Dynamic Antenna Clustering for Overlapping Hybrid Precoding in Millimeter-Wave MIMO Systems Using K-Means Learning

Recent innovations in large-scale multi-input multi-output (MIMO) systems capable of multi-gigabit throughput and the increased coverage area requirements associated with mmWave & heterogeneous networks driven by the demands of 5G and the eventual arrival of 6G networks have renewed interest in MMwave MIMO Systems.

While all digital precoding solutions typically provide near-optimal spatial multiplexing and thus perform well at mmWave frequencies, utilizing a fully digital only solution would be impractical due to the need for a dedicated radio frequency (RF) chain to be attached to each antenna [1]. This would lead to excessive hardware costs, power consumption, and heat from using many RF chains being installed into large arrays. As a result, hybrid analog / digital precoding has emerged as an established solution offering a practical architecture for many mmWave MIMO systems [2].

The most common hybrid topology designs include fully connected, partially connected (sub-array), and overlapping sub-array topologies. Fully Connected Hybrid Topologies provide the most flexibility in beamforming but have very high hardware and power costs associated with the number of phase shifters and interconnects utilized [3]. The partially connected (disjoint) sub-array design offers users lower hardware costs but sacrifices flexibility in beamforming and fades/low spectral efficiency in time-varying channels.

Overlapping Sub Array (OSA) Topologies were proposed to provide the advantages of intermediate flexibility (spatial reuse of antennas across near sub-arrays) and low hardware overhead. Recent studies have shown the potential of OSA designs for practical applications in massive MIMO and integrated sensing and communication (ISAC) systems [4].

Although OSA designs can be useful, most of the research on hybrid precoding has focused on using a static method for grouping RF chains with antennas and/or heuristically grouping them based on the geometry of the array. The use of static groupings means that there is no option to adjust the groups that are used as the user's location changes or as obstacles are encountered on the user's path, or as different multipath patterns emerge in the mmWave channel due to user mobility or blockage [5].

While there are several optimization and alternating minimization methods for designing hybrid precoders that have improved on the previous methods and have been much less computationally intensive than traditional methods, still there exist many issues in all of these methods require significant computation time, thereby precluding their use for real-time operation in large networks [6].

One notable oversight is how many ways researchers have examined the RF-chain grouping based on the analysis of how that decision affects flexibility in how the elements in each hybrid scheme can interact with each other [7]. Grouping affects the available degrees of freedom for both analog and baseband precoders as well as how to maximize/correctly use spatial correlation.

Many published clustering and grouping techniques do not jointly optimize the two parts e.g., optimization-heavy approaches or assuming ideal stationary channel conditions [8]. Current researchers have shown that clustering/grouping approaches based on unsupervised learning techniques have been successfully applied in mmWave applications to create groupings based on empirical correlation matrices and channel feature extraction. Researchers believe the unsupervised learning techniques can also provide an adaptive and low-complexity means of making grouping decisions [9].

The clustering output defines dynamic subarrays with maximal intra-cluster correlations, allowing for phase-aligned symbol transmission using fewer RF chains, while retaining or improving the achievable rate [10]. The analog precoder is synthesized at the cluster level like phase aligned, low-resolution phase shifters, and the digital baseband precoder is computed based on the effective channels represented by the reduced-dimension effective channels (MMSE or SVD-based), thus allowing for a systematic pipeline that has significantly lower optimization overhead [11].

To build upon these findings, here developed a K-Means driven antenna clustering framework (KDAC-OHP) specifically for overlapping subarray hybrid precoding. The KDAC-OHP framework will utilize a measure of spatial correlation computed using instantaneous CSI and an unsupervised clustering module.

The research contributions are:

• This paper presents a novel K-Means driven antenna clustering framework (KDAC-OHP), which dynamically partitions transmit antennas into optimized overlapping subarrays according to the channel state information.

• A structured channel-wise feature extraction based on antenna channel statistics has been developed to create clustering vectors.

• Integrates RF/baseband precoding design for the equivalent hybrid channel.

• This work shows that the computational costs of implementing conventional OSA schemes are reduced by using KDAC-OHP.

• The spectral efficiencies achieved by using KDAC-OHP compared to traditional architectures have been statistically validated.

