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单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Chat 6 z-tranform,Definition z-Transforms,Region of Convergence z-Transforms,The inverse,z,-Transforms,z,-Transforms Properties,The Transfer Function,Chat 6 z-tranform Definition,6.1 Definition and Properties,The,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 such frequency-domain characterization in these case.,6.1 Definition and Properties,6.1 Definition and Properties(p227),z,-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 important tool in the analysis and design of digital filters.,1.,Definition,6.1 Definition and Properties,6.1 Definition and Properties(p227),Re z,j Im z,z,=,r,e,j,r,1,1,j,j,Unit circle,0,6.1 Definition and Properties,For a given sequence,the set,R,of values of,z,for which its,z,-transform converges is called the,region of convergence(ROC),.,6.1 Definition and Properties(p227),The interpretation of the,z,-transform G(z)as the DTFT of sequence,g,n,r,-n,.,We can choose the value of,r,properly even though gn is not absolutely summable.,In general,ROC can be represented as,For a given sequence,the s,6.1 Definition and Properties(p227),Note:,The,z,-transform of the two sequence are identical even though the two parent sequence are different.,Only way a unique sequence can be associated with a,z,-transform is by specifying its ROC.,The DTFT,G,(,e,j,)of a sequence,g,n,converges uniformly if and only if the ROC of the,z,-transform,G,(,z,)of gn includes the unit circle.,6.1 Definition and Properties,6.1 Definition and Properties(p227),Table 6.1,6.1 Definition and Properties,6.2 Rational,z,-Transforms(p231),M-the degree of the numerator polynomial P(z),N-the degree of the denominator polynomial D(z),6.2 Rational z-Transforms(p2,6.2 Rational,z,-Transforms(p231),In Eq.(6.15),there are M finite zeros and N finite poles,If NM,there are additional N-M zeros at z=0.,If NM,there are additional M-N poles at z=0,.,6.2 Rational z-Transforms(p23,6.3 ROC of Rational,z,-Transforms,The ROC of a rational,z,-transform is bounded by the location of its poles.The ROC of a rational,z,-Transform cannot contain any poles,A sequence can be one of the following type:,finite-length,right-sided,left-sided,and,two-sided,.,If the rational,z,-transform has N poles with R distinct magnitudes,then it has R+1 ROCs,R+1 distinct sequence having the same rational,z,-transform.,6.3 ROC of Rational z-Transfo,a)The ROC of the,z,-transform of a finite-length sequence defined for Mn N is the entire z-plane except possibly z=0 and/or z=+,6.3 ROC of Rational,z,-Transforms,We have the following observation with regard to the ROC of a Rational,z,-Transform,a)The ROC of the z-transform,6.3,ROC of Rational,z,-Transforms,b)The ROC of the,z,-transform of a right-sided sequence defined for Mn is the exterior to a circle in the z-plane passing through the pole furthest from the origin z=0.,6.3 ROC of Rational z-Transfor,6.3 ROC of Rational,z,-Transforms,c)The ROC of the,z,-transform of a left-sided sequence defined for-n N 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-Transfor,6.3 ROC of Rational,z,-Transforms,d)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-Transfor,6.4 The Inverse,z,-Transform(p238),6.4.1 General Expression,-Cauchys integral theorem,6.4 The Inverse z-Transform(,6.4.1 General Expression,If the pole at,z=,0,of,G(z)z,n-1,is of multiplicity,m,.,6.4.1 General ExpressionIf th,6.4.3 Partial-Fraction Expansion Method,A rational,z,-transform G(z)with a causal inverse transform gn has an ROC that is exterior,-M N,P(z)/D(z)is an improper fraction,-M N,P,1,(z)/D(z)is a proper fraction,6.4.3 Partial-Fraction Expansi,6.4.3 Partial-Fraction Expansion Method,Simple Poles,6.4.3 Partial-Fraction Expansi,6.4.3 Partial-Fraction Expansion Method,Multiple Poles,If the pole at,z=v,is of multiplicity,L,and the remaining,N-L,poles are simple.,6.4.3 Partial-Fraction Expansi,6.5,z,-Transform Properties(p246),Conjugation Property,Time-Reversal Property,Linearity Property,6.5 z-Transform Properties(p,6.5,z,-Transform Properties(p246),Multiplication by an Exponential Sequence,Differentiation Property,Time-Shifting Property,6.5 z-Transform Properties(p,6.5,z,-Transform Properties(p246),Modulation theorem,Parsevals Relation,Convolution Property,6.5 z-Transform Properties(p,6.7 The Transfer Function(p258),hn,xn,yn,6.7.1 Definition,6.7 The Transfer Function(p25,6.7.1 Definition,-system function or transfer function,6.7.2 Transfer Function Expression,FIR Digital Filte
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