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IEEE COMMUNICATIONS LETTERS, VOL. 22, NO. 4, APRIL 2018
A Beam Selection Algorithm for Millimeter-Wave Multi-User MIMO Systems Rahul Pal , K. V. Srinivas, and A. Krishna Chaitanya Abstract— Millimeter wave (mmWave) frequencies offer huge transmission bandwidths and allow packing a large number of antennas within a given aperture area, enabling high-dimensional MIMO communications. A major bottleneck in realizing such systems is the requirement of a large number of RF chains. Working in the beamspace domain provides an attractive solution through beam selection. We propose a novel beam selection algorithm for downlink mmWave multi-user MIMO systems that selects K beams for K users. The proposed method attempts to maximize the sum-rate and nulls-out the multi-user interference. We show, through simulations, that the proposed method outperforms the existing ones. Index Terms— Beamspace, beam selection, high dimensional MIMO, mmWave communications, multiuser precoder.
I. I NTRODUCTION
C
URRENTLY, most of the commercial wireless systems operate at carrier frequencies below 6 GHz, where there is a scarcity for additional spectral bandwidth. Millimeter wave (mmWave) communications is expected to be an important component in 5G wireless networks to provide multigigabit wireless services [1], [2]. Millimeter waves occupy the spectrum from 30 GHz to 300 GHz, and can provide huge bandwidths, about 2 GHz, for wireless communications [3]. The shorter wavelengths of mmWaves enable us to pack more antenna elements into a given antenna aperture, making it feasible to have a high-dimensional multiple input multiple output (MIMO) operation [4]. Such high-dimensional mmWave MIMO systems hold high potential as they can be used for achieving higher beamforming/spatial multiplexing gains and/or for employing narrow beams to reduce the multiuser interference [5]. One of the main hurdles in realizing high-dimensional mmWave MIMO systems is the hardware complexity. Each antenna in the antenna array needs to be driven by a radio frequency (RF) chain, which consumes a significant portion of the total system cost. In addition, the power consumption of an RF chain at mmWave frequencies is significantly higher than that of at 6 GHz, making it practically prohibitive to have a large number of RF chains [3], [4]. Antenna selection, spatial modulation and analog beamforming are three prominent techniques that enable reducing the number of RF chains in MIMO systems. The first two techniques are shown to have severe performance degradation
Manuscript received January 4, 2018; accepted January 31, 2018. Date of publication February 8, 2018; date of current version April 7, 2018. The associate editor coordinating the review of this paper and approving it for publication was L. Dai. (Corresponding author: Rahul Pal.) R. Pal and K. V. Srinivas are with the Department of Electronics Engineering, IIT (BHU), Varanasi 221005, India (e-mail:
[email protected];
[email protected]). A. K. Chaitanya is with the Indian Institute of Information Technology, Sri City 517646, India (e-mail:
[email protected]). Digital Object Identifier 10.1109/LCOMM.2018.2803805
when the underlying MIMO channels are correlated, which is the case with mmWave channels due to the highly directional propagation at mmWave frequencies. Analog beamforming supports only single-stream transmission and can not exploit the multiplexing gain offered by the MIMO channels [6]. Recently, the concept of beamspace MIMO (B-MIMO) has gained much attention as it enables reducing the number of RF chains by exploiting the sparsity in the mmWave channels [7]–[10]. In B-MIMO, conventional spatial channel is transformed into beamspace (i.e., angular domain), by employing a lens antenna array at the transmitter. As the propagation of mmWaves is highly directional in nature, occupying only a smaller number of directions, the beamspace channel is sparse. Since each beam in B-MIMO corresponds to a single RF chain, we can reduce the number of RF chains without incurring considerable loss in the sum-rate performance, by appropriately selecting a small number of beams. The maximum magnitude beam selection [11], selects beams that maximize the received power at the users. However, it may result in multiple users selecting the same beam, making the number of simultaneously active RF chains dependent upon channel realization and user locations. The interferenceaware beam selection algorithm [12] circumvents this problem and achieves improved performance by taking care of multi-user interference among the interfering users. Based on the criteria of maximizing signal-to-interference plus noiseratio (SINR) and maximizing capacity, [13] has proposed a different set of beam selection algorithms. We propose a novel iterative beam selection algorithm that outperforms the existing ones in sum-rate performance and power efficiency. The proposed algorithm eliminates the beams, one-by-one, that contribute minimally to the sum-rate and retains K (out of N K ) beams for K users, resulting in a reduced-dimensional channel. To eliminate the multiuser interference in the reduced-dimensional system, rather than employing the commonly used zero-forcing precoder, we develop a simple precoder that diagonalizes the channel. Thus, our main contribution is a sum-rate maximizing beam selection algorithm and an associated precoder. Notation: Matrices and vectors are denoted by boldface uppercase and lowercase letters, respectively. ai j and x i denote (i j )th element of matrix A and i th element of vector x, respectively. A− j denotes A with its j th row removed. I N is N × N identity matrix. Superscripts −1, T, H indicate inverse, transpose and conjugate transpose, respectively. a, b denotes inner product of a and b. II. S YSTEM M ODEL We consider downlink communication from an access point (AP) to K users, each user having a single receive
