Feedback Capacity of MIMO Gaussian Channels

Oron Sabag*, Victoria Kostina, Babak Hassibi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Finding a computable expression for the feedback capacity of channels with colored Gaussian, additive noise is a long standing open problem. In this paper, we solve this problem in the scenario where the channel has multiple inputs and multiple outputs (MIMO) and the noise process is generated as the output of a time-invariant state-space model. Our main result is a computable expression for the feedback capacity in terms of a finite-dimensional convex optimization. The solution to the feedback capacity problem is obtained by formulating the finite-block counterpart of the capacity problem as a sequential convex optimization problem which leads in turn to a single-letter upper bound. This converse derivation integrates tools and ideas from information theory, control, filtering and convex optimization. A tight lower bound is realized by optimizing over a family of time-invariant policies thus showing that time-invariant inputs are optimal even when the noise process may not be stationary. The optimal time-invariant policy is used to construct a capacity-achieving and simple coding scheme for scalar channels, and its analysis reveals an interesting relation between a smoothing problem and the feedback capacity expression.

Original languageAmerican English
Pages (from-to)6121-6136
Number of pages16
JournalIEEE Transactions on Information Theory
Issue number10
StatePublished - 1 Oct 2023

Bibliographical note

Publisher Copyright:
© 1963-2012 IEEE.


  • Multiple inputs and multiple outputs (MIMO)
  • additive white Gaussian noise (AWGN)
  • channel capacity
  • colored noise
  • convex optimization
  • feedback


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