T. Laakso et al., ELIMINATION OF LIMIT-CYCLES IN FLOATING-POINT IMPLEMENTATIONS OF DIRECT-FORM RECURSIVE DIGITAL-FILTERS, IEEE transactions on circuits and systems. 2, Analog and digital signal processing, 41(4), 1994, pp. 308-313
This paper focuses on the limit cycle analysis of floating-point imple
mentations of direct form recursive digital filters. A sufficient crit
erion for the absence of zero-input limit cycles is derived for a dire
ct-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 tha
t the limit cycle behaviour of the filter is directly related to the m
aximum 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) boun
d for the underflow limit cycles is directly proportional to the recur
sive loop gain. The error feedback technique can be used to reduce the
peak gain and thus to save bits in the mantissa wordlength in critica
l applications, as shown by examples.