Joint Channel Direction Information Quantization For Spatially Correlated 3D MIMO Channels

0.58 MB·English

by  Fang Yuan

1
Joint Channel Direction Information Quantization
For Spatially Correlated 3D MIMO Channels
Fang Yuan, Chenyang Yang, Yang Song, Lan Chen, Yuichi Kakishima and Huiling Jiang
Abstract—This paper proposes a codebook for jointly e.g., rectangular and circular arrays, or even non-
quantizingchanneldirectioninformation(CDI)ofspatially planararrays,attheBS.
correlatedthree-dimensional(3D)multi-input-multi-output The non-linear arrays can provide both horizonal
(MIMO)channels.Toreducethedimensionforquantizing
5 and vertical spatial resolution, and thus can support
1 the CDI of large antenna arrays, we introduce a special
3D beamforming [3]. 3D beamforming can be user-
0 structure to the codewords by using Tucker decomposi-
2 tion to exploit the unique features of 3D MIMO channels. specific, which improves the signal to noise ratio
n Specifically, the codeword consists of four parts each with (SNR)meanwhilegeneratinglessinterferencetoad-
a low dimension individually targeting at a different type of jacentusers.Recently,a3DMIMOprototypesystem
J
information:statisticalCDIsinhorizontaldirectionandin operating at millimeter-wave bands was reported to
3
vertical direction, statistical power coupling, and instanta-
2 supportamulti-Gbpsdatarateserviceinmacrocells,
neousCDI.Theproposedcodebookavoidstheredundancy
which provides high array gain to compensate the
] led by existing independent CDI quantization. Analytical
T resultsprovideasufficientconditionon3DMIMOchannels severepathloss[5].
I to show that the proposed codebook can achieve the same In practice, the promised performance of 3D
.
s quantization performance as thewell-known rotated code- beamforming largely depends on how accurate the
c
[ bookappliedtotheglobalchannelCDI,butwithsignificant channel direction information (CDI) is obtained at

reduction in the required statistical channel information. the BS. In time division duplexing (TDD) systems,
1
Simulation results validate our analysis and demonstrate
v the CDI obtained by channel estimation in uplink
3 thattheproposedjointCDIquantizationprovidessubstan-
can be used for beamforming in downlink if the
9 tial performance gain over independent CDI quantization.
6 antennasattheBSareperfectlycalibrated[1],where
5 the performance is limited by pilot contamination
0 IndexTerms—Three-dimensional(3D),multi-input-and-
[6].In frequencydivisionduplexing (FDD)systems,
. multi-output(MIMO),limitedfeedback,codebookdesign
1
limited feedback is widely used, where the CDI is
0
5 firstly quantized at the user and then fed back to the
:1 I. INTRODUCTION BS [7]. Yet the feedback overhead is supposed to
v increase with the number of antennas [8], which is
Xi To meet the ever-growing data demand in future notacceptableforlargeantennaarraysystems.
r 5th Generation (5G) cellular networks, one of the Spatial correlation is observed very typical in
a
promisingwaysistoincreasethenumberofantennas MIMO channels [9], due to small spacing between
at the base station (BS) [1]. However, equipping adjacent antennas and low angular spreads. In [10],
large number of antennas at a BS is challenging due B. Clerckx, et al. found that spatial correlation can
to the physical space limitation. This naturally calls beexploitedtoreducetheoverheadforfeedingback
for three-dimensional (3D) multi-input-and-multi- CDI significantly. The same conclusion was drawn
output(MIMO)systems[2–5],whereactiveantenna in [11–13] for spatially correlated massive MIMO
elementsareplacedinatwo-dimensional(2D)array, channels,whichindicatesthatFDDisalsoapplicable
for large antenna array systems without heavy feed-
Fang Yuan and Chenyang Yang are with the School of Electron- backoverheadassupposedtobe.
ics and Information Engineering, Beihang University, Beijing China
Various codebooks have been proposed for spa-
(email:[email protected],[email protected]).
YangSongandHuilingJiangarewithDOCOMOBeijingCommu- tially correlated channels. Theoretically, Llyod al-
nications Laboratories Co., Ltd (email:{song,jiang}@docomolabs- gorithm [14] can be applied to generate codebooks
beijing.com.cn). Lan Chen and Yuichi Kakishima are with
for 3D MIMO channels, but they are difficult to
NTT DOCOMO, INC. (email: [email protected],
[email protected]). be used off-line in practice. There are some code-
2
books obtained from Llyod algorithms with reduced To design a desirable codebook in 3D MIMO
complexity, e.g., local-packing codebook [15], and channels under various angular spreads and reduce
gain and phase separated quantization [16]. A well- the amount of required spatial correlation informa-
known codebook, rotated codebook [17], transforms tion, we propose a joint CDI quantization strategy
the codewords optimized for uncorrelated channels to quantize the CDI of two directions jointly. The
(e.g., Grassmannian subspace packing (GSP) code- designed codeword has a special structure led by
words) by channel correlation matrix. It was proved the Tucker decomposition [25], which exploits the
that the rotated codebook is asymptotically optimal uniquefeaturesof3DMIMOchannels.
in quantizing any spatially correlated channels as Thefirstfeatureof3DMIMOchannelistheinher-
the codebook size becomes large [18]. An extension ent geometrical structure inherited from the regular-
of rotated codebook to multiuser MIMO system is ityofantennaarrays.Forexample,thearrayresponse
provided in [19], where not only each user’s own of 3D MIMO with URA for each ray in the channel
butalsotheotherusers’correlationmatricesareem- can be decomposed into two subarray responses of
ployed for the codeword rotation. When designing lower channel dimensions in horizontal and vertical
codebooks for real-world cellular systems, practi- directions.Similardecompositionscanbealsofound
cal limitations such as constant modulus and finite for those large arrays nested from several smaller
alphabet need to be taken into account. Discrete identicalsubarrays.Thisfeaturecanbeexploitedfor
Fourier transformation (DFT) codebook meets these coping with the high-dimension quantization prob-
limitations, which is suitable for highly correlated leminlargeantennaarrays.
