Alternating Minimization Algorithms for Hybrid Precoding ...€¦ · mmWave MIMO precoding can...

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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/JSTSP.2016.2523903, IEEE Journalof Selected Topics in Signal Processing

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Alternating Minimization Algorithms for HybridPrecoding in Millimeter Wave MIMO Systems

Xianghao Yu, Student Member, IEEE, Juei-Chin Shen, Member, IEEE, Jun Zhang, Senior Member, IEEE, andKhaled B. Letaief, Fellow, IEEE

Abstract—Millimeter wave (mmWave) communications hasbeen regarded as a key enabling technology for 5G networks, as itoffers orders of magnitude greater spectrum than current cellularbands. In contrast to conventional multiple-input-multiple-output(MIMO) systems, precoding in mmWave MIMO cannot beperformed entirely at baseband using digital precoders, as only alimited number of signal mixers and analog-to-digital converters(ADCs) can be supported considering their cost and powerconsumption. As a cost-effective alternative, a hybrid precodingtransceiver architecture, combining a digital precoder and ananalog precoder, has recently received considerable attention.However, the optimal design of such hybrid precoders has notbeen fully understood. In this paper, treating the hybrid precoderdesign as a matrix factorization problem, effective alternatingminimization (AltMin) algorithms will be proposed for twodifferent hybrid precoding structures, i.e., the fully-connectedand partially-connected structures. In particular, for the fully-connected structure, an AltMin algorithm based on manifoldoptimization is proposed to approach the performance of the fullydigital precoder, which, however, has a high complexity. Thus, alow-complexity AltMin algorithm is then proposed, by enforcingan orthogonal constraint on the digital precoder. Furthermore,for the partially-connected structure, an AltMin algorithm is alsodeveloped with the help of semidefinite relaxation. For practicalimplementation, the proposed AltMin algorithms are furtherextended to the broadband setting with orthogonal frequencydivision multiplexing (OFDM) modulation. Simulation resultswill demonstrate significant performance gains of the proposedAltMin algorithms over existing hybrid precoding algorithms.Moreover, based on the proposed algorithms, simulation compar-isons between the two hybrid precoding structures will providevaluable design insights.

Index Terms—Alternating minimization, hybrid precoding,low-complexity, manifold optimization, millimeter wave commu-nications, semidefinite relaxation.

I. INTRODUCTION

THE capacity of wireless networks has to exponentiallyincrease to meet the explosive demands for high-data-rate

multimedia access. In particular, the upcoming 5G networksaim at carrying out the projected 1000X increase in capacity

Manuscript received June 1, 2015; revised November 6, 2015; acceptedJanuary 13, 2016. This work was supported by the Hong Kong ResearchGrants Council under Grant No. 610212. The guest editor coordinating thereview of this paper and approving it for publication was Dr. Robert W. Heath.

This work was presented in part at IEEE Global Communications Confer-ence, San Diego, CA, Dec. 2015 [1].

X. Yu, J. Zhang, and K. B. Letaief are with the Department of Electronic andComputer Engineering, the Hong Kong University of Science and Technology(HKUST), Clear Water Bay, Kowloon, Hong Kong (email: xyuam, eejzhang,[email protected]). K. B. Letaief is also with Hamad Bin Khalifa University,Qatar (email: [email protected]).

J.-C. Shen is with MediaTek Inc., Hsinchu City 30078, Taiwan (e-mail:[email protected]).

by 2020 [2]. One way to boost the capacity is to improve thespectral efficiency through physical layer techniques, such asmassive multiple-input-multiple-output (MIMO) and advancedchannel coding [3]. Further improvement in area spectralefficiency can be achieved by network densification, suchas deploying small cells [4], [5] and allowing device-to-device (D2D) communications [6], and enabling advancedcooperation, such as Cloud-RANs [7], [8]. Nevertheless, thespectrum crunch in current cellular systems brings a funda-mental bottleneck for the further capacity increase. Thus, itis critical to exploit underutilized spectrum bands, includingthe bands that have not been used for cellular communicationsyet.

Millimeter wave (mmWave) bands from 30 GHz to 300GHz, previously only considered for outdoor point-to-pointbackhaul links [9] or for carrying indoor high-resolution mul-timedia streams [10], have now been put forward as a primecandidate for new spectrum in 5G cellular systems, with thepotential bandwidth reaching 10 GHz. This view is supportedby recent experiments in New York City that demonstrated thefeasibility of mmWave outdoor cellular communications [11],[12]. Originally, the main obstacles for the success of mmWavecellular systems are the huge path loss and rain attenuation,as a result of the ten-fold increase of the carrier frequency[11]. Thanks to the small wavelength of mmWave signals,mmWave MIMO precoding can leverage large-scale antennasat transceivers to provide significant beamforming gains tocombat the path loss and to synthesize highly directionalbeams. Moreover, spectral efficiency can be further increasedby transmitting multiple data streams via spatial multiplexing.

For traditional MIMO systems, precoding is typically ac-complished at baseband through digital precoders, which canadjust both the magnitude and phase of the signals. However,fully digital precoding demands radio frequency (RF) chains,including signal mixers and analog-to-digital converters (AD-Cs), comparable in number to the antenna elements. While thesmall wavelengths of mmWave frequencies facilitate the useof a large number of antenna elements, the prohibitive costand power consumption of RF chains make digital precodinginfeasible. Given such unique constraints in mmWave MIMOsystems, a hybrid precoding architecture has recently receivedmuch consideration, which only requires a small number of RFchains interfacing between a low-dimensional digital precoderand a high-dimensional analog precoder [13]. As the analogprecoders are still of high dimension, it is impractical toimplement them in the RF domain with power-hungry variablevoltage amplifiers (VGAs) [12]. This heuristic leads to a rule



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of thumb, i.e., realizing analog precoders with low-cost phaseshifters at the expense of sacrificing the ability to change themagnitude of the RF signals.

According to the mapping from RF chains to antennas,which determines the number of phase shifters in use, thehybrid precoding transceiver architectures can be categorizedinto the fully-connected and partially-connected structures,as illustrated in Fig. 1(b) and Fig. 1(c), respectively. Theformer structure enjoys the full beamforming gain for eachRF chain with a natural combination between RF chains andantenna elements, i.e., each RF chain is connected to allantennas. On the other hand, sacrificing some beamforminggain, the partially-connected structure significantly reduces thehardware implementation complexity by connecting each RFchain only with part of the antennas.

In [13], it has been pointed out that maximizing the spectralefficiency of mmWave systems can be approximated by min-imizing the Euclidean distance between hybrid precoders andthe fully digital precoder. This renders the hybrid precoderdesign as a matrix factorization problem with unit modulusconstraints imposed by the phase shifters. Although significantamounts of research efforts have been invested in solving var-ious matrix factorization problems in recent years [14], [15],with the unique unit modulus constraints, the optimal designof hybrid precoders remains unknown. Existing works oftenadd some extra constraints on analog precoders to simplifythe analog part design with unit modulus constraints, whichwill cause performance loss. This motivates us to reconsiderthe hybrid precoder design or, in other words, the matrixfactorization problem with unit modulus constraints on analogprecoders. In particular, a better way to deal with the unitmodulus constraint deserves further delicate investigations.

In this paper, by adopting alternating minimization (AltMin)as the main design approach, we will propose different hybridprecoding algorithms to approach the performance of the opti-mal fully digital precoder. Based on the principle of alternatingminimization, three novel algorithms will be proposed to findeffective hybrid precoding solutions for the fully-connectedand partially-connected structures.

A. Related Works

Hybrid precoding is a newly-emerged technique in mmWaveMIMO systems [16]–[20]. So far the main efforts are on thefully-connected structure [13], [21]–[28]. Orthogonal matchingpursuit (OMP) is the most widely used algorithm, which oftenoffers reasonably good performance. This algorithm requiresthe columns of analog precoding matrix to be picked fromcertain candidate vectors, such as array response vectors ofthe channel [13], [21], [22], and discrete Fourier transform(DFT) beamformers [23], [24]. Hence, the OMP-based hybridprecoder design can be viewed as a sparsity constrainedmatrix reconstruction problem. Though the design problemis greatly simplified in this way, restricting the space offeasible analog precoding solutions inevitably causes someperformance loss. Additionally, extra overhead will be broughtup for acquiring the information of array response vectors inadvance. More recent attention has mainly focused on reducing

the computation complexity of the OMP algorithm [21], [25],e.g., by reusing the matrix inversion result in each iteration.

There are works investigating some special hybrid precodingsystems. In [26], an optimal hybrid precoder design in a specialcase was identified, i.e., when the number of RF chains isat least twice that of the data streams. However, the optimalsolution for the general case is unknown. The authors of[28] investigated VGA-enabled hybrid precoding accordingto different design criteria. By removing VGAs from theRF domain, low-power analog precoders with phase shifterswere also considered in [28], whose phases are heuristicallyextracted from those of the VGA-enabled solution.

