## Abstract

In this paper, we consider the rate distortion problem of discrete-time, ergodic, and stationary sources with feed forward at the receiver. We derive a sequence of achievable and computable rates that converge to the feed-forward rate distortion. We show that for ergodic and stationary sources, the rate R-{n}(D)= {{1}\over {n}}\min I({\hat {X}}^{n} \rightarrow{} X^{n}) is achievable for any n, where the minimization is performed over the transition conditioning probability p({\hat {x}}^{n}\vert x^{n}) such that \BBE \left [{d(X^{n}, {\hat {X}}^{n})}\right]\leq D. We also show that the limit of R-{n}(D) exists and is the feed-forward rate distortion. We follow Gallager's proof where there is no feed forward and, with appropriate modification, obtain our result. We provide an algorithm for calculating R-{n}(D) using the alternating minimization procedure and present several numerical examples. We also present a dual form for the optimization of R-{n}(D) and transform it into a geometric programming problem.

Original language | English |
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Article number | 6320694 |

Pages (from-to) | 760-781 |

Number of pages | 22 |

Journal | IEEE Transactions on Information Theory |

Volume | 59 |

Issue number | 2 |

DOIs | |

State | Published - 24 Jan 2013 |

## Keywords

- Alternating minimization procedure
- Blahut-Arimoto (BA) algorithm
- causal conditioning
- concatenating code trees
- directed information
- ergodic and stationary sources
- ergodic modes
- geometric programming (GP)
- rate distortion with feed forward

## ASJC Scopus subject areas

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