channelswithuniformlineararrayattheBS[20].For The second feature is power coupling. When the
3D MIMO system with uniform rectangular array array responses of 3D MIMO channels are decom-
(URA), a Kronecker-product DFT codebook was posed into two subarray responses respectively in
proposed in [21]. However, as shown in [20], the horizontalandverticaldirections,acommonraygain
performanceofthesepureDFTbasedcodebooksde- is shared by the two subarray responses and not de-
grades severely when the angular spreads of channel composable. The power coupling is important in the
increase. codebookdesign,sinceitconnectswiththesubarray
Generally speaking, there are two straightforward responses from different lower channel dimensions.
strategiestoextendexistingcodebooksto3DMIMO Thefeaturecanbeexploitedtoavoidtheredundancy
systems. The first strategy is global channel quan- intheCDIquantization.
tization, which expresses the 3D MIMO channel as Although many research efforts have been made
a larger global vector, and reuse existing codebooks for the dimension reduction problem, they mainly
proposed for 2D MIMO channels. For example, the employ the singular value decomposition (SVD) of
codebooks in [15–17,22] can be applied to quantize correlation matrix to exploit the channel correlation,
the global 3D MIMO CDI directly. However, due to e.g., [26]. As far as the authors known, the unique
the high dimension of the resulting CDI vector, it featuresin3DMIMOchannelsarenotyetexploited.
is challenging for the BS to obtain accurate channel In this paper, we propose a codebook for jointly
statisticalinformationrequiredbytheglobalchannel quantizing the CDIs in horizontal and vertical direc-
quantization to reuse existing codebooks, e.g., chan- tions, where a new codeword structure is introduced
nel correlation matrix used by the rotated codebook, to reduce the dimension by using the features of 3D
which is known as the “curse of the dimension- MIMOchannels.Thecontributionsaretwo-fold:
ality” [23]. Moreover, large dimension incurs high 1) A joint CDI quantization codebook is pro-
complexity in matrix operations [24]. The second posed to quantize spatially correlated 3D
strategy is independent CDI quantization, which MIMO channels, which has a special code-
quantizes the CDI of 3D MIMO in horizontal and word structure. The spatial correlation infor-
vertical directions independently and reuses existing mation required by the cobook for general
codebooks in each direction [4]. This strategy is antennaarraysisprovidedbyusingtheTucker
simple and avoids the problem caused by the high decomposition. We show that the proposed
dimensionalchannelvector,whichhowevercomesat codebook provides better quantization perfor-
acostoflowquantizationaccuracywhentheangular mancethantheindependentCDIquantization,
spreadislarge,aswillbeclarifiedlater. and requires much less spatial correlation in-
3
with single-antenna users, and then extend the de-
sign to multi-antenna users. For simplicity, we will
not distinguish 3D MISO channel from 3D MIMO
channelhereafter.
Accordingtothegenericspatialchannelmodeling
methods [9,28,29], the 3D MIMO channel consists
of several scattering clusters distributed in the 3D
space, where in each cluster there are multiple rays
with small random angle offsets, which can be ex-
pressedasaN -dimensionalvector
t
(cid:88)N (cid:88)M
hhh (cid:44) g aaa(Θ ) (1)
n,m n,m
n=1 m=1
Fig.1. Anexampleof3DMIMOsystemwith8×8URAattheBS. where g ∈ C is the random gain of the mth
n,m
ray in the nth cluster with zero mean, the angle
vector Θ = [θ ,φ ]T is specified by the
formationthanthegloballyrotatedcodebook. n,m n,m n,m
angle coordinates θ and φ in the 3D space,
2) The performance of joint CDI quantization n,m n,m
and aaa(Θ ) ∈ CNt×1 is the corresponding array
codebook is analyzed. We show that the per- n,m
response.
formance is the same as the globally rotated
Theexpressionofarrayresponseaaa(Θ )depends
codebook for 3D MIMO systems with URA n,m
on the specific form of the array mounted at the BS.
under the channels with weak identically in-
The array responses of some well-structured arrays
dependentdistributed(i.i.d.)rays.
can be decomposed into the Kronecker product of
The rest of this paper is organized as follows.
subarray responses, while the others may not. Usu-
In Section II, we present the 3D MIMO channel
ally, the large antenna arrays nested from smaller
model and channel representations. In Section III,
identical subarrays are decomposable. For example,
we propose the joint CDI quantization codebook
as shown in [28,30], the array response of URA can
for spatially correlated channels. In Section IV, a
be decomposed into two subarray responses respec-
solution to find the spatial correlation information
tivelyinhorizontalandverticaldirectionsas
for general antenna arrays is provided by using the
Tucker decomposition. The performance of the joint aaa(Θ ) =aaa (φ )⊗aaa (θ ) (2)
n,m v n,m h n,m
CDIquantizationcodebookisanalyzedinSectionV.
with
SimulationresultsareprovidedinSectionVIandthe
paperisconcludedinSectionVII. aaah(θn,m)=[1,ej2πdλhcosθn,m,···,ej2πdλh(Nh−1)cosθn,m]T
Notations: (·)T, (·)H and (·)∗ are respectively the
transpose,Hermitianandconjugateoperation,(cid:12)and aaav(φn,m)=[1,ej2πdλvcosφn,m,···,ej2πdλv(Nv−1)cosφn,m]T
⊗ are respectively the element-wise and Kronecker whered andd arerespectivelytheantennaspacing
h v
matrix product, | · | and (cid:107) · (cid:107) are respectively the in horizontal and vertical directions, N and N are
h v
absolutevalueandnorm,EEE{}meanstheexpectation the number of antennas at the URA in horizontal
operation,diag(xxx)isthediagonalmatrixwithdiago- and vertical directions, λ is the carrier wavelength,
nal entries given in the vectorxxx, and diag(XXX) is the cosθ and cosφ are the direction cosines of
n,m n,m
diagonalmatrixwiththediagonalequaltothatofthe the mth ray in the nth cluster respectively in the
matrixXXX. horizontalandverticaldirections.
