Linear Minimum Mean Square Error Research Articles

Phase Noise Effects on OFDM Chirp Communication Systems: Characteristics and Compensation

Orthogonal frequency-division multiplexing (OFDM) chirp waveforms are an attractive candidate to be a dual-function signal scheme for the joint radar and communication systems. OFDM chirp signals can not only be employed to transmit communication data through classic phase modulation, but also can perform radar detection by applying linear frequency modulation for subcarriers. However, the performance of the OFDM chirp communication system under the phase noise environment still remains uninvestigated. This paper tries to discuss the influence of phase noise on OFDM chirp communication systems and proposes effective phase noise estimation and compensation methods. We find that the phase noise effect on OFDM chirp communication systems consists of a common phase error (CPE) and an inter-carrier interference (ICI). If not compensated, the performance of the dual-function systems can be seriously degraded. In particular, an exact expression for the signal-plus-interference to noise ratio (SINR) for the OFDM chirp communication system is derived and some critical parameters are analyzed to exhibit the phase noise effects on system performance. Moreover, two low-complexity estimation approaches, maximum likelihood (ML) and linear minimum mean square error (LMMSE), as well as two compensation approaches, the de-correlation and cancellation algorithms, are respectively utilized to eliminate the phase noise impairment. Finally, the phase noise effects and the effectiveness of the compensation approach are verified by extensive numerical results.

An inspired chaos-based estimation-theory optimization for low-density parity-check (LDPC) code decoding

Optimization techniques have emerged as powerful tools for solving complex engineering problems in real-time applications, surpassing traditional heuristic methods. This paper introduces a novel chaos optimization algorithm that integrates the bisection search method to enhance the convergence ability, solution accuracy, and search speed of conventional optimization problems. The proposed algorithm is applied to the design and analysis of a reduced complexity decoding scheme using Low-Density Parity-Check (LDPC) codes. By leveraging the Linear Minimum Mean Square Error (LMMSE) estimation criterion, the proposed method approximates the estimated parameters acquired during the message computation process. This combined approach effectively mitigates error propagation and premature convergence issues. The presented research offers a low-complexity decoding algorithm for LDPC codes, leveraging an optimized chaos bisection search method. The adoption of the LMMSE criterion overcomes data dependency and error propagation concerns during the estimation process. The modified chaos optimization technique, integrated with the bisection search, efficiently determines optimal values from perturbed observations, leading to improved convergence rates and data processing precision by reducing the observation dimension (search space). Simulation results validate the proposed approach, showcasing its ability to mitigate error propagation and premature convergence while maintaining coding gain accuracy. Experimental validation on an FPGA RF-SoC device (model XCZU27DR) demonstrates that the proposed method achieves a maximum throughput of 8.7 Gbps at a clock frequency of 386.72 MHz, with reduced hardware complexity. The experimental verification on an FPGA RF-SoC device highlights the algorithm's minimal hardware complexity, superior decoding throughput, and Bit Error Rate (BER) performance.

Channel reconstruction for mmWave Massive MIMO systems based on Channel Path Map

For the needs of high data rate and massive device access in current fifth-generation (5G) mobile communication networks and future sixth-generation (6G) mobile communication networks, Millimeter-Wave (mmWave) Massive Multiple-Input Multiple-Output (MIMO) is a crucial technical support. However, the conventional pilot based channel estimation methods are exceedingly time-consuming and substantially degrade system performance when the number of users or antennas is high. This paper proposes a Path Clustering based Channel Reconstruction (PCCR) scheme for channel reconstruction in order to solve this issue. By leveraging channel information (such as angle of arrival, angle of departure, and path gain) at a number of geographical locations that are recorded in the Channel Path Map (CPM), we aim to reconstruct the mmWave channel of any locations of interests. To be specific, the CPM is used to extract the neighborhood propagation path data, and an autonomous determination algorithm is used to find the number of path clusters at the location of interest. The recorded propagation path information is processed by the Equivalent Base Station (EBS) scheme, and the path clustering is accomplished by using the AGglomerative NESting (AGENS) algorithm. In each path cluster, Inverse Distance Weighted (IDW) algorithm is developed to forecast the path information at the user location. With all path information at user location are forecast, the channel can be reconstructed. The anticipated overhead is substantially lower than that of conventional methods because the computational cost of the procedure only depends on the total number of paths at the nearby locations. Ray-tracing based simulations are carried out to demonstrate the performance of the proposed algorithm. The results show that, when applied to a CPM with a spatial location precision of 5 m, the Normalized Mean Squared Error (NMSE) of the proposed algorithm is nearly identical to that of Linear Minimum Mean Square Error (LMMSE) estimation. The actual achievable date rate can match the needs of upcoming wireless communications and is superior over the conventional LMMSE.

