Multiplier-free polynomial-based FIR filters with an adjustable fractional delay

An efficient coefficient quantization scheme is described for minimizing the cost for implementing the 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/or subtracters in two cases required in the overall implementation to meet the given overall criteria. In the first case, the coefficients are implemented independently of each others, whereas in the second case, the common subexpressions within the coefficients are shared in order to reduce the implementation cost even further. The optimum finite-precision solution is found in four steps. First, the number of filters and their lengths are determined such that the given criteria are sufficiently exceeded in order to allow some coefficient quantization errors. Second, those impulse-response values of the subfilters having a negligible effect on the overall system performance are fixed to be zero-valued. Third, constrained nonlinear optimization is applied to determining for the remaining infinite-precision coefficients a parameter space including the feasible space where the given criteria are met. The fourth step involves finding in this space the finite-precision coefficient values for minimizing the given implementation cost. Examples are included illustrating the efficiency of the proposed synthesis scheme.