Multiplication-free Polynomial-based Fir Filters with an Adjustable Fractional Delay

An efficient coefficient quantization scheme is described for minimizing the costfor implementing the fixed parallel linear-phase finite-impulse response (FIR) filters in themodified Farrow structure introduced by Vesma and Saramäki for generating FIR filters withan adjustable fractional delay. The implementation costs under consideration are the minimumnumber of adders and subtracters when implementing these parallel subfilters as very-large-scale integration (VLSI) circuit. Two implementation costs are under consideration to meet thegiven criteria. In the first case, all the coefficient values are implemented independently of eachothers as a few signed powers-of-two terms, whereas in the second case, the common subex-pressions within all the coefficient values included in the overall implementation are properlyshared in order to reduce the overall implementation cost even further. The optimum finite-precision solution is found in four steps. First, the number of filters and their (common odd)order are determined such that the given criteria are sufficiently exceeded in order to allowsome coefficient quantization errors. Second, those coefficient values of the subfilters havinga negligible effect on the overall system performance are fixed to be zero-valued. In addition,the experimentally observed attractive connections between the coefficient values of the sub-filters, after setting some coefficient values equal to zero, are utilized in order to reduce boththe implementation cost and the parameters to be optimized even more. Third, constrainednonlinear optimization is applied to determining for the remaining infinite-precision coeffi-cients a parameter space that includes the feasible space where the given criteria are met. Thefourth step involves finding in this space the desired finite-precision coefficient values for min-imizing the given implementation costs to meet the stated overall criteria. Several examplesare included illustrating the efficiency of the proposed synthesis scheme.