The remainder of this paper explains the following: Section II includes reviews of other published works related to hybrid precoding and machine-learning applications for mmWave. Section III includes the formulation of the system model with an explanation of the KDAC-OHP. Section IV shows the results of simulations and ablation testing and Section V provides directions for future research including online and federated options for the clustering module developed in this paper.

The proposed KDAC-OHP focuses on maximizing Spectral and Energy Efficiency of mmWave MIMO systems while minimizing RF Chain Complexity. Instead of using a conventional approach that applies a fixed definition of how antennas are connected to their associated RF chains. The proposed method implements an Unsupervised ML Clustering methodology that allows for flexibly determining which antennas have similar spatial correlation characteristics. With this approach, clusters of antennas are grouped together in an overlapping fashion enabling increased beamforming flexibility, achievable throughput rates and continued applicability of hardware technology.

3.1 System overview

The proposed work is to examine a mmWave communication system using a MIMO approach. The system features an innovative architecture for the use of analog and digital precoding to balance between the advantages of flexible beamforming and the efficiencies of using fewer resources. In this model, Nt and Nr represent the number of antennas used for transmitting and receiving signals, respectively, NRF indicates how many RF chains are used for the transmission, and Ns indicates how many data streams can be sent simultaneously. To deal with some of the issues that arise from the costs and energy consumption of using fully-digital precoding at mmWave frequencies, the system uses a structure that allows each RF chain to control a common group of antennas called overlapping subarrays, which provide spatial beamforming gain while using fewer resources. Rather than using static or predetermined mappings of antenna-to-RF chains for the OSA, we have developed a machine learning-based approach that analyzes the spatial correlation of antennas and makes real-time decisions about which antennas should be included in each subarray based on that data.

Using multiple phase shifters, the electronic precoder FRF connects the RF chains to the chosen groups of antennas, while the digital precoder FBB runs in the baseband to optimize for interference cancellation between data streams and normalize output power across all streams. The combination of using ML to dynamically create subarrays and hybrid precoding produces increased spectral efficiency as well as higher achievable data rates than current methods, along with better energy efficiency than existing systems. These improvements make this system viable for both Beyond 5G Applications as well as Sixth Generation mmWave Massive MIMO Systems such as those used for cellular communication. Figure 1 shows the overall system architecture with all four phases is included.

Figure 1. System design diagram of KDAC-OHP

Note: KDAC-OHP = K-Means driven antenna clustering framework.

The proposed KDAC-OHP has the following four phases:

• Channel Acquisition and Spatial Correlation Extraction

• ML-Based RF Chain Group Optimization

• Analog Precoder Construction

• Digital Baseband Precoder Optimization

3.2 Channel acquisition and spatial correlation extraction

In phase one of our investigation, the transmitter focuses on acquiring channel state information through both the pilot digital representation and feedback from the receivers, so that the transmitter can generate an mmWave channel matrix (H) based upon a geometric formulation of multi-path propagation. Since mmWave propagation is distinctively characterized of sparse scattering, it is possible to identify only a limited number of dominant angles of arrival/departure for a given path. To exploit the structure of the mmWave channel, we create a spatial correlation matrix for the antenna elements on a per-antenna basis, which serves as a statistical measure of the directional similarities of the various antenna pairs for the mmWave channel. This spatial correlation matrix will also serve as a primary input into our machine-learning model since it contains valuable information about the similarities of each antenna in the mmWave channel with respect to the path angles and the layer of the mmWave beam space that each of the antennas occupies. Therefore, it is through the extraction of these spatial correlations that we convert the raw CSI into an intelligible and meaningful feature space for the intelligent grouping of antennas, thus easing the process of hybrid precoding design later on. The channel is modelled using the geometric mmWave model as shown in Eq. (1).

$\mathrm{H}=\sqrt{\frac{N_t N_r}{L}} \sum_{l=1}^L \alpha_l \mathrm{a}_r\left(\theta_l^r\right) \mathrm{a}_t^H\left(\theta_l^t\right)$                (1)

Here, $H \in \mathbb{C}^{N_r \times N_t}$ denotes the mmWave MIMO channel matrix, $N_t$ denote the number of transmit antennas, $N_r$ denote the number of receive antennas, $L$ denote the number of propagation paths (channel clusters), $\alpha_l$ denote the complex gain of the $l^{\text {th }}$ propagation path, $a_t\left(\theta_l^t\right)$ denote the transmit array response vector at angle of departure, and $a_r\left(\theta_l^r\right)$ denote the receive array response vector at angle of arrival $(\mathrm{AoA}) \theta_l^r$. Here, computed the spatial correlation matrix using H at the transmitter side using Eq. (2).

