Affine frequency division multiplexing (AFDM) has been considered as a promising waveform to enable high-reliable connectivity in the internet of vehicles. However, accurate channel estimation is critical and challenging to achieve the expected performance of the AFDM systems in doubly-dispersive channels. In this paper, we propose a sparse Bayesian learning (SBL) framework for AFDM systems and develop a dynamic grid update strategy with two off-grid channel estimation methods, i.e., grid-refinement SBL (GR-SBL) and grid-evolution SBL (GE-SBL) estimators. Specifically, the GR-SBL employs a localized grid refinement method and dynamically updates grid for a high-precision estimation. The GE-SBL estimator approximates the off-grid components via first-order linear approximation and enables gradual grid evolution for estimation accuracy enhancement. Furthermore, we develop a distributed computing scheme to decompose the large-dimensional channel estimation model into multiple manageable small-dimensional sub-models for complexity reduction of GR-SBL and GE-SBL, denoted as distributed GR-SBL (D-GR-SBL) and distributed GE-SBL (D-GE-SBL) estimators, which also support parallel processing to reduce the computational latency. Finally, simulation results demonstrate that the proposed channel estimators outperform existing competitive schemes. The GR-SBL estimator achieves high-precision estimation with fine step sizes at the cost of high complexity, while the GE-SBL estimator provides a better trade-off between performance and complexity. The proposed D-GR-SBL and D-GE-SBL estimators effectively reduce complexity and maintain comparable performance to GR-SBL and GE-SBL estimators, respectively.

Geo-Opportunistic Routing Networks (Geo-OR) within the Internet of Underwater Things (IoUT) are crucial for addressing the unique challenges of underwater communication environments. By leveraging geographic location data, these networks can opportunistically select forwarding nodes, a strategy that proves particularly effective in the dynamic and unpredictable nature of IoUT. However, uncertainty in forwarder selection and the inability to verify nodes’ claimed positions remain critical challenges. This paper investigates how uncertainty in next-hop selection arises in Geo-OR when viewed from the perspective of potential adversaries operating in a non-cooperative manner. By integrating an octree-based search strategy with an angle-constrained 3D localization algorithm, we accurately identify and locate “hot” nodes within Geo-OR. Geometric methods are employed to analyze vulnerable relay regions, pinpointing areas especially susceptible to malicious attacks. Based on this analysis, a location spoofing attack is proposed to disrupt legitimate node transmissions and degrade network performance. Simulation results reveal that the inherent uncertainties in Geo-OR expose the network to significant vulnerabilities, severely compromising throughput. This study highlights the critical weaknesses of Geo-OR, offering valuable insights into the most vulnerable regions and contributing to the design of targeted defense strategies to mitigate these risks.

Distributed Internet of Underwater Things (D-IoUT) has broad application prospects in marine monitoring, resource development, and security protection. However, the spatio-temporal uncertainty of underwater acoustic networks and the presence of multiple collision domains pose severe challenges to MAC-layer scheduling and interference management. To address these issues, this paper proposes an adaptive scheduling MAC protocol based on channel sensing and reinforcement decision-making (CSRD-ASMAC), aiming to provide an efficient and adaptive transmission scheduling scheme for D-IoUT. The protocol first introduces a hierarchical channel-state classification framework to accurately characterize the channel environment for each slot. Building on pre-divided basic slots, it integrates channel sensing and slot prediction into a reinforcement learning framework and employs a mixed-action exploration strategy to update Q-values, enabling each node to adaptively select the optimal transmission slot. Simulation results show that CSRD-ASMAC effectively improves overall network throughput in multi-collision domain environments.

In the complex, time-varying Internet of Underwater Things (IoUT), traditional hierarchical routing struggles to balance reliability, low delay, and energy efficiency due to measurement delays, routing voids and frequent transmission collisions. To address these challenges, we proposes HRT-TCOR, a high-reliability, time-reversal-based cross-layer opportunistic routing protocol that tightly integrates time reversal, reservation-based MAC, and opportunistic forwarding to collaboratively optimize the physical, data link, and network layers. First, lightweight beacons enable void sensing and preselection of candidate-forwarding sets (CFS). Next, a hybrid sender-receiver cooperation strategy forms the CFS, while a two-tier suppression mechanism eliminates redundant transmissions. Finally, pipelined handshakes merge reception acknowledgments with channel reservations to eliminate inter-hop idle gaps and ensure seamless end-to-end reliable delivery. Simulations show that, under both good and poor channel conditions, HRT-TCOR consistently achieves the highest packet delivery ratio, a lower delay, and competitive energy efficiency compared to other protocols.