On the other hand, much less attention has been paidon the partially-connected structure [29]–[34]. In [29], [30],codebook-based design of hybrid precoders was presented fornarrowband and orthogonal frequency division multiplexing(OFDM) systems, respectively. Although the codebook-baseddesign enjoys a low complexity, there will be certain perfor-mance loss, and it is not clear how much performance gaincan be further obtained. By utilizing the idea of successiveinterference cancellation (SIC), an iterative hybrid precodingalgorithm for the partially-connected structure was proposedin [31]. The algorithm is established based on the assumptionthat the digital precoding matrix is diagonal, which meansthat the digital precoder only allocates power to different datastreams, and the number of RF chains should be equal to thatof the data streams. However, using only analog precoders toprovide beamforming gains is obviously a suboptimal strategy[31], [32], which also deviates from the motivation of hybridprecoding. So far there is no study directly optimizing thehybrid precoders without extra constraints in the partially-connected structure, which will be pursued in this paper.

B. ContributionsIn this paper, we investigate the hybrid precoder design in

mmWave MIMO systems. We will adopt alternating minimiza-tion (AltMin) as the main design principle, which helps de-couple the precoder design problem into two subproblems, i.e.,the analog and digital precoder design. The proposed AltMinalgorithms will alternately optimize the digital precoder andthe analog precoder. Our major contributions are summarizedas follows:

• For the fully-connected structure, we shall show that theunit modulus constraints of the analog precoder define aRiemannian manifold. We will thus propose a manifoldoptimization based AltMin (MO-AltMin) algorithm. Thisalgorithm does not need any pre-determined candidateset for the analog precoder, and it is the first attempt todirectly solve the hybrid precoder design problem underthe unit modulus constraints.

• By imposing an orthogonal property of the digital pre-coder, we then develop an AltMin algorithm using phaseextraction (PE-AltMin) as a low-complexity counterpartof the MO-AltMin algorithm, which will also be morepractical for implementation.

• For the partially-connected structure, we propose asemidefinite relaxation based AltMin (SDR-AltMin) al-gorithm. This algorithm effectively designs the hybrid



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Digital

Baseband

Precoder

RF Chain

RF Chain

Mapping

Via Analog

RF

Precoder

(a) MmWave MIMO transmitter architecture.

Analog RF Precoder

(b) The mapping strategy for thefully-connected structure.

Analog RF Precoder

(c) The mapping strategy for thepartially-connected structure.

Fig. 1. Two structures of hybrid precoding in mmWave MIMO systems using different mapping strategies: each RF chain is connected to Nt antennas in(b) and to Nt/Nt

RF antennas in (c).

precoders by offering optimal solutions for both subprob-lems of analog and digital precoders in each alternatingiteration, and it is the first effort directly optimizing thehybrid precoders in such a structure.

• The three proposed AltMin Algorithms can be general-ly applied to both narrowband and broadband OFDMsystems. Simulation results will demonstrate that theMO-AltMin algorithm efficiently identifies a near-optimalsolution, while the PE-AltMin algorithm with practicalcomputational complexity outperforms the existing OMPalgorithm.

• With the proposed AltMin algorithms, extensive compar-isons are provided to reveal valuable design insights. Inparticular, the proposed AltMin algorithms for the fully-connected structure can help to approach the performanceof the fully digital precoder as long as the number ofRF chains is comparable to the number of data streams,which cannot be achieved by the widely applied OMPalgorithm. On the other hand, the SDR-AltMin algorithmfor the partially-connected structure provides significantgains over analog beamforming. Furthermore, by takingadvantage of its low-complexity hardware implementa-tion, the partially-connected structure provides a higherenergy efficiency than the fully-connected one with arelatively large number of RF chains implemented attransceivers.

Thus, our results firmly establish the effectiveness of thealternating minimization as a key design methodology forhybrid precoder design in mmWave MIMO systems.

C. Organization

The remainder of this paper is organized as follows. Weshall introduce the system model and channel model, followedby the problem formulation in Section II. Two alternatingminimization algorithms for the fully-connected structure aredemonstrated in Section III and Section IV, respectively.

In Section V, the hybrid precoder design for the partially-connected structure is investigated. Extensions of the proposedalgorithms to OFDM systems are provided in Section VI, andsimulation results will be presented in Section VII. Finally, wewill conclude this paper in Section VIII.

D. Notations

The following notations are used throughout this paper. aand 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 arerepresented by A∗, AT and AH ; det(A) and ∥A∥F denotethe determinant and Frobenius norm of A; A−1 and A† arethe inverse and Moore-Penrose pseudo inverse of A; Tr(A)and vec(A) indicate the trace and vectorization; Expectationand the real part of a complex variable is noted by E[·] andℜ[·]; and ⊗ denote the Hadamard and Kronecker productsbetween two matrices.

II. SYSTEM MODEL AND PROBLEM FORMULATION

In this section, we will first present the system model andchannel model of the considered mmWave MIMO system, andthen formulate the hybrid precoding problem.

A. System Model

Consider a single-user mmWave MIMO system1 as shownin Fig. 1(a), where Ns data streams are sent and collected byNt transmit antennas and Nr receive antennas. The numbersof RF chains at the transmitter and receiver are respectivelydenoted as N t

RF and NrRF, which are subject to constraints

Ns ≤ N tRF ≤ Nt and Ns ≤ Nr

RF ≤ Nr.The transmitted signal can be written as x = FRFFBBs,

where s is the Ns × 1 symbol vector such that E[ssH

]=

1Ns

INs . The hybrid precoders consist of an N tRF ×Ns digital

1The receiver side is omitted due to space limitation. More details can befound in [13].



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baseband precoder FBB and an Nt×N tRF analog RF precoder

FRF. The normalized transmit power constraint is given by∥FRFFBB∥2F = Ns. For simplicity, we first consider a narrow-band block-fading propagation channel, while the extension tobroadband OFDM systems will be treated in Section VI. Thus,the received signal after decoding processing is given as

y =√ρWH

BBWHRFHFRFFBBs+WH

BBWHRFn, (1)

where ρ stands for the average received power, H is the chan-nel matrix, WBB is the Nr

RF ×Ns digital baseband decoder,WRF is the Nr×Nr

RF analog RF decoder at the receiver, andn is the noise vector of independent and identically distributed(i.i.d.) CN (0, σ2

n) elements. In this paper, we assume thatperfect channel state information (CSI) is known at both thetransmitter and receiver. In practice, CSI can be accurately andefficiently obtained by channel estimation [18] at the receiverand further shared at the transmitter with effective feedbacktechniques [13], [35]. The achievable spectral efficiency whentransmitted symbols follow a Gaussian distribution can beexpressed as

R = log det

(INs +

ρ

σ2nNs

(WRFWBB)†HFRFFBB

×FHBBF

HRFH

H (WRFWBB)

).

(2)

Furthermore, the analog precoders are implemented with phaseshifters, which can only adjust the phases of the signals. Thus,all the nonzero entries of FRF and WRF should satisfy the unitmodulus constraints, namely |(FRF)i,j | = |(WRF)i,j | = 1 fornonzero elements.

According to different signal mapping strategies from RFchains to antennas, the transceiver architecture can be catego-rized into the fully-connected and partially-connected hybridprecoding structures, as illustrated in Fig. 1(b) and Fig. 1(c).For the fully-connected structure, the output signal of eachRF chain is sent to all the antennas through phase shifters,while the partially-connected structure only has Nt/N

tRF an-

tennas connected to each RF chain. Thus, the fully-connectedstructure enjoys full beamforming gain for each RF chainwith a natural combination between RF chains and antennas,whereas the hardware implementation complexity is lower inthe partially-connected one by sacrificing some beamforminggain for each RF chain. The structures of FRFs and WRFswill vary for different structures, which will be discussed inthe following sections in detail.

B. Channel Model

Due to high free-space path loss, the mmWave propagationenvironment is well characterized by a clustered channelmodel, i.e., the Saleh-Valenzuela model [12]. This modeldepicts the mmWave channel matrix as

H =

√NtNr

NclNray

Ncl∑i=1

Nray∑l=1

αilar(ϕril, θ

ril)at(ϕ

til, θ

til)

H , (3)

where Ncl and Nray represent the number of clusters andthe number of rays in each cluster, and αil denotes the gainof the lth ray in the ith propagation cluster. We assume

that αil are i.i.d. and follow the distribution CN (0, σ2α,i) and∑Ncl

i=1 σ2α,i = γ, which is the normalization factor to satisfy

E[∥H∥2F

]= NtNr. In addition, ar(ϕr

il, θril) and at(ϕ

til, θ

til)

represent the receive and transmit array response vectors,where ϕr

il(ϕtil) and θril(θ

til) stand for azimuth and elevation

angles of arrival and departure, respectively. In this paper,we consider the uniform square planar array (USPA) with√N ×

√N antenna elements. Therefore, the array response

vector corresponding to the lth ray in the ith cluster can bewritten as

a(ϕil, θil) =1√N

[1, · · · , ej 2π

λ d(p sinϕil sin θil+q cos θil),

· · · , ej2πλ d((

√N−1) sinϕil sin θil+(

√N−1) cos θil)

]T,

(4)

where d and λ are the antenna spacing and the signal wave-length, and 0 ≤ p <

√N and 0 ≤ q <

√N are the antenna

indices in the 2D plane. While this channel model will beused in simulations, our precoder design is applicable to moregeneral models.

