Optimum Equalization and the Effect of Timing and Carrier Phase on Synchronous Data Systems

In synchronous data systems, the transmission rates are frequently limited by the intersymbol interference which is caused by the ampli1671 1672 THE BELL SYSTEM TECHNICAL JOURNAL, M A Y - J U N E 1971 tude and phase distortion in the transmission channel. In order to reduce the effect of the intersymbol interference, it is necessary to equalize the channel before the data can be transmitted. Several automatic equalization schemes using transversal filters have been devised for such data systems. 1-5 Chang 0 has investigated the effect of the sampling instant and carrier phase on the minimum mean-square intersymbol interference for a noiseless system with a finite-length transversal equalizer. In principle, we can make the mean-square intersymbol interference arbitrarily small by using an infinitely long transversal filter, provided that the tap-gain settings can be made arbitrarily accurate. However, the equalizer which forces the intersymbol interference to zero may not be the most desirable one when noise is present. We have found, in this study, the optimum infinite-length meansquare equalizer for such synchronous data systems with a fixed channel. Two different cases are considered corresponding to the following optimality criteria: (i) the minimization of the output noise plus mean-square intersymbol interference, and (ii) the minimization of the output noise power subject to the constraint that the equalizer forces the intersymbol interference to zero. Explicit expressions for the optimum equalizer and the M.M.S.E.