Multiplication-Free Polynomial-Based FIR Filters with an Adjustable Fractional Delay

Abstract

An efficient coefficient quantization scheme is described for minimizing the cost of implementing fixed parallel linear-phase finite impulse response (FIR) filters in the modified Farrow structure introduced by Vesma and Saramaki for generating FIR filters with an adjustable fractional delay. The implementation costs under consideration are the minimum number of adders and subtracters when implementing these parallel subfilters as a very large-scale integration (VLSI) circuit. Two implementation costs are under consideration to meet the given criteria. In the first case, all the coefficient values are implemented independently of each other as a few signed-powers-of-two terms, whereas in the second case, the common subexpressions within all the coefficient values included in the overall implementation are properly shared 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 allow some coefficient quantization errors. Second, those coefficient values of the subfilters having a 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 subfilters, after setting some coefficient values equal to zero, are utilized to reduce both the implementation cost and the parameters to be optimized even more. Third, constrained nonlinear optimization is applied to determine for the remaining infinite-precision coefficients a parameter space that includes the feasible space where the given criteria are met. The fourth step involves finding in this space the desired finite-precision coefficient values for minimizing the given implementation costs to meet the stated overall criteria. Several examples are included illustrating the efficiency of the proposed synthesis scheme.