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Electronics 2019, 8(2), 133; https://doi.org/10.3390/electronics8020133
Hybrid Beamforming for Millimeter-Wave Heterogeneous Networks
Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, ON K7K 7B4, Canada
Received: 23 December 2018 / Accepted: 23 January 2019 / Published: 28 January 2019
Heterogeneous networks (HetNets) employing massive multiple-input multiple-output (MIMO) and millimeter-wave (mmWave) technologies have emerged as a promising solution to enhance the network capacity and coverage of next-generation 5G cellular networks. However, the use of traditional fully-digital MIMO beamforming methods, which require one radio frequency (RF) chain per antenna element, is not practical for large-scale antenna arrays, due to the high cost and high power consumption. To reduce the number of RF chains, hybrid analog and digital beamforming has been proposed as an alternative structure. In this paper, therefore, we consider a HetNet formed with one macro-cell base station (MBS) and multiple small-cell base stations (SBSs) equipped with large-scale antenna arrays that employ hybrid analog and digital beamforming. The analog beamforming weight vectors of the MBS and the SBSs correspond to the the best-fixed multi-beams obtained by eigendecomposition schemes. On the other hand, digital beamforming weights are optimized to maximize the receive signal-to-interference-plus-noise ratio (SINR) of the effective channels consisting of the cascade of the analog beamforming weights and the actual channel. The performance is evaluated in terms of the beampatterns and the ergodic channel capacity and shows that the proposed hybrid beamforming scheme achieves near-optimal performance with only four RF chains while requiring considerably less computational complexity.
Keywords:hybrid beamforming; massive MIMO; HetNets; mmWaves
Recently, heterogeneous networks (HetNets) that use massive multiple-input multiple-output (MIMO) and millimeter-wave (mmWave) technologies has emerged as a promising solution to enhance the network capacity and coverage of next-generation 5G cellular networks [1,2,3,4,5,6]. Small cell deployment in HetNets can achieve high signal to interference plus noise ratio (SINR) and dense spectrum reuse, mmWave can address the current challenge of bandwidth shortage, and the large number of antenna arrays [7,8,9,10] are essential for mmWaves to compensate for channel attenuation. In Reference  we applied the concept of massive multiuser (MU)-MIMO to enhance both the access and the backhaul links in HetNets, and it was shown that such a concept could significantly improve the system performance in terms of link reliability, spectral efficiency, and energy efficiency. Traditional MIMO-beamforming systems require a dedicated radio frequency (RF) chain for each antenna element, which becomes impractical with massive MIMO systems due to either cost or power consumption. To reduce the number of RF chains, hybrid beamforming (HBF), which combines RF analog and baseband digital beamformers, has been proposed as a promising solution [12,13,14,15,16,17]. Figure 1 shows a general hybrid configuration that connects antenna elements to RF chains, where , using an analog RF beamforming matrix built from only phase-shifters. Two widely-used analog beamformer architectures for hybrid beamforming are shown in Figure 2. The fully-connected hybrid beamforming structure of Figure 2a provides a full beamforming gain per transceiver—but with high complexity—by connecting each RF chain to all antennas through a network of phase shifters [12,13,14,15]. Figure 2b, on the other hand, shows a partially-connected structure, where each RF chain is connected to number of sub-arrays. Such a structure has a lower hardware complexity at the price of reduced beamforming gain.
Previous studies on massive hybrid MIMO mainly focused on single-user systems [12,13,14]. On the other hand, MU-MIMO cases were studied in References [15,16,17]. In Reference  a scheme called Joint Spatial Division Multiplexing (JSDM) was proposed to create multiple virtual sectors which reduce the overhead and computational complexity of downlink training and uplink feedback. In References [16,17] it was shown that the required number of RF chains only needs to be twice the number of data streams to achieve the same performance of any fully-digital beamforming scheme. These studies, however, did not consider HBF in the context of HetNets and focused primarily on macro-cellular systems. In this paper, we consider a HetNet where both the macro-cell base stations (MBSs) and small-cell base stations (SBSs) are equipped with fully-connected massive hybrid antenna arrays, while all mobile users have a single antenna. We propose a low-complexity HBF that is fully-based on eigenbeamforming. The MBSs and the SBSs select the best-fixed multi-beams by eigendecomposition of the access and backhaul channels. The selected beams are then used by the digital beamformers, which are based on the maximization of the receive SINR of the effective channels consisting of the cascade of the analog beamforming weights and the actual channel [18,19].
2. System Model
We consider the access and backhaul uplinks in the HetNet of Figure 3, where cognitive small cells and their small-cell users (SUs) are concurrently sharing the same frequency band with one MBS and their macro-cell primary users (PUs). It is assumed that both the MBS and SBSs are equipped with massive hybrid antenna arrays while the SUs and PUs are equipped with a single antenna. The SBSs are acting as smart relays between the SUs and the MBS with element transmitting/receiving antenna arrays and RF chains. The MBS is equipped with element antenna arrays and RF chains. It is also assumed that both the SBS and the MBS perform an OFDM-based transmission and that the analog beamformers are identical for all subcarriers while adapting digital beamformers in each subcarrier.
