Multi-rate transformation of directional affine recurrence equations

There has been an increased attention to the synthesis of algorithmic specific pipeline arrays such as systolic arrays. Most of the existing synthesis techniques are based on a transformation of the algorithm from a class of Recurrence Equations such as Uniform Recurrence Equations (UREs). However, many algorithms cannot be transformed to a URE and the temporal locality of systolic arrays results in additional delay time. The temporal locality constraint can be removed by using the multi-rate array (MRA) structure. In MRA the variables are propagated at different rates. By allowing data transmission at different clock rates, transparent data or ones with small delays are propagated. It is shown that using MRA, a broader class of REs termed directional affine recurrence equation (DARE) can be mapped onto pipeline arrays. The authors provide the definition and a synthesis technique for mapping DAREs on multi-rate array. Conditions for mapping AREs onto MRA is given and the corresponding timing and allocation functions are derived. Applications of multi-rate arrays for signal processing algorithms is also presented.