A Framework for Two-Dimensional Hyperstability Theory Based Provably Convergent Adaptive Two-Dimensional IIR Filtering

A set of results for two-dimensional quarter plane causal systems reminiscent of one-dimensional hyperstability theory have been reported. The key to this development is a little known result of Landau [Math. Ann., 62 (1906), p. 272], which asserts that a positive polynomial in two variables can be expressed as the sum of squares of polynomials in one variable whose coefficients are real rational functions of the other variable. The tools used are largely based on notions of passivity and the results obtained can be interpreted as a two-dimensional quarter plane causal generalization of the fact that if the total flow of energy into a dissipative system is upper bounded then both input and output asymptotically die out to zero. An adaptive two-dimensional recursive filtering scheme potentially useful in propagating wave type two-dimensional problems is considered next. It is then shown via our two-dimensional hyperstability results that the adaptive scheme converges in an appropriate sense.