A tight bound on Sigma_{n=0}^{infty} |h(n)| for general second-order H(z)
AHMAD I. ABU-EL-HAIJA · IEEE Transactions on Circuits and Systems · 1982
In the past, some bounds were derived on\sum_{n=0}^{\infty}|h(n)|when the transfer functionH(z)has only two poles and no zeros. These bounds are useful for determining bounds on limit cycles in certain digital filter structures. Recently, bounds were derived on the above summation whenH(z)has one or two zeros and for particular restricted locations of these zeros; namely atz = + 1. Such bounds were neither general nor tight. An upper bound on\sum_{n=0}^{\infty}|h(n)|is derived in this paper whenH(z)has two complex poles and two zeros located arbitrarily in the complexz-plane. The bound is compared with the actual summation and is found to be extremely tight. Moreover, closed formulas are derived giving the exact value of\sum_{n=0}^{\infty}|h(n)|whenH(z)has two real poles and two arbitrary zeros.