On limit cycle amplitudes in error-feedback digital filters

It has been indicated by several authors that an upper bound on limit cycles in digital filter structures, which are equivalent to error-feedback digital filters, is given by
\Sigma\min{n=0}\max{\infin} |h(n)|
, where h(n) is the unit pulse response from an error source due to quantization operation to the filter output. For such cases, H(z), the z-transform of h(n), has two poles and two zeros. Closed formulas for
\Sigma\min{n=0}\max{\infin} |h(n)|
are given in this paper for the cases when H(z) is first- or second-order and with or without zeros. It is then shown that such summation is not an upper bound on limit cycles in error-feedback digital filters. Several examples are given showing limit cycles of much larger amplitudes than the above summation.