General Message Sets Research Articles

Treating Content Delivery in Multi-Antenna Coded Caching as General Message Sets Transmission: A DoF Region Perspective

Coded caching can create coded multicasting thus significantly accelerates content delivery in broadcast channels with receiver caches. While the original delivery scheme in coded caching multicasts each coded message sequentially, it is not optimal for multiple-input multiple-output (MIMO) broadcast channels. This work aims to investigate the full spatial multiplexing gain in multi-antenna coded caching by transmitting all coded messages concurrently. In specific, we propose to treat the content delivery as the transmission problem with general message sets where all possible messages are present, each with different length and intended for different user set. We first obtain inner and outer bounds of the degrees of freedom (DoF) region of a $K$-user $(M,N)$ broadcast channel with general message sets, with $M$ and $N$ being the number of transmit and receive antennas, respectively. Then for any given set of coded messages, we find its minimum normalized delivery time (NDT) by searching the optimal DoF tuple in the DoF regions. The obtained minimum NDT is optimal at antenna configuration $\frac{M}{N} \in (0,1]\cup[K, \infty)$ and is within a multiplicative gap of $\frac{M}{N}$ to optimum at $\frac{M}{N} \in (1,K)$. Our NDT results can be evaluated for any user demand with both centralized and decentralized cache placement.

Hierarchical Successive Group Decoding Achieves Capacity in the Multiple Access Channel With General Message Sets

We establish that the capacity regions of the discrete memoryless and vector Gaussian multiple access channels with general message sets are achievable with hierarchical successive group decoding, where groups of messages are successively decoded in accordance with the following rule: for each pair of messages known to nested sets of transmitters, the message known to more transmitters is decoded prior to—and not jointly with—the message known to fewer transmitters. This conclusion requires neither rate-splitting nor time-sharing, and is agnostic to whether or not the messages were encoded dependently. For instance, with private messages, hierarchical successive group decoding includes jointly decoding all messages at once, while for degraded messages, it is successive decoding in only one order. A consequence of our main result is that the capacity region of the multiple access channel with general message sets is achievable by time-sharing between only those successive decoding vertices whose decoding order respects the rule described as earlier.

The $K$ -User Vector Gaussian Multiple-Access Channel With General Messages Sets: Capacity, Polymatroidal Structure, and Efficient Computation

The capacity region of the $K$ -user vector Gaussian multiple-access channel with general message sets is established. Furthermore, due to the presence of convexity, both in the discrete and continuous senses, it is shown that this capacity region is efficiently computable. The capacity result is obtained by specializing recent results by the authors for the discrete memoryless multiple-access channel and demonstrating that it suffices to only consider jointly Gaussian input and auxiliary random variables. For this second conclusion, it is shown that jointly Gaussian random variables maximize entropy subject to lattice conditional independence and covariance constraints, a result that is of interest in its own right. Discrete convexity arises since the capacity region is a union of polymatroids . Over each polymatroid, computing the maximal weighted sum rates is simple due to submodularity—a discrete analog of concavity—of the set function associated with the linear inequalities that define the polymatroid. Continuous convexity arises as the set of admissible covariance matrices is convex and the polymatroidal bounds are concave in these covariances.

A Unifying Order-Theoretic Framework for Superposition Coding: Polymatroidal Structure and Optimality in the Multiple-Access Channel With General Message Sets

Two different random coding techniques, both referred to as superposition coding in the literature, have been widely used to obtain inner bounds for the capacity regions of various communication networks. In one, auxiliary codewords are generated independently, and in the other, they are generated in a dependent manner. Using the multiple-access channel with general message sets as a case study, we place the two techniques under a common, order-theoretic framework. The key attribute of this framework is that it explicitly accounts for the acyclic direction and transitivity of the possible auxiliary codeword dependencies, leading to three significant discoveries. First, with respect to a fixed coding distribution, the set of rates achievable by superposition coding with dependent auxiliary codeword generation forms a polymatroid , thereby generalizing the same previously known result for superposition coding with independent auxiliary codewords. Second, we obtain a large class of superposition coding schemes by intermingling dependent and independent auxiliary codeword generation, and demonstrate that the constituent polyhedral achievable rate regions are also polymatroids in each case. The third discovery is that, in the multiple-access channel with general message sets, each associated superposition coding inner bound attains the capacity region. These results demonstrate a tradeoff between the complexity of dependencies in auxiliary codeword generation and that of the function that maps them into transmitted codewords.

