Structure of an Optimum Linear Precoder and its Application to ML Equalizer

The structure of an optimum linear precoder for a rotation-invariant performance measure is obtained subject to the constraint of limited input power for a block transmission scheme. It is shown that several known performance measures of a communication system are rotation invariant. The rotation invariant property provides a unified framework for obtaining the structure of an optimum linear precoder for several criteria such as: 1) maximization of minimum distance; 2) maximization of channel capacity; and 3) optimization of different performance measures, such as bit-error rate (BER), mean square error, signal-to-noise ratio of optimum equalizers (linear minimum mean square error, zero-forcing, maximum likelihood (ML), zero forcing-block decision feedback, minimum mean square error-block decision feedback, etc.). This framework provides a method for obtaining the structure of an optimum linear precoder for BER performance measure of the ML equalizer. The structure turns out to be a channel diagonalizing structure after a prerotation of the input constellation. Using this structure, the properties of the optimum precoder that minimize the BER of the ML equalizer for two binary phase-shift keying symbols transmission under input-power constraint are studied.