A decoding procedure for compress-and-forward and quantize-and-forward relaying

We present a natural and low-complexity technique for achieving the capacity of the Gaussian relay network in the high SNR regime. Specifically, we propose the use of end-to-end structured lattice codes with the amplify-and-forward strategy, where the source uses a nested lattice code to encode the messages and the destination decodes the messages by lattice decoding. All intermediate relays simply amplify and forward the received signals over the network to the destination. We show that the end-to-end lattice-coded amplify-and-forward scheme approaches the capacity of the layered Gaussian relay network in the high SNR regime. Next, we extend our scheme to non-layered Gaussian relay networks under the amplify-and-forward scheme, which can be viewed as a Gaussian intersymbol interference (ISI) channel. Compared with other schemes, our approach is significantly simpler and requires only the end-to-end design of the lattice precoding and decoding. It requires little knowledge of the network topology or the individual channel gains.

The problem of remotely stabilizing a noisy linear time invariant plant over a Gaussian relay network is addressed. The network is comprised of a sensor node, a group of relay nodes and a remote controller. The sensor and the relay nodes operate subject to an average transmit power constraint and they can cooperate to communicate the observations of the plant's state to the remote controller. The communication links between all nodes are modeled as Gaussian channels. Necessary as well as sufficient conditions for mean-square stabilization over various network topologies are derived. The sufficient conditions are in general obtained using delay-free linear policies and the necessary conditions are arrived at using information theoretic tools. Different settings where linear policies are optimal, asymptotically optimal (in certain parameters of the system) and suboptimal have been identified. For the case with noisy multidimensional sources controlled over scalar channels, it is shown that linear time varying policies lead to minimum capacity requirements, meeting the fundamental lower bound. For the case with noiseless sources and parallel channels, nonlinear policies which meet the lower bound have been identified.

We present a nested lattice-code-based strategy that achieves the random-coding based Compress-and-Forward (CF) rate for the three node Gaussian relay channel. To do so, we first outline a lattice-based strategy for the $(X+Z_1,X+Z_2)$ Wyner-Ziv lossy source-coding with side-information problem in Gaussian noise, a re-interpretation of the nested lattice-code-based Gaussian Wyner-Ziv scheme presented by Zamir, Shamai, and Erez. We use the notation $(X+Z_1,X+Z_2)$ Wyner-Ziv to mean that the source is of the form $X+ Z_1$ and the side-information at the receiver is of the form $X+ Z_2$, for independent Gaussian $X, Z_1$ and $Z_2$. We next use this $(X+Z_1,X+Z_2)$ Wyner-Ziv scheme to implement a "structured" or lattice-code-based CF scheme which achieves the classic CF rate for Gaussian relay channels. This suggests that lattice codes may not only be useful in point-to-point single-hop source and channel coding, in multiple access and broadcast channels, but that they may also be useful in larger relay networks. The usage of lattice codes in larger networks is motivated by their structured nature (possibly leading to rate gains) and decoding (relatively simple) being more practically realizable than their random coding based counterparts. We furthermore expect the proposed lattice-based CF scheme to constitute a first step towards a generic structured achievability scheme for networks such as a structured version of the recently introduced "noisy network coding". ; Comment: 7 pages, 2 figures

This paper focuses on the full-duplex Gaussian two-way diamond relay channel (TWDRC), where two users wish to exchange their messages with the help of two relays, when one relay performs a lattice-based Compress-and-Forward (CF) scheme. The other relay performs either Decode-and-Forward (DF), Amplify-and-Forward (AF) or a lattice-based CF scheme. We start by proposing a novel lattice-based CF/CF scheme, where both relays perform CF. As opposed to the single relay case, two main challenges arise: the quantization rate region is constrained by two Multiple Access Channel (MAC) rate regions, and each user has to combine two noisy observations to decode the sent message. These challenges are tackled by exploiting MAC techniques and Maximum Ratio Combining methods, which are used to combine the two noisy observations. We then characterize a general achievable rate region when one relay sends a version of both users' messages (eventually noisy), which encompasses both AF and DF, and the other relay performs CF and derive the corresponding achievable rate regions. Finally, the three proposed relaying schemes are compared via numerical evaluations. Clearly, the position of both relays has an impact on the best relaying strategy.

Short message noisy network coding (SNNC) transmits independent short messages in blocks rather than using long message repetitive encoding. SNNC is shown to achieve the same rates as noisy network coding (NNC) for discrete memoryless networks where each node transmits a multicast message. One advantage of SNNC is that backward decoding may be used which simplifies the analysis and understanding of the achievability proof. The analysis reveals that each decoder may ignore certain other nodes rather than including their message in the decoding procedure. Additionally, SNNC enables early decoding at nodes if the channel quality happens to be good.

