Reconfigurable Power Divider-Enabled Dynamic Hybrid m-MIMO Transmitter

A promising solution for massive Multiple-Input Multiple-Output (m-MIMO) systems is Hybrid digital-analogue (HDA) beamforming as it offers a balanced trade-off between energy efficiency (EE) and spectral efficiency (SE). The lack of practical demonstrations in the open literature is primarily because those most existing works require a large number of phase shifters (PSs), radio frequency (RF) switches, power dividers (PDs), and power combiners which contribute to high hardware complexity and energy consumption (due to insertion loss). In this paper, we introduce a practical dynamic subarray m-MIMO structure that is based on reconfigurable power dividers (RPDs). The extensive system simulation results, considering hardware imperfection extracted from a practical RPD implementation, indicate that the proposed RPD-based dynamic HDA m-MIMO outperforms the fixed subarray counterpart.


I. INTRODUCTION
T HE PROLIFERATION of data and wireless devices demands higher capacity of wireless communication networks. Multiple-Input Multiple-Output (MIMO) has become a prevailing technology to significantly enhance the channel capacity by employing multiple antennas at both transmitter (Tx) and receiver (Rx) to explore multipath radio propagation channels. This high dimension of Tx and Rx propagation pairs creates an extra multiplexing domain, namely spatial multiplexing, in addition to the conventional time and frequency domains. In order for further channel capacity enhancement, the massive-MIMO (m-MIMO) concept [1], [2], where hundreds of antenna elements are equipped at Tx and/or Rx ends, is becoming a reality.
A major challenge of the m-MIMO architecture, as shown in Fig. 1(a), is its high cost and complexity since every antenna is equipped with a dedicated radio frequency (RF) chain consisting of power-hungry power amplifiers, mixers, Digital-to-Analog Converters (DACs), etc. Furthermore, channel estimation, equalization, precoding, and baseband processing are all heavily burdens because of channel matrices of large dimensions. In order to alleviate this issue, various simplified m-MIMO architectures have been proposed in the literature [3], [4], [5], [6]. They all follow the same concept, i.e., employing analog RF precoding networks (like power splitting, power combining, and phase shifting network or lens-based analog beamforming networks [7]) to reduce the number of RF chains and thus the dimension of equivalent baseband channels. Hence, they are also labelled as hybrid digital-analog (HDA) m-MIMO.
In [8], a type of HDA m-MIMO structure, depicted in Fig. 1(b), was introduced. Here each RF chain, the number of which is less than the number of Tx antennas, is routed to all antennas via RF beamforming networks consisting of power dividers (PDs), power combiners, and phase shifters (PSs). This is commonly categorized as fully-connected HDA m-MIMO, which has been extensively investigated, for example in [9], [10]. The authors in [9] constructed an HDA beamformer by reducing the Euclidean distance between the HDA precoder and the optimal fully digital baseband precoder. By contrast, the HDA beamformer in [10] was designed using a codebook-based approach in which a series of analog precoders are pre-defined. The beamforming networks that enable this fully-connected m-MIMO, however, are still extremely complicated and costly when hardware implementation is considered, even with reducing the resolution of PSs [3], [10], [11], [12], [13], [14] or the resolution of DACs [15], [16].
To further simplify required RF beamforming hardware another type of HDA structures was studied [5], [6], [17], [18], [19], [20], [21], [22], [23], [24], [25] named as partiallyconnected HDA m-MIMO, in which each RF chain is connected to a subset of the Tx antenna elements, referred to as a subarray. This partially-connected HDA m-MIMO can be sorted into two groups: To begin, the fixed subarray HDA structure [5], [6], [17], [18], as shown in Fig. 1(c), wherein the pairing of each RF chain and subarray is fixed. While the fixed subarray structure aims to maximize energy efficiency (EE) with reduced hardware complexity [19], it suffers low spectral efficiency (SE), especially in some dynamic propagation environments [20]. The reason for this is that low-resolution PSs and the reduced number of antennas in each subarray (compared with the full array) result in low precision on beam steering and low beamforming gain. The other category is the dynamic subarray HDA structure [18], [19], [20], [21], [22], [23], [24], [25] which aims to strike a better trade-off between EE and SE. The dynamic subarray can be constructed with various degrees of reconfigurability of connecting RF chains and Tx antenna elements/subarrays. For example, in Fig. 1(d) a partially dynamic subarray [21], [22] allows each RF chain to dynamically connect to any subarrays where the antenna partition is fixed in each subarray. Alternatively, Fig. 1(e) shows a fully dynamic subarray [18], [19], [23], [24] that permits each RF chain to be adaptively linked to any arbitrary subset of the array antennas. The authors in [22], [23] developed an alternating-selection method by co-designing switching and phase-shifting networks with the objective to minimize the Euclidean distance between the ideal fully digital and HDA beamformers. In [25] the design of dynamic subarray beamformer was decoupled into two sub-problems for reduced computation complexity.
The majority of the dynamic subarray beamformer works discussed above, unfortunately, assume ideal RF components with little consideration of RF performance, insertion loss, and associated implementation cost, such as ideal mappers constructed using power dividers/combiners and/or switching networks [18], [19], [21], [22], [23]. And those reported structures require significantly more RF components for beamforming networks. More critically, the issue around impedance matching is not even considered. For example, in work [21] when a power combiner is used to combine power of signals from different RF chains of varied numbers, the power loss due to mismatch (or even matched) unused ports is not considered. These issues motivate us to construct a practical dynamic hybrid m-MIMO solution with low complexity and real hardware implementation. This will provide more in-depth insights into the impact of beamforming hardware candidates on the system performance Under this context, in this work we study a practical dynamic subarray HDA m-MIMO that is constructed using two types of low-complexity reconfigurable PDs (RPDs)power-ratio-tuneable RPDs (PT-RPDs) and port-selectable RPDs (PS-RPDs) -avoiding the use of lossy switching network, and low-resolution (3-bits) PSs with RF hardware imperfection involved. Two PS-RPD prototypes were designed, fabricated, and measured. Additionally, a PT-RPDs-assisted low-complexity algorithm for analog precoder designs are studied and evaluated for single-user (SU) m-MIMO and multi-user (MU) m-MIMO under various channel conditions. The results show that the 3-way PS-RPDs can consistently provide SNR gain of about 3.7 dB in both SU and MU m-MIMO system.
Notations: The following notations are used throughout this paper: Boldface upper-case and lower-case letters indicate matrices and vectors, respectively. Other lower-case letters represent coefficients or constants. (·) T , (·) H , (·) −1 , (·) † and donate transpose, conjugate transpose, inversion, and Moore-Penrose pseudoinverse operations respectively. ||·|| is the Euclidean norm of a vector. E(·) and Var(·) refer to statistical expectation and variance. I N represents an identity matrix with dimension N.