Enhanced MIMO-DCT-OFDM system using cosine domain equaliser

ABSTRACT The Discrete Cosine Transform (DCT) can be used instead of the conventional Discrete Fourier Transform (DFT) for the Orthogonal Frequency Division Multiplexing (OFDM) construction, which offers many advantages. In this paper, the Multiple-Input-Multiple-Output (MIMO) DCT-OFDM is enhanced using a proposed Cosine Domain Equaliser (CDE) instead of a Frequency Domain Equaliser (FDE). The results are evaluated through the Rayleigh fading channel with Co-Carrier Frequency Offset (Co-CFO) of different MIMO configurations. The average bit error probability and the simulated time of the proposed scheme and the conventional one are compared, which indicates the importance of the proposed scheme. Also, a closed formula for the number of arithmetic operations of the proposed equaliser is developed. The proposed equaliser gives a simulation time reduction of about 81.21%, 83.74% compared to that of the conventional Linear Zero Forcing (LZF)-FDE and Linear Minimum Mean Square Error (LMMSE)-FDE, respectively, for the case of a 4 × 4 configuration.

Achievable Rate Analysis of the STAR-RIS-Aided NOMA Uplink in the Face of Imperfect CSI and Hardware Impairments

Reconfigurable intelligent surfaces (RIS) are capable of beneficially ameliorating the propagation environment by appropriately controlling the passive reflecting elements. To extend the coverage area, the concept of simultaneous transmitting and reflecting reconfigurable intelligent surfaces (STAR-RIS) has been proposed, yielding supporting 360° coverage user equipment (UE) located on both sides of the RIS. In this paper, we theoretically formulate the ergodic sum-rate of the STAR-RIS assisted non-orthogonal multiple access (NOMA) uplink in the face of channel estimation errors and hardware impairments (HWI). Specifically, the STAR-RIS phase shift is configured based on the statistical channel state information (CSI), followed by linear minimum mean square error (LMMSE) channel estimation of the equivalent channel spanning from the UEs to the access point (AP). Afterwards, successive interference cancellation (SIC) is employed at the AP using the estimated instantaneous CSI, and we derive the theoretical ergodic sum-rate upper bound for both perfect and imperfect SIC decoding algorithm. The theoretical analysis and the simulation results show that both the channel estimation and the ergodic sum-rate have performance floor at high transmit power region caused by transceiver hardware impairments.

Nonlinear Equalization for Performance Enhancement of FBMC Systems Using Walsh Transform

ABSTRACT The Filter Bank Multi-Carrier (FBMC) is an interesting recent technology for next-generation wireless radio communication systems. Equalization is a fundamental technique for enhancing Bit Error Rate (BER) performance. The traditional Linear Zero Forcing (LZF) and Linear Minimum Mean Square Error (LMMSE) equalizers have large noise amplification and high complexity, respectively. As the estimated error of the Signal-to-Noise Ratio (SNR) grows, the BER performance of the LMMSE equalizer degrades. In this paper, we present a Joint Low Complexity Optimized ZF-Successive Interference Cancellation (JLCOZF-SIC) equalizer for FBMC communication systems that are based on the Walsh–Hadamard Transform (WHT) rather than the Fast Fourier Transform (FFT). As in the MMSE-Successive Interference Cancellation (MMSE-SIC) equalizer, the proposed equalizer avoids the estimating error induced by average power per symbol estimation. The simulation results show that the proposed algorithm improves the system performance more than the existing algorithms. Furthermore, the proposed equalization provides stability in the BER performance at high levels of the co-Carrier Frequency Offset (co-CFO). In terms of computational complexity, the proposed scheme has a lower computational complexity than traditional schemes.