C. Problem Formulation

As shown in [13], [21], the design of precoders and decoderscan be separated into two subproblems, i.e., the precodingand decoding problems. They have similar mathematical for-mulations except that there is an extra power constraint inthe former. Therefore, we will mainly focus on the precoderdesign in the remaining part of this paper and the algorithmsproposed in this paper can be equally applied for the decoder.The corresponding problem formulation is given by

minimizeFRF,FBB

∥Fopt − FRFFBB∥F

subject to

FRF ∈ A∥FRFFBB∥2F = Ns,

(5)

where Fopt stands for the optimal fully digital precoder, whileFRF and FBB are the analog and digital precoders to beoptimized. Additionally, A ∈ Af ,Ap is the feasible set ofthe analog precoder induced by the unit modulus constraints,which will be distinct for different hybrid precoding structures.

It has been shown in [13] that minimizing the objectivefunction in (5) approximately leads to the maximization ofthe spectral efficiency. It is also intuitively true that theoptimal hybrid precoders should be sufficiently “close” tothe unconstrained optimal digital precoder. In addition, theunconstrained optimal precoder and decoder are comprised ofthe first Ns columns of V and U respectively, which areunitary matrices derived from the channel’s singular valuedecomposition (SVD), i.e., H = UΣVH .

We will mainly treat problem (5) as a matrix factorizationproblem, for which alternating minimization will be adoptedas the main tool. Alternating minimization represents a widelyapplicable and empirically successful approach for the opti-mization problems involving different subsets of variables. Ithas been successfully applied to many applications such asmatrix completion [14], phase retrieval [36], image reconstruc-tion [37], blind deconvolution [38] and non-negative matrix



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factorization [39]. In this paper, the problem formulation (5)is intrinsically a matrix factorization problem involving twomatrix variables FRF and FBB. However, jointly optimizingthese two variables is highly complicated due to the element-wise unit modulus constraints of FRF. By decoupling theoptimization of these two variables, alternating minimizationstands out as an efficient method to obtain an effectivesolution. With the principle of alternating minimization, wewill alternately solve for FRF and FBB while fixing the other,which will be the essential idea throughout this paper.

III. MANIFOLD OPTIMIZATION BASED HYBRIDPRECODING FOR THE FULLY-CONNECTED STRUCTURE

The fully-connected structure, in which each RF chain isconnected to all the antenna elements, is frequently used inmmWave MIMO systems, as shown in Fig. 1(b). This structurerestricts every entry in the analog precoding matrix to be unitmodulus, and this element-wise constraint makes the precoderdesign problem intractable. In this section, by observing thatthe unit modulus constraints define a Riemannian manifold,we will propose an AltMin algorithm based on manifoldoptimization to directly solve (5).

For the fully-connected structure, inspired by [40], the au-thors of [26] have shown that the Frobenius norm in (5) can bemade exactly zero under the condition that N t

RF ≥ 2Ns. Thismeans that the hybrid precoders can achieve the performanceof the fully digital precoder in this special case, and theoptimal hybrid precoders were obtained in [26]. Thus, we willfocus on the region where Ns ≤ N t

RF < 2Ns in this paper.

A. Digital Baseband Precoder Design

We first consider to design the digital precoder FBB with afixed analog precoder FRF. Thus, problem (5) can be restatedas

minimizeFBB

∥Fopt − FRFFBB∥F , (6)

which has a well-known least squares solution given by

FBB = F†RFFopt. (7)

Note that the power constraint in (5) is temporarily removed,and it will be dealt with in Section III-C. Nevertheless, thesolution in (7) has already offered a globally optimal solutionto the counterpart design problem at the receiver side.

B. Analog RF Precoder Design via Manifold Optimization

For the fully-connected structure, the feasible set Af ofthe analog precoder can be specified by |(FRF)i,j | = 1, aseach RF chain is connected to all the antennas. In the nextalternating step, we fix FBB and seek an analog precoderwhich optimizes the following problem2:

minimizeFRF

∥Fopt − FRFFBB∥2Fsubject to |(FRF)i,j | = 1, ∀i, j.

(8)

2The square of the Frobenius norm makes the objective function quadraticand smooth, and will not affect the solution.

The main obstacles are the unit modulus constraints, whichare intrinsically non-convex. To the best of the authors’ knowl-edge, there is no general approach to solve (8) optimally. In thefollowing, we will propose an effective manifold optimizationalgorithm to find a near-optimal solution of problem (8).

ξx

TxM

x

γ

M

Fig. 2. The tangent space and tangent vector of a Riemannian manifold [41].

We will start with some definitions and terminologies inmanifold optimization. More background on manifolds andmanifold optimization can be found in [41]–[43], and thereare some recent applications in wireless communications [44].As shown in Fig. 2, a manifold M is a topological space thatresembles a Euclidean space near each point [43]. In otherwords, each point on a manifold has a neighborhood thatis homeomorphic to the Euclidean space. The tangent spaceTxM at a given point x on the manifold M is composed ofthe tangent vectors ξx of the curves γ through the point x.In most applications, manifolds fall into a special categoryof topological manifold, namely, a Riemannian manifold.A Riemannian manifold is equipped with an inner productdefined on the tangent spaces TxM, called the Riemannianmetric, which allows one to measure distances and angles onmanifolds. In particular, it is possible to use calculus on aRiemannian manifold with the Riemannian metric.

The rich geometry of Riemannian manifolds makes it pos-sible to define gradients of cost functions. More importantly,optimization over a Riemannian manifold is locally analo-gous to that over a Euclidean space with smooth constraints.Therefore, a well-developed conjugate gradient algorithm inEuclidean spaces can find its counterpart on the specified Rie-mannian manifolds. In the following, we will briefly introducethis counterpart.

We first endow the complex plane C with the Euclideanmetric

⟨x1, x2⟩ = ℜx∗1x2, (9)

which is equivalent to treating C as R2 with the canonicalinner product. Then we are able to denote the complex circleas

Mcc = x ∈ C : x∗x = 1. (10)

For a given point x on the manifold Mcc, the directionsalong which it can move are characterized by the tangent



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vectors. Then the tangent space at the point x ∈ Mcc canbe represented by

TxMcc = z ∈ C : z∗x+ x∗z = 2 ⟨x, z⟩ = 0. (11)

Note that the vector x = vec(FRF) forms a complex circlemanifold Mm

cc = x ∈ Cm : |x1| = |x2| = · · · = |xm| = 1,where m = NtN

tRF. Therefore, the search space of the

optimization problem (8) is over a product of m circles inthe complex plane, which is a Riemannian submanifold ofCm with the product geometry. Hence, the tangent space at agiven point x ∈Mm

cc can be expressed as

TxMmcc = z ∈ Cm : ℜz x∗ = 0m . (12)

Among all the tangent vectors, similar to the Euclideanspace, one of them that is related to the negative Riemanniangradient represents the direction of the greatest decrease ofa function. Because the complex circle manifold Mm

cc is aRiemannian submanifold of Cm, the Riemannian gradientat x is a tangent vector gradf(x) given by the orthogonalprojection of the Euclidean gradient ∇f(x) onto the tangentspace TxMm

cc [41]:

gradf(x) = Projx∇f(x)= ∇f(x)−ℜ∇f(x) x∗ x,

(13)

where the Euclidean gradient of the cost function in (8) is

∇f(x) = −2(F∗BB ⊗ INt)

[vec(Fopt)− (FT

BB ⊗ INt)x].

(14)Solving this Euclidean gradient involves some techniques oncomplex-valued matrix derivatives, the details can be found in[45].

Retraction is another key factor in manifold optimization,which maps a vector from the tangent space onto the manifolditself. It determines the destination on the manifold whenmoving along a tangent vector. The retraction of a tangentvector αd at point x ∈Mm

cc can be stated as

Retrx :TxMmcc →Mm

cc :

αd 7→ Retrx(αd) = vec

[(x+ αd)i|(x+ αd)i|

].

(15)

Equipped with the tangent space, Riemannian gradient andretraction of the complex circle manifold Mm

cc, a line searchbased conjugate gradient method [46], which is a classicalalgorithm in the Euclidean space, can be developed to designthe analog precoder as shown in Algorithm 1.

