Hybrid Precoding Design in Millimeter Wave MIMO ... - Semantic Scholar
Hybrid Precoding Design in Millimeter Wave MIMO Systems: An Alternating Minimization Approach Xianghao Yu∗ , Juei-Chin Shen† , Jun Zhang∗ , and K. B. Letaief∗ , Fellow, IEEE ∗
Dept. of ECE, The Hong Kong University of Science and Technology † Mediatek Inc., Hsinchu ∗ Email: {xyuam, eejzhang, eekhaled}@ust.hk, † [email protected]
Abstract—Millimeter wave (mmWave) communications holds a promise to offer an unprecedented capacity boost for 5G cellular networks. Due to the small wavelength of mmWave signals, multiple-input-multiple-output (MIMO) systems can leverage large-scale antennas to combat the path loss and rain attenuation via precoding. Different from conventional MIMO systems, mmWave MIMO cannot realize precoding entirely at baseband using digital precoders, as a result of the potentially high power consumed by the signal mixers and analog-to-digital converters (ADCs). As a cost-effective alternative, a hybrid precoding transceiver architecture for mmWave MIMO systems has received considerable attention. However, the optimal design of such hybrid precoding has not been fully understood. In this paper, an alternating minimization algorithm based on manifold optimization is proposed to design the hybrid precoder, thereby making it comparable in performance to the digital precoder. Numerical results show that our proposed algorithm can significantly outperform existing ones in terms of spectral efficiency and, more importantly, it can achieve the optimal performance in certain cases. The alternating minimization approach is also shown to be generally applicable to precoding design with different hybrid structures, and the corresponding comparison will show interesting design insights for hybrid precoding.
I.
I NTRODUCTION
The capacity of wireless networks has to increase exponentially to meet the ever-increasing demands for high-data-rate multimedia access. One promising way to boost the capacity is to exploit new spectrum bands for wireless communications. MmWave bands from 30 GHz to 300 GHz, previously only considered in indoor and fixed outdoor scenarios [1], have now been put forward as a prime candidate for new spectrum in 5G cellular systems. This view is supported by recent experiments in New York City that demonstrated the feasibility of mmWave outdoor cellular wireless communications [2]. The small wavelength of mmWave signals enables large-scale antenna arrays at transceivers, so that significant beamforming gains can be obtained to combat the path loss and rain attenuation resulted from the ten-fold increase of the carrier frequency. Moreover, spectral efficiency can also be improved by transmitting multiple data streams via spatial multiplexing. For traditional MIMO systems, precoding is typically accomplished at the baseband through digital precoders, which can change both the magnitude and phase of signals. However, fully digital precoding demands RF chains, including signal mixers and ADCs, comparable in number to antenna elements. While the small wavelengths of mmWave frequencies facilitate the use of a large number of antenna elements, the prohibitive cost and power consumption of RF chains make digital precoding infeasible. Given such unique constraints in mmWave MIMO systems, a hybrid precoding structure has recently been
proposed, which requires only a small number of RF chains interfacing between low-dimensional digital precoders and highdimensional analog precoders. In particular, each RF chain is connected to all the antenna elements via phase shifters, so this arrangement is referred to as the fully-connected structure, as illustrated in Fig. 1. There exist several previous studies on precoding design for the fully-connected hybrid structure [3]. In [4], the optimal design presented a special case, i.e., when the number of RF chains is at least twice the number of data streams. The major obstacle for solving the general case is to satisfy unit modulus constraints on analog precoders. Orthogonal matching pursuit (OMP), one of the widely used hybrid precoding algorithms [3], overcomes this obstacle by restricting columns of the analog precoding matrices to be the subset of predefined candidate vectors. Though the design problem is greatly simplified, imposing this extra constraint inevitably causes some performance loss. In other words, the general design problem of hybrid precoding has not yet been satisfactorily solved. In this paper, we will approach the performance of the optimal fully digital precoder by hybrid precoders without extra restrictions. A novel alternating minimization algorithm will be proposed, where the digital and analog parts are separately considered per iteration. Unlike the previous result in [3], the unit modulus constraints will be directly handled by manifold optimization. In particular, the manifold structure of the analog precoding pattern is exploited to efficiently identify a near-optimal solution. Simulation results will demonstrate significant throughput gains of the proposed algorithm over existing ones. The proposed alternating minimization algorithm will also be extended to consider a partially-connected structure, also called as the array of subarrays structure [5], that employs notably less phase shifters. Different from previous investigations focusing on codebook-based precoding [6], our study provides an effective algorithm for non-codebook-based precoding design of this structure. Furthermore, the comparison between these two structures indicates that the fullyconnected structure results in higher spectral efficiency while the partially-connected one is more energy efficient. The following notations are used throughout this paper. a and A stand for a column vector and a matrix respectively; Ai,j is the entry on the ith row and jth column of A; The conjugate, transpose and conjugate transpose of A are represented by A∗ , AT and AH ; det(A) and AF denote the determinant and Frobenius norm of A; A−1 and A† are the inverse and Moore-Penrose pseudo inverse of A; diag(A) generates a diagonal matrix with the entries of vector a; tr(A) and vec(A) indicate the trace and vectorization of A;
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Expectation and the real part of a complex variable is noted by E[·] and [·]; ◦ and ⊗ denote the Hadamard and Kronecker products between two matrices. II.
S YSTEM M ODEL AND P ROBLEM F ORMULATION
A. System Model Consider a single-user mmWave MIMO system1 as shown in Fig. 1, where Ns data streams are sent and collected by Nt transmit antennas and Nr receive antennas. The limitation t on the amount of RF chains is given by Ns ≤ NRF ≤ Nt r t t and Ns ≤ NRF ≤ Nr , where NRF and NRF denote the numbers of RF chains facilitated at the transmitter and receiver, respectively. Analog RF Precoder
RF Chain Digital Baseband Precoder
RF Chain
Fig. 1.
The fully-connected hybrid precoding structure.
The transmitted signal can be written as x = FRF FBB s, where s is the Ns × 1 symbol vector such that E ssH = 1 t Ns INs . The hybrid precoder consists of an NRF × Ns digt ital baseband precoder FBB and an Nt × NRF analog RF precoder FRF . The transmit power constraint is given by 2 FRF FBB F = Ns . For simplicity, we consider a block-fading propagation channel and the received signal after combining processing is √ H H H H y = ρWBB WRF HFRF FBB s + WBB WRF n, (1) where ρ stands for the average received power, H is the chanr × Ns digital baseband decoder, nel matrix, WBB is the NRF r WRF is the Nr × NRF analog RF decoder at the receiver and n is the noise vector of independent and identically distributed (i.i.d.) CN (0, σn2 ) elements. The achievable spectral efficiency when transmitted symbols follow a Gaussian distribution can be expressed as ρ −1 H H R = log det INs + Σ WBB WRF HFRF FBB Ns (2) H H ×FH F H W W , RF BB BB RF H H where Σ = σn2 WBB WRF WRF WBB is the noise covariance matrix. Furthermore, the analog precoders are implemented with phase shifters, which can only adjust the phases of the signals. Thus all the entries of FRF and WRF should satisfy the unit modulus constraint, namely |(FRF )i,j | = |(WRF )i,j | = 1. For the fully-connected structure in Fig. 1, the output signal of each RF chain is sent to all the antennas through phase shifters, while a different structure will be discussed in Section IV. 1 The receiver side is omitted due to space limitation. More details can be found in [3].
