Exploiting near-field spherical wavefronts can improve the performance of integrated sensing and communication (ISAC) systems. Existing studies on near-field ISAC mainly consider either fully-digital or static hybrid beamforming, which may not be able to efficiently exploit the distance-dependent degrees of freedom (DoFs) of near-field channels. In this paper, we propose a dynamic hybrid beamforming architecture that adaptively adjusts the number of active radio frequency (RF) chains and further utilize a reconfigurable intelligent surface (RIS) to enhance the performance in ISAC systems. We formulate an energy efficiency maximization problem which aims to jointly optimize the hybrid precoding matrices at the base station, phase-shifts at the RIS, and hybrid combining matrices at the user equipment, subject to the constraint on guaranteeing the minimum beam pattern gain toward sensing targets. To tackle this intractable problem, we propose an alternating optimization algorithm by leveraging fractional programming, matrix lifting, and semidefinite relaxation techniques. Simulation results demonstrate that our proposed algorithm outperforms three baseline schemes in both optimizing the number of active RF chains and maximizing the energy efficiency.

Extremely large-scale reconfigurable intelligent surface (XL-RIS) can effectively overcome severe fading and provide higher communication performance. However, current research on XL-RIS overlooks the discrete phase-shift characteristics of RIS in practical systems, which will result in significant performance degradation. In this paper, we investigate near-field communication schemes assisted by XL-RIS with discrete phase shifts. Specifically, we propose a hierarchical beam training method to obtain the user channel state information (CSI), and develop the jointly optimized codebook construction (JOCC) method and separately optimized codebook construction (SOCC) method for base station (BS) precoding and XL-RIS phase shifts, respectively. With JOCC, the most superior beam training performance can be obtained. With SOCC, higher performance than the single-antenna BS codebook can be obtained at a similar complexity. Further, we propose a flexible multiuser interference management (IM) method that is simple to solve. The IM method uses adaptive gain matrix approximation to take into account user fairness and can be solved in closed-form iterations. In addition, we extend the proposed method to a hybrid precoding design. Simulation results demonstrate that the proposed multi-resolution codebook construction method can obtain more accurate beam patterns and user CSI, and the proposed IM method obtains superior performance over the benchmark methods.

Stacked intelligent metasurfaces (SIM) provide a cost-effective and scalable solution for large-scale antenna communications. However, efficient channel state information acquisition and phase shift optimization remain critical challenges. In this paper, we develop a unified framework of low-complexity algorithms for SIM-assisted communication systems to address these issues. Specifically, we propose a generalized two-step codebook construction (TSCC) method that lever-ages two-dimensional angular-domain decoupling to transform planar array beamformer design into two independent one-dimensional linear array beamformer design problems, efficiently solved via the Gerchberg–Saxton algorithm and our proposed majorization–minimization-based proximal-distance (PDMM) algorithm. We further develop a two-stage coded-sliding beam training (TSCSBT) method for low-overhead and high-accuracy beam training, where error-correcting codes are embedded in the first-stage training to enhance robustness against noise, and sliding sampling is subsequently performed around the matched angular samples to improve angular resolution. The proposed framework is further extended to multi-path user channels. Finally, a variable decoupling-based block successive upper bound minimization (VD-BSUM) algorithm is proposed to directly solve the QoS-constrained sum-rate maximization problem through closed-form iterative updates with substantially reduced computational complexity. Simulation results demonstrate the effectiveness of the proposed methods in achieving precise beam pattern realization, improved beam training accuracy and angular resolution, and enhanced sum-rate performance.

This paper develops scheduling policies to optimize the information freshness, quantified by the age of information (AoI) metric, in an uplink multi-user SIMO status update system. The multi-user scheduling problem is formulated as a Markov decision process (MDP) to derive the optimal policy that minimizes the average AoI across devices. However, the optimal policy suffers from high complexity due to dimensionality in large networks. To address this, a low-complexity max-weight (MW) policy is developed for large-scale networks using the Lyapunov optimization framework. The MW policy dynamically determines the subset of devices to schedule in each time slot by maximizing the expected AoI drop of the subsequent time slot. Simulations are conducted to compare the performance of the optimal policy, the MW policy, and the baseline fixed scheduling (FS) policy that always schedules a fixed number of devices with the highest AoI. The results show that the MW policy achieves close-to-optimal performance. Moreover, for a given network setup, there exists an FS policy with a particular number of scheduled devices that can approach the MW policy. This observation inspired the development of another low-complexity scheduling policy, termed optimized FS (OFS). This policy further optimizes the number of devices scheduled under the FS policy based on specific network configurations. Closed-form expressions for the average peak AoI and the approximated average AoI of the FS policy with a given number of scheduled devices are derived to determine the optimal number of scheduled devices for the OFS policy under different network setups. Simulation results validate the theoretical analysis and show that the OFS policy achieves performance comparable to the MW policy while circumventing the need for per-slot optimization.

