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Hybrid Digital and Analog Beamforming Design for Large-Scale Antenna Arrays
Foad Sohrabi, Student Member, IEEE, and Wei Yu, Fellow, IEEE
Abstract The potential of using of millimeter wave (mmWave) frequency for future wireless cellular communication systems has motivated the study of large-scale antenna arrays for achieving highly directional beamforming. However, the conventional fully digital beamforming methods which require one radio frequency (RF) chain per antenna element is not viable for large-scale antenna arrays due to the high cost and high power consumption of RF chain components in high frequencies. To address the challenge of this hardware limitation, this paper considers a hybrid beamforming architecture in which the overall beamformer consists of a low-dimensional digital beamformer followed by an RF beamformer implemented using analog phase shifters. Our aim is to show that such an architecture can approach the performance of a fully digital scheme with much fewer number of RF chains. Speci?cally, this paper establishes that if the number of RF chains is twice the total number of data streams, the hybrid beamforming structure can realize any fully digital beamformer exactly, regardless of the number of antenna elements. For cases with fewer number of RF chains, this paper further considers the hybrid beamforming design problem for both the transmission scenario of a point-to-point multiple-input multiple-output (MIMO) system and a downlink multi-user multiple-input single-output (MU-MISO) system. For each scenario, we propose a heuristic hybrid beamforming design that achieves a performance close to the performance of the fully digital beamforming baseline. Finally, the proposed algorithms are modi?ed for the more practical setting in which only ?nite resolution phase shifters are available. Numerical simulations show that the proposed schemes are effective even when phase shifters with very low resolution are used.
Copyright c 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, by Ontario Centres of Excellence (OCE) and by BLiNQ Networks Inc. The materials in this paper have been presented in part at IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brisbane, Australia, April 2015, and in part at IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, June 2015. The authors are with The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, Ontario M5S 3G4, Canada (e-mails: {fsohrabi, weiyu}@comm.utoronto.ca).
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Index Terms Millimeter wave, large-scale antenna arrays, multiple-input multiple-output (MIMO), multi-user multipleinput single-output (MU-MISO), massive MIMO, linear beamforming, precoding, combining, ?nite resolution phase shifters.
I. I NTRODUCTION Millimeter wave (mmWave) technology is one of the promising candidates for future generation wireless cellular communication systems to address the current challenge of bandwidth shortage [1]– [3]. The mmWave signals experience severe path loss, penetration loss and rain fading as compared to signals in current cellular band (3G or LTE) [4]. However, the shorter wavelength at mmWave frequencies also enables more antennas to be packed in the same physical dimension, which allows for large-scale spatial multiplexing and highly directional beamforming. This leads to the advent of large-scale or massive multiple-input multiple-output (MIMO) concept for mmWave communications. Although the principles of the beamforming are the same regardless of carrier frequency, it is not practical to use conventional fully digital beamforming schemes [5]–[9] for large-scale antenna arrays. This is because the implementation of fully digital beamforming requires one dedicated radio frequency (RF) chain per antenna element, which is prohibitive from both cost and power consumption perspectives at mmWave frequencies [10]. To address the dif?culty of limited number of RF chains, this paper considers a two-stage hybrid beamforming architecture in which the beamformer is constructed by concatenation of a low-dimensional digital (baseband) beamformer and an RF (analog) beamformer implemented using phase shifters. In the ?rst part of this paper, we show that the number of RF chains in the hybrid beamforming architecture only needs to scale as twice the total number of data streams for it to achieve the exact same performance as that of any fully digital beamforming scheme regardless of the number of antenna elements in the system. The second part of this paper considers the hybrid beamforming design problem when the number of RF chains is less than twice the number of data streams for two speci?c scenarios: (i) the point-to-point multiple-input multiple-output (MIMO) communication scenario with large-scale antenna arrays at both ends; (ii) the downlink multi-user multiple-input single-output (MU-MISO) communication scenario with large-scale antenna array at the base station (BS), but single antenna at each user. For both scenarios, we propose heuristic algorithms to design the hybrid beamformers for the problem of overall spectral ef?ciency maximization under total power constraint at the transmitter, assuming perfect and instantaneous channel state information (CSI) at the BS and all user terminals. The numerical results suggest that hybrid beamforming can achieve spectral ef?ciency close to that of the fully digital solution with the number
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of RF chains approximately equal to the number of data streams. Finally, we present a modi?cation of the proposed algorithms for the more practical scenario in which only ?nite resolution phase shifters are available to construct the RF beamformers. It should be emphasized that the availability of perfect CSI is an idealistic assumption which rarely occurs in practice, especially for systems implementing large-scale antenna arrays. However, the algorithms proposed in the paper are still useful as a reference point for studying the performance of hybrid beamforming architecture in comparison with fully digital beamforming. Moreover, for imperfect CSI scenario, one way to design the hybrid beamformers is to ?rst design the RF beamformers assuming perfect CSI, and then to design the digital beamformers employing robust beamforming techniques [11]– [15] to deal with imperfect CSI. It is therefore still of interest to study the RF beamformer design problem in perfect CSI. To address the challenge of limited number of RF chains, different architectures are studied extensively in the literature. Analog or RF beamforming schemes implemented using analog circuitry are introduced in [16]–[19]. They typically use analog phase shifters, which impose a constant modulus constraint on the elements of the beamformer. This causes analog beamforming to have poor performance as compared to the fully digital beamforming designs. Another approach for limiting the number of RF chains is antenna subset selection which is implemented using simple analog switches [20]–[22]. However, they cannot achieve full diversity gain in correlated channels since only a subset of channels are used in the antenna selection scheme [23], [24]. In this paper, we consider the alternative architecture of hybrid digital and analog beamforming which has received signi?cant interest in recent work on large-scale antenna array systems [25]–[35]. The idea of hybrid beamforming is ?rst introduced under the name of antenna soft selection for a pointto-point MIMO scenario [25], [26]. It is shown in [25] that for a point-to-point MIMO system with diversity transmission (i.e., the number of data stream is one), hybrid beamforming can realize the optimal fully digital beamformer if and only if the number of RF chains at each end is at least two. This paper generalizes the above result for spatial multiplexing transmission for multi-user MIMO systems. In particular, we show that hybrid structure can realize any fully digital beamformer if the number of RF chains is twice the number of data streams. We note that the recent work of [35] also addressed the question of how many RF chains are needed for hybrid beamforming structure to realize digital beamforming in frequency selective channels. But, the architecture of hybrid beamforming design used in [35] is slightly different from the conventional hybrid beamforming structure in [25]–[34]. The idea of antenna soft selection is reintroduced under the name of hybrid beamforming for mmWave frequencies [27]–[29]. For a point-to-point large-scale MIMO system, [27] proposes an algorithm based
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on the sparse nature of mmWave channels. It is shown that the spectral ef?ciency maximization problem for mmWave channels can be approximately solved by minimizing the Frobenius norm of the difference between the optimal fully digital beamformer and the overall hybrid beamformer. Using a compressed sensing algorithm called basis pursuit, [27] is able to design the hybrid beamformers which achieve good performance when (i) extremely large number of antennas is used at both ends; (ii) the number of RF chains is strictly greater than the number of data streams; (iii) extremely correlated channel matrix is assumed. But in other cases, there is a signi?cant gap between the theoretical maximum capacity and the achievable rate of the algorithm of [27]. This paper devises a heuristic algorithm that reduces this gap for the case that the number of RF chains is equal to the number of data streams; it is also compatible with any channel model. For the downlink of K -user MISO systems, it is shown in [32], [33] that hybrid beamforming with
K RF chains at the base station can achieve a reasonable sum rate as compared to the sum rate of
fully digital zero-forcing (ZF) beamforming which is near optimal for massive MIMO systems [36]. The design of [32], [33] involves matching the RF precoder to the phase of the channel and setting the digital precoder to be the ZF beamformer for the effective channel. However, there is still a gap between the rate achieved with this particular hybrid design and the maximum capacity. This paper proposes a method to design hybrid precoders for the case that the number of RF chains is slightly greater than K and numerically shows that the proposed design can be used to reduce the gap to capacity. The aforementioned existing hybrid beamforming designs typically assume the use of in?nite resolution phase shifters for implementing analog beamformers. However, the components required for realizing accurate phase shifters can be expensive [37], [38]. More cost effective low resolution phase shifters are typically used in practice. The straightforward way to design beamformers with ?nite resolution phase shifters is to design the RF beamformer assuming in?nite resolution ?rst, then to quantize the value of each phase shifter to a ?nite set [33]. However, this approach is not effective for systems with very low resolution phase shifters [34]. In the last part of this paper, we present a modi?cation to our proposed method for point-to-point MIMO scenario and multi-user MISO scenario when only ?nite resolution phase shifters are available. Numerical results in the simulations section show that the proposed method is effective even for the very low resolution phase shifter scenario. This paper uses capital bold face letters for matrices, small bold face for vectors, and small normal face for scalars. The real part and the imaginary part of a complex scalar s are denoted by Re{s} and
Im{s}, respectively. For a column vector v, the element in the ith row is denoted by v(i) while for a
matrix M, the element in the ith row and the j th column is denoted by M(i, j ). Further, we use the superscript
H
to denote the Hermitian transpose of a matrix and superscript
?
