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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LWC.2015.2409268, IEEE Wireless Communications Letters

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An Iterative Hybrid Transceiver Design Algorithm for Millimeter Wave MIMO Systems Chiao-En Chen, Member, IEEE

Abstract—In this letter, a new algorithm for millimeter wave multiple-input-multiple-output hybrid (mixed RF and baseband) transceiver design is proposed. The proposed algorithm iteratively updates the phases of the phase-shifters in the RF precoder (or RF combiner) to minimize the weighted sum of squared residuals between the optimal full-baseband design and the hybrid design, and is guaranteed to converge to at least a local optimal solution. Simulation results show that the proposed iterative design can achieve almost the same performance as the optimal full-baseband design, in spite of using a much smaller number of RF chains. Index Terms—Multiple-input-multiple-output (MIMO), massive MIMO, millimeter wave, hybrid precoding, transceiver

I. I NTRODUCTION

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ECAUSE of the ever-increasing mobile traffic from smart phones, tablets, and other mobile devices, it is expected that the next-generation wireless networks will have to support 1000 times more capacity when compared with what we have today [1]. To achieve this goal, a number of promising technologies have been proposed [2, 3]. Millimeter wave (mmWave) multiple-input-multiple-output (MIMO) communication enables additional access to the 30–300 GHz bands, and hence, has drawn great research interest recently [4]. The transceiver design architecture for mmWave MIMO can be very different from conventional microwave MIMO. This is mainly because mmWave MIMO requires tens-to-hundreds of antennas to leverage the acquired array gain to compensate for the high path loss in the mmWave bands. Owing to the high hardware cost of RF chains, the conventional full-baseband design in which a dedicated RF chain is allocated for each antenna element becomes impractical [5]. To address this issue, hybrid transceiver architecture, which concatenates an RF precoder (or RF combiner) with a baseband precoder (or baseband combiner) is proposed [6, 7]. In [7] and later refined in [8], the authors formulate the design problem into a sparsity signal reconstruction problem and then solved approximately by using the orthogonal matching pursuit (OMP) algorithm [9]. A modification of the OMP-based hybrid transceiver was later proposed in [10] to enable efficient parallel computation without sacrificing the performance when compared with the OMP-based design [7]. In this letter, we propose an iterative hybrid transceiver design algorithm aimed at reducing the performance gap between the existing OMP-based hybrid transceiver [8] and the This work was supported by the Ministry of Science and Technology, Taiwan, under Grant Number MOST103-2622-E-194-008-CC1. Chiao-En Chen is with the Department of Electrical/Communications Engineering, National Chung Cheng University, Chiayi, Taiwan. (e-mail: [email protected]).

optimal full-baseband design. The main idea is to sequentially update the phases in the RF precoder (or RF combiner) in a greedy manner and hence, monotonically decrease the weighted sum of squared residuals between the two precoders (or combiners). Simulation results show that the proposed iterative design can achieve almost identical rate performance in the mmWave channel with a much smaller number of RF chains when compared with the full-baseband design. A complexity reduction scheme is also presented in this letter. Notations: Throughout this letter, matrices and vectors are set in boldface, with uppercase letters for matrices and lower case letters for vectors. The superscripts T , H , and −1 denote the transpose, conjugate transpose, and inverse, respectively. k · kF is used to represent the Frobenius norm of a matrix, and the notation [·]:,ℓ is used to denote the ℓth column of a matrix. The commas and semicolons in a matrix are used as the column and row separator, respectively. Finally, IN represents the N × N identity matrix. II. S YSTEM M ODEL We consider a single-user mmWave hybrid MIMO system [8] as shown in Fig. 1. The transmitter is assumed to transmit Ns spatial streams via its Nt antennas and NtRF RF chains, while the receiver is assumed to use its Nr receive antennas and NrRF RF chains for reception. To enable multi-stream transmission with lower implementation complexity, we assume that number of RF chains is smaller than the number of antennas, and hence Ns ≤ NτRF < Nτ for all τ ∈ {t, r}. √ The Ns × 1 symbol vector ρs is first linear precoded by the baseband precoder FBB followed by the RF precoder FRF . Here, ρ represents the average transmitted power per symbol vector, while the elements in s are assumed to be independent and have been normalized to have energy N1s . The precoded √ signal vector x = ρFRF FBB s is then transmitted over the millimeter wave channel. The RF precoder FRF is assumed to be implemented via low-cost analog phase shifters, and hence each element in FRF is constrained to have an identical norm. We assume | [FRF ]ℓ,m |2 = 1/Nt , for all ℓ = 1, . . . , Nt , and m = 1, . . . , NtRF . The concatenated precoder F= FRF FBB is assumed to follow the total power constraint tr FFH = Ns . Without loss of generality, we can assume that the millimeter wave channel is narrowband and block-faded1. Under this assumption, the equivalent baseband channel can be represented by the channel matrix H ∈ CNr ×Nt . Without loss of generality, the channel matrix is assumed to have been 1 In principle, a frequency-selective channel can always be converted into a parallel collection of frequency flat sub-channels using orthogonal-frequencydivision-multiplexing (OFDM).

