The compute-and-forward framework permits each receiver in a Gaussian network to directly decode a linear combination of the transmitted messages. The resulting linear combinations can then be employed as an end-to-end communication strategy for relaying, interference alignment, and other applications. Recent efforts have demonstrated the advantages of employing unequal powers at the transmitters and decoding more than one linear combination at each receiver. However, neither of these techniques fit naturally within the original formulation of compute-and-forward. This paper proposes an expanded compute-and-forward framework that incorporates both of these possibilities and permits an intuitive interpretation in terms of signal levels. Within this framework, recent achievability and optimality results are unified and generalized.
The original motivation for the compute-and-forward strategy was the possibility of relaying in a multi-hop network while avoiding the harmful effects of interference between users (by decoding linear combinations) as well as noise accumulation (by decoding at every relay). Several works have investigated and improved upon the performance of the original compute-and-forward framework in the context of multi-hop relaying @cite_73 @cite_25 @cite_85 @cite_42 @cite_62 @cite_76 . Our expanded framework can improve performance further by permitting relays to employ unequal powers and decode multiple linear combinations when appropriate. Following the conference publication of this work, Tan al @cite_9 noted that our proposed message representation may be inefficient for multi-hop relaying if each relay naively treats a linear combination over @math as @math information symbols for the next hop. They proposed a lattice-based solution to this issue. It is also possible to resolve this issue directly over the message representation by having each relay only use some of its @math received symbols as information symbols for the next hop. See [Section III.F] ncncarxiv for details.
@cite_62 @cite_9 @cite_85 @cite_42 @cite_76 @cite_73 @cite_25
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In this paper, a point-to-point orthogonal-frequency- division multiplexing (OFDM) system with a decode-and- forward (DF) relay is considered. The transmission consists of two hops. The source transmits in the first hop, and the relay transmits in the second hop. Each hop occupies one time slot. The relay is half-duplex, and capable of decoding the message on a particular subcarrier in one time slot, and re-encoding and forwarding it on a different subcarrier in the next time slot. Thus, each message is transmitted on a pair of subcarriers in two hops. It is assumed that the destination is capable of combining the signals from the source and the relay pertaining to the same message. The goal is to maximize the weighted sum rate of the system by jointly optimizing subcarrier pairing and power allocation on each subcarrier in each hop. The weighting of the rates is to take into account the fact that different subcarriers may carry signals for different services. Both total and individual power constraints for the source and the relay are investigated. For the situations where the relay does not transmit on some subcarriers because doing so does not improve the weighted sum rate, we further allow the source to transmit new messages on these idle subcarriers. To the best of our knowledge, such a joint optimization inclusive of the destination combining has not been discussed in the literature. The problem is first formulated as a mixed integer programming problem. It is then transformed to a convex optimization problem by continuous relaxation, and solved in the dual domain. Based on the optimization results, algorithms to achieve feasible solutions are also proposed. Simulation results show that the proposed algorithms almost achieve the optimal weighted sum rate and outperform the existing methods in various channel conditions. The compute-and-forward framework permits each receiver in a Gaussian network to directly decode a linear combination of the transmitted messages. The resulting linear combinations can then be employed as an end-to-end communication strategy for relaying, interference alignment, and other applications. Recent efforts have demonstrated the advantages of employing unequal powers at the transmitters and decoding more than one linear combination at each receiver. However, neither of these techniques fit naturally within the original formulation of compute-and-forward. This paper proposes an expanded compute-and-forward framework that incorporates both of these possibilities and permits an intuitive interpretation in terms of signal levels. Within this framework, recent achievability and optimality results are unified and generalized. In a distributed communications scenario, a relay applying the compute-and-forward strategy for the real-valued channel model aims to decode an integer linear combination of transmitted messages, a task of very complex nature for which general efficient algorithms for dimension n > 1 have not yet been developed. Nonetheless, the maximum-likelihood decoding metric related to solving for the desired lattice point exhibits interesting properties which lead to partial design criteria for lattice codes in compute-and-forward. This article generalizes maximum-likelihood decoding at the relay to allow for arbitrary real lattice codes at the transmitters, and studies the behavior of the resulting decoding metric using an approximation of the theta series of a lattice, which is itself derived in this article. For the first time, the resulting random sums of lattices over whose points the relay needs to perform a sum are analyzed and furthermore, previous related work is extended to the case of K > 2 transmitters. The specific cases K = 2 and K = 3 are studied empirically. 