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 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.
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 introduce the real, discrete-time Gaussian parallel relay network. This simple network is theoretically important in the context of network information theory. We present upper and lower bounds to capacity and explain where they coincide. 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. The problem of designing physical-layer network coding (PNC) schemes via nested lattices is considered. Building on the compute-and-forward (C&F) relaying strategy of Nazer and Gastpar, who demonstrated its asymptotic gain using information-theoretic tools, an algebraic approach is taken to show its potential in practical, nonasymptotic, settings. A general framework is developed for studying nested-lattice-based PNC schemes-called lattice network coding (LNC) schemes for short-by making a direct connection between C&F and module theory. In particular, a generic LNC scheme is presented that makes no assumptions on the underlying nested lattice code. C&F is reinterpreted in this framework, and several generalized constructions of LNC schemes are given. The generic LNC scheme naturally leads to a linear network coding channel over modules, based on which noncoherent network coding can be achieved. Next, performance complexity tradeoffs of LNC schemes are studied, with a particular focus on hypercube-shaped LNC schemes. The error probability of this class of LNC schemes is largely determined by the minimum intercoset distances of the underlying nested lattice code. Several illustrative hypercube-shaped LNC schemes are designed based on Constructions A and D, showing that nominal coding gains of 3 to 7.5 dB can be obtained with reasonable decoding complexity. Finally, the possibility of decoding multiple linear combinations is considered and related to the shortest independent vectors problem. A notion of dominant solutions is developed together with a suitable lattice-reduction-based algorithm. In this paper, we study a Gaussian relay-interference network, in which relay (helper) nodes are to facilitate competing information flows between different source-destination pairs. We focus on two-stage relay-interference networks where there are weak cross links, causing the networks to behave like a chain of Z Gaussian channels. Our main result is an approximate characterization of the capacity region for such ZZ and ZS networks. We propose a new interference management scheme, termed interference neutralization, which is implemented using structured lattice codes. This scheme allows for over-the-air interference removal, without the transmitters having complete access the interfering signals. This scheme in conjunction a new network decomposition technique provides the approximate characterization. Our analysis of these Gaussian networks is based on insights gained from an exact characterization of the corresponding linear deterministic model. A relay channel consists of an input x_ l , a relay output y_ 1 , a channel output y , and a relay sender x_ 2 (whose transmission is allowed to depend on the past symbols y_ 1 . The dependence of the received symbols upon the inputs is given by p(y,y_ 1 |x_ 1 ,x_ 2 ) . The channel is assumed to be memoryless. In this paper the following capacity theorems are proved. 1)If y is a degraded form of y_ 1 , then C : = : !_ p(x_ 1 ,x_ 2 ) , I(X_ 1 ,X_ 2 ;Y), I(X_ 1 ; Y_ 1 |X_ 2 ) . 2)If y_ 1 is a degraded form of y , then C : = : !_ p(x_ 1 ) x_ 2 I(X_ 1 ;Y|x_ 2 ) . 3)If p(y,y_ 1 |x_ 1 ,x_ 2 ) is an arbitrary relay channel with feedback from (y,y_ 1 ) to both x_ 1 x_ 2 , then C : = : p(x_ 1 ,x_ 2 ) , I(X_ 1 ,X_ 2 ;Y),I ,(X_ 1 ;Y,Y_ 1 |X_ 2 ) . 4)For a general relay channel, C : : p(x_ 1 ,x_ 2 ) , I ,(X_ 1 , X_ 2 ;Y),I(X_ 1 ;Y,Y_ 1 |X_ 2 ) . Superposition block Markov encoding is used to show achievability of C , and converses are established. The capacities of the Gaussian relay channel and certain discrete relay channels are evaluated. Finally, an achievable lower bound to the capacity of the general relay channel is established.  Recently, Etkin, Tse, and Wang found the capacity region of the two-user Gaussian interference channel to within 1 bit s Hz. A natural goal is to apply this approach to the Gaussian interference channel with an arbitrary number of users. We make progress towards this goal by finding the capacity region of the many-to-one and one-to-many Gaussian interference channels to within a constant number of bits. The result makes use of a deterministic model to provide insight into the Gaussian channel. The deterministic model makes explicit the dimension of signal level. A central theme emerges: the use of lattice codes for alignment of interfering signals on the signal level. We consider the Gaussian “diamond” or parallel relay network, in which a source node transmits a message to a destination node with the help of N relays. Even for the symmetric setting, in which the channel gains to the relays are identical and the channel gains from the relays are identical, the capacity of this channel is unknown in general. The best known capacity approximation is up to an additive gap of order N bits and up to a multiplicative gap of order N2, with both gaps independent of the channel gains. In this paper, we approximate the capacity of the symmetric Gaussian N-relay diamond network up to an additive gap of 1.8 bits and up to a multiplicative gap of a factor 14. Both gaps are independent of the channel gains and, unlike the best previously known result, are also independent of the number of relays N in the network. Achievability is based on bursty amplify-and-forward, showing that this simple scheme is uniformly approximately optimal, both in the low-rate as well as in the high-rate regimes. The upper bound on capacity is based on a careful evaluation of the cut-set bound. We also present approximation results for the asymmetric Gaussian N-relay diamond network. In particular, we show that bursty amplify-and-forward combined with optimal relay selection achieves a rate within a factor O(log4(N)) of capacity with preconstant in the order notation independent of the channel gains. In this work, new achievable rates are derived for the uplink channel of a cellular network with joint multicell processing (MCP), where unlike previous results, the ideal backhaul network has finite capacity per cell. Namely, the cell sites are linked to the central joint processor via lossless links with finite capacity. The new rates are based on compress-and-forward schemes combined with local decoding. Further, the cellular network is abstracted by