Heuristic optimization of the cascade realization of fixed-point digital filters

In the cascade realization of fixed-point digital filters under dynamic range constraints, the output noise due to accumulation of roundoff errors is highly dependent upon the order of the sections. For recursive filters it also depends on the pole-zero pairing that forms the individual second-order sections. The output noise may vary over several orders of magnitude for different cascade realizations of high-order filters. Therefore an optimization procedure to find a good ordering and pairing is very desirable. We propose a heuristic optimization procedure for finding a "near optimal" solution. The procedure is completely automatic and does not require any knowledgeable judgment. The number of function evaluations required for a filter of N-cascaded sections is proportional to N
²
. By using this procedure, "near optimal" solutions have been found for a 22nd-order recursive filter in 23 s, and for a 55th-order nonrecursive filter in 37.5 s, on an IBM 360-91 computer.