A systematic algorithm for designing multiplierless computationally efficient recursive decimators and interpolators

It has been shown by Renfors and Saramaki that if the phase linearity is not required, then the single-stage and multistage decimators and interpolators based on the use of the so-called recursive Nth-band filters provide the lowest computational complexities among the existing decimators and interpolators. This paper describes an efficient algorithm for designing these decimators and interpolators for both the single-stage and multistage implementations in such a way that the resulting filters become multiplierless with short coefficient wordlength. For single-stage filters, the coefficient optimization is performed in two steps. First, a nonlinear optimization algorithm is used for determining a parameter space of the infinite-precision coefficients including the feasible space, where the filter meets the given criteria. The second step involves finding the filter parameters in this space so that the resulting filter meets the given criteria with the simplest coefficient representation forms. For multistage decimators and interpolators, these two steps are performed independently for each filter stage by properly sharing their attenuation responsibilities. This considerably reduces the overall optimization time. An example is included in order to illustrate the benefits of the proposed synthesis scheme.