Hybrid Digital and Analog Beamforming Design for Large-Scale Antenna Arrays

3.1 MB·

by  Foad Sohrabi

1
Hybrid Digital and Analog Beamforming Design
for Large-Scale Antenna Arrays
Foad Sohrabi, Student Member, IEEE, and Wei Yu, Fellow, IEEE
Abstract—Thepotentialofusingofmillimeterwave(mmWave) more antennas to be packed in the same physical dimension,
frequencyforfuturewirelesscellularcommunicationsystemshas which allows for large-scale spatial multiplexing and highly
motivated the study of large-scale antenna arrays for achieving
directional beamforming. This leads to the advent of large-
highly directional beamforming. However, the conventional fully
scale or massive multiple-input multiple-output (MIMO) con-
digitalbeamformingmethodswhichrequireoneradiofrequency
(RF) chain per antenna element is not viable for large-scale ceptformmWavecommunications.Althoughtheprinciplesof
6 antennaarraysduetothehighcostandhighpowerconsumption the beamforming are the same regardless of carrier frequency,
1 of RF chain components in high frequencies. To address the itisnotpracticaltouseconventionalfullydigitalbeamforming
0 challenge of this hardware limitation, this paper considers a hy-
schemes[5]–[9]forlarge-scaleantennaarrays.Thisisbecause
2 bridbeamformingarchitectureinwhichtheoverallbeamformer
the implementation of fully digital beamforming requires one
consists of a low-dimensional digital beamformer followed by
n
an RF beamformer implemented using analog phase shifters. dedicated radio frequency (RF) chain per antenna element,
a
J Our aim is to show that such an architecture can approach which is prohibitive from both cost and power consumption
the performance of a fully digital scheme with much fewer perspectives at mmWave frequencies [10].
5
number of RF chains. Specifically, this paper establishes that
2 To address the difficulty of limited number of RF chains,
if the number of RF chains is twice the total number of data
thispaperconsidersatwo-stagehybridbeamformingarchitec-
streams, the hybrid beamforming structure can realize any fully
T] digitalbeamformerexactly,regardlessofthenumberofantenna ture in which the beamformer is constructed by concatenation
elements. For cases with fewer number of RF chains, this paper of a low-dimensional digital (baseband) beamformer and an
I
. further considers the hybrid beamforming design problem for RF (analog) beamformer implemented using phase shifters.
s
both the transmission scenario of a point-to-point multiple-
c In the first part of this paper, we show that the number of
input multiple-output (MIMO) system and a downlink multi-
[ RF chains in the hybrid beamforming architecture only needs
user multiple-input single-output (MU-MISO) system. For each
1 scenario,weproposeaheuristichybridbeamformingdesignthat to scale as twice the total number of data streams for it to
v achieves a performance close to the performance of the fully achievetheexactsameperformanceasthatofanyfullydigital
4 digital beamforming baseline. Finally, the proposed algorithms beamforming scheme regardless of the number of antenna
1 are modified for the more practical setting in which only finite
elements in the system.
8 resolution phase shifters are available. Numerical simulations
The second part of this paper considers the hybrid beam-
6 show that the proposed schemes are effective even when phase
0 shifters with very low resolution are used. forming design problem when the number of RF chains is
. less than twice the number of data streams for two specific
1 Index Terms—Millimeter wave, large-scale antenna arrays,
scenarios: (i) the point-to-point multiple-input multiple-output
0 multiple-input multiple-output (MIMO), multi-user multiple-
6 input single-output (MU-MISO), massive MIMO, linear beam- (MIMO) communication scenario with large-scale antenna
1 forming, precoding, combining, finite resolution phase shifters. arrays at both ends; (ii) the downlink multi-user multiple-
: inputsingle-output(MU-MISO)communicationscenariowith
v
i large-scale antenna array at the base station (BS), but sin-
X
I. INTRODUCTION gle antenna at each user. For both scenarios, we propose
ar Millimeter wave (mmWave) technology is one of the heuristic algorithms to design the hybrid beamformers for
promising candidates for future generation wireless cellular the problem of overall spectral efficiency maximization under
communication systems to address the current challenge of total power constraint at the transmitter, assuming perfect and
bandwidth shortage [1]–[3]. The mmWave signals experience instantaneous channel state information (CSI) at the BS and
severe path loss, penetration loss and rain fading as compared all user terminals. The numerical results suggest that hybrid
to signals in current cellular band (3G or LTE) [4]. However, beamforming can achieve spectral efficiency close to that
the shorter wavelength at mmWave frequencies also enables of the fully digital solution with the number of RF chains
approximately equal to the number of data streams. Finally,
ManuscriptacceptedandtoappearinIEEEJournalofSelectedTopicsin we present a modification of the proposed algorithms for the
Signal Processing, 2016. This work was supported by the Natural Sciences
more practical scenario in which only finite resolution phase
andEngineeringResearchCouncil(NSERC)ofCanada,byOntarioCentres
ofExcellence(OCE)andbyBLiNQNetworksInc.Thematerialsinthispaper shifters are available to construct the RF beamformers.
have been presented in part at IEEE International Conference on Acoustics, It should be emphasized that the availability of perfect CSI
Speech and Signal Processing (ICASSP), Brisbane, Australia, April 2015,
is an idealistic assumption which rarely occurs in practice,
and in part at IEEE International Workshop on Signal Processing Advances
inWirelessCommunications(SPAWC),Stockholm,Sweden,June2015. especially for systems implementing large-scale antenna ar-
The authors are with The Edward S. Rogers Sr. Department of rays. However, the algorithms proposed in the paper are still
Electrical and Computer Engineering, University of Toronto, 10 King’s
useful as a reference point for studying the performance of
College Road, Toronto, Ontario M5S 3G4, Canada (e-mails: {fsohrabi,
weiyu}@comm.utoronto.ca). hybrid beamforming architecture in comparison with fully
2
digital beamforming. Moreover, for imperfect CSI scenario, to the number of data streams; it is also compatible with any
one way to design the hybrid beamformers is to first design channel model.
theRFbeamformersassumingperfectCSI,andthentodesign For the downlink of K-user MISO systems, it is shown in
the digital beamformers employing robust beamforming tech- [32], [33] that hybrid beamforming with K RF chains at the
niques [11]–[15] to deal with imperfect CSI. It is therefore base station can achieve a reasonable sum rate as compared
still of interest to study the RF beamformer design problem tothesumrateoffullydigitalzero-forcing(ZF)beamforming
in perfect CSI. which is near optimal for massive MIMO systems [36]. The
To address the challenge of limited number of RF chains, design of [32], [33] involves matching the RF precoder to the
different architectures are studied extensively in the litera- phase of the channel and setting the digital precoder to be the
ture. Analog or RF beamforming schemes implemented using ZF beamformer for the effective channel. However, there is
analog circuitry are introduced in [16]–[19]. They typically stillagapbetweentherateachievedwiththisparticularhybrid
use analog phase shifters, which impose a constant modulus design and the maximum capacity. This paper proposes a
constraint on the elements of the beamformer. This causes methodtodesignhybridprecodersforthecasethatthenumber
analog beamforming to have poor performance as compared ofRFchainsisslightlygreaterthanK andnumericallyshows
tothefullydigitalbeamformingdesigns.Anotherapproachfor that the proposed design can be used to reduce the gap to
limiting the number of RF chains is antenna subset selection capacity.
