Design of FIR Precoders and Equalizers for Broadband MIMO Wireless Channels with Power Constraints

This paper examines the optimum design of FIR precoders or equalizers for multiple-input multiple-output (MIMO) frequencyselective wireless channels. For the case of a left-coprime FIR channel, which arises generically when the number nT of transmit antennas is larger than the number nR of receive antennas, the Bezout matrix identity can be employed to design an FIR MIMO precoder that equalizes exactly the channel at the transmitter. Similarly, for a right-coprime FIR channel, the Bezout identity yields an FIR zero-forcing MIMO equalizer. Unfortunately, Bezout precoders usually increase the transmit power, and Bezout equalizers tend to amplify the noise power. To overcome this problem, we describe in this paper a convex optimization technique for the optimal synthesis of MIMO FIR precoders subject to transmit power constraints, and of MIMO FIR equalizers with output noise power constraints. The synthesis problem reduces to the minimization of a quadratic objective function under convex quadratic inequality constraints, so it can be solved by employing Lagrangian duality. Instead of solving the primal problem, we solve the lower-dimensional dual problem for the Lagrange multipliers. When an FIR MIMO precoder has already been selected, we also describe a technique for adding a vector shaping sequence to the transmitted signal in order to reduce the transmit power. The selection of effective shaping sequences requires a search over a trellis of large dimensionality, which can be accomplished suboptimally by employing reduced-complexity search techniques.