A tight bound on for general second-order H(z)

In the past, some bounds were derived on when the transfer function 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 when has one or two zeros and for particular restricted locations of these zeros; namely at . Such bounds were neither general nor tight. An upper bound on is derived in this paper when has two complex poles and two zeros located arbitrarily in the complex -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 when has two real poles and two arbitrary zeros.