2D-OPC Subarray Structure for Efficient Hybrid Beamforming over Sparse mmWave Channels

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Introduction
e key techniques adopted by the fifth-generation (5G) wireless systems include the utilization of higher frequency bands to significantly increase the transmitted data rates using enlarged bandwidth resources [1,2]. Due to the shortage of frequency resources in conventional low frequency bands, the utilization of the millimeterwave (mmWave) spectrum is actively pursued [3][4][5]. e mmWave communication can transmit gigabit data per second but suffer from high path loss and low scattering [6,7]. To overcome these difficulties, large antenna arrays need to be applied in mmWave communication systems to perform high-gain beamforming. e short wavelength, one of the features of mmWave communication, enables a large number of antennas to be packed within a limited space.
is results in high directional beamforming gains, which compensates for the severe path loss.
In multiple-input multiple-output (MIMO) systems using a large-scale antenna array, conventional digital beamforming requires a separate radio frequency (RF) chain for each antenna element.
is causes high power consumption as well as high hardware cost, which makes both the operation cost and capital expenditures increase [8,9]. Investigations have been made to suggest various architectures aimed at reducing the number of RF chains by using the separate analog RF beamformer and baseband digital beamformer, known as the hybrid beamforming technique [10][11][12]. Unlike conventional digital beamforming, hybrid beamforming is performed by using phase shifters in the analog RF domain. Analog beamforming controls the direction of the signal by adjusting the phase shifters, with much lower complexity compared with digital beamforming. Performance degradation is experienced, however, due to the lack of precise signal processing capability at the baseband, including the lack of the controllability of the magnitude of the beamformer coefficients.
e FCS can achieve the full array gain because each RF chain is connected to all antennas, but it is not practically efficient because it requires a large number of phase shifters and power amplifiers. On the other hand, each RF chain in the PCS is connected to a group of antenna elements called a subarray. e PCS effectively reduces the number of phase shifters required, decreasing the beamforming gain of each RF chain in return.
In order to implement the mmWave hybrid beamforming architecture that achieves high beamforming gain while minimizing the number of phase shifters required, there have been some attempts to generalize the antenna structure including the possible overlaps among subarrays. Most of the existing overlapped subarray structures (OSSs), however, are based on one-dimensional uniform linear arrays (ULAs) [20,21] or targeted to single-user transmission environments [22]. In this paper, we generalize the overlapping structure using the UPAs to propose the twodimensional overlapped partially connected (2D-OPC) subarray structure that can support various subarray shapes. Each antenna element can be connected to either single or multiple RF chains to provide versatility in the utilization of subarrays. e 2D-OPC structure can reduce the required number of phase shifters while maintaining the data throughput performance. e proposed subarray structure includes both the FCS and PCS as special cases. e rest of the paper is organized as follows. In Section 2, signal and channel models are described. e 2D-OPC subarray structure is discussed in Section 3, followed by the performance evaluation results in Section 4. Conclusions are given in Section 5.

System Model
We consider a mmWave multiuser MIMO communication system. User equipments (UEs) are randomly distributed in hexagonal cells as shown in Figure 1, located at the height of h UE from the ground. e base station (BS) is equipped with M transmit antennas which are connected to N RF RF chains and performs simultaneous transmission to K UEs. e height of the BS antenna is denoted by h BS . e beamforming module consists of the analog beamformer F RF ∈ C M×K and the digital beamformer F BB ∈ C K×K . e transmitted signal vector x ∈ C K×1 from the antenna array can then be expressed as x � F RF F BB s, where s ∈ C K×1 represents the transmitted symbol containing K data streams in the baseband with E[ss H ] � I K . At the receiver, the received signal y ∈ C K×1 is written as.
where H ∈ C K×M is the channel matrix and n ∼ CN(0, σ 2 I K ) is the K × 1 Gaussian noise vector including independent identically distributed (i.i.d.) complex Gaussian variables. We assume that all UEs have the identical noise variance σ 2 . e achievable rate R k for the k th UE can be expressed using the received signal-to-noise ratio (SNR) as where f BB i is the K × 1 i-th column of the digital beamformer F BB . Sum-rate R of the system can be represented as R � K k�1 R k . We consider a V × H uniform planar array (UPA) with M antennas as the overall BS antenna structure, which includes multiple subarrays. e 2D-OPC subarray structure is shown in Figure 2, where the subarray has v vertical elements and h horizontal elements, i.e., the subarray has p � vh antennas. When the size of each subarray is identical to the overall antenna array, i.e., v � V and h � H, the corresponding structure becomes the FCS by connecting the full-size subarray of size p � M to each of N RF RF chains. When subarrays in utilization partition the overall array, i.e., each subarray has p � (M/N RF ) antennas, the corresponding structure becomes the PCS by connecting each subarray to a single RF chain. e normalized array response vector of a UPA in the direction with azimuth angle ϕ and zenith angle θ is written as. Also, λ is the signal wavelength and d is the spacing between two adjacent antenna elements. Since the mmWave propagation environment is well characterized by the channel model with limited scattering [23], we adopt a simplified clustered channel model, i.e., the Saleh-Valenzuela (S-V) model in [24]. e K × M channel matrix H under the S-V model can be written as follows [25]: where e k-th UE channel matrix h k has average power equal to the total number of antennas, i.e., where L represents the number of propagation paths between BS and UE. It is assumed that each scatterer contributes to a single propagation path, with complex gain coefficient α k,l for the l-th path for the k-th UE. e coefficient is modelled to follow the complex Gaussian distribution, i.e., α k,l ∼ CN(0, 1). Parameter ϕ k,l denotes the azimuth angle of arrival for the l-th path which follows the Laplacian distribution with mean ϕ k and variance σ 2 ϕ . Also, θ k,l denotes the zenith angle of arrival for the l-th path which follows the Laplacian distribution with mean θ k and variance σ 2 θ . Another channel model of interest in this paper is the threedimensional spatial channel model (3D-SCM) described in [26]. e 3D-SCM includes the propagation information for carrier frequency bands in the range of 0.5 GHz to 100 GHz. Statistical properties of radio propagation in this frequency range including the delay spread, angular spreads, Ricean K factor, and shadow fading are all captured by parameters applied to this model. e channel matrix in equation (4) can be generated according to this model to support different transmission scenarios such as urban macrocell (UMa), urban microcell (UMi), rural macrocell (RMa), and indoor hotspot (InH).