Algorithm 1 utilizes the well-known Armijo backtrackingline search step and Polak-Ribiere parameter to guaranteethe objective function to be non-increasing in each iteration[47]. In addition, since Steps 7 and 8 involve the operationsbetween two vectors in different tangent spaces Txk

Mmcc and

Txk+1Mm

cc, which cannot be combined directly, a mappingbetween two tangent vectors in different tangent spaces calledtransport is introduced. The transport of a tangent vector dfrom xk to xk+1 can be specified as

Transpxk→xk+1:TxkMm

cc → Txk+1Mm

cc :

d 7→ d−ℜd x∗k+1 xk+1,

(16)

Algorithm 1 Conjugate Gradient Algorithm for Analog Pre-coding Based on Manifold OptimizationInput: Fopt,FBB,x0 ∈Mm

cc

1: d0 = −gradf(x0) and k = 0;2: repeat3: Choose Armijo backtracking line search step size αk;4: Find the next point xk+1 using retraction in (15):

xk+1 = Retrxk(αkdk);

5: Determine Riemannian gradient gk+1 = gradf(xk+1)according to (13) and (14);

6: Calculate the vector transports g+k and d+

k of gradientgk 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+1d+k ;

9: k ← k + 1;10: until a stopping criterion triggers.

which is accomplished in Step 6. According to Theorem4.3.1 in [41], Algorithm 1 is guaranteed to converge to acritical point, i.e., the point where the gradient of the objectivefunction is zero.

C. Hybrid Precoder Design

With Algorithm 1 at hand, the hybrid precoder design viaalternating minimization for the fully-connected structure isdescribed in the MO-AltMin Algorithm by solving problems(6) and (8) iteratively. To satisfy the power constraint in (5),we normalize FBB by a factor of

√Ns

∥FRFFBB∥Fat Step 7. The

following lemma help reveal the effect of this normalization.

MO-AltMin Algorithm: Manifold Optimization Based Hy-brid Precoding for the Fully-connected StructureInput: Fopt

1: Construct F(0)RF with random phases and set k = 0;

2: repeat3: Fix F

(k)RF, and F

(k)BB = F

(k)†RF Fopt;

4: Optimize F(k+1)RF using Algorithm 1 when F

(k)BB is fixed;

5: k ← k + 1;6: until a stopping criterion triggers;7: For the digital precoder at the transmit end, normalize

FBB =√Ns

∥FRFFBB∥FFBB.

Lemma 1. If the Euclidean distance before normalization is∥Fopt − FRFFBB∥F ≤ δ, then after normalization we have∥∥∥Fopt − FRFFBB

∥∥∥F≤ 2δ.

Proof: Define the normalization factor√Ns

∥FRFFBB∥F= 1

λ

and thus ∥FRFFBB∥F = λ√Ns = λ ∥Fopt∥F .

By norm inequality, we have

∥Fopt − FRFFBB∥F ≥ | ∥Fopt∥F − ∥FRFFBB∥F |= |1− λ| ∥Fopt∥F ,

(17)

which is equivalent to ∥Fopt∥F ≤1

|λ−1|δ.



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When λ = 1, which indicates ∥Fopt − FRFFBB∥F = 0,∥∥∥Fopt − FRFFBB

∥∥∥F

=

∥∥∥∥Fopt − FRFFBB +

(1− 1

λ

)FRFFBB

∥∥∥∥F

≤ ∥Fopt − FRFFBB∥F +

∣∣∣∣1− 1

λ

∣∣∣∣ ∥FRFFBB∥F

≤ δ + |λ− 1| ∥Fopt∥F ≤ δ +|λ− 1||λ− 1|

δ = 2δ.

(18)

Lemma 1 shows that as long as we can make the Euclideandistance between the optimal digital precoder and the hybridprecoders sufficiently small when ignoring the power con-straint in (5), the normalization step will also achieve a smalldistance to the optimal digital precoder.

Since the objective function in problem (5) is minimizedat Steps 3 and 4, each iteration will never increase it. Inaddition, the objective function is non-negative. These twoproperties together guarantee that the MO-AltMin algorithmcan converge to a feasible solution. Although the optimalityof alternating minimization algorithms for general non-convexproblems is still an open problem [48], simulation results inSection VII will show that the proposed algorithm can providenear-optimal performance.

However, the complexity of the MO-AltMin algorithm isrelatively high. In each iteration, the update of the analogprecoder involves a line search algorithm, i.e., Algorithm1, so the nested loops in the MO-AltMin algorithm willslow down the whole solving procedure. Furthermore, theKronecker products in (14) will result in two matrices ofdimension N t

RFNt×NsNt, which scales with the antenna sizeand results in an exponential increase of the computationalcomplexity in the MO-AltMin algorithm. Despite the highcomplexity, we note that the MO-AltMin algorithm basedon manifold optimization directly solves the hybrid precoderdesign problem (5) under unit modulus constraints, which willimprove the spectral efficiency when compared to existingalgorithms. Therefore, this algorithm can serve as a benchmarkof the performance in terms of spectral efficiency, and we willseek a low-complexity algorithm in the next section.

IV. LOW-COMPLEXITY HYBRID PRECODING FOR THEFULLY-CONNECTED STRUCTURE

Although the MO-AltMin algorithm can directly handle theunit modulus constraints, the number of such constraints maybe substantially large due to the large-size antenna array. Thus,the high computational complexity will prevent its practicalimplementation. It motivates us to develop a hybrid precodingalgorithm with lower computational complexity and slightperformance loss. In this section, by utilizing the orthogonalproperty of the digital precoder, we will propose a low-complexity design for the analog precoder subject to unitmodulus constraints. Thanks to the orthogonal property ofthe digital precoder, the phases of the analog precoder canbe extracted from the phases of an equivalent precoder deter-mined by the digital precoder and the unconstrained optimal

digital precoder. Though it will incur some performance losscompared to the manifold based algorithm, simulations willdemonstrate its performance gains over existing algorithms.

A. Digital Baseband Precoder Structure

Note that the columns of the unconstrained optimal precod-ing matrix Fopt are mutually orthogonal in order to mitigatethe interference between the multiplexed streams. Inspiredby this structure of the unconstrained precoding solution, weimpose a similar constraint that the columns of the digitalprecoding matrix should be mutually orthogonal, i.e.,

FHBBFBB = αFH

DDαFDD = α2INs , (19)

where FDD is a unitary matrix with the same dimensionas FBB. Although there is no existing conclusion on theoptimal structure of the digital precoder in hybrid precoding,it is natural and intriguing to investigate the hybrid precoderdesign under such an orthogonal constraint of the digitalprecoder. More importantly, this orthogonal constraint createsthe potential for the analog precoder FRF to get rid of theproduct form with FBB, which will help significantly simplifythe analog precoder design.

B. Hybrid precoder design

By replacing FBB with αFDD, the objective function in (5)can be further recast as

∥Fopt − FRFFBB∥2F= Tr

(FH

optFopt

)− Tr

(FH

optFRFFBB

)−Tr

(FH

BBFHRFFopt

)+Tr

(FH

BBFHRFFRFFBB

)= ∥Fopt∥2F − 2αℜTr

(FDDF

HoptFRF

)+α2 ∥FRFFDD∥2F .

(20)

Obviously, when α =ℜTr(FDDFH

optFRF)∥FRFFDD∥2

F

, the objective func-

tion ∥Fopt − FRFFBB∥2F in (20) has the minimum value,

given by ∥Fopt∥2F−ℜTr(FDDFH

optFRF)2∥FRFFDD∥2

F

. Note that the square

of the Frobenius norm ∥FRFFDD∥2F has the following upperbound

∥FRFFDD∥2F = Tr(FH

DDFHRFFRFFDD

)= Tr

(INs

0

)KHFH

RFFRFK

≤ Tr

KHFH

RFFRFK

= ∥FRF∥2F ,

(21)

where FDDFHDD = K

(INs

0

)KH is the SVD of

FDDFHDD and the equality holds when N t

RF = Ns, i.e., FDD

is a square matrix. Hence, the objective function in (5) is upper

bounded by ∥Fopt∥2F−ℜTr(FDDFH

optFRF)2∥FRF∥2

F

. In order to makeFRF get rid of the product with FBB, we choose to add theconstant term

(1

2∥FRF∥2F

− 1)∥Fopt∥2F + 1

2 to the bound and



8

multiply it by the positive constant term 2 ∥FRF∥2F . Then wehave

∥Fopt∥2F − 2ℜTr(FDDF

HoptFRF

)+ ∥FRF∥2F

= Tr(FH

RFFRF

)− 2ℜTr

(FDDF

HoptFRF

)+Tr

(FDDF

HoptFoptF

HDD

)=∥∥FoptF

HDD − FRF

∥∥2F.

(22)

Since directly optimizing the objective function (20) willstill incur high complexity, we intend to adopt the upper bound(22) as the objective function rather than the original one. Inaddition, as we can satisfy the transmit power constraint bynormalization after updating the hybrid precoders alternately,which has been shown in the MO-AltMin algorithm and Lem-ma 1, here we also temporarily remove the power constraint.Thus, by adopting (22) as the objective function, the hybridprecoder design problem is given as

minimizeFRF,FDD

∥∥FoptFHDD − FRF

∥∥2F

subject to

|(FRF)i,j | = 1, ∀i, jFH

DDFDD = INs .

