Lattice codes are known to achieve capacity in the Gaussian point-to-point channel, achieving the same rates as independent, identically distributed (i.i.d.) random Gaussian codebooks. Lattice codes are also known to outperform random codes for certain channel models that are able to exploit their linearity. In this work, we show that lattice codes may be used to achieve the same performance as known i.i.d. Gaussian random coding techniques for the Gaussian relay channel, and show several examples of how this may be combined with the linearity of lattices codes in multi-source relay networks. In particular, we present a nested lattice list decoding technique, by which, lattice codes are shown to achieve the Decode-and-Forward (DF) rate of single source, single destination Gaussian relay channels with one or more relays. We next present two examples of how this DF scheme may be combined with the linearity of lattice codes to achieve new rate regions which for some channel conditions outperform analogous known Gaussian random coding techniques in multi-source relay channels. That is, we derive a new achievable rate region for the two-way relay channel with direct links and compare it to existing schemes, and derive another achievable rate region for the multiple access relay channel. We furthermore present a lattice Compress-and-Forward (CF) scheme for the Gaussian relay channel which exploits a lattice Wyner-Ziv binning scheme and achieves the same rate as the Cover-El Gamal CF rate evaluated for Gaussian random codes. These results suggest that structured lattice codes may be used to mimic, and sometimes outperform, random Gaussian codes in general Gaussian networks.
Relay channels. Two of our main results are the demonstration that nested lattice codes may be used to achieve the DF and CF rates achieved by random Gaussian codes @cite_27 . For the DF scheme, we mimic the Regular encoding Sliding window decoding DF strategy @cite_12 @cite_2 in which the relay decodes the message of the source, re-encodes it, and then forwards it. The destination combines the information from the source and the relay by intersecting two independent lists of messages obtained from the source and relayed links respectively, over two transmission blocks. We will re-derive the DF rate, but with lattice codes replacing the random i.i.d. Gaussian codes. Of particular importance is constructing and utilizing a lattice version of the list decoder. It is worth mentioning that the concurrent work @cite_59 uses a different lattice coding scheme to achieve the DF rate in the three-node relay channel which does not rely on list decoding but rather on a careful nesting structure of the lattice codes.
@cite_59 @cite_27 @cite_12 @cite_2
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In this paper, a Gaussian two-way relay channel, where two source nodes exchange messages with each other through a relay, is considered. We assume that all nodes operate in full-duplex mode and there is no direct channel between the source nodes. We propose an achievable scheme composed of nested lattice codes for the uplink and structured binning for the downlink. Unlike conventional nested lattice codes, our codes utilize two different shaping lattices for source nodes based on a three-stage lattice partition chain, which is a key ingredient for producing the best gap-to-capacity results to date. Specifically, for all channel parameters, the achievable rate region of our scheme is within 1 2 bit from the capacity region for each user and its sum rate is within log3 2 bit from the sum capacity. It has been conjectured that lattice codes are good for (almost) everything. As an additional bit of evidence for this claim, we offer a few results showing the utility of lattice codes for the AWGN relay channel. We show that the decode-and-forward rates of the relay channel can be achieved using lattice encoding and decoding. We present an encoding decoding technique that uses a doubly-nested lattice code. Encoding is accomplished using a combination of superposition encoding and block Markov encoding, while decoding is accomplished using a strategy reminiscent of Cover and El Gamal's list decoding. Our technique can be extended to a wide variety of relay topologies, including the half-duplex relay channel and the cooperative multiple-access channel. In the two-way relay channel, two sources exchange information with help from a relay. We study a more general relay network with additional pairs of sources that perform pair-wise information exchange. We obtain achievable rate regions for amplify-and-forward and decode-and-forward (DF) relaying schemes based on both Gaussian codes and lattice codes, assuming a two-phase protocol. The lattice-code-based DF scheme is near optimal, in that the achievable rate region lies within one bit of the capacity region for two pairs of information exchange. In this paper, we consider a discrete memoryless state-dependent relay channel with non-causal Channel State Information (CSI). We investigate three different cases in which perfect channel states can be known non-causally: i) only to the source, ii) only to the relay or iii) both to the source and to the relay node. For these three cases we establish lower bounds on the channel capacity (achievable rates) based on using Gel'fand-Pinsker coding at the nodes where the CSI is available and using Compress-and-Forward (CF) strategy at the relay. Furthermore, for the general Gaussian relay channel with additive independent and identically distributed (i.i.d) states and noise, we obtain lower bounds on the capacity for the cases in which CSI is available at the source or at the relay. We also compare our derived bounds with the previously obtained results which were based on Decode-and-Forward (DF) strategy, and we show the cases in which our derived lower bounds outperform DF based bounds, and can achieve the rates close to the upper bound.