Abstract
Millimeter waves and massive MIMO are a promising combination to achieve the multiGb/s required by future 5G wireless systems. However, fully digital architectures are not feasible due to hardware limitations, which means that there is a need to design signal processing techniques for hybrid analogdigital architectures. In this manuscript, we propose a hybrid iterative block multiuser equalizer for subconnected millimeter wave massive MIMO systems. The low complexity userterminals employ pureanalog random precoders, each with a single RF chain. For the base station, a subconnected hybrid analogdigital equalizer is designed to remove multiuser interference. The hybrid equalizer is optimized using the average biterrorrate as a metric. Due to the coupling between the RF chains in the optimization problem, the computation of the optimal solutions is too complex. To address this problem, we compute the analog part of the equalizer sequentially over the RF chains using a dictionary built from the array response vectors. The proposed subconnected hybrid iterative multiuser equalizer is compared with a recently proposed fully connected approach. The results show that the performance of the proposed scheme is close to the fully connected hybrid approach counterpart after just a few iterations.
1. Introduction
A new generation of cellular network (fifth generation, 5G) is coming, and some innovative technologies are needed to ensure better performance and quality of service (QoS). Two enabling technologies have been considered to meet the QoS requirements for future wireless communication, massive MIMO (mMIMO), and millimeter wave (mmWave) communications [1]. By using mmWave bands, several tens of GHz of bandwidth become available for wireless systems [2], while the mMIMO allows the continued increasing demand of higher data rates for future wireless networks [3]. Comparing the mMIMO with conventional MIMO approaches, the mMIMO can scale up the conventional MIMO by orders of magnitude [4]. A survey on mMIMO, also identified as largescale MIMO, with channel modelling, applications scenarios, and physical/networking techniques can be seen in [5]. The use of mmWave with mMIMO is very promising, because smaller wavelength compared to conventional communication systems allows the same volume to pack more antennas [6], which means that the terminals can support a large number of antennas.
The mmWave mMIMO combination can be used to exploit new efficient spatial processing techniques, such as beamforming/precoding and spatial multiplexing, at both transmitter and/or receiver terminals [7]. These techniques are different than those used for sub6 GHz bands due to limitations in hardware [8]. In these systems, it is not practical to have one fully dedicated radio frequency (RF) chain by antenna [9] as in sub6 GHz conventional MIMO systems [10] due to the power consumption and the high cost of mmWave mixedsignal components. Another issue is the mmWave propagation characteristics, which are quite different from sub6 GHz because the mmWave channels are not so rich in multipath propagation effects [11, 12], which should be taken into consideration in the techniques design for these systems. To overcome the limitation of the RFs chains number, purely analog beamforming can use phase shifters [8], with some schemes proposed in [13, 14], where statistical channel knowledge is used through phase shifters, to optimally adjust the arrays response in space, applying a beam steering solution.
The performance of the pureanalog techniques is limited by constraints on the amplitudes of phase shifters and due to the phases of the ones quantized. Therefore, analog beamforming is usually limited to singlestream transmission [15]. These limitations are overcome by doing some signal processing at an analog level and the rest at the digital level. These architectures are called hybrid analog/digital architectures and have been addressed in [15, 16]. Precoding and/or combining/equalization designs for singleuser systems have been addressed for fully connected hybrid architectures in [17–19]. In these architectures, each of the RF chains is connected to all receiveandtransmit antennas. In [17], a hybrid spatially sparse precoding/combining approach was designed for mmWave mMIMO systems. The spatial structure of mmWave channels was used to transform the singleuser multistream precoding and combining scheme into a sparse reconstruction problem. In [18], joint turbolike beamforming was designed to compute transmit/receiving analog beamforming coefficients; however the digital processing part was not considered. In [19], codebook design approaches were addressed for singlestream transmissions through an analog beamforming structure. For multiuser systems, some beamforming approaches have been proposed for fully connected hybrid architectures [20–22]. The authors of [20] proposed uplink receiving beamforming where they assume only single antenna userterminals (UTs), and at both stages analog and digital ones dealt with multiuser interference. Reference [21] proposed for the downlink a limited feedback analog/digital twostage precoding and combining algorithm. Transmit/receiving analog beamforming are jointly computed in the first stage to maximize the power of the desired signal, and then the interference is explicitly mitigated using conventional linear zeroforcing (ZF) precoding in the second stage, that is, in the digital domain. An efficient hybrid iterative block spacetime multiuser equalizer was proposed in [22]. This equalizer was designed based on the iterative block decision feedback equalization (IBDFE) principle [23]. IBDFE was originally proposed in [24]. It does not need the feedback loop of the channel decoder output, and it can be considered as a low complexity turbo equalizer. IBDFE has been extended to several scenarios, like diversity scenarios and conventional and cooperative MIMO systems, among many others [25–30].
