On determining optimum multirate structures for narrow-band FIR filters

Several lower bounds on the storage and multiplication rate required to implement a bandpass filter using multirate structures are developed. These lower bounds suggest hypothetical optimum structures that actual optimum structures are found to resemble. The hypothetical structures are useful for predicting the number of stages, decimation and interpolation factors, and cost of actual structures from knowledge of only the passband width, transition bandwidths, and tolerance specifications of the desired filter. The determination of the actual optimum structure for a given bandpass filter specification requires a search over the space of all possible solutions. The size of this space grows very rapidly as filter bandwidth decreases, and excessive computer time is required to search it exhaustively. A fast branch-and-bound algorithm to search this space is also presented. This algorithm, which is applicable for a large class of cost criteria, improves search time by one to two orders of magnitude for filters with passband width in the range 10^-3-^-5 cycles/sample