Vector Gaussian Channel Research Articles

Maximum Likelihood Detection in Single-Input Double-Output Non-Gaussian Barrage-Jammed Systems

We derive the likelihood functions and the maximum likelihood (ML) detectors for four classes of single-input double-output (SIDO) communication systems, i.e., systems with one transmit and two receive antennas. For all classes, the received signals are contaminated by a Gaussian noise component and a non-Gaussian component induced by the Gaussian transmissions of a proactive continuous single-antenna jammer over an unknown complex <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2\times 1$</tex-math></inline-formula> Gaussian vector channel. The considered classes correspond to whether full channel distribution information (CDI), or partial CDI about the transmitter channel and the jammer channel is available at the receiver. Unlike their scalar counterparts, the vector channels considered herein interweave the components of the received signal, rendering the derivation of the likelihood function a daunting task for more than two receive antennas. Furthermore, the interweaving of the received signal components in the vector channel case prevents the optimal ML detector for unit-norm constellations from reducing to the corresponding Gaussian approximation-based detector. This is in sharp contrast with the scalar case, wherein the two detectors are equivalent for unit-norm constellations. Confirming our analytical findings, experimental results show that the difference between the two detectors can be significant, especially when the transmitter-receiver and jammer-receiver channels have substantial line-of-sight components. Although the computational cost of performing optimal ML detection in the presence of non-Gaussian jamming is higher in the case of two receive antennas, the performance advantage over the single antenna case justifies it.

Heuristic Sensing Schemes for Four-Target Detection in Time-Constrained Vector Poisson and Gaussian Channels

Heuristic Sensing Schemes for Four-Target Detection in Time-Constrained Vector Poisson and Gaussian Channels

Heuristic Sensing Schemes for Four-Target Detection in Time-Constrained Vector Poisson and Gaussian Channels

In this work we investigate the different sensing schemes for detection of four targets as observed through a vector Poisson and Gaussian channels when the sensing time resource is limited and the source signals can be observed through a variety of sum combinations during that fixed time. For this purpose we can maximize the mutual information or the detection probability with respect to the time allocated to different sum combinations, for a given total fixed time. It is observed that for both Poisson and Gaussian channels; mutual information and Bayes risk with 0 − 1 cost are not necessarily consistent with each other. Concavity of mutual information between input and output, for certain sensing schemes, in Poisson channel and Gaussian channel is shown to be concave w.r.t given times as linear time constraint is imposed. No optimal sensing scheme for any of the two channels is investigated in this work.

Sensing Method for Two-Target Detection in Time-Constrained Vector Gaussian Channel

This paper considers a vector Gaussian channel of fixed identity covariance matrix and binary input signalling as the mean of it. A linear transformation is performed on the vector input signal. The objective is to find the optimal scaling matrix, under the total time constraint, that would: i) maximize the mutual information between the input and output random vectors, ii) maximize the MAP detection. It was found that the two metrics lead to different optimal solutions for our experimental design problem. We have used the Monte Carlo method for our computational work.

Differential Entropy of the Conditional Expectation Under Additive Gaussian Noise

The conditional mean is a fundamental and important quantity whose applications include the theories of estimation and rate-distortion. It is also notoriously difficult to work with. This paper establishes novel bounds on the differential entropy of the conditional mean in the case of finite-variance input signals and additive Gaussian noise. The main result is a new lower bound in terms of the differential entropies of the input signal and the noisy observation. The main results are also extended to the vector Gaussian channel and to the natural exponential family. Various other properties such as upper bounds, asymptotics, Taylor series expansion, and connection to Fisher Information are obtained. Two applications of the lower bound in the remote-source coding and CEO problem are discussed.

Linear MIMO Precoders With Finite Alphabet Inputs via Stochastic Optimization and Deep Neural Networks (DNNs)