(23)

The problem formulation (43) implies that we only needto seek a unitary precoding matrix FDD, and then a corre-sponding precoding matrix FBB with orthogonal columns canbe obtained. When applying alternating minimization, it turnsout that the objective function in (43) significantly simplifiesthe analog precoder design. More specifically, since the matrixFRF gets rid of the product form with FBB, it has a closed-form solution

arg (FRF) = arg(FoptF

HDD

), (24)

where arg(A) generates a matrix containing the phases of theentries of A. Thus, it shows that the phases of FRF can beextracted from the phases of an equivalent precoder FoptF

HDD.

This closed-form solution can also be viewed as the Euclideanprojection of FoptF

HDD on the feasible set Af of the analog

precoder.For the digital precoder design, regarding FRF as fixed, we

try to solve a digital precoder which optimizes the followingproblem

minimizeFDD

∥∥FoptFHDD − FRF

∥∥2F

subject to FHDDFDD = INs .

(25)

Since problem (25) only has one optimization variable FDD,it is equivalent to

maximizeFDD

ℜTr(FDDF

HoptFRF

)subject to FH

DDFDD = INs .(26)

According to the definition of the dual norm, we have

ℜTr(FDDF

HoptFRF

)≤∣∣Tr (FDDF

HoptFRF

)∣∣(a)

≤∥∥FH

DD

∥∥∞ ·∥∥FH

optFRF

∥∥1

=∥∥FH

optFRF

∥∥1

=

Ns∑i=1

σi,

(27)

where (a) follows the Holder’s inequality, ∥·∥∞ and ∥·∥1 standfor the infinite and one Schatten norms [49]. The equality isestablished only when

FDD = V1UH , (28)

where FHoptFRF = UΣVH = USVH

1 , which is the SVDof FH

optFRF, and S is a diagonal matrix whose elementsare the first Ns nonzero singular values σ1, · · · , σNs . Thisresult bears some similarity to the solution of the orthogonalProcrustes problem (OPP) [50], although the formulation isslightly different3.

PE-AltMin Algorithm: A Low-Complexity Algorithm forthe Fully-connected StructureInput: Fopt


2: repeat3: Fix F

(k)RF, compute the SVD: FH

optF(k)RF =

U(k)S(k)V(k)1

H;

4: F(k)DD = V

(k)1 U(k)H ;

5: Fix F(k)DD, and arg

F

(k+1)RF

= arg

(FoptF

(k)DD

H)

;

6: k ← k + 1;7: until a stopping criterion triggers;8: For the digital precoder at the transmit end, normalize

FBB =√Ns

∥FRFFDD∥FFDD.

Based on the two closed-form solutions for the analog anddigital precoders, we summarize our new design as the PE-AltMin Algorithm. There are several issues involved in thePE-AltMin algorithm that require some further remarks.

(1) Complexity: In both the MO-AltMin and PE-AltMinalgorithms, the updating rules of the digital precoders are givenby closed-form solutions and thus these two algorithms are ofcomparable complexity in the digital parts. Furthermore, in thehybrid precoding system, the dimension of the analog precoderis much higher than that of the digital precoder, which makesthe complexity of the algorithms predominated by the analogpart.

In each iteration of the MO-AltMin algorithm, a conju-gate gradient descent search is needed to update the analogprecoder. In particular, when updating the analog precoderin each iteration, we need to search on the complex circlemanifold repeatedly to find a local optimum with zero gradientof the cost function. Additionally, the computation of the large-size matrices, i.e., matrices of dimension N t

RFNt × NsNt,will be involved in the gradient descent procedure due tothe use of the Kronecker products. More importantly, theconjugate gradient descent, which is an iterative procedureitself, is nested into each alternating minimization iteration.This nested iteration structure will dramatically degrade thecomputational efficiency of the MO-AltMin algorithm. On thecontrary, in each iteration of the PE-AltMin algorithm, theupdate of the analog precoder is simply realized by a phase

3OPP tries to minimize ∥AΩ−B∥F , where the optimization variable Ωis a square unitary matrix.



9

extraction operation of the matrix FoptFHDD, whose dimension

is Nt×N tRF. Compared with the MO-AltMin algorithm, it is

safe to conclude that the PE-AltMin algorithm is with a muchlower complexity, which is also numerically observed in oursimulations.

(2) Accuracy of the approximation: In the formulation of(43), we try to minimize an upper bound given by (21)rather than directly minimizing the original objective function.Therefore, the effectiveness of this strategy depends on howtight the upper bound is when Ns ≤ N t

RF < 2Ns. Accordingto (21), we can quantify the gap between ∥FRFFDD∥2F and∥FRF∥2F as ∥FRFK1∥2F , where K1 is comprised of the right-most N t

RF −Ns columns of K. Note that when N tRF = Ns,

the upper bound is tight, i.e., the equality holds in (21).Furthermore, when increasing N t

RF from Ns to 2Ns − 1,the gap ∥FRFK1∥2F will become larger with increasing N t

RF,which will result in some performance loss. The impact ofadopting this upper bound as the objective will be shown inSection VII via simulations.

(3) Calculation of α: Although we have stated that a valueof α can be found to construct a matrix FBB corresponding toeach FDD, we do not need to actually calculate α in the PE-AltMin algorithm for the following reasons. At the transmitterside, even though we compute FBB = αFDD in the laststep of the PE-AltMin algorithm, this digital precoder shouldbe immediately normalized. Hence, the overall procedure isequivalent to directly normalizing FDD to satisfy the powerconstraint without knowing α. At the receiver side, we notethat the spectral efficiency (2) will not be influenced by theconstant factor α multiplied with WBB. That is because thedecoder WBB takes effects on both received signals and noise,and thus signal-to-noise ratio (SNR) will not change due tothe constant factor α. Moreover, the avoidance of calculating αwill further reduce the complexity of the PE-AltMin algorithm.

V. HYBRID PRECODING FOR THE PARTIALLY-CONNECTEDSTRUCTURE

Different from the fully-connected structure, the partially-connected structure shown in Fig. 1(c) [29], [33], [34], alsocalled the array of subarray structure, employs notably lessphase shifters and is advocated for energy-efficient mmWaveMIMO systems [31], [32]. Particularly, the output signal ofeach RF chain is only connected with Nt/N

tRF antennas,

which reduces the hardware complexity in the RF domain.Therefore, the analog precoder FRF in this structure belongsto a set of block matrices Ap, where each block is an Nt/N

tRF

dimension vector with unit modulus elements and the structureof FRF can be depicted as

FRF =

p1 0 · · · 00 p2 0...

. . ....

0 0 · · · pNtRF

, (29)

where pi =

[exp

(jθ

(i−1)Nt

NtRF

+1

), · · · , exp

(jθ

iNt

NtRF

)]Tand θi stands for the phase of the ith phase shifter. Inthis section, we will propose an AltMin algorithm for this

structure. Surprisingly, optimal solutions can be found for bothsubproblems of analog and digital precoders.

A. Analog RF Precoder Design

Due to the special structure of the constraint on FRF, in theproduct FRFFBB, each nonzero element of FRF is multipliedby a corresponding row extracted from FBB. Thus, the powerconstraint in (5) at the transmit side can be recast as

∥FRFFBB∥2F =Nt

N tRF

∥FBB∥2F = Ns. (30)

Therefore, the analog precoder design is formulated as

minimizeFRF

∥Fopt − FRFFBB∥2Fsubject to FRF ∈ Ap.

(31)

Also, due to the same property of FRF, problem (31) can bereformulated as

minimizeθiNt

i=1

∥∥(Fopt)i,: − ejθi(FBB)l,:∥∥22, (32)

where l =⌈iNt

RF

Nt

⌉. This is basically a vector approximation

problem using phase rotation, and there exists a closed-formexpression for nonzero elements in FRF, given by

arg (FRF)i,l = arg(Fopt)i,:(FBB)l,:

H,

1 ≤ i ≤ Nt, l =

⌈iN t

RF

Nt

⌉.

(33)

We note that the special characteristic of FRF simplifies theanalog precoder design and makes the unit modulus constraintno longer an intractable issue in the partially-connected struc-ture.

B. Digital Baseband Precoder Design

According to (30), the precoder design at the transmit sidecan be rewritten as the following problem

minimizeFBB

∥Fopt − FRFFBB∥2F

subject to ∥FBB∥2F =N t

RFNs

Nt.

