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1、Chat 6 z -tranform,Definition z-Transforms Region of Convergence z-Transforms The inverse z-Transforms z-Transforms Properties The Transfer Function,6.1 Definition and Properties,The DTFT provides a frequency-domain representation of discrete-time signals and LTI discrete-time systems.,Because of th
2、e 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 (p227),z-Transform may exist for many sequence for which the DTFT does not exist.,Moreover,
3、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 (p227),For a given sequence, the set R of values of z for which its z-transform conve
4、rges 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 gnr-n.,We can choose the value of r properly even though gn is not absolutely summable.,In general, ROC can be represented as,6.1 Definition and Pro
5、perties (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(ej) of a sequence gn converges uniformly if and only if the ROC of the z-tran
6、sform G(z) of gn includes the unit circle.,6.1 Definition and Properties (p227),Table 6.1,6.2 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(p231),In Eq.(6.15), there are M finite zeros and N finite
7、 poles,If NM, there are additional N-M zeros at z=0.,If NM, there are additional M-N poles at z=0.,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 follo
8、wing 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.,a) The ROC of the z-transform of a finite-length sequence defined for Mn N is the ent
9、ire 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,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-
10、plane passing through the pole furthest from the origin z=0.,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-Transforms,d)
11、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.,6.4 The Inverse z-Transform (p238),6.4.1 General Expression,-Cauchys integral theorem,6.4.1 General Expression,If the pole at
12、z=0 of G(z)zn-1 is of multiplicity m.,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, P1(z)/D(z) is a proper fraction,6.4.3 Partial-Fraction Expansion Method,Simple Pole
13、s,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.,6.5 z-Transform Properties (p246),Conjugation Property,Time-Reversal Property,Linearity Property,6.5 z-Transform Properties (p246),Multiplication by an Exponential
14、 Sequence,Differentiation Property,Time-Shifting Property,6.5 z-Transform Properties (p246),Modulation theorem,Parsevals Relation,Convolution Property,6.7 The Transfer Function (p258),6.7.1 Definition,6.7.1 Definition,-system function or transfer function,6.7.2 Transfer Function Expression,FIR Digit
15、al Filter,For a causal FIR filter, 0N1N2, the ROC of H(z) is the entire z-plane,excluding the point z=0.,Finite-Dimensional LTI IIR Discrete-Time System,6.7.2 Transfer Function Expression,6.7.2 Transfer Function Expression,For a causal IIR filter, hn is a causal , the ROC of H(z) is exterior to the
16、circle going through the pole furthest from the origin.,6.7.3 Frequency Response from Transfer Function,If the ROC of H(z) includes the circle,6.7.3 Frequency Response from Transfer Function,Magnitude function,6.7.3 Frequency Response from Transfer Function,Phase response,Magnitude-squared function for a real-coefficient rational transfer function,A causal LTI digital filter is BIBO stable if and only if its impulse response hn is absolutely