II. SYSTEM MODEL OF HDA m-MIMO, AND THE PROPOSED DYNAMIC SUBARRAY STRUCTURE
This section first discusses the generic HDA m-MIMO system model for SU and MU applications, followed by introducing our proposed dynamic subarray HDA m-MIMO based on RPDs

SU-MIMO:
In this sub-section, we present a SU-MIMO system with an HDA beamformer at Tx and a fully digital combiner at Rx as depicted in Fig. 2. Here it is assumed that the transmitter employs N t antennas that are served by N RF RF chains. N s data streams are to be conveyed to a single receiver equipped with N r Rx antennas and RF chains, where N s ≤ N r ≤ N RF < N t . At the transmitter side, as seen in Fig. 2, N s data streams, denoted as s C N s ×1 , are first digitally precoded through the matrix P B C N RF ×N s before being injected into N RF RF chains. They are then upconverted to RF domain and further processed by an analog RF precoder P A . The excitation signal vector x C N t ×1 at the Tx antenna array can thus be written as The digital precoder is power normalized, i.e., ||P B || 2 = 1. Assuming that the wireless channel is assumed to be timeinvariant, noted as H C N r ×N t , the received baseband signal y C N s ×1 , after being processed through a digital baseband combiner W B C N s ×N r , becomes where n C N s ×1 is independently and identically distributed (i.i.d.) zero-mean complex Gaussian noise vector with a variance of σ 2 .
Once the analog RF precoder P A is designed, the digital baseband precoder P B at the Tx end and the combiner matrix W B at the Rx end can be obtained from the Singular Vector Decomposition (SVD) of the estimated effective channel matrix H e of reduced dimension C N r ×N RF , see (3). The power allocation vector ρ can be found using water-filling on the singular value of the effective channel matrix.