to denote the complex
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WRF1
... ...
User 1
Analog Precoder VRF
M
...
RF Nr
d s1 Ns
...
Digital Precoder
RF Chain
H1
x(1)
W D1 d
...
...
...
...
...
...
y1
?1 y
s
d sK
...
HK
x (N )
M
...
...
...
Fig. 1. Block diagram of a multi-user MIMO system with hybrid beamforming architecture at the BS and the user terminals.
conjugate. The identity matrix with appropriate dimensions is denoted by I; Cm×n denotes an m by
n dimensional complex space; CN (0, R) represents the zero-mean complex Gaussian distribution with
covariance matrix R. Further, the notations Tr(·), log(·) and E[·] represent the trace, logarithmic and expectation operators, respectively; | · | represent determinant or absolute value depending on context. Finally,
?f ?x
Consider a narrowband downlink single-cell multi-user MIMO system in which a BS with N antennas
RF and NtRF transmit RF chains serves K users, each equipped with M antennas and Nr receive RF
chains. Further, it is assumed that each user requires d data streams and that Kd ≤ NtRF ≤ N and
RF d ≤ Nr ≤ M . Since the number of transmit/receive RF chains is limited, the implementation of
fully digital beamforming which requires one dedicated RF chain per antenna element, is not possible. Instead, we consider a two-stage hybrid digital and analog beamforming architecture at the BS and the user terminals as shown in Fig. 1. In hybrid beamforming structure, the BS ?rst modi?es the data streams digitally at baseband using an NtRF × Ns digital precoder, VD , where Ns = Kd, then up-converts the processed signals to the carrier frequency by passing through NtRF RF chains. After that, the BS uses an N × NtRF RF precoder,
VRF , which is implemented using analog phase shifters, i.e., with |VRF (i, j )|2 = 1, to construct the ?nal
transmitted signal. Mathematically, the transmitted signal can be written as
K
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... ...
...
...
NtRF
N
...
...
WRFK
RF Chain
... ...
User K
VD
...
RF Nr
x
yK
WDK d
...
is used to denote the partial derivative of the function f with respect to x. II. S YSTEM M ODEL
...
?K y
x = VRF VD s =
=1
VRF VD s ,
(1)
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where VD = [VD1 , . . . , VDK ], and s ∈ CNs ×1 is the vector of data symbols which is the concatenation
T T of each user’s data stream vector such as s = [sT 1 , . . . , sK ] , where s is the data stream vector for user
. Further, it is assumed that E[ssH ] = INs . For user k , the received signal can be modeled as
yk = Hk VRF VDk sk + Hk
=k
VRF VD s + zk ,
(2)
where Hk ∈ CM ×N is the matrix of complex channel gains from the transmit antennas of the BS to the k th user antennas and zk ? CN (0, σ 2 IM ) denotes additive white Gaussian noise. The user k ?rst processes
RF the received signals using an M × Nr RF combiner, WRFk , implemented using phase shifters such that RF |WRFk (i, j )|2 = 1, then down-converts the signals to the baseband using Nr RF chains. Finally, using
a low-dimensional digital combiner, WDk ∈ CNr
RF
×d ,
the ?nal processed signals are obtained as
Vt s + WtH zk , k
=k
effective noise
? k = WtH y Hk Vtk sk + WtH Hk k k
desired signals
(3)
effective interference
where Vtk = VRF VDk and Wtk = WRFk WDk . In such a system, the overall spectral ef?ciency (rate) of user k assuming Gaussian signalling is [39]
1 H H H Rk = log2 IM + Wtk C? k Wtk Hk Vtk Vtk Hk ,
(4)
where Ck = WtH Hk k
=k Vt
2 H VtH HH k Wtk + σ Wtk Wtk is the covariance of the interference plus
noise at user k . The problem of interest in this paper is to maximize the overall spectral ef?ciency under total transmit power constraint, assuming perfect knowledge of Hk , i.e., we aim to ?nd the optimal hybrid precoders at the BS and the optimal hybrid combiners for each user by solving the following problem:
K VRF ,VD WRF ,WD
maximize
βk Rk
k=1 H H Tr(VRF VD VD VRF ) ≤ P
(5a) (5b) (5c) (5d)
subject to
|VRF (i, j )|2 = 1, ?i, j |WRFk (i, j )|2 = 1, ?i, j, k,
where P is the total power budget at the BS and the weight βk represents the priority of user k ; i.e., the larger
k K =1
β
β
implies greater priority for user k .
The system model in this section is described for a general setting. In the next section, we characterize the minimum number of RF chains in hybrid beamforming architecture for realizing a fully digital beamformer for the general system model. The subsequent parts of the paper focus on two speci?c scenarios:
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1) Point-to-point MIMO system with large antenna arrays at both ends, i.e., K = 1 and min(N, M )
Ns .
2) Downlink multi-user MISO system with large number of antennas at the BS and single antenna at the user side, i.e., N
K and M = 1.
III. M INIMUM N UMBER OF RF C HAINS TO R EALIZE F ULLY D IGITAL B EAMFORMERS The ?rst part of this paper establishes theoretical bounds on the minimum number of RF chains that are required for the hybrid beamforming structure to be able to realize any fully digital beamforming schemes. Recall that without the hybrid structure constraints, fully digital beamforming schemes can be
RF easily designed with NtRF = N RF chains at the BS and Nr = M RF chains at the user side [5]–[9].
