The multiway relay channel is a multicast network where L users exchange data through a relay. In this paper, the capacity region of a class of multiway relay channels is derived, where the channel inputs and outputs take values over finite fields. The cut-set upper bound to the capacity region is derived and is shown to be achievable by our proposed functional-decode-forward coding strategy. More specifically, for the general case where the users can transmit at possibly different rates, functional-decode-forward, combined with rate splitting and joint source-channel decoding, is proved to achieve the capacity region; while for the case where all users transmit at a common rate, rate splitting and joint source-channel decoding are not required to achieve the capacity. That the capacity-achieving coding strategies do not utilize the users' received signals in the users' encoding functions implies that feedback does not increase the capacity region of this class of multiway relay channels.
A channel model similar to the finite field channel considered in this paper is the deterministic (noiseless) channel. In the deterministic model, the channel output is the arithmetic summation of the channel inputs, and there is no noise. The deterministic model has been used to construct coding strategies and to gain insights for more general channels. This approach has been applied to the multiple-access channel @cite_35 , the broadcast channel @cite_35 , the interference channel @cite_22 @cite_23 , the deterministic TWRC @cite_25 , and the deterministic multi-pair TWRC @cite_20 . For the deterministic TWRC and the deterministic multi-pair TWRC, it has been shown that linear coding achieves the capacities, an observation similar to that in this paper for the finite field MWRC.
@cite_35 @cite_22 @cite_23 @cite_25 @cite_20
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We characterize the capacity region to within 2 bits s Hz and the symmetric capacity to within 1 bit s Hz for the two-user Gaussian interference channel (IC) with feedback. We develop achievable schemes and derive a new outer bound to arrive at this conclusion. One consequence of the result is that feedback provides multiplicative gain at high signal-to-noise ratio: the gain becomes arbitrarily large for certain channel parameters. This finding is in contrast to point-to-point and multiple-access channels where feedback provides no gain and only bounded additive gain respectively. The result makes use of a linear deterministic model to provide insights into the Gaussian channel. This deterministic model is a special case of the El Gamal-Costa deterministic model and as a side-generalization, we establish the exact feedback capacity region of this general class of deterministic ICs. An interference alignment example is constructed for the deterministic channel model of the K-user interference channel. The deterministic channel example is then translated into the Gaussian setting, creating the first known example of a fully connected Gaussian K-user interference network with single antenna nodes, real, nonzero and constant channel coefficients, and no propagation delays where the degrees of freedom outerbound is achieved. An analogy is drawn between the propagation delay based interference alignment examples and the deterministic channel model which also allows similar constructions for the two-user X channel as well. 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. SUMMARY This paper explores the two-user Gaussian interference channel through the lens of a natural deterministic channel model. The main result is that the deterministic channel uniformly approximates the Gaussian channel, the capacity regions differing by a universal constant. The problem of finding the capacity of the Gaussian channel to within a constant error is therefore reduced to that of finding the capacity of the far simpler deterministic channel. Thus, the paper provides an alternative derivation of the recent constant gap capacity characterisation of Etkin, Tse and Wang. Additionally, the deterministic model gives significant insight towards the Gaussian channel. Copyright © 2008 John Wiley & Sons, Ltd. 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 propose and design a practical modulation-coded (MC) physical-layer network coding (PNC) scheme to approach the capacity limits of Gaussian and fading two-way relay channels (TWRCs). In the proposed scheme, an irregular repeat–accumulate (IRA) MC over @math with the same random coset is employed at two users, which directly maps the message sequences into coded PAM or QAM symbol sequences. The relay chooses appropriate network coding coefficients and computes the associated finite-field linear combinations of the two users' message sequences using an iterative belief propagation algorithm. For a symmetric Gaussian TWRC, we show that, by introducing the same random coset vector at the two users and a time-varying accumulator in the IRA code, the MC-PNC scheme exhibits symmetry and permutation-invariant properties