Consider a Gaussian relay network where a source node communicates to a destination node with the help of several layers of relays. Recent work has shown that compress-and-forward-based strategies can achieve the capacity of this network within an additive gap. Here, the relays quantize their received signals at the noise level and map them to random Gaussian codebooks. The resultant gap to capacity is independent of the SNRs of the channels in the network and the topology, but is linear in the total number of nodes. In this paper, we provide an improved lower bound on the rate achieved by the compress-and-forward-based strategies (noisy network coding in particular) in arbitrary Gaussian relay networks, whose gap to capacity depends on the network not only through the total number of nodes but also through the degrees of freedom of the min cut of the network. We illustrate that for many networks, this refined lower bound can lead to a better approximation of the capacity. In particular, we demonstrate that it leads to a logarithmic rather than linear capacity gap in the total number of nodes for certain classes of layered networks. The improvement comes from quantizing the received signals of the relays at a resolution decreasing with the total number of nodes in the network. This suggests that the rule-of-thumb in the literature of quantizing the received signals at the noise level can be highly suboptimal.

We study a number of two-user interference networks with multiple-antenna transmitters/receivers (MIMO), transmitter side information in the form of linear combinations (over an appropriate finite-field) of the information messages, and two-hop relaying. We start with a cognitive interference channel (CIC) where one of the transmitters (noncognitive) has knowledge of a rank-1 linear combination of the two information messages, while the other transmitter (cognitive) has access to a rank-2 linear combination of the same messages. This is referred to as the network-coded CIC, since such linear combination may be the consequence of some random linear network coding scheme implemented in the backbone wired network. For such channel we develop an achievable region based on a few novel concepts: precoded compute-and-forward (PCoF) with channel integer alignment (CIA), combined with standard dirty-paper coding. We also develop a capacity region outer bound and find the symmetric generalized degrees of freedom (GDoF) region of the network-coded CIC. Through the GDoF characterization, we show that knowing mixed data (linear combinations of the information messages) provides a unbounded spectral efficiency gain over the classical CIC counterpart, if the ratio (in decibel) of signal-to-noise (SNR) to interference-to-noise is larger than certain threshold. Then, we consider a Gaussian relay network having the two-user MIMO IC as the main building block. We use PCoF with CIA to convert the MIMO IC into a deterministic finite-field IC. Then, we use a linear precoding scheme over the finite-field to eliminate interference in the finite-field domain. Using this unified approach, we derive the symmetric sum rate of the two-user MIMO IC with coordination, cognition, and two-hops. We also provide finite-SNR results (not just DoF) which show that the proposed coding schemes are competitive against state-of-the-art interference avoidance scheme based on orthogonal access, for standard randomly generated Rayleigh fading channels.

This paper investigates the problem of information transmission over the simultaneous relay channel with two users (or two possible channel outcomes) where for one of them the more suitable strategy is Decode-and-Forward (DF) while for the other one is Compress-and-Forward (CF). In this setting, it is assumed that the source wishes to send common and private informations to each of the users (or channel outcomes). This problem is relevant to: (i) the transmission of information over the broadcast relay channel (BRC) with different relaying strategies and (ii) the transmission of information over the conventional relay channel where the source is oblivious to the coding strategy of relay. A novel coding that integrates simultaneously DF and CF schemes is proposed and an inner bound on the capacity region is derived for the case of general memoryless BRCs. As special case, the Gaussian BRC is studied where it is shown that by means of the suggested broadcast coding the common rate can be improved compared to existing strategies. Applications of these results arise in broadcast scenarios with relays or in wireless scenarios where the source does not know whether the relay is collocated with the source or with the destination.

In practical large scale relay networks, it is often an unfeasible task to adapt the strategy of the relays to each source/sink pair communicating through the network. In this paper we consider the strategy of the relays as fixed and chosen in advance. In this context, it is a crucial problem for the source node transmitting information, to know the strategy of the relays in the network. Therefore we optimize the transmit strategy of the source node, assuming only partial relay information knowledge, i.e., channel state information and relay strategies. We also study the impact of getting complete relay information from certain relays only, derive the reduction in the problem uncertainty due to this knowledge and evaluate numerically the performance of the network.