Underwater acoustic sensor networks (UASNs) have attracted significant attention due to their potential applications in military surveillance and disaster early warning. However, due to the low data rate, high latency, and instability of underwater acoustic communication, UASNs are highly vulnerable to cascading failures triggered by the malfunction of key nodes, which can lead to network collapse. To address this challenge, we establish a cascading failure model tailored for dynamic traffic UASNs and propose a multi-criteria key node prediction (MC-KNP) algorithm based on the novel finding that nodes exhibit varying importance levels under different network traffic conditions. Although experimental results demonstrate that MC-KNP algorithm outperforms others approaches (e.g., degree, betweenness, and load) in accurately predicting key nodes for dynamic traffic UASNs, it suffers from high computational complexity. To address this limitation, we propose a key node prediction framework named KNP-GL, which integrates a graph convolutional network (GCN) to extract features that reflect both the structural roles of nodes and their potential impact on cascading failures, and a long short-term memory (LSTM) module to capture the temporal dynamics of cascading failures. Furthermore, based on the prediction results of KNP-GL framework, we design a mitigation strategy leveraging capacity expansion to improve network resilience against cascading failures. Experimental results show that KNP-GL framework achieves approximately 90% accuracy while reducing execution time from tens of seconds to tens of milliseconds. The proposed mitigation strategy further enhances network robustness, providing both theoretical insights and practical guidance for the development of high-reliability UASNs.

The recently proposed affine frequency division multiplexing (AFDM) modulation has been considered as a promising technology for narrowband doubly-dispersive channels. However, the time-scaling effects, i.e., pulse widening and pulse shortening phenomena, in extreme wideband doubly-dispersive channels have not been considered in the literatures. In this paper, we investigate such wideband transmission and develop an efficient transmission structure with chirp-periodic prefix (CPP) and chirp-periodic suffix (CPS) for AFDM system. We derive the input-output relationship of AFDM system under time-scaled wideband doubly-dispersive channels and demonstrate the sparsity in discrete affine Fourier (DAF) domain equivalent channels. We further optimize the AFDM chirp parameters to accommodate the time-scaling characteristics in wideband doubly-dispersive channels and verify the superiority of the derived chirp parameters by pairwise error probability (PEP) analysis. We also develop an efficient cross domain distributed orthogonal approximate message passing (CD-D-OAMP) algorithm for AFDM symbol detection and analyze its corresponding state evolution. By analyzing the detection complexity of CD-D-OAMP detector and evaluating the error performance of AFDM systems based on simulations, we demonstrate that the AFDM system with our optimized chirp parameters outperforms the existing competitive modulation schemes in time-scaled wideband doubly-dispersive channels. Moreover, our proposed CD-D-OAMP detector can achieve the desirable trade-off between the complexity and performance, while supporting parallel computing to significantly reduce the computational latency.

The development of the Internet of Underwater Things (IoUT) is of great significance to marine scientific research, deep-sea exploration and interdisciplinary data fusion. However, the existing medium access control (MAC) protocols usually face serious network congestion and unfair channel resource allocation challenges in large-scale IoUT, resulting in a decline in the overall performance of the system. To address these problems, this paper proposes a dynamic slot management MAC protocol based on POMDP (P-DSM-MAC). The protocol integrates network load estimation, ACB scheme optimization, dynamic slot allocation, and the sensing and multiplexing of idle/collision sub-slots through slot division to achieve efficient network resource management and sub-slot contention collision control. Simulation results show that P-DSM-MAC is significantly superior to existing protocols in key performance indicators such as network throughput, delay, and sub-slot contention collision rate, providing a feasible solution for the intelligent and dynamic optimization of IoUT in the future.

