A Proof of the Weak Simplex Conjecture

We solve a long-standing open problem about the optimal codebook structure of codes in $n$-dimensional Euclidean space that consist of $n+1$ codewords subject to a codeword energy constraint, in terms of minimizing the average decoding error probability. The conjecture states that optimal codebooks are formed by the $n+1$ vertices of a regular simplex (the $n$-dimensional generalization of a regular tetrahedron) inscribed in the unit sphere. A self-contained proof of this conjecture is provided that hinges on symmetry arguments and leverages a relaxation approach that consists in jointly optimizing the codebook and the decision regions, rather than the codeword locations alone.Comment: 6 pages, submitted to a conference for peer revie

A framework for joint design of pilot sequence and linear precoder

Most performance measures of pilot-assisted multiple-input multiple-output systems are functions of the linear precoder and the pilot sequence. A framework for the optimization of these two parameters is proposed, based on a matrix-valued generalization of the concept of effective signal-to-noise ratio (SNR) introduced in the famous work by Hassibi and Hochwald. Our framework aims to extend the work of Hassibi and Hochwald by allowing for transmit-side fading correlations, and by considering a class of utility functions of said effective SNR matrix, most notably including the well-known capacity lower bound used by Hassibi and Hochwald. We tackle the joint optimization problem by recasting the optimization of the precoder (resp. pilot sequence) subject to a fixed pilot sequence (resp. precoder) into a convex problem. Furthermore, we prove that joint optimality requires that the eigenbases of the precoder and pilot sequence be both aligned along the eigenbasis of the channel correlation matrix. We finally describe how to wrap all studied subproblems into an iteration that converges to a local optimum of the joint optimization.Peer ReviewedPostprint (author's final draft

A Rate-Splitting Approach to Fading Channels with Imperfect Channel-State Information

As shown by M\'edard, the capacity of fading channels with imperfect channel-state information (CSI) can be lower-bounded by assuming a Gaussian channel input $X$ with power $P$ and by upper-bounding the conditional entropy $h(X|Y,\hat{H})$ by the entropy of a Gaussian random variable with variance equal to the linear minimum mean-square error in estimating $X$ from $(Y,\hat{H})$. We demonstrate that, using a rate-splitting approach, this lower bound can be sharpened: by expressing the Gaussian input $X$ as the sum of two independent Gaussian variables $X_1$ and $X_2$ and by applying M\'edard's lower bound first to bound the mutual information between $X_1$ and $Y$ while treating $X_2$ as noise, and by applying it a second time to the mutual information between $X_2$ and $Y$ while assuming $X_1$ to be known, we obtain a capacity lower bound that is strictly larger than M\'edard's lower bound. We then generalize this approach to an arbitrary number $L$ of layers, where $X$ is expressed as the sum of $L$ independent Gaussian random variables of respective variances $P_{\ell}$, $\ell = 1,\dotsc,L$ summing up to $P$. Among all such rate-splitting bounds, we determine the supremum over power allocations $P_\ell$ and total number of layers $L$. This supremum is achieved for $L\to\infty$ and gives rise to an analytically expressible capacity lower bound. For Gaussian fading, this novel bound is shown to converge to the Gaussian-input mutual information as the signal-to-noise ratio (SNR) grows, provided that the variance of the channel estimation error $H-\hat{H}$ tends to zero as the SNR tends to infinity.Comment: 28 pages, 8 figures, submitted to IEEE Transactions on Information Theory. Revised according to first round of review