This section aims to show that hybrid beamforming architecture can realize fully digital beamforming schemes with potentially smaller number of RF chains. We begin by presenting a necessary condition on the number of RF chains for implementing a fully digital beamformer, VFD ∈ CN ×Ns . Proposition 1: To realize a fully digital beamforming matrix, it is necessary that the number of RF chains in the hybrid architecture (shown in Fig. 1) is greater than or equal to the number of active data streams, i.e., N RF ≥ Ns . Proof: It is easy to see that rank(VRF VD ) ≤ N RF and rank(VFD ) = Ns . Therefore, hybrid beamforming structure requires at least N RF ≥ Ns RF chains to implement VFD . We now address how many RF chains are suf?cient in the hybrid structure for implementing any fully digital VFD ∈ CN ×Ns . It is already known that for the case of Ns = 1, the hybrid beamforming structure can realize any fully digital beamformer if and only if N RF ≥ 2 [25]. Proposition 2 generalizes this result for any arbitrary value of Ns . Proposition 2: To realize any fully digital beamforming matrix, it is suf?cient that the number of RF chains in hybrid architecture (shown in Fig. 1) is greater than or equal to twice the number of data streams, i.e., N RF ≥ 2Ns . Proof: Let N RF = 2Ns and denote VFD (i, j ) = νi,j ejφi,j and VRF (i, j ) = ejθi,j . We propose the following solution to satisfy VRF VD = VFD . Choose the k th column of the digital precoder as
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vD = [0T v2k?1 v2k 0T ]T . Then, satisfying VRF VD = VFD is ? 0 ? ? . ? . . ? ? ? v2k?1 . . . ejθi,2k?1 ejθi,2k . . . ? ? ? v2k ? ? . ? . ? . 0
(k)
equivalent to ? ? ? ? ? ? ? ? = νi,j ejφi,j , ? ? ? ? ? ?
or
v2k?1 ejθi,2k?1 + v2k ejθi,2k = νi,k ejφi,k ,
(6)
for all i = 1, . . . , N and k = 1, . . . , Ns . This non-linear system of equations has multiple solutions [25]. If we further choose v2k?1 = v2k = νmax where νmax = max{νi,k }, it can be veri?ed after several
i (k) (k)
algebraic steps that the following is a solution to (6):
θi,2k?1 = φi,k ? cos?1 θi,2k = φi,k + cos?1 νi,k 2νmax νi,k 2νmax
(k) (k)
,
.
(7)
Thus for the case that N RF = 2Ns , a solution to VRF VD = VFD can be readily found. The validity of the proposition for N RF > 2Ns is obvious since we can use the same parameters as for N RF = 2Ns by setting the extra parameters to be zero in VD . Remark 1: The solution given in Proposition 2 is one possible set of solutions to the equations in (6). The interesting property of that speci?c solution is that as two digital gains of each data stream are identical; i.e., v2k?1 = v2k , it is possible to convert one realization of the scaled data symbol to RF signal and then use it twice. Therefore, it is in fact possible to realize any fully digital beamformer using the hybrid structure with Ns RF chains and 2Ns N phase shifters. This leads us to the similar result (but with different design) as in [35] which considers hybrid beamforming for frequency selective channels. However, in the rest of this paper, we consider the conventional con?guration of hybrid structure in which the number of phase shifters are N RF N . We show that near optimal performance can be obtained with
N RF ≈ Ns , thus further reducing the number of phase shifters as compared to the solution above.
Remark 2: Proposition 2 is stated for the case that VFD is a full-rank matrix, i.e., rank(VFD ) = Ns . In the case that VFD is a rank-de?cient matrix (which is a common scenario in the low signal-to-noise-ratio (SNR) regime), it can always be decomposed as VFD = AN ×r Br×Ns where r = rank(VFD ). Since A is a full-rank matrix, it can be realized using the procedure in the proof of Proposition 2 as A = VRF VD
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with hybrid structure using 2r RF chains. Therefore, VFD = VRF (VD B) can be realized by hybrid structure using 2r RF chains with VRF as RF beamformer and VD B as digital beamformer. IV. H YBRID B EAMFORMING D ESIGN FOR S INGLE -U SER L ARGE -S CALE MIMO S YSTEMS The second part of this paper considers the design of hybrid beamformers. We ?rst consider a pointto-point large-scale MIMO system in which a BS with N antennas sends Ns data symbols to a user with M antennas where min(N, M )
Ns . Without loss of generality, we assume identical number of
RF transmit/receive RF chains, i.e., NtRF = Nr = N RF , to simplify the notation. For such a system with
hybrid structure, the expression of the spectral ef?ciency in (4) can be simpli?ed to
R = log2 IM + 1 Wt (WtH Wt )?1 WtH HVt VtH HH . σ2
(8)
where Vt = VRF VD and Wt = WRF WD . In this section, we ?rst focus on hybrid beamforming design for the case that the number of RF chains is equal to the number of data streams; i.e., N RF = Ns . This critical case is important because according to Proposition 1, the hybrid structure requires at least Ns RF chains to be able to realize the fully digital beamformer. For this case, we propose a heuristic algorithm that achieves rate close to capacity. At the end of this section, we show that by further approximations, the proposed hybrid beamforming design algorithm for N RF = Ns , can be used for the case of Ns < N RF < 2Ns as well. The problem of rate maximization in (5) involves joint optimization over the hybrid precoders and combiners. However, the joint transmitter-receive matrix design, for similarly constrained optimization problem is usually found to be dif?cult to solve [40]. Further, the non-convex constraints on the elements of the analog beamformers in (5c) and (5d) make developing low-complexity algorithm for ?nding the exact optimal solution unlikely [27]. So, this paper considers the following strategy instead. First, we seek to design the hybrid precoders, assuming that the optimal receiver is used. Then, for the already designed transmitter, we seek to design the hybrid combiner. The hybrid precoder design problem can be further divided into two steps as follows. The transmitter design problem can be written as
VRF ,VD
max
log2 IM +
1 H H HVRF VD VD VRF HH σ2
(9a) (9b) (9c)
s.t.
H H Tr(VRF VD VD VRF ) ≤ P,
|VRF (i, j )|2 = 1, ?i, j.
This problem is non-convex. This paper proposes the following heuristic algorithm for obtaining a good solution to (9). First, we derive the closed-form solution of the digital precoder in problem (9) for a ?xed
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RF precoder, VRF . It is shown that regardless of the value of VRF , the digital precoder typically satis?es
H ∝ I. Then, assuming such a digital precoder, we propose an iterative algorithm to ?nd a local VD VD
optimal RF precoder. A. Digital Precoder Design for N RF = Ns The ?rst part of the algorithm considers the design of VD assuming that VRF is ?xed. For a ?xed RF precoder, Heff = HVRF can be considered as an effective channel and the digital precoder design problem can be written as
max log2 IM +
VD
1 H H Heff VD VD Heff σ2
(10a) (10b)
s.t.