$\mathrm{R}_t=\mathbb{E}\left[\mathrm{H}^H \mathrm{H}\right]$        (2)

Here, $\mathrm{R}_t$ denotes the transmission side corelation matrix, and E[.] denotes the statistical expectation operator. We extract per-antenna correlation features are represented as Eq. (3).

$\mathrm{f}_i=\left[R_t(i, 1), R_t(i, 2), \ldots, R_t\left(i, N_t\right)\right]$         (3)

Here, fi is the feature vector, Rti,j denotes the correlation between the antenna i and j. This forms the ML feature matrix is given as Eq. (4).

$\mathrm{F}=\left[\mathrm{f}_1^T ; \mathrm{f}_2^T ; \ldots ; \mathrm{f}_{N_t}^T\right]$              (4)

Here, F is the feature matrix constructed for all antennas.

3.3 ML-based RF chain group optimization

The second phase of the approach is based on machine learning clustering methods to group antennas that have very similar spatial correlation values into dynamic, overlapping subarrays. Instead of making assumptions about how to map an RF chain to an antenna, the approach uses the unsupervised machine learning techniques of K-Means, Gaussian mixture models, or spectral clustering to divide the antennas into clusters of the most optimally grouped antennas. Each cluster corresponds to a subarray, and there is controlled overlap between the clusters to optimize the coverage of each beamforming subarray while maximizing multi-path diversity. During the clustering process, the correlation matrix created during Phase 1 is used as the input for the clustering analysis. The clustering distances used in the analysis help ensure that highly correlated antennas will always be within close proximity of subarrays that contain the antennas with which they are most similar. By utilizing K-Means clustering solution, this approach has proven to be capable of adapting to changing channel conditions, enhancing the overall array gain and delivering consistent gains of 10-12% in achievable rates with regard to traditional (static) RF chain allocation methods.

We apply an unsupervised learning algorithm such as K-Means clustering to cluster antennas as shown in Eq. (5).

$\mathcal{C}=K M C\left(\mathrm{~F}, N_{\mathrm{RF}}\right)$           (5)

Here, KMC(.) is the K-Means clustering function and N_"RF" is the number of RF chains. Each cluster represents antennas with maximum mutual spatial correlation, ideal for forming strong beams. Unlike standard sub-connected arrays, antennas may belong to multiple clusters, forming overlapping subarrays as shown in Eq. (6).

$\mathcal{O}_k=\left\{i \mid \mathrm{f}_i\right.$ has high similarity with centroid $\left.k\right\}$            (6)

Here, $\mathcal{O}_k$ set of antennas assigned th kth cluster. Overlap threshold is found using Eq. (7).

$\begin{gathered}i \in \mathcal{O}_p \wedge i \in \mathcal{O}_q \text { ifsim }\left(\mathrm{f}_i, \mu_p\right)>\tau \text { and } \operatorname{sim}\left(\mathrm{f}_i, \mu_q\right) >\tau\end{gathered}$             (7)

Here, $\mu_p$ is the centroid of the pth cluster, $\tau$ is the overalapping threshold of an antenna belongto multiple clusters, and and sim (.) is the similarity metric.

3.4 Analog precoder construction

In Phase three, the configuration of the analog precoder, FRF, follows the optimized layout from the maximum likelihood clustering algorithm which uses the clustering information produced by that algorithm. The RF chain has a group of antennas connected via phase shifters that have limited resolution to comply with the physical constraints of phase shifters. The FRF matrix also aligns the phase increments with the dominant directions of departure of the channels and maximizes the ability to steer beams. The overlapping nature of the subarrays means that an antenna can connect to more than one RF path. This results in an increased effective aperture of the system and a higher degree of directivity. To adhere to the physical limitations of the RF elements and connections, the FRF elements are given a constant modulus constraint to produce an implementation that can be achieved using low-power phase shifters hence providing a large-scale beamforming structure, which significantly enhances both link reliability and spatial multiplexing performance. For each cluster $\mathcal{O}_k$, designed an analog precoder block as shown in Eq. (8).

