Optimization Algorithms for the Multiplierless Realization of Linear Transforms

Abstract

This article addresses the problem of finding the fewest numbers of addition and subtraction operations in the multiplication of a constant matrix with an input vector---a fundamental operation in many linear digital signal processing transforms. We first introduce an exact common subexpression elimination (CSE) algorithm that formalizes the minimization of the number of operations as a 0-1 integer linear programming problem. Since there are still instances that the proposed exact algorithm cannot handle due to the NP-completeness of the problem, we also introduce a CSE heuristic algorithm that iteratively finds the most common 2-term subexpressions with the minimum conflicts among the expressions. Furthermore, since the main drawback of CSE algorithms is their dependency on a particular number representation, we propose a hybrid algorithm that initially finds promising realizations of linear transforms using a numerical difference method, and then applies the proposed CSE algorithm to utilize the common subexpressions iteratively. The experimental results on a comprehensive set of instances indicate that the proposed approximate algorithms find competitive results with those of the exact CSE algorithm and obtain better solutions than the prominent, previously proposed, heuristics. It is also observed that our solutions yield significant area reductions in the design of linear transforms after circuit synthesis, compared to direct realizations of linear transforms.