B. Channel Model Due to high free-space path losses, the mmWave propagation environment is well characterized by a clustered channel model, i.e., the Saleh-Valenzuela model [2]. This model depicts the mmWave channel matrix as Ncl N ray Nt Nr r t H H= αil ar (φril , θil )at (φtil , θil ) , (3) Ncl Nray i=1 l=1
where Ncl and Nray represent the number of clusters and the number of rays in each cluster, and αil denotes the gain of the lth ray in the ith propagation cluster. We assume 2 that α are i.i.d. and follow the distribution CN (0, σα,i ) and Ncl il 2 i=1 σα,i = γ, which is the normalization factor to satisfy 2
r t ) and at (φtil , θil ) E HF = Nt Nr . In addition, ar (φril , θil represent the receive and transmit array response vectors, r t r t (θil ) and θil (θil ) stand for azimuth and elevation where θil angles of arrival and departure, respectively. In this paper, we consider √ √ the uniform square planar array (USPA) with Nx × Nx (x ∈ {t, r}) antenna elements. Therefore, the ith element of array response vector can be written as 1 2π x x x ax (φ, θ)i = √ exp j d(p sin φ sin θ + q cos θ ) , λ Nx (4) where d and λ are√the antenna spacing√and the signal wavelength, 0 ≤ p < Nx and 0 ≤ q < Nx are the antenna indices in the 2D plane. While this channel model will be used in simulations, our precoder design is applicable to more general models.
C. Problem Formulation As shown in [3], the design of precoders and decoders can be separated into two sub-problems, i.e., the precoding and decoding problems. They have similar mathematical formulations except that there is an extra power constraint in the former. Therefore, we will mainly focus on the precoder design in the remaining part of this paper. The corresponding problem formulation is given by minimize FRF ,FBB
subject to
Fopt − FRF FBB F |(FRF )i,j | = 1, ∀i, j, 2 FRF FBB F = Ns ,
(5)
where Fopt stands for the optimal fully digital precoder, while FRF and FBB are the analog and digital precoders to be optimized. It has been shown in [3] that minimizing the objective function in (5) leads to the maximization of spectral efficiency. In addition, the matrix Fopt is comprised of the first Ns columns of V, which is a unitary matrix derived from the channel’s singular value decomposition (SVD), i.e., H = UΣVH . III.
H YBRID P RECODING D ESIGN T HROUGH A LTERNATING M INIMIZATION
Alternating minimization, which reduces the objective function with respect to different subsets of variables in each alternating step, has been successfully applied to many applications such as matrix completion, phase retrieval and dictionary learning [7]. Here we take this approach to solve problem (5),
by alternating between locating the best FBB and the best FRF . The main difficulty will be in the analog precoder design, for which we will propose a manifold optimization algorithm. A. Digital Baseband Precoder Design To design the digital precoder FBB , we fix the analog precoder FRF and restate problem (5) as minimize FBB
Fopt − FRF FBB F ,
(6)
which has a well-known least squares solution FBB =
F†RF Fopt .
(7)
Note that the power constraint in (5), temporarily removed, will be dealt with in Section III-C. Despite this, (7) has already offered a global optimal solution to the counterpart design problem at the receiver side. B. Analog RF Precoder Design via Manifold Optimization In the next alternating step, regarding FBB as fixed, we seek an analog precoder which optimizes the following problem2 : 2 minimize Fopt − FRF FBB F FRF (8) subject to |(FRF )i,j | = 1, ∀i, j. The main obstacle of is the unit modulus constraints, i.e., |(FRF )i,j | = 1, which are intrinsically non-convex. To the best of the authors’ knowledge, there is no general approach to solve (8) optimally, which is the major difficulty in alternating minimization. It has been pointed out in [3] that the unit modulus constraints make the original problem (5) mathematically intractable. However, we will show that the sub-problem (8) can be effectively solved by manifold minimization, as the unit modulus constraints define a Riemannian manifold. More importantly, the minimization over this manifold is locally analogous to that over a Euclidean space with smooth constraints. Therefore, a well-developed conjugate gradient algorithm on Euclidean spaces can have its counterpart on the specified Riemannian manifold. Manifold optimization has recently been applied for topological interference management in wireless networks [8]. In the following, we will briefly introduce the main idea. Note that the vector x = vec(FRF ) forms a complex circle manifold Mm = {x ∈ Cm : |x1 | = |x2 | = · · · = |xm | = 1}, t where m = Nt NRF . Therefore, the search space of the optimization problem (8) is over a product of m circles in the complex plane, which is a Riemannian submanifold of Cm with the product geometry. Please refer to [9] for background on Riemannian manifold and manifold optimization. For a given point x on the manifold Mm , the directions it can move along are characterized by the tangent vectors, ensuring the destination is also on the manifold. All the tangent vectors construct a tangent space at the point x ∈ Mm , which can be represented by Tx Mm = {y ∈ Cm : {y ◦ x∗ } = 0m } .
(9)