whichisimplementedusingsimpleanalogswitches[20]–[22]. The aforementioned existing hybrid beamforming designs
However, they cannot achieve full diversity gain in correlated typicallyassumetheuseofinfiniteresolutionphaseshiftersfor
channels since only a subset of channels are used in the implementing analog beamformers. However, the components
antenna selection scheme [23], [24]. requiredforrealizingaccuratephaseshifterscanbeexpensive
In this paper, we consider the alternative architecture of [37],[38].Morecosteffectivelowresolutionphaseshiftersare
hybrid digital and analog beamforming which has received typically used in practice. The straightforward way to design
significantinterestinrecentworkonlarge-scaleantennaarray beamformers with finite resolution phase shifters is to design
systems[25]–[35].Theideaofhybridbeamformingisfirstin- the RF beamformer assuming infinite resolution first, then to
troduced under the name of antenna soft selection for a point- quantize the value of each phase shifter to a finite set [33].
to-point MIMO scenario [25], [26]. It is shown in [25] that However, this approach is not effective for systems with very
for a point-to-point MIMO system with diversity transmission lowresolutionphaseshifters[34].Inthelastpartofthispaper,
(i.e., the number of data stream is one), hybrid beamforming we present a modification to our proposed method for point-
can realize the optimal fully digital beamformer if and only if to-point MIMO scenario and multi-user MISO scenario when
the number of RF chains at each end is at least two. This only finite resolution phase shifters are available. Numerical
paper generalizes the above result for spatial multiplexing results in the simulations section show that the proposed
transmission for multi-user MIMO systems. In particular, method is effective even for the very low resolution phase
we show that hybrid structure can realize any fully digital shifter scenario.
beamformer if the number of RF chains is twice the number This paper uses capital bold face letters for matrices, small
of data streams. We note that the recent work of [35] also bold face for vectors, and small normal face for scalars. The
addressedthequestionofhowmanyRFchainsareneededfor real part and the imaginary part of a complex scalar s are
hybrid beamforming structure to realize digital beamforming denoted by Re{s} and Im{s}, respectively. For a column
infrequencyselectivechannels.But,thearchitectureofhybrid vector v, the element in the ith row is denoted by v(i) while
beamformingdesignusedin[35]isslightlydifferentfromthe for a matrix M, the element in the ith row and the jth column
conventional hybrid beamforming structure in [25]–[34]. is denoted by M(i,j). Further, we use the superscript H to
The idea of antenna soft selection is reintroduced under denote the Hermitian transpose of a matrix and superscript
the name of hybrid beamforming for mmWave frequencies ∗ to denote the complex conjugate. The identity matrix with
[27]–[29].Forapoint-to-pointlarge-scaleMIMOsystem,[27] appropriate dimensions is denoted by I; Cm×n denotes an
proposesanalgorithmbasedonthesparsenatureofmmWave m by n dimensional complex space; CN(0,R) represents
channels.Itisshownthatthespectralefficiencymaximization the zero-mean complex Gaussian distribution with covariance
problem for mmWave channels can be approximately solved matrixR.Further,thenotationsTr(·),log(·)andE[·]represent
by minimizing the Frobenius norm of the difference between the trace, logarithmic and expectation operators, respectively;
the optimal fully digital beamformer and the overall hybrid | · | represent determinant or absolute value depending on
beamformer. Using a compressed sensing algorithm called context. Finally, ∂f is used to denote the partial derivative
∂x
basis pursuit, [27] is able to design the hybrid beamformers of the function f with respect to x.
which achieve good performance when (i) extremely large
number of antennas is used at both ends; (ii) the number of
II. SYSTEMMODEL
RF chains is strictly greater than the number of data streams;
(iii) extremely correlated channel matrix is assumed. But in Consider a narrowband downlink single-cell multi-user
other cases, there is a significant gap between the theoretical MIMO system in which a BS with N antennas and NRF
t
maximum capacity and the achievable rate of the algorithm transmitRFchainsservesK users,eachequippedwithM an-
of [27]. This paper devises a heuristic algorithm that reduces tennas and NRF receive RF chains. Further, it is assumed that
r
this gap for the case that the number of RF chains is equal each user requires d data streams and that Kd ≤ NRF ≤ N
t
3
User 1
WRF1
Analog Precoder VRF
... ...
H M... ...NrRF WD1 ...d
d... Digital CRhaFin ... ... x(1) 1 y...1 ... y˜1
s1
Ns... Precoder NtRF ... ...N ... Use...r K
WRFK
d
... VD RF
s sK Chain ... ... xx(N) HK M......... ...NrRF WDK ...d
... ... y˜K
yK
Fig.1. Blockdiagramofamulti-userMIMOsystemwithhybridbeamformingarchitectureattheBSandtheuserterminals.