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PAL et al.: BEAM SELECTION ALGORITHM FOR MILLIMETER-WAVE MULTI-USER MIMO SYSTEMS
antenna. The AP is equipped with a uniform linear array (ULA) of N antenna elements spaced d = λ2 meters apart, where λ is the carrier wavelength in meters. The discrete-time input-output relation of such a multi-user MIMO (MU-MIMO) system, in spatial domain, can be expressed as y = H H Ps + w,
(1)
where H = [h1 , . . . , h K ] ∈ C N×K is the channel matrix and hk ∈ C N×1 contains the channel gains from N elements of the ULA to the k th user. P ∈ C N×K is the precoding matrix, s ∈ C K ×1 is the symbol vector with E[ss∗ ] = I K and the average power constraint is given by E[ x 2 ] ≤ ρ, where x = Ps. w denotes additive white Gaussian noise vector with w ∼ CN (0, σ 2 I K ). Following Saleh-Valenzuela channel model for mmWave communications [14], [15], L N () () (2) βk a θ k , hk = L +1 =0
(0) βk
() βk ,
where and = 1, . . . , L, denote the complex-valued path gain of the line-of-sight (LoS) component, and t h nonline-of-sight (NLoS) component, respectively. For the critically sampled ULA of N elements considered here, the array response vector corresponding to the LoS component is given by, 1 , (3) a θk(0) = √ exp −j 2πθk(0)i i∈I (N) N where I(N) = {m − (N − 1)/2, m = 0, 1, . . . , N − 1} is (0) (0) an index set and θk = 0.5 sin φk is the spatial frequency (0) induced by the physical angle of arrival φk ∈ − π2 , π2 of the LoS component at k th user. The array response vector cor () th responding to the NLoS component, a θk , is obtained by using θk() = 0.5 sin φk() , where φk() is the physical angle of arrival of the th NLoS component at k th user.
A. Beamspace Representation The discussion above presents the channel model in the conventional spatial domain. Due to the highly directional nature of propagation at mmWave frequencies, LoS component dominates over NLoS components and the channel is sparse. Beamspace, i.e., angular domain, representation enables us to exploit the inherent sparsity in such channels. The spatial channel can be transformed into the beamspace channel by employing a discrete lens array (DLA) at the transmitter. DLA plays the role of a spatial discrete Fourier transform, which can be represented by the matrix U ∈ C N×N . The columns of U are array response vectors corresponding to N fixed spatial frequencies (equivalently, N orthogonal predefined directions) given by θi = Ni , i ∈ I(N), covering the entire angular space;
Thus, U = √1 a θi = Ni i∈I (N) [7]–[10]. The beamspace N representation of the system is given by, y = HbH Pb s + w,
(4)
where Hb = U H H is the beamspace channel and Pb = U H P. Each hb,k = U H hk , k = 1, . . . , K , will have few dominant (0) entries (significantly less than N) around the LoS direction θk
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and thus, Hb captures the inherent sparsity in the mmWave channel. III. P ROPOSED B EAM S ELECTION A LGORITHM Our aim is to select K beams out of N without incurring K Rk , where Rk = considerable loss in the sum-rate Rs = k=1 log2 (1 + SINRk ) bits/s/Hz is the data rate achieved by k th user and SINRk is the k th user’s SINR, after beam selection. Such a reduced-dimensional system requires only K RF chains. The K -dimensional system can be expressed as ˜ bH P˜ b s + w, y˜ = H
(5)
˜ H ∈ C K ×K is the reduced-dimensional beamspace where H b channel matrix corresponding to the K selected beams and P˜ b ∈ C K ×K is a reduced-dimensional precoder. A straightforward method to select K out of N beams is to try all the possible KN combinations, which is prohibitively complex even for moderate values of N and K . We propose an iterative beam selection algorithm and an associated simple precoder that cancels the multi-user interference. The following discussion outlines the proposed method and helps in understanding the motivation behind the algorithms (for com˜ b and precoder P˜ b ), puting the reduced-dimensional channel H presented in Section III-A. ˜ H = (Q ˜ R) ˜ H, ˜ b is given by H Let the QR decomposition of H b K ×K ˜ = [q˜ 1 , . . . , q˜ K ] ∈ C is a unitary matrix and where Q ˜ ∈ C K ×K is an upper triangular matrix [16]. The columns R ˜ provide us an orthonormal basis, obtained through of Q Gram-Schmidt procedure, for the vector space spanned by the ˜ b. K column vectors of H ˜ ˜ By choosing Pb = Q, Eqn. (5) becomes, ˜ H s + w. y˜ = R
(6)
Thus, when K data streams are multiplexed in the coordinate ˜ the received signal at user k becomes, system specified by Q, (7) y˜k = r˜kk sk + Ik + wk , k = 1, . . . , K , where Ik = k> j r˜kj s j is the interference signal. Interference can be made equal to zero for all the users by diagonaliz˜ in Eqn. (5), ˜ H . We achieve this by having P˜ b = QG ing R where we compute (in the next sub-section) G ∈ C K ×K such ˜ H G is a diagonal matrix, with its i th diagonal entry that R equal to r˜ii . With Ik = 0, Rk depends only on r˜kk , the effective channel gain for user k, and the sum-rate is given by
1 ρ 2 r˜ Rs = log2 1 + bits/s/Hz, (8) N0 K kk k
where N0 is the noise variance of the additive white Gaussian noise at each user. ˜ b and P˜ b A. Algorithms for Computing H The algorithm consists of (N − K ) iterations. In each iteration, we eliminate a beam (i.e., a row of Hb ) that contributes minimally towards Rs . To make a decision on which row to be removed, we follow another iterative process.