In a relay network with a single source-destination pair, we examine the achievable rates with amplify-and-forward (AF) relaying strategy. Motivated by applications in sensor networks, we consider power-constrained networks with large bandwidth resources and a large number of nodes. We show that the AF strategy does not necessarily benefit from the large available bandwidth. We characterize the optimum AF bandwidth and show that transmitting in the optimum bandwidth allows the network to operate in the linear regime where the achieved rate increases linearly with the available network power. We then present the optimum power allocation among the AF relays. The solution, which can be viewed as a form of maximum ratio combining, indicates the favorable relay positions in the network. Motivated by the large bandwidth resources we further consider a network that uses orthogonal transmissions at the nodes. While the above result for the optimum bandwidth still holds, a different set of relays should optimally be employed. In this case, the relay power solution can be viewed as a form of water-filling. The optimum AF bandwidth and the relay powers can be contrasted to the decode-and-forward (DF) solution. In a network with unconstrained bandwidth, the DF strategy will operate in the wideband regime to minimize the energy cost per information bit (S. Verdu, 2002), (O. Oyman et al., 2004). The wideband DF strategy requires again a different choice of relays; in the case of orthogonal signaling, the data should be sent through only one DF relay (I. Maric et al., 2004). Thus, in general, in a large scale network, a choice of a coding strategy goes beyond determining a coding scheme at a node; it also determines the operating bandwidth as well as the best distribution of the relay power.

In this chapter, the fundamental limits of relay networks are introduced under different channel settings, including Gaussian channels and wireless fading channels. For each scenario, we will study the information-theoretic capacities, diversity-multiplexing tradeoffs, and scaling laws of large networks accordingly. In Section 5.1, we first examine the case of the single-relay Gaussian channel. When the relay is full-duplex, it can be shown that decode-and-forward (DF) and compress-and-forward (CF) schemes achieve capacity under certain scenarios; however, such results are not enjoyed in the more practical half-duplex relay network. In Section 5.2, we take the channel fading into consideration and discuss the fundamental limits under fast and slow fading scenarios.

The short message service (SMS), a bidirectional service for short alphanumeric (up to 160 bytes) messages, is a unique feature of GSM not found in older analog systems. Multireceiver short message traffic has increased amazingly over the years. We propose a multicast SMS architecture over our backbone network. Then, we demonstrate some approaches to maintain the SMS center location information consistent with the home location register and illustrate our scheme for mobile terminated short message transfer. Finally, we simulate and evaluate this architecture. Our simulation indicates that the proposed strong consistency approaches have optimal efficiency by adjusting parameters, and the proposed architecture efficiently provides self-routing capability and multicast functionality in our cellular backbone network. The study also provides a further insight on the issues of multicast wireless cellular backbone network and demonstrates a referable methodology to propose and analyze a multicast cellular backbone network, which can promote the technology of a personal communication network.

We consider probabilistic network codes over Gaussian relay network from coding theoretic perspective. Network coding by relay nodes is performed based on reliable received signals from source nodes. Effect of the signal selection and the relaying error on the performance is addressed by deriving upper bound on the expected error probability at the destination.

Computing optimal half-duplex schedules in Gaussian relay networks is a challenging problem due to the lack of an exact capacity characterization and the large number of transmit-receive configurations that must be considered. We approach the problem using a constant-gap capacity approximation based on the cut-set bound with independent encoding at the nodes. We formulate an optimization problem to obtain the cut-set optimal half-duplex schedule and find that it is hard to solve in general. This is because it involves an exponential number of variables, since the number of ways to assign each node to either transmitter or receiver mode is exponential in the number of nodes. We present a general technique that takes advantage of specific structures in the topology of a given network and allows us to reduce the complexity of this problem. In certain classes of network topologies, our approach yields polynomial time algorithms for finding half-duplex schedules that achieve capacity within a constant gap. We use simulations to show running time improvements over alternative methods and compare the performance of various half-duplex scheduling approaches in different SNR regimes.

In ISIT'12 Brahma, \"{O}zg\"{u}r and Fragouli conjectured that in a half-duplex diamond relay network (a Gaussian noise network without a direct source-destination link and with $N$ non-interfering relays) an approximately optimal relay scheduling (achieving the cut-set upper bound to within a constant gap uniformly over all channel gains) exists with at most $N+1$ active states (only $N+1$ out of the $2^N$ possible relay listen-transmit configurations have a strictly positive probability). Such relay scheduling policies are said to be simple. In ITW'13 we conjectured that simple relay policies are optimal for any half-duplex Gaussian multi-relay network, that is, simple schedules are not a consequence of the diamond network's sparse topology. In this paper we formally prove the conjecture beyond Gaussian networks. In particular, for any memoryless half-duplex $N$-relay network with independent noises and for which independent inputs are approximately optimal in the cut-set upper bound, an optimal schedule exists with at most $N+1$ active states. The key step of our proof is to write the minimum of a submodular function by means of its Lov\'{a}sz extension and use the greedy algorithm for submodular polyhedra to highlight structural properties of the optimal solution. This, together with the saddle-point property of min-max problems and the existence of optimal basic feasible solutions in linear programs, proves the claim.