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We present new Lyapunov stability results for discrete-time interconnected dynamical systems and we apply these results in the qualitative analysis of certain classes of high-order fixed-point digital filters. We consider two classes of dynamical systems: those which are endowed with arbitrary interconnections and those which can be transformed into a block lower triangular form. To circumvent difficulties which often arise in the qualitative analysis of complex dynamical systems, we phrase our results in terms of the qualitative properties of the subsystems and in terms of the properties of the interconnecting structure of the classes of systems which we consider. In doing so, the concept of stability preserving mappings arises naturally in the characterization of the interconnections of systems in block lower triangular form. Although the development presented herein is fairly general, we emphasize results which make use of computer-generated Lyapunov functions, using the constructive algorithm of Brayton and Tong.