Degrees of Freedom of the MIMO $2 \times 2$ Interference Network with General Message Sets

We establish the degrees of freedom (DoF) region for the multiple-input multiple-output (MIMO) two-transmitter, and two-receiver ( $2 \times 2$ ) interference network with a general message set consisting of nine messages, one for each pair of a subset of transmitters at which that message is known and a subset of receivers where that message is desired. An outer bound on the general nine-message $2 \times 2$ interference network is obtained and it is shown to be tight, thereby establishing the DoF region for the most general antenna setting wherein the four nodes have an arbitrary number of antennas each. The DoF-optimal scheme is applicable to the MIMO $2 \times 2$ interference network with constant channel coefficients, and hence, a fortiori , to time/frequency varying channel scenarios. In particular, a linear precoding scheme is proposed that can achieve all the DoF tuples in the DoF region. In it, the precise roles played by transmit zero-forcing, interference alignment, random beamforming, symbol extensions, and asymmetric complex signaling (ACS) are delineated. For instance, we identify a class of antenna settings, in which ACS is required to achieve the fractional-valued corner points. Evidently, the DoF regions of all previously unknown cases of the $2 \times 2$ interference network with a subset of the nine-messages are newly established as special cases of the general result of this paper. For instance, the DoF region of the well-known four-message (and even three-message) MIMO $X$ channel is newly established. This problem had remained open despite previous studies, which had found inner and outer bounds that were not tight in general. Hence, the DoF regions of all special cases obtained from the general DoF region of the nine-message $2\times 2$ interference network of this paper that include at least three of the four $X$ channel messages are new, among many others. This paper sheds light on how the same physical $2 \times 2$ interference network could be used by a suitable choice of message sets to take most advantage of the channel resource in a flexible and efficient manner.

Degrees of Freedom of the Two-User MIMO Broadcast Channel with Private and Common Messages Under Hybrid CSIT Models

We establish the degree of freedom (DoF) regions of the two-user multiple-input multiple-output (MIMO) broadcast channel with a general message set (BC-GM), that includes private and common messages, under fast fading. Nine different channel state knowledge assumptions—collectively known as hybrid channel state information at the transmitter (CSIT) models—are considered wherein the transmitter has either perfect/instantaneous, delayed, or no CSI from each of the two receivers. General antenna configurations are addressed wherein the three terminals have arbitrary numbers of antennas. The DoF regions are established for the five hybrid CSIT models in which either both channels are unknown at the transmitter or each of the two channels is known perfectly or with delay. In the four remaining cases in which exactly one of the two channels is unknown at the transmitter, the DoF regions under the restriction of linear encoding strategies—also known as the linear DoF (LDoF) regions—are established. As the key to the converse proofs of the LDoF region of the MIMO BC-GM under such hybrid CSIT assumptions, we show that, when only considering linear encoding strategies, the CSI from the receiver with more antennas does not help if there is no CSI available from the receiver with fewer antennas. This result is conjectured to be true even without the restriction on the encoding strategies to be linear. If true, the LDoF regions obtained for the four hybrid CSIT cases herein will also be the DoF regions for those cases. Many of the results of this paper when specialized to even the two-message problems are new. These include the LDoF regions, when one of the two channels is not known, of the MIMO BC-GM when specialized to the MIMO BC with private messages. They also include the DoF/LDoF regions for all the hybrid CSIT models obtained by specializing the corresponding regions for the MIMO BC-GM to the case with degraded messages.

Interference Alignment Cancellation in Compounded MIMO Broadcast Channels With General Message Sets

This paper proposes an alignment based interference cancellation scheme for a multi-cell network with multiple-input multiple-output users under a Gaussian Interference compounded broadcast channel scenario. The basic idea is to split up the role of the transmit beamforming matrix according to the type of interference caused by each base station. It will be shown that, for a network of $L$ base stations, the proposed method would result in achieving $(L-1)$ degrees of freedom per user and convert the multi-cell multi-user MIMO channel into a set of single-cell single-user MIMO channels. Finally, we derive the relationship between all the relevant network elements as $N_t=(L-d_s+1)K^{\prime}N_r+1$ , where $N_t$ and $N_r$ are the numbers of transmit and receive antennas, respectively, $L$ and $d_s$ denote the number of cells and the number of data streams transmitted from each base station, and $K^{\prime}$ is the number of users per cell defined under the proposed system model. Hence, we show that the proposed scheme makes it possible to implement interference alignment using less antenna resources.

