外文翻譯----利用梳狀濾波器設(shè)計(jì)多速率濾波器(存儲(chǔ)版)
【正文】 actory frequency response. Let H(z) and D be the transfer function and decimation ratio of one stage of a multistage decimator. We propose to design H(z) such that H(z) = f(z)g(zD). In the implementation, by the mutative rule [5], the transfer function g(zD) can be implemented at the lower rate (after decimation) as g(z). This implementation reduces the filter order, storage requirement, and the arithmetic. In this paper, to simplify arithmetic, further requirements are put on H(z) to allow only simple integer coefficients. This is feasible because there are no passband specifications on the frequency response. A cascade of b filters is a particular case of these filters where the coefficients are only 1or1 and, therefore, no multiplications are needed. Hogenauer [6] had also used a cascade of b filters as a onestage decimator or interpolator but with a limited frequencyresponse characteristic. Here the cascade of b filters is used as one stage of a multistage multirate filter with just the right frequency response. More b filter structures are easily derived using the mutative rule. The FIR filter optimizing procedure used in this paper minimizes the Chebyshev norm of the approximation error and this is done using the Remez exchange algorithm. The IIR filter optimizing procedure used minimizes the lp error norm which approaches the Chebyshev norm when p is large. The New Multistage Multirate Digital Filter Design Method In a paper for limited range DFT putation using decimation [7], Cooley and Winograd pointed out that the passband response of a decimator can be neglected and be taken care of after decimation. A multistage multirate digital filter design method which has no passband specification but using passband and stopband gain difference as an aliasing attenuation criterion for each stage is described in [5]. The design method and equations used in that paper which are needed for the b filter structure are outlined in this section. The mutative rule introduced in [5] states that the filter structures in Fig. l(a) and (b) are equivalent. It means that a filter can mute with a rate changing switch provided that the filter has its transfer function changed from H(z) to H(zD) or vice versa. Fig. 1 illustrates the case for decimation, and it is also true for interpolation. This rule is very useful in finding equivalent multirate filter structures and in deriving the transfer function of a multistage multirate filter. For example, Fig. 2(a) shows the filter structure of a multistage decimator where frk, k = 0, 1, . . . , K, is the sampling rate at each stage, and a onestage equivalent decimator shown in Fig. 2(b) is found by repeatedly applying the mutative rule to move the latter stages forward. From the onestage equivalent, it is clear that the transfer function and frequency response of the multistage decimator are 1 1 21 2 3( ) ( ) ( ) ( ) .. . ( )D D D DcH z H z H z H z H z? (1) and 1 2 1 3 1 2( ) ( ) ( ) ( ) .. . ( )cH w H w H D w H D D w H D w? (2) where D = D1D2 . . . Dk. The filtering function of Hc(z) does not involve a sampling rate change. It is used to pensate the passband frequency responses of previous stages, and hence, is called the pensator. Each decimation stage is designed successively. At the time of designing the i th stage filter, all the previous i1 stages have already been designed and the transfer functions known. The requirement on Hi(z) is that the posite frequency response HDi (w) of the first stage to the i th stage have enough aliasing attenuation where 1 2 11 1 2 1 1( ) ( ) ( ) . . . ( ) ( ). . . . . .iD i ii i iw w wH w H H H H wD
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