Mini-Batch Gradient-Based MCMC for Decentralized Massive MIMO Detection

Consider an uplink massive MIMO system with $U$ single-antenna users and a BS equipped with $B\geq U$ receiving antennas, as shown in Fig. 1. Each user independently encodes its information bit stream, modulates the coded bits using the quadrature amplitude modulation (QAM) scheme, and transmits the modulated symbols over the wireless channels to the BS. The input-output relationship¹¹1We consider a flat-fading channel in the system model; however, the proposed method can be readily generalized to handle frequency-selectivity by using orthogonal frequency division multiplexing (OFDM). In this approach, MIMO detection is independently conducted on each flat-fading subcarrier. of this uplink transmission is given by

where $\mathbf{y}\in\mathbb{C}^{B\times 1}$ is the received signal at the BS, $\mathbf{H}\in\mathbb{C}^{B\times U}$ is the uplink channel matrix, $\mathbf{x}\in\mathcal{A}^{U\times 1}$ consists of transmit symbols from each user, with $\mathcal{A}$ representing the QAM lattice set having a cardinality of $M$ and unit symbol energy, and $\mathbf{n}\in\mathbb{C}^{B\times 1}$ represents the IID circular complex Gaussian noise with zero-mean and variance $\sigma^{2}$ entries. The signal-to-noise ratio (SNR) is given by $\text{SNR}=\mathbb{E}[\|\mathbf{Hx}\|^{2}]/\mathbb{E}[\|\mathbf{n}\|^{2}]$. The objective of uplink MIMO detection is to estimate the transmit vector $\mathbf{x}$ given the received signal $\mathbf{y}$ and the knowledge of the uplink channel matrix $\mathbf{H}$. The optimal ML solution is given by

Note that the finite set of $\mathbf{x}$ makes the optimal MIMO detection an integer least-squares problem, known to be non-deterministic polynomial-time hard (NP-hard) [3].

To deal with the heavy interconnection and computation burdens due to the use of extra-large-scale antenna arrays, DBP has been proposed as a promising alternative to traditional centralized processing [6]. Throughout the paper, we assume that the massive MIMO BS possesses the DBP architecture as shown in Fig. 1. The BS antenna array is partitioned into $C\geq 1$ antenna clusters, where the $c$-th cluster is associated with $B_{c}$ antennas ($c\in[C]$), and $\sum_{c=1}^{C}B_{c}=B$. Without loss of generality, we assume that each cluster is of equal size, i.e., $B_{c}=B/C$. Each cluster is equipped with local RF chains that are connected to dedicated hardware referred to as the DU.

We consider two widely accepted DBP topologies, namely star and daisy-chain topologies [6, 7, 9, 29, 30], which are depicted in Fig. 1. We assume data links among the CU and DUs are bidirectional for both topologies. The star topology is featured by all DUs connected to the CU (solid lines). Thus, the information exchange between the CU and each DU is in parallel. The daisy-chain topology is featured by DUs connected sequentially [7] (dashed lines), with a single connection to the CU at the end. Hence, communication is limited solely to adjacent units in the daisy-chain topology and the information exchange is performed in a sequential manner. Both these topologies have their distinct strengths in the trade-off among throughput, latency, and interconnection bandwidth [9]. The star topology is well recognized for the low data transfer latency owing to the simultaneous information exchange between the CU and each DU, while the daisy-chain topology is known for the constant interconnection bandwidth regardless of the number of DUs, as will be shown in Sec. IV-B.

Under the DBP architecture, the uplink channel matrix, received signal, and noise vector are partitioned row-wise as $\mathbf{H}=[\mathbf{H}_{1}^{T},\ldots,\mathbf{H}_{C}^{T}]^{T}$, $\mathbf{y}=[\mathbf{y}_{1}^{T},\ldots,\mathbf{y}_{C}^{T}]^{T}$, and $\mathbf{n}=[\mathbf{n}_{1}^{T},\ldots,\mathbf{n}_{C}^{T}]^{T}$, respectively. Hence, the signal model at the $c$-th antenna cluster can be written as

where $\mathbf{H}_{c}\in\mathbb{C}^{B_{c}\times U}$ is the channel matrix between the $U$ users and the $c$-th antenna cluster. $\mathbf{y}_{c}\in\mathbb{C}^{B_{c}\times 1}$ and $\mathbf{n}_{c}\in\mathbb{C}^{{B}_{c}\times 1}$ represent the local received signal and noise vector of the $c$-th antenna cluster, respectively.

