A multiplication-free solution for linear minimum mean-square estimation and equalization using the branch-and-bound principle

An optimal linear mean-square estimation algorithm is derived under the constraint that the algorithm be multiplication-free. A classical linear estimation problem with block length generally requires multiplications. For many on-line signal processing situations a large number of multiplications is objectionable. This class of estimation problems includes the classical linear filtering of a random signal in random noise, as well as the linear equalization of digital data over a dispersive channel with additive noise. Here we consider the linear estimation problem on a binary computer where the estimation parameters are constrained to be powers of two and thus all multiplications are replaced by shifts. Then the optimal constrained linear estimation problem resembles an integer-programming problem except that the allowable discrete points are nonintegers. The branch-and-bound principle is used to convert this minimization problem to a series of convex programming problems. An algorithm is given for the solution as well as numerical results for filtering and data equalization. These examples show that the multiplication-free constraint does not generally increase the mean-square error significantly compared with the classical optimal solution. Furthermore, the intuitive "round to the nearest power of two" procedure for the estimation parameters can be inferior to the optimal brunch-and-bound solution.