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The major contribution of this work is a transformational model of systolic design. In this design model, software transformations are first applied to put the algorithm to be implemented into a regular form conducive to systolic implementation. The steps of allocating operations to hardware, scheduling their execution, and optimizing the design are then performed bottom-up, starting with the innermost blocks of the algorithm. We have successfully used this model to rederive several published designs, and it appears suitable for designing complex systolic arrays. This model may help guide manual design, explain systolic algorithms, or capture the design process in the machine where it can benefit from effective automated support.