Information rates of Wiener processes

Abstract

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, for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0 \leq D \leq \sigma^2 /4</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(D)</tex> for the Wiener process is identical to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(D)</tex> for the sequence of zero mean independent normally distributed increments of variance \sigma^2 whose partial sums form the Wiener process. In the time continuous case, we derive the explicit formula <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(D) = 2 \sigma^2 / ( \pi^2 D)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma^2</tex> is the variance of the increment daring a one-second interval. The resuiting <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(D)</tex> 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 our <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(D)</tex> curves are physically significant for information transmission purposes even though Wiener processes are nonstationary.