MU-MIMO:
In the downlink MU-MIMO system, a base station (BS), which uses the same structure as that in the SU-MIMO, transmits signals to K users simultaneously. In order to eliminate the inter-user interference, linear precoding methods like channel inversion or Zeroforcing [27], [28], [29] can be adopted at transmitter end with a minimal level of complexity in baseband processing [29].
When each user in an MU-MIMO system is equipped with multiple receive antennas, it is feasible to transmit several parallel data streams to each user. In this scenario, the method of channel inversion, though workable, becomes inefficient since it essentially treats a user with a multiple antenna receiver as multiple independent users, each with a single receiving antenna. To tackle this issue, approaches like block channel inversion or block diagonalization are commonly used [27]. It allows transferring the transmit signal to be transferred to a group of antennas rather than a single antenna with zero-interference constraints. Here we use N k r to represent the number of receive antennas for user k out of a total of K users, and N k s as the corresponding number of data streams for the same user. The received signals y k for user k can be mathematically expressed as where H k C N k r ×N t and n k C N k r ×1 are, respectively, the channel matrix and associated noise for user k.
A combiner, as in the SU-MIMO scenario, is needed to post-process the received signals. When determining the baseband precoder and combiner using block diagonalization available to calculate the power allocation vector using water-filling method.

B. PROPOSED DYNAMIC SUBARRAY m-MIMO
Following the antenna structure reported in [26], in this paper we study a transmitter employing a planar antenna panel consisting of N RF subarrays.
Each subarray consists of the same number of antenna elements with the size of M s = 2N. In each subarray, there are N antennas for each orthogonal linear polarization, corresponding to red and blue elements in Fig. 3(a). Each subarray of 2N elements is excited with only one RF chain with the analog beamforming network comprised of a 2-way PT-RPD, two M b -bit PSs, and two N-way PS-RPDs, see illustrations in Fig. 3(b).
The details on the two types of RPDs used in our proposed HRA m-MIMO transmitters are now explained.
2-way PT-RPD: The authors in [30] proposed a 2-way PD with unequal power division between output Ports 2 and 3, as illustrated in Fig. 4(a). And it also includes the special case of turning on or off one of the ports, achieved by two Switch Elements with Matching Networks (SEMNs) are added for port matching and selection, and one Reconfigurable Complex Impedance Transformer Network (RCITN) is inserted at input Port 1 for impedance matching. Readers are referred to [30] for more details. Since the power splitting ratio in this design is determined by the electrical lengths ψ, φ, θ , and the characteristic impedance Z A , Z B . Here we propose to replace these highlighted transmission lines in Fig. 4(a) with tunable ones described in [31] and depicted in Fig. 4(b) in order to reconfigure the power radio between Port 2 and Port 3. See equation (5)-(11) below. The p-i-n diodes MADP04230513060 from MACOM are used in our design simulation and implementation hereafter, and the varactor diode MA46H120 from MACOM is employed here for simulation.
where ω is angular frequency, characteristic impedance and electrical length of the tunable transmission line are Z ttl , θ ttl respectively. N-way PS-RPD: The N-way PS-RPD introduced in [32] is able to control the 'On-Off' states of each output port using a switch-like component called SEMN which acts like a switch but with a functionality to convert port impedance for matching. For the output ports in 'Off' state, the associated p-i-n diode is reverse-biased, and its parasitic inductance and capacitance are compensated through open and shortterminated microstrip stubs for enhanced port isolation. The power injected into these 'Off' ports is thus reflected back and re-routed to other ports in 'On' state, i.e., those with forward-biased diodes. This energy re-routing, however, alters the impedance of other ports. This issue is addressed by adding reconfigurable multiple-impedance transformer networks (RMITNs) at those ports. This RMITN is capable of transforming different load impedances into matched load impedance Z 0 , through different combinations of forward and reverse biasing a set of p-i-n diodes. Thus, this N-way PS-RPD can greatly reduce the insertion loss. To facilitate system simulation in the next section, prototypes of 3-way and 4-way PS-RPDs are designed and fabricated, see Fig. 5, with all S-parameters at 3.5 GHz measured.