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IEEE COMMUNICATIONS LETTERS, VOL. 22, NO. 4, APRIL 2018
Reduced-dimensional channel matrix at the end of (i − 1)th (i) iteration, denoted by Hb , can be expressed as T T (i) (i) T (i) T (i) . . . c N−i , (9) c2 H b = c1 1×K is the j th row of H(i) , for j = 1, . . . , N −i . where c(i) j ∈C b (i)
From Hb , we delete one row (out of N −i rows), by following another iterative process, which can be explained as follows. (i) (i) Let T(i) = Hb . For j = 1, . . . , N − i , we remove c j from T(i) to obtain the matrix T(i) − j , find its QR decomposition T(i) −j u∈{1,...,N−i}
(u, v)th
=
log2
element of
(i) Q(i) then compute, γ j(i) = − j R− j and 2 (i) (i) 1 + r− , where r− denotes the j j uu
(i) R− j .
uv
Please note that, we have ignored (i)
the interference terms while computing γ j . This will not be a problem as our precoder P˜ b will completely cancel the multi-user interference. We remove row j from Hb(i) (i+1) (i) to obtain Hb , where j = arg max j ∈{1,...,N−i} γ j . Thus, we eliminate a beam whose effect on the sum-rate is the least. Please refer to Algorithm 1, given below. ˜b Algorithm 1 Algorithm to Compute H Initialize Hb(0) = Hb for i = 0 → N − K − 1 do (i) T(i) = Hb for j = 1 → N − i do (i) Remove row j from T(i) to obtain T− j (i) (i) T(i) of T(i) − j = Q− j R− j (QR-decomposition −j)
2 (i) (i) N−i γ j = u=1 log2 1 + r− j uu
end for j = arg max j ∈{1,...,N−i} γ j(i) (i)
(i)
follows: For l = 1, . . . , K , m ⎧ 1, ⎪ ⎪ ⎨ r˜ ij (i, j ) elm = − , ⎪ r ˜ ⎪ ⎩ ii 0,
= 1, . . . , K , if l = m, if l = i, m = j,
(10)
otherwise.
