Optimal Word-Length Allocation for the Fixed-Point Implementation of Linear Filters and Controllers

This article presents a word-length optimization problem under accuracy constraints for the hardware implementation of linear signal processing systems with fixed-point arithmetic. For State-Space systems (describing a linear filter or a controller), a complete error analysis is exhibited, where the final output error bound depends on the word-lengths and the fixed-point formats chosen for each variable. The Most Significant Bit of each one can be determined in order to guarantee that no overflow occurs. Thus, it is possible to obtain a hardware implementation minimizing resource use. This leads to a convex nonlinear integer optimization problem where the resources to minimize and the accuracy constraints depend on the internal word-lengths. This problem can then be solved with appropriate heuristics. Finally, a global approach is proposed and illustrated by some examples.