In two recent papers, Brayton and Tong developed some significant results which are the basis of a constructive approach in the stability analysis of dynamical systems. We apply this constructive stability algorithm to the examination of the global asymptotic stability of four different classes of second order fixed-point digital filters: direct form, coupled form, wave filters and lattice filters. With this method we obtain new stability results for many of these filters, and we improve upon several existing stability results. Descriptions and listings of the computer programs used in our investigation are also included in this dissertation.