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 main concept underlying compute-and-forward is that the superposition property of the wireless medium can be exploited for network coding @cite_31 @cite_71 @cite_43 . This phenomenon was independently and concurrently discovered by @cite_72 @cite_8 @cite_86 , with the latter coining the phrase Subsequent efforts @cite_17 @cite_22 @cite_60 developed lattice coding strategies for communicating the sum of messages to a single receiver. This lead to the compute-and-forward framework @cite_54 for multiple receivers that recover linear combinations of the messages (albeit with equal power constraints, unlike the single receiver framework of @cite_60 ).
@cite_22 @cite_8 @cite_60 @cite_54 @cite_43 @cite_72 @cite_71 @cite_86 @cite_31 @cite_17
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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. 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. We consider a wireless device-to-device (D2D) network where communication is restricted to be single-hop. Users make arbitrary requests from a finite library of files and have pre-cached information on their devices, subject to a per-node storage capacity constraint. A similar problem has already been considered in an infrastructure setting, where all users receive a common multicast (coded) message from a single omniscient server (e.g., a base station having all the files in the library) through a shared bottleneck link. In this paper, we consider a D2D infrastructureless version of the problem. We propose a caching strategy based on deterministic assignment of subpackets of the library files, and a coded delivery strategy where the users send linearly coded messages to each other in order to collectively satisfy their demands. We also consider a random caching strategy, which is more suitable to a fully decentralized implementation. Under certain conditions, both approaches can achieve the information theoretic outer bound within a constant multiplicative factor. In our previous work, we showed that a caching D2D wireless network with one-hop communication, random caching, and uncoded delivery (direct file transmissions) achieves the same throughput scaling law of the infrastructure-based coded multicasting scheme, in the regime of large number of users and files in the library. This shows that the spatial reuse gain of the D2D network is order-equivalent to the coded multicasting gain of single base station transmission. It is, therefore, natural to ask whether these two gains are cumulative, i.e., if a D2D network with both local communication (spatial reuse) and coded multicasting can provide an improved scaling law. Somewhat counterintuitively, we show that these gains do not cumulate (in terms of throughput scaling law). This fact can be explained by noticing that the coded delivery scheme creates messages that are useful to multiple nodes, such that it benefits from broadcasting to as many nodes as possible, while spatial reuse capitalizes on the fact that the communication is local, such that the same time slot can be reused in space across the network. Unfortunately, these two issues are in contrast with each other. We study the approximate message-passing decoder for sparse superposition coding on the additive white Gaussian noise channel and extend our preliminary work. We use heuristic statistical-physics-based tools, such as the cavity and the replica methods, for the statistical analysis of the scheme. While superposition codes asymptotically reach the Shannon capacity, we show that our iterative decoder is limited by a phase transition similar to the one that happens in low density parity check codes. We consider two solutions to this problem, that both allow to reach the Shannon capacity: 1) a power allocation strategy and 2) the use of spatial coupling, a novelty for these codes that appears to be promising. We present, in particular, simulations, suggesting that spatial coupling is more robust and allows for better reconstruction at finite code lengths. Finally, we show empirically that the use of a fast Hadamard-based operator allows for an efficient reconstruction, both in terms of computational time and memory, and the ability to deal with very large messages. 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. A novel construction of lattices is proposed. This construction can be thought of as Construction A with codes that can be represented as the Cartesian product of L linear codes over Fp1 , . . . ,FpL , respectively; hence, is referred to as the product construction. The existence of a sequence of such lattices that are good for quantization and Poltyrev-good under multistage decoding is shown. This family of lattices is then used to generate a sequence of nested lattice codes which allows one to achieve the same computation rate of Nazer and Gastpar for computeand-forward under multistage decoding, which is referred to as lattice-based multistage compute-and-forward. Motivated by the proposed lattice codes, two families of signal constellations are then proposed for the separation-based compute-and-forward framework proposed by together with a multilevel coding multistage decoding scheme tailored specifically for these constellations. This scheme is termed separation-based multistage compute-and-forward and is