Autoencoder assisted subcarrier optimization for nonlinear frequency division multiplexing.

Nonlinear frequency division multiplexing (NFDM) is a novel optical communication technique that can achieve nonlinear free transmission. However, current design of NFDM is analogous to orthogonal frequency division multiplexing (OFDM), where sinc function is utilized as subcarriers, which may not be optimal for nonlinear spectrums. In this paper, we propose an auto-encoder (AE) assisted subcarrier optimization scheme for dual-polarized (DP) NFDM systems. Numerical verifications show that our scheme can improve the Q-factor by 1.54 dB and 0.62 dB compared to sinc subcarrier and linear minimum mean square error (LMMSE) equalization, respectively, in a 960 km transmission scenario. We also analyze the characteristics of the optimized subcarriers and discuss how they enhance the performance. Furthermore, we demonstrate the robustness of the optimized subcarriers to different modulation formats, transmission distances and bandwidth. Our work provides a new idea in subcarrier design for NFDM.

Data-Aided CSI Estimation Using Affine-Precoded Superimposed Pilots in Orthogonal Time Frequency Space Modulated MIMO Systems

An orthogonal affine-precoded superimposed pilot (AP-SIP)-based architecture is developed for the cyclic prefix (CP)-aided single input single output (SISO) and multiple input multiple output (MIMO) orthogonal time frequency space (OTFS) systems relying on arbitrary transmitter-receiver (Tx-Rx) pulse shaping. The data and pilot symbol matrices are affine-precoded and superimposed in the delay Doppler (DD)-domain followed by the development of an end-to-end DD-domain relationship for the input-output symbols. At the receiver, the decoupled pilot and data symbol are extracted by employing orthogonal precoder matrices, which eliminates the mutual interference. Furthermore, a novel pilot-aided Bayesian learning (PA-BL) technique is conceived for the channel state information (CSI) estimation of SISO OTFS systems based on the expectation-maximization (EM) technique. Subsequently, a data-aided Bayesian learning (DA-BL)-based joint CSI estimation and data detection technique is proposed, which beneficially harnesses the estimated data symbols for improved CSI estimation. In this scenario our sophisticated data detection rule also integrates the CSI uncertainty of channel estimation into our the linear minimum mean square error (LMMSE) detectors. The AP-SIP framework is also extended to MIMO OTFS systems, wherein the DD-domain input matrix is affine-precoded for each transmit antenna (TA). Then an EM algorithm-based PA-BL scheme is derived for simultaneous row-group sparse CSI estimation for this system, followed also by our data-aided DA-BL scheme that performs joint CSI estimation and data detection. Moreover, the Bayesian Cramer-Rao bounds (BCRBs) are also derived for both SISO as well as MIMO OTFS systems. Finally, simulation results are presented for characterizing the performance of the proposed CSI estimation techniques in a range of typical settings along with their bit error rate (BER) performance in comparison to an ideal system having perfect CSI.

Deep Learning OFDM Receivers for Improved Power Efficiency and Coverage

In this article, we propose multiple machine learning (ML) based physical-layer receiver solutions for demodulating orthogonal frequency-division multiplexing (OFDM) signals that are subject to high level of nonlinear distortion. Specifically, three novel deep learning based convolutional neural network receivers are devised, containing layers in time- and/or frequency-domains, allowing to demodulate and decode the transmitted bits reliably despite the high error vector magnitude (EVM) in the transmit signal. Applicable training procedures are also described, such that the learned layers in the receiver processing properly generalize over different nonlinear distortion and multipath channel characteristics. Extensive set of numerical results is provided, in the context of 5G NR uplink (UL) incorporating also measured terminal power amplifier (PA) characteristics. The obtained results show that the proposed receiver systems are able to clearly outperform the classical linear minimum mean-squared error (LMMSE) receiver as well as the existing ML receiver approaches, especially when the EVM is high compared to modulation order. This is particularly so when the devised ML receiver is of hybrid nature with layers both in time and frequency. The proposed ML receivers can thus facilitate pushing the terminal PA systems deeper into saturation, and thereon improve the terminal power-efficiency, radiated power and network coverage. Through combining the obtained radio link performance results with link budget calculations, all carried out at the 28 GHz mmWave band, it is shown that the proposed ML receivers can enhance the network coverage in terms of maximum UL link distances by close to 100%, when compared to classical LMMSE receiver based networks.