and d ≤ NrRF ≤ M. Since the number of transmit/receive low-dimensional digital combiner, WDk ∈ CNrRF×d, the final
RF chains is limited, the implementation of fully digital processed signals are obtained as
beamforming which requires one dedicated RF chain per (cid:88)
y˜ =WHH V s +WHH V s + WHz , (3)
antenna element, is not possible. Instead, we consider a two- k tk k tk k tk k t(cid:96) (cid:96) tk k
stage hybrid digital and analog beamforming architecture at (cid:124) (cid:123)(cid:122) (cid:125) (cid:96)(cid:54)=k (cid:124) (cid:123)(cid:122) (cid:125)
desiredsignals (cid:124) (cid:123)(cid:122) (cid:125) effectivenoise
the BS and the user terminals as shown in Fig. 1. effectiveinterference
In hybrid beamforming structure, the BS first modifies where V = V V and W = W W . In such a
the data streams digitally at baseband using an NtRF × Ns system,thtkeoveralRlFspeDcktralefficietkncy(rateR)FokfusDekrkassuming
digital precoder, VD, where Ns = Kd, then up-converts the Gaussian signalling is [39]
processed signals to the carrier frequency by passing through
(cid:12) (cid:12)
NRF RF chains. After that, the BS uses an N × NRF RF R =log (cid:12)I +W C−1WHH V VHHH(cid:12), (4)
t t k 2(cid:12) M tk k tk k tk tk k (cid:12)
precoder, V , which is implemented using analog phase
shifters, i.e.,RFwith |VRF(i,j)|2 = 1, to construct the final where Ck = WtHkHk(cid:0)(cid:80)(cid:96)(cid:54)=kVt(cid:96)VtH(cid:96)(cid:1)HHk Wtk +σ2WtHkWtk
transmitted signal. Mathematically, the transmitted signal can is the covariance of the interference plus noise at user k. The
be written as problem of interest in this paper is to maximize the overall
spectral efficiency under total transmit power constraint, as-
K suming perfect knowledge of Hk, i.e., we aim to find the
(cid:88)
x=V V s= V V s , (1) optimal hybrid precoders at the BS and the optimal hybrid
RF D RF D(cid:96) (cid:96)
combiners for each user by solving the following problem:
(cid:96)=1
K
where VD = [VD1,...,VDK], and s ∈ CNs×1 is the vector maximize (cid:88)βkRk (5a)
of data symbols which is the concatenation of each user’s VRF,VDWRF,WD k=1
data stream vector such as s = [sT1,...,sTK]T, where s(cid:96) is subject to Tr(VRFVDVDHVRHF)≤P (5b)
the data stream vector for user (cid:96). Further, it is assumed that
|V (i,j)|2 =1, ∀i,j (5c)
E[ssH]=I .Foruserk,thereceivedsignalcanbemodeled RF
Ns |W (i,j)|2 =1, ∀i,j,k, (5d)
as RFk
where P is the total power budget at the BS and the weight
yk =HkVRFVDksk+Hk(cid:88)(cid:96)(cid:54)=kVRFVD(cid:96)s(cid:96)+zk, (2) iβmkprleiepsregsreenattserthperioprriitoyriftoyroufseursekr. k; i.e., the larger (cid:80)K(cid:96)β=k1β(cid:96)
The system model in this section is described for a general
where H ∈ CM×N is the matrix of complex channel gains setting. In the next section, we characterize the minimum
k
from the transmit antennas of the BS to the kth user antennas number of RF chains in hybrid beamforming architecture for
and z ∼ CN(0,σ2I ) denotes additive white Gaussian realizing a fully digital beamformer for the general system
k M
noise. The user k first processes the received signals using model.Thesubsequentpartsofthepaperfocusontwospecific
an M ×NRF RF combiner, W , implemented using phase scenarios:
r RFk
shifters such that |W (i,j)|2 = 1, then down-converts the 1) Point-to-point MIMO system with large antenna arrays
RFk
signals to the baseband using NRF RF chains. Finally, using a at both ends, i.e., K =1 and min(N,M)(cid:29)N .
r s
4
(cid:18) (cid:19)
ν
2) Downlinkmulti-userMISOsystemwithlargenumberof θ =φ +cos−1 i,k . (7)
antennas at the BS and single antenna at the user side, i,2k i,k 2ν(k)
max
i.e., N (cid:29)K and M =1. Thus for the case that NRF = 2N , a solution to V V =
s RF D
V can be readily found. The validity of the proposition for
FD
III. MINIMUMNUMBEROFRFCHAINSTOREALIZE NRF >2N is obvious since we can use the same parameters
s
FULLYDIGITALBEAMFORMERS as for NRF =2N by setting the extra parameters to be zero
s
Thefirstpartofthispaperestablishestheoreticalboundson in V .
D
the minimum number of RF chains that are required for the Remark 1: The solution given in Proposition 2 is one
hybrid beamforming structure to be able to realize any fully possiblesetofsolutionstotheequationsin(6).Theinteresting
digital beamforming schemes. Recall that without the hybrid propertyofthatspecificsolutionisthatastwodigitalgainsof
structure constraints, fully digital beamforming schemes can each data stream are identical; i.e., v =v , it is possible
2k−1 2k
be easily designed with NRF = N RF chains at the BS and to convert one realization of the scaled data symbol to RF
t
NRF =M RFchainsattheuserside[5]–[9].Thissectionaims signalandthenuseittwice.Therefore,itisinfactpossibleto
r
toshowthathybridbeamformingarchitecturecanrealizefully realizeanyfullydigitalbeamformerusingthehybridstructure
digital beamforming schemes with potentially smaller number with N RF chains and 2N N phase shifters. This leads us
s s
of RF chains. We begin by presenting a necessary condition to the similar result (but with different design) as in [35]
on the number of RF chains for implementing a fully digital which considers hybrid beamforming for frequency selective
beamformer, VFD ∈CN×Ns. channels. However, in the rest of this paper, we consider
Proposition1:Torealizeafullydigitalbeamformingmatrix, the conventional configuration of hybrid structure in which
it is necessary that the number of RF chains in the hybrid the number of phase shifters are NRFN. We show that near
architecture (shown in Fig. 1) is greater than or equal to the optimal performance can be obtained with NRF ≈ N , thus
s
number of active data streams, i.e., NRF ≥Ns. further reducing the number of phase shifters as compared to
Proof: It is easy to see that rank(VRFVD) ≤ NRF and the solution above.
rank(VFD) = Ns. Therefore, hybrid beamforming structure Remark 2: Proposition 2 is stated for the case that VFD
requires at least NRF ≥Ns RF chains to implement VFD. is a full-rank matrix, i.e., rank(VFD) = Ns. In the case
We now address how many RF chains are sufficient in the that V is a rank-deficient matrix (which is a common
FD
hybrid structure for implementing any fully digital VFD ∈ scenario in the low signal-to-noise-ratio (SNR) regime), it
CN×Ns. It is already known that for the case of Ns = 1, can always be decomposed as VFD = AN×rBr×Ns where
the hybrid beamforming structure can realize any fully digital r = rank(V ). Since A is a full-rank matrix, it can be
FD
beamformer if and only if NRF ≥ 2 [25]. Proposition 2 realized using the procedure in the proof of Proposition 2
generalizes this result for any arbitrary value of Ns. as A = VRFVD(cid:48) with hybrid structure using 2r RF chains.
Proposition 2: To realize any fully digital beamforming Therefore, V = V (V(cid:48) B) can be realized by hybrid
FD RF D
matrix, it is sufficient that the number of RF chains in hybrid structure using 2r RF chains with V as RF beamformer
RF
architecture(showninFig.1)isgreaterthanorequaltotwice and V(cid:48) B as digital beamformer.
D
the number of data streams, i.e., NRF ≥2N .
s
Proof:LetNRF =2Ns anddenoteVFD(i,j)=νi,jejφi,j IV. HYBRIDBEAMFORMINGDESIGNFORSINGLE-USER
and VRF(i,j) = ejθi,j. We propose the following solution to LARGE-SCALEMIMOSYSTEMS
satisfy V V = V . Choose the kth column of the digital
RF D FD Thesecondpartofthispaperconsidersthedesignofhybrid
precoder as v(k) = [0T v v 0T]T. Then, satisfying
D 2k−1 2k beamformers. We first consider a point-to-point large-scale
V V =V is equivalent to
RF D FD MIMO system in which a BS with N antennas sends N data
s
 
0 symbolstoauserwithM antennaswheremin(N,M)(cid:29)N .
s
 ..  Without loss of generality, we assume identical number of
(cid:2) ... ejθi,2k−1 ejθi,2k ... (cid:3) v2vk2.−k1 =νi,jejφi,j, tstrhiamenpesxmlipfiyrte/rtsehsceioeninvooetaftRtihoFen.scpFheoacritnrsasu,lcehif.fiea.c,siyeNnstctReymFinw=(i4th)NchraRynFbbri=edssiNtmrupRclFti,ufireteod,
 ...  to
0 (cid:12) 1 (cid:12)
R=log (cid:12)I + W(WHW)−1WHHVVHHH(cid:12). (8)
or 2(cid:12) M σ2 t t t t t t (cid:12)
v2k−1ejθi,2k−1 +v2kejθi,2k =νi,kejφi,k, (6) where Vt =VRFVD and Wt =WRFWD.