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where ℓ = 0, . . . , Ny − 1 and m = 0, . . . , Nz − 1. With the aforementioned system architecture, the achieved rate can be H H H expressed as R = log2 det{INs + Nρs R−1 n W HFF H W}, where Rn = σn2 WH W.

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Hybrid Combiner

Fig. 1: Block diagram of a hybrid transceiver for mmWave MIMO communications. normalized to satisfy E kHk2F = Nt Nr . Using the above notations, the received data vector y can then be represented √ as y = ρHFRF FBB s + n, where n ∈ CNr ×1 is the noise vector, modelled as CN (0, σ 2 INr ) representing a circularly symmetric complex Gaussian random vector with zero-mean and covariance matrix σ 2 INr . At the receiver side, the data vector y is processed by the concatenated combiner matrix H H H H WH = WBB WRF , where WRF and WBB are the RF combiner and the baseband combiner matrices, respectively. Again, the RF combiner is implemented by analog phase shifters, and hence we can assume | [WRF ]ℓ,m |2 = 1/Nr , for all ℓ = 1, . . . , Nr , and m = 1, . . . , NrRF . To take the channel characteristics of the millimeter wave channel into account, we follow the model adopted in [8] which is based on the widely accepted extended SalehValenzuela geometric channel model [11]. In this model the channel matrix H is expressed as H =γ

Ncl N ray X X p=1 q=1

t r Λt φtp,q , θp,q αp,q Λr φrp,q , θp,q

H t r , at φtp,q , θp,q · ar φrp,q , θp,q

(1)

where γ is a normalization factor such that E{kHk2F} = t r ) denote the complex ), and φtp,q (θp,q Nt Nr . αp,q , φrp,q (θp,q gain, the azimuth (elevation) angle of arrival (AOA), and the azimuth (elevation) angle of departure (AOD) associated with the qth propagationpath (ray) from the pth cluster, respecτ τ tively. Λτ φτp,q , θp,q and aτ φτp,q , θp,q represent the antenna gain and array response vector, respectively, for all τ ∈ {t, r}, p = 1, . . . , Ncl and q = 1, . . . , Nray . The complex gain αp,q is 2 2 denotes the average modeled as CN (0, σα,p INr ), where σα,p power associated with the propagation paths in the pth cluster. τ ) within each The azimuth angles φτp,q (elevation angles θp,q cluster are assumed to be Laplacian random variables with angle spread σφτ (θθτ ) centered at a uniformly distributed mean cluster angle φ¯τp (θ¯pτ ), where τ ∈ {t, r}. For simplicity, we assume ideal sectored transmit and receive antenna elements whose antenna gain is modeled as Λτ (φτ , θτ ) = 1 for all φτ ∈ [φτmin , φτmax ], and Λτ (φτ , θτ ) = 0, otherwise. We consider the case of a uniform planar array (UPA) as the array geometry. It follows that the normalized array response for an Ny Nz -element UPA in which Ny and Nz elements are placed on the y-axis and z-axis can be expressed as h 1 aτ (φ, θ) = p 1, . . . , ejkd(ℓ sin φ sin θ+m cos θ) , Ny Nz iT · · · , ejkd((Ny −1) sin φ sin θ+(Nz −1) cos θ) , (2)