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-and-Forward (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. Lattice codes used under the Compute-and-Forward paradigm suggest an alternative strategy for the standard Gaussian multiple-access channel (MAC): The receiver successively decodes integer linear combinations of the messages until it can invert and recover all messages. In this paper, this strategy is developed and analyzed. For the two-user MAC, it is shown that without time-sharing, the entire capacity region can be attained with a single-user decoder as soon as the signal-to-noise ratios are above @math . A partial analysis is given for more than two users. Lastly the strategy is extended to the so-called dirty MAC where two interfering signals are known non-causally to the two transmitters in a distributed fashion. Our scheme extends the previously known results and gives new achievable rate regions. Energy harvesting (EH) two-hop communications are considered. The transmitter and the relay harvest energy from the environment and use it exclusively for transmitting data. A data arrival process is assumed at the transmitter. At the relay, a finite data buffer is used to store the received data. We consider a realistic scenario in which the EH nodes have only local causal knowledge, i.e., at any time instant, each EH node only knows the current value of its EH process, channel state, and data arrival process. Our goal is to find a power allocation policy to maximize the throughput at the receiver. We show that because the EH nodes have local causal knowledge, the two-hop communication problem can be separated into two point-to-point problems. Consequently, independent power allocation problems are solved at each EH node. To find the power allocation policy, reinforcement learning with linear function approximation is applied. Moreover, to perform function approximation two feature functions which consider the data arrival process are introduced. Numerical results show that the proposed approach has only a small degradation as compared to the offline optimum case. Furthermore, we show that with the use of the proposed feature functions a better performance is achieved compared to standard approximation techniques. We consider a two-hop relaying network in which orthogonal frequency division multiplexing (OFDM) is employed for the source-to-destination, the source-to-relay and the relay- to-destination links. Amplify-and-forward (AF) and decode-and- forward (DF) policies are both discussed with or without two-hop diversity, respectively, for the relaying network with a sum-power constraint. An unified approach is used for optimal power allocation in the four different relaying scenarios. First, equivalent channel gains are developed for any given subcarrier pair in each scenario, and then optimal power allocation can be obtained by applying the classic water-filling method. Moreover, we provide the proof to the optimality of sorted subcarrier pairing for AF and DF relaying without diversity, which, combined with optimal power allocation, can offer further performance gain. We analyze the asymptotic behavior of compute-and-forward relay networks in the regime of high signal-to-noise ratios. We consider a section of such a network consisting of K transmitters and K relays. The aim of the relays is to reliably decode an invertible function of the messages sent by the transmitters. An upper bound on the capacity of this system can be obtained by allowing full cooperation among the transmitters and among the relays, transforming the network into a K times K multiple-input multiple-output (MIMO) channel. The number of degrees of freedom of compute-and-forward is hence at most K. In this paper, we analyze the degrees of freedom achieved by the lattice coding implementation of compute-and-forward proposed recently by Nazer and Gastpar. We show that this lattice implementation achieves at most 2 (1+1 K) 2 degrees of freedom, thus exhibiting a very different asymptotic behavior than the MIMO upper bound. This raises the question if this gap of the lattice implementation to the MIMO upper bound is inherent to compute-and-forward in general. We answer this question in the negative by proposing a novel compute-and-forward implementation achieving K degrees of freedom. Consider several source nodes communicating across a wireless network to a destination node with the help of several layers of relay