Let and denote, respectively, the set of SUs signals and PUs signals transmitted on each subcarrier, and , , where denotes the number of subcarriers per OFDM symbol in the system. The analysis is done separately on each subcarrier. For brevity therefore, we drop the frequency index .
2.1. Access Link
The received signal vector at the SBS is given bywhere is the channel matrix between the SBS and its users, is the channel matrix between the SBS and the PUs, is the transmitted signal vector of users in the small-cell, is the transmitted signal vector of users in the HetNet, and is the received complex additive white Gaussian noise (AWGN) vector at the SBS.
It should be noted that in Equation (1), the interference between small-cells was neglected. This was justified by the fact that small-cell base stations are using a large number of antennas, which enables sharp beamforming towards their users without harming neighboring small-cells.
The SBS received signal, , is first applied to an receive analog beamforming weight matrix, , whose output is directed to an receive digital beamforming weight vector. If we denote the combined digital-analog receive beamformer for the user as , then the detection of the user signal by its SBS can be expressed aswhere is the column of that represents the channel between the SBS and its user.
If we denote as the effective access channel between the SBS and its user, then for a set of selected beams, i.e. known , the SINR can be expressed in terms of the digital beamformer, , aswhere is the covariance matrix of the interference-plus-noise given by
2.2. Backhaul Link
The hybrid beamforming weights at the backhaul link are obtained based on orthogonal pilot signals transmitted from each SBS to the MBS. The SBS applies its pilot signal to an transmit digital beamforming weight vector followed by an transmit analog beamforming matrix . If we denote the combined digital-analog transmit beamformer for the SBS as , then the array output of the MBS can be written aswhere is the vector containing the outputs of the element antenna array of the MBS, is the channel matrix representing the transfer functions from the element antenna array of the SBS to the element antenna array of the MBS, is the channel matrix from the PUs to the MBS’s element antenna array, and is the received complex AWGN vector at the MBS.
The MBS detects the SBS signal by applying the output of the array to the receive analog weight matrix followed by the receive digital beamforming weight vector. If we denote the combined digital-analog receive beamformer for the SBS as , then the detection of the SBS signal by the MBS can be expressed aswhere is the SBS signal, is the interference from other SBSs, and is the interference from PUs.
Assuming that are complex-valued random variables with unit power, i.e., , and denoting as the effective channel between the SBS and the MBS, the SINR at the MBS for the SBS can be expressed aswhere is the covariance matrix of the interference-plus-noise at the backhaul link.
2.3. End-to-End SINR and Channel Capacity
Once the hybrid beamforming weights of the backhaul link are obtained, they can be used to forward the SUs signals to the MBS. The SBS applies the received user signal, to the hybrid beamformer, . The transmitted signal at the output of the antenna array can then be expressed asand the expression for the array output of the MBS can be written aswhere is the array output of the MBS from the SBS.
Using Equation (2), can be expressed in terms of the user signal, , as follows:
When the MBS applies the output of the array to the hybrid weight, , the detection of the user signal of the SBS by the MBS can be expressed aswhere
- is the user signal of the SBS,
- is the interference from the other SUs of SBS,
- is the interference from the other SBSs.
The end-to-end SINR at the MBS for the user of the SBS can be expressed aswhere is the covariance matrix of the interference-plus-noise for the user end-to-end link.
The ergodic channel capacity for each user, , is given by where denotes the expectation operator.
2.4. Channel Model
For the access and backhaul links, we consider mmWave propagation channels with limited scattering which can be modelled at each subcarrier by the clustered channel representation . We assume a scattering environment with scattering clusters randomly distributed in space and within each cluster, there are closely located scatterers.
For the backhaul link, the channel matrix at subcarrier between the SBS and the MBS can be expressed aswhere are the complex gains of the ray in the scattering cluster and are assumed i.i.d CN. With representing the average power of the cluster, are the azimuth angles of arrival and departure, respectively, are the elevation angles of arrival and departure, respectively, and represent, respectively, the normalized receive and transmit array response vectors of the MBS and the SBS.
For the access link, the channel matrix at subcarrier between the SBS and its users can be written aswhere represents the spatial signature vector of the single antenna users.
The azimuth and elevation angles, , within the cluster i are assumed to be randomly distributed with a uniformly-random mean cluster angle of , respectively, and a constant angular spread of , respectively.
For simplicity, the access links between the MBS and its users () and between PUs and the SBS () are modeled by convetional i.i.d MIMO channels.