C. PROBLEM FORMULATION FOR PROPOSED DYNAMIC SUBARRAY HDA m-MIMO
In some previous works [9], [12], the HDA precoder design was formulated as finding solutions to minimizing the SE drop over fully digital systems. In SU-MIMO system this is written as in (12).
Here P FD is the ideal precoder in fully digital MIMO systems. P B 2 = 1 is added for power normalization, while P is a set representing the phase and modulus constraints determined by the analog beamforming network, In the proposed system studied in this paper, the precoder consists of M b -bits PSs and two types of RPDs. However, in practical HDA m-MIMO systems not all entries of P FD can be measured, because there are only limited number of RF chains. Therefore, the HDA precoder design problem is instead formulated to maximizing system SE as in (13).
Here HP A , as discussed in (3) In (14) λ e C min(N RF ,N r )×min(N RF ,N r ) is a diagonal singular matrix, V e and U e are left and right singular matrices, respectively. Note that the power allocation vector ρ is obtained by applying the water-filling algorithm to λ e . It is worth emphasizing that they are also unitary matrices, indicating that they will not affect the value of the objective function in (13). With (14), the formulation in (13) can be simplified into (15).
Therefore, the optimization problem in (15) is equivalent to finding an optimal RF precoder to maximize the sum of squared singular values of the effective channel matrix, see (16), where (·) denotes SVD operation which returns singular values of the enclosed matrix. The algorithm to find the optimal RF precoder will be introduced in the following section. There the configurations of N-way PS-RPDs are first determined using an iteration method with the criterion to maximize the received signal power. Then the channel covariance matrix [18], [33] approach is employed to determine the settings of the 2-way PT-RPDs and quantized PSs.
In MU-MIMO scenarios the design problem can be regarded as a combination of several independent SU-MIMO systems but using different baseband precoding methods to manage inter-user interference and maximize the sum rate.

III. DESIGN ALGORITHMS FOR THE PROPOSED DYNAMIC SUBARRAY HDA m-MIMO
In this section, the algorithms for implementing (16) are described and which find optimal settings for two types of RPDs and M b -bit PSs. The design of the analog precoder in Fig. 3(b) in the dynamic subarray can be separated into two subproblems: (i) searching for the optimal antenna combination in each subarray and (ii) determining the low-resolution PSs and 2-way PT-RPDs output power ratios among two polarization sections. The methods we use in our work to tackle these two subproblems are summarized as below, and it is assumed that the full channel matrix H is unknown to both Tx and Rx.
Antenna Combination Selection: The reconfigurability here is employed to control the 'On-Off' states of the output ports in N-way PS-RPDs in order to allocate more signal power to the antenna element(s) that have lower channel loss to the targeted Rx. When considering exhaustive search method to experience all combinations, the iteration space needed for 2×N RF N-way PS-RPDs is of size (2 N − 1) 2×N RF . For example, when N = 3 and N RF = 32, the total iteration would be over 10 54 , making this exhaustive search method infeasible. While it is found that the use of 2-way PT-RPDs can make each antenna subarray be optimized independently. Thus, this reduces the search space to (2 N − 1)×2×N RF . For the same example, the size of the iteration space dramatically drops to 448. With the help of Rx feedback, each N-way PS-RPD configuration corresponding to the maximum Rx signal power can be identified. The essence that these N-way PS-RPDs can be treated independently is that the signal magnitudes and phases can be further adjusted by the remaining 2-way PT-RPDs and PSs in-between.
Power Splitting and Phase Shifting between Two antenna Sections in Subarrays: Author Sungwoo proposed the channel covariance matrix method to design analog precoders in [18], summarized in Algorithm 1. Here the largest singular value of the channel covariance matrix is used to determine the values of PSs in a fixed subarray structure, where each RF chain connects to M s antennas via M s PSs. In our dynamic subarray structure, after determining the Nway PS-RPDs, the subset of each subarray is fixed as well. Therefore, the channel covariance matrix method can also be used to determine the values of PSs when the number of paths in the wireless channel is roughly known. The detailed method to estimate the channel covariance is described in [33]. When applying the channel covariance estimation method in our work, the 2-way PT-RPDs are set up as the equal power mode, and the N-way PS-RPDs are configured with optimal setting obtained in the first step. The next step is to determine the power ratio e 2 of the 2-way PT-RPDs. Recall the definition of the channel covariance matrix [33], and re-write it in (17), from which the diagonal elements in R can be extracted as in (18).
Each diagonal element can be regarded as the path gain associated with the corresponding transmit antenna. The concept of finding optimal settings for the 2-way PT-RPDs is similar to the water-filling algorithm when determining the power allocation vector ρ. The power ratio of the 2-way PT-RPDs e 2 can thus be computed as sum R diaS r,a2 for r = 1, 2, . . . , N RF , where R diaS r,a{1,2} is the diagonal element of channel covariance matrix associated with the antenna section connected with the {1 st , 2 nd } output port of the 2-way PT-RPD in the r th subarray. After obtaining the optimal settings of PSs, 2-way PT-RPDs and N-way PS-RPDs, the analog precoder can be constructed. The baseband precoder and combiner can then be derived using (14).