2 The square of the Frobenius norm makes the objective function smooth, and will not affect the solution.
Among all the tangent vectors, similar to the Euclidean space, one of them related to the negative Riemannian gradient represents the direction of the greatest decrease of a function. Because the complex circle manifold Mm is a Riemannian submanifold of Cm , the Riemannian gradient at x is a tangent vector gradf (x) given by the orthogonal projection of the Euclidean gradient ∇f (x) onto the tangent space Tx Mm : gradf (x) = Projx ∇f (x) = ∇f (x) − {diag [∇f (x) ◦ x∗ ]}x,
(10)
where the Euclidean gradient of the cost function in (8) is ∇f (x) = −2(F∗BB ⊗ INt ) vec(Fopt ) − (FTBB ⊗ INt )x . (11) Retraction is another key factor in manifold optimization, which maps a vector from the tangent space onto the manifold itself, determining the destination when moving along a tangent vector. The retraction of a tangent vector αd at point x ∈ Mm can be stated as Retrx :Tx Mm → Mm : αd → Retrx (αd) = vec
(x + αd)i . |(x + αd)i |
(12)
Equipped with the tangent space, Riemannian gradient and retraction of the complex circle manifold Mm , a line search based conjugate gradient method [10], which is a classical algorithm in Euclidean space, can be developed to design the analog precoder as shown in Algorithm 1. Algorithm 1 Conjugate Gradient Algorithm for Analog Precoder Design Based on Manifold Optimization Require: Fopt , FBB , x0 ∈ Mm 1: d0 = −gradf (x0 ) and k = 0; 2: repeat 3: Choose Armijo backtracking line search step size αk ; 4: Find the next point xk+1 using retraction in (12): xk+1 = Retrxk (αk dk ); 5: Determine Riemannian gradient gk+1 = gradf (xk+1 ) according to (10) and (11); 6: Calculate the vector transports gk+ and d+ k of gradient gk and conjugate direction dk from xk to xk+1 ; 7: Choose Polak-Ribiere parameter βk+1 ; 8: Compute conjugate direction dk+1 = −gk+1 +βk+1 d+ k; 9: k ← k + 1; 10: until |gradf (xk+1 )| ≤ . Algorithm 1 utilizes the well-known Armijo backtracking line search step and Polak-Ribiere parameter to guarantee the objective function to be non-increasing in each iteration. In addition, since Steps 7 and 8 involve the operations between two vectors in different tangent spaces Txk Mm and Txk+1 Mm , which can not be combined directly, a mapping between two tangent vectors in different tangent spaces called transport is introduced. The transport of a tangent vector d from xk to xk+1 can be specified as Transpxk →xk+1 :Txk Mm → Txk+1 Mm : (13) d → d − {diag d ◦ x∗k+1 }xk+1 , which is accomplished in Step 6.
C. Hybrid Precoder Design
IV.
The hybrid precoder design via alternating minimization is described in Algorithm 2 by solving problems (6) and (8) iteratively. To reduce the complexity of the algorithm and satisfy the power constraint in (5), we √ normalize FBB by s having it multiplied by a factor of FRF FNBB F at Step 7. The following lemma help reveal the effect of this normalization. Algorithm 2 Hybrid Precoder Design through Alternating Minimization Require: Fopt (0) (0) (0)† 1: Construct FRF with random phases, FBB = FRF Fopt and k = 0; 2: repeat (k+1) (k) 3: Optimize FRF using Algorithm 1 when FBB is fixed; (k+1) (k+1) (k+1)† 4: Fix FRF , and FBB = FRF Fopt ; 5: k ← k + 1; (k+1) (k) (k+1) (k+1) 6: until Fopt −FRF FBB 2F −Fopt −FRF FBB 2F ≤ ; 7: For the digital precoder at the transmit end, normalize √ Ns
BB = F FRF FBB FBB .
A LTERNATING M INIMIZATION FOR THE PARTIALLY- CONNECTED STRUCTURE
Besides the fully-connected hybrid precoding structure, another structure with relatively simple hardware implementation called the partially-connected structure has also been proposed recently [5], [6], as illustrated in Fig. 2. In the partiallyconnected structure, each RF chain is just connected with t Nt /NRF antennas, which reduces the hardware complexity in the RF domain. In the partially-connected structure, the analog precoder FRF belongs to a set of block diagonal matrices ARF , t where each block is an Nt /NRF dimension vector with unit modulus elements. Analog RF Precoder
RF Chain Digital Baseband Precoder
RF Chain
F
Lemma 1. If the Euclidean distance before normalization is F opt − FRF FBB F ≤ δ, then after normalization we have
BB ≤ 2δ. Fopt − FRF F F
√
s Proof: Define the normalization factor FRF FNBB F = and thus FRF FBB F = λ Fopt F . By norm inequality,
Fopt − FRF FBB F ≥ | Fopt F − FRF FBB F | = |1 − λ| Fopt F , which is equivalent to Fopt F ≤
1 |λ−1| δ.
1 λ
(14)
The partially-connected hybrid precoding structure.
Alternating minimization can also be utilized under this structure and the design procedure is presented below. More importantly, thanks to the characteristic of FRF , the optimality can be guaranteed for the digital and analog precoders respectively for this partially-connected structure. A. Analog RF Precoder Design The problem formulation of the analog precoder is given as
When λ = 1,
BB Fopt − FRF F F 1 F )F = − F F + (1 − F RF BB RF BB opt λ F 1 ≤ Fopt − FRF FBB F + (1 − ) FRF FBB F λ λ−1 δ ≤ 2δ. ≤ δ + (λ − 1) Fopt F ≤ δ + |λ − 1|
Fig. 2.
Fopt − FRF FBB F
subject to
FRF ∈ ARF .
FRF
(15)
Lemma 1 shows that as long as we can make the Euclidean distance between the optimal digital precoder and the hybrid precoder sufficiently small without considering the power constraint in (5), the normalization step will also achieve a small distance to the optimal digital precoder. Since the objective function in problem (5) is minimized at Steps 3 and 4, each iteration will never increase it. Also, the objective function is nonnegative. These two properties together guarantee that our algorithm converges to a feasible solution. The optimality of alternating minimization algorithms for non-convex problems is still an open problem [11].
2
minimize
(16)
Due to special structure of the constraint on FRF , in the product FRF FBB , each nonzero element of FRF is multiplied by a corresponding row extracted from FBB . This special characteristic simplifies the analog precoder design and there exists a closed-form expression for nonzero elements in FRF H
(FRF )i,j = arg{(Fopt )i,: (FBB )j,: }, t N 1 ≤ i ≤ Nt , j = i RF . Nt
(17)
B. Digital Baseband Precoder Design Since there is no power constraint with respect to the decoder design at receiver, similar to the design in Section III-A, the global optimal solution of the digital decoder can be determined by (7). On the other hand, also due to the block diagonal and unit modulus structures, the power constraint in (5) at the transmit side can be recast as the following problem 2
minimize
Fopt − FRF FBB F
subject to
FBB F =
FBB
2
t Ns NRF . Nt
(18)
Basically, problem (18) is a non-convex quadratic constraint quadratic programming (QCQP) problem, which can be reformulated as a homogeneous QCQP problem: Y∈H
subject to
tr(CY) ⎧ N t Ns ⎪ ⎨tr(A1 Y) = RF Nt , tr(A2 Y) = 1, ⎪ ⎩Y 0, rank(Y) = 1,
(19)
14
t with Hn being the set of n = NRF Ns + 1 dimension complex T Hermitian matrices. In addition, y = [ vec(FBB ) t ] with H an auxiliary variable t, Y = yy , f = vec(Fopt ) and In−1 0 0n−1 0 A1 = , A2 = , 0 0 0 1 (INs ⊗ FRF )H (INs ⊗ FRF ) −(INs ⊗ FRF )H f . C= −f H (INs ⊗ FRF ) vec(Fopt )H f
12
Spectral Efficiency (bits/s/Hz)
minimize n
angles of departure and arrival (AoDs and AoAs) follow the Laplacian distribution with uniformly distributed mean angles and angular spread of 7.5 degrees. Signal-to-noise ratio (SNR) is defined as σρ2 . The antenna elements in USPA are separated n by a half wavelength distance and all simulation results are averaged over 1000 channel realizations. Digital Optimal Proposed Algorithm in Section III OMP Algorithm [3] Beam Steering [14]
10
8
6
4
2
In fact, the most difficult part in problem (19) is the rank constraint, which is non-convex with respect to Y. Thus we may drop it to obtain a relaxed version of (19), i.e., a semidefinite relaxation (SDR) problem. It turns out that the SDR is tight when the number of constraints is less than three for a complex-valued homogeneous QCQP problem [12]. Consequently, problem (19) without the rank-one constraint reduces into a semidefinite programming (SDP) problem and can be solved by standard convex optimization algorithms, from which we can obtain the optimal solution of (18). C. Comparison Between Two Hybrid Precoding Structures The main difference between the two hybrid precoding structures considered in this paper is the number of phase shifters NPS in use for given numbers of data streams, RF chains, and antennas. For spectral efficiency, the fully-connected structure provides more design degrees of freedom (DoFs) in the RF domain and thus outperforms the partially-connected one. However, when taking power consumption into consideration, it is intriguing to know which structure has better energy efficiency. Energy efficiency is defined as the ratio between spectral efficiency and total power consumption [13] E=
R , t P Pcommon + NRF RF + NPS (PPS + PPA )
(20)
where the unit of E is bits/Hz/J, Pcommon is the common power of the transmitter, and PRF , PPS and PPA are the powers of each RF chain, phase shifter and power amplifier. t The number of phase shifters NPS is equal to Nt NRF for the fully-connected structure and NPS equals Nt for the partiallyconnected one. The numerical comparison will be provided in the next section. V.