MSTGAN: A Multi-Slot Conditional Generative Adversarial Network Based on Swin Transformer for Channel Estimation

Accurate channel estimation is the key to enhance the transmission performance of communication systems. In this letter, we propose a multi-slot conditional generative adversarial network (cGAN) based on Swin Transformer called MSTGAN for channel estimation in a single-input single-output (SISO) scenario. Specifically, the proposed MSTGAN could learn the temporal correlation feature from continuous channel slots by the 3D convolution and extract the deep feature well by the Swin Transformer to improve the accuracy of channel estimation. The model is trained with data augmentation to reduce the computational cost for offline training. The simulation results demonstrate that the proposed method outperforms the linear minimum mean square error (LMMSE) method and other deep learning methods. Furthermore, extension schemes of the MSTGAN to the multiple-input multiple-output (MIMO) case are also provided.

Super resolution direction finding technique of vortex electromagnetic wave radar in missing mode

Vortex electromagnetic waves have superior performance over electromagnetic waves. In order to improve the radar super-resolution lateralization technique in its missing modes, this study proposes to start from the perspective of mode missing. The missing modes are reconstructed using the Adaptive Step Size Gradient Descent (ASSGD) method by exploiting the features of the missing modes. The linear minimum mean square error (LMMSE) estimation method is also used to solve the problem of poor reconstruction accuracy due to the Missing Modes. The Missing Modes Iterative Adaptive Approach (MMIAA) algorithm and Missing Modes Sparse Learning via Iterative Minimization (MMSLIM) algorithm are then used. Minimization (MMSLIM) algorithm to recover missing modes. The results showed that the RMSEs of the recovery errors of MMSLIM, MMIAA and ASSGD were 0.16, 0.31 and 0.82 respectively at a modal missing ratio of 0.7, while ASSGD fails to recover the missing modal data at a modal missing ratio of 0.9. The overall data quality of the azimuthally estimated RMSE was average when the signal-to-noise ratio was at [–5, 10] dB. And the curve becomes flatter when it reaches 15 dB or more, indicating that MMSLIM, MMIAA has important theoretical and practical value.

Channel Estimation for RIS-Assisted MIMO Systems in Millimeter Wave Communications

The large number of estimated parameters in a reconfigurable intelligent surface (RIS) makes it difficult to achieve accurate channel estimation accuracy in 6G. Therefore, we suggest a novel two-phase channel estimation framework for uplink multiuser communication. In this context, we propose an orthogonal matching pursuit (OMP)-based linear minimum mean square error (LMMSE) channel estimation approach. The OMP algorithm is used in the proposed algorithm to update the support set and pick the columns of the sensing matrix that are most correlated with the residual signal, which successfully reduces pilot overhead by removing redundancy. Here, we use LMMSE’s advantages for handling noise to address the problem of inadequate channel estimation accuracy when the signal-to-noise ratio (SNR) is low. Simulation findings demonstrate that the proposed approach outperforms least-squares (LS), traditional OMP, and other OMP-based algorithms in terms of estimate accuracy.

Intelligently Wireless Batteryless RF-Powered Reconfigurable Surface: Theory, Implementation & Limitations

This work exploits ultra-low cost, commodity, radio frequency identification (RFID) tags as elements of a reconfigurable intelligent surface (RIS). Such batteryless tags are powered and controlled by a software-defined radio reader, with properly modified software, so that a source-destination link is assisted, operating at a different band. Signal model includes small-scale and large-scale fading, direct link, as well as specific parameters relevant to reflection (i.e., backscatter) radio, such as antenna structural mode and reflection efficiency, typically overlooked in the literature. An algorithm is offered that computes the optimal RIS configuration with complexity of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> log <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> ) in number of elements <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> , instead of intractable exponential complexity of exhaustive search, while accommodating any number <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> ≥ 2 of loads. With the proposed algorithm, it is shown that performance gains reach a plateau for constant element spacing and increasing number of elements, suggesting that the weak, passive nature of backscattered links limits the performance gains, even with perfect channel estimation. Channel estimation with linear minimum mean squared error (LMMSE) estimator is shown to be effective, provided that there are sufficient number of pilot symbols. A concrete way is offered to design and prototype a wireless, batteryless, RF-powered, reconfigurable surface and a proof-of-concept is experimentally demonstrated.

