A lattice compress-and-forward scheme

We present a nested lattice-code-based strategy that achieves the random-coding based Compress-and-Forward (CF) rate for the three node Gaussian relay channel. To do so, we first outline a lattice-based strategy for the (X + Z₁, X + Z₂) Wyner-Ziv lossy source-coding with side-information problem in Gaussian noise, a re-interpretation of the nested lattice-code-based Gaussian Wyner-Ziv scheme presented by Zamir, Shamai, and Erez. We use the notation (X + Z₁, X + Z₂) Wyner-Ziv to mean that the source is of the form X + Z₁ and the side-information at the receiver is of the form X+Z₂, for independent Gaussian X, Z₁ and Z₂. We use this (X + Z₁, X + Z₂) Wyner-Ziv scheme to implement a “structured” or lattice-code-based CF scheme for the Gaussian relay channel which achieves the same rate as the Cover-El Gamal CF rate achieved by random Gaussian codebooks.