H Tr(QVD VD ) ≤ P,
H V . This problem has a well-known water-?lling solution as where Q = VRF RF
VD = Q?1/2 Ue Γe ,
(11)
where Ue is the set of right singular vectors corresponding to the Ns largest singular values of Heff Q?1/2 and Γe is the diagonal matrix of allocated powers to each stream. Note that for large-scale MIMO systems, Q ≈ N I with high probability [27]. This is because the
HV diagonal elements of Q = VRF RF are exactly N while the off-diagonal elements can be approximated
as a summation of N independent terms which is much less than N with high probability for large N . This
H ∝ I. property enables us to show that the optimal digital precoder for N RF = Ns typically satis?es VD VD
The proportionality constant can be obtained with further assumption of equal power allocation for all streams, i.e., Γe ≈
P/N RF I. So, optimal digital precoder is VD ≈ γ Ue where γ 2 = P/(N N RF ). Since
H ≈ γ 2 I. Ue is a unitary matrix for the case that N RF = Ns , we have VD VD
B. RF Precoder Design for N RF = Ns
H ≈ γ 2 I. Under this assumption, the Now, we seek to design the RF precoder assuming VD VD
transmitter power constraint (9b) is automatically satis?ed for any design of VRF . Therefore, the RF precoder can be obtained by solving
max log2 I +
VRF
γ2 H V F1 VRF σ 2 RF
(12a) (12b)
s.t.
|VRF (i, j )|2 = 1, ?i, j,
where F1 = HH H. This problem is still non-convex, since the objective function of (12) is not concave in VRF . However, the decoupled nature of the constraints in this formulation enables us to devise an iterative coordinate descent algorithm over the elements of the RF precoder.
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In order to extract the contribution of VRF (i, j ) to the objective function of (12), it is shown in [34], [41] that the objective function in (12) can be rewritten as
? log2 Cj + log2 2 Re VRF (i, j )ηij + ζij + 1 ,
(13)
where
Cj = I + γ2 ? j H ? j (V ) F1 VRF , σ 2 RF
? j is the sub-matrix of VRF with j th column removed, and V RF ηij ζij =
=i
Gj (i, )VRF ( , j ),
= Gj (i, i) ? ? +2 Re ?
m=i,n=i
? ? ? VRF (m, j )Gj (m, n)VRF (n, j ) , ?
and Gj =
γ σ 2 F1
2
?
γ ? j ?1 ? j H σ 4 F1 VRF Cj (VRF ) F1 .
4
Since Cj , ζij and ηij are independent of VRF (i, j ), if we
assume that all elements of the RF precoder are ?xed except VRF (i, j ), the optimal value for the element of the RF precoder at the ith row and j th column is given by ? ? ?1, if ηij = 0, VRF (i, j ) = ? ? ηij , otherwise. |ηij |
(0)
(14)
This enables us to propose an iterative algorithm that starts with an initial feasible RF precoder satisfying (12b), i.e., VRF = 1N ×N RF , then sequentially updates each element of RF precoder according to (14) until the algorithm converges to a local optimal solution of VRF of the problem (12). Note that since in each element update step of the proposed algorithm, the objective function of (12) increases (or at least does not decrease), therefore the convergence of the algorithm is guaranteed. The proposed algorithm for designing the RF beamformer in (12) is summarized in Algorithm 1. We mention that the proposed algorithm is inspired by the algorithm in [41] that seeks to solve the problem of transmitter precoder design with per-antenna power constraint which happens to have the same form as the problem in (12). C. Hybrid Combining Design for N RF = Ns Finally, we seek to design the hybrid combiners that maximize the overall spectral ef?ciency in (8) assuming that the hybrid precoders are already designed. For the case that N RF = Ns , the digital combiner is a square matrix with no constraint on its entries. Therefore, without loss of optimality, the design of
WRF and WD can be decoupled by ?rst designing the RF combiner assuming optimal digital combiner
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Algorithm 1 Design of VRF by solving (12) Require: F1 , γ 2 , σ 2
1: 2: 3: 4: 5: 6:
Initialize VRF = 1N ×N RF . for j = 1 → N RF do Calculate Cj = I + Calculate Gj =
γ2 ? j H ?j σ 2 (VRF ) F1 VRF . γ2 γ4 ? j ?1 ? j H σ 2 F1 ? σ 4 F1 VRF Cj (VRF ) F1 .
for i = 1 → N do Find ηij =
VRF (i, j ) =
7: 8: 9: 10:
?=i ? ?1, ? ?
Gj (i, )VRF ( , j ).
if ηij = 0, otherwise.
ηij |ηij | ,
end for end for Check convergence. If yes, stop; if not go to Step 2.
and then ?nding the optimal digital combiner for that RF combiner. As a result, the RF combiner design problem can be written as
max log2 I +
WRF
1 H H (WRF WRF )?1 WRF F2 WRF σ2
(15a) (15b)
s.t.
|WRF (i, j )|2 = 1, ?i, j,
where F2 = HVt VtH HH . This problem is very similar to the RF precoder design problem in (12), except
H W )?1 . Analogous to the argument made in Section IV-A for the RF precoder, the extra term (WRF RF HW it can be shown that the RF combiner typically satis?es WRF RF ≈ M I, for large M . Therefore, the
problem (15) can be approximated in the form of RF precoder design problem in (12) and Algorithm 1 can be used to design WRF by substituting F2 and
max log2 I +
WRF 1 M
by F1 and γ 2 , respectively, i.e., (16a) (16b)
1 WH F2 WRF M σ 2 RF
s.t.
|WRF (i, j )|2 = 1, ?i, j.
Finally, assuming all other beamformers are ?xed, the optimal digital combiner is the MMSE solution as
H WD = J?1 WRF HVt , H HV VH HH W 2 H where J = WRF t t RF + σ WRF WRF .
(17)
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D. Hybrid Beamforming Design for Ns < N RF < 2Ns In Section III, we show how to design the hybrid beamformers for the case N RF ≥ 2Ns for which the hybrid structure can achieve the same rate as the rate of optimal fully digital beamforming. Earlier in this section, we propose a heuristic hybrid beamforming design algorithm for N RF = Ns . Now, we aim to design the hybrid beamformers for the case of Ns < N RF < 2Ns . For Ns < N RF < 2Ns , the transmitter design problem can still be formulated as in (9). For a ?xed RF precoder, it can be seen that the optimal digital precoder can still be found according to (11), however
H ≈ γ 2 [I now it satis?es VD VD Ns 0]. For such a digital precoder, the objective function of (9) that should
be maximized over VRF can be rewritten as
Ns
log2
i=1
1+
γ2 λi , σ2
(18)
where λi is the i
th
largest eigenvalues of
N RF i=1 (1 γ2 σ 2 λi ),
H HH HV . VRF RF
Due to the dif?culties of optimizing over a
function of subset of eigenvalues of a matrix, we approximate (18) with an expression including all of the eigenvalues, i.e., log2
+
or equivalently,
γ2 H H V H HVRF , σ 2 RF
log2 IN RF +
(19)
which is a reasonable approximation for the practical settings where N RF is in the order of Ns . Further, by this approximation, the RF precoder design problem is now in the form of (12). Hence, Algorithm 1 can be used to obtain the RF precoder. In summary, we suggest to ?rst design the RF precoder assuming that the number of data streams is equal to the number of RF chains, then for that RF precoder, to obtain the digital precoder for the actual Ns . At the receiver, we still suggest to design the RF combiner ?rst, then set the digital combiner to the MMSE solution. This decoupled optimization of RF combiner and digital combiner is approximately optimal for the following reason. Assume that all the beamformers are already designed except the digital
HW combiner. Since WRF RF ≈ M I, the effective noise after the RF combiner can be considered as an
uncolored noise with covariance matrix σ 2 M I. Under this condition, by choosing the digital combiner as the MMSE solution, the mutual information between the data symbols and the processed signals before digital combiner is approximately equal to the mutual information between the data symbols and the ?nal processed signals. Therefore, it is approximately optimal to ?rst design the RF combiner using Algorithm 1, then set the digital combiner to the MMSE solution. The summary of the overall proposed procedure for designing the hybrid beamformers for spectral ef?ciency maximization in a large-scale point-to-point MIMO system is given in Algorithm 2. Assuming the number of antennas at both ends are in the same range, i.e., M = O(N ), it can be shown that
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Algorithm 2 Design of Hybrid Beamformers for Point-to-Point MIMO systems Require: σ 2 , P
1:
H = γ I where γ = Assuming VD VD
P/(N N RF ), ?nd VRF by solving the problem in (12) using
Algorithm 1.
