This section presents approaches for pipelining higher-order recursive digital filter topologies using clustered and scattered look-ahead techniques. (For 1st- order IIR filters, these two look-ahead approaches reduce to the same form.) With clustered look-ahead, pipelined realizations require a linear complexity in the number of loop pipeline stages and are not always guaranteed to be stable. We illustrate the use of scattered look-ahead approach to derive stable pipelined filters. Then the decomposition technique is revisited in this section for higher-order IIR filters and is used to obtain area-efficient implementations for higher-order filters. In addition to the clustered and the scattered lookahead techniques, constrained filter design techniques that achieve pipelining without pole-zero cancellation are discussed.
Let the transfer function of an N-th order direct-form recursive filter be described by
Equivalently, the output sample y(n) can be described in terms of the input
sample u(n), and the past input and output samples, and is given by