Oneexampleofnon-decomposablearrayresponse
II. SYSTEM AND CHANNEL MODELS
is the uniform concentric circular array (UCCA).
Consider a downlink 3D MIMO system where an Denote J and L respectively as the number of rings
array with N antennas is mounted at the BS [2,27], and the number of antennas equally placed on each
t
andallantennasareomni-directional.Anexampleof ringinthearray.Asshownin[30],thearrayresponse
3DMIMOsystemwithURAisshowninFig.1. ofUCCAcanbeexpressedas
For easy exposition, we start by considering the
aaa(Θ ) = [aaa(ϕ )T,···aaa(ϕ )T]T (3)
3Dmultiple-input-and-single-output(MISO)system n,m 1 L
4
with When tens or hundreds of antennas are placed
in the array, say N = 64 or 256, the dimension
aaa(ϕl)=[ej2πdλ1cos(φn,m−ϕl)cosθn,m, of vectors aaa(Θ )tand hhh can be very large, and
n,m
···,ej2πdλJcos(φn,m−ϕl)cosθn,m]T the dimension of corresponding channel correlation
matrixismuchlargersince
where d and ϕ = 2lπ/L are respectively the radius
j l
of the jth ring and the lth radial direction in the RRR =EEE{hhhhhhH} (5)
array, j = 1,...,J, l = 1,...,L, θ and φ are
n,m n,m which is of size N × N . In fact, the dimension
respectively the directions of the mth ray in the nth t t
N ×N is large even for small value of N and N ,
t t h v
cluster with respect to the positive x- and y-axis in
say N = N N = 82 for N = N = 8. Such a
t h v h v
thearray.
largedimensionnotonlyincreasesthecomputational
¯
The CDI of 3D MIMO channel is hhh = hhh/|hhh|,
complexity for MIMO signal processing but also
whichisofunit-norm[7].InlimitedfeedbackMIMO
resultsinthedifficultlyofCDIfeedback[31].
systems, the CDI is quantized at the user by using a
The channel correlation matrixRRR is important for
pre-determined codebook and then fed back to the
many codebooks [17,19,26]. For example, when RRR
BS. The quality of CDI available at the BS largely
is available, the rotated codebook can immediately
depends on the codebook. A desirable codebook be applied to quantize the 3D MIMO CDI vector,h¯h¯h¯.
shouldbejudiciouslydesignedbytakingthechannel
Specifically, the globally rotated codeword can be
featuresintoaccount.Toobservetheuniquefeatures
constructed from an instantaneous codeword ggg of
0
of3DMIMOchannels,besidesthevectorexpression
sizeN ×1as[17]
hhh ∈ CNt×1, we also express the channel in matrix t
form. ccc =RRR1/2ggg (6)
g 0
Taking the URA as an example, the 3D MIMO
wherethecodewordccc isofunit-norm.
channelandthearrayresponsecanbeexpressedas g
The globally rotated codebook has been proved
HHH ∈ CNh×Nv andAAA =aaa (θ )aaa (φ )T (4) to be asymptotical optimal to quantize arbitrary spa-
n,m h n,m v n,m
tially correlated channels [18]. Yet it is challenging
where hhh = vec(HHH), aaa(Θ ) = vec(AAA ), and
n,m n,m to obtain the channel correlation matrixRRR at the BS
vec(·) denotes the operation of vectorizing a matrix
sinceN islargein3DMIMOsystems.Inthesequel,
t
into a vector. Then, the CDI can be expressed as
we strive to reduce the dimension of quantization by
¯
HHH =HHH/(cid:107)HHH(cid:107).
exploitinguniquefeaturesof3DMIMOchannels.
The matrix representation helps us to identify the
different types of channel information in 3D MIMO
channels with the URA, since the columns and rows A. DifferentTypesofCDIof3DMIMOChannels
of HHH respectively stand for the sub-responses at the We begin with the following proposition to iden-
horizontalandverticaldirections. tify different types of CDI in 3D MIMO channels.
Iftherayresponsesofanarrayaredecomposable,
i.e.,aaa(Θn,m) = xxx⊗yyy, we can easily find the matrix Proposition1: Any 3D MIMO channel matrixHHH
expression for each array response as xxxyyyT and the can be decomposed into HHH = UUU HHH UUUT such that
h t v
matrixexpressionforthechannel,wherexxxandyyy are EEE{HHH HHHH} = diag(λλλ ) andEEE{HHHTHHH∗} = diag(λλλ ),
t t h t t v
not necessarily the sub-responses in the horizontal whereUUU andUUU areunitarymatrices,λλλ andλλλ are
h v h v
and vertical directions. However, the matrix expres- vectorswithnonnegativeentries.
sionsforarbitraryantennaarrayscannotbeobtained
Proof:Thepropositioniseasytoshowbytakingthe
straightforwardlyandwillbediscussedinSectionIV.
SVD to channel matrix, which is given here simply
forintroducingnotations.