In this paper, we investigate designs of linear precoders for vector Gaussian channels via stochastic optimizations and deep neural networks (DNNs). We assume that the channel inputs are drawn from practical finite alphabets, and we search for precoders maximizing the mutual information between channel inputs and outputs. Though the problem is generally non-convex, we prove that when the right singular matrix of precoder is fixed, any local optima of this problem is a global optima. Based on this fact, an efficient projected stochastic gradient descent (PSGD) algorithm is designed to search the optimal precoders. Moreover, to reduce the complexity of calculating a posterior means involved in gradients calculation, K-best algorithm is adopted to make approximations of a posterior means with negligible loss of accuracy. Furthermore, to avoid explicit calculation of mutual information and its gradients, DNN-based autoencoders (AEs) are constructed for this precoding task, and an efficient training algorithm is proposed. We also prove that the AEs, with ‘softmax’ activation function and ‘categorical cross entropy’ loss, maximize the mutual information under reasonable assumptions. Then, in order to extend the AE methods to large scale systems, ‘sigmoid’ activation function and ‘binary cross entropy’ loss are used such that the size of AEs will not grow prohibitively large. We prove that this maximizes a lower bound of the mutual information under reasonable assumptions. Finally, to make the precoders practical for high speed wireless scenarios, we propose an offline training paradigm which trains DNNs to infer optimal precoders given channel state information instead of training online for every different channel. Simulation results show that all the proposed methods work well in maximizing mutual information and improving bit error rate (BER) performance.

Capacity Limits of Full-Duplex Cellular Network

This paper aims to characterize the capacity limits of a wireless cellular network with a full-duplex (FD) base-station (BS) and half-duplex user terminals, in which three independent messages are communicated: the uplink message $m_1$ from the uplink user to the BS, the downlink message $m_2$ from the BS to the downlink user, and the device-to-device (D2D) message $m_3$ from the uplink user to the downlink user. From an information theoretical perspective, the overall network can be viewed as a generalization of the FD relay broadcast channel with a side message transmitted from the relay to the destination. We begin with a simpler case that involves the uplink and downlink transmissions of $(m_1,m_2)$ only, and propose an achievable rate region based on a novel strategy that uses the BS as a FD relay to facilitate the interference cancellation at the downlink user. We also prove a new converse, which is strictly tighter than the cut-set bound, and characterize the capacity region of the scalar Gaussian FD network without a D2D message to within a constant gap. This paper further studies a general setup wherein $(m_1,m_2,m_3)$ are communicated simultaneously. To account for the D2D message, we incorporate Marton's broadcast coding into the previous scheme to obtain a larger achievable rate region than the existing ones in the literature. We also improve the cut-set bound by means of genie and show that by using one of the two simple rate-splitting schemes, the capacity region of the scalar Gaussian FD network with a D2D message can already be reached to within a constant gap. Finally, a generalization to the vector Gaussian channel case is discussed. Simulation results demonstrate the advantage of using the BS as relay in enhancing the throughput of the FD cellular network.

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.

Capacity Theorems for Broadcast Channels With Two Channel State Components Known at the Receivers

We establish the capacity region of several classes of broadcast channels with random state in which the channel to each user is selected from two possible channel state components and the state is known only at the receivers. When the channel components are deterministic, we show that the capacity region is achieved via Marton coding. This channel model does not belong to any class of broadcast channels for which the capacity region was previously known and is useful in studying wireless communication channels when the fading state is known only at the receivers. We then establish the capacity region when the channel components are ordered, e.g., degraded. In particular, we show that the capacity region for the broadcast channel with degraded Gaussian vector channel components is attained via Gaussian input distribution. Finally, we extend the results on ordered channels to two broadcast channel examples with more than two channel components, but show that these extensions do not hold in general.

On the Capacity of Vector Gaussian Channels With Bounded Inputs

The capacity of a deterministic multiple-input multiple-output channel under the peak and average power constraints is investigated. For the identity channel matrix, the approach of Shamai et al. is generalized to the higher dimension settings to derive the necessary and sufficient conditions for the optimal input probability density function. This approach prevents the usage of the identity theorem of the holomorphic functions of several complex variables which seems to fail in the multi-dimensional scenarios. It is proved that the support of the capacity-achieving distribution is a finite set of hyper-spheres with mutual independent phases and amplitude in the spherical domain. Subsequently, it is shown that when the average power constraint is relaxed, if the number of antennas is large enough, the capacity has a closed-form solution and constant amplitude signaling at the peak power achieves it. Moreover, it will be observed that in a discrete-time memoryless Gaussian channel, the average power constrained capacity, which results from a Gaussian input distribution, can be closely obtained by an input where the support of its magnitude is a discrete finite set. Finally, we investigate some upper and lower bounds for the capacity of the non-identity channel matrix and evaluate their performance as a function of the condition number of the channel.

