1、附录 A Multirate Filter Designs Using Comb Filters SHUN1 CHU, MEMBER, IEEE, AND C. SIDNEY BURRUS, FELLOW, IEEE Abstract-Results on multistage multirate digital filter design indicate most of the stages can be designed to control aliasing with only slight regard for the passband which is controlled by
2、a single stage compensator. Because of this, the aliasing controlling stages can be made very simple. This paper considers comb filter structures for decimators and interpolators in multistage structures. Design procedures are developed and examples shown that have a very low multiplication rate, ve
3、ry few filter coefficients, low storage requirements, and a simple structure. Introduction Multirate filters are members of a class which has different sampling rates in various stages of the filtering operation. This class of filters includes decimators, interpolators, and narrow-band low-pass filt
4、ers implemented with decimation, low-pass filtering, and interpolation. A multistage implementation of these filters has the sample rate changed in several steps where each step is a combined filtering and sample rate change operation. Crochiere and Rabiner 1-4 gave the standard multistage design me
5、thod for these filters which has each stage as a low-pass filter where one optimally chooses the decimation(or interpolation) ratio at each stage. A design method was presented in 5 which uses a different design criterion for each stage. It only requires that each stage have enough aliasing attenuat
6、ion but has no passband specifications. Using the design described in 5 with no passband specifications for each stage allows simple filters to be employed and gives a satisfactory frequency response. Let H(z) and D be the transfer function and decimation ratio of one stage of a multistage decimator
7、. We propose to design H(z) such that H(z) = f(z)g(zD). In the implementation, by the commutative 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,
8、 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 comb filters is a particular case of these filters where the coefficients are only 1or-1 and,
9、 therefore, no multiplications are needed. Hogenauer 6 had also used a cascade of comb filters as a one-stage decimator or interpolator but with a limited frequency-response characteristic. Here the cascade of comb filters is used as one stage of a multistage multirate filter with just the right fre
10、quency response. More comb filter structures are easily derived using the commutative 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 min
11、imizes 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 computation using decimation 7, Cooley and Winograd pointed out that the passband response of a decimator can be neglected and be t
12、aken 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 n
13、eeded for the comb filter structure are outlined in this section. The commutative rule introduced in 5 states that the filter structures in Fig. l(a) and (b) are equivalent. It means that a filter can commute with a rate changing switch provided that the filter has its transfer function changed from
14、 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
15、structure of a multistage decimator where frk, k = 0, 1, . . . , K, is the sampling rate at each stage, and a one-stage equivalent decimator shown in Fig. 2(b) is found by repeatedly applying the commutative rule to move the latter stages forward. From the one-stage equivalent, it is clear that the
16、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 ra
17、te change. It is used to compensate the passband frequency responses of previous stages, and hence, is called the compensator. Each decimation stage is designed successively. At the time of designing the i th stage filter, all the previous i-1 stages have already been designed and the transfer funct
18、ions known. The requirement on Hi(z) is that the composite 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 D D D D (3) referenced to fr(i- 1) = 1. Enough aliasing a
19、ttenuation means that those frequency components which will alias into the passband at the current decimation process will have adequate attenuation with respect to the corresponding passband components. Fig. 3 shows an example frequency response of HDi (w) which has an aliasing attenuation exceedin
20、g 60 dB. In Fig. 3, the passband response is repeated in the stopbands but has been moved down by 60dB. They are used as the atttenuation bounds for the stopbands. If the stopband response is below these bounds, it will have enough aliasing attenuation. The overall filter frequency response is Hc(w)
21、HDK( w/DK) referenced to frK = 1. The design of the compensator transfer function is to make the overall frequency response approximate one in the passband. The frequency-response error in the passband is ( ) 1 ( ) ( )1( ) ( ) ()KKKcDkDCKDKwE w H w HDwH H wwD HD (4) for 0, .Pww To give attenuation t
22、o the first band that will alias to the transition band, it is required that | ( ) ( / )KC D K SH w H w D for , 2 pww, or equivalently, | ( ) ( ( 2 ) / ) |KC D K SH w H w Dfor , pww . The frequency band , pw can be considered as the stopband of the compensator and the frequency-response error is 2(
23、) 0 ( ) ( )2( ) 0 ( ) KKCDKDCKwE w H w HDwH H wD (5) for , Pww . Equations (4) and (5) can be combined to give an error function of ( ) ( ) ( ) ( ) esDCE w W w H w H w (6) and /r P SW , which is the error weighting of the stopband with respect to the passband. The optimal HC(z) is obtained by minimi
24、zing the error norm |E| of (6). The solution depends on the definition of the norm. The multistage interpolator design is the same as the multistage decimator design but with the filter structure reversed. The multirate low-pass filter structure is a multistage decimator followed by a multistage int
25、erpolator and, in between, there is a compensator operated at the lowest sampling rate with no rate change. If the aliasing attenuation requirement for the decimator is the same as the imaging attenuation requirement for the interpolator, the design of the multistage decimator part and that of the interpolator part can be the same. The overall frequency response is 2( ) ( ) ( ) ( m o d )CwH w G w H D G w D (9) where 1 2 1 1 1( ) ( ) ( ) . . . ( . . . )KKG w H w H D w H D D w (10)