Each DU $c$ exclusively utilizes its local channel matrix $\mathbf{H}_{c}$ and the received signal $\mathbf{y}_{c}$ to compute intermediate detection results.²²2We assume that the local channel matrix $\mathbf{H}_{c}$ is perfectly known at each DU since we focus on MIMO detector design in this paper. In realistic scenarios, this knowledge can be obtained by performing local channel estimation. The CU works as a coordinator for periodically aggregating the intermediate information from the DUs and forming the global consensus [6], which is then broadcast to the DUs for subsequent updates, and finally producing the estimate. In this decentralized scheme, the heavy computation burden at the CU can be offloaded to DUs and be performed in a parallel and distributed fashion. Hence, the overall computation latency would be significantly reduced. Meanwhile, the DUs can process the locally stored data to obtain low-dimensional intermediate information instead of directly transmitting full-dimensional data, thus saving interconnection bandwidth.

The MCMC method samples from the target posterior distribution using a Markov chain given by $\mathbf{x}_{0},\mathbf{x}_{1},\ldots,\mathbf{x}_{t},\ldots,\mathbf{x}_{S}$, where $S$ is the number of sampling iterations. The state transition $\mathbf{x}_{t-1}\to\mathbf{x}_{t}$ in this chain is regulated by a transition kernel $T(\mathbf{x}|\mathbf{x}_{t-1})$, specifically designed to align the chain’s stationary distribution with the target distribution. Hence, the realization of this chain can be utilized to (approximately) generate samples from the target distribution, facilitating statistical inference. Notably, the integration of MCMC and the GD method has recently emerged as a promising approach for resolving nonconvex optimization challenges [23]. The underlying concept of this gradient-based MCMC approach involves leveraging gradients and the Metropolis-Hastings (MH) criterion [31, 25] to design the transition kernel, expediting the Markov chain’s convergence to the target distribution. When using gradient-based MCMC for MIMO detection, each state of the Markov chain corresponds to a sample of $\mathbf{x}$ over the discrete space $\mathcal{A}^{U\times 1}$. The initial sample $\mathbf{x}_{0}$ is either randomly generated or transferred from another detector. The transition from $\mathbf{x}_{t-1}$ to $\mathbf{x}_{t}$, i.e., the generation of the $t$-th sample $\mathbf{x}_{t}$, includes the following steps [25, 27]:

Next, we present the decentralized gradient-based MCMC algorithm tailored for the DBP architecture. The central concept revolves around distributing the resource-intensive gradient computations across multiple DUs, thereby diminishing computation latency and bolstering overall efficiency.

We first provide some useful notations for ease of illustration. The local objective function at each DU $c\in[C]$ is defined as

Therefore, the local gradient is given by

The optimal MIMO detection problem in (2) is rewritten as

where the global objective function $f(\mathbf{x})$ is given by

The decomposition of the global objective function into multiple local objective functions allows the calculation of gradients on each DU as shown in (10). These local gradients can then be collected at the CU to derive the full gradient $\nabla f(\mathbf{x})=\sum_{c=1}^{C}\nabla f_{c}(\mathbf{x})$, which enables the GD-based optimization in (4). The distributed computations alleviate the computation burden of directly calculating the full gradient at the CU.

However, the decentralization of the standard full-batch GD in (4) does not lead to a reduction in the overall computation cost. Moreover, this decentralized full-batch GD requires information exchange between the CU and all DUs in each iteration, resulting in high interconnection costs. To further mitigate the computation and interconnection requirements associated with the full-batch GD, we turn to the mini-batch stochastic GD method [32], which shows promise in efficiently optimizing objective functions in finite-sum form as (12). This method serves as an efficient alternative to the full-batch GD in (4), offering the potential for rapid convergence and overhead reduction. Specifically, we introduce a mini-batch-based gradient updating rule tailored to the DBP architecture, as detailed below.

