Distortion Rate Function of Sub-Nyquist Sampled Gaussian Sources

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

Research output: Contribution to journalArticlepeer-review

48 Scopus citations


The amount of information lost in sub-Nyquist sampling of a continuous-time Gaussian stationary process is quantified. We consider a combined source coding and sub-Nyquist reconstruction problem in which the input to the encoder is a noisy sub-Nyquist sampled version of the analog source. We first derive an expression for the mean squared error in the reconstruction of the process from a noisy and information rate-limited version of its samples. This expression is a function of the sampling frequency and the average number of bits describing each sample. It is given as the sum of two terms: minimum mean square error in estimating the source from its noisy but otherwise fully observed sub-Nyquist samples, and a second term obtained by reverse waterfilling over an average of spectral densities associated with the polyphase components of the source. We extend this result to multi-branch uniform sampling, where the samples are available through a set of parallel channels with a uniform sampler and a pre-sampling filter in each branch. Further optimization to reduce distortion is then performed over the pre-sampling filters, and an optimal set of pre-sampling filters associated with the statistics of the input signal and the sampling frequency is found. This results in an expression for the minimal possible distortion achievable under any analog-to-digital conversion scheme involving uniform sampling and linear filtering. These results thus unify the Shannon-Whittaker-Kotelnikov sampling theorem and Shannon rate-distortion theory for Gaussian sources.

Original languageEnglish (US)
Article number7286824
Pages (from-to)401-429
Number of pages29
JournalIEEE Transactions on Information Theory
Issue number1
StatePublished - Jan 1 2016
Externally publishedYes

All Science Journal Classification (ASJC) codes

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


  • Gaussian processes
  • Rate-distortion
  • Remote source coding
  • Source coding
  • Sub-Nyquist sampling


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