Editing Wiener process (section)

=== Information rate ===
The [[information rate]] of the Wiener process with respect to the squared error distance, i.e. its quadratic [[rate-distortion function]], is given by <ref>T. Berger, "Information rates of Wiener processes," in IEEE Transactions on Information Theory, vol. 16, no. 2, pp. 134-139, March 1970.
doi: 10.1109/TIT.1970.1054423</ref>
<math display="block">R(D) = \frac{2}{\pi^2 D \ln 2} \approx 0.29D^{-1}.</math>
Therefore, it is impossible to encode <math>\{w_t \}_{t \in [0,T]}</math> using a [[binary code]] of less than <math>T R(D)</math> [[bit]]s and recover it with expected mean squared error less than <math>D</math>. On the other hand, for any <math> \varepsilon>0</math>, there exists <math>T</math> large enough and a [[binary code]] of no more than <math>2^{TR(D)}</math> distinct elements such that the expected [[mean squared error]] in recovering <math>\{w_t \}_{t \in [0,T]}</math> from this code is at most <math>D - \varepsilon</math>.

In many cases, it is impossible to [[binary code|encode]] the Wiener process without [[Sampling (signal processing)|sampling]] it first. When the Wiener process is sampled at intervals <math>T_s</math> before applying a binary code to represent these samples, the optimal trade-off between [[code rate]] <math>R(T_s,D)</math> and expected [[mean square error]] <math>D</math> (in estimating the continuous-time Wiener process) follows the parametric representation <ref>Kipnis, A., Goldsmith, A.J. and Eldar, Y.C., 2019. The distortion-rate function of sampled Wiener processes. IEEE Transactions on Information Theory, 65(1), pp.482-499.</ref>
<math display="block"> R(T_s,D_\theta) =  \frac{T_s}{2} \int_0^1 \log_2^+\left[\frac{S(\varphi)- \frac{1}{6}}{\theta}\right] d\varphi, </math>
<math display="block"> D_\theta =  \frac{T_s}{6} + T_s\int_0^1 \min\left\{S(\varphi)-\frac{1}{6},\theta \right\} d\varphi, </math>
where <math>S(\varphi) = (2 \sin(\pi \varphi /2))^{-2}</math> and <math>\log^+[x] = \max\{0,\log(x)\}</math>. In particular, <math>T_s/6</math> is the mean squared error associated only with the sampling operation (without encoding).