III. R EVIEW

ON THE S PATIALLY S PARSE H YBRID T RANSCEIVER D ESIGN

In [8], a spatially sparse transceiver design is proposed to maximize an approximation of the achieved rate. After some mathematical approximations, the following optimization problems are formulated [8]: 2

opt (Fopt RF , FBB ) = argmin kFopt − FRF FBB kF FBB ,FRF

subject to [FRF ]:,ℓ ∈ {at (φp,q , θp,q ), for all p, q}, for all ℓ = 1, . . . , NtRF , 2

kFRF FBB kF = Ns ,

(3)

2

1

2 opt opt (WMMSE − WRF WBB ) (WRF , WBB ) = argmin Ryy

F

WBB ,WRF

subject to [WRF ]:,ℓ ∈ {ar (φp,q , θp,q ), for all p, q}, for all ℓ = 1, . . . , NrRF ,

(4)

1 1 2 where Ryy = (ρ/Ns ) HFFH HH + σn2 INr 2 , WMMSE is the optimal MMSE combiner given by WMMSE = √ Nt ×Ns ( ρ/Ns )R−1 is the optimal semiyy HF, and Fopt ∈ C unitary precoder obtained as the right singular vectors of H associated with the largest Ns singular values. Fopt and WMMSE are also referred to as the optimal full-baseband precoders and combiners, respectively. In [8], the optimization problems (3) and (4) are then formulated into optimization problems with sparsity constraints, and then solved approximately using the OMP algorithm [9]. The OMP-based hybrid transceiver enjoys low computational complexity and has been shown to achieve rate performance close to that of the optimal full-baseband design in the simulated mmWave channel [8].

IV. P ROPOSED I TERATIVE H YBRID T RANSCEIVER D ESIGN A. Derivation of the proposed algorithm In this subsection, we propose an iterative algorithm for the hybrid transceiver design problem in mmWave MIMO systems. We start from the optimization problems in (3) and (4), and then express the objective functions in the following common form: 1

J(Ψ, X) = kΛ 2 (Y − AX)k2F .

(5)

Here, Λ = INt , Y = Fopt , A = FRF , X = FBB in the precoder design problem, and Λ = Ryy , Y = WMMSE, A = WRF , X = WBB in the combiner design problem. The√(ℓ, m)th element of A can be denoted as [A]ℓ,m = (1/ Nτ )exp (jψℓ,m ) due to the norm constraints of the RF precoders and combiners, where ψℓ,m = [Ψ]ℓ,m . Here τ equals t in the precoder design problem, and τ equals r in the combiner design problem, respectively.



