Finite word-length effects of pipelined recursive digital filters

Scattered look-ahead (SLA) pipelining is a new IIR filter structure that can achieve very high throughput, regardless of multiplier latency. However, the numerical properties of SLA have been largely unexplored. In this paper we analyze the finite word-length (FWL) performance of SLA filters under fixed-point arithmetic. To support this analysis, two new state variable descriptions (SVD) are introduced. First, a state variable based description especially suited for analysis of certain SLA structures, called the sectioned K and W description (SKWD), is defined by sectioning the noise contributions of the nonrecursive nodes. Second, a noncanonic state variable description (NCSVD) is introduced, which explicitly includes the pipelined delay variables in the state space. Roundoff noise (RON) and statistical coefficient quantization noise (SCQN) are derived under an independent pseudonoise source model. SCQN is shown to be interpretable as RON under the single length accumulation model, which enables the unification of RON and SCQN analysis. Analytic closed form solutions for firstand second-order direct form (DF) SLA filters are derived and compared to results from SLA minimum roundoff noise (MRON) structures. The SLA structures are all found to have generally good numerical properties, except that the DF SLA structure performs poorly in certain regions near the unit circle. However, the DF SLA structure actually performs better than the MRON form over much of the unit disk at a far smaller implementation cost.