nodes. Recent work by Avestimehr has approximated the capacity of this network up to an additive gap. The communication scheme achieving this capacity approximation is based on compress-and-forward, resulting in noise accumulation as the messages traverse the network. As a consequence, the approximation gap increases linearly with the network depth. This paper develops a computation alignment strategy that can approach the capacity of a class of layered, time-varying wireless relay networks up to an approximation gap that is independent of the network depth. This strategy is based on the compute-and-forward framework, which enables relays to decode deterministic functions of the transmitted messages. Alone, compute-and-forward is insufficient to approach the capacity as it incurs a penalty for approximating the wireless channel with complex-valued coefficients by a channel with integer coefficients. Here, this penalty is circumvented by carefully matching channel realizations across time slots to create integer-valued effective channels that are well suited to compute-and-forward. Unlike prior constant gap results, the approximation gap obtained in this paper also depends closely on the fading statistics, which are assumed to be i.i.d. Rayleigh. We consider a network where each route comprises a backlogged source, a number of relays and a destination at a finite distance. The locations of the sources and the relays are realizations of independent Poisson point processes. Given that the nodes observe a TDMA ALOHA MAC protocol, our objective is to determine the number of relays and their placement such that the mean end-to-end delay in a typical route of the network is minimized.We first study an idealistic network model where all routes have the same number of hops, the same distance per hop and their own dedicated relays. Combining tools from queueing theory and stochastic geometry, we provide a precise characterization of the mean end-to-end delay. We find that the delay is minimized if the first hop is much longer than the remaining hops and that the optimal number of hops scales sublinearly with the source-destination distance. Simulating the original network scenario reveals that the analytical results are accurate, provided that the density of the relay process is sufficiently large. We conclude that, given the considered MAC protocol, our analysis provides a delay-minimizing routing strategy for random, multihop networks involving a small number of hops. We analyze the asymptotic behavior of compute-and-forward relay networks in the regime of high signal-to-noise ratios. We consider a section of such a network consisting of K transmitters and K relays. The aim of the relays is to reliably decode an invertible function of the messages sent by the transmitters. An upper bound on the capacity of this system can be obtained by allowing full cooperation among the transmitters and among the relays, transforming the network into a K × K multiple-input multiple-output (MIMO) channel. The number of degrees of freedom of compute-and-forward is hence at most K. In this paper, we analyze the degrees of freedom achieved by the lattice coding implementation of compute-and-forward proposed recently by Nazer and Gastpar. We show that this lattice implementation achieves at most 2 (1+1 K) ≤ 2 degrees of freedom, thus exhibiting a very different asymptotic behavior than the MIMO upper bound. This raises the question if this gap of the lattice implementation to the MIMO upper bound is inherent to compute-and-forward in general. We answer this question in the negative by proposing a novel compute-and-forward implementation achieving K degrees of freedom. We consider a three-node network where a relay node establishes a bidirectional communication between the two other nodes using a spectrally efficient decode-and-forward protocol. In the first phase we have the classical multiple-access channel where both nodes transmit a message to the relay node, which then decodes the messages. In the second phase the relay broadcasts a re-encoded composition of them based on the network coding idea. This means that each receiving node uses the same data stream to infer on its intended message. We characterize the optimal transmit strategy for the broadcast phase where either the relay node or the two other nodes are equipped with multiple antennas. Our main result shows that beamforming into the subspace spanned by the channels is always an optimal transmit strategy for the multiple-input single-output bidirectional broadcast channel. Thereby, it shows that correlation between the channels is advantageous. Moreover, this leads to a parametrization of the optimal transmit strategy which specifies the whole capacity region. In retrospect the results are intuitively clear since the single-beam transmit strategy reflects the single stream processing due to the network coding approach. We consider virtual full-duplex relaying by means of half-duplex relays. In this configuration, each relay stage in a multihop relaying network is formed by at least two relays, used alternatively in transmit and receive modes, such that while one relay transmits its