Pipeline interleaving and parallelism in recursive digital filters. I. Pipelining using scattered look-ahead and decomposition

A look-ahead approach (referred to as scattered look-ahead) to pipeline recursive loops is introduced in a way that guarantees stability. A decomposition technique is proposed to implement the nonrecursive portion (generated due to the scattered look-ahead process) in a decomposed manner to obtain concurrent stable pipelined realizations of logarithmic implementation complexity with respect to the number of loop pipeline stages (as opposed to linear). The upper bound on the roundoff error in these pipelined filters is shown to improve with an increase in the number of loop pipeline stages. Efficient pipelined realizations are studied of both direct-form and state-space-form recursive digital filters. Based on the scattered look-ahead technique, fully pipelined and fully hardware efficient linear bidirectional systolic arrays for recursive digital filters are presented. The decomposition technique is extended to time-varying recursive systems