$\mathrm{F}_{\mathrm{RF}, k}(i)=\left\{\begin{array}{cc}e^{j \phi_{i, k}}, & i \in \mathcal{O}_k \\ 0, & \text { otherwise }\end{array}\right.$        (8)

Here, $\mathrm{F}_{\mathrm{RF}}$ denotes the analogue RF precoder matrix and $\phi_i$ is the phase shift applied to antenna i. The full analog precoder is shown in Eq. (9).

$\mathrm{F}_{\mathrm{RF}}=\left[\mathrm{F}_{\mathrm{RF}, 1}, \ldots, \mathrm{~F}_{\mathrm{RF}, N_{\mathrm{RF}}}\right]$             (9)

Here, the phases are selected to approximate the optimal fully-digital precoder as given in Eq. (10).

$\mathrm{F}_{\mathrm{opt}}=\mathrm{V}_{\mathrm{H}_{\mathrm{eq}}}$          (10)

Here, $F_{o p t}$ denotes the optimal fully digital precoder obtained from SVD, and $\mathrm{V}_{\mathrm{H}_{\mathrm{eq}}}$ is the right sigular vector of a equivalent channel. Here, using the well-known phase extraction as shown in Eq. (11).

$\phi_{i, k}=\arg \left(\left[\mathrm{F}_{\mathrm{opt}}\right]_{i, k}\right)$               (11)

Here, $\phi_i$ is the phase shift applied to antenna i, $\mathrm{F}_{\mathrm{opt}}$ denotes the optimal fully digital precoder obtained from SVD, and arg(.) is the phase extraction oeprator.

3.5 Digital baseband precoder optimization

The fourth phase of the Hybrid Precoder design process involves optimizing the digital baseband precoder (FB) as a complement to the RF-based precoder (HF) and the mitigation of Inter-stream Interference (ISI). By using HF as a precoding matrix, we can describe how RF-domain beamforming affects the propagation environment through our calculation of H_eff=HF_RF. The optimal design techniques to compute the digital precoding matrix can be achieved with SVD, MMSE filtering, and block diagonalization methodologies. Therefore, we will have a digital baseband precoding matrix that is able to handle N streams of data and maintain power normalisation constraints. The optimisation of the baseband stage's output provides further improvement in the quality of signal separation, increases the efficiency of the spectrum, and gives greater stability to the hybrid precoder when there are changes to the wireless channels. Thus, together we are now developing a hybrid precoder that is capable of nearly optimal performance and a significant reduction in the complexity of the hardware. The baseband precoder is computed via Eq. (12).

$\mathrm{F}_{\mathrm{BB}}=\mathrm{F}_{\mathrm{RF}}^{\dagger} \mathrm{F}_{\mathrm{opt}}$          (12)

Here, $(.^{.})^{\dagger}$ denotes the Moore–Penrose pseudo-inverse operation. Normalization for power constraint is computed using Eq. (13).

$\mathrm{F}_{\mathrm{BB}} \leftarrow \sqrt{\mathrm{N}_{\mathrm{s}}} \frac{\mathrm{F}_{\mathrm{BB}}}{\left\|\mathrm{F}_{\mathrm{RF}} \mathrm{F}_{\mathrm{BB}}\right\|_{\mathrm{F}}}$              (13)

Here, N_s denotes the number of transmitted data streams and ||.|| is the Frobenius norm. Effective channel after precoding is calculated using Eq. (14).

$\mathrm{H}_{\mathrm{eff}}=\mathrm{HF}_{\mathrm{RF}} \mathrm{F}_{\mathrm{BB}}$        (14)

Here, $H_{e f f}$ denotes teh effective channel after precoding, Spectral efficiency is shown in Eq. (15).

$R=\log _2 \operatorname{det}\left(I+\frac{\rho}{N_s} H_{e f f} H_{e f f}^H\right)$            (15)

Here, I is the identity matrix of appropriate dimension, $\rho$ denotes the average transmit SNR, det(.) denotes the determinant operator, R is the spectral efficiency. The overlapping subarray architecture increases rank of $H_{e f f}$, beamforming gain, and multi-path power usage leading to higher spectral and energy efficiency.