Compute-and-forward (CF) harnesses interference in a wireless network by allowing relays to compute combinations of source messages. The computed message combinations at relays are correlated, and so directly forwarding these combinations to a destination generally incurs information redundancy and spectrum inefficiency. To address this issue, we propose a novel relay strategy, termed compute-compress-and-forward (CCF). In CCF, source messages are encoded using nested lattice codes constructed on a chain of nested coding and shaping lattices. A key difference of CCF from CF is an extra compressing stage inserted in between the computing and forwarding stages of a relay, so as to reduce the forwarding information rate of the relay. The compressing stage at each relay consists of two operations: first to quantize the computed message combination on an appropriately chosen lattice (referred to as a quantization lattice), and then to take modulo on another lattice (referred to as a modulo lattice). We study the design of the quantization and modulo lattices and propose successive recovering algorithms to ensure the recoverability of source messages at destination. Based on that, we formulate a sum-rate maximization problem that is in general an NP-hard mixed integer program. A low-complexity algorithm is proposed to give a suboptimal solution. Numerical results are presented to demonstrate the superiority of CCF over the existing CF schemes.Index Terms-Compute-and-forward, compute-compress-andforward, modulo, nested lattice codes, physical-layer network coding, quantization, wireless relaying. 1053-587X

Physical-layer network coding (PNC) is a communications paradigm that exploits overlapped transmissions to boost the throughput of wireless relay networks. A high point of PNC research was a theoretical proof that PNC that makes use of nested lattice codes could approach the information-theoretic capacity of a two-way relay network (TWRN), where two end nodes communicate via a relay node. The capacity cannot be achieved by conventional methods of time-division or straightforward network coding. Many practical challenges, however, remain to be addressed before the full potential of lattice-coded PNC can be realized. Two major challenges are: (1) for good performance in lattice-coded PNC, channels of simultaneously transmitting nodes must be aligned; (2) for lattice-coded PNC to be practical, the complexity of lattice encoding at the transmitters and lattice decoding at the receiver must be reduced. We address these challenges and implement a first lattice-coded PNC system on a software-defined radio (SDR) platform. Specifically, we design and implement a low-overhead channel precoding system that accurately aligns the channels of distributed nodes. In our implementation, the nodes only use low-cost temperature-compensated oscillators (TCXO)-a consequent challenge is that the channel alignment must be done more frequently and more accurately compared with the use of expensive oscillators. The low overhead and accurate channel alignment are achieved by (1) a channel precoding system implemented over FPGA to realize fast feedback of channel state information; (2) a highly-accurate carrier frequency offset (CFO) estimation method; and (3) a partial-feedback channel estimation method that significantly reduces the amount of feedback information from the receiver to the transmitters for channel precoding at the transmitters. To reduce lattice encoding and decoding complexities, we adapt the low-density lattice code (LDLC) for use in PNC systems. Experiments show that our implemented lattice-coded PNC achieves better bit error rate performance compared with time-division and straightforward network coding systems. It also has good throughput performance in mobile non-LoS scenarios.

We consider a two-hop relay model in which multiple sources communicate with a single destination via multiple distributed relays. We propose an asymmetric Compute-andForward (CoF) scheme that allows lattice coding with different coarse and fine lattices at the sources. The proposed scheme is motivated by the observation that, in an asymmetric CoF system, a higher transmission power at a source does not necessarily translate to a higher achievable information rate. We show that significant performance enhancement can be achieved by optimizing the transmission powers of the sources below their respective budgets. Further, the asymmetric construction of lattice coding allows the relays to conduct different modulo operations to reduce their forwarding rates, thereby supporting higher rates at the sources. However, modulo operations in general incur information loss, and so need to be carefully designed to ensure that the destination can successfully recover the source messages. As such, we propose a novel successive recovering algorithm for decoding at the destination, and establish sufficient conditions to guarantee successful recovery. Numerical results are provided to verify the superiority of our proposed scheme over other schemes.

Compute-and-forward (CF) harnesses interference in wireless communications by exploiting structured coding. The key idea of CF is to compute integer combinations of codewords from multiple source nodes, rather than to decode individual codewords by treating others as noise. Compute-compress-and-forward (CCF) can further enhance the network performance by introducing compression operations at receivers. In this paper, we develop a more general compression framework, termed generalized computecompress-and-forward (GCCF), where the compression function involves multiple quantization-and-modulo lattice operations. We show that GCCF achieves a broader compression rate region than CCF. We also compare our compression rate region with the fundamental Slepian-Wolf (SW) region. We show that GCCF is optimal in the sense of achieving the minimum total compression rate. We also establish the criteria under which GCCF achieves the SW region. In addition, we consider a two-hop relay network employing the GCCF scheme. We formulate a sum-rate maximization problem and develop an approximate algorithm to solve the problem. Numerical results are presented to demonstrate the performance superiority of GCCF over CCF and other schemes.

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