S IMULATION R ESULTS
In this section, we numerically evaluate the performance of our proposed algorithms. Assume that two data streams are sent from a transmitter with Nt = 64 to a receiver with Nr =16, while both are equipped with USPA. The channel parameters are given by Ncl = 8 clusters, Nray = 10 rays and the average 2 power of each cluster is σα,i = 1. The azimuth and elevation
0 -30
-25
-20
-15
-10
-5
0
SNR (dB)
Fig. 3. Spectral efficiency achieved by different precoding algorithms when t r = 2 for fully-connected structure. NRF = NRF
First we investigated the spectral efficiency under the fullyconnected structure when the number of RF chains is equal to t r that of data streams NRF = NRF = Ns = 2. This is the worst case since RF chains can not be less under the assumption in t r Section II-A that Ns ≤ NRF ≤ Nt and Ns ≤ NRF ≤ Nr . In this case, as shown in Fig. 3, the existing OMP and beam steering algorithms developed in [3], [14] achieve significantly lower spectral efficiency than the optimal digital precoding at higher SNRs. On the contrary, our proposed alternating minimization algorithm achieves near-optimal performance over the whole SNR range in consideration. This means the proposed algorithm can more accurately approximate the optimal digital precoder than existing algorithms. t r It has been shown that when NRF ≥ 2Ns and NRF ≥ 2Ns , there exists a closed-form solution to the design problem of the fully-connected hybrid precoding, which leads to the same spectral efficiency provided by the optimal digital precoding [4]. Thus it is interesting to examine if our proposed algorithm can achieve the comparable performance in this special case. Fig. 4 compares the performance of different precoding schemes for different NRF . We see that the proposed algorithm for the fully-connected structure starts to coincide t r with the optimal digital precoding when NRF = NRF ≥ t 4. This result demonstrates that when NRF ≥ 2Ns and r NRF ≥ 2Ns , our proposed algorithm can actually achieve the optimal spectral efficiency, which can not be achieved by the OMP algorithm. Furthermore, the comparison between two hybrid precoding structures shows that the partially-connected structure, using less phase shifters, does entail some nonnegligible performance loss when compared with the fullyconnected structure. However, the spectral efficiency achieved by this structure is still higher than that achieved by the fullyconnected structure using the beam steering approach. For the t partially-connected structure, we choose NRF ∈ {2, 4, 8} to t guarantee that Nt /NRF is an integer.
14
VI.
Spectral Efficiency (bits/s/Hz)
13.5
Fully-connected structure
13 12.5 Partially-connected structure 12 11.5 11
Digital Optimal Alternating Minimization in Section III OMP Algorithm [3] Alternating Minimization in Section IV Beam Steering [14]
10.5 10 9.5 9
2
3
4
5
6
7
8
NRF
Fig. 4. Spectral efficiency achieved by different precoding algorithms given t r =N = NRF NRF RF and SNR = 0dB. 0.3
Energy Efficiency (bits/Hz/J)
0.25
Partially-connected Fully-connected
0.15
0.1
2
3
4
5
6
7
This paper proposed an alternating minimization approach for the hybrid precoding design in mmWave MIMO systems. By alternating between the digital baseband precoder design and the analog RF precoder design, much higher spectral efficiency is achieved compared to existing methods. One particular contribution is the identification of the manifold structure in the analog precoder, and thus powerful manifold optimization can be applied. Numerical results have demonstrated that the proposed algorithm performs favorably in comparison to several existing algorithms with regard to spectral efficiency. When it comes to the partially-connected structure, the precoder design solution is obtained by using the SDRenabled alternating minimization algorithm. It turns out that the fully-connected structure has a higher spectral efficiency while the partially-connected structure, taking advantage of its low-complexity hardware architecture, achieves a higher energy efficiency. The proposed approach can be extended to other hybrid precoder design problems in mmWave MIMO systems. Meanwhile, the computation efficiency will need to be improved to make it practical. R EFERENCES
0.2
0.05
C ONCLUSION
8
NtRF
Fig. 5. Energy efficiency of fully-connected and partially-connected structures when SNR = −5dB.
Fig. 5 illustrates the achievable energy efficiency of different hybrid precoding structures. The simulation parameters are set as follows: Pcommon = 10 W, PRF = 100 mW, PPS = 10 mW and PPA = 300 mW [2]. Since the number of phase t shifters scales linearly with NRF and Nt in the fully-connected structure, the power consumption will increase substantially t when increasing NRF . As shown in Fig. 4, however, the spectral efficiency achieved by the proposed algorithm in Section III is sufficiently close or exactly equal to the optimal digital one. Based on these two facts, the power consumption grows much faster than the spectral efficiency, which gives rise to the dramatic decrease of the energy efficiency. For the partially-connected structure, as the number of t phase shifters is independent of NRF , the dominant part of total power consumption remains almost unchanged over the investigated range of RF chain numbers. Meanwhile, the spectral efficiency will gradually approach the optimal digital t precoder when increasing NRF . The improvement of the spectral efficiency and the almost unchanged power consumption t together account for the rise in energy efficiency when NRF goes up in the partially-connected structure. Thus, we may conclude that the fully-connected structure is superior in terms of spectral efficiency while the partially-connected structure has an advantage of being more energy efficient.