signal to the next stage, the other relay receives a signal from the previous stage. With such a pipelined scheme, the source is active and sends a new information message in each time slot. We consider the achievable rates for various coding schemes and compare them with a cut-set upper bound, which is tight in certain conditions. We show that both lattice-based compute-and-forward (CoF) and quantize-map-and-forward (QMF) yield attractive performance and implementation. In particular, QMF in this context does not require long messages and joint (non-unique) decoding, if the quantization mean-square distortion at the relays is chosen appropriately. In addition, in the multihop case the gap of QMF from the cut-set upper bound grows logarithmically with the number of stages, and not linearly as in the case of noise level quantization. Furthermore, we show that CoF is particularly attractive in the case of multihop relaying, when the channel gains have fluctuations not larger than 3 dB, yielding a rate that does not depend on the number of relaying stages. In particular, we argue that such architecture may be useful for a wireless backhaul with line-of-sight propagation between the relays. This paper studies a new design of a wireless-powered relay network to improve its throughput performance by opportunistic energy harvesting (EH) and information relaying (IR) at the relay node. To enable the relay to implement opportunistic EH and IR, a new protocol, namely, the adaptive harvest-store-forward (AHSF) protocol, is proposed. In the AHSF protocol, the relay is allowed to adaptively switch between an EH mode and a power-splitting (PS)-based IR mode at the beginning of each time frame; moreover, the relay can adaptively determine the amount of energy for IR based on the wireless channel conditions over one finite time horizon, which consists of multiple consecutive time frames. Then, under the proposed protocol, a throughput maximization problem is formulated to optimize the relay operation mode (i.e., EH mode or IR mode) and resource allocation (i.e., time slot, PS ratio, and power allocation) in the IR mode in a finite time horizon. The formulated problem is coupled over time and, thus, is intractable. To address the intractability, dynamic programming is employed to transform the problem into a series of Bellman equations. Based on the Bellman equations, the optimal decision criteria for the relay operation mode is derived, and a time-coupled optimization problem for the resource allocation in the IR mode is addressed by decoupling into two subproblems for two adjacent time frames. By solving the subproblems via convex programming, an optimization policy with causal channel-state information (CSI), which can be implemented online in real time, and an optimization policy with full CSI, which is used to reveal the upper bound of the achieved throughput, are achieved. The simulation results verify that the proposed design can yield significant throughput gains over the existing wireless-powered relaying schemes. Coding strategies that exploit node cooperation are developed for relay networks. Two basic schemes are studied: the relays decode-and-forward the source message to the destination, or they compress-and-forward their channel outputs to the destination. The decode-and-forward scheme is a variant of multihopping, but in addition to having the relays successively decode the message, the transmitters cooperate and each receiver uses several or all of its past channel output blocks to decode. For the compress-and-forward scheme, the relays take advantage of the statistical dependence between their channel outputs and the destination's channel output. The strategies are applied to wireless channels, and it is shown that decode-and-forward achieves the ergodic capacity with phase fading if phase information is available only locally, and if the relays are near the source node. The ergodic capacity coincides with the rate of a distributed antenna array with full cooperation even though the transmitting antennas are not colocated. The capacity results generalize broadly, including to multiantenna transmission with Rayleigh fading, single-bounce fading, certain quasi-static fading problems, cases where partial channel knowledge is available at the transmitters, and cases where local user cooperation is permitted. The results further extend to multisource and multidestination networks such as multiaccess and broadcast relay channels. We consider a multiuser two-way relay network where multiple pairs of users communicate with their pre-assigned partners, using a common intermediate relay node, in a two-phase communication scenario. In this system, a pair of partners transmit to the relay sharing a common spreading signature in the first phase, and the relay broadcasts an estimate of the XORed symbol for each user pair in the second phase employing the relaying scheme termed jointly demodulate-and-XOR forward (JD-XOR-F) in [1]. We investigate the joint power control and receiver optimization problem for this multiuser two-way relay system with JD-XOR-F relaying. We show that the total power optimization problem decouples