Traditional angle of arrival (AOA) localization in 3-D space typically requires sensors equipped with planar arrays, which incurs additional hardware costs. This limitation restricts its application in systems such as the Internet of Underwater Things (IoUT). Recent studies have shown that localization can also be achieved using sensor networks composed solely of linear arrays. However, most existing methods impose strict constraints on the orientation or placement of sensor arrays. Although some studies have proposed localization solutions free from these two limitations, such methods still exhibit two notable shortcomings: 1) high computational complexity that scales with network size, making them unsuitable for resource-constrained scenarios or large-scale sensor network deployments and 2) poor robustness to sensor position errors, where nodal deviations can significantly degrade localization accuracy. To address these challenges, this study proposes a novel computationally efficient 3-D source localization method based on 1-D AOA measurements from multiple linear arrays. Significantly, we innovatively incorporate a weighted least squares (WLSs) compensation model that effectively enhances the method’s robustness against sensor position errors. Experimental results demonstrate that: 1) while achieving the theoretical optimum localization accuracy as defined by the Cramér-Rao lower bound (CRLB), the proposed method shows significantly lower computational complexity than existing methods, with complexity independent of sensor network scale and 2) in practical scenarios with node position errors, our method outperforms other state-of-the-art methods in localization accuracy.

This paper aims to address the issue of low accuracy in underwater Direction of Arrival (DOA) estimation using Deep Learning (DL) methods, which arises due to the scarcity of underwater data caused by the difficulties in conducting underwater experiments. For multi-snapshot sampled signals, we segment the snapshots and reconstruct the task into a problem of processing few-snapshot data within an expanded dataset. By utilizing the new task, we employ a deep mutual learning (DML) model to enhance the accuracy of the original task's DOA estimates. Experimental results demonstrate that under conditions of small and limited datasets, our approach effectively improves the accuracy of DL-based DOA estimation methods.

Latest advances in multi-agent technology and hardware architecture design have made multivariate or multichannel data, often supported by graphs or networks, ubiquitous in recent scientific and engineering applications. Examples include smart grids management and sensor networks monitoring to cite a few. Multivariate empirical mode decomposition (MEMD), as a fully data-driven technique, has been shown to be effective in the multiscale analysis of non-stationary signals across multiple channels. However, it still lacks the capability to capture the dependency structure of signals over channels when supported by a graph, which limits the relevance of the provided analyses. This work aims to extend MEMD to temporal graph signals. To achieve this, non-uniform projections are processed to preserve smoothness relative to the topology of the graph. A noise-assisted mechanism is also proposed in order to adapt to the randomness of signals on vertices, eliminating mode mixing and misalignment phenomena. To further demonstrate the performance of the proposed graph multivariate empirical mode decomposition (GMEMD), and in particular of its noise-assisted counterpart, we validate its mode alignment property among same-index intrinsic mode functions and its efficacy as a filter bank. Simulations on both synthetic temporal graph signals and real-world electroencephalogram data support the analysis.

Data generated by network-structured applications, such as sensor networks or communication networks, typically reside on complex and irregular structures. These data necessitate specific graph signal processing tools to harness their characteristics. Detecting anomalous events in graph signals is significant in enhancing reliability of systems, where anomalies often activate localized groups of vertices. In this paper, we introduce a novel approach, the Joint Graph Wavelet Canonical Correlation Analysis, for detecting anomalies in graph signals through cooperative filtering while identifying their locations. This approach conducts canonical correlation analysis on graph signals to achieve data fusion within the wavelet domain while accounting for the graph topology. Subsequently, we devise an optimization algorithm specifically tailored for anomaly detection in graph signals. Finally, we illustrate its effectiveness through numerical simulations on synthetic data and by presenting test results from a multi-microphone network.

Underwater object detection tasks face significant challenges due to the complexity of aquatic environments and underwater conditions, coupled with the limited computational resources of current underwater equipment. The characteristics of online knowledge distillation, including its ability to compress models and enhance the performance and generalization capability of lightweight models, make it a highly suitable approach for underwater object detection tasks. Despite this, the key parameter setting of online knowledge distillation is still challenging in underwater image object detection, because of the difficulties in capturing the key information that differentiates different underwater objects. Therefore, we propose an online decoupled distillation framework based on prototype contrastive learning (PCD). The core idea of PCD is to facilitate knowledge transfer among a group of networks through prototype contrastive learning and to design decoupled distillation to enable the student model to learn more comprehensive and fine-grained knowledge. In the PCD model, we employ a contrastive loss to align the distributions of the student model and the teacher model in the feature space, enhancing its semantic structural properties. This enables the student model to achieve better feature representation. Innovatively, we use the average feature vector as the prototype to execute prototype contrastive learning to ensure model stability, thereby enhancing the detection capability of lightweight models in complex environments. On detectors, we designed cross-knowledge transfer and decoupled distillation loss to make the distillation of the logit output more comprehensive and effective. Our extensive experimental results demonstrate significant improvements in underwater object detection tasks and applicability to various types of dense detectors. Our PCD can improve the average precision of ResNet-based GFL detectors by 3.0–5.3, proving the effectiveness of our PCD model.