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Dept. of ECE, The Hong Kong University of Science and Technology † Mediatek Inc., Hsinchu ∗ Email: {xyuam, eejzhang, eekhaled}@ust.hk, † [email protected]
Abstract—Millimeter wave (mmWave) communications holds a promise to offer an unprecedented capacity boost for 5G cellular networks. Due to the small wavelength of mmWave signals, multiple-input-multiple-output (MIMO) systems can leverage large-scale antennas to combat the path loss and rain attenuation via precoding. Different from conventional MIMO systems, mmWave MIMO cannot realize precoding entirely at baseband using digital precoders, as a result of the potentially high power consumed by the signal mixers and analog-to-digital converters (ADCs). As a cost-effective alternative, a hybrid precoding transceiver architecture for mmWave MIMO systems has received considerable attention. However, the optimal design of such hybrid precoding has not been fully understood. In this paper, an alternating minimization algorithm based on manifold optimization is proposed to design the hybrid precoder, thereby making it comparable in performance to the digital precoder. Numerical results show that our proposed algorithm can significantly outperform existing ones in terms of spectral efficiency and, more importantly, it can achieve the optimal performance in certain cases. The alternating minimization approach is also shown to be generally applicable to precoding design with different hybrid structures, and the corresponding comparison will show interesting design insights for hybrid precoding.
I.
I NTRODUCTION
The capacity of wireless networks has to increase exponentially to meet the ever-increasing demands for high-data-rate multimedia access. One promising way to boost the capacity is to exploit new spectrum bands for wireless communications. MmWave bands from 30 GHz to 300 GHz, previously only considered in indoor and fixed outdoor scenarios [1], have now been put forward as a prime candidate for new spectrum in 5G cellular systems. This view is supported by recent experiments in New York City that demonstrated the feasibility of mmWave outdoor cellular wireless communications [2]. The small wavelength of mmWave signals enables large-scale antenna arrays at transceivers, so that significant beamforming gains can be obtained to combat the path loss and rain attenuation resulted from the ten-fold increase of the carrier frequency. Moreover, spectral efficiency can also be improved by transmitting multiple data streams via spatial multiplexing. For traditional MIMO systems, precoding is typically accomplished at the baseband through digital precoders, which can change both the magnitude and phase of signals. However, fully digital precoding demands RF chains, including signal mixers and ADCs, comparable in number to antenna elements. While the small wavelengths of mmWave frequencies facilitate the use of a large number of antenna elements, the prohibitive cost and power consumption of RF chains make digital precoding infeasible. Given such unique constraints in mmWave MIMO systems, a hybrid precoding structure has recently been
proposed, which requires only a small number of RF chains interfacing between low-dimensional digital precoders and highdimensional analog precoders. In particular, each RF chain is connected to all the antenna elements via phase shifters, so this arrangement is referred to as the fully-connected structure, as illustrated in Fig. 1. There exist several previous studies on precoding design for the fully-connected hybrid structure [3]. In [4], the optimal design presented a special case, i.e., when the number of RF chains is at least twice the number of data streams. The major obstacle for solving the general case is to satisfy unit modulus constraints on analog precoders. Orthogonal matching pursuit (OMP), one of the widely used hybrid precoding algorithms [3], overcomes this obstacle by restricting columns of the analog precoding matrices to be the subset of predefined candidate vectors. Though the design problem is greatly simplified, imposing this extra constraint inevitably causes some performance loss. In other words, the general design problem of hybrid precoding has not yet been satisfactorily solved. In this paper, we will approach the performance of the optimal fully digital precoder by hybrid precoders without extra restrictions. A novel alternating minimization algorithm will be proposed, where the digital and analog parts are separately considered per iteration. Unlike the previous result in [3], the unit modulus constraints will be directly handled by manifold optimization. In particular, the manifold structure of the analog precoding pattern is exploited to efficiently identify a near-optimal solution. Simulation results will demonstrate significant throughput gains of the proposed algorithm over existing ones. The proposed alternating minimization algorithm will also be extended to consider a partially-connected structure, also called as the array of subarrays structure [5], that employs notably less phase shifters. Different from previous investigations focusing on codebook-based precoding [6], our study provides an effective algorithm for non-codebook-based precoding design of this structure. Furthermore, the comparison between these two structures indicates that the fullyconnected structure results in higher spectral efficiency while the partially-connected one is more energy efficient. The following notations are used throughout this paper. a and A stand for a column vector and a matrix respectively; Ai,j is the entry on the ith row and jth column of A; The conjugate, transpose and conjugate transpose of A are represented by A∗ , AT and AH ; det(A) and AF denote the determinant and Frobenius norm of A; A−1 and A† are the inverse and Moore-Penrose pseudo inverse of A; diag(A) generates a diagonal matrix with the entries of vector a; tr(A) and vec(A) indicate the trace and vectorization of A;
978-1-4799-5952-5/15/$31.00 ©2015 IEEE
Expectation and the real part of a complex variable is noted by E[·] and [·]; ◦ and ⊗ denote the Hadamard and Kronecker products between two matrices. II.
S YSTEM M ODEL AND P ROBLEM F ORMULATION
A. System Model Consider a single-user mmWave MIMO system1 as shown in Fig. 1, where Ns data streams are sent and collected by Nt transmit antennas and Nr receive antennas. The limitation t on the amount of RF chains is given by Ns ≤ NRF ≤ Nt r t t and Ns ≤ NRF ≤ Nr , where NRF and NRF denote the numbers of RF chains facilitated at the transmitter and receiver, respectively. Analog RF Precoder
RF Chain Digital Baseband Precoder
RF Chain
Fig. 1.
The fully-connected hybrid precoding structure.
The transmitted signal can be written as x = FRF FBB s, where s is the Ns × 1 symbol vector such that E ssH = 1 t Ns INs . The hybrid precoder consists of an NRF × Ns digt ital baseband precoder FBB and an Nt × NRF analog RF precoder FRF . The transmit power constraint is given by 2 FRF FBB F = Ns . For simplicity, we consider a block-fading propagation channel and the received signal after combining processing is √ H H H H y = ρWBB WRF HFRF FBB s + WBB WRF n, (1) where ρ stands for the average received power, H is the chanr × Ns digital baseband decoder, nel matrix, WBB is the NRF r WRF is the Nr × NRF analog RF decoder at the receiver and n is the noise vector of independent and identically distributed (i.i.d.) CN (0, σn2 ) elements. The achievable spectral efficiency when transmitted symbols follow a Gaussian distribution can be expressed as ρ −1 H H R = log det INs + Σ WBB WRF HFRF FBB Ns (2) H H ×FH F H W W , RF BB BB RF H H where Σ = σn2 WBB WRF WRF WBB is the noise covariance matrix. Furthermore, the analog precoders are implemented with phase shifters, which can only adjust the phases of the signals. Thus all the entries of FRF and WRF should satisfy the unit modulus constraint, namely |(FRF )i,j | = |(WRF )i,j | = 1. For the fully-connected structure in Fig. 1, the output signal of each RF chain is sent to all the antennas through phase shifters, while a different structure will be discussed in Section IV. 1 The receiver side is omitted due to space limitation. More details can be found in [3].
