The distortion-rate function of sampled wiener processes

Alon Kipnis, Andrea J. Goldsmith, Yonina C. Eldar

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


We consider the recovery of a continuous-time Wiener process from a quantized or a lossy compressed version of its uniform samples under limited bitrate and sampling rate. We derive a closed-form expression for the optimal tradeoff among sampling rate, bitrate, and quadratic distortion in this setting. This expression is given in terms of a reverse waterfilling formula over the asymptotic spectral distribution of a sequence of finite-rank operators associated with the optimal estimator of the Wiener process from its samples. We show that the ratio between this expression and the standard distortion rate function of the Wiener process, describing the optimal tradeoff between bitrate and distortion without a sampling constraint, is only a function of the number of bits per sample. We also consider a sub-optimal lossy compression scheme in which the continuous-time process is estimated from the output of an encoder that is optimal with respect to the discrete-time samples. We show that the latter is strictly greater than the distortion under optimal encoding but only by at most 3%. We, therefore, conclude that near optimal performance is attained even if the encoder is unaware of the continuous-time origin of the samples.

Original languageEnglish (US)
Article number8513858
Pages (from-to)482-499
Number of pages18
JournalIEEE Transactions on Information Theory
Issue number1
StatePublished - Jan 2019
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


  • Brownian bridge
  • Brownian motion
  • Wiener process
  • analog to digital conversion
  • compress and estimate
  • lossy compression
  • remote source coding
  • sampling


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