Elimination of limit cycles in floating-point implementations of direct-form recursive digital filters

This paper focuses on the limit cycle analysis of floating-point implementations of direct form recursive digital filters. A sufficient criterion for the absence of zero-input limit cycles is derived for a direct-form implementation with a single quantizer in the recursive loop. Both the unlimited exponent range (UER) limit cycles and those due to exponent underflow are considered. The main result of the paper is that the limit cycle behaviour of the filter is directly related to the maximum gain of the recursive loop. The higher the recursive loop gain, the longer mantissa wordlength is required to eliminate the possible large-amplitude UER limit cycles. Also the root-mean-square (RMS) bound for the underflow limit cycles is directly proportional to the recursive loop gain. The error feedback technique can be used to reduce the peak gain and thus to save bits in the mantissa wordlength in critical applications, as shown by examples