Underwater acoustic signal recognition (UASR) is significant for marine life and ecological environment protection. However, 2-D fixed-parameter inputs are inadequate for adapting to the variable underwater acoustic environment, and learnable-parameter inputs with inductive bias priors in transformers lead to difficulties in model convergence. Additionally, the differing optimization objectives between noise reduction and recognition methods can cause signal distortion, hindering recognition accuracy. To address these issues, this article proposes an innovative end-to-end soft threshold Swin Transformer model based on a multiloss self-distillation training strategy (ESTMST-ST) for robust recognition of weak underwater acoustic targets. Building on the previously proposed time-frequency Swin Transformer (TFST), we design a learnable dual filter module (LDFM) that decomposes underwater acoustic signals in the frequency direction, with parameters obtained through model training. To improve the model’s antinoise performance, we incorporate a soft threshold strategy within TFST to reduce nonstationary interference in underwater acoustic signals. For enhanced robustness and training efficiency, we introduce a self-distillation training strategy with four specific loss functions in selected stage in TFST. Using publicly available datasets, ShipsEar and DeepShip, we conduct three experiments: fixed signal-to-noise ratio (SNR) UASR, multi-SNR UASR, and model generalization ability tests. The experimental results demonstrate that ESTMST-ST achieves superior performance (at least a 1.6 improvement in${F}1$scores and a 2.2 improvement in kappa coefficients) compared to five state-of-the-art methods across two open-source datasets.

In underwater acoustic sensor networks (UANs), reliable packet delivery is critical in data collection and monitoring of the oceans. It primarily relies on the design of routing protocols to guarantee network durability and connectivity. However, utilizing routing design to achieve enhanced network longevity and reliable packet forwarding is challenging due to the complex underwater environment, unstable link connectivity, high transmission power, and high propagation latency. Thus, we propose a novel routing strategy called the longevity-oriented and reliable forwarding percolation (LRP) routing protocol. The goal of LRP is to ensure reliability by exploring multipath percolation-based routing and extend network longevity by energy control and residual energy optimization. Network reliability can be estimated using a built-in calculation model, and the source node controls energy and records the residual energy to extend the network lifetime. Utilizing an optimization of the network reliability requirement and residual energy, we develop a routing strategy to select the activated link set and node set for each hop in an energy-saving and reliable way. Moreover, a recursive approach is used to avoid the occurrence of void regions. Simulation results exhibit the effectiveness of the power control and routing strategy and demonstrate its superiority over the benchmarks in terms of network longevity and reliability.

Detecting weak nonlinear time series is critical in various applications, such as ocean monitoring, port security, coastal operations, and offshore activities. However, traditional methods for detecting such signals often require informative priors, leading to inefficiencies. This study proposes a novel approach that transforms nonlinear time series detection into network characterization through a weighted undirected similarity network construction method. The method integrates symmetric Kullback-Leibler divergence and complex network theory, transforming the node similarity measurement problem into a geometric problem on matrix manifolds. This method constructs a network representation of the time series data by measuring the similarity between data at different time scales. To demonstrate the effectiveness of our proposed approach, we conducted simulations and applied it to actual recorded data collected in the South China Sea. The synthetic data study showed that our method has a significant advantage in detecting weak nonlinear time series from ambient noise. Additionally, our approach successfully distinguished ship signals from marine ambient noise by comparing the network spectral values.

The motion of sensors during the measurement period, if not accounted for, can degrade significantly the localization accuracy. This paper investigates the sensor motion effect for the positioning of an object, using angles of arrival only or together with time of arrival measurements. The biases from the AOA and TOA data models when ignoring the motion effect are examined. Positioning algorithms for AOA localization and mixed AOA-TOA localization that account for the motion effect are developed by the pseudo-linear formulation. The algorithms derived include the computationally attractive closed-form estimators and the noise resilient semidefinite programming solutions. The bias coming from the pseudo-linear formulation is analyzed in detail, and it can be subtracted from the closed-form solution to obtain a bias-suppressed estimate. Simulation validates the effectiveness of the proposed solutions in achieving the CRLB performance under Gaussian noise before the thresholding effect occurs.