III. CODEBOOK DESIGN Denote the SVD of the left and right correlation
matricesof3DMIMOchannelrespectivelyas
In this section, we propose a joint quantization
codebook for spatially correlated 3D MIMO chan-
RRR =EEE{HHHHHHH} =UUU diag(λλλ )UUUH (7)
nels, given that the channel matrix expression HHH is h h h h
RRR =EEE{HHHTHHH∗} =UUU diag(λλλ )UUUH (8)
available. v v v v
5
¯
where the entries of λλλ and λλλ are in a descending channelinmatrixform,HHH =HHH/(cid:107)HHH(cid:107),byseparating
h v
order.ConsideringthatHHH =UUUHHHHUUU∗,andbyusing the statistical sub-directions, the average and instan-
t h v
(7)and(8),wecanobtaintheproposition.(cid:4) taneouschannelgains,whichis
This proposition suggests that for arbitrary 3D
ˆ ˆ ˆT
MIMO systems under arbitrary channels, we can CCC =UUU (ΛΛΛ(cid:12)GGG)UUU (10)
J h v
always obtain two unitary matrices shown in (7) and
where the codeword CCC is normalized to have unit
(8). Although this is nothing but taking the SVD to J
norm. By using vec(ABCT) = (C ⊗ A)vec(B),
the 3D MIMO channel matrix, but the geometrical
thecodewordstructureforquantizingtheCDIof3D
structureofthearrayhasbeenexplicitlyexploited.
¯
Note that we can also obtain a unitary matrix UUU MIMOchannelinvectorformhhhcanbeexpressedas
forthefullchannelcorrelationmatrixbytheSVDas
ˆ ˆ ˆ
ccc = (UUU ⊗UUU )diag(λλλ)ggg (11)
RRR = UUUdiag(λλλ)UUUH. To emphasize the difference, the J v h
unitarymatrixUUU isreferredtoasstatisticaldirection ˆ ˆ
whereccc = vec(CCC ),λλλ = vec(ΛΛΛ)andggg = vec(GGG).
information,andthetwounitarymatricesUUU ,UUU are J J
h v Theroleofeachpartofthecodewordisasfollows,
referredtoasstatisticalsub-directioninformation.
ˆ
Thestatisticaldirectioninformationcantransform • unitarymatrixUUUh isofsizeNh×rh,whichtar-
the 3D MIMO channel matrixHHH into at most N N gets at the statistical sub-direction information
h v
UUU inhorizontaldirection(1 ≤ r ≤ N ),
uncorrelatedchannelgainsexpressedinthediagonal h h h
ˆ
matrix diag(λλλ). In contrast, the two statistical sub- • unitary matrixUUUv is of size Nv ×rv, which tar-
gets at the statistical sub-direction information
direction information independently transform the
channel matrix HHH into at most N and N uncor- UUUv inverticaldirection(1 ≤ rv ≤ Nv),
h v ˆ
related channel gains respectively expressed in di- • nonnegative scaler matrix ΛΛΛ is of size rh × rv,
agonal matrices diag(λλλ ) and diag(λλλ ). One desir- which targets at the statistical power coupling
h v
informationΛΛΛbetweenthehorizontalandverti-
able property of statistical sub-direction information
caldirections,
is that the dimensions of the correlation matrices
of N × N and N × N are much smaller than • instantaneous codeword GGG is of size rh × rv,
h h v v
N N × N N for statistical direction information. which quantizes the instantaneous power cou-
h v h v
pling information. In fact, the instantaneous
In the sequel, we refer to the two sub-directions as
power coupling information reflects the instan-
“horizontal”and“vertical”directions,althoughsuch
taneousCDIofHHH inProposition1.
notions only agree with their physical meanings for t
theURA. With the codeword structure in (10), the CDIs in
Denote ΛΛΛ as the the average channel gain of horizontal and vertical directions are jointly quan-
i,j
the (i,j)th element in HHH given by Proposition 1, tized together with the power coupling. Therefore,
t
e.g., ΛΛΛ2 = EEE{|HHH |2}. By expressing uuu and vvv we refer the codebook of codewords with this struc-
i,j t,i,j i i
respectively as the ith column of matrixUUU andUUU , tureasthejointCDIquantizationcodebook.
h v
thenΛΛΛ canbealsoexpressedby Compared with the codeword structure of the
i,j
ΛΛΛi,j=EEE12{|uuuHi HHHvvv∗j|2}. (9) gcolodbeawlloyrdrotsatrteudctcuordeeibnoo(1k1s)honweendisn(lo6w),-tdhiempernospioosneadl
which means ΛΛΛ is the common average channel statisticalchannelinformationforrotation.
i,j
gain shared by the ith statistical horizontal direction In practice, the statistical informationUUU ,UUU and
h v
andjthstatisticalverticaldirection. ΛΛΛcanbeobtainedattheBSeitherbyuplinkchannel
Different from λλλh in (7) and λλλv in (8) which are estimationorbyfeedback.
obtained separately, ΛΛΛ is interacted jointly with UUUh In FDD systems where the downlink and the
andUUUv,andcanbeinterpretedasthestatisticalpower uplink are operated at separated frequency bands,
couplinginformation. estimating the downlink channel correlation matrix
from uplink training symbols is still possible [32].
B. TheProposedCodewordStructure
However,suchamethodmaynotbeusedtoestimate
Based on Proposition 1 and the observations in the channel correlation matrix RRR for 3D MIMO
lastsubsection,weproposeajointquantizationcode- systems,whichisofhighdimension.Theproblemof
word structure for quantizing the CDI of 3D MIMO estimatingthecorrelationmatrixforhigh-dimension
6
random vectors is recognized as “curse of dimen- associatedwithdifferentreceiveantennasmaydiffer.
sionality” and far from trivial [24]. This is because Based on this observation, we can easily extend the
the dimension of the channel correlation matrix may codebook design to the systems where each user is
be comparable with the number of collected training equippedwithN antennas.