Communication rates for fading channels with imperfect channel-state information

The present thesis studies information rates for reliable transmission of information over fading channels in the realistic situation where the receiver has only imperfect channel-state knowledge. Of particular interest are analytical expressions of achievable transmission rates under imperfect and no CSI, that is, lower bounds on the mutual information and on the Shannon capacity. A well-known mutual information lower bound for Gaussian codebooks is obtained when conflating the additive (thermal) noise with the multiplicative noise due to the imperfections of the CSIR into a single effective noise term, and then assuming that this term is independent Gaussian. This so-called worst-case-noise approach allows to derive a strikingly simple and well-known lower bound on the mutual information of the channel. A first part of this thesis proposes a simple way to improve this worst-case-noise bound by means of a rate-splitting approach: by expressing the Gaussian input as a sum of several independent Gaussian inputs, and by assuming that the receiver performs successive decoding of the corresponding information streams, we show how to derive a larger mutual information lower bound. On channels with a single transmit antenna, the optimal allocation of transmit power across the different inputs is found to be approached as the number of inputs (so-called layers) tends to infinity, and the power assigned to each layer tends to zero. This infinite-layering limit gives rise to a mutual information bound expressible as an integral. On channels with multiple transmit antennas, an analogous result is derived. However, since multiple transmit antennas open up more possibilities for spatial multiplexing, the rate-splitting approach gives rise to a whole family of infinite-layering bounds. This family of bounds is closely studied for independent and identically zero-mean Gaussian distributed fading coefficients (so-called i.i.d. Rayleigh fading). Most notably, it is shown that for asymptotically perfect CSIR, any bound from the family is asymptotically tight at high signal-to-noise ratios (SNR). Specifically, this means that the difference between the mutual information and its lower bound tends to zero as the SNR tends to infinity, provided that the CSIR tends to be exact as the SNR tends to infinity. A second part of this thesis proposes a framework for the optimization of a class of utility functions in black-Rayleigh fading multiple-antenna channels with transmit-side antenna correlation, and no CSI at the receiver. A fraction of each fading block is reserved for transmitting a sequence of training symbols, while the remaining time instants are used for transmission of data. The receiver estimates the channel matrix based on the noisy training observation and then decodes the data signal using this channel estimate. For utilities that are symmetric functions of the eigenvalues of the matrix-valued effective SNR (such as, e.g., the worst-case-noise bound), the problems consisting in optimizing the pilot sequence and the linear precoder are cast into convex (or quasi-convex) problems for concave (or quasi-concave) utility functions. We also study an important subproblem of the joint optimization, which consists in computing jointly Pareto-optimal pilot sequences and precoders. By wrapping these optimization procedures into a cyclic iteration, we obtain an algorithm which converges to a local joint optimum for any utility.Aquesta tesi estudia les taxes d'informació per la transmissió fiable d'informació en canals amb esvaïments sota la hipòtesi realista de que el receptor té un coneixement tan sols imperfecte de l'esvaïment aleatori. De particular interès són les expressions analítiques de les taxes de transmissió assolibles amb coneixement imperfecte i sense coneixement de l'estat del canal, és a dir, cotes inferiors de la informació mútua i de la capacitat de Shannon. Una cota inferior de la informació mútua per a codis gaussians ben coneguda s'obté combinant el soroll additiu (tèrmic) amb el terme de soroll multiplicatiu causat per les imperfeccions del coneixement de l'estat del canal en un únic soroll efectiu, i assumint que el soroll és gaussià i independent. Aquesta aproximació del pitjor soroll permet obtenir una expressió molt simple i ben coneguda de la informació mútua del canal. Una primera part d'aquesta tesi proposa un procediment senzill per a millorar aquesta cota associada al pitjor cas mitjançant una estratègia de repartiment de taxa: expressant l'entrada gaussiana del canal com a la suma de diverses entrades gaussianes independents i suposant que el receptor realitza una descodificació seqüencial dels fluxos d'informació, es mostra com obtenir una major cota inferior de la informació mútua del canal. En canals amb una única antena en transmissió, la distribució òptima de potència als diferents fluxos s'obté quan el seu nombre (capes) tendeix a infinit, i la potència associada a cada capa tendeix a zero. El límit associat a un nombre infinit de capes dóna lloc a una expressió integral de la cota de la informació mútua. En canals amb múltiples antenes s'obté un resultat similar. No obstant això, atès que la utilització de múltiples antenes proporciona més possibilitats de multiplexat espacial, el procediment dóna lloc a tota una família de cotes inferiors de la informació mútua associades a una combinació de capes infinita. S'estudia en detall aquesta família de cotes per al cas de coeficients d'esvaïments gaussians de mitjana zero, independents i idènticament distribuïts (conegut com esvaïment i.i.d. Rayleigh). S'obtenen diverses propietats de la família de cotes. És important destacar que per a coneixement asimptòtic perfecte del canal en recepció, qualsevol membre de la família de cotes és asimptòticament ajustat per alta relació senyal a soroll (SNR). En concret, la diferència entre la informació mútua i la seva cota inferior tendeix a zero quan la SNR tendeix a infinit sempre que el coneixement del canal tendeixi a ser exacte a mesura que la SNR tendeix a infinit. Una segona part d'aquesta tesi proposa un marc per a l'optimització d'una classe de funcions d'utilitat en canals amb múltiples antenes i esvaïments Rayleigh per blocs amb correlació en transmissió i sense informació sobre el canal a recepció. Una fracció temporal de cada bloc d'esvaïment es reserva per transmetre una seqüència de símbols d'entrenament mentre que la resta de mostres temporals s'utilitzen per transmetre informació. El receptor estima la matriu del canal partint de la seva observació sorollosa i descodifica la informació mitjançant la seva estimació del canal. Per a una classe de funcions d'utilitat que són funcions simètriques dels autovalors de la SNR matricial efectiva, els problemes consistents en optimitzar la seqüència pilot i el precodificador lineal són transformats en problemes convexos (o quasi-convexos) per a funcions d'utilitat còncaves (o quasi-còncaves). També s'estudia un subproblema important de l'optimització conjunta, que consisteix en el càlcul de les seqüències d'entrenament i dels precodificadors conjuntament Pareto-òptims. Integrant aquests procediments d'optimització en una iteració cíclica, s'obté un algoritme que convergeix a un òptim local conjunt per a qualsevol utilitat quasi-còncav

