Error spectrum shaping in state-space digital filters with L2 -scaling constraints

The optimal closed-form solution of an error-feedback diagonal matrix is developed to minimize the output noise due to roundoff error for the state-space model with l₂ scaling constraints on the state variables. It is then shown that the error-feedback diagonal matrix can be derived directly from the diagonal elements of the A matrix in the input-balanced realization. When all the diagonal elements are equal in the error-feedback matrix, a condition is also explored which provides a smaller output noise than that of the optimal filter structures with minimum roundoff noise. Finally, a numerical example illustrates the utility of the proposed technique