Underwater acoustic sensor networks (UANs) have been shown as a promising technology to monitor and explore the oceans. Nevertheless, the routing design for data gathering of UANs considering the acoustic channel communication characteristics and limited energy is a pressing, open issue. To address this challenge, we propose the long-lifetime and reliable percolation routing protocol (LRP) for UANs to ensure the reliability of the network and prolong the network lifetime. The proposed protocol adaptively selects the forwarders to deliver each message. By estimating the reliability of the next-hop and considering the remaining energy of candidates, the proposed protocol takes a recursive approach to avoid trapping in a locally optimal solution. Simulations results validate the feasibility of the proposed protocol and demonstrate its superiority over the existing routing algorithms by prolonging the lifetime of LRP by up to 35%.

Despite the remarkable progress of deep learning on speech recognition and music processing, it is still challenging to classify general audio signals due to the high cost of collection and annotation of the samples. The ability to learn discriminative features from a small dataset makes deep metric learning a promising method for general audio classification. However, because of the difficulty in mining informative sample pairs, it usually suffers from slow convergence or even poor local minima. In this letter, to improve classification performance by exploiting the advantages of both the weight-based loss and the metric-based loss, we proposed a multi-positive metric loss and a framework to joint it with the common softmax loss. The proposed method eliminates the need for sub-loss weighting by measuring the similarity between samples in a consistent probabilistic form. It also enhances the classification performance by improving the estimation of the intra-class and inter-class relationships from multiple positive samples. Finally, we evaluated the proposed method on the ShipsEar dataset and the Ocean Networks Canada dataset, and the results verified its effectiveness.

Accurate location information is critical for many engineering applications (e.g., radar, sonar, autonomous robots, intelligent transportation systems). In traditional source localization algorithms, the perfect knowledge of noisy Time-of-Arrival (ToA) measurements are assumed to be obtained by the fusion center in a sensor network. This assumption is not practical for wireless sensor networks, especially for a resource-limited sensor network with stringent power and communication bandwidth constraints. In this paper, we propose a novel channel-aware source localization method based on quantized asynchronous ToA measurements, where the quantization errors as well as the imperfect communication link between each sensor and the fusion center are considered. The maximum-likelihood (ML) source localization by jointly estimating the signal transmission instant and source location is formulated. An efficient relaxation is provided to transform the non-convex ML optimization problem into a convex problem. The Cramér-Rao lower bounds (CRLBs) for the quantized ToA measurements with the uncertainty of data exchange are derived. Furthermore, a Fisher information based heuristic quantization scheme is proposed to design quantized thresholds for asynchronous ToA measurements. The simulation and experimental results demonstrate that our proposed method can yield an efficient estimate under different scenarios.

Accurate localization of an autonomous underwater vehicle (AUV) is essential in many applications. The motion of an AUV during the measurement acquisition period can be significant and the localization performance can suffer considerably if it is neglected. A new time delay model that accounts for the motion is proposed for moving AUV localization. The non-recursive form of the proposed model is next derived. An algebraic explicit positioning solution based on the non-recursive model is developed when the measurement noise and transponder location errors are present. Simulation results illustrate the importance of accounting for AUV motion in localization, and validate the theoretical analysis that the proposed solution can reach the Cramér-Rao lower bound (CRLB) accuracy over the small error region under Gaussian noise.

We study the problem of distributed estimation over adaptive networks where communication delays exist between nodes. In particular, we investigate the diffusion Least-Mean-Square (LMS) strategy where delayed intermediate estimates (due to the communication channels) are employed during the combination step. One important question is: Do the delays affect the stability condition and performance? To answer this question, we conduct a detailed performance analysis in the mean and in the mean-square-error sense of the diffusion LMS with delayed estimates. Stability conditions, transient and steady-state mean-square-deviation (MSD) expressions are provided. One of the main findings is that diffusion LMS with delays can still converge under the same step-sizes condition of the diffusion LMS without delays. Finally, simulation results illustrate the theoretical findings.

Bursty data links can benefit directly from the removal of pilot symbol transmission for channel estimation by improving the spectral efficiency. For such networking scenarios including data or paging signals, blind equalization for joint data detection and channel estimation with few or no pilot can improve spectrum efficiency. Though some existing works typically have attempted to take advantage of the sparsity of multipath channels, substantial performance improvement remains elusive. In this work, we develop an iterative Markov chain Monte Carlo algorithm based on Gibbs sampling designed for sparse channels. We incorporate the channel sparsity in the form of an l1type prior probability distribution, and derive the posterior channel distribution via stochastic sampling. Furthermore, we propose transmitter and receiver structures that could resolve unknown phase ambiguity in frequency-selective channels. This algorithm is also generalizable to non-sparse channels.

