system function:ROC contains unit circlesystem stableall poles: | pi | 1the anticausal system is stableAn DT LTI system has difference equation(1) Determine H (z) and sketch the pole-zero plot.(2) If the system is causal, determin

31、e hn.(3) If the system is anticausal, determine hn.(4) If the system is stable, determine hn. ExampleSolution: (1)The system function is(2) If the system is causal, the ROC is (3) If the system is anticausal, the ROC is (4) If the system is stable, the ROC isExampleThe response to the input isDeterm

32、ine the system function and its ROC.Solution：H(z) has two poles, 3 and -1/2 . Only when the ROC of H(z) is 1/2 |z| 3, can we get the ROC of H(z) being 1 |z| 0right shifting if n0 0Proof: Ex: Analyzing LTI System Using ZT and UZTCalculating system response using ZTCalculating system total response us

33、ing UZTSystem response to complex exponentialsCalculating System Response using ZTExample: The difference equation of a causal LTI system isFind the system impulse response hn and the response to input signalSolution: Take ZT to get(causal system)fn two-sideduse BLTCalculating Total Response using U

34、ZTExample: For a causal LTI system suppose the input is fn = nun and the initial conditions are y-2 = 5/2 and y-1 = 1/2, determine(1) Total response yn.(2) Zero-state response yfn.(3) Zero-input response ysn.Solution: (1) Take UZT for the diff eqnPlugging in Fu(z) and initial conditions y-2 = 5/2 and