IRS-Assisted MISO System With Phase Noise: Channel Estimation and Power Scaling Laws

Recent studies have shown that intelligent reflecting surfaces (IRS) can significantly improve the spectral and energy efficiency of wireless communication links. However, most works assume perfect transceivers and IRS, which is impractical in real communication systems. In this work, we study the effect of the hardware impairments in IRS-assisted MISO systems with single user, where we consider both phase noise caused by the imperfect transceivers and IRS. To this end, we propose a linear minimum mean square error (LMMSE) channel estimation algorithm that takes the phase noise into account. Furthermore, we study the impact of phase noise on the downlink performance of an IRS-assisted system. Both analytical and numerical results are presented, where we prove that the transceiver phase noise can be compensated with the optimized IRS, while the IRS phase noise remains but is not exacerbated. As the number of reflective elements <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> approaches infinity, the IRS phase noise results in a constant loss in terms of the ergodic rate. Moreover, we find that the phase noise has no impact on the scaling laws. If the direct channel is blocked and as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> → ∞, the transmit power can be scaled down by 1/ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> and 1/ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , respectively, for random and optimized IRS, without compromising the received signal to noise ratio (SNR).

81 Gb/s Dual-Polarization Discrete Eigenvalue Transmission Based on Nonlinear Fourier Transform

We investigated dual-polarization (DP) discrete eigenvalue transmissions and propose to shift the solitons in one polarization by half a symbol period with respect to the soliton train in the other polarization. Simulation results are conducted to illustrate its advantages over conventional schemes. Experiments of a net 81 Gb/s dual-polarization discrete eigenvalue transmissions over 650 km are also demonstrated with the help of DP multi-symbol linear minimum mean-squared error (DP-MS-LMMSE) filters for eigenvalues and b-coefficients.

Two-Timescale Design for Reconfigurable Intelligent Surface-Aided Massive MIMO Systems With Imperfect CSI

This paper investigates the two-timescale transmission scheme for reconfigurable intelligent surface (RIS)-aided massive multiple-input multiple-output (MIMO) systems, where the beamforming at the base station (BS) is adapted to the rapidly-changing instantaneous channel state information (CSI), while the nearly-passive beamforming at the RIS is adapted to the slowly-changing statistical CSI. Specifically, we first consider a system model with spatially independent Rician fading channels, which leads to tractable expressions and offers analytical insights on the power scaling laws and on the impact of various system parameters. Then, we analyze a more general system model with spatially correlated Rician fading channels and consider the impact of electromagnetic interference (EMI) caused by any uncontrollable sources present in the considered environment. For both case studies, we apply the linear minimum mean square error (LMMSE) estimator to estimate the aggregated channel from the users to the BS, utilize the low-complexity maximal ratio combining (MRC) detector, and derive a closed-form expression for a lower bound of the achievable rate. Besides, an accelerated gradient ascent-based algorithm is proposed for solving the minimum user rate maximization problem. Numerical results show that, in the considered setup, the spatially independent model without EMI is sufficiently accurate when the inter-distance of the RIS elements is sufficiently large and the EMI is mild. In the presence of spatial correlation, we show that an RIS can better tailor the wireless environment. Furthermore, it is shown that deploying an RIS in a massive MIMO network brings significant gains when the RIS is deployed close to the cell-edge users. On the other hand, the gains obtained by the users distributed over a large area are shown to be modest.

Information Rates for Channels with Fading, Side Information and Adaptive Codewords.

