Decision feedback equalization is presently of interest as a technique for reducing intersymbol interference in high‐rate PAM data communications systems. The basic principle is to cancel out intersymbol interference arising from previously decided data symbols at the receiver, leaving remaining intersymbol interference components to be handled by linear equalization. In this work we consider the application of decision feedback equalization to quadrature‐amplitude modulation (QAM) transmission, in which two independent information streams modulate quadrature carriers. Extending Salz's treatment in a companion paper of decision feedback for a baseband channel, we derive the form of the optimum receiver filters via a matrix Wiener‐Hopf analysis. We obtain explicit analytical expressions for minimum mean‐square error and optimum transmitting filters. The optimization is subject to a constraint on the transmitted signal power and assumes no prior decision errors. The class of QAM transmitter and receiver structures treated here is actually much larger than the class usually considered for QAM systems. However, our results for decision feedback equalization show that, for nonexcess bandwidth systems, optimum performance is achievable without taking advantage of the most general structure. If the transmitter is required to have the conventional QAM structure, study of the time continuous system that gives rise to the sampled data system considered here demonstrates that under quite general assumptions a nonexcess bandwidth system is optimum. Finally, the explicit description of the optimum transmitting matrix filter follows from an information‐theoretic “water‐pouring” algorithm in conjunction with the determination of the form of the points of maxima of a determinant extremal problem.

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Journal Bell System Technical Journal
Falconer, D.D, & Foschini, G.J. (1973). Theory of Minimum Mean‐Square‐Error QAM Systems Employing Decision Feedback Equalization. Bell System Technical Journal, 52(10), 1821–1849. doi:10.1002/j.1538-7305.1973.tb02710.x