Optimal and suboptimal error spectrum shaping for cascade-form digital filters

Error spectrum shaping (ESS) can significantly reduce finite-wordlength error in recursive fixed-point digital filters. Most past work on ESS has focused on second-order filters. In this paper, expressions for the ESS coefficients minimizing the roundoff error are derived for high-order filters composed of cascaded second-order sections. A comparison of the noise gains is made between optimal and suboptimal ESS structures and the section-optimal structure; the ESS structures perform well in comparison. A heuristic section-ordering strategy, based on ideas underlying ESS, is proposed and deemed useful, particularly for elliptic filters.