On the tightness of linear policies for stabilization of linear systems over Gaussian networks

In this paper, we consider stabilization of multi-dimensional linear systems driven by Gaussian noise controlled over parallel Gaussian channels. For such systems, it has been recognized that for stabilization in the sense of asymptotic stationarity or stability in probability, Shannon capacity of a channel is an appropriate measure on characterizing whether a system can be made stable when controlled over the channel. However, this is in general not the case for quadratic stabilization. On a related problem of joint-source channel coding, in the information theory literature, the source-channel matching principle has been shown to lead to optimality of uncoded or analog transmission and when such matching conditions occur, it has been shown that capacity is also a relevant figure of merit for quadratic stabilization. A special case of this result is applicable to a scalar LQG system controlled over a scalar Gaussian channel. In this paper, we show that even in the absence of source-channel matching, to achieve quadratic stability, it may suffice that information capacity (in Shannon’s sense) is greater than the sum of the logarithm of unstable eigenvalue magnitudes. In particular, we show that periodic linear time varying coding policies are optimal in the sense of obtaining a finite second moment for the state of the system with minimum transmit power requirements for a large class of vector Gaussian channels. Our findings also extend the literature which has considered noise-free systems.

Causal Relay Networks

In this paper, we study causal discrete memoryless relay networks. The network consists of multiple nodes, each of which can be a source, a relay, and/or a destination. In the network, there are two types of relays: 1) relays with one sample delay (strictly causal) and 2) relays without delay (causal) whose transmit signals depend not only on the past received symbols but also on the current received symbols. For this network, we derive two new cut-set bounds, one when every node has a message and the other when only the strictly causal relays have messages. Using the examples of a causal vector Gaussian two-way relay channel and a causal vector Gaussian relay channel, we show that the new cut-set bounds can be achieved by a simple amplify-and-forward type relaying. Our result for the causal relay channel strengthens the previously known capacity result for the same channel by El Gamal, Hassanpour, and Mammen.

A Bregman Matrix and the Gradient of Mutual Information for Vector Poisson and Gaussian Channels

A generalization of Bregman divergence is developed and utilized to unify vector Poisson and Gaussian channel models, from the perspective of the gradient of mutual information. The gradient is with respect to the measurement matrix in a compressive-sensing setting, and mutual information is considered for signal recovery and classification. Existing gradient-of-mutual-information results for scalar Poisson models are recovered as special cases, as are known results for the vector Gaussian model. The Bregman-divergence generalization yields a Bregman matrix, and this matrix induces numerous matrix-valued metrics. The metrics associated with the Bregman matrix are detailed, as are its other properties. The Bregman matrix is also utilized to connect the relative entropy and mismatched minimum mean squared error. Two applications are considered: 1) compressive sensing with a Poisson measurement model and 2) compressive topic modeling for analysis of a document corpora (word-count data). In both of these settings, we use the developed theory to optimize the compressive measurement matrix, for signal recovery and classification.

Analysis of Mismatched Estimation Errors Using Gradients of Partition Functions

We consider the problem of signal estimation (denoising) from a statistical-mechanical perspective, in continuation to a recent work on the analysis of mean-square error (MSE) estimation using a direct relationship between optimum estimation and certain partition functions. This paper consists of essentially two parts. In the first part, using the aforementioned relationship, we derive single-letter expressions of the asymptotic mismatched MSE of a codeword (from a randomly selected code), corrupted by a Gaussian vector channel. In the second part, we provide several examples to demonstrate phase transitions in the behavior of the MSE. These examples enable us to understand more deeply and to gather intuition regarding the roles of the real and the mismatched probability measures in creating these phase transitions.

On the Precoder Design of a Wireless Energy Harvesting Node in Linear Vector Gaussian Channels with Arbitrary Input Distribution

A Wireless Energy Harvesting Node (WEHN) operating in linear vector Gaussian channels with arbitrarily distributed input symbols is considered in this paper. The precoding strategy that maximizes the mutual information along N independent channel accesses is studied under non-causal knowledge of the channel state and harvested energy (commonly known as offline approach). It is shown that, at each channel use, the left singular vectors of the precoder are equal to the eigenvectors of the Gram channel matrix. Additionally, an expression that relates the optimal singular values of the precoder with the energy harvesting profile through the Minimum Mean-Square Error (MMSE) matrix is obtained. Then, the specific situation in which the right singular vectors of the precoder are set to the identity matrix is considered. In this scenario, the optimal offline power allocation, named Mercury Water-Flowing, is derived and an intuitive graphical representation is presented. Two optimal offline algorithms to compute the Mercury Water-Flowing solution are proposed and an exhaustive study of their computational complexity is performed. Moreover, an online algorithm is designed, which only uses causal knowledge of the harvested energy and channel state. Finally, the achieved mutual information is evaluated through simulation.