IV. SYSTEM PERFORMANCE EVALUATION
In this section, the simulation results are presented to evaluate the performance of our proposed RPDs-based dynamic subarray HDA m-MIMO.
In the simulation, the CDL-B wireless channel model [34] is employed with the number of paths L = N c × M r where N c is number of clusters and M r of 20 is the number of rays in each cluster. The system simulation is performed at 3.5 GHz and Orthogonal Frequency Division Multiplexing (OFDM) signals with 64 subcarriers of 15 kHz spacing are used. The data streams, where N s = 2 for each user, are Quadrature Phase Shift Keying (QPSK) modulated in each subcarrier. In our system simulation, we assume PSs are 3-bit with 0.5 dB insertion loss, and the insertion loss for N-way fixed power dividers (FPDs) are 1 dB when N = 3 and 1.2 dB when N = 4, which are typical performance of some commercial devices, e.g., ZB3PD-63+ and ZN4PD1-63HP-S+ from Mini-Circuits. Measured S-parameters of N-way PS-RPDs are used in the link simulation. The 2-way PT-RPD is simulated in ADS with practical lossy diodes and lossy dielectric materials included.

A. BER PERFORMANCE ANALYSIS
Firstly, the SU m-MIMO system is considered to evaluate the performance of our proposed dynamic subarray HDA beamformer (labelled as PT&PS-RPD), depicted in Fig. 3(b). For performance comparison and to reveal the benefits of those RPDs, we also simulate the systems with the following beamforming structures (taking N = 3 as an example): i) 64 RF chains, each of which connects to a 3-element Tx subarray via a 3-way FPD (labelled as digital-FPD); ii) 32 RF chains, each of which connects to a 6-element Tx subarray via a 2-way fixed PD, PSs, and two 3-way fixed PDs (labelled as fixed-subarray); and finally iii) 32 RF chains, each of which connects to a 6-element Tx subarray via a 2way fixed PD, PSs, and two 3-way PS-RPDs (labelled as FPD&PS-RPD). These benchmarking Tx architectures are illustrated in Fig. 6.
Following the design procedures and algorithms described in Section III, the simulated bit error rate (BER) curves versus SNR t (signal to noise ratio normalized at Tx end) for the proposed 2-way PT-RPDs and 3-way PS-RPDs based dynamic subarray SU m-MIMO (namely PT&PS-RPD) against other benchmarking systems introduced above are depicted in Fig. 7. The CDL-B channel with 23 clusters (N c = 23) is assumed in the simulation. It can be observed that the proposed dynamic subarray PT&PS-RPD outperforms the fixed-subarray by 4 dB, in which the 3-way PS-RPDs and the 2-way PT-RPD contribute to 3 dB and 1 dB respectively. Here the gain in this section is defined  as the gap in SNR t when BER equals 10 −3 , seen in Fig. 7. When considering different channel conditions, for example different number of clusters N c , the total SNR gains obtained, compared with the fixed-subarray system, as well as gains contributed by the 3-way PS-RPDs and 2-way PT-RPD, are plotted in Fig. 8. In average 4.5 dB gain can be achieved. It is also important to highlight that with half number of RF chains the proposed dynamic subarray only compromises less than 1 dB SNR when comparing with the corresponding digital-FPD system. When the antenna panel is extended from 192 antenna elements to 256 elements, the 4-way PS-RPDs, see the prototype in Fig. 5(b), can be employed to  keep the number of RF chains unchanged. Fig. 9 shows BER curve versus SNR t for the 2-way PT-RPD and 4-way PS-RPDs based dynamic subarray SU m-MIMO system under CDL-B channel with N c = 23. The results in Fig. 9 show that the overall achievable gain is reduced from 4 dB to 1.5 dB, and, in particular, the gain provided by the 2-way PT-RPD diminishes by comparing FPD&PS-RPD BER curve with PT&PS-RPD.