Inthissection,wefirstfocusonhybridbeamformingdesign
for all i = 1,...,N and k = 1,...,N . This non-linear
s for the case that the number of RF chains is equal to the
system of equations has multiple solutions [25]. If we further
number of data streams; i.e., NRF = N . This critical case
choose v = v = ν(k) where ν(k) = max{ν }, it can s
2k−1 2k max max i,k is important because according to Proposition 1, the hybrid
i
be verified after several algebraic steps that the following is a
structurerequiresatleastN RFchainstobeabletorealizethe
s
solution to (6):
fullydigitalbeamformer.Forthiscase,weproposeaheuristic
(cid:18) (cid:19)
ν algorithmthatachievesrateclosetocapacity.Attheendofthis
θ =φ −cos−1 i,k ,
i,2k−1 i,k 2ν(k) section,weshowthatbyfurtherapproximations,theproposed
max
5
hybrid beamforming design algorithm for NRF =N , can be where γ2 =P/(NNRF). Since U is a unitary matrix for the
s e
used for the case of N <NRF <2N as well. case that NRF =N , we have V VH ≈γ2I.
s s s D D
The problem of rate maximization in (5) involves joint
optimization over the hybrid precoders and combiners. How- B. RF Precoder Design for NRF =N
s
ever, the joint transmitter-receive matrix design, for similarly
Now, we seek to design the RF precoder assuming
constrained optimization problem is usually found to be diffi-
V VH ≈ γ2I. Under this assumption, the transmitter power
cult to solve [40]. Further, the non-convex constraints on the D D
constraint(9b)isautomaticallysatisfiedforanydesignofV .
elements of the analog beamformers in (5c) and (5d) make RF
Therefore, the RF precoder can be obtained by solving
developing low-complexity algorithm for finding the exact
optimal solution unlikely [27]. So, this paper considers the (cid:12) γ2 (cid:12)
max log (cid:12)I+ VHF V (cid:12) (12a)
following strategy instead. First, we seek to design the hybrid VRF 2(cid:12) σ2 RF 1 RF(cid:12)
precoders, assuming that the optimal receiver is used. Then, s.t. |V (i,j)|2 =1, ∀i,j, (12b)
RF
for the already designed transmitter, we seek to design the
hybrid combiner. whereF1 =HHH.Thisproblemisstillnon-convex,sincethe
The hybrid precoder design problem can be further divided objectivefunctionof(12)isnotconcaveinVRF.However,the
into two steps as follows. The transmitter design problem can decouplednatureoftheconstraintsinthisformulationenables
be written as us to devise an iterative coordinate descent algorithm over the
(cid:12) 1 (cid:12) elements of the RF precoder.
VmRFa,VxD log2(cid:12)(cid:12)IM + σ2HVRFVDVDHVRHFHH(cid:12)(cid:12) (9a) In order to extract the contribution of VRF(i,j) to the
objective function of (12), it is shown in [34], [41] that the
s.t. Tr(V V VHVH)≤P, (9b)
RF D D RF objective function in (12) can be rewritten as
|V (i,j)|2 =1, ∀i,j. (9c)
RF log2(cid:12)(cid:12)Cj(cid:12)(cid:12)+log2(cid:0)2Re(cid:8)VR∗F(i,j)ηij(cid:9)+ζij +1(cid:1), (13)
Thisproblemisnon-convex.Thispaperproposesthefollowing
heuristic algorithm for obtaining a good solution to (9). First, where
we derive the closed-form solution of the digital precoder in γ2
C =I+ (V¯j )HF V¯j ,
problem (9) for a fixed RF precoder, VRF. It is shown that j σ2 RF 1 RF
regardless of the value of V , the digital precoder typically
RF and V¯j is the sub-matrix of V with jth column removed,
satisfies V VH ∝I. Then, assuming such a digital precoder, RF RF
D D
we propose an iterative algorithm to find a local optimal RF (cid:88)
η = G (i,(cid:96))V ((cid:96),j),
ij j RF
precoder.
(cid:96)(cid:54)=i
ζ = G (i,i)
ij j
A. Digital Precoder Design for NRF =N  
s
 (cid:88) 
The first part of the algorithm considers the design of V +2Re V∗ (m,j)G (m,n)V (n,j) ,
D RF j RF
assuming that VRF is fixed. For a fixed RF precoder, Heff = m(cid:54)=i,n(cid:54)=i 
HV canbeconsideredasaneffectivechannelandthedigital
precoRdFer design problem can be written as and Gj = σγ22F1 − σγ44F1V¯RjFC−j1(V¯RjF)HF1. Since Cj, ζij
and η are independent of V (i,j), if we assume that all
mVaDx log2(cid:12)(cid:12)IM + σ12HeffVDVDHHHeff(cid:12)(cid:12) (10a) eolpetmimeainjltsvaolfuethefoRr FthepreelceomdeerntaRroFef fithxeedReFxcperpectoVdeRrF(ait,jth),ethiteh
s.t. Tr(QVDVDH)≤P, (10b) row and jth column is given by
where Q = VRHFVRF. This problem has a well-known water- (cid:40)1, if ηij =0,
filling solution as VRF(i,j)= ηij , otherwise. (14)
V =Q−1/2U Γ , (11) |ηij|
D e e This enables us to propose an iterative algorithm that starts
where U is the set of right singular vectors corresponding with an initial feasible RF precoder satisfying (12b), i.e.,
e
to the N largest singular values of H Q−1/2 and Γ is the V(0) = 1 , then sequentially updates each element of
s eff e RF N×NRF
diagonal matrix of allocated powers to each stream. RF precoder according to (14) until the algorithm converges
Note that for large-scale MIMO systems, Q ≈ NI with to a local optimal solution of V of the problem (12).