In addition to the fully connected architectures, there are subconnected architectures that allow us to reduce the number of phase shifters from to , when compared with fully connected counterparts, where and are the number of antennas and the number of RF chains [31]. Thus, the power consumption used to excite and to compensate the insertion loss of phase shifters is reduced, and the computational complexity is also lower [31]. There are two types of subconnected architectures, dynamic and fixed [32]. In the dynamic subconnected case, each RF chain can dynamically connect to a different set of antennas, and in the fixed subconnected one, each RF chain is always physically connected to the same set of antennas. Precoding schemes for dynamic subconnected hybrid architectures have been proposed in [32, 33]. Reference [32] uses a relaxation of the mutual information maximization problem to design a technique that adapts the subarray structure according to the channel covariance matrix for frequency selective channels. The authors of [33] proposed a twostep algorithm for singleuser narrowband systems that iteratively optimizes the hybrid precoder for spectral efficiency maximization, obtaining an extra data stream via the index of the active antenna set without any extra RF chain. Fixed subconnected hybrid architectures were addressed in [34–37]. In [34], precoder and combiner schemes for narrowband singleuser systems are proposed, where a twolayer optimization method jointly exploiting the interference alignment and fractional programming was employed. First, the analog precoder and combiner are optimized via the alternatingdirection optimization method, and then the digital precoder and combiner are optimized based on an effective channel matrix. The authors of [35] designed a precoder and a combiner for a wideband singleuser system where the overall spectral efficiency is maximized considering a power budget constraint for each subcarrier. The works in [36, 37] are focused on multiuser downlink systems. In [36], the total achievable rate optimization problem with nonconvex constraints is decomposed to a series of subrate optimization problems for each subantenna array, and then an algorithm is implemented to perform a successive interference cancelationbased hybrid precoding. Precoding and combining schemes are performed in [37] for downlink, where virtual path selection maximizes the channel gain of the analog effective channel, and then a zeroforcing precoding in the digital domain is applied to mitigate the interference.
In this manuscript, we propose an efficient hybrid iterative block multiuser equalizer for subconnected mmWave mMIMO systems. The limitation that each RF chain is only physically connected to a subset of antennas makes the design of the proposed subconnected hybrid iterative multiuser equalizer harder than for the fully connected based approaches. To the best of our knowledge, iterative block detection designed for subconnected mmWave mMIMO architectures has not been addressed in the literature. We propose low complexity UTs without access to the Channel State Information (CSI), using a single RF chain and pureanalog random precoding. A time encoder/precoder is applied to guarantee that the transmit signal and thus the noise plus interference at the receiver side are Gaussian distributed. This not only simplifies the receiver optimization problem but also increases the diversity effects on the mmWave mMIMO system. The designed subconnected hybrid equalizer is optimized by using the average biterrorrate (BER) as a metric. We compute the analog part of the equalizer sequentially over the RF chains using a dictionary built by the array response vectors. Finally, we show that the BER performance of the proposed subconnected hybrid iterative multiuser equalizer tends to the BER performance of a fully connected hybrid equalizer as the number of iterations increases.
This paper is organized as follows. Section 2 presents the subconnected hybrid multiuser mmWave mMIMO system model. Section 3 begins with the description of a channel model for mmWave mMIMO systems, followed by the description of userterminals (UT) and finally the design of the proposed subconnected hybrid iterative multiuser equalizer. Section 4 shows some BER performance results, and in Section 5, major conclusions are presented.
Notations. Matrices are denoted in boldface capital letters and column vectors in boldface lowercase letters. The operations , , , and represent the trace, the conjugate, the transpose, and the Hermitian transpose of a matrix. is the operator that represents the sign of a real number and if is a complex number. It can also be employed elementwise to matrices. The functions and represent the real part and imaginary part of . The functions and , where is a vector and is a square matrix, correspond to a diagonal matrix where the entries of diagonal are equal to and to a vector equal to the diagonal entries of . The element of row and column of matrix is denoted by . The identity matrix is .
2. System Characterization
In this section, we describe the channel model, the UTs, and the receiver signals for the mmWave mMIMO system. We consider a multiuser system with users, each one with transmit antennas, that sends one data stream per time slot to the base station with receiving antennas.