r
symbols such that the widely-used “sample covari- Denotethesuper-scriptinparenthesesastheindex
ance”estimationbecomesinvalid. of receiver antenna, the joint quantization codeword
The matricesUUU ,UUU andΛΛΛ can also be quantized in(11)formulti-antennauserscanbegivenas
h v
and fed back [33,34]. Since the spatial correlation
information can be fed back in wide-band and in ccc(1,...,Nr)=[ccc(1),···,ccc(Nr)]
J J J
long-term, the feedback overhead can be almost ig- = (UUUˆ ⊗UUUˆ )diag(λλλˆ)[ggg(1),···,ggg(Nr)]
v h
nored compared with the feedback for instantaneous
CDI. This is especially true when the array feature where ggg(1),··· ,ggg(Nr) are respectively the instanta-
is taken into account. For example, for the URA, neouscodewordsfordifferentreceiveantennas.
according to Szego’s theory of Toeplitz matrices,
the statistical sub-direction information UUU and UUU
h v IV. STATISTICAL INFORMATION FOR
become a subset of DFT matrix when the size of the
ARBITRARY ANTENNA ARRAY
array increases [11]. This will significantly simplify
thefeedbackforthecorrelationinformation. Theproposedcodewordstructureisapplicablefor
Since our focus is to reduce the dimension of any antenna array if the channel can be expressed
quantizationbyintroducingnewcodewordstructure, in matrix form as HHH. This is because Proposition 1
we assume that the statistical channel information is valid for arbitrary 3D MIMO channels. Although
UUU ,UUU andΛΛΛ are available at the BS. Then, the ma- quite natural to the decomposable arrays such as the
h v
tricesusedtoconstructthecodewodwithstructurein URA,thewaytoexpressthechannelsinmatrixform
(10)UUUˆ ,UUUˆ andΛΛΛˆ canbeobtainedas for arbitrary antenna array is not obvious because of
h v
twoissues.
ˆ ˆ
UUU = [uuu ,··· ,uuu ], UUU = [vvv ,··· ,vvv ]
h 1 rh v 1 rv The first issue is the choice of dimensionality for
ˆ
ΛΛΛ =ΛΛΛ ,with1 ≤ i ≤ r ,1 ≤ j ≤ r (12) the 3D MIMO channel matrix, i.e., N and N with
i,j i,j h v h v
N ×N = N .TakingtheUCCAarraywithN = 32
The codeword quantizing the instantaneous CDI h v t t
as an example, we may express the channel vectorhhh
of 3D MIMO channel, i.e., GGG, can be obtained by
inamatrixformHHH ofsize8×4,4×8,ortheothers,
usingthecodebooksdesignedforuncorrelatedchan-
but we are not clear which expression achieves a
nels with dimension of r × r . For example, GSP
h v better quantization performance because the array is
codebooks proposed in [7] can be used, which is
notrectangular.
optimizedforuncorrelatedRayleighchannels.
The second issue is the “antenna grouping” in
The dimensions r and r in (10) can be designed
h v the channel matrix, i.e., which subset of antenna
to further reduce the dimension of codeword by
responses should be in the same row or column in
discarding trivial statistical sub-directions in strong
HHH for a given dimensionality of N and N . The
correlated channel. We leave this topic for future h v
antenna grouping determines the values of UUU , UUU
study, since similar idea has already been developed h v
andΛΛΛ, and eventually affects the performance of the
in [26], and moreover, the dimension parameter can
proposedcodebook.
be optimized by considering various system param-
As a consequence, we need to solve these two
eters, e.g., the error tolerance, storage space, and
issues when designing the joint codebook for 3D
overallquantizationperformance.
MIMO systems with arbitrary antenna array. The
challengesforaddressingthetwoissuesaredifferent.
C. ExtensiontoMulti-antennaUsers
The choices satisfying N = N × N are always
t h v
Whenconsideringmultiplereceiveantennasatthe limited, and thus exhaustive searching is efficient to
user, it is reasonable to assume that the statistical find the best dimensionality. However, the possible
channel information in (10) associated with differ- choices of antenna grouping exponentially increases
ent receive antennas are identical due to the small with N , which belongs to a combinational prob-
t
space separation. However, the instantaneous CDI lem and is of prohibitive computational complexity.
7
Therefore, we focus on the second issue in the fol- Problem (14) belongs to a classic approximation
lowing, i.e., to find properUUU ,UUU andΛΛΛ for a given problem of Tucker decomposition [25], where λλλ is
h v t
N andN . thecoretensor,UUU andUUU arerespectivelythefactor
h v h v
matriceswithcolumnsuuu andvvv astensors.Themain
i i
objective of Tucker decomposition is to decompose
A. OptimizingUUU ,UUU andΛΛΛ
h v
a higher dimensional matrix into low dimensional
To find the desirable statistical information, be-
factor matrices, and the tensor core encompass all
fore solving the antenna grouping problem, we first
the possible interactions among the low dimensional
reconsider the proposed codebook structure. Recall
tensorsinthefactormatrices.Inotherwords,Tucker
that the entries ofHHH defined in Proposition 1 reflect
t decompositionistoreducethedimensioninthelarge
the instantaneous channel gains, and the GSP code-
matrixbyfindingthestructureproperties.Moreover,
book can be used to quantize the instantaneous CDI,
Tucker decomposition is a generation of the matrix
which is optimal for i.i.d. channels. Therefore, the
SVD with several desirable features, such as orthog-
proposed codebook will be optimal if the entries of
onality, decorrelation and computational tractability.