Note that in this per-subcarrrier representation, it is assumed that for each subcarrier the carrier frequency is much larger than and that can approximately be considered equal for all subcarriers. Consequently, the channel covariance matrices are approximately similar with the same set of eigenvectors for all subcarriers and can be replaced by the average of the covariance matrices, i.e., , , and .
3. Proposed Hybrid Beamforming
3.1. Access Link
The SBS communicates with each SU through a set of selected beams that corresponds to a set of weight vectors. These weight vectors are obtained using the eigenbeamforming scheme and can be expressed aswhere denote the selected eigenvector corresponding to the maximum eigen value of .
Since the analog beamforming matrix is realized using phase shifters only, its elements, satisfy . It should be noted that each SBS is using a different analog matrix for each user and that the system model shown in Figure 1 focuses on the detection of the user of the SBS and shows the analog beamformer and the RF chains for one user only. The analog beamformer can be implemented using the Butler matrix as shown in Figure 4, where four users and four RF chains per user are assumed. Depending upon which 4 ports are activated, 4 beams are produced in specified directions. Since we are assuming 4 different channels, we should expect 4 different ports for each user.
Once the analog beams are selected, the received optimal digital weights, , are obtained based on the maximization of the access link receive SINR, , given by Equation (3):where denote the eigen vector corresponding to the maximum eigenvalue of the effective access channel, .
3.2. Backhaul Link
The transmit analog weights of the SBS are based on the eigenbeamforming scheme and are given bywhere denote the selected eigenvector corresponding to the maximum eigenvalue of .
Assuming channel reciprocity with , the receive analog weight vectors of the MBS are given by . It should also be noted that the MBS is using a different analog matrix for each SBS, which can be implemented using the Butler matrix of Figure 4, where mobile users are replaced by SBSs.
For fixed analog beamforming weights, and, the transmit optimal digital weight vector of the SBS, , and the receive optimal digital weight vector of the MBS, , are obtained base on the maximization of the backhaul link receive SINR and are given bywhere is the eigenvector corresponding to the maximum eigenvalue of , with representing the effective channel given by .
4. Simulation Results
In our simulation setups, we considered a HetNet organized into four SBSs () and one macro-cell. The SBSs and the MBS used the same number of antennas, , and the same number of RF chains, . Each SBS is serving users and the macro-cell is serving 4 users, each transmitting with a single antenna. We assumed QPSK modulation. For the OFDM configurations, we assumed the 256-OFDM system (), which is widely deployed in broadband wireless access services.
Figure 5 shows the beampattern of the proposed HBF with four RF chains and the optimal fully-digital one for the access link. It is noted that the optimal beamformer has about five dominant beams, three of which are similar to the selected beams of the proposed HBF. This beampattern means that the data streams can be successfully transmitted through those three beams using the proposed HBF and that near optimal performance could be achieved if we were to bring the number of RF chains close to the number of dominant beams of the optimal beamformer. For the backhaul link, Figure 6 shows very similar beampatterns with more dominant beams.
Figure 7, on the other hand, compares the ergodic channel capacity of the proposed HBF and the optimal fully-digital one. It is observed that for both cases the optimal beamformer is outperforming the proposed HBF. However, as we increase the number of RF chains, the performance gap between the two schemes was reduced, and a near-optimal solution was achieved by the proposed HBF using four RF chains. On the other hand, for the single cell MU-MIMO case presented in References [12,13,14], near optimal performance was achieved with only five RF chains, and for the MU-MIMO case in [16,17], it was shown that the required number RF chains could be reduced to two to achieve fully digital beamforming performance. However, unlike our case, where we have assumed a HetNet with a macro cell and multiple small cognitive cells, these studies focused primarily on macro-cellular systems and did not consider HBF in the context of HetNets.
In this paper, we employed hybrid beamforming at the access and backhaul links of a mmWave HetNet system. We proposed a low-complexity HBF that was fully-based on MRT/MRC Eigen-beamforming schemes. The performance evaluation in terms of the beam patterns and the ergodic channel capacity showed that the proposed HBF scheme achieved near-optimal performance with only four RF chains and required considerably less computational complexity.
This research received no external funding.
The author would like to thank the Canadian Microelectronics Corporation (CMC) for providing the Heterogeneous Parallel Platform to run the computationally-intensive Monte-Carlo Simulations.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1. Hybrid beamforming.
Figure 2. The architecture of analog beamformers: (a) Fully-connected structure; (b) partially-connected structure.
Figure 3. System model: hybrid beamforming-based HetNet with one macro-cell and K small-cells.
Figure 4. Hybrid beamforming based on Butler matrix for the access link.
Figure 5. Beampattern of the access link: (a) Proposed HBF, 4 RF chains; (b) fully-digital beamforming (optimal).
Figure 6. Beampattern of the backhaul link: (a) Proposed HBF, 4 RF chains; (b) fully-digital beamforming (optimal).
Figure 7. Ergodic channel capacity of the proposed HBF for different number of RF chains.
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