Sum Rate Computation for the Gaussian Many-Help-One Problem

In this paper we consider the separate coding problem for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L+1$</tex> correlated Gaussian memoryless sources. We deal with the case where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L$</tex> separately encoded data of sources work as side information at the decoder for the reconstruction of the remaining source. In this paper we study the case where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L+1$</tex> sources satisfy a kind of tree structure on their correlation. This tree structure of information sources is called the TS condition. In his previous work the author determine the sum rate part of the rate distortion region for the case where information sources satisfy the TS condition. In this paper we derive an explicit recursive formula of this sum rate part. This result completes the author's previous work in which the formula is established for some limited case.