【文章內容簡介】 radians. 410? ?2 5000???1 , . .,ie??? ? ? ? ? ?111 1 ,j pHe ?? ? ? ?? ? ? ? ?1 ? ?2 , . .,ie??? ? 42 30 00 / 10 0. 6s? ? ???2 ? ?? ? 42 2022 / 10 ? ? ???? ? 2 ,j sHe ? ? ? ? ?? ? ? Figure (a) Speciffications for effective frequency response of overall system in for the case of a lowpass filter. (b) Corresponding specifications for the discretetime system in 1 163。 171。 ?11 163。 ?1)(ej ?HP a s s b andT r a n s it i o ns t o p h a n d?2o11 163。 171。 ?11 163。 ?1)(ej ?HP a s s b andT r a n s it i o ns t o p h a n d?2o1 (a) (b) DESIGN OF DISCRETETIME IIR FILTERS FROM CONTINUOUSTIME FILTERS The traditional approach to the design of discretetime IIR filters involves the transformation of a continuoustime filter into a discrete filter meeting prescribed specifications . This is a reasonable approach for several reasons: The art of continuoustime IIR filter design is highly advanced , and since useful results can be achieved, it is advantageous to use the design procedures already developed for continuoustime filters. Many useful continuoustime IIR design methods have relatively simple closedform design formulas. Therefore, discretetime IIR filter design methods based on such standard continuoustime design formulas are rather simple to carry out. The stand approximation methods that work well for continuoustime IIR filters do not lead to simple closedform design formulas when these methods are applied directly to the discretetime IIR case. Filter Design by Impulse Invariance Impulse invariance provides a direct means of puting samples of the output of a bandlimited continuoustime system for bandlimited input signals . Alternatively , in the context of filter design, we can think of impulse invariance as a method for obtaining a discretetime system. In the impulse invariance design procedure for transforming continuoustime filters into discretetime filters , the impulse response of the discretetime filter is chosen proportional to equally spaced samples of the impulse response of the continuoustime filter。 ., where represents a sampling interval. ? ? ? ?d c dh n T h nT?dTWhen impulse invariance is used as a means for designing a discretetime filter with a specified frequency response , we are especially interested in the relationship between the frequency response of the discretetime and continuoustime filters. If the continuoustime filter is bandlimited, so that then ., the discretetime and continuoustime frequency response are related by a linear scaling of the frequency axis , namely , ? ? 2 , ( 7 . 5 )j ck ddH e H j j kTT? ???? ? ????? ?????? ? 0 , / , ( )cdH j T?? ? ? ?? ? , 。 ( 7 . 7 )j cdH e H j T? ? ??????????dT for? ? ?? ? ?Unfortunately, any practical continuoustime filter cannot be exactly bandlimited, and consequently , interference between successive terms in Eq.() occurs , causing aliasing, as illustrated in Figure . － 3? － 2?… …)j(aΩH?oo－ ? 2 ? 3 ?? ? ＝ ? ? T)(ej ?HTπ2TπTπTπ2－While the impulse invariance transform from continuous time to discrete time is defined in terms of timedomain sampling , it is easy to carry out as a transformation on the system functions. The corresponding impulse response is The impulse response of the discretetime filter obtained by sampling is ? ?1, ( 7 . 9 )Nkck kAHsss?? ??? ? 1,0, ( 7 .1 0 )0 , 0kNstkkcA e thtt?? ????? ???? ? ? ? ? ? ? ?11( ) , ( 7 . 1 1 )k d k dNN ns n T s Td c d d k d kkkh n T h n T T A e u n T A e u n??? ? ???()dcT h t The system function of the discretetime filter is therefore given by In paring Eqs.() and () , we observe that a pole at in the splane transforms to a pole at in the zplane and the coefficients in the partial fraction expansions of and are equal , except for the scaling multiplier Td. ? ?Hz? ?cHskdsTze?kss?? ? 11, ( 7 .1 2 )1kdNdksTkTAHzez ??? ??Example Impulse Invariance with a Butterworth Filter Let us consider the design of a lowpass discretetime filter by applying impulse invariance to an appropriate Butterworth continuoustime filter. The specifications for the discretetime filter are Since the parameter Td cancels in the impulse invariance procedure , we can choose Td=1, so that . ? ?? ?0. 89 12 5 1 , 0 0. 20. 17 78 3 , 0. 3jjHeHe????? ? ?? ? ? ?? ? ????Because of the preceding considerations , we want to design a continuoustime Butterworth filter with magnitude function for which Since the magnitude response of an analog Butterworth filter is a monotonic functions of frequency. ? ?? ? 91 25 1 , 0 , ( 4 ) 77 83 , , ( 4 )ccH j aH j b???? ? ? ? ? ?? ? ? ? ?Eqs.() and () will be satisfied if and ? ?0 .2 0 .8 9 1 2 5 , ( 7 .1 5 )cH j a? ?? ?0 .3 0 .1 7 7 8 3 , ( 7 .1 5 )cH j b? ? Specifically , the magnitudesquared function of a Butterworth filter is of the form So that the filter design process consists of determining the parameters N and to meet the desired specifications. Using Eq.() in Eqs.() with equality leads to the equations and The solution of these two equations is N= and ? ? ? ?2 2/1 , ( 7 . 1 6 )1 cc NHj ???? ?c?2 20 . 2 11 , ( 7 . 1 7 )0 . 8 9 1 2 5Nca??? ?????? ??? ????2 20 . 3 11 ,