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Several recent publications have called attention to a substantial number of new digital filter topologies to be added to designers' "catalogs." Because of this growing list of available structures, it is worthwhile to code a general-purpose computer analysis and design program that is not limited to any single filter topology. We derive here the mathematical background required for such a program, as well as some of its various programming considerations. For tutorial purposes, a number of numerically based analysis techniques are unified under a common notation and approach. New results include the application of symbolic analysis techniques to digital filters. Digital filter design optimization with symbolic second-derivative information is also discussed. An optimization example is given.