Elimination of Limit Cycles in Direct Form Digital Filters Using Error Feedback

摘要

The elimination of zero-input limit cycles in direct form filter sections usingeither a rounding or a magnitude truncation fixed-point quantizer is considered. The well-known criterion of Chang is reformulated for second-order filter sections in a more practical form using the bilinear transformation. This enables graphical interpretation and qualitative analysis of the possible error feedback solutions for different pole locations. The synthesis of limit-cycle free error feedback for second-order filter sections is addressed in detail and several solutions are proposed. The error feedback coefficients are constrained to power-of-two values or to symmetric values so that the implementation is efficient, e.g., with signal processors. Different limit-cycle-free solutions are discussed with design examples and their roundoff noise properties are compared. The absence of limit cycles is related to the roundoff noise properties of the filter section. The main conclusion of the paper is that when the EF coefficients are appropriately chosen, low roundoff noise and the absence of limit cycles can always be accomplished at the same time. A particularly close correlation between limit cycles and roundoff noise is demonstrated with a rounding quantizer: a limit-cycle-free implementation always guarantees low roundoff noise. For most complex-conjugate pole locations the converse relation also holds, i.e., low roundoff noise implies the absence of limit cycles.