In each GD iteration $k$, the estimate is not updated using the full gradient $\nabla f(\cdot)$. Instead, a subset of DUs, termed a mini-batch and denoted as $\mathcal{I}$, is randomly selected for the update. The mini-batch contains $m=|\mathcal{I}|$ elements. Here, $m$ represents the batch size, and we assume that the number of antenna clusters $C$ is divisible by $m$. The mini-batch-based GD iteration operates as

where $\nabla f_{\mathcal{I}}(\cdot)$ denotes the estimated gradient based on the mini-batch $\mathcal{I}$. This is expressed as

where the scaling factor $C/m$ ensures the unbiasedness of the mini-batch gradient as an estimator of the full gradient [32], denoted as,

with the expectation taken across any possible selections of mini-batch $\mathcal{I}$. This guarantees the asymptotic convergence to the minimum of the continuous-relaxed version of problem (11) via the mini-batch GD method.

With the proposed mini-batch-based gradient updating rule in mind, we next introduce the Mini-NAG-MCMC algorithm for decentralized MIMO detection. For ease of presentation, in the remainder of this paper, we consider the star topology as displayed in Fig. 2 unless noted otherwise. We assume that a total of $S$ samples are generated to perform the final inference for MIMO detection. The generation of each sample $\mathbf{x}_{t}$ is divided into two stages as detailed in the following.

Table I presents the computational complexity of Mini-NAG-MCMC compared with various centralized and decentralized detection algorithms. The decentralized baselines include the decentralized ADMM (DeADMM) [6], LAMA (DeLAMA)³³3We consider the fully decentralized version of the LAMA detector proposed in [12]. [12], EP (DeEP) [14], and Newton (DeNewton) [9] detectors. Centralized LMMSE [38], $K$-best sphere decoding [40], and NAG-MCMC [27] algorithms are also compared. For the considered decentralized schemes, operations in the CU and each DU are counted separately. We only present the count of a single DU to reflect the parallel processing of the DBP architecture. We remark that while this count is lower than the total computational cost from all local processing units, it reflects the computation latency of the algorithm.

In the comparison, the complexity of a $U\times U$ matrix inversion is $\mathcal{O}(U^{3})$, where $\mathcal{O}(\cdot)$ denotes the standard asymptotic notation [39, Ch. 3.1] to indicate the order of complexity. The complexity of Mini-NAG-MCMC is dominated by gradient computations and objective function evaluation, costing $\mathcal{O}(N_{\rm g}SB_{c}U)$ and $\mathcal{O}(\sqrt{M}B_{c}U)$ at each DU, respectively. The other operations in the MCMC sampling stage, such as the random walk, QAM mapping, and acceptance test, incur a modest cost of $\mathcal{O}\big{(}SU(N_{\rm g}+\sqrt{M})\big{)}$ multiplications at the CU. The comparison in Table I shows that Mini-NAG-MCMC is highly efficient due to avoiding high-complexity matrix multiplications and inversions.

Table I also reveals that the proposed Mini-NAG-MCMC effectively relieves the computation burden at the CU compared to the centralized algorithms by shifting the computationally intensive gradient computations to the DUs. As a result, the computational complexity at each processing unit is independent of the number of BS antennas $B$, which is beneficial for reducing computation latency. Furthermore, the proposed Mini-NAG-MCMC possesses the advantage that only a portion of the DUs is required to perform computations in each mini-batch GD iteration of the algorithm. Thus, the total computational cost can be saved to a large extent compared to other decentralized schemes.

Note that the number of iterations also has a significant impact on the complexity of the proposed method and other iterative detection schemes in Table I. However, it is practically difficult to provide an analytical estimate of the number of iterations required to reach a certain performance level. Hence, we resort to empirically evaluating the computational complexity of the detectors listed in Table I and meanwhile taking into account the achieved performance. Detailed results are shown in Sec. V.

The interconnection bandwidth is measured by the number of bits transferred from and to the CU within a symbol period. Denote by $\omega$ the bit-width of each real value (real/imaginary parts of a complex scalar number) in the implementation. Note that the representation of each QAM symbol in the derived samples only requires $\log_{2}M$ bits, which can be significantly smaller than $\omega$. For centralized schemes, the entities required to be uploaded to the CU include the full channel matrix $\mathbf{H}$ and received signal $\mathbf{y}$, resulting in the bandwidth

For the star topology-based Mini-NAG-MCMC, the CU receives $UC$ and $2N_{\rm g}SUm+(S+1)C$ real values during preprocessing and sampling iterations, respectively. The CU also broadcasts $2N_{\rm g}SUm$ continuous-valued real numbers and $(S+1)UC$ QAM symbols during the sampling. For the daisy-chain topology-based Mini-NAG-MCMC, the numbers that are required to be transferred are identical to those in the star topology. However, the CU requires communication with only one directly connected DU, thereby significantly reducing interconnection costs. Hence, the interconnection bandwidth of Mini-NAG-MCMC with star and daisy-chain topologies are respectively given by

The results of both topologies do not depend on the BS antenna count $B$, a notable advantage over centralized schemes.

