On Maxentropic Discrete Stationary Processes

Let X denote a stationary time-discrete random process whose variables, · · · , X-x , X0 , X, , · · · , take values from t h e finite set of real numbers {xi , x 2 , · · · , Let X have mean zero and a given covariance sequence p k = EXjXi+k , j, k = 0, ± 1 , ± 2 , ··· . W h a t is the largest entropy t h a t X can have and what is the probability structure of this most random process of given second moments? Our interest in this question arose from the consideration of certain pulse-type communication systems used for the transmission of digital data. In such systems, a customer provides d a t a in t h e form of an infinite sequence of binary digits t h a t can be represented by a stationary process Y whose variables, · · · , , Y0 , Yy , · · · , are independent random variables each taking values zero and one with equal probabilities. An encoder transforms Y into a K-level process X of the sort described above, whose random variables are then used as amplitudes for successive pulses of a train. T h e transmitted signal is thus of the form s(t) = Z Xng{t -nT + d) (1)