The combination of reconfigurable intelligent surface (RIS) and non-orthogonal multiple access (NOMA) is a promising technique to enhance spectral efficiency and energy efficiency. However, the imperfect channel state information (CSI) of the wireless channel and hardware impairments (HWI) of the radio frequency chain will cause significant performance loss on RIS-aided NOMA communication. In this paper, we focus on the effect of the imperfect CSI and generalized HWI on RIS-aided NOMA communication and propose an efficient robust precoding and phase shift design scheme to enhance the energy efficiency of systems. Specifically, we minimize the transmit power with quality of service constraints of users, where the imperfect CSI and generalized HWI are modeled as Gaussian error models, and the successful successive interference cancellation condition is characterized by the variance criterion. The minimization problem is greatly challenging due to the non-closed form outage constraints and non-explicit phase shift design. To solve it, we develop an efficient probabilistic technique to reformulate the outage constraints and restrict an explicit objective function to obtain a more efficient phase shift design. Numerical results demonstrate that our scheme has superior convergence performance, energy efficiency, and feasibility rate compared to existing schemes.

Federated learning (FL) enables multiple users to collaboratively train a shared model while protecting user privacy. In this paper, we investigate the transmission delay minimization problem for non-orthogonal multiple access (NOMA)-assisted FL. We analyze the convergence rate of heterogeneous quantized FL to demonstrate that the minimum quantization level among scheduled users is crucial in controlling the trade-off between the number of training rounds and the transmission delay of each round. Based on the convergence analysis, we formulate a delay minimization problem for NOMA-assisted FL and propose a communication-efficient heterogeneous compression NOMA scheme for FL. Subsequently, we develop a block coordinate descent (BCD)-based algorithm that jointly optimizes the sub channel allocation, power allocation, and quan-tization level for each scheduled user. Results reveal that our proposed algorithm significantly reduces the transmission delay while achieving the same learning performance compared with conventional FL algorithms.

Abstract In next‐generation wireless communications, reconfigurable intelligent surface (RIS) has emerged as a cost‐effective technique for enhancing physical layer security (PLS) in millimeter‐wave (mmWave) communications, especially under challenging scenarios with adversarial entities and obstructions. However, the primary studies for RIS incorporation in mmWave communication systems utilized static deployments, lacking adaptability and efficiency in complex environments. To address this problem, a dynamic RIS deployment design framework is introduced for PLS enhancement in mmWave systems against jamming and eavesdropping attacks. For the design, it is aimed to maximize the secrecy rate by jointly optimizing the RIS selection with the beamforming design. The resulting optimization problem is challenging to solve due to the coupling of the RIS control factor, joint beamforming design, and non‐convex constraints. To tackle these issues, an efficient multi‐RIS‐aided PLS enhancement algorithm is proposed. It transforms the objective into a series of subproblems and employs the fractional programming technique and prox‐linear block coordinate descent updating method to solve them alternatively and obtain the optimal solution. The simulations demonstrate the advantage of the dynamic deployment, which exhibits enhanced security performance with reduced complexity compared with benchmarks. Further examinations also provide insight into optimal RIS activation configurations, achieving optimal balance for securing mmWave communications against emerging threats while maintaining system efficiency.