We note that if we consider the interconnection cost on all links in the daisy-chain topology, the total interconnection bandwidth would be comparable to that of the star topology. However, the daisy-chain architecture can uniformly allocate the bandwidth demand among all links, resulting in a significant reduction in the individual requirements on each link [7]. Notwithstanding, the price to pay is extended processing latency as a result of the sequential processing architecture.

Both IID Rayleigh fading and realistic 3GPP channel models are considered in the simulation. The Rayleigh fading MIMO channel has elements that follow the Gaussian distribution $\mathcal{CN}(0,1/B)$. For realistic channels, the 3D channel model outlined in the 3GPP 36.873 technical report [41] is utilized. This model has been widely adopted for its accurate representation of real-world channel environments. We consider both uncoded and coded MIMO systems. For uncoded systems, we evaluate the bit error rate (BER) performance until $5\times 10^{7}$ bits are transmitted or the number of bit errors exceeds 1000. Different independent channel realizations are considered for each symbol transmission. For coded systems, we employ a rate-$3/4$ low-density parity-check (LDPC) code and the OFDM scheme, wherein each user’s coded bits are converted to QAM symbols and mapped onto 216 subcarriers for data transmission. The detector calculates soft outputs using max-log approximation [20]. Belief propagation decoding is then utilized with a maximum number of 10 iterations. The block error rate (BLER) is evaluated until $10^{5}$ coded blocks are transmitted with independent channel realizations.

We compare our method with the centralized LMMSE and NAG-MCMC [27] schemes and representative decentralized detector baselines as listed in Table I. The optimal ML [3] and near-optimal $K$-best [40] detectors are used as performance benchmarks. For parameter settings of NAG-MCMC and the proposed Mini-NAG-MCMC, the number of NAG iterations is set to $N_{\rm g}=4$. The choice of momentum factor $\rho_{k}$ follows the recommended setting in [28]:

The random walk step size is empirically set to $\gamma=0.05$. To ensure computational efficiency, we set $\mathbf{M}_{\rm c}=\mathbf{I}$ in (27). The initial sample $\mathbf{x}_{0}$ for both NAG-MCMC and Mini-NAG-MCMC is randomly selected from $\mathcal{A}^{U\times 1}$.

Fig. 3 presents the convergence performance of Mini-NAG-MCMC in terms of the BER versus the number of sampling iterations.⁴⁴4Note that conducting a rigorous theoretical analysis of convergence is challenging for gradient-based MCMC methods in discrete search spaces [22], [26]. Therefore, we have attempted to empirically evaluate the convergence property via numerical simulations, aiming to provide some valuable insights. The MIMO system is configured with $B=128$ and $U=8$ or $32$. The system employs 16-QAM modulation and operates under IID Rayleigh fading channels.

Fig. 3(a) shows the convergence behavior under different choices of mini-batch sizes. We also provide the convergence performance of the centralized NAG-MCMC algorithm as a reference. The results indicate that Mini-NAG-MCMC converges faster as the mini-batch size $m$ increases in both $U=8$ and $U=32$ cases. This result is in accordance with expectation because the variance of the mini-batch gradient approximation in (14) is a decreasing function of $m$ [42]. Meanwhile, the convergence speed is virtually unaffected by the number of users, revealing the robustness of Mini-NAG-MCMC to varying user loads.

Remarkably, the Mini-NAG-MCMC with $m=16$ achieves virtually the same convergence performance as the centralized NAG-MCMC that uses full gradients. This phenomenon can be attributed to the sufficient accuracy of the mini-batch approximation when $m$ is sufficiently large (e.g., $m=C/2$ in this simulation) and the synergistic effect of stochastic gradients and MCMC in escaping local minima. Consequently, the proposed method achieves equivalent performance to the centralized counterpart that computes full gradients, while offering the benefit of reduced computational cost since only half of all the DUs are utilized for computations per iteration.