for the soft information distribution of the network-coded message sequence (NCMS). We explore these properties in analyzing the convergence behavior of the scheme and optimizing the MC to approach the capacity limit of a TWRC. For a block fading TWRC, we present a new MC linear PNC scheme and an algorithm used at the relay for computing the NCMS. We demonstrate that our developed schemes achieve near-capacity performance in both Gaussian and Rayleigh fading TWRCs. For example, our designed codes over GF(7) and GF(3) with a code rate of 3 4 are within 1 and 1.2 dB of the TWRC capacity, respectively. Our method can be regarded as a practical embodiment of the notion of compute-and-forward with a good nested lattice code, and it can be applied to a wide range of network configurations. We study the problem of multicasting over a network of multiple-access channels (MACs). The separation-based solution to this problem is to reduce each MAC to a set of noiseless bit pipes via a channel code and then employ network coding. Sometimes, however, the physical-layer structure of the MAC can be exploited more advantageously. In many cases of interest, the MAC output is a (deterministic) function of its inputs, corrupted by noise. We develop structured codes to exploit the natural function of a MAC to reliably compute functions as part of a network code and show that in many scenarios of interest our scheme outperforms the separation-based solution. If each MAC can be written as a sum over some finite field plus noise, then our achievable rate coincides with the max-flow min-cut bound. In this paper, we present an achievable rate region for double-unicast networks by assuming that the intermediate nodes perform random linear network coding, and the source and sink nodes optimize their strategies to maximize the achievable region. Such a setup can be modeled as a deterministic interference channel, whose capacity region is known. For the particular class of linear deterministic interference channels of our interest, in which the outputs and interference are linear deterministic functions of the inputs, we show that the known capacity region can be achieved by linear strategies. As a result, for a given set of network coding coefficients chosen by the intermediate nodes, the proposed linear precoding and decoding for the source and sink nodes will give the maximum achievable rate region for double-unicast networks. We further derive a suboptimal but easy-to-compute rate region that is independent of the network coding coefficients used at the intermediate nodes, and is instead specified by the min-cuts of the network. It is found that even this suboptimal region is strictly larger than the existing achievable rate regions in the literature. In this paper, we study the impact of different channel output feedback architectures on the capacity of the two-user interference channel. For a two-user interference channel, a feedback link can exist between receivers and transmitters in nine canonical architectures (see Fig. 3 ), ranging from only one feedback link to four feedback links. We derive the exact capacity region for the symmetric deterministic interference channel and the constant-gap capacity region for the symmetric Gaussian interference channel for all of the nine architectures. We show that for a linear deterministic symmetric interference channel, in the weak interference regime, all models of feedback, except the one, which has only one of the receivers feeding back to its own transmitter, have the identical capacity region. When only one of the receivers feeds back to its own transmitter, the capacity region is a strict subset of the capacity region of the rest of the feedback models in the weak interference regime. However, the sum-capacity of all feedback models is identical in the weak interference regime. Moreover, in the strong interference regime, all models of feedback with at least one of the receivers feeding back to its own transmitter have the identical sum-capacity. For the Gaussian interference channel, the results of the linear deterministic model follow, where capacity is replaced with approximate capacity. We present a framework to study linear deterministic interference networks over finite fields. Unlike the popular linear deterministic models introduced to study Gaussian networks, we consider networks where the channel coefficients are general scalars over some extension field Fpm (scalar m-th extensionfield models), m×m diagonal matrices over Fp (m-symbol extension ground-field models), and m×m general non-singular matrices (MIMO ground field models). We use the companion matrix representation of the extension field to convert m-th extension scalar models into MIMO ground-field models where the channel matrices have special algebraic structure. For such models, we consider the 2× 2× 2 topology (two-hops two-flow) and the 3-user interference network topology. We derive achievability results and feasibility conditions for certain schemes based on the Precoding-Based Network