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A criterion of minimum mean square error is used to specify the optimum, linear, time-invariant equalizer for a system in which data are transmitted by pulse amplitude modulation through a randomly selected channel. The results are applied to the case in which the received pulse is a synchronous demodulation of a passband pulse, with the phase of the demodulating oscillator a random variable and with a random time delay. The results can be applied to automatic equalization. During adjustment intervals, an automatic equalizer can use its channel measurement to specify a set of possible channel measurements instead of using it as a single exact measurement. The parameters of the automatic equalizer are then set in such a way that performance will be satisfactory over the "smeared" measurement set, reducing the sensitivity of the equalizer's performance to measurement error.