# Synthesizing systolic arrays with control signals from recurrence equations

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## Abstract

We present a technique for synthesizing systolic arrays which have non-uniform data flow governed by control signals. The starting point for the synthesis is an*Affine Recurrence Equation*—a generalization of the simple recurrences encountered in mathematics. A large class of programs, including most (single and multiple) nested-loop programs can be described by such recurrences. In this paper we extend our earlier work (Rajopadhye and Fujimoto 1986) in two principal directions. Firstly, we characterize a class of transformations called*data pipelining* and show that they yield recurrences that have*linear conditional expressions* governing the computation. Secondly, we discuss the synthesis of systolic arrays that have non-uniform data flow governed by control signals. We show how to derive the control signals in such arrays by applying similar pipelining transformations to these*linear conditional expressions*. The approach is illustrated by deriving the Guibas-Kung-Thompson architecture for computing the cost of optimal string parenthesization.

## Key words

Synthesizing systolic arrays Control signals Recurrence equations## Preview

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