On the Optimality of Treating Interference as Noise: General Message Sets

In a K-user Gaussian interference channel, it has been shown that if for each user the desired signal strength is no less than the sum of the strengths of the strongest interference from this user and the strongest interference to this user (all values in decibel scale), then treating interference as noise (TIN) is optimal from the perspective of generalized degrees of freedom (GDoF) and achieves the entire channel capacity region to within a constant gap. In this paper, we show that for such TINoptimal interference channels, even if the message set is expanded to include an independent message from each transmitter to each receiver, operating the new channel as the original interference channel and treating interference as noise is still optimal for the sum capacity up to a constant gap. Furthermore, we extend the result to the sum-GDoF optimality of TIN in the general setting of X channels with arbitrary numbers of transmitters and receivers.

Degrees of Freedom Region of a Class of Multisource Gaussian Relay Networks

We study a layered <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> -user <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -hop Gaussian relay network consisting of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Km</i> nodes in the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> layer, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> ≥ 2 and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">KM</i> +1. We observe that the time-varying nature of wireless channels or fading can be exploited to mitigate the interuser interference. The proposed amplify-and-forward relaying scheme exploits such channel variations and works for a wide class of channel distributions including Rayleigh fading. We show a general achievable degrees of freedom (DoF) region for this class of Gaussian relay networks. Specifically, the set of all ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,..., <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dK</i> ) such that <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">di</i> ≤ 1 for all <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> and Σ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> =1 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K di</i> ≤ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Σ</sub> is achievable, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">di</i> is the DoF of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> source-destination pair and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Σ</sub> is the maximum integer such that <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Σ</sub> ≤ min <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> { <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Km</i> } and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> / <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Σ</sub> is an integer. We show that surprisingly the achievable DoF region coincides with the cut-set outer bound if <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> / min <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> { <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Km</i> } is an integer; thus, interference-free communication is possible in terms of DoF. We further characterize an achievable DoF region assuming multi-antenna nodes and general message set, which again coincides with the cut-set outer bound for a certain class of networks.

Guaranteeing the timely transmission of periodic messages with arbitrary deadline constraints using the timed token media access control protocol

Synchronous bandwidth, defined as the maximum time a node is allowed to send its synchronous messages while holding the token, is a sensitive parameter for deadline guarantees of synchronous messages in a timed token network. In order to offer such guarantees, synchronous bandwidth has to be allocated carefully. The allocation of synchronous bandwidths to a general synchronous message set with the minimum message deadline (Dmin) larger than the target token rotation time is studied. A new approach for allocating synchronous bandwidth, which can be easily implemented in practice, is proposed. It is demonstrated, through simulations and numerical examples, that the proposed approach performs better than any of previously proposed local synchronous bandwidth allocation schemes, in terms of its ability in guaranteeing hard real-time traffic.

Guaranteeing hard real-time traffic with legitimately short deadlines with the timed token protocol

Synchronous bandwidth, defined as the maximum time a node is allowed to send its synchronous messages while holding the token, is a sensitive parameter for deadline guarantees of synchronous messages in any timed token network. In order to offer such guarantees, synchronous bandwidth has to be allocated carefully to each individual node. This paper studies the synchronous bandwidth allocated to those synchronous message streams whose deadlines are less than twice the Target Token Rotation Time (TTRT). A new approach for allocating synchronous bandwidth to such streams, which can be used with any previously published local synchronous bandwidth allocation (SBA) for guaranteeing a general synchronous message set with its minimum deadline ( D min) no less than 2 · TTRT, is proposed. The proposed scheme can be applied efficiently in practice to any general synchronous message set with D min > TTRT. Numerical examples are presented to demonstrate the enhanced performance of this new local scheme over any of the previously published local SBA schemes.