2.1. Channel Model
We consider a sized block fading channel, that is, a channel that remains constant during a block but varies independently between blocks. The channel follows the clustered sparse mmWave channel model discussed in [17] where is the channel matrix, which contributes the sum of clusters, each one with the contribution of propagation paths. The channel matrix may be expressed to the th user as is a normalization factor, and is the complex gain of the th ray in the th scattering cluster. The functions and represent transmit and receiving antenna element gain for and , that is, the azimuth (elevation) angles of arrival and departure. The vectors and represent the normalized receiving and transmit array response vectors for the corresponding angles. Reference [17] addressed the random distributions used to generate the path gains and the angles of channel, such that .
2.2. UserTerminal Model Description
We assume that each user has only a single RF chain and sends only one data stream per time slot over the transmit antennas, as shown in Figure 1. We also consider that the UTs have no access to CSI, simplifying the overall system design. The analog precoder of th UT at the instant is mathematically modelled by and is physically realized using a vector of analog phase shifters, where all elements of vector have equal norm . Therefore, the analog precoder vector of the th UT at the instant is generated randomly according towhere with and are i.i.d. uniform random variables such that .
To guarantee that the transmit signal and then the noise plus interference are Gaussian distributed at the receiver side, the transmit signal is built by using a spacetime block code (STBC). A Discrete Fourier Transform (DFT) performs the time and the analog precoder performs the space encoding. The STBC simplifies receiver optimization and increases the diversity of the mmWave mMIMO system. Mathematically, the operation is expressed bywhere denotes a point DFT matrix and is the time encoded version of . designates a complex data symbol selected from a QAM constellation with , where . The transmitter total power constraint is . For the sake of simplicity, and without loss of generality, in this manuscript, only QPSK constellations are considered.
2.3. Receiver Model Description
For a given sized block, the received signal is given bywhere denotes the received signal vector and is the zero mean Gaussian noise vector with variance . This signal is processed at the receiver by an iterative block decoder based in subconnected hybrid architecture, as seen in Figure 2. First, the received signal is processed through the phase shifters in the analog part, modelled by the vector , where represents the index of the RF chain connected to a set of antennas. The global analog matrix has a blockdiagonal structure:
Baseband processing follows that contains chains, each one connected to a subset of antennas, and a closedloop comprising a digital forward and feedback path. All elements of the vectors have equal norms . For the digital forward path, the signal first passes by a linear filter and then follows a time equalizer and decoding. In the digital feedback path, the recovered data from the forward path first passes through the time precoder and then the feedback matrix . The time encoding and decoding of the data symbols obey (4) andwhere the codeword matrix is a soft estimate of transmitted symbols. Both feedback and feedforward paths are combined; the signal output of the feedback path is subtracted from the filtered received signal .
This proposed receiver structure is a subconnected hybrid iterative feedback multiuser equalizer, where the main difference from the conventional iterative block decision feedback based equalizers is the analog frontend of constant amplitude phase shifters. The limitation that all elements of each analog vector must have the same norm makes the design of the proposed equalizer harder than for the conventional fully digital one. On the other hand, it assumes a subconnected hybrid architecture, where only a set of antennas is connected to each RF chain, which simplifies the physical implementation when compared with the fully connected hybrid architecture. In the following sections the analog and digital forward matrices, and , are designed, as well as the digital feedback matrix, .
3. Iterative Receiver Design
In this section, the proposed hybrid iterative multiuser equalizer is derived for subconnected mmWave mMIMO based systems, discussed in the previous section. We assume a decoupled joint transmitterreceiver optimization problem, with the focus on the design of the subconnected hybrid multiuser equalizer.
3.1. MSE Calculation
A block diagram of the proposed iterative multiuser equalizer is shown in Figure 2. The received signal corresponding to user at the th time slot for the th iteration is given by where is the element of th row and th column of , denotes the analog part of the feedforward vector of user , , is the feedback vector of user , and is the estimate of the transmitted symbols. The matrix is the DFT of the detector output :where is the hard decision related to the QPSK data symbols . At iteration
From the central limit theorem, , , approximately are Gaussian distributed. In addition, since the inputoutput relationship between and , , is memoryless, by the Bussgang theorem [38], we havewhere is a diagonal matrix given by and is an error vector with zero mean and uncorrelated with , with For practical systems, the transmitted signals are not known a priori and definition (13) cannot be used to compute matrix , which must be estimated using other methods. Please refer to work [39] where several methods are presented to estimate matrix .
Define , ; therefore, from (8) and (11), it follows thatwhere . Define and which denotes the overall error. Then, by (15), the overall error isAs we can see in (16), the error has three terms: the residual ISI term; the error stemming from the estimate made by of ; and the term corresponding to the channel noise. Let us assume that error is complex Gaussian distributed. From the Bussgang [38] theorem, the zero mean error variable is uncorrelated with , and then, from (16), the average error power, that is, the MSE at time slot , is given bywhere , with the only nonzero value of in the th position, and then .