HHH are uncorrelated and the statistical information is
t However, in general the problem of finding the solu-
ˆ ˆ ˆ
perfect(i.e.,UUU =UUU ,UUU =UUU andΛΛΛ =ΛΛΛ).
h h h v tion of Tucker decomposition is NP-hard, and there
ˆ
We can reconstruct a full channel matrix RRR from
arefewefficientalgorithmsinuse.
thematriceswithlowerdimensionsas
Note that although the problem in (14) contains
RRRˆ =EEE{vec(UUU HHH UUUT)vec(UUU HHH UUUT)H} three variables, it can be simplified by only optimiz-
h t v h t v
ing UUU and UUU without losing the optimality. This
= (UUU ⊗UUU )EEE{vec{HHH }vec{HHH }H}(UUU ⊗UUU )H h v
v h t t v h is because given UUU and UUU , the objective function
= (UUU ⊗UUU )diag(vec{ΛΛΛ(cid:12)ΛΛΛ})(UUU ⊗UUU )H (13) h v
v h v h satisfies
where (13) is achieved when the entries of HHH are
t min(cid:107)RRR−(UUU ⊗UUU )diag(λλλ )(UUU ⊗UUU )H(cid:107)2
uncorrelated. λλλt v h t v h F
ˆ
If RRR = RRR, the proposed codebook is identical to = min(cid:107)(UUU ⊗UUU )HRRR(UUU ⊗UUU )−diag(λλλ )(cid:107)2 (15)
v h v h t F
the globally rotated codebook, since the codeword λλλt
(cid:0) (cid:1)
in (11) becomes ccc = RRRˆ12ggg and the same as in (6). = min(cid:107)diag (UUUv⊗UUUh)HRRR(UUUv⊗UUUh) −diag(λλλt)(cid:107)2F
J λλλt
Unfortunately, in general cases, RRRˆ (cid:54)= RRR, and the +(cid:107)off(cid:0)(UUU ⊗UUU )HRRR(UUU ⊗UUU )(cid:1)(cid:107)2
v h v h F
proposed codebook becomes inferior to the globally (cid:0) (cid:1)
= (cid:107)off (UUU ⊗UUU )HRRR(UUU ⊗UUU ) (cid:107)2 (16)
rotated codebook, which is asymptotically optimal. v h v h F
To provide good quantization performance, it is rea-
where (15) is due to fact that the norm is unitarily
ˆ
sonable to findUUUh,UUUv andΛΛΛ such thatRRR is as close invariant, off(XXX) is the operation to the matrix XXX
toRRR aspossible. with all zeros on the diagonal, (16) is achieved by
For simplicity, we define λλλt = vec{ΛΛΛ (cid:12)ΛΛΛ}. The theoptimalλλλt foragivenUUUh andUUUv,whichis
problem of finding the optimal UUU , UUU and λλλ for
h v t (cid:0) (cid:1)
arbitraryantennaarraycanbemodeledas diag(λλλ ) = diag (UUU ⊗UUU )HRRR(UUU ⊗UUU ) (17)
t v h v h
min (cid:107)RRR−(UUU ⊗UUU )diag(λλλ )(UUU ⊗UUU )H(cid:107)2 (14) Itcanbeverifiedthattheoperationforcomputingλλλ
v h t v h F t
UUUh,UUUv,λλλt in(17)isidenticaltothatforcomputingΛΛΛwith(9).
s.t. UUUHUUU =III ,
v v Nv Since (16) does not depend on the parameter λλλt,
UUUHUUU =III , we only need to find UUU and UUU from a new opti-
h h Nh h v
mizationproblembyreplacingtheobjectivefunction
λλλ (cid:31) 0
t
in(14)with(16).WiththeoptimizedUUU andUUU ,we
h v
where IIIn is the identity matrix of size n × n, and canimmediatelyobtainingoptimalλλλt byusing(17).
xxx (cid:31) 0 means each element in xxx is larger than 0.
The problem in (14) does not directly optimize the
B. Closed-formSolutionforUUU andUUU
antenna grouping in the channel matrix, which is h v
unnecessary since when given the optimal solution Since a closed-form solution is more desirable for
ofUUU ,UUU andλλλ ,wecanobtainthematricesusedto practical use, we consider a modified problem to
h v t
constructthejointquantizationcodewods. find UUU and UUU . Specifically, we impose an extra
h v
8
constraint on λλλ into the new optimization problem, semi-definite [35]. Thus, the matrices BBB and CCC can
t
whichisgivenby beregardedasthecorrelationmatrices.
By letting RRR = BBB, RRR = CCC, using the SVD, we
λλλ =λλλ ⊗λλλ (18) v h
t v h can obtain UUU and UUU immediately. Then, by using
v h
whereλλλ (cid:31) 0 andλλλ (cid:31) 0. Then, the term inside the (17),wecanobtaintheoptimalλλλ orΛΛΛ.
v h t
objectivefunctionofproblem(14)becomes It is easy to validate that the statistical direction
informationUUU is approximated by the lower dimen-
(UUU ⊗UUU )diag(λλλ )(UUU ⊗UUU )H =BBB ⊗CCC (19)
v h t v h sion matrices as UUU ⊗UUU , and the average channel
v h
whereBBB =UUU diag(λλλ )UUUH andCCC =UUU diag(λλλ )UUUH gains in λλλ are approximated by λλλ . By introduc-
v v v h h h t
arepositivesemi-definitematrices. ing the structure of joint quantization codebook, we
With the constraint in (18) and considering (19), employ a new correlation matrix constructed with
theoptimizationproblemtofindUUU andUUU becomes reduced dimension that approximates the full corre-
h v
lationmatrixascloseaspossible.Therefore,withthe
min (cid:107)RRR−BBB ⊗CCC(cid:107)2 (20)
F obtained correlation matrices, we can not only solve
B,C
s.t. BBB ∈ SNv×Nv,andCCC ∈ SNh×Nh thedimension problemfor rotatedcodebook in[17],
but also for the codebook in [19], both rely on the
where Sn×n is the space of positive semi-definite channelcorrelationmatrices.
matriceswiththedimensionalityofn×n.