RF
highprobability[27].Thisisbecausethediagonalelementsof Note that since in each element update step of the proposed
Q=VHV are exactly N while the off-diagonal elements algorithm, the objective function of (12) increases (or at least
RF RF
can be approximated as a summation of N independent terms doesnotdecrease),thereforetheconvergenceofthealgorithm
which is much less than N with high probability for large is guaranteed. The proposed algorithm for designing the RF
N. This property enables us to show that the optimal digital beamformer in (12) is summarized in Algorithm 1. We men-
precoder for NRF = N typically satisfies V VH ∝ I. tion that the proposed algorithm is inspired by the algorithm
s D D
The proportionality constant can be obtained with further in[41]thatseekstosolvetheproblemoftransmitterprecoder
assumption of equal power allocation for all streams, i.e., design with per-antenna power constraint which happens to
(cid:112)
Γ ≈ P/NRFI. So, optimal digital precoder is V ≈ γU have the same form as the problem in (12).
e D e
6
Algorithm 1 Design of V by solving (12) Algorithm 2 Design of Hybrid Beamformers for Point-to-
RF
Require: F , γ2, σ2 Point MIMO systems
1
1: Initialize VRF =1N×NRF. Require: σ2, P
(cid:112)
2: for j =1→NRF do 1: Assuming VDVDH = γI where γ = P/(NNRF), find
3: Calculate Cj =I+ σγ22(V¯RjF)HF1V¯RjF. VRF by solving the problem in (12) using Algorithm 1.
4: Calculate Gj = σγ22F1− σγ44F1V¯RjFC−j1(V¯RjF)HF1. 2: Calculate VD = (VRHFVRF)−1/2UeΓe where Ue and Γe
5: for i=1→N do are defined as following (11).
6: Find ηij =(cid:80)(cid:96)(cid:54)=iGj(i,(cid:96))VRF((cid:96),j). 3: Find WRF by solving the problem in (16) using Algo-
(cid:40) rithm 1.
1, if η =0,
87:: endVfoRrF(i,j)= |ηηiijj|, otheirjwise. 4: CWalRHcFuHlaVteRFWVDDVDH=VRHFJH−1HWWRHRFFH+VσR2FVWDRHFWwhReFr.e J =
9: end for
10: Check convergence. If yes, stop; if not go to Step 2. we aim to design the hybrid beamformers for the case of
N <NRF <2N .
s s
For N <NRF <2N , the transmitter design problem can
s s
C. Hybrid Combining Design for NRF =N still be formulated as in (9). For a fixed RF precoder, it can
s
beseenthattheoptimaldigitalprecodercanstillbefoundac-
Finally, we seek to design the hybrid combiners that max-
cordingto(11),howevernowitsatisfiesV VH ≈γ2[I 0].
imize the overall spectral efficiency in (8) assuming that D D Ns
For such a digital precoder, the objective function of (9) that
the hybrid precoders are already designed. For the case that
NRF = N , the digital combiner is a square matrix with no should be maximized over VRF can be rewritten as
s
constraint on its entries. Therefore, without loss of optimality, (cid:89)Ns (cid:18) γ2 (cid:19)
the design of WRF and WD can be decoupled by first log2 1+ σ2λi , (18)
designingtheRFcombinerassumingoptimaldigitalcombiner i=1
and then finding the optimal digital combiner for that RF where λ is the ith largest eigenvalues of VHHHHV . Due
i RF RF
combiner. As a result, the RF combiner design problem can to the difficulties of optimizing over a function of subset of
be written as eigenvalues of a matrix, we approximate (18) with an expres-
(cid:12) 1 (cid:12) sionincludingalloftheeigenvalues,i.e.,log (cid:81)NRF(1+γ2λ ),
max log (cid:12)I+ (WHW )−1WHF W (cid:12)(15a) 2 i=1 σ2 i
WRF 2(cid:12) σ2 RF RF RF 2 RF(cid:12) or equivalently,
s.t. |W (i,j)|2 =1, ∀i,j, (15b) (cid:12) γ2 (cid:12)
RF log (cid:12)I + VHHHHV (cid:12), (19)
2(cid:12) NRF σ2 RF RF(cid:12)
where F = HVVHHH. This problem is very similar to
2 t t which is a reasonable approximation for the practical settings
the RF precoder design problem in (12), except the extra
where NRF is in the order of N . Further, by this approxi-
term (WHW )−1. Analogous to the argument made in s
RF RF mation, the RF precoder design problem is now in the form
Section IV-A for the RF precoder, it can be shown that the
of (12). Hence, Algorithm 1 can be used to obtain the RF
RF combiner typically satisfies WHW ≈ MI, for large
RF RF precoder. In summary, we suggest to first design the RF
M. Therefore, the problem (15) can be approximated in the
precoderassumingthatthenumberofdatastreamsisequalto
form of RF precoder design problem in (12) and Algorithm 1
the number of RF chains, then for that RF precoder, to obtain
can be used to design W by substituting F and 1 by F
RF 2 M 1 the digital precoder for the actual N .
and γ2, respectively, i.e., s
At the receiver, we still suggest to design the RF combiner
(cid:12) 1 (cid:12) first,thensetthedigitalcombinertotheMMSEsolution.This
max log (cid:12)I+ WHF W (cid:12) (16a)
WRF 2(cid:12) Mσ2 RF 2 RF(cid:12) decoupled optimization of RF combiner and digital combiner
s.t. |W (i,j)|2 =1, ∀i,j. (16b) is approximately optimal for the following reason. Assume
RF
that all the beamformers are already designed except the
Finally, assuming all other beamformers are fixed, the digital combiner. Since WHW ≈ MI, the effective noise
RF RF
optimal digital combiner is the MMSE solution as after the RF combiner can be considered as an uncolored
noise with covariance matrix σ2MI. Under this condition,
W =J−1WHHV, (17)
D RF t by choosing the digital combiner as the MMSE solution,
where J=WHHVVHHHW +σ2WHW . the mutual information between the data symbols and the
RF t t RF RF RF
processed signals before digital combiner is approximately
equal to the mutual information between the data symbols
D. Hybrid Beamforming Design for N <NRF <2N
s s and the final processed signals. Therefore, it is approximately
InSectionIII,weshowhowtodesignthehybridbeamform- optimal to first design the RF combiner using Algorithm 1,
ers for the case NRF ≥ 2N for which the hybrid structure then set the digital combiner to the MMSE solution.
s
can achieve the same rate as the rate of optimal fully digital Thesummaryoftheoverallproposedprocedurefordesign-
beamforming. Earlier in this section, we propose a heuristic ing the hybrid beamformers for spectral efficiency maximiza-
hybrid beamforming design algorithm for NRF = N . Now, tion in a large-scale point-to-point MIMO system is given in
s
7
Algorithm 2. Assuming the number of antennas at both ends can be found for a fixed RF precoder. In addition, for a fixed
are in the same range, i.e., M = O(N), it can be shown powerallocation,anapproximatelylocal-optimalRFprecoder
that the overall complexity of Algorithm 2 is O(N3) which can be obtained. By iterating between those designs, a good
is similar to the most of the existing hybrid beamforming solution of the problem (5) for MU-MISO can be found.
designs, i.e., the hybrid beamforming designs in [25], [27].