Fig. 3(b) compares the convergence performance under different numbers of clusters, with the mini-batch size fixed as $m=C/2$. The figure reveals that the convergence speed shows minimal variation with respect to the number of clusters, which is a notable advantage of the proposed Mini-NAG-MCMC. Hence, the proposed method can accommodate to different levels of decentralization, allowing for trade-offs between the computational complexity at each DU and the number of interconnection interfaces (as well as bandwidth) by flexibly adjusting the system parameter $C$ according to different requirements.

In this subsection, we investigate the proposed method under Rayleigh fading MIMO channels in uncoded systems. Fig. 4 provides the BER performance and computational complexity for a massive MIMO system with $B=128$ BS antennas, $U=32$ users, $C=32$ clusters, and 16-QAM modulation. The mini-batch size for Mini-NAG-MCMC is $m=16$. We compare the proposed method with centralized LMMSE [38], $K$-best [40], and NAG-MCMC [27] detectors and two representative decentralized baselines, namely the DeADMM [6] and DeEP [14] detectors.⁵⁵5For the implementation of the DeADMM scheme, we use the code released at https://github.com/VIP-Group/DBP. The list size for the $K$-best detector is set to $K=4$, serving as a near-optimal baseline. The number of iterations for DeADMM and DeEP is set as $T=50$ and $T=5$, respectively, which is necessitated by the convergence of the algorithm in the considered system setup.

Fig. 4(a) shows that the LMMSE and DeADMM detectors have a significant performance gap to the ML detector. In contrast, Mini-NAG-MCMC achieves a noticeable performance gain over LMMSE and DeADMM when the number of sampling iterations is $S=6$ and further approaches the ML performance with a slight increase in $S$. Furthermore, Mini-NAG-MCMC obtains comparable BER performance to NAG-MCMC when both methods are equipped with the same $S$.

Fig. 4(b) illustrates the complexity bar chart of different detectors in terms of the number of real-valued multiplications required per detection. The number of multiplications at the CU for Mini-NAG-MCMC is much lower than the counterparts at the CU for the centralized LMMSE, $K$-best, and NAG-MCMC detectors. We remark that despite the total computational cost of Mini-NAG-MCMC being comparable to the centralized LMMSE and NAG-MCMC, the proposed decentralized algorithm offers dual advantages: First, the complexity at each DU is notably low, effectively leading to a reduced computation latency. Second, the distribution of complexity among many DUs allows the use of cheap computing hardware within each processing unit, in contrast to the centralized algorithms that exert the entire complexity burden on the CU. Furthermore, the computational cost of Mini-NAG-MCMC is also similar to that of the DeADMM and substantially lower than that of the DeEP.

Considering its near-ML performance and low complexity at each unit, as shown in Fig. 4, our proposed Mini-NAG-MCMC demonstrates high promise in achieving an attractive trade-off between performance and complexity.

Fig. 5 shows the results when the number of users is increased to $U=48$, with the other system settings remaining unchanged compared to Fig. 4. Fig. 5(a) indicates that the performance of LMMSE, DeADMM, and DeEP further degrades as compared to the near-optimal $K$-best detector when the user number increases. However, Mini-NAG-MCMC demonstrates significant performance improvement over LMMSE when $S=6$ and achieves comparable performance to the $K$-best when $S=12$, consistent with the results presented in Fig. 4(a). This finding reflects that the proposed Mini-NAG-MCMC and the associated learning rate approximation are robust to the increased user load. Moreover, upon comparing Fig. 5(b) to Fig. 4(b), it is observed that the complexity increase of Mini-NAG-MCMC is subtle in contrast to the LMMSE, $K$-best, and DeEP detectors. Specifically, the complexity increase is 50% for Mini-NAG-MCMC, whereas the corresponding values for LMMSE, $K$-best, and DeEP are 165%, 125%, and 230%, respectively, demonstrating the outstanding scalability of Mini-NAG-MCMC as revealed in Sec. IV-A.

