Limit cycle bounds for floating point implementations of second-order recursive digital filters

It is shown that floating point realizations of linearly stable systems can exhibit four fundamental types of free responses. Sufficient conditions for the existence or nonexistence of some of these periodic response types in a given system are presented. Explicit closed form conditions on the mantissa length to guarantee certain limit cycle bounds are provided. The effect of various floating point arithmetic reformatting schemes on the convergence of recursive difference equations is also addressed. Truncation and rounding quantization schemes as well as double- and single-length product mantissa schemes are analyzed and compared. Although the method introduced is applicable to any digital filter realization implemented in floating point format, the analysis focuses on the zero-input behavior of second-order direct-form digital filters