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The minimization of (5) corresponds to the weighted nonlinear least squares problem, where Ψ and X are the nonlinear and linear parameters, respectively. Following the concentration technique [12], the weighted least squares solution for ˆ = AH ΛA −1 AH ΛY. X given A(Ψ) is obtained as X ˆ as X in the objective function (5), we then Substituting X obtain the concentrated objective function n o −1 H J˜(Ψ) = tr YH ΛY − tr YH ΛA AH ΛA A ΛY which only depends on the nonlinear parameter Ψ. Since tr YH ΛY is independent of the parameter Ψ, the optimal weighted least squares solution for Ψ can be obtained as ˜˜ ˆ = argmaxΨ J(Ψ), Ψ where n o −1 H ˜˜ J(Ψ) = tr YH ΛA AH ΛA A ΛY . (6) (6) is known to be a non-convex function of Ψ and no closedform solution is available to date. As a result, we propose to search for a local optimal solution through alternating minimization. The algorithm is briefly summarized as follows. Initialization: The proposed algorithm starts with some lowcost initial solution A, such as the solution obtained from the OMP-based design. Iteration: In each iteration, the algorithm performs a local search in each element of Ψ and then updates [A]ℓ,m . When ψℓ,m is being optimized, the rest of the parameters in Ψ are fixed to their latest updated values. Since ψℓ,m is merely a real variable with bounded ranges [−π, π], a local optimal point of (6) around the latest updated solution can be found efficiently by performing 1-D search using the Nelder-Mead simplexmethod [13]. The algorithm performs Nτ NτRF element-wise local searches per iteration as ℓ proceeds from 1 to Nτ , and m proceeds from 1 to NτRF , respectively. The algorithm iterates the above procedure until the prescribed number of iterations has been reached. Construction of X: After A has been obtained, the linear ˆ For the precoding problem, parameter X is estimated as X. √ the obtained parameter X is further scaled by Ns X/kAXkF in order to satisfy the total power constraint described in (3). ˜ ˜ Since the objective function J(Ψ) monotonically increases in each update and is upper bounded by tr YH ΛY , one can conclude that the proposed algorithm is guaranteed to converge to at least a local optimal solution. ˜ ˜ B. Complexity reduction method for evaluating J(Ψ) The Nelder-Mead simplex-method used in the proposed ˜˜ iterative algorithm requires multiple computations of J(Ψ) as the algorithm searches for optimized variables. Consequently, multiple matrix inverses (AH ΛA)−1 or associated QR factorizations need to be recomputed in (6) even though only a single element in A has been changed. In this subsection, we derive a complexity reduction method by exploiting the structure in AH ΛA via the low-rank update technique. Without loss of generality, we consider the 1-D optimization problem where the phase in [A]ℓ,m is the optimization variable. We can construct two permutation matrices P ∈ CNτ ×Nτ RF RF ˇ = PAQ and [A] ˇ 1,1 = [A]ℓ,m . and Q ∈ CNτ ×Nτ so that A

With the above n permutations, (6) can be equivalently expressed o ˜ ˇ H PΛY , where ˇ H ΦA ˇ −1 A ˜ ˇ ˇ A as J(Ψ) = tr YH ΛPT A ˇ = PΨQ, and Φ = PΛPT . In this new formulaΨ ˇ 1,1 = ψˇ1,1 is the optimization variable and has tion, [Ψ] ˜˜ evaluations in the optimizato be varied for different J(·) ˇ tion algorithm. For notational convenience, √ we denote A as H ˇ ˇ ˇ ˇ A = a ˇ, b ; ˇ c, D , where a ˇ = (1/ Nτ )exp(j ψ1,1 ) ∈ ˇ ˇ ˇ are the corresponding vectors and C, while b, c, and D matrix that will be held constant during the optimization. ˇ A ˇH ˇ Using the notation in A, as ΦA can then∗ beH expressed H H ˇ ˇ ˇ ˇ ˇ A ΦA = eˇ, f ; f , G , where eˇ = [ˇ a ,ˇ c ]Φ[ˇ aT , ˇ cT ]T , ˇ ˇ H ]Φ[ˇ f = [ˇ b, D aT , ˇ cT ]T are both functions of a ˇ, whereas ∗ H ˇ = [ˇ ˇ ]Φ[ˇ ˇ T ]T is independent of a G b, D b ,D ˇ. By using the matrix inversion Lemma and also the Woodbury’s identity ˇ H ΦA) ˇ −1 can be computed as (A ˇ H ΦA) ˇ −1 = (A hresults [14], i H H 1 −ν −ν ˇ −1ˇ ˇ −1 − νν , where ν = G f and η = η, η ; η , G η ˇ H ΦA) ˇ −1 can now be replaced eˇ−ˇ f H ν. The matrix inverse (A by matrix-vector and vector-vector multiplications. The algorithm is performed as follows. Φ1/2 , the hermitian matrix square root of Φ, is first computed. The resulting complexity overhead is negligible when compared with the overall complexity of the algorithm and hence will not be counted in the complexity analysis. At the beginning of each 1-D optimization, the proposed algorithm computes ˇ D ˇ −1 only once because all ˇ H ]H , Φ1/2 c, and G Φ1/2 [b, 1:Nτ ,2:Nτ ˇ these quantities are independent of the current optimization variable. For each function evaluation, the algorithm then com˜˜ Ψ) ˇ using putes Φ1/2 [ˇ aT , ˇ cT ]T , eˇ, ˇ f , η, and then computes J( these newly computed quantities. With the proposed reduction scheme, it can be shown that the average computational complexity per function evaluation is reduced from the original O Nτ2 NτRF to O(max{Nτ2 , Nτ NτRF Ns , Nτ3 /Neval }), where Neval is the average number of function evaluations needed in the 1-D Nelder-Mead optimization. The complexity reduction method is therefore most effective when Nτ3 /Neval ≪ NτRF . V. S IMULATION R ESULTS Here, we present some simulation results of the proposed iterative hybrid transceiver design simulated under an mmWave MIMO channel model. For better comparison and crossreferencing, we follow the channel parameters used in [8] for our simulation. Throughout the simulation, the sector angles at the transmitter are assumed to be 60◦ wide in the azimuth and 20◦ wide in the elevation domain, while the sector angle at the receiver is assumed to be the whole angle space. The mean azimuth angle and mean elevation angle are set to be 0◦ and 90◦ , respectively. The channel is assumed to consist of Ncl = 8 clusters and Nray = 10 rays per cluster. The angular spreads are assumed to be σφt = σθt = σφr = σθr = σ, while the average power associated with the propagation paths is assumed to be 2 σα,p = 1 for all clusters. The true angles-of-departure and angles-of-arrival are assumed to be available when performing the OMP-based design as in [8]. The received SNR is defined as SNR = ρ/σn2 , and each simulation point is averaged over 5000 independent Monte Carlo runs. In Fig. 2, the achieved rates of various transceiver designs are simulated under the scenario where the transmitter is



