Chapter 5 Advanced Systolic Design 5.1 Introduction 5.2 Systolic Design by Recurrence Transformations 5.2.1 Recurrence Equations : Deenitions and Notation Deenition 5.2.1 a Recurrence Equation Deening a Function (variable) X At

Systolic arrays are locally connected parallel architectures, whose structure is well-suited to the implementation of many algorithms, in scientiic computation, signal and image processing, biological data analysis, etc. The nature of systolic algorithms makes it possible to synthesize architectures supporting them, using correctness preserving transformations, in a theoretical framework that has a deep relationship with loop parallelization techniques. This opens the way to new very high-level architecture synthesis techniques, which will be a major step in mastering the use of IC technologies in the future. This chapter has two parts. First, we present the current state of the art in systolic synthesis techniques, and the second part surveys various ways of implementing systolic algorithms and architectures using programmable architectures, fpgas, or dedicated architectures. The term systolic arrays was coined by Kung and Leiserson in 1978 to describe application speciic vlsi architectures that were regular, locally connected and massively parallel with simple processing elements (PEs). The idea of using such regular circuits was even present in von Neuman's cellular automata in the fties, Hennie's iterative logic arrays in the sixties, and also in specialized arithmetic circuits (Lyon's bit-serial multiplier 1] is clearly a linear systolic array). However, the emergence of vlsi technology in the late seventies and early eighties made the time ripe for introducing such architectures in order to highlight the characteristics appropriate to the technology. Systolic arrays immediately caught on, since they involved a fascinating interplay between algorithm and architecture design. When researchers started in-1 2 Chapter 5 vestigating automatic synthesis, a third stream joined this connuence, namely the analysis, manipulation and transformation of programs. Some of the early designs are classics. The Guibas et al. array for optimal string parenthesization 2] is one of our all time favorites. The early eighties was a period of intense activity in this area. A large number of \paper designs" were proposed for a wide variety of algorithms from linear algebra , graph theory, searching, sorting, etc. There was also much work on automatic synthesis methods, using dependency analysis, space-time transformations of inner loops, and also other formalisms such as recurrence equations. Then, the technological evolutions in the late eighties seemed to invalidate the assumptions of the systolic model, namely that (i) locality, regularity and simplicity were primordial for vlsi and (ii) an elementary computation could be performed in the same time that it took to perform an elementary communication. The rst …