Reconfigurable intelligent surfaces (RIS) can dynamically reconstruct wireless environments to enhance spectral efficiency. However, most existing studies have ignored the impact of the phase error and imperfect amplitude gain of the RIS. In this paper, we investigate the practical RIS-aided multiuser communication systems by maximizing the sum of users’ average achievable rate, filling the current research gap. Specifically, a novel RIS phase shift design approach, namely amplitude feedback (AF), is proposed by utilizing the coupling relationship between the amplitude and phase to derive the optimal phase shift under the worst phase error. The feasibility of AF is demonstrated by proving the measurability of amplitude response through the electromagnetic theory. We propose a channel estimation design with low pilot overhead and provide an effective closed-form achievable rate to approximate the average achievable rate. Moreover, an efficient optimization algorithm is proposed to achieve the optimal closed-form precoding at the base station and the optimal trade-off design between the amplitude and phase of RIS, and the algorithm is extended to active RIS systems. Numerical results demonstrate that our proposed AF method and optimization algorithm can efficiently improve the performance in terms of achievable sum rate compared to existing methods.

Integrating reconfigurable intelligent surface (RIS) and non-orthogonal multiple access (NOMA) provides a promising solution to support large-scale secure communications. However, reducing the impact of transceiver hardware impairments (HWI) on the performance of RIS-aided NOMA secure networks remains a challenge. In this paper, we propose a robust transmission scheme for RIS-aided NOMA secure networks with transceiver HWI under two proposed eavesdropping scenarios and two artificial noise methods. A closed-form expression for the distortion noise power caused by transceiver HWI in NOMA networks is derived to quantify transceiver HWI. The sum secrecy rate of NOMA users with imperfect successive interference cancellation (SIC) is maximized under distortion noise for more robust security. Further, to tackle the resulting non-convex problem, we decompose it into two sub-problems. Both of them are converted into convex problems using the multi-dimensional quadratic transform (MDQT) method, semi-definite relaxation (SDR) technique, and the successive convex approximation (SCA) algorithm, and then employing the alternate optimization (AO) algorithm solve them. Numerical results demonstrate that the proposed scheme has superior security performance over orthogonal multiple access (OMA) networks, time division multiple access (TDMA) networks, and traditional NOMA networks, achieving more robust performance than the conventional scheme that ignores HWI and imperfect SIC.

In this paper, we concern the uplink of a massive single-input multiple-output enabled ultra-reliable low-latency communication system, in which a single-antenna transmitter aims to timely and reliably send data to a receiver equipped with a large number of antennas over Rayleigh fading channels. For such a scenario, to eliminate the considerable overhead caused by channel estimation, we adopt a noncoherent maximum-likelihood (ML) receiver, which is known to be optimal in terms of average symbol-error rate for equiprobable discrete input signals. We propose a two-dimensional noncoherent constellation design framework to enhance the reliability of the considered system. Specifically, our design principle is to maximize the minimum Kullback-Leibler divergence between the conditional distributions induced by different transmitted signals under average power constraint for any given transmission rate. The resulting optimization problem is shown to be a challenging mixed discrete-continuous problem. We manage to solve the problem by deliberately designing optimal bit allocation and optimal constellation structure as a function of signal-to-noise ratios. We then unveil that the proposed constellation can facilitate efficient ML detection with low computational complexity. Finally, simulation results illustrate that the proposed scheme has a superior error performance than conventional training-based schemes and existing energy detection designs.

In this paper, a hybrid adaptive MCB-PSO node localization algorithm is proposed for three-dimensional mobile wireless sensor networks (MWSNs), which considers the random mobility of both anchor and unknown nodes. An improved particle swarm optimization (PSO) approach is presented with Monte Carlo localization boxed (MCB) to locate mobile nodes. It solves the particle degeneracy problem that appeared in traditional MCB. In the proposed algorithm, a random waypoint model is incorporated to describe random movements of anchor and unknown nodes based on different time units. An adaptive anchor selection operator is designed to improve the performance of standard PSO for each particle based on time units and generations, to maintain the searching ability in the last few time units and particle generations. The objective function of standard PSO is then reformed to make it obtain a better rate of convergence and more accurate cost value for the global optimum position. Furthermore, the moving scope of each particle is constrained in a specified space to improve the searching efficiency as well as to save calculation time. Experiments are made in MATLAB software, and it is compared with DV-Hop, Centroid, MCL, and MCB. Three evaluation indexes are introduced, namely, normalized average localization error, average localization time, and localization rate. The simulation results show that the proposed algorithm works well in every situation with the highest localization accuracy, least time consumptions, and highest localization rates.