into two subproblems, one for each phase. We construct the distributed power control and receiver updates in each phase which converge to the corresponding unique optimum. Simulation results are presented to demonstrate the significant power savings of the multiuser two-way relay system with the proposed iterative power control and receiver optimization algorithms, as compared to the designs with a iquestone-wayiquest communication perspective. In this paper, we study resource allocation and relay selection in a decode-and-forward downlink OFDMA cooperative network assuming imperfect channel state information (CSI) at the base station (BS). We assume that due to feedback delay, the BS has only outdated CSI of BS to users and inter-user links. We propose a centralized optimization framework in which the BS takes decisions on the basis of available outdated CSI given the conditional probability distribution of the current CSI. We approach the problem by using dual optimization framework, decompose it into per-subcarrier optimization problem, and develop an optimal resource allocation and relay selection algorithm. The proposed algorithm minimizes the total transmit power of the system by jointly selecting the best relay node for each source-destination pair, and optimally allocating power and subcarriers to the source and relay nodes under constraints on users' conditional expected data rates. The duality gaps of the solution are virtually zero. Furthermore, we show that the per-subcarrier decomposition of the problem reduces the complexity of the solution from exponential to polynomial. Simulation results show that the system performance may be badly affected by feedback delay. Simulation results also show that users' cooperation significantly reduces the power consumption compared to non-cooperative communication. We focus on energy harvesting (EH) two-hop communications since they are the essential building blocks of more complicated multi-hop networks. The scenario consists of three nodes, where an EH transmitter wants to send data to a receiver through an EH relay. The harvested energy is used exclusively for data transmission and we address the problem of how to efficiently use it. As in practical scenarios, we assume only causal knowledge at the EH nodes, i.e., in each time interval, the transmitter and the relay know their own current and past amounts of incoming energy, battery levels, data buffer levels and channel coefficients for their own transmit channels. Our goal is to find transmission policies which aim at maximizing the throughput considering that the EH nodes fully cooperate with each other to exchange their causal knowledge during a signaling phase. We model the problem as a Markov game and propose a multi-agent reinforcement learning algorithm to find the transmission policies. Furthermore, we show the trade-off between the achievable throughput and the signaling required, and provide convergence guarantees for the proposed algorithm. Results show that even when the signaling overhead is taken into account, the proposed algorithm outperforms other approaches that do not consider cooperation among the nodes. We consider a multiuser two-way relay network where multiple pairs of users communicate with their preassigned partners, using a common intermediate relay node, in a two-phase communication scenario employing code division multiple access (CDMA). By taking advantage of the bidirectional communication structure, we first propose that each pair of partners share a common spreading signature and design a jointly demodulate-and-XOR forward (JD-XOR-F) relaying scheme, where all users transmit to the relay simultaneously followed by the relay broadcasting an estimate of the XORed symbol for each user pair. We derive the decision rules and the corresponding bit error rates (BERs) at the relay and at the users' receivers. We then investigate the joint power control and receiver optimization problem for each phase for this multiuser two-way relay network with JD-XOR-F relaying. We solve each optimization problem by constructing the iterative power control and receiver updates that converge to the corresponding unique optimum. Simulation results are presented to demonstrate the performance of the proposed multiuser two-way JD-XOR-F relaying scheme in conjunction with the joint power control and receiver optimization algorithms. Specifically, we observe significant power savings and user capacity improvement with the proposed communication scheme as compared to the designs with a "one-way" communication perspective. We consider multi-hop OFDM relaying systems employing either decode-and-forward (DF) or amplify-and-forward (AF) relaying protocols in this paper. We propose adaptive power allocation (PA) algorithms under joint transmit power constraint at source and relays in order to maximize system capacity with channel state information (CSI) known at all nodes. The paring of subcarriers is also discussed. Simulation results show that the proposed adaptive PA algorithms achieve higher system capacity than uniform PA algorithms. If paring techniques are employed, system capacity can be further enhanced.