3.2. Design of Subconnected Hybrid Iterative Multiuser Equalizer
We start by considering the optimal case, that is, a full digital linear feedforward filter, without the sparsity constraint. We design the iterative multiuser equalizer based on the IBDFE principles, and we use the average BER as the metric. The average BER minimization, according to [22], is equivalent to the MSE minimization:where is the linear feedforward filter, the feedback filter, and a vector with ones. The solution to optimization problem (18) is (see the Appendix)
The previous optimization problem of (18) does not reflect the analog domain constraints. Designate as the set of vectors with constantmagnitude entries, that is, the set of feasible RF equalizers. Thus, the reformulated optimization problem for the subconnected hybrid iterative equalizer is as follows:
The feedback equalizer for the subconnected hybrid iterative equalizer is analogous to the fully digital iterative equalizer referred to in the previous section, because the analog domain constraints do not impose any restriction on the vector and thus are given byFrom (17) and (21), the MSE expression is
According to the constraint of (18) and therefore (22) can be rewritten as
Because the feasible set has a nonconvex nature, an analytical solution to problem (20) is very hard to obtain. However, an approximate solution is possible by assuming that the vector is a version of vector . Let , with , forming the matrix with the array response vectors of the receiver, corresponding to user . As each RF chain is not connected to all antennas, in the selection process of the vector for the th RF chain, the entries corresponding to the unconnected antennas are removed. For the th RF chain we use the dictionary , which is a submatrix of . Then, the vector is selected from dictionary . Therefore, optimization problem (20) can be approximated as follows:where andThe constraint enforces that only one element of vector is nonzero, representing the sparsity constraint. The optimum digital feedforward vector is computed from the solution of the optimization problem in (24), by removing the zero elements. The optimum analog feedforward vector is given by the selected column from , corresponding to the nonzero element of . Consider optimization problem (24), without the sparsity constraint: The associated Lagrangian to problem (26) iswhere , , are the Lagrange multipliers. Taking the derivate in order to obtain , we obtain the optimality conditionwhere (28) is the orthogonality principle relative to MSE estimator (24) and is the residue vector given byAn iterative greedy method is used to select the best column of the dictionary allowing enforcement of the sparsity constraint. Due to the mmWave channel nature, the number of the paths is small and, consequently, the complexity of this selection process is low.
Denote and as the digital and analog terms of the feedforward vector, corresponding to the selected indices from . For example, if the selected indices are in the first positions, then . Thus, from (29) the orthogonality condition simplifies toThe solution to (30) iswith , andThe Lagrangian multiplier is computed by the power constraint of problem (24), given by
After obtaining , the optimum value of the digital feedforward part, the previous steps are repeated until . All steps are synthesized in Algorithm 1, pseudocode of the proposed subconnected hybrid iterative block multiuser equalizer.

4. Performance Results
In this section, we show the performance of the proposed hybrid iterative block multiuser equalizer for subconnected millimeter wave massive MIMO systems. The BER is considered the performance metric, presented as a function of , where is the average bit energy and is the onesided noise power spectral density. The power of each UT is normalized to and the channel matrix, as previously mentioned, is normalized, such that . Then, the average for all users is identical and given by .
The carrier frequency was set to 72 GHz and was considered a clustered narrowband channel model [40] for each user with clusters, all with the same average power, and rays per cluster. The azimuth angles of arrival and departure of the channel model are Laplacian distributed as in [21]. We assumed an angle spread equal to 8° at both the transmitter and receiver and uniform linear arrays (ULAs) with antenna element spacing equal to halfwavelength; however, the subconnected hybrid equalizer developed in this paper can be applied to any antenna arrays. The channel remains constant during a block with size but varies independently between them. We assume perfect synchronization and CSI knowledge at the receiver side. All results were obtained for a QPSK modulation. We present results for two main scenarios, all with . The other parameters are as follows.
Scenario 1 (). (1.a) .(1.b) .(1.c) .
Scenario 2 (). (2.a) .(2.b) .(2.c) .
Here represents the number of antennas per RF chain. In both scenarios, we assume the full load case (worst case), where the number of users and RF chains is equal. To compute the proposed subconnected hybrid equalizer, we consider that , the analog precoder for the th user, is generated according to (2). As mentioned before, the motivation is to keep the UTs with very low complexity without the knowledge of CSI before transmission. The proposed subconnected hybrid iterative multiuser equalizer is referred to here as subconnected. The results are compared with both fully digital, referred to here as digital, and fully connected, referred to as fully connected, which was recently proposed in [22], approaches.
