Reliable Physical Layer Network Coding

When two or more users in a wireless network transmit simultaneously, their electromagnetic signals are linearly superimposed on the channel. As a result, a receiver that is interested in one of these signals sees the others as unwanted interference. This property of the wireless medium is typically viewed as a hindrance to reliable communication over a network. However, using a recently developed coding strategy, interference can in fact be harnessed for network coding. In a wired network, (linear) network coding refers to each intermediate node taking its received packets, computing a linear combination over a finite field, and forwarding the outcome towards the destinations. Then, given an appropriate set of linear combinations, a destination can solve for its desired packets. For certain topologies, this strategy can attain significantly higher throughputs over routing-based strategies. Reliable physical layer network coding takes this idea one step further: using judiciously chosen linear error-correcting codes, intermediate nodes in a wireless network can directly recover linear combinations of the packets from the observed noisy superpositions of transmitted signals. Starting with some simple examples, this survey explores the core ideas behind this new technique and the possibilities it offers for communication over interference-limited wireless networks.Comment: 19 pages, 14 figures, survey paper to appear in Proceedings of the IEE

Compute-and-Forward: Harnessing Interference through Structured Codes

Interference is usually viewed as an obstacle to communication in wireless networks. This paper proposes a new strategy, compute-and-forward, that exploits interference to obtain significantly higher rates between users in a network. The key idea is that relays should decode linear functions of transmitted messages according to their observed channel coefficients rather than ignoring the interference as noise. After decoding these linear equations, the relays simply send them towards the destinations, which given enough equations, can recover their desired messages. The underlying codes are based on nested lattices whose algebraic structure ensures that integer combinations of codewords can be decoded reliably. Encoders map messages from a finite field to a lattice and decoders recover equations of lattice points which are then mapped back to equations over the finite field. This scheme is applicable even if the transmitters lack channel state information.Comment: IEEE Trans. Info Theory, to appear. 23 pages, 13 figure

The Multi-way Relay Channel

The multiuser communication channel, in which multiple users exchange information with the help of a relay terminal, termed the multi-way relay channel (mRC), is introduced. In this model, multiple interfering clusters of users communicate simultaneously, where the users within the same cluster wish to exchange messages among themselves. It is assumed that the users cannot receive each other's signals directly, and hence the relay terminal in this model is the enabler of communication. In particular, restricted encoders, which ignore the received channel output and use only the corresponding messages for generating the channel input, are considered. Achievable rate regions and an outer bound are characterized for the Gaussian mRC, and their comparison is presented in terms of exchange rates in a symmetric Gaussian network scenario. It is shown that the compress-and-forward (CF) protocol achieves exchange rates within a constant bit offset of the exchange capacity independent of the power constraints of the terminals in the network. A finite bit gap between the exchange rates achieved by the CF and the amplify-and-forward (AF) protocols is also shown. The two special cases of the mRC, the full data exchange model, in which every user wants to receive messages of all other users, and the pairwise data exchange model which consists of multiple two-way relay channels, are investigated in detail. In particular for the pairwise data exchange model, in addition to the proposed random coding based achievable schemes, a nested lattice coding based scheme is also presented and is shown to achieve exchange rates within a constant bit gap of the exchange capacity.Comment: Revised version of our submission to the Transactions on Information Theor

Achievable Rate Regions for Two-Way Relay Channel using Nested Lattice Coding

This paper studies Gaussian Two-Way Relay Channel where two communication nodes exchange messages with each other via a relay. It is assumed that all nodes operate in half duplex mode without any direct link between the communication nodes. A compress-and-forward relaying strategy using nested lattice codes is first proposed. Then, the proposed scheme is improved by performing a layered coding : a common layer is decoded by both receivers and a refinement layer is recovered only by the receiver which has the best channel conditions. The achievable rates of the new scheme are characterized and are shown to be higher than those provided by the decode-and-forward strategy in some regions.Comment: 27 pages, 13 figures, Submitted to IEEE Transactions on Wireless Communications (October 2013

Weak Secrecy in the Multi-Way Untrusted Relay Channel with Compute-and-Forward

We investigate the problem of secure communications in a Gaussian multi-way relay channel applying the compute-and-forward scheme using nested lattice codes. All nodes employ half-duplex operation and can exchange confidential messages only via an untrusted relay. The relay is assumed to be honest but curious, i.e., an eavesdropper that conforms to the system rules and applies the intended relaying scheme. We start with the general case of the single-input multiple-output (SIMO) L-user multi-way relay channel and provide an achievable secrecy rate region under a weak secrecy criterion. We show that the securely achievable sum rate is equivalent to the difference between the computation rate and the multiple access channel (MAC) capacity. Particularly, we show that all nodes must encode their messages such that the common computation rate tuple falls outside the MAC capacity region of the relay. We provide results for the single-input single-output (SISO) and the multiple-input single-input (MISO) L-user multi-way relay channel as well as the two-way relay channel. We discuss these results and show the dependency between channel realization and achievable secrecy rate. We further compare our result to available results in the literature for different schemes and show that the proposed scheme operates close to the compute-and-forward rate without secrecy.Comment: submitted to JSAC Special Issue on Fundamental Approaches to Network Coding in Wireless Communication System

Lattice Coding for the Two-way Two-relay Channel

Lattice coding techniques may be used to derive achievable rate regions which outperform known independent, identically distributed (i.i.d.) random codes in multi-source relay networks and in particular the two-way relay channel. Gains stem from the ability to decode the sum of codewords (or messages) using lattice codes at higher rates than possible with i.i.d. random codes. Here we develop a novel lattice coding scheme for the Two-way Two-relay Channel: 1 2 3 4, where Node 1 and 4 simultaneously communicate with each other through two relay nodes 2 and 3. Each node only communicates with its neighboring nodes. The key technical contribution is the lattice-based achievability strategy, where each relay is able to remove the noise while decoding the sum of several signals in a Block Markov strategy and then re-encode the signal into another lattice codeword using the so-called "Re-distribution Transform". This allows nodes further down the line to again decode sums of lattice codewords. This transform is central to improving the achievable rates, and ensures that the messages traveling in each of the two directions fully utilize the relay's power, even under asymmetric channel conditions. All decoders are lattice decoders and only a single nested lattice codebook pair is needed. The symmetric rate achieved by the proposed lattice coding scheme is within 0.5 log 3 bit/Hz/s of the symmetric rate capacity.Comment: submitted to IEEE Transactions on Information Theory on December 3, 201