(34)

Problem (34) is a non-convex quadratic constraint quadraticprogramming (QCQP) problem, which can be reformulated asa homogeneous QCQP problem:

minimizeY∈Hn

Tr(CY)

subject to

Tr(A1Y) =

NtRFNs

Nt

Tr(A2Y) = 1

Y ≽ 0, rank(Y) = 1,

(35)

with Hn being the set of n = N tRFNs+1 dimension complex

Hermitian matrices. In addition, y =[vec(FBB) t

]Twith

an auxiliary variable t, Y = yyH , f = vec(Fopt), and

A1 =

[In−1 00 0

],A2 =

[0n−1 00 1

],



10

C =

[(INs ⊗ FRF)

H(INs ⊗ FRF) −(INs ⊗ FRF)Hf

−fH(INs ⊗ FRF) fHf

].

The derivation and the formulation of the homogeneous QCQPproblem can be found in Appendix A.

In fact, the most difficult part in problem (35) is the rankconstraint, which is non-convex with respect to Y. Thus,we first drop it to obtain a relaxed version of (35), i.e., asemidefinite relaxation (SDR) problem as follows.

minimizeY∈Hn

Tr(CY)

subject to

Tr(A1Y) =

NtRFNs

Nt

Tr(A2Y) = 1

Y ≽ 0.

(36)

It has been established that the SDR is tight when the numberof constraints is less than three for a complex-valued ho-mogeneous QCQP problem [51]. Consequently, problem (36)without the rank-one constraint reduces into a semidefiniteprogramming (SDP) problem and it can be solved by standardconvex optimization algorithms [52], from which we canobtain the globally optimal solution of the digital precoderdesign problem (34). Therefore, a step-by-step summary isprovided below as the SDR-AltMin Algorithm.

SDR-AltMin Algorithm: SDR Based Hybrid Precoding forthe Partially-connected StructureInput: Fopt


2: repeat3: Fix F

(k)RF, solving F

(k)BB using SDR (36);

4: Fix F(k)BB, and update F

(k+1)RF by (33);

5: k ← k + 1;6: until a stopping criterion triggers.

C. Comparison Between Two Hybrid Precoding Structures

The main difference between two hybrid precoding struc-tures considered in this paper is the number of phase shiftersNPS in use for given numbers of data streams, RF chains, andantennas.

In terms of spectral efficiency, the fully-connected structureprovides more design degrees of freedom (DoFs) in the RFdomain and thus will outperform the partially-connected one.However, when taking power consumption into consideration,it is intriguing to know which structure has better energyefficiency. Energy efficiency is defined as the ratio betweenspectral efficiency and total power consumption

η =R

Pcommon +N tRFPRF +NtPPA +NPSPPS

, (37)

where the unit of η is bits/Hz/J and Pcommon is the commonpower of the transmitter. PRF, PPS, and PPA are the power ofeach RF chain, phase shifter, and power amplifier, respectively.The number of phase shifters NPS can be expressed as follows,

NPS =

NtN

tRF fully-connected

Nt partially-connected . (38)

The numerical comparison will be provided in the Section VII.

VI. HYBRID PRECODING IN MMWAVE MIMO-OFDMSYSTEMS

In previous sections, we designed hybrid precoders fornarrowband mmWave systems. On the other hand, the largeavailable bandwidth is one of the unique characteristics ofmmWave systems, and therefore the design of the hybridprecoders should be investigated when multicarrier techniquessuch as OFDM are utilized to overcome the multipath fading.In this section, we will extend the proposed AltMin algorithmsto mmWave MIMO-OFDM systems.

In conventional MIMO-OFDM systems with sub-6 GHzcarrier frequencies, digital precoding is performed in thefrequency domain for every subcarrier, which can also beadopted in mmWave MIMO-OFDM systems. Futhermore,the digital precoding is followed by an inverse fast Fouriertransform (IFFT) operation, which combines the signals of allthe subcarriers together. However, since the analog precodingis a post-IFFT processing, the signals of all the subcarriers canonly share one common analog precoder in mmWave MIMO-OFDM systems [22], [30]. Under this new restriction, thereceived signal of each subcarrier after the decoding processcan then be expressed as

y[k] =√ρWH

BB[k]WHRFH[k]FRFFBB[k]s+WH

BB[k]WHRFn,(39)

where k ∈ [0,K − 1] is the subcarrier index. H[k] is thefrequency domain channel matrix for the kth subcarrier, whichis given by [22]

H[k] = γ

Ncl−1∑i=0

Nray∑l=1

αilar(ϕril, θ

ril)at(ϕ

til, θ

til)

He−j2πik/K ,

(40)where γ =

√NtNr

NclNrayis the normalization factor and K is the

total number of subcarriers. Then the hybrid precoder designin mmWave MIMO-OFDM systems can be formulated as [22]

minimizeFRF,FBB[k]

K−1∑k=0

∥Fopt[k]− FRFFBB[k]∥2F

subject to

FRF ∈ A∥FRFFBB[k]∥2F = Ns,

(41)

where Fopt[k] is the optimal digital precoder for the kthsubcarrier. While it does not directly maximize the spectralefficiency, similar to the narrowband case, as shown in [13],[22], the objective is a good surrogate and it will make theproblem tractable.

The alternating minimization framework can be adopted tosolve problem (41). In particular, the digital precoders of allthe subcarriers can be updated in a parallel fashion, sincewe can get rid of the summation in (41) when optimizingthe digital precoder for each subcarrier. Hence, the solutionsfrom (7), (28) and (36) still hold for problem (41). Nextwe will focus on the analog precoder design, which is themain difference from narrowband systems. Then the proposedAltMin algorithms can be applied to the hybrid precoding inOFDM systems.

For the MO-AltMin algorithm, based on the principles ofmanifold optimization, as mentioned in Section III, we first



11

derive the Euclidean gradient of the objective function in (41)as

∇f(x) = −2K−1∑k=0

(F∗BB[k]⊗ INt)×[

vec(Fopt[k])− (FTBB[k]⊗ INt)x

].

(42)

After calculating this Euclidean gradient, we can still usethe projection (13) and the retraction (15) to obtain theRiemannian gradient and the mapped gradient vector on themanifold, which are key elements in Algorithm 1.

For the PE-AltMin algorithm, after we adopt a similarupper bound and manipulations in Section IV, the problemformulation for the analog precoder design is given as

minimizeFRF

K−1∑k=0

∥∥Fopt[k]FHDD[k]− FRF

∥∥2F

subject to |(FRF)i,j | = 1, ∀i, j,(43)

which has a closed-form solution

arg (FRF) = arg

(K−1∑k=0

Fopt[k]FHDD[k]

). (44)

By substituting Step 5 in the PE-AltMin algorithm, thissolution enables the extension of the PE-AltMin algorithm toOFDM systems.

Similar to the previous two AltMin Algorithms, the SDR-AltMin algorithm can also be realized in OFDM systems. Wecan get the closed-form solution for the analog precoder as

arg (FRF)i,l = arg

K−1∑k=0

(Fopt[k])i,:(FBB[k])l,:H

,

1 ≤ i ≤ Nt, l =

⌈iN t

RF

Nt

⌉,

(45)

which can be added in Step 4 in the SDR-AltMin algorithm.The aforementioned solutions demonstrate that the proposed

AltMin algorithms can be directly extended to mmWaveMIMO-OFDM systems, and the performance of the proposedAltMin algorithms will be evaluated in the next section.

VII. SIMULATION RESULTS

In this section, we will numerically evaluate the perfor-mance of our proposed algorithms. Data streams are sent froma transmitter with Nt = 144 to a receiver with Nr = 36antennas, while both are equipped with USPA. The channelparameters are set as Ncl = 5 clusters, Nray = 10 rays and theaverage power of each cluster is σ2

α,i = 1. The azimuth andelevation angles of departure and arrival (AoDs and AoAs)follow the Laplacian distribution with uniformly distributedmean angles and angular spread of 10 degrees. The antennaelements in the USPA are separated by a half wavelengthdistance and all simulation results are averaged over 1000channel realizations. For all the proposed AltMin algorithms,the initial phases of the analog precoder FRF follow a uniformdistribution over [0, 2π).

A. Spectral Efficiency Evaluation

Firstly, we investigate the spectral efficiency achieved bydifferent algorithms when the number of RF chains is equalto that of the data streams, i.e., N t

RF = NrRF = Ns. This

-35 -30 -25 -20 -15 -10 -5 0 5

SNR (dB)

0

5

10

15

20

25

30

Sp

ectr

al E

ffic

ien

cy (

bits/s

/Hz)

Optimal Digital Precoder

MO-AltMin

OMP Algorithm

SDR-AltMin

SIC-Based Method

Analog Beamforming

Fully-connected structure

Partially-connected structure

Fig. 3. Spectral efficiency achieved by different precoding algorithms whenNt

RF = NrRF = Ns = 3.

is the worst case since the number of RF chains cannotbe smaller under the assumptions in Section II-A. In thiscase, as shown in Fig. 3, for the fully-connected structure,the existing OMP algorithm [13] achieve significantly lowerspectral efficiency than the optimal digital precoder. On thecontrary, our proposed alternating minimization algorithm4,i.e., the MO-AltMin algorithm, achieves near-optimal perfor-mance over the whole SNR range in consideration. This meansthat the proposed algorithm can more accurately approxi-mate the optimal digital precoder than existing algorithms,even though the RF chains are limited. In this scenario, thepartially-connected structure with the proposed SDR-AltMinalgorithm provides substantial performance gains over analogbeamforming, especially at high SNRs. For comparison5, theSIC-Based method proposed in [31] is adopted as a benchmarkfor the partially-connected structure. Fig. 3 shows that theSDR-AltMin algorithm outperforms the benchmark. This ismainly because the SDR-AltMin algorithm takes full use ofthe digital precoder, while the SIC-Based method only uses thedigital precoder to allocate power to the data streams underthe extra diagonal constraint.