The new problem in (20) offers a suboptimal so- V. PERFORMANCE ANALYSIS
lutionofUUUh andUUUv withclosed-formfortheTucker Since the globally rotated codebook is asymptot-
decomposition,andisknownastheKroneckerprod- ical optimal and can serve as a performance upper
uct decomposition [35]. There are many other bene- bound for other codebooks, we analyze the per-
fitstoconsidertheKroneckerproductdecomposition formance of the proposed joint CDI quantization
here. By using such an decomposition, many struc- codebook by comparing with the globally rotated
ture properties ofRRR, such as symmetry, definite, and codebookinthissection.
permutations,canbeinheritedbythematricesBBB and As addressed in Section IV.A, if the entries ofHHH
t
CCC. It is worthy to note that these structure properties are uncorrelated and the statistical information used
are usually led by the regularity of antenna array, in the codeword with structure in (10) is perfect, the
e.g.,symmetry,andnestedsubarrays. joint CDI quantization codebook will perform the
Thesolutiontotheproblemin(20)isgivenin[35]. same as the globally rotated codebook. However, it
To obtain the solution, according to [35], we need to is unclear whether the entries ofHHH are uncorrelated
t
rearrange the matrixRRR. Specifically, the matrixRRR is or not. In what follows, we show that for 3D MIMO
dividedintoNv ×Nv blocksas systems with URA such an uncorrelated property is
 RRR ··· RRR  validunderverygeneralchannelconditions.
1,1 1,Nv
RRR =  ... ... ...  (21) To facilitate analysis and gain useful insights, we
consider weak i.i.d. rays in 3D MIMO channels,
RRR ··· RRR
Nv,1 Nv,Nv whichassume,
where the (i,j)th block denoted as RRR is of size 1) theanglesθ arei.i.d.for∀nandm,
i,j n,m
N ×N .Thenarearrangedmatrixisgeneratedby 2) theanglesφ arei.i.d.for∀nandm,
h h n,m
3) thegainsg areuncorrelatedfor∀nandm,
RRR˜= [vec(RRR ),vec(RRR ),··· ,vec(RRR )]T (22) n,m
1,1 2,1 Nv,Nv where the word “weak” comes from the fact that
whichisofsizeN2 ×N2. the third condition only requires the gains being
v h
Denote the largest singular value, the correspond- uncorrelatedbutnotbeingidenticallydistributed.
˜
ing left and right eigenvectors of the SVD to RRR The above assumption does not weaken our per-
respectively as σ2, uuu and vvv. Then, as shown in [35], formance analysis for realistic scenarios. First of
thematricesBBB andCCC areobtainedas all, the assumption of uncorrelated ray gains follows
from the commonly-used uncorrelated scattering as-
vec(BBB) = σuuu,andvec(CCC) = σvvv (23)
sumption for multi-path fading channels, which is
Moreover, Since RRR is symmetric and positive semi- validated by many channel models [9,28,29]. Sec-
definite, BBB and CCC are also symmetric and positive ond, i.i.d. angles are observed very typical and thus
9
are adopted in the 3GPP channel models [29]. Fi- example, UCCA) are used. This means for general
nally, no specific probability density distribution as- conditions, the two codebooks are not identical. In
sumptionisimposedonthegainsandangles.Forex- these cases, the joint CDI quantization codebook,
ample, the angles can be uniformly distributed [28], thoughstill applicable,becomesinferiorto theglob-
Gaussian distributed [9], or log-normal distributed allyrotatedcodebook.
[29].
In the following, we show that the proposed joint
A. Further Comparison With Globally Rotated
CDI quantization codebook achieves the same per-
Codebook
formance as the globally rotated codebook when the
3DMIMOchannelshaveweaki.i.d.raysandtheBS As shown in (5), the globally rotated codebook
isequippedwithURA. employs the channel correlation matrix, which is
of size N N × N N . By contrast, the joint CDI
Lemma1: Fora3DMIMOsystemwithURA,the h v h v
quantization codebook employs three lower dimen-
channel with weak i.i.d. rays can be expressed as
sion correlation matrices, each respectively of size
HHH = HHH´UUUT, where UUU is a unitary matrix and the
v v N × N for RRR given by (7), of size N × N for
columnsofHHH´ areuncorrelatedwitheachother. h h h v v
RRR given by (8), and of size N × N for ΛΛΛ given
v h v
Proof:SeeAppendixA.(cid:4)
by (12). Apparently, a high dimension matrix leads
Lemma2: Fora3DMIMOsystemwithURA,the to a high complexity in generating the codewords to
channel with weak i.i.d. rays can be expressed as quantize the statistical information or transforming
HHH = UUU HHH`, where UUU is a unitary matrix and the the instantaneous codeword. By using the joint CDI
h h
rowsofHHH` areuncorrelatedwitheachother. quantization,thecomputationalcomplexityissignif-
Proof:TheproofissimilartoLemma1.(cid:4) icantlyreduced.
In addition, different size of correlation matrix
Lemma3: For a 3D MIMO system with URA
needs different amount of information to estimate or
under the channel with weak i.i.d. rays, the entries
feedback. Taking an 8×8 3D MIMO channel as an
of HHH in the channel decomposition HHH = UUU HHH UUUT
t h t v example, the globally rotated codebook requires a
givenbyProposition1areuncorrelated.
correlation matrix with 642 = 4096 entries, while
Proof:SeeAppendixB.(cid:4)
the joint CDI quantization requires the correlation
By using these lemmas for the 3D MIMO system matrix only with 64 × 3 = 192 entries. Even when
with URA under channels with weak i.i.d. rays, we the correlation matrices RRR, RRR , RRR are Hermitian
h v
can immediately find the relationship between the andToeplitz,thegloballyrotatedcodebookneeds63
joint CDI quantization codebook and the globally complex elements inRRR, while the joint quantization
rotatedcodebook. only needs 7 complex elements in RRR , 7 in RRR , and
h v
r × r scaler elements in ΛΛΛ. This implies that the
Theorem1: For a 3D MIMO system with URA h v
proposedjointCDIquantizationbookrequiresmuch
underthechannelwithweaki.i.d.rays,theproposed
less channel correlation information to be estimated
joint quantization codebook with perfect correlation
orfedbackthanthegloballyrotatedcodebook.
matrices performs the same as the globally rotated
codebook.