Alignment (PBNA) approach, where intermediate nodes use random linear network coding (i.e., propagate random linear combinations of their incoming messages) and non-trivial precoding decoding is performed only at the network edges, at the sources and destinations. Furthermore, we apply this approach to the scalar 2×2×2 complex Gaussian IC with fixed channel coefficients, and show two competitive schemes outperforming other known approaches at any SNR, where we combine finite-field linear precoding decoding with lattice coding and the Compute and Forward approach at the signal level. As a side result, we also show significant advantages of vector linear network coding both in terms of feasibility probability (with random coding coefficients) and in terms of coding latency, with respect to standard scalar linear network coding, in PBNA schemes. In this paper we study the capacity region of the multi-pair bidirectional (or two-way) wireless relay network, in which a relay node facilitates the communication between multiple pairs of users. This network is a generalization of the well known bidirectional relay channel, where we have only one pair of users. We examine this problem in the context of the deterministic channel interaction model, which eliminates the channel noise and allows us to focus on the interaction between signals. We characterize the capacity region of this network when the relay is operating at either full-duplex mode or half-duplex mode (with non adaptive listen-transmit scheduling). In both cases we show that the cut-set upper bound is tight and, quite interestingly, the capacity region is achieved by a simple equation-forwarding strategy. We study the capacity of the full-duplex bidirectional (or two-way) relay channel with two nodes and one relay. The channels in the forward direction are assumed to be different (in general) than the channels in the backward direction, i.e. channel reciprocity is not assumed. We use the recently proposed deterministic approach to capture the essence of the problem and to determine a good transmission and relay strategy for the Gaussian channel. Depending on the ratio of the individual channel gains, we propose to use either a simple amplify-and-forward or a particular superposition coding strategy at the relay. We analyze the achievable rate region and show that the scheme achieves to within 3 bits the cut-set bound for all values of channel gains. We present a deterministic channel model which captures several key features of multiuser wireless communication. We consider a model for a wireless network with nodes connected by such deterministic channels, and present an exact characterization of the end-to-end capacity when there is a single source and a single destination and an arbitrary number of relay nodes. This result is a natural generalization of the max-flow min-cut theorem for wireline networks. Finally to demonstrate the connections between deterministic model and Gaussian model, we look at two examples: the single-relay channel and the diamond network. We show that in each of these two examples, the capacity-achieving scheme in the corresponding deterministic model naturally suggests a scheme in the Gaussian model that is within 1 bit and 2 bit respectively from cut-set upper bound, for all values of the channel gains. This is the first part of a two-part paper; the sequel [1] will focus on the proof of the max-flow min-cut theorem of a class of deterministic networks of which our model is a special case. We study memoryless channels with synchronization errors as defined by a stochastic channel matrix allowing for symbol drop-outs or symbol insertions with particular emphasis on the binary and non-binary deletion channels. We offer a different look at these channels by considering equivalent models by fragmenting the input sequence where different subsequences travel through different channels. The resulting output symbols are combined appropriately to come up with an equivalent input-output representation of the original channel which allows for derivation of new upper bounds on the channel capacity. We consider both random and deterministic types of fragmentation processes applied to binary and nonbinary deletion channels. With two specific applications of this idea, a random fragmentation applied to a binary deletion channel and a deterministic fragmentation process applied to a nonbinary deletion channel, we prove certain inequality relations among the capacities of the original channels and those of the introduced subchannels. The resulting inequalities prove useful in deriving tighter capacity upper bounds for: 1) independent identically distributed (i.i.d.) deletion channels when the deletion probability exceeds 0.65 and 2) nonbinary deletion channels. Some extensions of these results, for instance, to the case of deletion substitution channels are also explored. 