On MMSE Crossing Properties and Implications in Parallel Vector Gaussian Channels

The scalar additive Gaussian noise channel has the “single crossing point” property between the minimum mean square error (MMSE) in the estimation of the input given the channel output, assuming a Gaussian input to the channel, and the MMSE assuming an arbitrary input. This paper extends the result to the parallel vector additive Gaussian channel in three phases. The channel matrix is the identity matrix, and we limit the Gaussian input to a vector of Gaussian i.i.d. elements. The “single crossing point” property is with respect to the signal-to-noise ratio (as in the scalar case). The channel matrix is arbitrary, and the Gaussian input is limited to an independent Gaussian input. A “single crossing point” property is derived for each diagonal element of the MMSE matrix. The Gaussian input is allowed to be an arbitrary Gaussian random vector. A “single crossing point” property is derived for each eigenvalue of the difference matrix between the two MMSE matrices. These three extensions are then translated to new information theoretic properties on the mutual information, using the I-MMSE relationship, a fundamental relationship between estimation theory and information theory revealed by Guo and coworkers. The results of the last phase are also translated to a new property of Fisher information. Finally, the applicability of all three extensions on information theoretic problems is demonstrated through a proof of a special case of Shannon's vector entropy power inequality, a converse proof of the capacity region of the parallel degraded broadcast channel (BC) under an input per-antenna power constraint and under an input covariance constraint, and a converse proof of the capacity region of the compound parallel degraded BC under an input covariance constraint.

Weighted Sum Rate Maximization for the Two-User Vector Gaussian Broadcast Channel

In this letter, the problem of determining the maximum weighted sum rate (WSR) for given arbitrary weights is investigated for the vector Gaussian broadcast channel (GBC). Although the solution to the problem in general involves an iterative algorithm, we focus on the important special case of the two-user channel to provide a closed-form expression for the maximum WSR. The achievable rate region is known to be identical for both the vector GBC and its dual uplink Gaussian multiple access channel (GMAC) due to the uplink-downlink duality. We first derive the optimal power allocation in the dual uplink channel, which is then applied to the capacity formula to obtain the desired result. The presented result is a generalization of the (unweighted) sum rate maximization of the two-user GBC provided by Caire and Shamai .

Non-linear coding and decoding strategies exploiting spatial correlation in wireless sensor networks

The authors consider the acquisition of measurements from a source, representing a physical phenomenon, by means of sensors deployed at different distances, and measuring random variables (r.v.&apos;s) that are correlated with the source output. The acquired values are transmitted over a wireless channel to a sink, where an estimation of the source has to be constructed, according to a given distortion criterion. In the presence of Gaussian random variables (r.v.&apos;s) and a Gaussian vector channel, the authors are seeking optimum real-time joint source-channel encoder–decoder pairs that achieve a distortion sufficiently close to the theoretically optimal one, under a global resource constraint, by activating only a subset of the sensors. The problem is posed in a team decision theoretic framework, and the optimal strategies are approximated by means of neural networks. The analysis investigates the generalisation capabilities of the proposed approach, by showing insights into the structure of the problem. The surprising outcome is that a quasi-static application of the approach reveals to be sufficient to maintain quasi-optimal performance under a dynamic environment (e.g. with respect to nodes&apos; positions).

Globally Optimal Linear Precoders for Finite Alphabet Signals Over Complex Vector Gaussian Channels

We study the design optimization of linear precoders for maximizing the mutual information between finite alphabet input and the corresponding output over complex-valued vector channels. This mutual information is a nonlinear and non-concave function of the precoder parameters, posing a major obstacle to precoder design optimization. Our work presents three main contributions: First, we prove that the mutual information is a concave function of a matrix which itself is a quadratic function of the precoder matrix. Second, we propose a parameterized iterative algorithm for finding optimal linear precoders to achieve the global maximum of the mutual information. The proposed iterative algorithm is numerically robust, computationally efficient, and globally convergent. Third, we demonstrate that maximizing the mutual information between a discrete constellation input and the corresponding output of a vector channel not only provides the highest practically achievable rate but also serves as an excellent criterion for minimizing the coded bit error rate. Our numerical examples show that the proposed algorithm achieves mutual information very close to the channel capacity for channel coding rate under 0.75, and also exhibits a large gain over existing linear precoding and/or power allocation algorithms. Moreover, our examples show that certain existing methods are susceptible to being trapped at locally optimal precoders.