To enable ultrareliable low-latency wireless communications required in the Industrial Internet of Things, in this paper we develop an energy-based modulation [i.e., non-negative pulse amplitude modulation (PAM)] constellation design framework for noncoherent detection in massive single-input multiple-output (SIMO) systems. We consider that one single-antenna transmitter communicates to a receiver with a large number of antennas over a Rayleigh fading channel, and the receiver decodes the transmitted information at the end of every symbol. For such an SIMO system with non-negative PAM modulation, we first propose a fast noncoherent maximum-likelihood decoding algorithm and derive a closed-form expression of its symbol error probability (SEP). We then enhance the system energy efficiency by finding the optimal PAM constellation that minimizes the exact SEP subject to a total signal power constraint for such a system with an arbitrary number of receiver antennas, signal-to-noise ratio (SNR), and constellation size. Furthermore, the closed-form upper and lower bounds on the optimal SEP are derived. Based on these bounds, the exact expression for coding gain of the dominant term of the SEP is presented for such an optimal massive SIMO system. We also present an asymptotic SEP expression at a high SNR regime and the approximate diversity gain of the system. Simulation results for the proposed optimal PAM constellation validate the theoretical analysis, and show that our presented optimal constellation attains significant performance gains over the currently available minimum-distance-based constellation systems.

This paper investigates the symbol error probability (SEP) of point-to-point massive multiple-input multiple-output (MIMO) systems using equally likely PAM, PSK, and square QAM signallings in the presence of transmitter correlation. The receiver has perfect knowledge of the channel coefficients, while the transmitter only knows first- and second-order channel statistics. With a zero-forcing (ZF) detector implemented at the receiver side, we design and derive closed-form expressions of the optimal precoders at the transmitter that minimizes the average SEP over channel statistics for various modulation schemes. We then unveil some nice structures on the resulting minimum average SEP expressions, which naturally motivate us to explore the use of two useful mathematical tools to systematically study their asymptotic behaviors. The first tool is the Szegö's theorem on large Hermitian Toeplitz matrices and the second tool is the well-known limit: limx→∞(1 + 1/x)x= e. The application of these two tools enables us to attain very simple expressions of the SEP limits as the number of the transmitter antennas goes to infinity. A major advantage of our asymptotic analysis is that the asymptotic SEP converges to the true SEP when the number of antennas is moderately large. As such, the obtained expressions can serve as an effective SEP approximations for massive MIMO systems even when the number of antennas is not very large. For the widely used exponential correlation model, we derive closed-form expressions for the SEP limits of both optimally precoded and uniformly precoded systems. Extensive simulations are provided to demonstrate the effectiveness of our asymptotic analysis and compare the performance limit of optimally precoded and uniformly precoded systems.

We consider the non-orthogonal multiple access (NOMA) design for a classical two-user multiple access channel (MAC) with finite-alphabet inputs. In contrast to the majority of existing NOMA schemes using continuous Gaussian distributed inputs, we consider practical quadrature amplitude modulation (QAM) constellations at both transmitters, whose sizes are not necessarily the same. By adjusting the scaling factors (i.e., instantaneous transmitting powers) of both users, we aim to maximize the minimum Euclidean distance of the received sum-constellation for a maximum likelihood (ML) receiver. The formulated problem is a mixed continuous-discrete optimization problem and in general it is nontrivial to resolve. By carefully examining the structure of the objective function, we discover that Farey sequence can be employed to tackle the formulated problem. However, the existing Farey sequence is not applicable when the constellation sizes of the two users are different. To address this challenge, we define a new type of Farey sequence, termed punched Farey sequence. Based on this new definition and its properties, we manage to attain a closed-form optimal solution to the original problem by first dividing the entire feasible region into a finite number of Farey intervals and then taking the maximum over all the subintervals. Finally, computer simulations are carried out to verify our theoretical analysis, and to demonstrate the advantages of the proposed NOMA over known orthogonal and non-orthogonal designs.

In this study, the authors consider the physical‐layer security problem in the relay networks. The relay nodes are employed not only to aid the information transmission but also to improve the network security in the physical layer by sending artificial noise to resist the potential malicious eavesdropper. By allowing shared randomness between the jamming nodes, it is shown that the maximum degrees of freedom (DoF) of the considered network is almost surely. The necessary and sufficient conditions for the optimal DoF setup are established. Moreover, a simple DoF‐optimal collaborative beamforming algorithm is proposed, and it works very well in the high signal‐to‐noise ratio regime, which is verified by computer simulations.