2: 3: 4:
H V )?1/2 U Γ where U and Γ are de?ned as following (11). Calculate VD = (VRF RF e e e e
Find WRF by solving the problem in (16) using Algorithm 1.
H HV V where J = WH HV V VH VH HH W 2 H Calculate WD = J?1 WRF RF D RF D D RF + σ WRF WRF . RF RF
the overall complexity of Algorithm 2 is O(N 3 ) which is similar to the most of the existing hybrid beamforming designs, i.e., the hybrid beamforming designs in [25], [27]. Numerical results presented in the simulation part of this paper suggest that for the case of N RF = Ns and in?nite resolution phase shifters, the achievable rate of the proposed algorithm is very close the maximum capacity. The case of Ns < N RF < 2Ns is of most interest when the ?nite resolution phase shifters are used. It is shown in the simulation part of this paper that the extra number of RF chains can be used to trade off the accuracy of the phase shifters. V. H YBRID B EAMFORMING D ESIGN FOR M ULTI -U SER M ASSIVE MISO S YSTEMS Now, we consider the design of hybrid precoders for the downlink MU-MISO system in which a BS with large number of antennas N , but limited number of RF chains N RF , supports K single-antenna users where N
K . For such a system with hybrid precoding architecture at the BS, the rate expression
for user k in (4) can be expressed as
Rk = log2 1+
2 |hH k VRF vDk | H 2 σ2 + =k |hk VRF vD |
,
th
(20) column of the digital
th where hH k is the channel from the BS to the k user and vD denotes the
precoder VD . The problem of overall spectral ef?ciency maximization for the MU-MISO systems differs from that for the point-to-point MIMO systems in two respects. First, in the MU-MISO case the receiving antennas are not collocated, therefore we cannot use the rate expression in (8), which assumes cooperation between the receivers. The hybrid beamforming design for MU-MISO systems must account for the effect of inter-user interference. Second, the priority of the streams may be unequal in a MU-MISO system, while different streams in a point-to-point MIMO systems always have the same priority. This section considers the hybrid beaforming design of a MU-MISO system to maximize the weighted sum rate. In [32], [33], it is shown for the case N RF = K and N → ∞, that by matching the RF precoder to the overall channel (or the strongest paths of the channel) and using a low-dimensional zero-forcing (ZF)
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digital precoder, the hybrid beamforming structure can achieve a reasonable sum rate as compared to the sum rate of fully digital ZF scheme (which is near optimal in massive MIMO systems [36]). However, for practical values of N , there is still a gap between the achievable rates and the capacity. This section proposes a design for the scenarios where N RF > K with practical N and show numerically that adding a few more RF chains can increase the overall performance of the system and reduce the gap to capacity. Solving the problem (5) for such a system involves a joint optimization over VRF and VD which is challenging. We again decouple the design of VRF and VD by considering ZF beamforming with power allocation as the digital precoder. We show that the optimal digital precoder with such a structure can be found for a ?xed RF precoder. In addition, for a ?xed power allocation, an approximately local-optimal RF precoder can be obtained. By iterating between those designs, a good solution of the problem (5) for MU-MISO can be found. A. Digital Precoder Design We consider ZF beamforming with power allocation as the low-dimensional digital precoder part of the BS’s precoder to manage the inter-user interference. For a ?xed RF precoder, such a digital precoder can be found as [6]
1 ZF H H ? DP 1 2, VD = VRF HH (HVRF VRF HH )?1 P 2 = V
(21)
? D = VH HH (HVRF VH HH )?1 and P = diag(p1 , . . . , pK ) with pk where H = [h1 , . . . , hK ]H , V RF RF
denoting the received power at the k th user. For a ?xed RF precoder, the only design variables of ZF digital precoder are the received powers, [p1 , . . . , pk ]. Using the properties of ZF beamforming; i.e., √ ZF ZF = k , problem (5) for designing those powers pk and |hH |hH k VRF vD | = 0 for all k VRF vDk | = assuming a feasible RF precoder is reduced to
K p1 ,...,pK ≥0
max
βk log2 1 +
k=1
pk σ2
(22a) (22b)
s.t.
? ) ≤ P, Tr(QP
? =V ? H VH VRF V ? D . The optimal solution of this problem can be found by water-?lling as where Q D RF pk = 1 max q ?kk βk ?q ?kk σ 2 , 0 , λ
K βk k=1 max{ λ
(23)
?q ?kk σ 2 , 0} = P .
? and λ is chosen such that where q ?kk is k th diagonal element of Q
B. RF Precoder Design Now, we seek to design the RF precoder assuming the ZF digital precoding as in (21). Our overall strategy is to iterate between the design of ZF precoder and the RF precoder. Observe that the achievable
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Algorithm 3 Design of Hybrid Precoders for MU-MISO systems Require: βk , P , σ 2
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14:
Start with a feasible VRF and P = IK . for j = 1 → N RF do
1 ? j (V ? j )H HH P? 1 2. Calculate Aj = P? 2 HV RF RF
for i = 1 → N do
B , ζ D , η B , η D as de?ned in Appendix A. Find ζij ij ij ij (1) (2)
Calculate θi,j and θi,j according to (27).
opt ?(θ(1) ), f ?(θ(2) ) . Find θij = argmin f i,j i,j
Set VRF (i, j ) = e?jθij . end for end for Check convergence of RF precoder. If yes, continue; if not go to Step 2. Find P = diag[p1 , . . . , pk ] using water-?lling as in (23). Check convergence of the overall algorithm. If yes, stop; if not go to Step 2.
H HH (HV VH HH )?1 P 2 . Set VD = VRF RF RF
1
opt
weighted sum rate with ZF precoding in (22) depends on the RF precoder VRF only through the power constraint (22b). Therefore, the RF precoder design problem can be recast as a power minimization problem as
min f (VRF )
VRF
(24a) (24b)
s.t.
? D PV ? H VH ). where, f (VRF ) = Tr(VRF V D RF
|VRF (i, j )|2 = 1, ?i, j.
This problem is still dif?cult to solve since the expression f (VRF ) in term of VRF is very complicated.