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. Abstract Molecular communication enables nanomachines to exchange information with each other by emitting molecules to their surrounding environment. Molecular nanonetworks are envisioned as a number of nanomachines that are deployed in an environment to share specific molecular information such as odor, flavor, or any chemical state. In this paper, using the stochastic model of molecular reactions in biochemical systems, a realistic channel model is first introduced for molecular communication. Then, based on the realistic channel model, we introduce a deterministic capacity expression for point-to-point, broadcast, and multiple-access molecular channels. We also investigate information flow capacity in a molecular nanonetwork for the realization of efficient communication and networking techniques for frontier nanonetwork applications. The results reveal that molecular point-to-point, broadcast, and multiple-access channels are feasible with a satisfactorily high molecular communication rate, which allows molecular information flow in nanonetworks. Furthermore, the derived molecular channel model with input-dependent noise term also reveals that unlike a traditional Gaussian communication channel, achievable capacity is affected by both lower and upper bounds of the channel input in molecular communication channels. This paper investigates the fundamental building blocks of physical-layer network coding (PNC). Most prior work on PNC focused on its application in a simple two-way-relay channel (TWRC) consisting of three nodes only. Studies of the application of PNC in general networks are relatively few. This paper is an attempt to fill this gap. We put forth two ideas: 1) a general network can be decomposed into small building blocks of PNC, referred to as the PNC atoms, for scheduling of PNC transmissions; and 2) we identify nine PNC atoms, with TWRC being one of them. Three major results are as follows. First, using the decomposition framework, the throughput performance of PNC is shown to be significantly better than those of the traditional multi-hop scheme and the conventional network coding scheme. For example, under heavy traffic volume, PNC can achieve 100 throughput gain relative to the traditional multi-hop scheme. Second, PNC decomposition based on a variety of different PNC atoms can yield much better performance than PNC decomposition based on the TWRC atom alone. Third, three out of the nine atoms are most important to good performance. Specifically, the decomposition based on these three atoms is good enough most of the time, and it is not necessary to use the other six atoms. We study the discrete memoryless Z-interference channel where the transmitter of the pair that suffers from interference is cognitive. We first provide an outer bound on the capacity region of this channel. We then show that, when the channel of the transmitter-receiver pair that does not experience interference is deterministic and invertible, our proposed outer bound matches the best known inner bound. The obtained results imply that in the considered channel, superposition encoding at the noncognitive transmitter as well as Gel'fand-Pinsker encoding at the cognitive transmitter is needed in order to minimize the impact of interference. As a byproduct of the obtained capacity region, we obtain the capacity under the generalized Gel'fand-Pinsker setting where a transmitter-receiver pair communicates in the presence of interference noncausally known at the encoder. The capacity region of a class of deterministic discrete memoryless interference channels is established. In this class of channels the outputs Y_ 1 and Y_ 2 are (deterministic) functions of the inputs X_ 1 and X_ 2 such that H(Y_ 1 |X_ 1 )=H(V_ 2 ) and H(Y_ 2 |X_ 2 )=H(V_ l ) for all product probabiliW distributions on X_ 1 X_ 2 , where V_ 1 is a function of X_ 1 and V_ 2 a function of X_ 2 . The capacity, region for the case in which V_ 2 0 and Y_ 1 depends randomly on X_ 1 is also obtained and illustrated with an example. A wireless channel is always composed of both deterministic and stochastic multi-path components. A high Rician K-factor increases the contribution of deterministic channel components, thereby reducing the significance of stochastic parts of a channel. This paper focus at the investigative analysis of K-factor and fading depth to analyze the deterministic behavior of a channel under a certain bandwidth. In this paper, 4 different propagation setups have been studied including a LOS, reflections from black board and wall are considered due to their different surface roughness and double bounce reflections from both surfaces. It has been observed that in all propagation cases, small scale fading depth asymptotically converges towards zero dB whereby K-factor increases with bandwidth. Results also show that the de-polarization of a signal increases its amplitude fading. This effect is much more significant at lower bandwidths but an increase in bandwidth reduces the difference in fade depths between polarized and depolarized signals. These observations lead to a conclusion that channel tend to be more deterministic at higher bandwidths.