2D Overlapped Partially Connected Subarray Structure
In the FCS, each antenna element is connected to all RF chains. Taking advantage of a network that is fully connected between the antenna elements and the RF chains, the FCS can achieve the full beamforming gain for each RF chain. However, the FCS requires a substantial number of power amplifiers and phase shifters which makes its practical implementation less attractive. e number of required phase shifters is as many as MN RF . For the PCS, the whole array is divided into disjoint subarrays, and the antenna elements in each subarray are connected to a single RF chain. e total number of phase shifters in this case is M, significantly reducing the hardware complexity. Despite this complexity reduction, the utilization of the PCS is limited due to the degraded transmission performance.
In order to solve the weaknesses of the existing hybrid beamforming, the OSS was introduced to achieve a higher beamforming gain while maintaining the hardware complexity at a reasonable level. e 2D-OPC subarray structure is a generalization of the OSS for the ULA to the planer 2D space. Each subarray in the 2D-OPC structure is connected to multiple RF chains and can overlap in both horizontal and vertical directions with other subarrays. e proposed the 2D-OPC subarray structure is shown in Figure 3. e set of antenna indices included in the subarray connected to the k-th RF chain is defined as.
which includes p � vh elements. In Figure 3, c k,m is the parameter indicating whether the m-th antenna element is connected to the k-th RF chain and is defined as If c k,m � 1, the k-th RF chain and the m-th antenna are connected with a phase shifter. If c k,m � 0, on the other hand, the k-th RF chain and the m-th antenna are not connected, and no phase shifter exists. We assume there are S subarrays, each including p � vh antenna elements. We further allow one subarray being connected q different RF chains, with index sets satisfying I 1 � I 2 � · · · � I q , for example. Parameter q is called the multiplicity factor for subarrays. Accordingly, the total number of RF chains satisfies N RF � qS and the total number of phase shifters satisfies N PS � pN RF � pqS. As a special case, the FCS has index sets I k � 1, 2, . . . , M { } for k � 1, 2, . . . , N RF , and multiplicity factor is q � N RF . In Figure 4  to one of 4 RF chains in the antenna structure when multiplicity factor is q � 1. In Figure 4(b), S � 3 subarrays of size v � 8 and h � 4 are placed in the overall array. e number of RF chains is N RF � S � 3 if the multiplicity factor is q � 1. If the multiplicity factor is q � 2, two RF chains are connected to the same subarray with separate sets of phase shifters, and the total number of RF chains in use becomes N RF � qS � 6. By changing the subarray size, the number of subarrays, and the multiplicity factor, various different ways of constituting the antenna arrays can be arranged, providing a significantly increased degree of freedom in array design.
Hybrid beamforming is performed in two steps. First, analog beamformer F RF is applied using RF phase shifters with the constant modulus constraint on beamforming elements. e M × K analog beamformer is constructed as follows.where f k � [f k,1 , f k,2 , . . . , f k,M ] T is the M × 1 analog beamforming vector for the k-th UE with its elements defined as for k � 1, 2, . . . , K and m � 1, 2, . . . , M. Here, [.] m,n denotes the (m, n)-th element of a matrix. Secondly, the digital beamformer is determined. For example, the zero-forcing (ZF) precoder can be applied to the effective channel H eff � HF RF to eliminate the multiuser interference. e dimension for the effective channel is K × K, obtained by multiplying the K × M channel H and M × K analog beamformer F BB . In this case, the K × K digital beamformer has the form of For equal power allocation to all users, the column-wise power normalization may be applied to the overall beamformer F � F RF F BB . e dimension for the overall beamformer is M × K, obtained by multiplying the M × K analog beamformer F RF and K × K digital beamformer F BB .