B. Channel Model Due to high free-space path losses, the mmWave propagation environment is well characterized by a clustered channel model, i.e., the Saleh-Valenzuela model [2]. This model depicts the mmWave channel matrix as Ncl N ray Nt Nr r t H H= αil ar (φril , θil )at (φtil , θil ) , (3) Ncl Nray i=1 l=1
where Ncl and Nray represent the number of clusters and the number of rays in each cluster, and αil denotes the gain of the lth ray in the ith propagation cluster. We assume 2 that α are i.i.d. and follow the distribution CN (0, σα,i ) and Ncl il 2 i=1 σα,i = γ, which is the normalization factor to satisfy 2
r t ) and at (φtil , θil ) E HF = Nt Nr . In addition, ar (φril , θil represent the receive and transmit array response vectors, r t r t (θil ) and θil (θil ) stand for azimuth and elevation where θil angles of arrival and departure, respectively. In this paper, we consider √ √ the uniform square planar array (USPA) with Nx × Nx (x ∈ {t, r}) antenna elements. Therefore, the ith element of array response vector can be written as 1 2π x x x ax (φ, θ)i = √ exp j d(p sin φ sin θ + q cos θ ) , λ Nx (4) where d and λ are√the antenna spacing√and the signal wavelength, 0 ≤ p < Nx and 0 ≤ q < Nx are the antenna indices in the 2D plane. While this channel model will be used in simulations, our precoder design is applicable to more general models.
C. Problem Formulation As shown in [3], the design of precoders and decoders can be separated into two sub-problems, i.e., the precoding and decoding problems. They have similar mathematical formulations except that there is an extra power constraint in the former. Therefore, we will mainly focus on the precoder design in the remaining part of this paper. The corresponding problem formulation is given by minimize FRF ,FBB
subject to
Fopt − FRF FBB F |(FRF )i,j | = 1, ∀i, j, 2 FRF FBB F = Ns ,
(5)
where Fopt stands for the optimal fully digital precoder, while FRF and FBB are the analog and digital precoders to be optimized. It has been shown in [3] that minimizing the objective function in (5) leads to the maximization of spectral efficiency. In addition, the matrix Fopt is comprised of the first Ns columns of V, which is a unitary matrix derived from the channel’s singular value decomposition (SVD), i.e., H = UΣVH . III.
H YBRID P RECODING D ESIGN T HROUGH A LTERNATING M INIMIZATION
Alternating minimization, which reduces the objective function with respect to different subsets of variables in each alternating step, has been successfully applied to many applications such as matrix completion, phase retrieval and dictionary learning [7]. Here we take this approach to solve problem (5),
by alternating between locating the best FBB and the best FRF . The main difficulty will be in the analog precoder design, for which we will propose a manifold optimization algorithm. A. Digital Baseband Precoder Design To design the digital precoder FBB , we fix the analog precoder FRF and restate problem (5) as minimize FBB
Fopt − FRF FBB F ,
(6)
which has a well-known least squares solution FBB =
F†RF Fopt .
(7)
Note that the power constraint in (5), temporarily removed, will be dealt with in Section III-C. Despite this, (7) has already offered a global optimal solution to the counterpart design problem at the receiver side. B. Analog RF Precoder Design via Manifold Optimization In the next alternating step, regarding FBB as fixed, we seek an analog precoder which optimizes the following problem2 : 2 minimize Fopt − FRF FBB F FRF (8) subject to |(FRF )i,j | = 1, ∀i, j. The main obstacle of is the unit modulus constraints, i.e., |(FRF )i,j | = 1, which are intrinsically non-convex. To the best of the authors’ knowledge, there is no general approach to solve (8) optimally, which is the major difficulty in alternating minimization. It has been pointed out in [3] that the unit modulus constraints make the original problem (5) mathematically intractable. However, we will show that the sub-problem (8) can be effectively solved by manifold minimization, as the unit modulus constraints define a Riemannian manifold. More importantly, the minimization over this manifold is locally analogous to that over a Euclidean space with smooth constraints. Therefore, a well-developed conjugate gradient algorithm on Euclidean spaces can have its counterpart on the specified Riemannian manifold. Manifold optimization has recently been applied for topological interference management in wireless networks [8]. In the following, we will briefly introduce the main idea. Note that the vector x = vec(FRF ) forms a complex circle manifold Mm = {x ∈ Cm : |x1 | = |x2 | = · · · = |xm | = 1}, t where m = Nt NRF . Therefore, the search space of the optimization problem (8) is over a product of m circles in the complex plane, which is a Riemannian submanifold of Cm with the product geometry. Please refer to [9] for background on Riemannian manifold and manifold optimization. For a given point x on the manifold Mm , the directions it can move along are characterized by the tangent vectors, ensuring the destination is also on the manifold. All the tangent vectors construct a tangent space at the point x ∈ Mm , which can be represented by Tx Mm = {y ∈ Cm : {y ◦ x∗ } = 0m } .
(9)
