An algebraic view of multidimensional multiple-input multiple-output finite impulse response equalizers

The problem of computing equalizers for multidimensional multiple-input multiple-output (MIMO) finite impulse response (FIR) filters using MIMO FIR filters is considered. Prior results are reviewed establishing conditions for the equalizability of a given MIMO FIR filter and of a generic MIMO FIR filter. Algorithms for the computation of MIMO FIR equalizers are derived and their relative strengths and weaknesses are discussed. A parametrization of the set of all equalizers for a given MIMO FIR filter is derived. Known bounds on the orders of the equalizer are doubly exponential in the order of the given MIMO FIR filter; alternative generic bounds are derived which are linear in the order of the given MIMO FIR filter, and the behavior of the bound for large number of outputs is characterized. A fast algorithm (generalizing the Levinson algorithm to the multidimensional case) is proposed for multidimensional Wiener filtering and for computing equalizers for single-input multiple-output (SIMO) systems. A minimal set of test targets is established for polarimetric calibration of ultra-wide band synthetic aperture radar systems, and equalizer orders for multiple antenna systems are computed. A brief overview of the applicability of the results to blind equalization problems is provided.