It has been shown in [26] that when N tRF ≥ 2Ns and

NrRF ≥ 2Ns, there exists a closed-form solution to the design

problem of the fully-connected hybrid precoding, which leadsto the same spectral efficiency provided by the optimal digitalprecoding. Although the hybrid precoder design in this paperaims at cases of Ns ≤ N t

RF < 2Ns, it is interesting to

4When optimizing the analog precoder, Algorithm 1 converges within 20iterations for almost all the channel realizations.

5Since the SIC-Based method in [31] can only design hybrid precoders atthe transmitter, we assume that the optimal digital decoder is adopted at thereceiver. Furthermore, the same setting is also employed for the SDR-AltMinalgorithm for fair comparison in Fig. 3. Nevertheless, in the remainder of thissection, both hybrid precoder and decoder are assumed at the transmitter andreceiver for the partially-connected structure.



12

examine if our proposed algorithm can achieve the comparableperformance as the case considered in [26]. Fig. 4 compares

2 2.5 3 3.5 4 4.5 5 5.5 6

NRF

13.5

14

14.5

15

15.5

16

16.5

17

17.5

18

18.5

19

Spectr

al E

ffic

iency (

bits/s

/Hz)


MO-AltMin

OMP Algorithm

SDR-AltMin

Fully-connected structure

Partially-connected structure

Fig. 4. Spectral efficiency achieved by different precoding algorithms givenNs = 2, N t

RF = NrRF = NRF and SNR = 0 dB.

the performance of different precoding schemes for differentNRF. We see that the proposed MO-AltMin algorithm for thefully-connected structure starts to coincide with the optimaldigital precoding when N t

RF = NrRF ≥ 4. This result

demonstrates that when N tRF ≥ 2Ns and Nr

RF ≥ 2Ns, ourproposed algorithm can actually achieve the optimal spectralefficiency, which, however, cannot be achieved by the OMPalgorithm. Furthermore, the comparison between two hybridprecoding structures shows that the partially-connected struc-ture, using less phase shifters, does entail some non-negligibleperformance loss when compared with the fully-connectedstructure.

B. Energy Efficiency Evaluation

In this part, we will compare the performance of the twohybrid precoding structures in terms of energy efficiency, asdefined in (37). The simulation parameters are set as follows:Pcommon = 10 W, PRF = 100 mW, PPS = 10 mW andPPA = 100 mW [12]. The simulation results are shownin Fig. 5, which shows substantially different behaviors forthe two structures. Since the number of phase shifters scaleslinearly with N t

RF and Nt in the fully-connected structure, thepower consumption will increase substantially when increasingN t

RF. As shown in Fig. 4, however, the spectral efficiencyachieved by the proposed MO-AltMin algorithm is sufficientlyclose or exactly equal to the optimal digital one, and will notincrease further as NRF increases. Based on these two facts,the power consumption grows much faster than the spectralefficiency, which gives rise to the dramatic decrease of theenergy efficiency.

For the partially-connected structure, as the number ofphase shifters is independent of N t

RF, the dominant partof total power consumption remains almost unchanged overthe investigated range of RF chain numbers. Meanwhile, thespectral efficiency will gradually approach the optimal digital

2 4 6 8 10 12 14 16 18

NRF

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Energ

y E

ffic

iency (

bits/H

z/J

)

Partially-connected

Fully-connected

Fig. 5. Energy efficiency of the fully-connected and partially-connectedstructures when Ns = 2, Nt


precoder when increasing N tRF. The improvement of the spec-

tral efficiency and the almost unchanged power consumptiontogether account for the rise in the energy efficiency whenN t

RF goes up in the partially-connected structure.More importantly, Fig. 5 shows that there is an intersection

point, i.e., when NRF = 5, of the energy efficiency for twohybrid precoding structures. In particular, the fully-connectedstructure enjoys higher energy efficiency with a small numberof RF chains, while the partially-connected one is moreenergy efficient when a relatively large number of RF chainsare implemented at transceivers. This phenomenon will offervaluable insights for the RF chain implementation in hybridprecoding. As shown in Fig. 4, the fully-connected structurecan approach the performance of the optimal digital precoderwhen the number of RF chains is slightly larger than thatof the data streams. Therefore, there is no need to implementmore RF chains considering the energy efficiency. On the otherhand, with a low-complexity hardware implementation, it isbeneficial for the partially-connected structure to leverage thelarger size of RF chains to improve both spectral and energyefficiency.

C. Low-Complexity Design for the Fully-connected Structure

As mentioned in Section IV-B, there are two intermediatesteps taken for developing the PE-AltMin algorithm, i.e.,constructing the digital precoder with orthogonal columns andreplacing the objective function with an upper bound. Firstly,we test the influence of the additional constraint on the digitalprecoder, for which we eliminate the impact of the second stepby setting the numbers of RF chains and data streams to beequal, i.e., to make the bound tight.

Fig. 6 plots the spectral efficiency achieved by both the MO-AltMin and PE-AltMin algorithms when there are 2, 4 and 8data streams transmitted. We see that the curves of the low-complexity PE-AltMin algorithm nearly coincide with thoseof the MO-AltMin algorithm. This phenomenon implies thatthe orthogonal column structure of the digital precoder hasnegligible impact on spectral efficiency, which justifies the



13

-35 -30 -25 -20 -15 -10 -5 0 5

SNR (dB)

0

10

20

30

40

50

60S

pe

ctr

al E

ffic

ien

cy (

bits/s

/Hz)


MO-AltMin

PE-AltMin

Ns=8

Ns=4

Ns=2

Fig. 6. Spectral efficiency achieved by the MO-AltMin and PE-AltMinalgorithms given Nt

RF = NrRF = NRF = Ns.

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11

NRF

32

33

34

35

36

37

38

39

40

Sp

ectr

al E

ffic

ien

cy (

bits/s

/Hz)


MO-AltMin

PE-AltMin

OMP Algorithm

Fig. 7. Spectral efficiency achieved by different precoding algorithms givenNs = 6, N t


rationality of the digital precoder design in Section IV-B. Fig.6 also indicates that we can achieve the performance of thehigh complexity MO-AltMin algorithm by adopting the low-complexity PE-AltMin algorithm when NRF = Ns. Under thissystem setting, the PE-AltMin algorithm serves as an excellentcandidate for the hybrid precoder design, achieving both goodperformance and low complexity. On the contrary, the OMPalgorithm works poorly when NRF = Ns.

Next we investigate the impact of the number of RF chains.Fig. 7 compares different algorithms assuming 6 data streamsare transmitted. Since the optimal solution in the N t

RF ≥ 2Ns

region has been fully developed in [26], here we focus on theremaining region, i.e., N t

RF = NrRF = NRF ∈ [6, 11]. From

Fig. 7, we find that the PE-AltMin algorithm has a small gapcompared with the MO-AltMin algorithm. That is because thelow-complexity algorithm tries to minimize an upper boundinstead of the original objective function. As explained inSection IV-B, the upper bound is tight when NRF = Ns andgets looser when NRF increases, which determines the gap

-15 -10 -5 0 5 10

SNR (dB)

10

15

20

25

30

35

40

45

Sp

ectr

al E

ffic

ien

cy (

bits/s

/Hz)


MO-AltMin

PE-AltMin

OMP Algorithm

Ns=4, N

RF=5

NRF

=Ns=2

Fig. 8. Spectral efficiency achieved by different precoding algorithms inmmWave MIMO-OFDM systems given Nt

RF = NrRF = NRF.

between the MO-AltMin and PE-AltMin algorithms. However,the spectral efficiency provided by the PE-AltMin algorithm isfar higher than the existing OMP algorithm, especially whenthe RF chain number is small. With the proposed AltMinalgorithms, we see that the performance of the fully-connectedstructure approaches that of the fully digital precoder when thenumber of RF chains is comparable with the number of datastreams, which cannot be revealed from the OMP algorithm.

D. Hybrid Precoding in MmWave MIMO-OFDM Systems

In this part, we will show the performance of the proposedAltMin algorithms when applied to mmWave MIMO-OFDMsystems. We assume that the number of subcarriers is K =128.