shown having a complexity of the channel coding dominated by the greatest common divisor of the constellation size (may not be a prime number) instead of the constellation size itself. We consider a wireless Device-to-Device (D2D) network where communication is restricted to be single-hop. Users make arbitrary requests from a finite library of files and have pre-cached information on their devices, subject to a per-node storage capacity constraint. A similar problem has already been considered in an infrastructure'' setting, where all users receive a common multicast (coded) message from a single omniscient server (e.g., a base station having all the files in the library) through a shared bottleneck link. In this work, we consider a D2D infrastructure-less'' version of the problem. We propose a caching strategy based on deterministic assignment of subpackets of the library files, and a coded delivery strategy where the users send linearly coded messages to each other in order to collectively satisfy their demands. We also consider a random caching strategy, which is more suitable to a fully decentralized implementation. Under certain conditions, both approaches can achieve the information theoretic outer bound within a constant multiplicative factor. In our previous work, we showed that a caching D2D wireless network with one-hop communication, random caching, and uncoded delivery, achieves the same throughput scaling law of the infrastructure-based coded multicasting scheme, in the regime of large number of users and files in the library. This shows that the spatial reuse gain of the D2D network is order-equivalent to the coded multicasting gain of single base station transmission. It is therefore natural to ask whether these two gains are cumulative, i.e.,if a D2D network with both local communication (spatial reuse) and coded multicasting can provide an improved scaling law. Somewhat counterintuitively, we show that these gains do not cumulate (in terms of throughput scaling law). In a recent work, Nazer and Gastpar proposed the compute-and-forward strategy as a physical-layer network coding scheme. They described a code structure based on nested lattices whose algebraic structure makes the scheme reliable and efficient. In a more recent paper, Niesen and Whiting revealed a fundamental limitation of the decoder used by Nazer and Gastpar. In this work, we consider maximum-likelihood decoding of compute-and-forward, aiming to overcome its limitation. By examining the decoding metric from the viewpoint of Gaussian measures over lattices, we present a design criterion of the lattice code based on the flatness factor, and propose a new decoding algorithm based on inhomogeneous Diophantine approximation. A novel solution is proposed to undertake a frequent task in wireless networks, which is to let all nodes broadcast information to and receive information from their respective one-hop neighboring nodes. The contribution in this paper is twofold. First, as each neighbor selects one message-bearing codeword from its unique codebook for transmission, it is shown that decoding their messages based on a superposition of those codewords through the multiaccess channel is fundamentally a problem of compressed sensing. In the case where each message is designed to consist of a small number of bits, an iterative algorithm based on belief propagation is developed for efficient decoding. Second, to satisfy the half-duplex constraint, each codeword consists of randomly distributed on-slots and off-slots. A node transmits during its on-slots and listens to its neighbors only through its own off-slots. Over one frame interval, each node broadcasts a message to its neighbors and simultaneously receives the superposition of neighbors' signals through its own off-slots and then decodes all messages. The proposed solution fully exploits the multiaccess nature of the wireless medium and addresses the half-duplex constraint at the fundamental level. In a network consisting of Poisson distributed nodes, numerical results demonstrate that the proposed scheme often achieves several times the rate of slotted ALOHA and CSMA with the same packet error rate. 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 paper explores the core ideas behind this new technique and the possibilities it offers for communication over interference-limited wireless networks. 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 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. In this paper, we consider the use of lattice codes over Eisenstein integers for implementing a compute-and-forward protocol in wireless networks when channel state information is not available at the transmitter. We extend the compute-and-forward paradigm of Nazer and Gastpar to decoding Eisenstein integer combinations of transmitted messages at relays by proving the existence of a sequence of pairs of nested lattices over Eisenstein integers in which the coarse lattice is good for covering and the fine lattice can achieve the Poltyrev limit. Using this result, we show that both the outage performance and error-correcting performance of the nested lattice codebooks over Eisenstein integers surpass those of lattice codebooks over integers considered by Nazer and Gastpar with no additional computational complexity. 