2 The square of the Frobenius norm makes the objective function smooth, and will not affect the solution.
Among all the tangent vectors, similar to the Euclidean space, one of them related to the negative Riemannian gradient represents the direction of the greatest decrease of a function. Because the complex circle manifold Mm is a Riemannian submanifold of Cm , the Riemannian gradient at x is a tangent vector gradf (x) given by the orthogonal projection of the Euclidean gradient ∇f (x) onto the tangent space Tx Mm : gradf (x) = Projx ∇f (x) = ∇f (x) − {diag [∇f (x) ◦ x∗ ]}x,
(10)
where the Euclidean gradient of the cost function in (8) is ∇f (x) = −2(F∗BB ⊗ INt ) vec(Fopt ) − (FTBB ⊗ INt )x . (11) Retraction is another key factor in manifold optimization, which maps a vector from the tangent space onto the manifold itself, determining the destination when moving along a tangent vector. The retraction of a tangent vector αd at point x ∈ Mm can be stated as Retrx :Tx Mm → Mm : αd → Retrx (αd) = vec
(x + αd)i . |(x + αd)i |
(12)
Equipped with the tangent space, Riemannian gradient and retraction of the complex circle manifold Mm , a line search based conjugate gradient method [10], which is a classical algorithm in Euclidean space, can be developed to design the analog precoder as shown in Algorithm 1. Algorithm 1 Conjugate Gradient Algorithm for Analog Precoder Design Based on Manifold Optimization Require: Fopt , FBB , x0 ∈ Mm 1: d0 = −gradf (x0 ) and k = 0; 2: repeat 3: Choose Armijo backtracking line search step size αk ; 4: Find the next point xk+1 using retraction in (12): xk+1 = Retrxk (αk dk ); 5: Determine Riemannian gradient gk+1 = gradf (xk+1 ) according to (10) and (11); 6: Calculate the vector transports gk+ and d+ k of gradient gk and conjugate direction dk from xk to xk+1 ; 7: Choose Polak-Ribiere parameter βk+1 ; 8: Compute conjugate direction dk+1 = −gk+1 +βk+1 d+ k; 9: k ← k + 1; 10: until |gradf (xk+1 )| ≤ . Algorithm 1 utilizes the well-known Armijo backtracking line search step and Polak-Ribiere parameter to guarantee the objective function to be non-increasing in each iteration. In addition, since Steps 7 and 8 involve the operations between two vectors in different tangent spaces Txk Mm and Txk+1 Mm , which can not be combined directly, a mapping between two tangent vectors in different tangent spaces called transport is introduced. The transport of a tangent vector d from xk to xk+1 can be specified as Transpxk →xk+1 :Txk Mm → Txk+1 Mm : (13) d → d − {diag d ◦ x∗k+1 }xk+1 , which is accomplished in Step 6.
C. Hybrid Precoder Design
IV.
The hybrid precoder design via alternating minimization is described in Algorithm 2 by solving problems (6) and (8) iteratively. To reduce the complexity of the algorithm and satisfy the power constraint in (5), we √ normalize FBB by s having it multiplied by a factor of FRF FNBB F at Step 7. The following lemma help reveal the effect of this normalization. Algorithm 2 Hybrid Precoder Design through Alternating Minimization Require: Fopt (0) (0) (0)† 1: Construct FRF with random phases, FBB = FRF Fopt and k = 0; 2: repeat (k+1) (k) 3: Optimize FRF using Algorithm 1 when FBB is fixed; (k+1) (k+1) (k+1)† 4: Fix FRF , and FBB = FRF Fopt ; 5: k ← k + 1; (k+1) (k) (k+1) (k+1) 6: until Fopt −FRF FBB 2F −Fopt −FRF FBB 2F ≤ ; 7: For the digital precoder at the transmit end, normalize √ Ns
BB = F FRF FBB FBB .
A LTERNATING M INIMIZATION FOR THE PARTIALLY- CONNECTED STRUCTURE
Besides the fully-connected hybrid precoding structure, another structure with relatively simple hardware implementation called the partially-connected structure has also been proposed recently [5], [6], as illustrated in Fig. 2. In the partiallyconnected structure, each RF chain is just connected with t Nt /NRF antennas, which reduces the hardware complexity in the RF domain. In the partially-connected structure, the analog precoder FRF belongs to a set of block diagonal matrices ARF , t where each block is an Nt /NRF dimension vector with unit modulus elements. Analog RF Precoder
RF Chain Digital Baseband Precoder
RF Chain
F
Lemma 1. If the Euclidean distance before normalization is F opt − FRF FBB F ≤ δ, then after normalization we have
BB ≤ 2δ. Fopt − FRF F F
√
s Proof: Define the normalization factor FRF FNBB F = and thus FRF FBB F = λ Fopt F . By norm inequality,
Fopt − FRF FBB F ≥ | Fopt F − FRF FBB F | = |1 − λ| Fopt F , which is equivalent to Fopt F ≤
1 |λ−1| δ.
1 λ
(14)
The partially-connected hybrid precoding structure.
Alternating minimization can also be utilized under this structure and the design procedure is presented below. More importantly, thanks to the characteristic of FRF , the optimality can be guaranteed for the digital and analog precoders respectively for this partially-connected structure. A. Analog RF Precoder Design The problem formulation of the analog precoder is given as
When λ = 1,
BB Fopt − FRF F F 1 F )F = − F F + (1 − F RF BB RF BB opt λ F 1 ≤ Fopt − FRF FBB F + (1 − ) FRF FBB F λ λ−1 δ ≤ 2δ. ≤ δ + (λ − 1) Fopt F ≤ δ + |λ − 1|
Fig. 2.
Fopt − FRF FBB F
subject to
FRF ∈ ARF .
FRF
(15)
Lemma 1 shows that as long as we can make the Euclidean distance between the optimal digital precoder and the hybrid precoder sufficiently small without considering the power constraint in (5), the normalization step will also achieve a small distance to the optimal digital precoder. Since the objective function in problem (5) is minimized at Steps 3 and 4, each iteration will never increase it. Also, the objective function is nonnegative. These two properties together guarantee that our algorithm converges to a feasible solution. The optimality of alternating minimization algorithms for non-convex problems is still an open problem [11].
2
minimize
(16)
Due to special structure of the constraint on FRF , in the product FRF FBB , each nonzero element of FRF is multiplied by a corresponding row extracted from FBB . This special characteristic simplifies the analog precoder design and there exists a closed-form expression for nonzero elements in FRF H
(FRF )i,j = arg{(Fopt )i,: (FBB )j,: }, t N 1 ≤ i ≤ Nt , j = i RF . Nt
(17)
B. Digital Baseband Precoder Design Since there is no power constraint with respect to the decoder design at receiver, similar to the design in Section III-A, the global optimal solution of the digital decoder can be determined by (7). On the other hand, also due to the block diagonal and unit modulus structures, the power constraint in (5) at the transmit side can be recast as the following problem 2
minimize
Fopt − FRF FBB F
subject to
FBB F =
FBB
2
t Ns NRF . Nt
(18)
Basically, problem (18) is a non-convex quadratic constraint quadratic programming (QCQP) problem, which can be reformulated as a homogeneous QCQP problem: Y∈H
subject to
tr(CY) ⎧ N t Ns ⎪ ⎨tr(A1 Y) = RF Nt , tr(A2 Y) = 1, ⎪ ⎩Y 0, rank(Y) = 1,
(19)
14
t with Hn being the set of n = NRF Ns + 1 dimension complex T Hermitian matrices. In addition, y = [ vec(FBB ) t ] with H an auxiliary variable t, Y = yy , f = vec(Fopt ) and In−1 0 0n−1 0 A1 = , A2 = , 0 0 0 1 (INs ⊗ FRF )H (INs ⊗ FRF ) −(INs ⊗ FRF )H f . C= −f H (INs ⊗ FRF ) vec(Fopt )H f
12
Spectral Efficiency (bits/s/Hz)
minimize n
angles of departure and arrival (AoDs and AoAs) follow the Laplacian distribution with uniformly distributed mean angles and angular spread of 7.5 degrees. Signal-to-noise ratio (SNR) is defined as σρ2 . The antenna elements in USPA are separated n by a half wavelength distance and all simulation results are averaged over 1000 channel realizations. Digital Optimal Proposed Algorithm in Section III OMP Algorithm [3] Beam Steering [14]
10
8
6
4
2
In fact, the most difficult part in problem (19) is the rank constraint, which is non-convex with respect to Y. Thus we may drop it to obtain a relaxed version of (19), i.e., a semidefinite relaxation (SDR) problem. It turns out that the SDR is tight when the number of constraints is less than three for a complex-valued homogeneous QCQP problem [12]. Consequently, problem (19) without the rank-one constraint reduces into a semidefinite programming (SDP) problem and can be solved by standard convex optimization algorithms, from which we can obtain the optimal solution of (18). C. Comparison Between Two Hybrid Precoding Structures The main difference between the two hybrid precoding structures considered in this paper is the number of phase shifters NPS in use for given numbers of data streams, RF chains, and antennas. For spectral efficiency, the fully-connected structure provides more design degrees of freedom (DoFs) in the RF domain and thus outperforms the partially-connected one. However, when taking power consumption into consideration, it is intriguing to know which structure has better energy efficiency. Energy efficiency is defined as the ratio between spectral efficiency and total power consumption [13] E=
R , t P Pcommon + NRF RF + NPS (PPS + PPA )
(20)
where the unit of E is bits/Hz/J, Pcommon is the common power of the transmitter, and PRF , PPS and PPA are the powers of each RF chain, phase shifter and power amplifier. t The number of phase shifters NPS is equal to Nt NRF for the fully-connected structure and NPS equals Nt for the partiallyconnected one. The numerical comparison will be provided in the next section. V.
