Maximum amplitude zero-input limit cycles in digital filters

With zero input, the output of a recursive digital filter may be nonzero due to arithmetic rounding or truncation. Nonzero outputs are periodic, and the involved cycles are called zero-input limit cycles. For direct form second-order filters with fixed-point arithmetic, many limit-cycle properties have been discovered and upper bounds on limit-cycle amplitudes have been derived. Unfortunately, no algorithm has been suggested for finding exact maximum amplitude limit cycles. In cases where limit-cycle bounds indicate the possibility of large limit cycles, such an algorithm would help filter designers determine whether to use a low-error filter structure or a special arithmetic. The present investigation is concerned with the development of a practical algorithm for finding maximum amplitude limit cycles.