Synchronization of pipelines

In this paper we apply a recently formulated general timing model of synchronous operation to the special case of latch-controlled pipelined circuits. The model accounts for multiphase synchronous clocking, correctly captures the behavior of level-sensitive latches, handles both shortand longpath delays, accommodates wave pipelining, and leads to a comprehensive set of timing constraints. Pipeline circuits are important because of their frequent use in computer systems. We define their concurrency as a function of the clock schedule and degree of wave pipelining. We then identify a special class of clock schedules, coincident multiphase clocks, which provide a lower bound on the value of the optimum cycle time. We show that the region of feasible solutions for single-phase clocking can be nonconvex or even disjoint, and derive a closed-form expression for the minimum cycle time of a restricted but practical form of single-phase clocking. We compare these forms of clocking on three pipeline examples and highlight some of the issues in pipeline synchronization.