Jointly Optimal High-Order Error Feedback and Realization for Roundoff Noise Minimization in 1-D and 2-D State-Space Digital Filters

Joint optimization of high-order error feedback and state-space realization for minimizing roundoff noise at filter output subject to l₂-scaling constraints is investigated for one- and two-dimensional state-space digital filters. Linear algebraic techniques that convert the problems at hand into an unconstrained optimization problem are explored, and an efficient quasi-Newton algorithm is then applied to solve the unconstrained optimization problem iteratively. In this connection, closed-form formulas are derived for fast and accurate gradient evaluation. Finally, case studies are presented to demonstrate that the high-order error feedback does offer much improved performance and that the proposed joint optimization is superior relative to a sequentially optimized system where the state-space coordinate transformation and high-order error feedback matrices are optimized separately.