4

15 14.8 14.6

Rate[bits/sec/Hz]

14.4 14.2 14 13.8 13.6

Optimal full−baseband OMP−based hybrid + 1 iteration (NRF=6)

13.4

OMP−based hybrid + 1 iteration (N =4)

13.2

OMP−based hybrid (NRF=6)

13 0

RF

OMP−based hybrid (NRF=4) 5 10 Angle Spread (degrees)

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Fig. 3: Achieved rate versus angle spread achieved by various transceiver designs in several mmWave MIMO channel configurations. Fig. 2: Achieved rate versus SNR achieved by various transceiver designs in an mmWave MIMO channel (σ = 7.5◦ ). assumed to have Nt = 64 antennas and NtRF = 4 RF chains, while the receiver is assumed to have Nr = 16 antennas and NrRF = 4 RF chains. The proposed iterative designs using various initializations such as the OMP initialization, random phase initialization, and dominant steering vectors [15] initialization have been simulated. The simulation results suggest that the proposed iterative design is not very sensitive to different initializations (at least in the simulated scenario) as all iterative designs can achieve near-optimal rate performance in two iterations. Among the three initializations, the OMPbased iterative design appears to have the best performance. Running the OMP-based iterative algorithm for just one iteration appears to have acquired most of the performance gain, while further performance improvement can be achieved by running more iterations at the expense of higher complexity. In Fig. 3, the achieved rates of the proposed OMP-based iterative design is simulated at SNR=0 dB with NRF = 4 and NRF = 6. The simulation setting is the same as the one used in Fig. 2, except that the angle spread σ now varies from 0◦ to 15◦ . It is observed from the figure that the proposed OMP-based iterative design consistently provides improved rate when compared with the OMP-based design within the whole range of σ. It is further observed that the proposed OMP-based iterative design equipped with 4 RF chains can even outperform the OMP-based design equipped with 6 RF chains. VI. C ONCLUSION In this letter, an iterative hybrid transceiver design algorithm for mmWave MIMO systems is presented. The proposed algorithm is based on the nonlinear least-squares formulation in which the residual is minimized iteratively in a greedy manner. A complexity reduction method is also proposed to speed up the optimization involved in each iteration. Simulations show that the performance gap between the optimal full-baseband and the existing OMP-based hybrid transceiver designs can be substantially reduced by the proposed algorithm.

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