S IMULATION R ESULTS
In this section, we numerically evaluate the performance of our proposed algorithms. Assume that two data streams are sent from a transmitter with Nt = 64 to a receiver with Nr =16, while both are equipped with USPA. The channel parameters are given by Ncl = 8 clusters, Nray = 10 rays and the average 2 power of each cluster is σα,i = 1. The azimuth and elevation
0 -30
-25
-20
-15
-10
-5
0
SNR (dB)
Fig. 3. Spectral efficiency achieved by different precoding algorithms when t r = 2 for fully-connected structure. NRF = NRF
First we investigated the spectral efficiency under the fullyconnected structure when the number of RF chains is equal to t r that of data streams NRF = NRF = Ns = 2. This is the worst case since RF chains can not be less under the assumption in t r Section II-A that Ns ≤ NRF ≤ Nt and Ns ≤ NRF ≤ Nr . In this case, as shown in Fig. 3, the existing OMP and beam steering algorithms developed in [3], [14] achieve significantly lower spectral efficiency than the optimal digital precoding at higher SNRs. On the contrary, our proposed alternating minimization algorithm achieves near-optimal performance over the whole SNR range in consideration. This means the proposed algorithm can more accurately approximate the optimal digital precoder than existing algorithms. t r It has been shown that when NRF ≥ 2Ns and NRF ≥ 2Ns , there exists a closed-form solution to the design problem of the fully-connected hybrid precoding, which leads to the same spectral efficiency provided by the optimal digital precoding [4]. Thus it is interesting to examine if our proposed algorithm can achieve the comparable performance in this special case. Fig. 4 compares the performance of different precoding schemes for different NRF . We see that the proposed algorithm for the fully-connected structure starts to coincide t r with the optimal digital precoding when NRF = NRF ≥ t 4. This result demonstrates that when NRF ≥ 2Ns and r NRF ≥ 2Ns , our proposed algorithm can actually achieve the optimal spectral efficiency, which can not be achieved by the OMP algorithm. Furthermore, the comparison between two hybrid precoding structures shows that the partially-connected structure, using less phase shifters, does entail some nonnegligible performance loss when compared with the fullyconnected structure. However, the spectral efficiency achieved by this structure is still higher than that achieved by the fullyconnected structure using the beam steering approach. For the t partially-connected structure, we choose NRF ∈ {2, 4, 8} to t guarantee that Nt /NRF is an integer.
14
VI.
Spectral Efficiency (bits/s/Hz)
13.5
Fully-connected structure
13 12.5 Partially-connected structure 12 11.5 11
Digital Optimal Alternating Minimization in Section III OMP Algorithm [3] Alternating Minimization in Section IV Beam Steering [14]
10.5 10 9.5 9
2
3
4
5
6
7
8
NRF
Fig. 4. Spectral efficiency achieved by different precoding algorithms given t r =N = NRF NRF RF and SNR = 0dB. 0.3
Energy Efficiency (bits/Hz/J)
0.25
Partially-connected Fully-connected
0.15
0.1
2
3
4
5
6
7
This paper proposed an alternating minimization approach for the hybrid precoding design in mmWave MIMO systems. By alternating between the digital baseband precoder design and the analog RF precoder design, much higher spectral efficiency is achieved compared to existing methods. One particular contribution is the identification of the manifold structure in the analog precoder, and thus powerful manifold optimization can be applied. Numerical results have demonstrated that the proposed algorithm performs favorably in comparison to several existing algorithms with regard to spectral efficiency. When it comes to the partially-connected structure, the precoder design solution is obtained by using the SDRenabled alternating minimization algorithm. It turns out that the fully-connected structure has a higher spectral efficiency while the partially-connected structure, taking advantage of its low-complexity hardware architecture, achieves a higher energy efficiency. The proposed approach can be extended to other hybrid precoder design problems in mmWave MIMO systems. Meanwhile, the computation efficiency will need to be improved to make it practical. R EFERENCES
0.2
0.05
C ONCLUSION
8
NtRF
Fig. 5. Energy efficiency of fully-connected and partially-connected structures when SNR = −5dB.
Fig. 5 illustrates the achievable energy efficiency of different hybrid precoding structures. The simulation parameters are set as follows: Pcommon = 10 W, PRF = 100 mW, PPS = 10 mW and PPA = 300 mW [2]. Since the number of phase t shifters scales linearly with NRF and Nt in the fully-connected structure, the power consumption will increase substantially t when increasing NRF . As shown in Fig. 4, however, the spectral efficiency achieved by the proposed algorithm in Section III is sufficiently close or exactly equal to the optimal digital one. Based on these two facts, the power consumption grows much faster than the spectral efficiency, which gives rise to the dramatic decrease of the energy efficiency. For the partially-connected structure, as the number of t phase shifters is independent of NRF , the dominant part of total power consumption remains almost unchanged over the investigated range of RF chain numbers. Meanwhile, the spectral efficiency will gradually approach the optimal digital t precoder when increasing NRF . The improvement of the spectral efficiency and the almost unchanged power consumption t together account for the rise in energy efficiency when NRF goes up in the partially-connected structure. Thus, we may conclude that the fully-connected structure is superior in terms of spectral efficiency while the partially-connected structure has an advantage of being more energy efficient.
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