Proof:SeeAppendixC.(cid:4) B. ComparisonWithIndependentCDIQuantization
It follows that for 3D MIMO systems with URA Another strategy, independent CDI quantization,
under the channels with i.i.d. rays, the proposed can be applied to 3D MIMO systems, where the
joint CDI quantization codebook is asymptotically CDIs in horizontal and vertical directions are quan-
optimal as the codebook size increases, which is tizedindependently[4].Bynature,jointquantization
promisedbythegloballyrotatedcodebook[18]. is not a simple aggregation of independent quantiza-
It is worthy to note that the Theorem 1 provides tionintwodirections.
a sufficient condition on the 3D MIMO channels Denotethecodewordsindependentlydesignedfor
wherethejointCDIquantizationandgloballyrotated horizontal and vertical directions as ccc and ccc ,
Ih Iv
codebook are identical. The entries in HHH may not respectively. In [4], the CDI of 3D MIMO is recon-
t
be uncorrelated when other antenna arrays ( for structed by two independently designed codewords
10
asCCCI = cccIhcccTIv. Taking independently rotated code- VI. SIMULATION RESULTS
book as an example, ccc and ccc are respectively
Ih Iv In this section, we compare the performance of
obtained by normalizing RRR1h/2ccch and RRR1v/2cccv, where joint CDI quantization with existing codebooks for
ccch andcccv areinstantaneouscodewordsforhorizontal 3DMIMOsystemswithplanararraybysimulations.
and vertical directions. The independent CDI quan- We consider multi-user MIMO transmission,
tization is easy to implement and is compatible to whereK single-antennausersareservedbytheBSat
existing 2D MIMO systems, but at a cost of quanti- the same time-frequency resource with zero-forcing
zationperformanceloss. beamforming [36]. In the simulation, the users are
randomly selected, which is equivalent to be se-
First, independent CDI quantization will not be
lected by Round-Robin scheduling. The users are
optimaltoquantizethe3DMIMOchanneliftherank
with homogeneous SNR but with different azimuth
of channel matrix HHH exceeds one. The rank of HHH,
and elevation angles. In order to put the results of
denotedasrank(HHH),hasacloserelationshipwiththe
differentantennaarraysizewithinonefigure,theX-
scattering environment. If there are multiple well-
axis is set as the receive SNR. For transmit SNR,
separated clusters or rays in the channel, it is easy
an extra log N dB should be considered due to the
to see that rank(HHH) is far larger than one. However, 2 t
array gain. The gains of each ray are i.i.d. complex
the independent CDI quantization codebook always
Gaussian distributed with zero mean. The sum rates
yieldsrank(CCC ) = 1,whichfailstomatchtherankof
I
are computed by the Shannon formula by averaging
generic 3D MIMO channel matrix. This implies that
over103 channelrealizations.
independentCDIquantizationisonlyappropriatefor
Inthefollowing,weconsidertwodifferentangular
quantizing the rank one 3D MIMO channels with
spread modelings for the channel given by (1). One
a single ray. By contrast, the row rank and column
is simplified, which allows us to run simulations
rank of the joint CDI quantization codewordCCC are
J
for large scale system efficiently. The other is more
respectively r and r . By selecting proper values
h v
realistic, which is given in [29] and allows us to
of r and r , we can design joint CDI quantization
h v
obtain more reliable results. It should be noted that
codeword for any 3D MIMO channel matrix HHH. In
ourconclusionsholdforbothchannelmodels.
practice, the values of r and r may be selected
h v
Inthesimulations,alltheinstantaneousCDIcode-
lessthan therankofHHH forthe purposeofdimension
words are generated using random vector quantiza-
reduction. In these cases, the performance loss can
tion (RVQ), which is easy to generate while with
be minimized by reserving the statistical directions
performanceclosetotheGSPcodebook[37].Specif-
in(10)withnontrivialaveragechannelgains.
ically, two B-bit RVQ codebooks are used for quan-
Second, independent CDI quantization results in
tizing the instantaneous horizonal and vertical chan-
redundant quantization on the channel information,
neldirectionsintheindependentlyrotatedcodebook.
since the codebooks are separately designed for hor-
A 2B-bit RVQ codebook are used for quantizing the
izontal and vertical directions from their own per-
instantaneous channel direction information respec-
spective. To see this, we can find that the power
tively in the joint CDI quantization codebooks, and
profile of uncorrelated clusters seen at horizontal
inthegloballyrotatedcodebook.
direction and at vertical direction in the transformed
3D MIMO channel after SVD are respectively the
A. ResultsinSimplifiedScenarios
eigenvalues ofRRR andRRR given by (7) and (8). The
h v
In the simplified angular spread model, the az-
two power profiles are generally not independent
imuthandelevationanglesin(1)are,
with each other. However, with independent CDI
quantization the two power profiles are quantized θ = θ +θ +δθ (24)
n,m 0 n n,m
separately, which leads to a redundancy in the CDI
φ = φ +φ +δφ (25)
quantization. By contrast, when using joint CDI n,m 0 n n,m
quantization, such a redundancy can be completely where θ = UUU(−60◦, 60◦) and φ = UUU(−45◦, 45◦)
0 0
removed by quantizing the power coupling informa- are respectively the mean of azimuth and elevation
tion. As a consequence, joint CDI quantization is clusters, and UUU(a,b) denotes uniform distribution
more efficient to quantize the 3D MIMO channels withrangefromatob,θ andφ arethetheazimuth
n n
thanindependentCDIquantization. and elevation deviation of the nth cluster from their