HV But, using the fact that the RF precoder typically satis?es VRF RF ≈ N I when N is large [27], this can
be simpli?ed as
H ? D PV ? H) f (VRF ) = Tr(VRF VRF V D
? HV ? DP 2 ) ≈ N Tr(P 2 V D ?(VRF ), ? RF VH H ? H )?1 = f = N Tr (HV RF
1
1
(25)
? = P? 1 2 H. Now, analogous to the procedure for the point-to-point MIMO case, we aim to where H
extract the contribution of VRF (i, j ) in the objective function (here the approximation of the objective
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?(VRF ), then seek to ?nd the optimal value of VRF (i, j ) assuming all other elements are ?xed. function), f
For N RF > Ns , it is shown in Appendix A that
?(VRF ) = N Tr(A?1 ) ? N f j
B + 2 Re V? (i, j )η B ζij RF ij D + 2 Re V? (i, j )η D 1 + ζij RF ij
,
(26)
B , ζ D , η B and η D are de?ned as in Appendix A and are independent of V (i, j ). If we where Aj , ζij RF ij ij ij
assume that all elements of the RF precoder are ?xed except VRF (i, j ) = e?jθi,j , the optimal value for
θi,j should satisfy
?(VRF ) ?f ?θi,j
= 0. Using the results in Appendix B, it can be seen that it is always the case
that only two θi,j ∈ [0, 2π ) satisfy this condition:
θi,j = ?φi,j + sin?1 θi,j = π ? φi,j ? sin?1
(2) (1)
zij |cij |
, ,
(27a) (27b)
zij |cij |
D )η B ? ζ B η D , z = Im{2(η B )? η D } and where cij = (1 + ζij ij ij ij ij ij ij ? ? ?sin?1 ( Im{cij } ), if Re{cij } ≥ 0, |cij | φi,j = ? ?π ? sin?1 ( Im{cij } ), if Re{cij } < 0. |cij |
(28)
?(VRF ) is periodic over θi,j , only one of those solutions is the minimizer of f ?(VRF ). The optimal Since f θi,j can be written as
opt ?(θ(1) ), f ?(θ(2) ) . θij = argmin f i,j i,j
(29)
θi,j ,θi,j
(1)
(2)
Now, we are able to devise an iterative algorithm starting from an initially feasible RF precoder and sequentially updating each entry of RF precoder according to (29) until the algorithm converges to a
?(VRF ). local minimizer of f
The overall algorithm is to iterate between the design of VRF and the design of P. First, starting with a feasible VRF and P = I, the algorithm seeks to sequentially update the phase of each element of RF precoder according to (29) until convergence. Then, assuming the current RF precoder, the algorithm ?nds the optimal power allocation P using (23). The iteration between these two steps continues until convergence. The overall proposed algorithm for designing the hybrid digital and analog precoder to maximize the weighted sum rate in the downlink of a multi-user massive MISO system is summarized in Algorithm 3. VI. H YBRID B EAMFORMING WITH F INITE R ESOLUTION P HASE S HIFTERS Finally, we consider the hybrid beamforming design with ?nite resolution phase shifters for the two scenarios of interest in this paper, the point-to-point large-scale MIMO system and the multi-user MISO
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system with large arrays at the BS. So far, we assume that in?nite resolution phase shifters are available in the hybrid structure, so the elements of RF beamformers can have any arbitrary phase angles. However, components required for accurate phase control can be expensive [38]. Since the number of phase shifters in hybrid structure is proportional to the number of antennas, in?nite resolution phase shifter assumption is not always practical for systems with large antenna array terminals. In this section, we consider the impact of ?nite resolution phase shifters with VRF (i, j ) ∈ F and WRF (i, j ) ∈ F where F =
{1, ω, ω 2 , . . . ω nPS ?1 } and ω = e
2π jn
PS
and nPS is the number of realizable phase angles which is typically
nPS = 2b , where b is the number of bits in the resolution of phase shifters.
With ?nite resolution phase shifters, the general weighted sum rate maximization problem can be written as
K VRF ,VD WRF ,WD
maximize
βk Rk
k=1 H H Tr(VRF VD VD VRF ) ≤ P
(30a) (30b) (30c) (30d)
subject to
VRF (i, j ) ∈ F , ?i, j WRFk (i, j ) ∈ F , ?i, j, k.
For a set of ?xed RF beamformers, the design of digital beamformers is a well-studied problem in the literature. However, the combinatorial nature of optimization over RF beamformers in (30) makes the design of RF beamformers more challenging. Theoretically, since the set of feasible RF beamformers are ?nite, we can exhaustively search over all feasible choices. But, as the number of feasible RF beamfomers is exponential in the number of antennas and the resolution of the phase shifters, this approach is not practical for systems with large number of antennas. The other straightforward approach for ?nding the feasible solution for (30) is to ?rst solve the problem under the in?nite resolution phase shifter assumption, then to quantize the elements of the obtained RF beamformers to the nearest points in the set F . However, numerical results suggest that for low resolution phase shifters, this approach is not effective. This section aims to show that it is possible to account for the ?nite resolution phase shifter directly in the optimization procedure to get better performance. For hybrid beamforming design of a single-user MIMO system with ?nite resolution phase shifters, Algorithm 2 for solving the spectral ef?ciency maximization problem can be adapted as follows. According to the procedure in Algorithm 2, assuming all of the elements of the RF beamformer are ?xed
? (i, j )η except VRF (i, j ), we need to maximize Re VRF ij
for designing VRF (i, j ). This is equivalent to
minimizing the angle between VRF (i, j ) and ηij on the complex plane. Since VRF (i, j ) is constrained to
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be chosen from the set F , the optimal design is
MIMO VRF (i, j ) = Q (ψ (ηij )) ,
(31)
where for a non-zero complex variable a, ψ (a) =
a |a|
and for a = 0, ψ (a) = 1, and the function Q(·)
quantizes a complex unit-norm variable to the nearest point in the set F . Assuming that the number of antennas at both ends in the same range, i.e., M = O(N ), it can be shown that the complexity of the proposed algorithm is polynomial in the number of antennas, O(N 3 ), while the complexity of ?nding the optimal beamformers using exhaustive search method is exponential, O(N 2 2bN ). Similarly, for hybrid beamforming design of a MU-MISO system with ?nite resolution phase shifters, Algorithm 3 can likewise be modi?ed as follows. Since the set of feasible phase angles are limited, instead
?(VRF ) in (26) using one-dimensional of (29), we can ?nd VRF (i, j ) in each iteration by minimizing f
exhaustive search over the set F , i.e.,
MU-MISO ?(VRF ). VRF (i, j ) = argmin f
(32)
VRF (i,j )∈F
The overall complexity of the proposed algorithm for hybrid beamforming design of a MU-MISO system with ?nite resolution phase shifters is O(N 2 2b ), while the complexity of ?nding the optimal beamforming using exhaustive search method is O(N 2bN ). Note that accounting for the effect of phase quantization is most important when low resolution phase shifters are used, i.e., b = 1 or b = 2. Since in these cases, the number of possible choices for each element of RF beamformer is small, the proposed one-dimensional exhaustive search approach is not computationally demanding. VII. S IMULATIONS In this section, simulation results are presented to show the performance of the proposed algorithms for point-to-point MIMO systems and MU-MISO systems and also to compare them with the existing hybrid beamforming designs and the optimal (or nearly-optimal) fully digital schemes. In the simulations, the propagation environment between each user terminal and the BS is modeled as a geometric channel with
L paths [33]. Further, we assume uniform linear array antenna con?guration. For such an environment,
the channel matrix of the k th user can be written as
Hk = NM L
L
αk ar (φrk )at (φtk )H ,
=1 th
(33)
where αk ? CN (0, 1) is the complex gain of the
path between the BS and the user k , and φrk ∈ [0, 2π )
and φtk ∈ [0, 2π ). Further, ar (.) and at (.) are the antenna array response vectors at the receiver and the transmitter, respectively. In a uniform linear array con?guration with N antenna elements, we have
1 ? ? a(φ) = √ [1, ejkd sin(φ) , . . . , ejkd(N ?1) sin(φ) ]T , N
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20
40
Optimal Fully?Digital Beamforming Proposed Hybrid Beamforming Algorithm Hybrid beamforming in [25] Hybrid beamforming in [27]
35
Spectral Efficiency (bits/s/Hz)
30
25
20
15
10 ?10 ?8 ?6 ?4 ?2 0 2 4 6
SNR(dB)
Fig. 2. Spectral ef?ciencies achieved by different methods in a 64 × 16 MIMO system where N RF = Ns = 6. For hybrid beamforming methods, the use of in?nite resolution phase shifters is assumed.
