On large moduli residue number system recursive digital filters

The three moduli set
\{2^{n} - 1, 2^{n},2^{n} + 1\}
residue number system, or RNS, has recently been shown to possess several attractive properties. In particular, the problem of scaling is much simplified through the use of an autoscale algorithm. Using this approach, a class of RNS recursive digital filter structures is studied in the context of their precision. The filter forms differ from one another in the placement of the autoscale unit. The error introduced by the scaling algorithm, called the scaling error, is the counterpart of roundoff error in fixed-point arithmetic. Formulas for predicting the scaling error variance for the different filter architectures are developed and optimal values for the various parameters of the RNS filters are derived. A model for predicting the scaling error is also developed and is used to reconstruct an error free filter output. The tradeoffs between the various architectures are analyzed both qualitatively and quantitatively.