Performance Evaluation
We present performance evaluation results for the proposed 2D-OPC array structure under different parameter settings. ree types of channel models are used for the evaluation. e first channel of interest is the pure line-of-sight (LoS) channel with L � 1 single path. e second channel is based on the multipath S-V model with L � 8 paths. e azimuth and zenith angles for each path follow the Laplacian distribution with mean values ϕ k and θ k determined according to the UE's location, which is randomly assigned within the cell with the intersite distance of 200 m. e angular spread of 5°is used for the Laplacian distribution. e third channel model is the SCM UMi adopted by 3GPP for link-level performance evaluation for urban microcells [26]. We first apply the L � 1 LoS channel for performance evaluation results in Figures 5-8 and then extend the evaluation to all channel models in Figures 9-11. UEs are assumed to have a single antenna. e antenna heights for the BS and the UEs are, respectively, h BS � 10 m and h UE � 1.5 m. Unless otherwise stated, the received signal-to-noise (SNR) at the UEs is assumed to be 20 dB. e UPA at the BS has M � 64 antennas with V � 8 vertical elements and H � 8 horizontal elements. Although the 2D-OPC array structure of any sizes can be used for proposed hybrid beamforming, we choose the 8 × 8 UPA for the performance verification. e 8 × 8 UPA is an array type adopted by the 5G NR standard, as specified in 3GPP TR 38.901 [26]. Many other related investigation results for hybrid beamforming are given using the array dimension V � H � 8 for this reason, including those in [9,22]. e spacing between adjacent antenna elements is set to be halfwavelength. Subarray dimensions are chosen to be in even numbers v � 2, 4, 6, 8 and h � 2, 4, 6, 8 which do not exceed the maximum size of V � 8 and H � 8. Using these subarray parameters, symmetric placements of subarrays within the whole 8 × 8 array can be made as further explained below.
We first evaluate the sum-rate versus the number of RF chains and present the corresponding result in Figure 5. Each RF chain is connected to a single subarray with multiplicity factor q � 1 in this figure. As can be seen in the figure, the sum-rate performance for all array structures tend to increase as the number RF chains increases. After the number of RF chains exceeds 16, however, interuser interference becomes more severe to overcompensate the spatial multiplexing gain. e sum-rate performance for the 2D-OPC structure significantly exceeds that of the conventional PCS, and the performance improves as the subarray size increases. However, a larger subarray can support a limited number of locational shifts within the overall array, thereby providing a limited range of support for the RF chains. is limitation can be removed by introducing the multiplicity factor, by connecting the same subarray to q RF chains. Figure 6 shows the sum-rate performance with different multiplicity factors, and it can be observed that a wider range of RF chain numbers can be supported by increasing the value of q. It is also noted from the figure that there exists a value of q which gives the best sum-rate performance for given N RF . erefore, it is recommended that we use this optimal q value for the 2D-OPC subarray design. e optimal choice of q depends on the subarray sizes, and the figure shows such choices for subarrays with parameters (v � 6, h � 6), (v � 8, h � 2), and (v � 2, h � 4). All subsequent performance figures are based on these optimal values for q.
Although the FCS outperforms the 2D-OPC subarray structure in Figure 5, it uses a significantly larger number of phase shifters due to the full connections to RF chains from each antenna element. erefore, we compute the number of all phase shifters in use for each array structure for given numbers of RF chains and then present the sum-rate versus the number of phase shifters N PS in Figure 7. e figure exhibits the fact that there exists a certain range of N PS values over which the proposed structure outperforms the conventional FCS. e result can be applied to choose a desirable antenna structure under the constraint of strict limitations on hardware cost.
We further normalize the performance by dividing the achievable sum-rate by the number of phase shifters in use and then plot this normalized sum-rate as a function of N RF in Figure 8. Again, we observe that the proposed scheme can outperform the FCS in this measure, confirming the strong cost-effectiveness of the 2D-OPC structure.
We extend our performance evaluation results to a wider class of channels. In Figures 9 and 10, the sum-rate performance is evaluated and compared for three different types of channel models in consideration. e multipath channel with enhanced spatial diversity exhibits improved throughput performance over LoS channel. On the other hand, the SCM with a large number of clusters and rays within each cluster generates received power fluctuations and reduces the transmission performance. Figure 11 shows the sum-rate performance for these channel types for increasing values of the signal-to-noise ratio (SNR). e 6-by-6 subarray is used with the number of    International Journal of Antennas and Propagation 7

Conclusions
We presented a generalized description of the overlapped subarray structure for UPAs, which includes the existing FCS, PCS, and 1D OSS as special cases. By adjusting the parameters for the 2D-OPC subarray structure, the hardware complexity of the overall antenna array can be adequately controlled. Performance evaluation results show that the array parameters which maximize the throughput can be chosen for given conditions such as the supported number of users for simultaneous data transmission, i.e., the number of RF chains employed in the antenna system. A good balance between the hardware cost and the sum-rate performance can be achieved by employing the proposed structure. Although the presented results are mainly targeted for the mmWave channel, the same approach can also be adopted at lower frequencies with a dominant directional path, for which hybrid beamforming effectively applies.

Data Availability
e data used to support this study are available upon request to the authors.

Conflicts of Interest
e authors declare that they have no conflicts of interest.