This paper proposes a low-complexity solution for the data protection problem without newcomer nodes in Internet of Things (IoT) scenarios, i.e., when device losses cannot be replaced by new devices. Application scenarios include environmental monitoring, data collection, and industrial automation. Although the optimal solution and optimization framework have been studied in previous work to minimize the network costs and storage capacity requirements, this paper shows that the optimal solution has a high complexity as the number of devices increases. Given the massive number of IoT devices, we propose a relaxation to the cut capacity constraints that (a) guarantees data recoverability, (b) achieves the minimum network use, and (c) reduces the problem's complexity dramatically. Our numerical results show that the proposed relaxation allows us to change the computational scaling of the problem. More specifically, we show that the time taken to compute the optimal transmission policy with the relaxation for a system with 800 devices is the same as the time it takes the optimal solution to solve the case of 15 devices.

We study the problem of parametric modeling of network-structured signals with graph filters. To benefit from the properties of several graph shift operators simultaneously, and to enhance interpretability, we investigate combinations of parallel graph filters with different shift operators. Due to their extra degrees of freedom, these models might suffer from over-fitting. We address this problem through a weighted ℓ2-norm regularization formulation to perform model selection by encouraging group sparsity. What makes this formulation interesting is that it is actually a smooth convex optimization problem. Experiments on real-world data structured by undirected and directed graphs show the effectiveness of this method.

The large propagation delay of underwater acoustic signals significantly affects the throughput performance of underwater communication networks. While past research focused on mitigating the impact of large propagation delays, recent work has suggested exploiting large delays. In this paper, we consider an underwater linear unicast network which employs a time- division based scheduling strategy to exploit large propagation delays to improve network throughput. We assume the protocol model in a network with partially overlapping collision domains, where the transmission range is normalized as 1 and the interference range is an integer k. We systematically discuss the throughput of the linear networks with single traffic flow, showing that the average throughput of an N-node linear network with single traffic flow cannot exceed (N-1)/k. We then propose a general transmission scheduling strategy that can achieve the throughput upper bound and also give some examples of the optimal schedules.

This study investigates the blind data detection in two‐way relay networks (TWRN) that employ amplify‐and‐forward (AF) relay strategy. To blindly detect the data in TWRN in the presence of uncertain time‐frequency offsets and phase noise, the authors develop a new Bayesian‐based approximate Gibbs algorithm based on truncated Taylor series expansion approximation. In addition, the authors exploit available constraint information on parameters of interest. The authors present three receivers based on three different parameter estimation approaches. The authors further discuss the implementation issue and present diagnostic convergence analysis. The authors’ numerical results demonstrate the performance and efficacy of their proposed algorithm.

This work investigates joint estimation of symbol timing synchronization and channel response in two-way relay networks (TWRN) that utilize amplify-and-forward (AF) relay strategy. With unknown relay channel gains and unknown timing offset, the optimum maximum likelihood (ML) algorithm for joint timing recovery and channel estimation can be overly complex. We develop a new Bayesian based Markov chain Monte Carlo (MCMC) algorithm in order to facilitate joint symbol timing recovery and effective channel estimation. In particular, we present a basic Metropolis-Hastings algorithm (BMH) and a Metropolis-Hastings-ML (MH-ML) algorithm for this purpose. We also derive the Cramer-Rao lower bound (CRLB) to establish a performance benchmark. Our test results of ML, BMH, and MH-ML estimation illustrate near-optimum performance in terms of mean-square errors (MSE) and estimation bias. We further present bit error rate (BER) performance results.

We present investigation results on joint estimation of symbol timing synchronization and channel response in two-way multiple antenna relay networks that utilize amplify-and-forward (AF) relay strategy. With practically unknown relay channel gains and timing offset, optimum maximum likelihood (ML) estimation for joint timing recovery and channel estimation can be prohibitively complex. We develop a new Bayesian based Markov chain Monte Carlo (MCMC) algorithm and generalize our previous principle to a multiple antenna relay network to accomplish joint symbol timing recovery and effective channel estimation. Simulation results are provided to demonstrate the performance of the proposed algorithm.