Variational Bayesian Channel Estimation and Data Detection for Cell-Free Massive MIMO with Low-Resolution Quantized Fronthaul Links

We study the joint channel estimation and data detection (JED) problem in a cell-free massive multiple-input multiple-output (CF-mMIMO) network, where access points (APs) communicate with a central processing unit (CPU) over fronthaul links. However, the bandwidth of these links is limited, and thus, presents challenges to the applicability of CF-mMIMO, especially with an ever-increasing number of users. To address this, we propose a method based on variational Bayesian (VB) inference for performing the JED process, where the APs forward low-resolution quantized versions of the signals to the CPU. We consider two approaches: \emph{quantization-and-estimation} (Q-E) and \emph{estimation-and-quantization} (E-Q). In the Q-E approach, each AP uses a low-bit quantizer to quantize the signal before forwarding it to the CPU, while in the E-Q approach, each AP first performs local channel estimation and then sends a low-bit quantized version of the estimated channel to the CPU. We evaluate the performance of our VB-based approach under perfect fronthaul link (PFL) with unquantized received signals, Q-E, and E-Q in terms of symbol error rate (SER), normalized mean square error (NMSE) of the channel estimation, computational complexity, and fronthaul signaling overhead. We also compare these results with those of the linear minimum mean squared error (LMMSE) method under the PFL scenario. Our numerical results show that both the VB(Q-E) and VB(E-Q) approaches achieve superior performance compared to LMMSE(PFL), benefiting from the nonlinear modeling inherent in VB. Furthermore, the VB(Q-E) method outperforms VB(E-Q) due to errors in the local channel estimation process at the APs within the VB(E-Q) approach.

Variational Bayesian Inference for Time-Varying Massive MIMO Channels: Estimation and Detection

Massive multiple-input multiple-output (MIMO) stands as a key technology for advancing performance metrics such as data rate, reliability, and spectrum efficiency in the fifth generation (5G) and beyond of wireless networks. However, its efficiency depends greatly on obtaining accurate channel state information. This task becomes particularly challenging with increasing user mobility. In this paper, we focus on an uplink scenario in which a massive MIMO base station serves multiple high-mobility users. We leverage variational Bayesian(VB) inference for joint channel estimation and data detection(JED), tailored for time-varying channels. In particular, we use the VB framework to provide approximations of the true posterior distributions. To cover more real-world scenarios, we assume the time correlation coefficients associated with the channels are unknown. Our simulations demonstrate the efficacy of our proposed VB-based approach in tracking these unknown time correlation coefficients. We present two processing strategies within the VB framework: online and block processing strategies. The online strategy offers a low-complexity solution for a given time slot, requiring only the knowledge of the parameters/statistics within that time slot. In contrast, the block processing strategy focuses on the entire communication block and processes all received signals together to reduce channel estimation errors. Additionally, we introduce an interleaved structure for the online processing strategy to further enhance its performance. Finally, we conduct a comparative analysis of our VB approach against the linear minimum mean squared error(LMMSE), the Kalman Filter(KF), and the expectation propagation(EP) methods in terms of symbol error rate(SER) and channel normalized mean squared error(NMSE). Our findings reveal that our VB framework surpasses these benchmarks across the performance metrics.

MIMO Detection with Spatial Sigma-Delta ADCs: A Variational Bayesian Approach

The spatial Sigma-Delta ($\Sigma\Delta$) architecture can be leveraged to reduce the quantization noise and enhance the effective resolution of few-bit analog-to-digital converters (ADCs) at certain spatial frequencies of interest. Utilizing the variational Bayesian (VB) inference framework, this paper develops novel data detection algorithms tailored for massive multiple-input multiple-output (MIMO) systems with few-bit $\Sigma\Delta$ ADCs and angular channel models, where uplink signals are confined to a specific angular sector. We start by modeling the corresponding Bayesian networks for the $1^{\mathrm{st}}$- and $2^{\mathrm{nd}}$-order $\Sigma\Delta$ receivers. Next, we propose an iterative algorithm, referred to as Sigma-Delta variational Bayes (SD-VB), for MIMO detection, offering low-complexity updates through closed-form expressions of the variational densities of the latent variables. Simulation results show that the proposed $2^{\mathrm{nd}}$-order SD-VB algorithm delivers the best symbol error rate (SER) performance while maintaining the same computational complexity as in unquantized systems, matched-filtering VB with conventional quantization, and linear minimum mean-squared error (LMMSE) methods. Moreover, the $1^{\mathrm{st}}$- and $2^{\mathrm{nd}}$-order SD-VB algorithms achieve their lowest SER at an antenna separation of one-fourth wavelength for a fixed number of antenna elements. The effects of the steering angle of the $\Sigma\Delta$ architecture, the number of ADC resolution bits, and the number of antennas and users are also extensively analyzed.