Information rates of Wiener processes

Rate distortion functions are calculated for time discrete and time continuous Wiener processes with respect to the mean squared error criterion. In the time discrete case, we find the interesting result that, for0 ≤ D ≤ σ^2 /4,R(D)for the Wiener process is identical toR(D)for the sequence of zero mean independent normally distributed increments of variance σ^2 whose partial sums form the Wiener process. In the time continuous case, we derive the explicit formulaR(D) = 2 σ^2 / ( π^2 D), whereσ^2is the variance of the increment daring a one-second interval. The resuitingR(D)curves are compared with the performance of an optimum integrating delta modulation system. Finally, by incorporating a delta modulation scheme in the random coding argument, we prove a source coding theorem that guarantees ourR(D)curves are physically significant for information transmission purposes even though Wiener processes are nonstationary.