On the overflow problem in a second order digital filter

In this paper, we study the adder overflow problem of a second order IIR filter in a more direct way by considering the probability of state overflow. Equations which relate the scale factor and the state overflow probability are derived. This scaling policy is applicable to different transfer function properties and various implementation structures, e.g., the canonical form realization, the uniform grid form realization, and the low sensitivity form realization. Numerical examples and simulation results covering a broad scope of filter implementation problems are presented. We also give quantitative comparisons for various schemes of handling the overflow problem. The results of this paper will offer the filter designer a realistic and efficient way to deal with the forced response overflow problem.