Learning Random Access Schemes for Massive Machine-Type Communication with MARL

In this paper, we explore various multi-agent reinforcement learning (MARL) techniques to design grant-free random access (RA) schemes for low-complexity, low-power battery operated devices in massive machine-type communication (mMTC) wireless networks. We use value decomposition networks (VDN) and QMIX algorithms with parameter sharing (PS) with centralized training and decentralized execution (CTDE) while maintaining scalability. We then compare the policies learned by VDN, QMIX, and deep recurrent Q-network (DRQN) and explore the impact of including the agent identifiers in the observation vector. We show that the MARL-based RA schemes can achieve a better throughput-fairness trade-off between agents without having to condition on the agent identifiers. We also present a novel correlated traffic model, which is more descriptive of mMTC scenarios, and show that the proposed algorithm can easily adapt to traffic non-stationaritiesComment: 15 pages, 10 figure

A Weighted Autoencoder-Based Approach to Downlink NOMA Constellation Design

End-to-end design of communication systems using deep autoencoders (AEs) is gaining attention due to its flexibility and excellent performance. Besides single-user transmission, AE-based design is recently explored in multi-user setup, e.g., for designing constellations for non-orthogonal multiple access (NOMA). In this paper, we further advance the design of AE-based downlink NOMA by introducing weighted loss function in the AE training. By changing the weight coefficients, one can flexibly tune the constellation design to balance error probability of different users, without relying on explicit information about their channel quality. Combined with the SICNet decoder, we demonstrate a significant improvement in achievable levels and flexible control of error probability of different users using the proposed weighted AE-based framework.Comment: 5 pages, 5 figures, to appear at SPAWC 202

Space-time rate splitting for the MISO BC with magnitude CSIT

A novel coding strategy is proposed for a broadcast setting with two transmitter (TX) antennas and two single-antenna receivers (RX). The strategy consists of using space-time block coding to send a common message (to be decoded by both RXs) across the two TX antennas, while each TX antenna also sends a private message to one of the RXs. The relative weight of the private and common messages from each TX antenna is tuned to maximize the instantaneous achievable sum-rate of the channel. Closed-form expressions for the optimal weight factors are derived. In terms of the generalized degrees of freedom (GDoF) metric, the new scheme is able to achieve the sum-GDoF with finite precision channel state information at the transmitter (CSIT) of the two user broadcast channel. Moreover, as opposed to the existing rate-splitting schemes, the proposed scheme yields instantaneous achievable rates that are independent of the channel phases. This property is instrumental for link adaptation when only magnitude CSIT is available. Our numerical results indeed demonstrate the superiority of the scheme for the 2-user setting in case of magnitude CSIT. Extension to a more general K -user scenario is briefly discussed.Grant numbers : SatNEx IV - Satellite Network of Experts IV project. © 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works