This paper focuses on the non-orthogonal multiple access (NOMA) design for a classical two-user multiple access channel (MAC) with finite-alphabet inputs. In contrast to most of the existing NOMA designs using continuous Gaussian input distributions, we consider practical quadrature amplitude modulation (QAM) constellations at both transmitters, the sizes of which are assumed to be not necessarily identical. We propose maximizing the minimum Euclidean distance of the received sum constellation with a maximum likelihood (ML) detector by adjusting the scaling factors (i.e., instantaneous transmitted powers and phases) of both users. The formulated problem is a mixed continuous-discrete optimization problem, which is nontrivial to resolve in general. By carefully observing the structure of the objective function, we define a new type of Farey sequence, termed punched Farey sequence to tackle the formulated problem. Based on this, we manage to achieve a closed-form optimal solution to the original problem by first dividing the entire feasible region into a finite number of Farey intervals and then taking the maximum over all possible intervals. The resulting sum constellation is proved to be a regular QAM constellation of a larger size, and hence, a simple quantization receiver can be implemented as the ML detector for the demodulation. Moreover, the superiority of NOMA over time-division multiple access in terms of minimum Euclidean distance is rigorously proved. We subsequently address how to extend our design framework intended for the two-user MAC to systems with multiple users and multiple antennas. Finally, simulation results are provided to verify our theoretical analysis and demonstrate the merits of the proposed NOMA over existing orthogonal and non-orthogonal designs.

In this paper, we consider the design of multi-users space-time modulation (MUSTM) for an uplink MIMO system with one base station equipped with the massive number of antennas and N single-antenna users, where it is assumed that only large scale channel coefficients are available at both the transmitter and the receiver. For such a system, a novel concept called uniquely factorable (UF) MUSTM is introduced. Then, using our recently developed framework on uniquely decomposable constellation group with energy-efficient quadrature amplitude modulation (QAM), and properly and timely assigning each sub-constellation to each user at each time slot, we develop a machinery method for systematically designing a family of invertible UF-MUSTM with flexible data rates in order to assure the reliable estimation of the transmitted signal as well as of the channel for the massive MIMO system. In addition, a simple cross-correlation receiver is proposed to efficiently and effectively detect such UF-MUSTM. Its pair-wise error probability (PEP) is derived, showing that our proposed invertible UF-MUSRM enables full receiver diversity. Furthermore, the optimal closed-form power allocation and the optimal user constellation assignment are found to maximize the worst-case coding gain under a peak power constraint on each user and each time slot.

This paper focuses on the design of non-orthogonal multiple access in a classical two-transmitter two-receiver Z-channel, wherein one transmitter sends information to its intended receiver from the direct link while the other transmitter sends information to both receivers from the direct and cross links. Unlike most existing designs using (continuous) Gaussian input distribution, we consider the practical finite-alphabet (i.e., discrete) inputs by assuming that the widely used quadrature amplitude modulation constellations are adopted by both transmitters. To balance the error performance of two receivers, we apply the max-min fairness design criterion in this paper. More specifically, we propose to jointly optimize the scaling factors at both transmitters, which control the minimum Euclidean distance of transmitting constellations, to maximize the smaller minimum Euclidean distance of two resulting constellations at the receivers, subject to an individual average power constraint at each transmitter. The formulated problem is a mixed continuous-discrete optimization problem and is thus intractable in general. By resorting to the Farey sequence, we manage to attain the closed-form expression for the optimal solution to the formulated problem. This is achieved by dividing the overall feasible region of the original optimization problem into a finite number of sub-intervals and deriving the optimal solution in each sub-interval. Through carefully observing the structure of the optimal solutions in all sub-intervals, we obtain compact and closed-form expressions for the optimal solutions to the original problem in three possible scenarios defined by the relative strength of the cross link. Simulation studies are provided to validate our analysis and demonstrate the merits of the proposed design over existing orthogonal or non-orthogonal schemes.