Numerical results presented in the simulation part of this
A. Digital Precoder Design
paper suggest that for the case of NRF = N and infinite
s
We consider ZF beamforming with power allocation as the
resolution phase shifters, the achievable rate of the proposed
low-dimensional digital precoder part of the BS’s precoder to
algorithm is very close the maximum capacity. The case of
N < NRF < 2N is of most interest when the finite manage the inter-user interference. For a fixed RF precoder,
s s
such a digital precoder can be found as [6]
resolutionphaseshiftersareused.Itisshowninthesimulation
puasertdotfotthriasdepaopfefrththeaatctchueraecxytraofntuhmebpehraosef RshFiftcehras.ins can be VDZF =VRHFHH(HVRFVRHFHH)−1P21 =V˜DP12, (21)
where H = [h ,...,h ]H, V˜ =
1 K D
V. HYBRIDBEAMFORMINGDESIGNFORMULTI-USER VRHFHH(HVRFVRHFHH)−1 and P = diag(p1,...,pK)
MASSIVEMISOSYSTEMS with pk denoting the received power at the kth user. For a
fixed RF precoder, the only design variables of ZF digital
Now, we consider the design of hybrid precoders for the
precoder are the received powers, [p ,...,p ]. Using the
downlinkMU-MISOsysteminwhichaBSwithlargenumber 1 k √
properties of ZF beamforming; i.e., |hHV vZF|= p and
ofantennasN,butlimitednumberofRFchainsNRF,supports k RF Dk k
|hHV vZF| = 0 for all (cid:96) (cid:54)= k, problem (5) for designing
K single-antennauserswhereN (cid:29)K.Forsuchasystemwith k RF D(cid:96)
those powers assuming a feasible RF precoder is reduced to
hybridprecodingarchitectureattheBS,therateexpressionfor
user k in (4) can be expressed as K
R =log (cid:32)1+ |hHk VRFvDk|2 (cid:33), (20) p1,.m..,apKx≥0 k(cid:88)=1βklog2(cid:16)1+ σpk2(cid:17) (22a)
k 2 σ2+(cid:80)(cid:96)(cid:54)=k|hHk VRFvD(cid:96)|2 s.t. Tr(Q˜P)≤P, (22b)
where hH is the channel from the BS to the kth user and where Q˜ = V˜HVHV V˜ . The optimal solution of this
k D RF RF D
vD(cid:96) denotes the (cid:96)th column of the digital precoder VD. The problem can be found by water-filling as
problem of overall spectral efficiency maximization for the (cid:26) (cid:27)
1 β
MU-MISO systems differs from that for the point-to-point p = max k −q˜ σ2,0 , (23)
k q˜ λ kk
MIMO systems in two respects. First, in the MU-MISO case kk
the receiving antennas are not collocated, therefore we cannot where q˜ is kth diagonal element of Q˜ and λ is chosen such
kk
use the rate expression in (8), which assumes cooperation that (cid:80)K max{βk −q˜ σ2,0}=P.
k=1 λ kk
between the receivers. The hybrid beamforming design for
MU-MISO systems must account for the effect of inter-
B. RF Precoder Design
user interference. Second, the priority of the streams may be
unequal in a MU-MISO system, while different streams in a Now, we seek to design the RF precoder assuming the ZF
point-to-point MIMO systems always have the same priority. digital precoding as in (21). Our overall strategy is to iterate
ThissectionconsidersthehybridbeaformingdesignofaMU- between the design of ZF precoder and the RF precoder. Ob-
MISO system to maximize the weighted sum rate. servethattheachievableweightedsumratewithZFprecoding
In [32], [33], it is shown for the case NRF = K and in (22) depends on the RF precoder V only through the
RF
N → ∞, that by matching the RF precoder to the overall power constraint (22b). Therefore, the RF precoder design
channel (or the strongest paths of the channel) and using a problem can be recast as a power minimization problem as
low-dimensionalzero-forcing(ZF)digitalprecoder,thehybrid
beamforming structure can achieve a reasonable sum rate as
min f(V ) (24a)
compared to the sum rate of fully digital ZF scheme (which RF
VRF
is near optimal in massive MIMO systems [36]). However, s.t. |V (i,j)|2 =1, ∀i,j. (24b)
RF
for practical values of N, there is still a gap between the
achievable rates and the capacity. This section proposes a where, f(V )=Tr(V V˜ PV˜HVH).
RF RF D D RF
design for the scenarios where NRF > K with practical N This problem is still difficult to solve since the expression
and show numerically that adding a few more RF chains can f(V ) in term of V is very complicated. But, using the
RF RF
increase the overall performance of the system and reduce the fact that the RF precoder typically satisfies VHV ≈ NI
RF RF
gap to capacity. when N is large [27], this can be simplified as
Solving the problem (5) for such a system involves a joint
f(V )=Tr(VHV V˜ PV˜H)
optimization over V and V which is challenging. We RF RF RF D D
RF D
again decouple the design of VRF and VD by considering ZF ≈NTr(P12V˜DHV˜DP12)
beamformingwithpowerallocationasthedigitalprecoder.We =NTr(cid:16)(H˜V VHH˜H)−1(cid:17)=fˆ(V ), (25)
show that the optimal digital precoder with such a structure RF RF RF
8
Algorithm 3 Design of Hybrid Precoders for MU-MISO The overall algorithm is to iterate between the design of
systems V and the design of P. First, starting with a feasible V
RF RF
Require: β , P, σ2 and P = I, the algorithm seeks to sequentially update the
k
1: Start with a feasible VRF and P=IK. phase of each element of RF precoder according to (29) until
2: for j =1→NRF do convergence. Then, assuming the current RF precoder, the
3: Calculate Aj =P−12HV¯RjF(V¯RjF)HHHP−21. algorithmfindstheoptimalpowerallocationPusing(23).The
4: for i=1→N do iterationbetweenthesetwostepscontinuesuntilconvergence.
5: Find ζB, ζD, ηB, ηD as defined in Appendix A. Theoverallproposedalgorithmfordesigningthehybriddigital
ij ij ij ij
6: Calculate θ(1) and θ(2) according to (27). andanalogprecodertomaximizetheweightedsumrateinthe
i,j (cid:16)i,j (cid:17) downlinkofamulti-usermassiveMISOsystemissummarized
7: Find θiojpt =argmin fˆ(θi(,1j)),fˆ(θi(,2j)) . in Algorithm 3.
8: Set VRF(i,j)=e−jθiojpt.