where k =
2π λ ,
? is the antenna spacing. λ is the wavelength and d
In the following simulations, we consider an environment with L = 15 scatterers between the BS and
?= each user terminal assuming uniformly random angles of arrival and departure and d
λ 2. P σ2 )
For each over 100
simulation, the average spectral ef?ciency is plotted versus signal-to-noise-ratio (SNR = channel realizations. A. Performance Analysis of a MIMO System with Hybrid Beamforming
In the ?rst simulation, we consider a 64 × 16 MIMO system with Ns = 6. For hybrid beamforming schemes, we assume that the number of RF chains at each end is N RF = Ns = 6 and in?nite resolution phase shifters are used at both ends. Fig. 2 shows that the proposed algorithm has a better performance as compared to hybrid beamforming algorithms in [27] and [25]: about 1.5dB gain as compared to the algorithm of [27] and about 1dB improvement as compared to the algorithm of [25]. Moreover, the performance of the proposed algorithm is very close to the rate of optimal fully digital beamforming scheme. This indicates that the proposed algorithm is nearly optimal. Now, we analyze the performance of our proposed algorithm when only low resolution phase shifters are available. First, we consider a relatively small 10 × 10 MIMO system with hybrid beamforming architecture where the RF beamformers are constructed using 1-bit resolution phase shifters. Further, it is assumed that N RF = Ns = 2. The number of antennas at each end is chosen to be relatively small in order to be able to compare the performance of the proposed algorithm with the exhaustive search method. We also compare the performance of the proposed algorithm in Section VI, which considers the
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21
26 24 22
Exhaustive Search Proposed Algorithm for b=1 Quantized?Proposed Algorithm for b=∞ Quantized?Hybrid beamforming in [25] Quantized?Hybrid beamforming in [27]
Spectral Efficiency (bits/s/Hz)
20 18 16 14 12 10 8 6 4 0
5
10
15
20
25
30
SNR(dB)
Fig. 3. Spectral ef?ciencies versus SNR for different methods in a 10 × 10 system whereN RF = Ns = 2 and b = 1.
34 32 30 28
Optimal Fully?Digital Beamforming Proposed Algorithm for b=∞, NRF = Ns Proposed Algorithm for b=1, NRF = Ns Proposed Algorithm for b=1, NRF = Ns+1 Proposed Algorithm for b=1, NRF = Ns+3 Quantized?Proposed Algorithm for b=∞
Spectral Efficiency (bits/s/Hz)
26 24 22 20 18 16 14 12 10 8 6 4 ?10
?8
?6
?4
?2
0
2
4
6
SNR(dB)
Fig. 4. Spectral ef?ciencies versus SNR for different methods in a 64 × 16 system where Ns = 4.
?nite resolution phase shifter constraint in the RF beamformer design, to the performance of the quantized version of the algorithms in Section IV, and in [25], [27], where the RF beamformers are ?rst designed under the assumption of in?nite resolution phase shifters, then each entry of the RF beamformers is quantized to the nearest point of the set F . Fig. 3 shows that the performance of the proposed algorithm for b = 1 has a better performance: at least 1.5dB gain, as compared to the quantized version of the other algorithms that design the RF beamformers assuming accurate phase shifters ?rst. Moreover, the spectral efticiency achieved by the proposed algorithm is very close to that of the optimal exhaustive search method, con?rming that the proposed methods is near to optimal.
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22
50
Fully?Digital ZF
45 40 35
Proposed Algorithm for NRF=9 Hybrid beamforming in [33], NRF=8 Hybrid beamforming in [32], NRF=8
Sum Rate (bits/s/Hz)
30 25 20 15 10 5 0 ?10
?8
?6
?4
?2
0
2
4
6
8
10
SNR(dB)
Fig. 5. Sum rate achieved by different methods in an 8-user MISO system with N = 64. For hybrid beamforming methods, the use of in?nite resolution phase shifters is assumed.
30
Fully?Digital ZF Proposed Algorithm for b=∞, NRF=5 Proposed Algorithm for b=1, NRF=5 Quantized?Proposed Algorithm for b=∞, NRF=5 Quantized?Hybrid beamforming in [33], NRF=4 Quantized?Hybrid beamforming in [32], NRF=4
25
Sum Rate (bits/s/Hz)
20
15
10
5
0 ?10
?8
?6
?4
?2
0
2
4
6
8
10
SNR(dB)
Fig. 6. Sum rate achieved by different methods in a 4-user MISO system with N = 64. For the methods with ?nite resolution phase shifters, b = 1.
Finally, we consider a 64 × 16 MIMO system with Ns = 4 to investigate the performance degradation of the hybrid beamforming with low resolution phase shifters. Fig. 4 shows that the performance degradation of a MIMO system with very low resolution phase shifters as compared to the in?nite resolution case is signi?cant—about 5dB in this example. However, Fig. 4 veri?es that this gap can be reduced by increasing the number of RF chains, and by using the algorithm in Section IV-D to optimize the RF and digital beamformers. Therefore, the number of RF chains can be used to trade off the accuracy of phase shifters in hybrid beamforming design.
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B. Performance Analysis of a MU-MISO System with Hybrid Beamforming To study the performance of the proposed algorithm for MU-MISO systems, we ?rst consider an 8user MISO system with N = 64 antennas at the BS. Further, it is assumed that the users have the same priority, i.e, βk = 1, ?k . Assuming the use of in?nite resolution phase shifters for hybrid beamforming schemes, we compare the performance of the proposed algorithm with K + 1 = 9 RF chains to the algorithms in [33] and [32] using K = 8 RF chains. In [33] and [32] each column of RF precoder is designed by matching to the phase of the channel of each user and matching to the strongest paths of the channel of each user, respectively. Fig. 5 shows that the approach of matching to the strongest paths in [32] is not effective for practical value of N ; (here N = 64). Moreover, the proposed approach with one extra RF chain are very close to the sum rate upper bound achieved by fully digital ZF beamforming. It improves the method in [33] by about 1dB in this example. Finally, we study the effect of ?nite resolution phase shifters on the performance of the hybrid beamforming in a MU-MISO system. Toward this aim, we consider a MU-MISO system with N = 64,
K = 4 and βk = 1, ?k . Further, it is assumed that only very low resolution phase shifters, i.e., b = 1, are
available at the BS. Fig. 6 shows that the performance of hybrid beamforming with ?nite resolution phase shifters can be improved by using the proposed approach in Section VI; it improves the performance about
1dB, 2dB and 8dB respectively as compared to the quantized version of the algorithms in Section IV,
[33] and [32] . VIII. C ONCLUSION This paper considers the hybrid beamforming architecture for wireless communication systems with large-scale antenna arrays. We show that hybrid beamforming can achieve the same performance of any fully digital beamforming scheme with much fewer number of RF chains; the required number RF chains only needs to be twice the number of data streams. Further, when the number of RF chains is less than twice the number of data streams, this paper proposes heuristic algorithms for solving the problem of overall spectral ef?ciency maximization for the transmission scenario over a point-to-point MIMO system and over a downlink MU-MISO system. The numerical results show that the proposed approaches achieve a performance close to that of the fully digital beamforming schemes. Finally, we modify the proposed algorithms for systems with ?nite resolution phase shifters. The numerical results suggest that the proposed modi?cations can improve the performance signi?cantly, when only very low resolution phase shifters are available. Although the algorithms proposed in this paper all require perfect CSI, they nevertheless serve as benchmark for the maximum achievable rates of the hybrid beamforming architecture.