This paper considers the asymptotic analysis of symbol error probability (SEP) for either optimally precoded or uniformly precoded large correlated MIMO fading channels using the zero-forcing (ZF) detector. For such systems, we reveal some very nice structures which naturally lead us to the exploration of two very strong and very useful mathematical tools for the systematic study of asymptotic behaviors on their error performance. The first tool is the Szegö's theorem on large Hermitian Toeplitz matrices and the second tool is the well known limit: limx→∞(1 + 1/x)x= e. This new approach enables us to attain a very simple expression for the SEP limit as the number of the transmitter antennas goes to infinity. One of the major advantages for this method is that its convergence rate is very fast. Hence, this expression is very efficient and effective SEP approximation for the large MIMO systems. Due to the constraint on allowable space and limited (sparse) multi-path scattering, fading correlation between neighbouring antenna elements is almost inevitable in a large MIMO architecture. By specifically examining an exponential correlation matrix model, we show that channel fading correlation can lead to significant performance loss. The optimal precoding technique can yield substantial power gain over the uniform power allocation strategy.

This paper develops a novel uniquely-factorable constellation pair (UFCP) by carefully factorizing phase-shift keying (PSK) constellation. With this unique factorization, a new distributed concatenated Alamouti code is proposed for a one-way relaying network consisting of two single-antenna terminals and one relay having two antennas. This design allows the relay to transmit its own information while forwarding the source information which it has received to the destination. By making use of the Alamouti coding scheme twice and jointly processing the signals from the two antennas at the relay node, such a code design also renders the equivalent channel between source and destination be a product of the two Alamouti channels and therefore, is called distributed concatenated Alamouti space-time block code (STBC). In addition, the asymptotic symbol error probability (SEP) formula is derived with the maximum likelihood (ML) detector, showing that the optimal diversity gain function is achieved and proportional to ln SNR/SNR2.

In this paper, we consider a three-hop multi-relay network with one transmitter, one receiver and two clusters of relay nodes. With the aid of perfect channel state information (CSI), two different design approaches are devised. In the first approach, we aim to maximize the received signal to noise ratio (SNR) under different transmit power constraint at the relay nodes by performing optimization on the beamforming vectors of the two clusters jointly, whilst in the second design, the total transmit power of the relay nodes is minimized under the QoS constraint at the receiver. It is shown that, in both approaches, upon the coefficients of either one cluster being determined, the other one could be optimized, and therefore the overall optimization problem can be resorted to an efficient iteration process. Simulation results show that our proposed approaches outperform the existing solutions for improved energy efficiency and increased receiving SNR.

In this paper, we design an outage probability constrained robust collaborative beamforming approach for the distributed multi-relay network in the downlink, where the channel state information (CSI) available is imperfect. We aim to minimize the total transmit power of the relay nodes whilst keeping the outage probability at the destination node below the predefined threshold. Assuming that the CSI mismatches follow Gaussian distribution, the equivalent counterpart for the outage probability constraint on the required signal-to-noise-ratio (SNR) is given explicitly. We show that though the original optimization problem is non-convex thus very intractable, it could be optimally solved by using the well-known interior-point method together with an efficient one-dimension search. Simulation results reveal that our proposed approaches can guarantee the quality-of-service (QoS) in term of outage probability in statistical sense while the non-robust scheme fails to do so.

In this study, the performances of selection cooperation are investigated in a scenario based on decode-and-forward and where equal gain combining (EGC) technique is adopted at the destination. Assuming that the channels suffer from independent non-identical Rayleigh fading, we first derive the cumulative distribution function, probability density function and moment generating function for the total instantaneous signal-to-noise ratio (SNR) at the destination after EGC. Then, these statistical functions are used to derive closed-form expressions for average SNR output, outage probability and average symbol error rate (SER) of selection cooperation. The results hold for arbitrary number of relays and refer to multiple-phase shift keying (M-PSK) modulations. Finally, simulations are carried out to verify the correctness of our theoretical analysis. A random network model is introduced to investigate the effect of relay number and path loss exponent on outage probability and average SER. In addition, based on this random network, the comparison between the performance of selection cooperation with EGC and that of selection cooperation with maximal ratio combining are performed.

In this paper, we propose a new model for the slow-start phase based on the discrete evolutions on the congestion window, and we use this slow-start model together with our improved TCP steady-state model to develop an extensive stochastic model which can more accurately predict the throughput and latency of short-lived TCP connections as functions of loss rate, round-trip time (RTT), and file size. The result from simulation experiments show that our model's performance predictions are up to 20% more accurate than the predictions obtained from the models proposed in [N. Cardwell et al., Mar. 2002] and [B. Sikdar et al., 2001].