It can be verified that ri j = 0. We compute G as a product of elementary matrices defined in (10) taken in a specific order. Algorithm 2 details how to compute G. We observe that the matrix G is a lower triangular matrix. Algorithm 2 Algorithm to Compute G Initialize G = I K , where I K is identity matrix of order K ˜H Initialize R = R for i = 1 → K do if rii = 0 then for j = 1 → i − 1 do Define E(i, j ) as in (10) Compute R = R E(i, j ) G = GE(i, j ) end for end if end for ˜ H is not diagonalizable if r˜kk = 0 for some k ∈ R {1, . . . , K }. In such a case, y˜k does not contain the message sk intended for user k and the particular user need not be served. By considering the effective channel matrix for the remaining K − 1 users, we can still use Algorithm 2 to compute G that avoids interference at all users with r˜kk = 0. Note that G need not be unitary and, to satisfy the transmit power constraint, we need to normalize G to have unit norm. ˜ G, ˜ where G ˜ = G , Thus, P˜ b in Eqn. (5) is chosen as P˜ b = Q G F ˜ is obtained from the QR decomposition of H ˜ b. Q
(i+1)
Remove c j from Hb to obtain Hb end for ˜ b = H(N−K ) H b
The main complexity of the above algorithm is due to the QR decomposition in the inner loop. In i th iteration, i = 0, . . . , (N − K − 1), the algorithm computes (N − i ) QR decompositions and the complexity of each decomposition is O(2(N − i )K 2 ) [16]. Thus, the overall complexity is N−K −1 (N − i )O(2(N − i )K 2 ). This complexity is higher i=0 than that of the interference aware beam selection [12] and lower than maximizing capacity beam selection [13]. ˜ b given by the above algorithm, and by Now, by using H ˜ (with Q ˜ obtained from the QR decomchoosing P˜ b = Q, ˜b = Q ˜ R), ˜ we obtain the system described by position of H Eqn. (7). Interference Ik , k = 1, . . . , K , can be eliminated by ˜ H . We propose to accomplish this by having diagonalizing R ˜ and by computing G ∈ C K ×K such that R ˜ H G is a P˜ b = QG th diagonal matrix, with its k diagonal entry given by r˜kk . ˜ H can be made equal to zero An element r˜i j , i > j , in R H ˜ with a matrix. Let R = R ˜ H E(i, j ) , by post-multiplying R (i, j ) where E is a K × K lower triangular matrix obtained as
IV. S IMULATION R ESULTS In this section, we evaluate the performance of the proposed algorithm through simulations, and compare it with existing beam selection algorithms. We focus on two performance metrics, namely, spectral efficiency and power efficiency. We consider downlink communication from an access point equipped with N = 256 element ULA and K = 16 users, each user having a single receive antenna. Spatial channel between the AP and user k, k ∈ {1, . . . , K }, is assumed to be (0) having one LoS component with path gain βk ∼ CN (0, 1) () and two NLoS components, each having a path gain βk ∼ () −2 CN (0, 10 ), = 1, 2. The path gains βk are assumed to be independent of each other. The spatial frequencies, () θk , k = 0, 1, 2, of user k, are uniformly distributed in the interval − 12 , 12 and independent of each other. Figure 1 plots the achievable sum-rate of the proposed algorithm and compares its performance with “maximum magnitude” beam selection with one beam per user [11], [13], “maximizing capacity” and “maximizing SINR” beam selection [13] and “interference-aware” beam selection [12]. For maximizing capacity, [13] proposed an incremental algorithm
PAL et al.: BEAM SELECTION ALGORITHM FOR MILLIMETER-WAVE MULTI-USER MIMO SYSTEMS
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V. C ONCLUSION
Fig. 1.
Sum-rate performance comparison.
Transforming the mmWave channel from spatial domain to the beamspace domain and selecting few beams would significantly reduce the number of required RF chains. Considering downlink mmWave communication from an access point having a large number of antenna elements to multiple users each having a single receive antenna, we have proposed a sumrate maximizing beam selection algorithm and an associated precoder that eliminates the multi-user interference by diagonalizing the effective channel matrix. The achievable sum-rate and the power efficiency of the proposed method are evaluated through simulations and shown to outperform other beam selection algorithms, including the full dimensional ZF precoding. Reducing the complexity of the beam selection algorithm and power allocation across users, which may further improve its performance, will be pursued in the future work. R EFERENCES
Fig. 2.
Comparing power efficiency of different beam selection algorithms.
and a decremental algorithms. Performance of both these algorithms is almost the same, and, here, we have compared with the decremental algorithm. It can be observed that the proposed algorithm outperforms the other algorithms. In “maximizing capacity” beam selection [13], the beams are selected by maximizing the capacity given by Eqn. (24) in Section 3C of [13]. It should be noted that Eqn. (24) is the maximum achievable rate of a point-to-point MIMO channel and maximizing Eqn. (24) may not maximize the sum-rate of a mmWave multi-user MIMO system that is under consideration. “Full dimensional ZF” is the sum-rate achieved by full system zero-forcing precoding that uses all the N beams. Note that, full dimensional ZF is one of the upper bounds on the achievable performance of a full dimensional system. Full dimensional Wiener filtering, full dimensional matched filtering and the idealistic upper bound, proposed in [11], provide us with other upper bounds on the full dimensional system performance. We note that the sum-rate achieved by our algorithm is considerably lower than the idealistic upper bound given by [11, eq. (13)], but higher than the performance of full dimensional ZF. Figure 2 compares the power efficiency of different beam selection algorithms, including the proposed one. Power efficiency is defined as [12], [13], and [17], η p = Rs ρ+NRF PRF bits/s/Hz/Watt, where NRF is the number of RF chains and PRF is the power consumed by an RF chain. As in [12] and [13], we evaluated η p at SNR = 20dB, with PRF = 34.4mW and ρ = 32mW. As it is evident from the figure, the proposed method achieves considerably higher power efficiency than the other beam selection algorithms. Note that the full dimensional ZF has very high power consumption. It is worth noting that, with the proposed algorithm, different users see different effective channel gains and optimal power allocation across users might further improve the sum-rate.
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