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A PPENDIX A D ERIVATION OF (26)
?V ? j (V ? j )H H ? H where V ? j is the sub-matrix of VRF with j th column v(j ) removed. It is Let Aj = H RF RF RF RF ?(VRF ) in (25) can be written as N Tr (Aj +Hv ? (j ) v(j ) H ? H )?1 , where Hv ? (j ) v(j ) H ?H easy to see that f RF RF RF RF
H H
is a rank one matrix and Aj is a full-rank matrix for N RF > Ns . This enables us to write ? ? ?1 ? (j ) (j ) H ? H ?1 ? A Hv v H A f (VRF ) (a) ? ?1 RF RF j j ? = Tr Aj ? N ?1 ? (j ) (j ) H ? H 1 + Tr(A Hv v H )
j
RF RF (b)
1 = Tr(A? j )?
1 ? (j ) (j ) H ? H ?1 Tr(A? j HvRF vRF H Aj ) 1 ? (j ) (j ) H ? H 1 + Tr(A? j HvRF vRF H )
(c)
=
1 Tr(A? j )
?
vRF Bj vRF
(j ) H
(j ) H
(j ) (j )
1 + vRF Dj vRF
(d)
=
1 Tr(A? j )
?
B + 2 Re V? (i, j )η B ζij RF ij D + 2 Re V? (i, j )η D 1 + ζij RF ij
(35)
where
B ζij = Bj (i, i) ? ? +2 Re ? D ζij
m=i,n=i
? ? ? VRF (m, j )Bj (m, n)VRF (n, j ) , ? ? ? ? VRF (m, j )Dj (m, n)VRF (n, j ) , ?
= Dj (i, i) ? ? +2 Re ?
m=i,n=i
B ηij = =i D ηij = =i
Bj (i, )VRF ( , j ), Dj (i, )VRF ( , j ),
th
j th where bj i and di are the i row and
? H A?2 H ? and Dj = H ? H A?1 H ?, column element of Bj = H j j
respectively. In (35), the ?rst equality, (a), is written using the Sherman Morrison formula [42]; i.e.,
(A + B)?1 = A?1 ?
A?1 BA?1 1+Tr(A-1 B)
for a full-rank matrix A and a rank-one matrix B. Since Tr(·) is a
linear function, equation (b) can be obtained. Equation (c) is based on the fact that the trace is invariant under cyclic permutations; i.e., Tr (AB) = Tr (BA) for any arbitrary matrices A and B with appropriate dimensions. Finally, (d) is obtained by expanding the terms.
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A PPENDIX B D ERIVATION OF (27) Consider the following function of θ,
g (θ) =
?jθ a1 + b1 ejθ + b? a1 + 2 Re{b1 ejθ } 1e , = ?jθ a2 + 2 Re{b2 ejθ } a2 + b2 ejθ + b? 2e
(36)
where a1 and a2 are real constants and b1 and b2 are complex constants. The maximums and minimums of g (θ) can be found by solving
?g (θ) ?θ
= 0 or equivalently
?jθ )(a + b ejθ + b? e?jθ ) (jb1 ejθ ? jb? ?g (θ) 2 2 1e 2 = ? jθ ?θ (a2 + b2 e + b2 e?jθ )2
?
?jθ )(a + b ejθ + b? e?jθ ) (jb2 ejθ ? jb? 1 1 2e 1 = 0. ? jθ (a2 + b2 e + b2 e?jθ )2
(37)
By some further algebra, it can be shown that (37) is equivalent to
Im{cejθ } = Im{c} cos(θ) + Re{c} sin(θ) = z,
(38)
where z = Im{2b? 1 b2 } and c = a2 b1 ? a1 b2 . The equation (38) can be further simpli?ed to
|c| sin(θ + φ) = z,
(39)
where
φ=
? ? ?sin?1 ( Im{c} ), |c|
if Re{c} ≥ 0, (40)
? ?π ? sin?1 ( Im{c} ), if Re{c} < 0. |c| It is easy to show that the (39) has only two solutions over one period of 2π as follows:
θ(1) = ?φ + sin?1 θ(2) = π ? φ ? sin?1 z |c| , z |c| .
(41a) (41b)
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Foad Sohrabi (S’13) received his B.A.Sc. degree in 2011 from the University of Tehran, Tehran, Iran, and his M.A.Sc. degree in 2013 from McMaster University, Hamilton, ON, Canada, both in Electrical and Computer Engineering. Since September 2013, he has been a Ph.D student at University of Toronto, Toronto, ON, Canada. Form July to December 2015, he was a research intern at Bell-Labs, Alcatel-Lucent, in Stuttgart, Germany. His main research interests include MIMO communications, optimization theory, wireless communications, and signal processing.
Wei Yu (S’97-M’02-SM’08-F’14) received the B.A.Sc. degree in Computer Engineering and Mathematics from the University of Waterloo, Waterloo, Ontario, Canada in 1997 and M.S. and Ph.D. degrees in Electrical Engineering from Stanford University, Stanford, CA, in 1998 and 2002, respectively. Since 2002, he has been with the Electrical and Computer Engineering Department at the University of Toronto, Toronto, Ontario, Canada, where he is now Professor and holds a Canada Research Chair (Tier 1) in Information Theory and Wireless Communications. His main research interests include information theory, optimization, wireless communications and broadband access networks. Prof. Wei Yu currently serves on the IEEE Information Theory Society Board of Governors (2015-17). He is an IEEE Communications Society Distinguished Lecturer (2015-16). He served as an Associate Editor for IEEE T RANSACTIONS ON I NFORMATION T HEORY (2010-2013), as an Editor for IEEE T RANSACTIONS ON C OMMUNICATIONS (2009-2011), as an Editor for IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS (2004-2007), and as a Guest Editor for a number of special issues for the IEEE J OURNAL ON S ELECTED A REAS IN C OMMUNICATIONS and the EURASIP J OURNAL ON A PPLIED S IGNAL P ROCESSING. He was a Technical Program co-chair of the IEEE Communication Theory Workshop in 2014, and a Technical Program Committee co-chair of the Communication Theory Symposium at the IEEE International Conference on Communications (ICC) in 2012. He was a member of the Signal Processing for Communications and Networking Technical Committee of the IEEE Signal Processing Society (2008-2013). Prof. Wei Yu received a Steacie Memorial Fellowship in 2015, an IEEE Communications Society Best Tutorial Paper Award in 2015, an IEEE ICC Best Paper Award in 2013, an IEEE Signal Processing Society Best Paper Award in 2008, the McCharles Prize for Early Career Research Distinction in 2008, the Early Career Teaching Award from the Faculty of Applied Science and Engineering, University of Toronto in 2007, and an Early Researcher Award from Ontario in 2006. He is recognized as a Highly Cited Researcher by Thomson Reuters.
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