symmetric models, which render analytical treatment plausible. For this family of idealistic models, achievable rates are presented for both Gaussian and fading channels. The rates are given in closed form for the classical Wyner model and the soft-handover model. These rates are then demonstrated to be rather close to the optimal unlimited backhaul joint processing rates, even for modest backhaul capacities, supporting the potential gain offered by the joint MCP approach. Particular attention is also given to the low-signal-to-noise ratio (SNR) characterization of these rates through which the effect of the limited backhaul network is explicitly revealed. In addition, the rate at which the backhaul capacity should scale in order to maintain the original high-SNR characterization of an unlimited backhaul capacity system is found. The capacity of a particular large Gaussian relay network is determined in the limit as the number of relays tends to infinity. Upper bounds are derived from cut-set arguments, and lower bounds follow from an argument involving uncoded transmission. It is shown that in cases of interest, upper and lower bounds coincide in the limit as the number of relays tends to infinity. Hence, this paper provides a new example where a simple cut-set upper bound is achievable, and one more example where uncoded transmission achieves optimal performance. The findings are illustrated by geometric interpretations. The techniques developed in this paper are then applied to a sensor network situation. This is a network joint source-channel coding problem, and it is well known that the source-channel separation theorem does not extend to this case. The present paper extends this insight by providing an example where separating source from channel coding does not only lead to suboptimal performance-it leads to an exponential penalty in performance scaling behavior (as a function of the number of nodes). Finally, the techniques developed in this paper are extended to include certain models of ad hoc wireless networks, where a capacity scaling law can be established: When all nodes act purely as relays for a single source-destination pair, capacity grows with the logarithm of the number of nodes. 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. A noisy network coding scheme for communicating messages between multiple sources and destinations over a general noisy network is presented. For multi-message multicast networks, the scheme naturally generalizes network coding over noiseless networks by Ahlswede, Cai, Li, and Yeung, and compress-forward coding for the relay channel by Cover and El Gamal to discrete memoryless and Gaussian networks. The scheme also extends the results on coding for wireless relay networks and deterministic networks by Avestimehr, Diggavi, and Tse, and coding for wireless erasure networks by Dana, Gowaikar, Palanki, Hassibi, and Effros. The scheme involves lossy compression by the relay as in the compress-forward coding scheme for the relay channel. However, unlike previous compress-forward schemes in which independent messages are sent over multiple blocks, the same message is sent multiple times using independent codebooks as in the network coding scheme for cyclic networks. Furthermore, the relays do not use Wyner-Ziv binning as in previous compress-forward schemes, and each decoder performs simultaneous decoding of the received signals from all the blocks without uniquely decoding the compression indices. A consequence of this new scheme is that achievability is proved simply and more generally without resorting to time expansion to extend results for acyclic networks to networks with cycles. The noisy network coding scheme is then extended to general multi-message networks by combining it with decoding techniques for the interference channel. For the Gaussian multicast network, noisy network coding improves the previously established gap to the cutset bound. We also demonstrate through two popular Gaussian network examples that noisy network coding can outperform conventional compress-forward, amplify-forward, and hash-forward coding schemes.  A centrifugating device for biological liquids, e.g. blood, in which a rotatable container carries a specially shaped seal that surrounds and bears on a fixed assembly with a minimum area of interface between the fixed and rotating parts. This seal is disposed outside the path of the liquid to be treated. The fixed assembly, in turn, is releasably carried by a bracket, the bracket being selectively longitudinally extensible as well as selectively adjustably swingable about a vertical axis of oscillation eccentric to the centrifuge, thereby to permit exact positioning of the fixed assembly coaxially of the rotatable container. The parts are so simple and inexpensive in construction that at least some of them can be used once and thrown away. Moreover, the fixed assembly is easily insertable in sealed relationship in any of a variety of containers, by the simplest of manual assembly and disassembly operations. The multicast capacity is determined for networks that have deterministic channels with broadcasting at the transmitters and no interference at the receivers. The multicast capacity is shown to have a cut-set interpretation. It is further shown that one cannot always layer channel and network coding in such networks. The proof of the latter result partially generalizes to discrete memoryless broadcast channels and is used to bound the common rate for problems where one achieves a cut bound on throughput. 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. The capacity of the two-user Gaussian interference channel has been open for 30 years. The understanding on this problem has been limited. The best known achievable region is due to Han and Kobayashi but its characterization is very complicated. It is also not known how tight the existing outer bounds are. In this work, we show that the existing outer bounds can in fact be arbitrarily loose in some parameter ranges, and by deriving new outer bounds, we show that a very simple and explicit Han-Kobayashi type scheme can achieve to within a single bit per second per hertz (bit s Hz) of the capacity for all values of the channel parameters. We also show that the scheme is asymptotically optimal at certain high signal-to-noise ratio (SNR) regimes. Using our results, we provide a natural generalization of the point-to-point classical notion of degrees of freedom to interference-limited scenarios. 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. United States. Defense Advanced Research Projects Agency. Information Theory for Mobile Ad-Hoc Networks Program (grant 1105741-1-TFIND)