Relaxed Simultaneous Congruences

Abstract.

 In certain digital signal processing applications, residues of an input integer signal x are measured with respect to integer moduli . The signal x is then recovered by solving the system of simultaneous linear congruences . Since the residues r _i are measured quantities, they are subject to noise contamination. To provide noise protection, the moduli m _i may be chosen to possess common factors. Accurate approximations for x may then be determined by solving approximate or ``relaxed" simultaneous congruences. This paper presents a coherent mathematical theory for the approximate solution of simultaneous congruences with inaccurate residues when no exact solution exists. After precisely formulating the notion of relaxed congruences, it is found that, under nonrestrictive technical assumptions, unique solutions to these congruences always exist. A variety of examples illustrating characteristics of solutions of relaxed congruences are provided, and a fast, efficient algorithm for solving them numerically is presented. The problem of finding an optimal approximate solution is then discussed. Several optimality criteria are proposed and procedures for finding optimal approximate solutions are outlined. Error bounds are derived which specify the maximum amount that an approximate solution based upon inaccurate residues may differ from the corresponding true solution based upon the exact residues.