【正文】 D D D D?? ? ?? (3) referenced to fr(i 1) = 1. Enough aliasing attenuation means that those frequency ponents which will alias into the passband at the current decimation process will have adequate attenuation with respect to the corresponding passband ponents. Fig. 3 shows an example frequency response of HDi (w) which has an aliasing attenuation exceeding 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)HDK( w/DK) referenced to frK = 1. The design of the pensator transfer function is to make the overall frequency response approximate one in the passband. The frequencyresponse error in the passband is ( ) 1 ( ) ( )1( ) [ ( ) ]()KKKcDkDCKDKwE w H w HDwH H wwD HD???? (4) for [0, ].Pww? To give attenuation to 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 D????for [ , ]pww?? . The frequency band[ , ]pw? can be considered as the stopband of the pensator and the frequencyresponse error is 2( ) 0 ( ) ( )2( ) [ 0 ( ) ]KKCDKDCKwE w H w HDwH H wD???????? (5) for [ , ]Pww?? . Equations (4) and (5) can be bined 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 minimizing 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 lowpass filter structure is a multistage decimator followed by a multistage interpolator and, in between, there is a pensator 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) Hi(w) is the frequency response of each decimator (or interpolator) stage and “mod” means a modulo operation. The frequency response of (9) is the output baseband response due to the whole input in terms of the input frequency as in the case of decimator. It is also the output response due to the baseband input in terms of the output frequency as in the case of interpolator. In the multirate lowpass filter design, each decimation or interpolation stage design is the same as that in a multistage decimator design. The pensator is to give the desired frequency response in the baseband where the baseband is the frequency band that never aliases. Its design is to minimize ||E|| of (6) with the weighting and desired functions given by In the case where there is not a full decimation, ., sw ?? referenced to frK =1, there is a stopband for the pensator design. The transition region can also be viewed as the stopband of the pensator with requirement to limit the transition region aliasing. Comb filter structures as decimators or interpolat