Space-Time Equations for Non-Unimodular Mappings

Abstract. The class of systems of uniform recurrence equations (UREs) is closed under unimodular transformations. As a result, every systolic array described by a unimodular mapping can be specified by a system of space-time UREs, in which the time and space coordinates are made explicit. As non-unimodular mappings are frequently used in systolic designs, this paper presents a method that derives space-time equations for systolic arrays described by nonunimodular mappings. The space-time equations for non-unimodular mappings are known elsewhere as sparse UREs (SUREs) because the domains of their variables are sparse and their data dependences are uniform. Our method is compositional in that space-time SUREs can be further transformed by unimodular and non-unimodular mappings, allowing a straightforward implementation in systems like ALPHA. Specifying a systolic design by space-time equations has two advantages. First, the space-time equations exhibit all useful properties about the design, allowing the design to be formally verified. Second, depending on the application area and performance requirement, the space-time equations can be realised as custom VLSI systems, FPGAs, or programs to be run on a parallel computer.