数字信号处理英文版课件：Chap 6 z-Transform

1、DSP group 2007 Chap6-ed1 1Chapter 6 z-Transform Definition of z-transform Transfer function of LTI discrete-time system Frequency response from transfer function Region of convergence (ROC) of z-transform The inverse z-transform z-transform theorems The transfer function Partial-fraction expansionDS

2、P group 2007 Chap6-ed1 2 6.1 Definition and PropertiesThe DTFT provides a frequency-domain representation of discrete-time signals and LTI discrete-time systems.Because of the convergence condition, in many case, the DTFT of a sequence may not exist.As a result, it is not possible to make use of suc

3、h frequency-domain characterization in these cases.DSP group 2007 Chap6-ed1 3 6.1 Definition and Propertiesz-Transform may exist for many sequence for which the DTFT does not exist.Moreover, use of z-Transform techniques permits simple algebraic manipulation.Consequently, z-Transform has become an i

4、mportant tool in the analysis and design of digital filters. DefinitionDSP group 2007 Chap6-ed1 4 z is a complex variable, expressed as Denotation of z-transform pairOr z = r e jRe zIm zz = r e jr11jjUnit circle06.1 Definition and PropertiesDSP group 2007 Chap6-ed1 5 6.1 Definition and Properties If

5、 r =1 Region of Convergence (ROC )DSP group 2007 Chap6-ed1 6Example 6.1Find the z-transform of sequence x n= n u nZIf |z|a, the above power series converge to 6.1 Definition and Properties exampleSolution:Similarly:DSP group 2007 Chap6-ed1 7Example 6.2x n= nun1ZIf | z | M: there are additional N M z

6、eros at z = 0. Na, b0, a0Solution:DSP group 2007 Chap6-ed1 13 6.3 ROC of Rational z-Transform DSP group 2007 Chap6-ed1 14Re(z)Im(z) 6.3 ROC of Rational z-Transform x2n=anun+bnun?xn=-anun1bnun1 ?15a) The ROC of the z-transform of a right-sided sequence defined for n1n is the exterior to a circle in t

7、he z-plane passing through the pole furthest from the origin z=0. 6.3 ROC of Rational z-Transform 16b) The ROC of the z-transform of a left-sided sequence defined for n n2 is the interior to a circle in the z-plane passing through the pole nearest from the origin z=0. 6.3 ROC of Rational z-Transform

8、 17c) The ROC of the z-transform of a two-sided sequence of infinite length is a ring bounded by two circle in the z-plane passing through two poles with no poles inside the ring. 6.3 ROC of Rational z-Transform P257 example 6.8DSP group 2007 Chap6-ed1 18d) The ROC of the z-transform of a finite-len

9、gth sequence defined for Mn N is the entire z-plane except possibly z=0 and/or z=+ 6.3 ROC of Rational z-Transform nNMx nDSP group 2007 Chap6-ed1 19 6.3 ROC of Rational z-Transform exampleExample-A3: Find the z-Transform of sequence:xn= un+(0.5)n un2nun1Solution:DSP group 2007 Chap6-ed1 20 6.3 ROC o

10、f Rational z-Transform example Re(z)Im(z)DSP group 2007 Chap6-ed1 21 6.4 The Inverse z-Transform 6.4.1 General Expressiongn = residues of G(z)z n1 at the poles inside C (6.30)6.4.3 Partial-Fraction ExpansionDSP group 2007 Chap6-ed1 22 M N, P(z)/D(z) is an improper fraction M| l |, For | z | l |,DSP

11、group 2007 Chap6-ed1 24Example 6.4: Let the z-Transform of a causal sequence hn be 6.4.3 PFE ExampleSolution:DSP group 2007 Chap6-ed1 25 6.4.3 PFE ExampleDSP group 2007 Chap6-ed1 26 Multiple Poles If the pole at z=v is of multiplicity L and the remaining N-L poles are simple. 6.4.3 PFEmultiple poles

12、DSP group 2007 Chap6-ed1 27 6.4.3 PFE ExampleExample A4: Let the z-Transform of a causal sequence gn be bySolution:DSP group 2007 Chap6-ed1 28 6.4.3 PFE ExampleP253 Table 6.1 Property Sequence z-Transform ROC LinearityTime-reversalConjugationTime-shiftingParsevals Relation 6.5 z-Transform Properties

