An algorithm for constrained roundoff noise minimization in digital filters with application to two-dimensional filters

An algorithm is presented for minimizing roundoff noise effects in state-space implementations of recursive digital filters while constraining certain coefficients to be zero. Beginning with a direct form structure, the algorithm uses the scaling and noise matrices to iteratively construct an upper or lower triangular transformation matrix to apply to the original state-space model. The result is a structure with a saving of N(N-1)/2 multiples over the minimum noise structure. The technique is of great utility in the realization of two-dimensional filters that cannot, in general, be factored into the parallel or cascade connection of lower-order subfilters. Because of the sparse nature of the local state-space structure coefficient matrices in two-dimensional direct- and controller-form realizations, transforming with an upper and lower triangular block diagonal transformation matrix retains many zeros in the structure and results in a low roundoff noise, computationally efficient state-space realization. A novel controller-form state-space structure for a general two-dimensional function is presented to which the noise minimization procedure is directly applicable. Numerical examples are included