An absolute bound on limit cycles due to roundoff errors in digital filters

An absolute bound on limit cycle oscillations in fixed-point digital filter implementations due to roundoff errors is presented. Periodicity of the limit cycles is assumed in the derivation. Useful design results are explicitly given for the case of second-order filter sections. In addition it is shown that this bound is equal to the rms bound of Sandberg and Kaiser for real roots, and is never more than a factor of two greater than the rms bound for complex roots for second-order filters.