In this thesis the problem of digital communication through noisy dispersive time-invariant known channels is investigated. The channel model is a known time-invariant linear filter followed by a source of additive noise. The Bayesian demodulator is determined for any digit-by-digit modulation scheme for m-ary digits, any specified set of m-ary digit error costs, any set of known digit a priori probabilities, and any linear noisy dispersive channel that introduces intersymbol interference between a finite number of digits. Decision-theoretic methods are used to determine the Bayesian decision procedure for the case where a finite number of digits are sent and also for the case where the digit sequence is infinite. These general results are applied to the situation where the additive noise is white and gaussian, and where an infinite sequence of digits is sent by transmitting one of m possible waveforms every T seconds. It is required that a final decision on each digit be made within a specified time after that digit is transmitted. The received signal in the infinite past is used by a finite size demodulator to minimize the average risk associated with that decision. The demodulator contains a bank of linear filters, followed by a nonlinear sampled data system with memory. Both the number of filters and the complexity of the sampled-data system are exponentially related to the number of interfering digits. The problem is then made yet more specific by assuming that pulse amplitude modulation is used. In this case the size of the filter bank varies linearly, rather than exponentially, with the amount of intersymbol interference, although the remainder of the demodulator retains the exponential relationship. Knowledge of this demodulator can be used to evaluate any suboptimum demodulator that is designed to overcome additive noise and intersymbol Interference. Qualitative comparisons are made between the Bayesian demodulator and a large number of suboptimum demodulators proposed in the literature. The results of a computer simulation study are used to quantitatively compare the Bayesian demodulator with the optimum linear equalizer and the decision feedback equalizer, two attractive practical demodulators. It is shown that the decision feedback equalizer performs almost as well as the Bayesian demodulator, and much better than the linear equalizer in many cases.