Start with the results obtained for scenario (1.b) presented in Figures 3 and 4. In Figure 3, we present results for iterations 1, 2, 3, and 4 of the proposed subconnected hybrid iterative multiuser equalizer. It can be seen that the performance improves as the number of iterations increases, as expected. Additionally, the gaps between the 1st and 2nd iterations are much higher than between the 3rd and 4th iterations. This occurs because most of the residual intersymbol and multiuser interferences are removed from the 1st to the 2nd iteration. For this scenario, the results with the fully digital and fully connected hybrid multiuser equalizer are presented in Figure 4. In this figure, we verify that the subconnected performance is worse than both digital and fully connected approaches, mainly for the first iteration. However, for the 4th iteration we can see that the performance of the proposed subconnected hybrid multiuser equalizer is close to the digital and fully connected counterparts. Therefore, we can argue that the dictionary approximation and the sequential optimization are quite precise.
Compare scenario (1.b) with scenario (2.b) presented in Figure 5, where the number of users and RF chains changes, but the number of antennas per RF chain is the same; that is, . In scenario (1.b), at a target BER of , the penalties of the proposed subconnected equalizer for the fully connected and digital approaches are 2.3 dB and 3.8 dB, respectively, at the 4th iteration. In scenario (2.b), for the same target BER and iteration, the penalties for fully connected and digital are approximately 1.6 dB and 3.2 dB, slightly decreasing for lower We can also see that for scenario (1.b) a lower is needed to achieve a BER of compared with scenario (2.b). This is because scenario (1.b) can achieve higher diversity and array gain given by the larger dimension of the receiving antenna array. When we reduce the number of antennas per RF chain by half, that is, for , which corresponds to scenario (1.a) presented in Figure 6, we verify that the penalty for fully connected and digital approaches is approximately 1.2 dB and 1.9 dB, thus decreasing for lower . This occurs because when we reduce , the subconnected architecture tends toward the fully digital case. In the extreme case where , we have one antenna per RF chain, that is, the digital architecture. As expected, by reducing the degrees of freedom of the subconnected architecture increase, and the gap with the fully connected and digital approaches decreases. For the digital equalizer the curve for iteration 1 is not presented in Figures 4, 5, and 6 to improve the intelligibility since it almost overlaps with the curve for iteration 4 of the proposed scheme.
In Figures 7 and 8, we compare the performance of the subconnected approach for Scenarios 1 and 2 by considering , and antennas per RF chain. Although the penalty of the proposed subconnected multiuser equalizer relative to the fully connected and digital increases with , as previously verified, the BER performance of the subconnected multiuser equalizer improves with because the diversity and antenna gain are higher due to a larger number of receiving antennas.
Notably, the above was considered the worst case, where . Improved BER performance would be achieved for , that is, for a number of users (streams) lower than the number of RF chains. For that case, the performance gaps between the proposed subconnected hybrid multiuser equalizer and the digital/fully connected approaches would be even smaller.
5. Conclusions
We designed a new hybrid iterative multiuser equalizer for a subconnected mmWave massive MIMO architecture. We considered low complexity UTs employing randomly analogonly precoding and a single RF chain. First, it was verified that the proposed hybrid iterative multiuser equalizer converges and that it achieves a performance close to the digital and fully connected counterparts, requiring very few iterations. This structure efficiently manages multiuser interference; the dictionary approximation and the sequential optimization are quite precise. We observed that when we reduce the number of receiving antennas per RF chain, that is, , the BER performance gets worse due to the reduction of both the antenna gain and diversity. We also observed that the gap between the proposed subconnectedscheme and the digital/fully connected approaches decreases when decreases, because the architecture becomes closer to the digital one.
The performance of the proposed receiver structure shows that it is interesting for practical mmWave massive MIMO based systems, that is, the systems where hardware constraints are considered. This means that the number of transmit/receiving RF chains is much lower than the number of transmit/receiving antennas, and each RF chain is only connected to a reduced number of antennas, ensuring good performance at a low cost.
Appendix
Solution for Optimization Problem (18)
The Lagrangian associated with problem (18) iswhere , , are the Lagrange multipliers. For the optimization variable , the correspondent firstorder optimality condition [41, 42] iswhose solution gives the optimum value of :From (A.3), the Lagrangian function simplifies toBy taking the derivate in relation to and setting it equal to zero, we obtain
The Lagrangian multipliers may be redefined to . By considering (A.6) then reduces toThe optimum feedforward vector