Generalized mutual information (GMI) is used to compute achievable rates for fading channels with various types of channel state information at the transmitter (CSIT) and receiver (CSIR). The GMI is based on variations of auxiliary channel models with additive white Gaussian noise (AWGN) and circularly-symmetric complex Gaussian inputs. One variation uses reverse channel models with minimum mean square error (MMSE) estimates that give the largest rates but are challenging to optimize. A second variation uses forward channel models with linear MMSE estimates that are easier to optimize. Both model classes are applied to channels where the receiver is unaware of the CSIT and for which adaptive codewords achieve capacity. The forward model inputs are chosen as linear functions of the adaptive codeword's entries to simplify the analysis. For scalar channels, the maximum GMI is then achieved by a conventional codebook, where the amplitude and phase of each channel symbol are modified based on the CSIT. The GMI increases by partitioning the channel output alphabet and using a different auxiliary model for each partition subset. The partitioning also helps to determine the capacity scaling at high and low signal-to-noise ratios. A class of power control policies is described for partial CSIR, including a MMSE policy for full CSIT. Several examples of fading channels with AWGN illustrate the theory, focusing on on-off fading and Rayleigh fading. The capacity results generalize to block fading channels with in-block feedback, including capacity expressions in terms of mutual and directed information.

Channel estimation for OFDM-based indoor broadband power line communication systems

State-of-the-art indoor broadband power line communications (PLC) systems use orthogonal frequency division multiplexing (OFDM) signals with constellations up to 10 bits/symbol, which makes channel estimation a key aspect. This paper focuses on the initial channel estimation used in the payload, which might be further adapted by means of a decision-directed strategy. This initial estimate has to be computed on a per-frame basis and can be accomplished from the preamble, the header symbol (assuming that it has been correctly decoded) or from a combination of both. This work proposes simple channel estimation techniques for this problem and derives their most appropriate parameters. They are compared with the linear minimum mean squared-error (LMMSE) estimator and others commonly used in wireless scenarios by considering both their performance and computational complexity. The factors limiting the performance of the estimators based on the preamble and the header symbols are analyzed and the implications of the differences between PLC and wireless channels, such as the absence of fading, in the design of estimation techniques that require knowledge of the channel statistics are discussed. A performance analysis of the considered techniques is accomplished in a set of 171 measured indoor PLC channels. Obtained results indicate that estimators from the header perform better than those based on the preamble symbols and we provide a computationally simple estimator that gives nearly optimum performance by combining estimates from both the preamble and the header symbols.

A Best Linear Empirical Bayes Method for High-Dimensional Covariance Matrix Estimation

Covariance matrix estimation plays a significant role in both in the theory and practice of portfolio analysis and risk management. This paper deals with the available data prior to developing a factor model to enhance covariance matrix estimation. Our work has two main outcomes. First, for a general linear model with unknown prior parameters, a class of best linear empirical Bayes estimators is established through two kinds of architectures to improve the estimation accuracy by utilizing additional data prior. The theoretical results indicate two key points: the proposed estimators are equivalent to the linear minimum mean-square error estimator when complete or sufficient partial data prior are provided; and the proposed estimators perform better than the optimal weighted least squares method, which ignores the data prior in each situation. Second, the proposed estimators are used for calculating a high-dimensional covariance matrix through factor models. The numerical example and the simulation results verify the effectiveness of our methods.

Bayesian Over-the-Air Computation

As an important piece of the multi-tier computing architecture for future wireless networks, over-the-air computation (OAC) enables efficient function computation in multiple-access edge computing, where a fusion center aims to compute a function of the data distributed at edge devices. Existing OAC relies exclusively on the maximum likelihood (ML) estimation at the fusion center to recover the arithmetic sum of the transmitted signals from different devices. ML estimation, however, is much susceptible to noise. In particular, in the misaligned OAC where there are channel misalignments among received signals, ML estimation suffers from severe error propagation and noise enhancement. To address these challenges, this paper puts forth a Bayesian approach by letting each edge device transmit two pieces of statistical information to the fusion center such that Bayesian estimators can be devised to tackle the misalignments. Numerical and simulation results verify that, 1) For the aligned and synchronous OAC, our linear minimum mean squared error (LMMSE) estimator significantly outperforms the ML estimator. In the low signal-to-noise ratio (SNR) regime, the LMMSE estimator reduces the mean squared error (MSE) by at least 6 dB; in the high SNR regime, the LMMSE estimator lowers the error floor of MSE by 86.4%; 2) For the asynchronous OAC, our LMMSE and sum-product maximum a posteriori (SP-MAP) estimators are on an equal footing in terms of the MSE performance, and are significantly better than the ML estimator. Moreover, the SP-MAP estimator is computationally efficient, the complexity of which grows linearly with the packet length.