In MU m-MIMO systems, the similar conclusion can be obtained, see simulation results for 192 antennas (with 3-way PS-RPDs) and 256 antennas (with 4-way PS-RPDs) in Fig. 10 and Fig. 11.
The SNR gains from RPDs in both SU and MU m-MIMO systems are averaged over different channel condition (NO. clusters) and are summarized in Table 1. Overall, N-way PS-RPDs deliver more performance improvement than 2-way PT-RPDs, while the m-MIMO system benefits at least 2 dB more gain from 3-way PS-RPDs than 4-way PS-RPDs. To demonstrate performance gain from our proposed dynamic structure, the comparison with the peer's work is carried out. However, the existing dynamic HDA m-MIMO structures [18], [19], [21], [22], [23] have issues in their complex beamforming networks and the lack of information on the performance of employed RF components, such as insertion loss, impedance matching, etc. Among those reported HAD m-MIMO candidates, a few assumptions are made to compare their performance to our proposed system. For example, the one employed for multi-user/multi-cellular system in [35], re-drawn in Fig. 12, is constructed, in each RF chain, using a 6-way power divider, 6 switches, and 6 antennas. This is the counterpart for our proposed solution when N = 3. Here we assume the insertion loss of the switches is 0.5 dB, and that of 6-way power dividers is 2 dB. The BER simulation results are presented in Fig. 13, to compare the performance of our proposed PT&PS-RPD against this adaptive beamformer in [35], and fixed subarray. There is around 2.3 dB performance gain available from our proposed dynamic structure.

B. HARDWARE COMPLEXITY ANALYSIS
The energy consumption of the analog beamformers we used is tiny, in the order of microwatt, since only p-i-n and varactor diodes were exploited for reconfigurability. Thus, this energy consumption, comparing with tens or even hundreds of W consumed by RF chains, can be safely ignored. However, the complexity of the employed beamforming networks comes from the number of signals required for control/reconfiguration.   Fig. 5(b).
Here the hardware complexity is analyzed and quantized with regard to the number of control signals required by analog beamformers associated with a single RF chain. For our proposed structure, there exist one 2-way PT-RPD of 15 control signals (12 analog signals and 3 digital signals), two N-way PS-RPDs of (3N-1) digital control signals each, and two 3-bit PSs (3 digital control signals per PS). For an antenna panel of 192 Tx antennas (N = 3), a total of 34 digital and 12 analog control signals are needed. While for the structure using ideal mappers [18], [19], [21], [22], [23], each RF chain requires power dividers (PDs) and power combiners (PCs) with much greater number of ports, e.g., 192-way for those in [18], [19], [23], and 32-way for those in [21], [22]. A practical structure called adaptive beamformer [35] employs only one 6-way PD and 6 switches for each RF chain output. A detailed complexity comparison against some reported dynamic hybrid m-MIMO structures is presented in Table 2.

V. CONCLUSION
In this paper, we proposed a novel and practical dynamic subarray HDA m-MIMO beamforming structure which is realized by two types of RPDs (PT-RPDs and PS-RPDs) and 3-bit PSs. The extensive system simulation results, considering hardware imperfection extracted from practical RPD implementation, indicate that the proposed RPD-based  Fig. 5(a).
dynamic HDA m-MIMO outperforms the fixed subarray counterpart. For instance, a 4.9 dB SNR gain was obtained in MU m-MIMO system which was constructed with 2-way PT-RPDs and 3-way PS-RPDs.