Fig. 8 plots the spectral efficiency achieved by the MO-AltMin and PE-AltMin algorithms compared to the OMP-based method proposed in [22]. It turns out that the MO-AltMin algorithm always has the highest spectral efficiencyunder different system parameters. With a large number of RFchains, similar to what we have observed in Fig. 7, the MO-AltMin algorithm will quickly get close to the performance ofthe optimal digital precoder. Interestingly, we discover that thelow-complexity PE-AltMin algorithm can achieve almost thesame spectral efficiency as that of the MO-AltMin algorithmwhen the numbers of RF chains and data streams are equal.This phenomenon is the same as that in narrowband systems,and it demonstrates that the extra orthogonality constraint onthe digital precoder also has negligible impact on spectralefficiency in mmWave OFDM systems. It is also found in Fig.8 that the PE-AltMin algorithm outperforms the OMP-basedmethod in mmWave OFDM systems. This indicates that thePE-AltMin algorithm can serve as an outstanding candidatefor the low-complexity hybrid precoding, both in narrowbandand broadband OFDM systems, when the transceivers onlyhave limited RF chains available.



14

VIII. CONCLUSIONS

Built on the principle of alternating minimization, in thispaper, we proposed an innovative design methodology forhybrid precoding in mmWave MIMO systems. Effective al-gorithms were proposed for the fully-connected and partially-connected hybrid precoding structures, and simulation resultshelped reveal the following valuable design insights:

• The hybrid precoders with the fully-connected structurecan approach the performance of the fully digital precoderwhen the number of RF chains is slightly larger thanthe number of data streams. Considering the increasingcost and power consumption, there is no need to furtherincrease the number of RF chains.

• For the partially-connected structure, in terms of spectralefficiency, hybrid precoders provide substantial gains overanalog beamforming. Furthermore, it is profitable toimplement a relatively large number of RF chains, inorder to enhance both spectral and energy efficiency.

Finally, our results have clearly demonstrated the effectivenessof alternating minimization in designing hybrid precoders inmmWave MIMO systems. It will be interesting to extend thealternating minimization techniques to other hybrid precoderdesign problems, as well as to consider the hybrid precoderdesign combined with channel training and feedback. Also afiner convergence analysis and optimality characterization ofthe proposed algorithms will require further investigation.

APPENDIX AFORMULATION OF THE HOMOGENEOUS QCQP PROBLEM

The original problem is a non-homogeneous QCQP problem

minimizeFBB

∥Fopt − FRFFBB∥2F

subject to ∥FBB∥2F =N t

RFNs

Nt.

(46)

According to the vectorization property, the objective functionin (46) can be rewritten as

∥Fopt − FRFFBB∥2F = ∥vec(Fopt − FRFFBB)∥22= ∥vec(Fopt)− vec(FRFFBB)∥22= ∥vec(Fopt)− (INs ⊗ FRF)vec(FBB)∥22 .

(47)

To simplify the notation, denote f = vec (Fopt), b =vec (FBB) and E = INs ⊗ FRF. In order to apply SDR, weintroduce an auxiliary variable t to homogenize the originalproblem as

minimizeb

∥tf −Eb∥22

subject to

∥b∥22 =

NtRFNs

Nt

t2 = 1.

(48)

The objective function in (48) can be further rewritten as

∥tf −Eb∥22 =[bH t

] [ EHE −EHf−fHE fHf

] [bt

].

(49)

Furthermore, the two constraints in (48) can also be manipu-lated as

∥b∥22 =[bH t

] [ INtRFNs

0

0 0

] [bt

]=

N tRFNs

Nt,

(50)

t2 =[bH t

] [ 0NtRFNs

0

0 1

] [bt

]= 1. (51)

Consequently, if we define

y =

[bt

],Y = yyH ,C =

[EHE −EHf−fHE fHf

],

A1 =

[INt

RFNs0

0 0

],A2 =

[0Nt

RFNs0

0 1

],

the original problem (46) can be reformulated as

minimizeY∈Hn

Tr(CY)

subject to

Tr(A1Y) =

NtRFNs

Nt

Tr(A2Y) = 1

Y ≽ 0, rank(Y) = 1.

(52)

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Xianghao Yu (S’15) received the B.Eng. degree inInformation Engineering from Southeast University(SEU), Nanjing, China, in 2014. He is currentlyworking towards the Ph.D. degree in Electronic andComputer Engineering at the Hong Kong Universityof Science and Technology (HKUST), under thesupervision of Prof. Khaled B. Letaief. His researchinterests include millimeter wave communications,MIMO systems, mathematical optimization, and s-tochastic geometry.

Juei-Chin Shen (S’10–M’14) received the Ph.D.degree in communication engineering from Univer-sity of Manchester, United Kingdom, in 2013. Hewas with the Hong Kong University of Scienceand Technology (HKUST) from 2013 to 2015 as aResearch Associate. Since 2015, he has been withMediatek Inc. as a Senior Engineer, working onthe research and standardization of future wirelesstechnologies. He is a co-recipient of IEEE PIMRC2014 Best Paper Award. His research interests in-clude compressed sensing, massive MIMO systems,

and millimeter-wave communications.



16

Jun Zhang (S’06–M’10–SM’15) received theB.Eng. degree in Electronic Engineering from theUniversity of Science and Technology of China in2004, the M.Phil. degree in Information Engineeringfrom the Chinese University of Hong Kong in 2006,and the Ph.D. degree in Electrical and Computer En-gineering from the University of Texas at Austin in2009. He is currently a Research Assistant Professorin the Department of Electronic and Computer Engi-neering at the Hong Kong University of Science andTechnology (HKUST). Dr. Zhang co-authored the

book Fundamentals of LTE (Prentice-Hall, 2010). He received the 2014 BestPaper Award for the EURASIP Journal on Advances in Signal Processing, andthe PIMRC 2014 Best Paper Award. He is an Editor of IEEE Transactionson Wireless Communications, and served as a MAC track co-chair for IEEEWCNC 2011. His research interests include wireless communications andnetworking, green communications, and signal processing.

Khaled B. Letaief (S’85–M’86–SM’97–F’03) re-ceived the B.S. degree with distinction in ElectricalEngineering (1984) from Purdue University, USA.He has also received the M.S. and Ph.D. Degreesin Electrical Engineering from Purdue University in1986 and 1990, respectively.

From 1990 to 1993, he was a faculty memberat the University of Melbourne, Australia. He hasbeen with the Hong Kong University of Science &Technology. While at HKUST, he has held numerousadministrative positions, including the Head of the

Department of Electronic and Computer Engineering, Director of the Centerfor Wireless IC Design, Director of Huawei Innovation Laboratory, andDirector of the Hong Kong Telecom Institute of Information Technology.He has also served as Chair Professor and Dean of HKUST School ofEngineering. Under his leadership, the School of Engineering has dazzledin international rankings (ranked #14 in the world in 2015 according to QSWorld University Rankings).

From September 2015, he joined Hamad Bin Khalifa University (HBKU) asProvost to help establish a research-intensive university in Qatar in partnershipwith strategic partners that include Northwestern University, Carnegie MellonUniversity, Cornell, and Texas A&M.

Dr. Letaief is a world-renowned leader in wireless communications andnetworks. In these areas, he has over 500 journal and conference papers andgiven invited keynote talks as well as courses all over the world. He has made6 major contributions to IEEE Standards along with 13 patents including 11US patents.

He served as consultants for different organizations and is the foundingEditor-in-Chief of the prestigious IEEE Transactions on Wireless Communi-cations. He has served on the editorial board of other prestigious journalsincluding the IEEE Journal on Selected Areas in Communications – WirelessSeries (as Editor-in-Chief). He has been involved in organizing a number ofmajor international conferences.

Professor Letaief has been a dedicated educator committed to excellencein teaching and scholarship. He received the Mangoon Teaching Award fromPurdue University in 1990; HKUST Engineering Teaching Excellence Award(4 times); and the Michael Gale Medal for Distinguished Teaching (Highestuniversity-wide teaching award at HKUST).

He is also the recipient of many other distinguished awards including2007 IEEE Joseph LoCicero Publications Exemplary Award; 2009 IEEEMarconi Prize Award in Wireless Communications; 2010 Purdue UniversityOutstanding Electrical and Computer Engineer Award; 2011 IEEE HaroldSobol Award; 2011 IEEE Wireless Communications Technical CommitteeRecognition Award; and 11 IEEE Best Paper Awards.

He had the privilege to serve IEEE in many leadership positions includingIEEE ComSoc Vice-President, IEEE ComSoc Director of Journals, andmember of IEEE Publications Services and Products Board, IEEE ComSocBoard of Governors, IEEE TAB Periodicals Committee, and IEEE FellowCommittee.

Dr. Letaief is a Fellow of IEEE and a Fellow of HKIE. He is also recognizedby Thomson Reuters as an ISI Highly Cited Researcher.

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