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. A wireless network with fading and a single source-destination pair is considered. The information reaches the destination via multiple hops through a sequence of layers of single-antenna relays. At high signal-to-noise ratio (SNR), the simple amplify-and-forward strategy is shown to be optimal in terms of degrees of freedom, because it achieves the degrees of freedom equal to a point-to-point multiple-input multiple-output (MIMO) system. Hence, the lack of coordination in relay nodes does not reduce the achievable degrees of freedom. The performance of this amplify-and-forward strategy degrades with increasing network size. This phenomenon is analyzed by finding the tradeoffs between network size, rate, and diversity. A lower bound on the diversity-multiplexing tradeoff for concatenation of multiple random Gaussian matrices is obtained. Also, it is shown that achievable network size in the outage formulation (short codes) is a lot smaller than the ergodic formulation (long codes). 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. We consider the 2 2 2 multiple-input multipleoutput interference channel where two source-destination pairs wish to communicate with the aid of two intermediate relays. In this paper, we propose a novel lattice strategy called Aligned Precoded Compute-and-Forward (PCoF). This scheme consists of two phases: 1) Using the CoF framework based on signal alignment we transform the Gaussian network into a deterministic finite field network. 2) Using linear precoding (over finite field) we eliminate the end-to-end interference in the finite field domain. Further, we exploit the algebraic structure of lattices to enhance the performance at finite SNR, such that beyond a degree of freedom result (also achievable by other means). We can also show that Aligned PCoF outperforms time sharing in a range of reasonably moderate SNR, with increasing gain as SNR increases. This paper investigates the compress-and-forward scheme for an uplink cloud radio access network (C-RAN) model, where multi-antenna base stations (BSs) are connected to a cloud-computing-based central processor (CP) via capacity-limited fronthaul links. The BSs compress the received signals with Wyner-Ziv coding and send the representation bits to the CP; the CP performs the decoding of all the users’ messages. Under this setup, this paper makes progress toward the optimal structure of the fronthaul compression and CP decoding strategies for the compress-and-forward scheme in the C-RAN. On the CP decoding strategy design, this paper shows that under a sum fronthaul capacity constraint, a generalized successive decoding strategy of the quantization and user message codewords that allows arbitrary interleaved order at the CP achieves the same rate region as the optimal joint decoding. Furthermore, it is shown that a practical strategy of successively decoding the quantization codewords first, then the user messages, achieves the same maximum sum rate as joint decoding under individual fronthaul constraints. On the joint optimization of user transmission and BS quantization strategies, this paper shows that if the input distributions are assumed to be Gaussian, then under joint decoding, the optimal quantization scheme for maximizing the achievable rate region is Gaussian. Moreover, Gaussian input and Gaussian quantization with joint decoding achieve to within a constant gap of the capacity region of the Gaussian multiple-input multiple-output (MIMO) uplink C-RAN model. Finally, this paper addresses the computational aspect of optimizing uplink MIMO C-RAN by showing that under fixed Gaussian input, the sum rate maximization problem over the Gaussian quantization noise covariance matrices can be formulated as convex optimization problems, thereby facilitating its efficient solution. We consider a wireless Device-to-Device (D2D) network where communication is restricted to be single-hop, users make arbitrary requests from a finite library of possible files and user devices cache information in the form of linear combinations of packets from the files in the library (coded caching). We consider the combined effect of coding in the caching and delivery phases, achieving "coded multicast gain", and of spatial reuse due to local short-range D2D communication. Somewhat counterintuitively, we show that the coded multicast gain and the spatial reuse gain do not cumulate, in terms of the throughput scaling laws. In particular, the spatial reuse gain shown in our previous work on uncoded random caching and the coded multicast gain shown in this paper yield the same scaling laws behavior, but no further scaling law gain can be achieved by using both coded caching and D2D spatial reuse. For the additive white Gaussian noise channel with average codeword power constraint, sparse superposition codes are developed. These codes are based on the statistical high-dimensional regression framework. In a previous paper, we investigated decoding using the optimal maximum-likelihood decoding scheme. Here, a fast decoding algorithm, called the adaptive successive decoder, is developed. For any rate R less than the capacity C, communication is shown to be reliable with nearly exponentially small error probability. Specifically, for blocklength n, it is shown that the error probability is exponentially small in n logn.