13、 Table 6.2DSP group chap3-ed130 Property Sequence z-Transform ROC Multiplication by an exponential sequenceDifferentiation of G(z) 6.5 z-Transform Properties Table 6.2ConvolutionModulation DSP group 2007 Chap6-ed1 31 6.7 The Transfer FunctionDiscrete-Time SystemxnynInput sequenceOutput sequence6.7.1

14、 Definition of transfer function in z-transformDSP group 2007 Chap6-ed1 32system function or transfer function6.7.2 Transfer Function Expression FIR Digital Filter 6.7. 1 Definition of The Transfer Function33 Finite-Dimensional LTI IIR Discrete-Time System 6.7.2 Transfer Function ExpressionDSP group

15、 2007 Chap6-ed1 34For a causal IIR filter, hn is a causal sequence, the ROC of H(z) is exterior to the circle going through the pole furthest from the origin. 6.7.2 Transfer Function ExpressionDSP group 2007 Chap6-ed1 35 6.7.2 Transfer Function Expression ExampleExample 6.34: A causal LTI system cha

16、racterized by the difference equationSolution:DSP group 2007 Chap6-ed1 36 If the ROC of H(z) includes the unit circle For LTI DT system: 6.7.3 Frequency Response from Transfer FunctionDSP group 2007 Chap6-ed1 37 Magnitude response function 6.7.3 Frequency Response from Transfer FunctionPhase respons

17、eDSP group 2007 Chap6-ed1 38Magnitude-squared function for a real-coefficient rational transfer function 6.7.3 Frequency Response from Transfer Function6.7.4 Geometric Interpretation of Frequency Response ComputationDSP group 2007 Chap6-ed1 39A typical factor in the factored from of the frequency re

18、sponse is given byWhere is a zero if it is zero factor or is a pole if it is a pole factor 6.7.4 Geometric Interpretation of Frequency Response ComputationRe(z)jIm(z) Zero factor: Pole factorDSP group 2007 Chap6-ed1 40 6.7.4 Geometric Interpretation of Frequency Response Computation Magnitude respon

19、se functionPhase responseDSP group 2007 Chap6-ed1 41 As is varied from 0 to 2, the tip of the vector moves counterclockwise from z=1 tracing the unit circle and back to z=1 6.7.4 Geometric Interpretation of Frequency Response ComputationDSP group 2007 Chap6-ed1 42A zero vectors has the smallest magn

20、itude when =To highly attenuate signal components in a specified frequency range, we need to place zeros very close to or on the unit circle in this range.To highly emphasize signal components in a specified frequency range, we need to place poles very close to or on the unit circle in this range. 6

21、.7.4 Geometric Interpretation of Frequency Response ComputationDSP group 2007 Chap6-ed1 43Example A5: Find the frequency response of the following causal system, where |a1|0freqz(1,1 -0.5,whole)DSP group 2007 Chap6-ed1 45a1Re(z)jIm(z) 0+102）a1 00 6.7.4 Geometric Interpretation of Frequency Response

22、Computation Examplefreqz(1,1 0.5,whole)DSP group 2007 Chap6-ed1 46 A LTI digital filter is BIBO stable if and only if its impulse response hn is absolutely summable. In transfer function H(z) if the ROC includes the unit circle | z |=1, then the digital filter is stable, and vice versa. 6.7.5 Stabil

23、ity Condition in terms of pole locationDSP group 2007 Chap6-ed1 47A FIR digital filter with bounded impulse response is always stable.On the other hand, an IIR filter may be unstable if not designed properly. An originally stable IIR filter characterized by in finite precision coefficients may becom

24、e unstable when coefficients get quantized due to implementation 6.7.5 Stability Condition in terms of pole locationDSP group 2007 Chap6-ed1 48Example 6.38: consider the causal IIR transfer function. 6.7.5 Stability Condition in terms of pole location ExampleStableDSP group 2007 Chap6-ed1 49 Quantiz

25、ation: the transfer function coefficients are rounded to values with 2 digits after the decimal point: 6.7.5 Stability Condition in terms of pole location ExampleUnstableDSP group 2007 Chap6-ed1 502) All poles of a causal stable transfer function H(z) must be strictly inside the unit circle.1) The ROC of transfer function of an LTI digital filter includes the unit circle, then the filter is BIBO stable.3) All poles of a anticausal stable transfer function H(z) must be strictly outside the unit circl