Fig. 6 presents the BER performance under high-order 64-QAM/256-QAM modulation schemes. The other system configurations are the same as Fig. 5. In this experimental setup, the $K$-best detector is configured with $K=8$ to provide a near-ML baseline. The DeADMM and DeEP detectors execute for $T=50$ and $T=10$ iterations, respectively. The proposed Mini-NAG-MCMC is set up with $S=24$ iterations and a mini-batch size of $m=16$. The figure illustrates that the proposed method achieves a performance gain of more than 1.5 dB when compared to the LMMSE detector and the decentralized baselines and approaches the performance of the near-optimal $K$-best. These results demonstrate that the proposed decentralized detector scales to high-order modulation.

We further investigate the proposed method under the more realistic 3GPP 3D MIMO channel model [41] with coded transmissions. We examine the urban macrocell non-line-of-sight scenario. For the simulation of this scenario, the QuaDRiGa simulator [43] is employed. In the considered scenario, the BS is equipped with $B=128$ antennas, which are uniformly partitioned into $C=32$ clusters, to serve $U=8$ users. The BS adopts a uniform planar antenna array with half-wavelength antenna spacing. The users are evenly dropped within a $120^{\circ}$ cell sector centered around the BS, spanning a radius ranging from 10 m to 500 m. The carrier frequency is set as 2.53 GHz, and the channel spans over 20 MHz bandwidth. To cope with the complicated channel environments, the $K$-best detector is set up with $K=8$. The number of iterations for DeADMM and DeEP is selected as $T=50$ and $T=10$, respectively. The number of iterations and mini-batch size for Mini-NAG-MCMC are $S=64$ and $m=16$, respectively.

As shown in Fig. 7, Mini-NAG-MCMC demonstrates stable performance, exhibiting significant robustness to the realistic channel model. Specifically, Mini-NAG-MCMC outperforms LMMSE and other decentralized detectors, achieving an approximate 1 dB gain under both 16-QAM and 64-QAM modulation. Furthermore, the gap between Mini-NAG-MCMC and the near-optimal performance achieved under centralized schemes by the $K$-best detector is a mere 0.4 dB. Considering the distribution of complexity across DUs, achieving near-optimal performance in this context is indeed promising.

We evaluate the interconnection cost of the proposed Mini-NAG-MCMC in this subsection. The bit-width is selected as $\omega=16$ in the evaluation. The BER performance versus interconnection bandwidth of Mini-NAG-MCMC with star and daisy-chain topologies for a MIMO system with $B=256$, $U=8$, and $C=8$ is presented in Fig. 8. The mini-batch size for Mini-NAG-MCMC is set to $m=2$. In the figure, the $x$-axis represents the percentage of interconnection bandwidth of Mini-NAG-MCMC relative to the centralized algorithm, calculated according to (31)-(33) by setting the number of sampling iterations $S$ to $[2,4,6,8]$. The $y$-axis represents the performance gap between Mini-NAG-MCMC and the centralized ML algorithm in terms of the SNR required to achieve a target $\text{BER}=10^{-3}$. From the figure, we have the following observations:

• •

  Mini-NAG-MCMC achieves a flexible trade-off between performance and interconnection bandwidth.

• •

  The daisy-chain topology provides lower interconnection bandwidth than the star topology and is more appreciated in interconnection bandwidth-constrained scenarios, despite a larger processing delay.

Furthermore, Fig. 9 presents the interconnection bandwidth of Mini-NAG-MCMC compared with the centralized algorithm as a function of the number of BS antennas $B$ when $U=8$ and $C=8$. The mini-batch size and the number of sampling iterations for Mini-NAG-MCMC are $m=2$ and $S=4$, respectively. The figure shows that the proposed method with both star and daisy-chain topologies exhibits significantly reduced interconnection cost compared to the centralized processing as $B$ increases. This result confirms the scalability of the proposed method to massive MIMO systems with a large number of BS antennas.

Fig. 10 presents the performance-complexity trade-off of various detectors for a massive MIMO system with $B=128$, $U=32$, $C=4$, 16-QAM modulation, and Rayleigh fading channels. All the detectors are implemented using the star DBP topology. The figure reveals that the proposed Mini-NAG-MCMC ($m=2$) achieves the most promising results among the considered detectors in the trade-off between BER performance, computational complexity, and interconnection bandwidth. Despite the higher interconnection bandwidth of Mini-NAG-MCMC compared to DeLAMA and DeEP, the gap can be narrowed by deploying the proposed detector using the daisy-chain topology. This result highlights the potential of the proposed method as a highly implementable and desirable solution for massive MIMO detectors under DBP architectures.