9: end for VI. HYBRIDBEAMFORMINGWITHFINITERESOLUTION
10: end for PHASESHIFTERS
11: Check convergence of RF precoder. If yes, continue; if Finally, we consider the hybrid beamforming design with
not go to Step 2.
finiteresolutionphaseshiftersforthetwoscenariosofinterest
12: Find P=diag[p1,...,pk] using water-filling as in (23). inthispaper,thepoint-to-pointlarge-scaleMIMOsystemand
13: Check convergence of the overall algorithm. If yes, stop; themulti-userMISOsystemwithlargearraysattheBS.Sofar,
if not go to Step 2.
we assume that infinite resolution phase shifters are available
14: Set VD =VRHFHH(HVRFVRHFHH)−1P12. in the hybrid structure, so the elements of RF beamformers
can have any arbitrary phase angles. However, components
required for accurate phase control can be expensive [38].
where H˜ =P−12H. Now, analogous to the procedure for the Since the number of phase shifters in hybrid structure is pro-
point-to-point MIMO case, we aim to extract the contribution portional to the number of antennas, infinite resolution phase
of VRF(i,j) in the objective function (here the approximation shifter assumption is not always practical for systems with
of the objective function), fˆ(VRF), then seek to find the large antenna array terminals. In this section, we consider the
optimal value of VRF(i,j) assuming all other elements are impact of finite resolution phase shifters with VRF(i,j) ∈ F
fixed. For NRF >Ns, it is shown in Appendix A that and WRF(i,j) ∈ F where F = {1,ω,ω2,...ωnPS−1} and
fˆ(VRF)=NTr(A−j1)−N1+ζiBjζD++2R2eR(cid:8)eV(cid:8)VR∗F∗(i,(ji,)ηj)iBjη(cid:9)D(cid:9), ωwh=ichejisn2PπtSypaincdallnyPSnPiSs =the2bn,uwmhbeerreobfirseathliezanbulmebpehrasoef banitgsleins
ij RF ij(26) the resolution of phase shifters.
where A , ζB, ζD, ηB and ηD are defined as in Appendix A With finite resolution phase shifters, the general weighted
j ij ij ij ij
and are independent of V (i,j). If we assume that all ele- sum rate maximization problem can be written as
RF
tmthheeenrotespstouifmlttsahlienvRaAFluppepreefnocdoridxθeirB,ja,rsiehtoficuaxlneddbseeaxtsciseefepyntV∂thf∂ˆRa(θtFVi(,iRjitF,)ijs)=a=lw0.ea−yUsjsθitinh,jge, VRmF,VaxDWimRiFz,We D k(cid:88)K=1βkRk (30a)
case that only two θi,j ∈[0,2π) satisfy this condition: subject to Tr(VRFVDVDHVRHF)≤P (30b)
θ(1) =−φ +sin−1(cid:18) zij (cid:19), (27a) VRF(i,j)∈F, ∀i,j (30c)
i,j i,j |cij| WRFk(i,j)∈F, ∀i,j,k. (30d)
(cid:18) (cid:19)
z
θ(2) =π−φ −sin−1 ij , (27b) For a set of fixed RF beamformers, the design of digi-
i,j i,j |cij| tal beamformers is a well-studied problem in the literature.
where c =(1+ζD)ηB −ζBηD, z =Im{2(ηB)∗ηD} and However, the combinatorial nature of optimization over RF
ij ij ij ij ij ij ij ij
beamformers in (30) makes the design of RF beamformers
(cid:40)sin−1(Im{cij}), if Re{c }≥0, more challenging. Theoretically, since the set of feasible RF
φ = |cij| ij (28)
i,j π−sin−1(Im{cij}), if Re{c }<0. beamformers are finite, we can exhaustively search over all
|cij| ij feasible choices. But, as the number of feasible RF beam-
Since fˆ(V ) is periodic over θ , only one of those fomers is exponential in the number of antennas and the
RF i,j
solutions is the minimizer of fˆ(V ). The optimal θ can resolution of the phase shifters, this approach is not practical
RF i,j
be written as for systems with large number of antennas.
(cid:16) (cid:17) The other straightforward approach for finding the feasible
θopt =argmin fˆ(θ(1)),fˆ(θ(2)) . (29)
ij i,j i,j solution for (30) is to first solve the problem under the
θ(1),θ(2)
i,j i,j infinite resolution phase shifter assumption, then to quantize
Now, we are able to devise an iterative algorithm starting the elements of the obtained RF beamformers to the nearest
from an initially feasible RF precoder and sequentially up- points in the set F. However, numerical results suggest that
dating each entry of RF precoder according to (29) until the forlowresolutionphaseshifters,thisapproachisnoteffective.
algorithm converges to a local minimizer of fˆ(V ). This section aims to show that it is possible to account for
RF
9
the finite resolution phase shifter directly in the optimization
procedure to get better performance. 40
For hybrid beamforming design of a single-user MIMO Optimal Fully−Digital Beamforming
Proposed Hybrid Beamforming Algorithm
system with finite resolution phase shifters, Algorithm 2 for 35 Hybrid beamforming in [25]
solving the spectral efficiency maximization problem can be Hybrid beamforming in [27]
aaexdsscauepmptetidnVgasRafFlol(liol,ofjw)th,s.ewAeeclecmonreedenidntsgtootof tmthheaexpRirmFoiczbeeedaumRreefo(cid:8)inrVmAR∗elFrg(oairr,iejth)fiηmxije2(cid:9)d, ncy (bits/s/Hz)2350
oVfaonprRgtiFldme(eiasb,lijegd)tnweiisnesieggncnVoVinsRsRFtF(r(ai,iin,jje)).dTatnhodisbηieisjceohqnousitvehanelecfnortommtoplemthxienpilsmaenitzei.FnSg,intthhceee Spectral Efficie20
15
VMIMO(i,j)=Q(ψ(η )), (31)
RF ij
10
where for a non-zero complex variable a, ψ(a) = a and
|a| −1 0 −8 −6 −4 −2 0 2 4 6
for a = 0, ψ(a) = 1, and the function Q(·) quantizes a SNR(dB)
complex unit-norm variable to the nearest point in the set F.
Assumingthatthenumberofantennasatbothendsinthesame Fig. 2. Spectral efficiencies achieved by different methods in a 64×16
range, i.e., M = O(N), it can be shown that the complexity MIMOsystemwhereNRF=Ns=6.Forhybridbeamformingmethods,the
useofinfiniteresolutionphaseshiftersisassumed.
of the proposed algorithm is polynomial in the number of
antennas,O(N3),whilethecomplexityoffindingtheoptimal
beamformers using exhaustive search method is exponential, where α(cid:96) ∼ CN(0,1) is the complex gain of the (cid:96)th path
O(N22bN). betweenkthe BS and the user k, and φ(cid:96) ∈[0, 2π) and φ(cid:96) ∈
Similarly, for hybrid beamforming design of a MU-MISO [0, 2π).Further,a (.)anda (.)aretherkantennaarrayrespotknse
r t
system with finite resolution phase shifters, Algorithm 3 can vectors at the receiver and the transmitter, respectively. In a
likewisebemodifiedasfollows.Sincethesetoffeasiblephase uniform linear array configuration with N antenna elements,
angles are limited, instead of (29), we can find VRF(i,j) we have
in each iteration by minimizing fˆ(V ) in (26) using one-
RF 1
dimensional exhaustive search over the set F, i.e., a(φ)= √ [1,ejkd˜sin(φ),...,ejkd˜(N−1)sin(φ)]T, (34)
N
VRMFU-MISO(i,j)= argmin fˆ(VRF). (32) where k = 2π, λ is the wavelength and d˜ is the antenna
VRF(i,j)∈F spacing. λ
The overall complexity of the proposed algorithm for hy- In the following simulations, we consider an environment
brid beamforming design of a MU-MISO system with finite withL=15scatterersbetweentheBSandeachuserterminal
resolution phase shifters is O(N22b), while the complexity assuming uniformly random angles of arrival and departure
of finding the optimal beamforming using exhaustive search andd˜= λ.Foreachsimulation,theaveragespectralefficiency
2
method is O(N2bN). Note that accounting for the effect of is plotted versus signal-to-noise-ratio (SNR = P ) over 100
σ2
phase quantization is most important when low resolution channel realizations.
phase shifters are used, i.e., b = 1 or b = 2. Since in these
cases, the number of possible choices for each element of RF
A. Performance Analysis of a MIMO System with Hybrid
beamformerissmall,theproposedone-dimensionalexhaustive
Beamforming
search approach is not computationally demanding.
In the first simulation, we consider a 64 × 16 MIMO
system with N = 6. For hybrid beamforming schemes,
s
VII. SIMULATIONS we assume that the number of RF chains at each end is
In this section, simulation results are presented to show NRF = Ns = 6 and infinite resolution phase shifters are
the performance of the proposed algorithms for point-to- used at both ends. Fig. 2 shows that the proposed algorithm
point MIMO systems and MU-MISO systems and also to has a better performance as compared to hybrid beamforming
compare them with the existing hybrid beamforming designs algorithms in [27] and [25]: about 1.5dB gain as compared to
and the optimal (or nearly-optimal) fully digital schemes. In thealgorithmof[27]andabout1dBimprovementascompared
the simulations, the propagation environment between each to the algorithm of [25]. Moreover, the performance of the
user terminal and the BS is modeled as a geometric channel proposed algorithm is very close to the rate of optimal fully
with L paths [33]. Further, we assume uniform linear array digital beamforming scheme. This indicates that the proposed
antenna configuration. For such an environment, the channel algorithm is nearly optimal.
matrix of the kth user can be written as Now,weanalyzetheperformanceofourproposedalgorithm
when only low resolution phase shifters are available. First,
(cid:114) L
NM (cid:88) we consider a relatively small 10×10 MIMO system with
H = α(cid:96)a (φ(cid:96) )a (φ(cid:96) )H, (33)
k L k r rk t tk hybrid beamforming architecture where the RF beamformers
(cid:96)=1
10
26 34
Optimal Fully−Digital Beamforming
24 EPrxohpaoussetidv Ae Slgeoarricthhm for b=1 32 Proposed Algorithm for b=∞, NRF = Ns
22 QQuuaannttiizzeedd−−PHryobproidse bde Aamlgfoorrimthimng f ionr [b2=5∞] 2380 PPrrooppoosseedd AAllggoorriitthhmm ffoorr bb==11,, NNRRFF == NNs+1
Hz)20 Quantized−Hybrid beamforming in [27] Hz)26 Proposed Algorithm for b=1, NRF = Ns+3
bits/s/18 bits/s/2224 Quantized−Proposed Algorithm for b=s∞
ncy (16 ncy (20
Efficie14 Efficie1168
Spectral 1102 Spectral 1124
8 10
8
6
6
40 5 10 15 20 25 30 −41 0 −8 −6 −4 −2 0 2 4 6
SNR(dB) SNR(dB)
Fig.3. SpectralefficienciesversusSNRfordifferentmethodsina10×10 Fig.4. SpectralefficienciesversusSNRfordifferentmethodsina64×16
systemwhereNRF=Ns=2andb=1. systemwhereNs=4.
are constructed using 1-bit resolution phase shifters. Further,
it is assumed that NRF =Ns =2. The number of antennas at 50 Fully−Digital ZF
eachendischosentoberelativelysmallinordertobeableto 45 Proposed Algorithm for NRF=9
compare the performance of the proposed algorithm with the Hybrid beamforming in [33], NRF=8
40 Hybrid beamforming in [32], NRF=8
exhaustive search method. We also compare the performance
35
of the proposed algorithm in Section VI, which considers the
fidalengsioitgerintr,hemstooslutithnioenSpepechrtfaioosrenmsIahVnifc,teearnocdfontihsntera[2qin5ut]a,innt[i2tzh7ee]d,RvwFehrbseeiroaenmtfhooefrmRtheFer Rate (bits/s/Hz)2350
beamformersarefirstdesignedundertheassumptionofinfinite m 20
u
S
resolutionphaseshifters,theneachentryoftheRFbeamform- 15
ersisquantizedtothenearestpointofthesetF.Fig.3shows
10
that the performance of the proposed algorithm for b=1 has
a better performance: at least 1.5dB gain, as compared to the 5
quantized version of the other algorithms that design the RF
−01 0 −8 −6 −4 −2 0 2 4 6 8 10
beamformersassumingaccuratephaseshiftersfirst.Moreover, SNR(dB)
the spectral efticiency achieved by the proposed algorithm is
very close to that of the optimal exhaustive search method,
Fig. 5. Sum rate achieved by different methods in an 8-user MISO system
confirming that the proposed methods is near to optimal. withN =64.Forhybridbeamformingmethods,theuseofinfiniteresolution
Finally, we consider a 64×16 MIMO system with N =4 phaseshiftersisassumed.
s
toinvestigatetheperformancedegradationofthehybridbeam-
forming with low resolution phase shifters. Fig. 4 shows that
theusershavethesamepriority,i.e,β =1,∀k.Assumingthe
theperformancedegradationofaMIMOsystemwithverylow k
use of infinite resolution phase shifters for hybrid beamform-
resolutionphaseshiftersascomparedtotheinfiniteresolution
case is significant—about 5dB in this example. However, ing schemes, we compare the performance of the proposed
algorithm with K + 1 = 9 RF chains to the algorithms in
Fig. 4 verifies that this gap can be reduced by increasing
[33] and [32] using K =8 RF chains. In [33] and [32] each
the number of RF chains, and by using the algorithm in
column of RF precoder is designed by matching to the phase
Section IV-D to optimize the RF and digital beamformers.
of the channel of each user and matching to the strongest
Therefore, the number of RF chains can be used to trade off
paths of the channel of each user, respectively. Fig. 5 shows
the accuracy of phase shifters in hybrid beamforming design.
that the approach of matching to the strongest paths in [32]
is not effective for practical value of N; (here N = 64).
B. Performance Analysis of a MU-MISO System with Hybrid Moreover, the proposed approach with one extra RF chain
Beamforming are very close to the sum rate upper bound achieved by fully
To study the performance of the proposed algorithm for digital ZF beamforming. It improves the method in [33] by
MU-MISO systems, we first consider an 8-user MISO system about 1dB in this example.
with N = 64 antennas at the BS. Further, it is assumed that Finally,westudytheeffectoffiniteresolutionphaseshifters