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The Wireless Networking and Communications Group
Impact of Spatial Correlation and Distributed Antennas
for Massive MIMO SystemsKien T. Truong* and Robert W. Heath Jr.
Wireless Networking & Communication Group Department of Electrical & Computer Engineering
The University of Texas at Austin http://www.profheath.org
!* MIMO Wireless Inc.
This work was supported by Huawei Technologies
�2
What is Massive MIMO?
A very large antenna array at each base station An order of magnitude more antenna elements in conventional systems
A large number of users are served simultaneously
An excess of base station (BS) antennas
�2
Essentially multiuser MIMO with lots of base station antennas
Uhundreds of BS antennas
Nt � U � 1Nt � U � 1
Nt � U � 1 Nt � U � 1Nt � U � 1
Nt � U � 1Nt � U � 1 Nt � U � 1Nt � U � 1
Nt � U � 1
tens of users
Pioneered by Marzetta, Larsson, others see e.g. [Mar10], [LarEtAl13], [RusekEtAl13]
Problem StatementMassive MIMO requires many antennas for best performance
Yet co-locating large antennas is challenging
Massive MIMO analysis often assumes uncorrelated antennas
Yet spatial correlation is likely present when the arrays are packed
Objectives
Explore potential gains achieved with MRC & MMSE strategies
Establish the potential of massive MIMO in distributed antenna systems
Investigate performance improvement with remote radio unit selection
�3
The Wireless Networking and Communications Group
System Model
Nt � U � 1
Nt � U � 1
Nt � U � 1Nt � U � 1
Nt � U � 1Nt � U � 1
UUU
U UU
Central station c
�5
System Model
cells share the same frequency, i.e. universal frequency reuse Each cell has one BS with antennas & single-antenna users
Time-division duplexing (TDD) protocol
MIMO operation is multiuser rather than single-user transmission
BSs estimate inst. channels based on UL (training) signals�5
UNt � U � 1
Nt � U � 1
Nt � U � 1UEU
gbcu[n] ⇠ CN (0, INt)
UE
UE ⇤ = IU
AMRCr,bu = |tr�bbu|2
BMRCr,bu =
�2b
prtr �bbu
CMRCr,bu =
X
(c,k) 6=(,bu)
trRbck�bbu
DMRCr,bu =
X
c 6=b
|tr�bcu|2
uyr,b =
ppr
CX
c=1
Hbcxr,c + nr,b
U UCentral station b
Pilot-based: Uplink Training
�6
Orthogonal pilot sequences within a cell
Cells share a common set of pilot sequences
MMSE channel estimation based on the observation at BS b
xr,c := [xr,c1, · · · , xr,cU ]
C
Training stage in pilot-based methods causes spectral efficiency loss
Nt � U � 1Nt � U � 1Nt � U � 1 UU U
Central station c
UNt � U � 1
Nt � U � 1
Nt � U � 1 U UCentral station b
xr,ck
Rbcu
Nt
The downlink for each user is organized in three phases:first, a user sends a pilot sequence of length K to its BTS.Using this pilot the BTS estimates the corresponding channelvector and generates a beamforming vector. We assume thata BTS needs a time interval of N OFDM symbols for thisprocessing. Finally, the BTS uses the beamforming vector totransmit data during T � K � N OFDM symbols. Figure 1depicts this protocol for T = 9, K = 3, and N = 1.
Fig. 1. TDD downlink transmission protocol
III. ASYMPTOTIC BEHAVIOR OF SINRIn this section, we analyze the asymptotic behavior of the
SINR as the number of BTS antennas M tends to infinity whilethe number of users Nu remains finite and constant. We firstanalyze the case where all users send the pilots simultaneously.In [5] SINR values for this regime are obtained. We revisitthem by taking into account all constants that do not vanishin the asymptotic regime, pilot powers, and base stationtransmit powers. Next, we propose and analyze a scheme withtime-shifted pilots and show that this scheme allows one tosignificantly reduce the interference in the asymptotic regime.
In both cases, we assume that in all cells the sameset of K orthogonal pilots of length K is used. The k-th users in all cells use the same pilot sequence k =
( k1, . . . , kK), | kj | = 1. Since the pilots are orthogonalwe have | H
k0 k| = K�k,k0 .
A. Aligned Pilots
Here, we focus on the case where pilots are sent simulta-neously by all users in the system. At the first phase, the i-thbase station receives the signal
yBi=
LX
l=1
KX
k=1
p⇢kl �iklhikl k + zi, (1)
where zi 2 CM⇥K is the additive noise. Without loss ofgenerality, we assume that the entries of zi are i.i.d. CN (0, 1)
random variables and that all gains are scaled accordingly.The i-th base station estimates the vectors hik0i for users
located in the same cell as ˆ
hik0i =yBi
Hk0
K
, which results in
ˆ
hik0i =
p⇢k0i�ik0i hik0i +
LX
l=1,l 6=i
p⇢k0l�ik0l hik0l + z
0i, (2)
where z
0i =
zi Hk
K ⇠ CN (0,
1K IM ). The base station computes
the beamforming vector to its k
0-th user as the normalizedversion of (2).
Let us define the normalization factor ↵k0i =khik0ikp
M, so that
wk0i =hik0i
↵k0ipM
. The k
0-th user of the i-th cell then receives
yUk0i =
LX
l=1
KX
k=1
pPkl�lk0ih
Hlk0iwklskl + vk0i, (3)
where skl is the signal intended to the k-th user in the l-th celland vk0i is the unit variance additive white Gaussian noise. Tocompute the SINR as M ! 1, we use the following lemma.
Lemma 1. Let x,y 2 CM⇥1be two independent vectors with
distribution CN (0, c I). Then
lim
M!1
x
Hy
M
a.s.
= 0 and lim
M!1
x
Hx
M
a.s.
= c. (4)
Using the fact that the channel vectors of different users areindependent, and applying the above lemma, we can derive theasymptotic behavior of ↵2
k0i:
↵
2k0i =
1
M
LX
l=1
⇢k0l�ik0lkhik0lk2 + kz0ik2 + o(M)
!
a.s.=
LX
l=1
⇢k0l �ik0l +1
K
The k
0-th user of the i-th cell then receives
yUk0i =
LX
l=1
KX
k=1
pPkl�lk0ih
Hlk0iwklskl + vk0i, (5)
where skl is the signal intended to the k-th user in the l-th celland vk0i is the unit variance additive white Gaussian noise.
In the following theorem, we present the behavior of theSINR in the asymptotic regime.
Theorem 1. Consider the downlink of a celllular system
composed of BTSs with M antennas and users with 1 antenna
each, in which users receive signals according to (5). Then,
as M ! 1, the SINR of the k
0-th user in the i-th cell is
&ik0=
Pk0i�2ik0i/↵
2k0iPL
l=1,l 6=i Pk0l�2lk0i/↵
2k0l
, (6)
with ↵k0l =PL
j=1 ⇢k0j �lk0j +1K .
Proof: Due to lack of space, we refer the reader toan extended version of this paper, available in [13], for thedetailed proof of the above theorem.
Note that additive noise impacts only the normalizationconstants ↵k0l. Therefore base station transmit powers Pkl arescalable. This allows for smaller power levels, resulting in amore power-efficient systems.
The expression (6) involves the pilot powers ⇢k0l and onemay think about possible SINR optimization by choosingoptimal pilot powers. The following theorem shows that infact this is not the case.
Theorem 2. If a set of SINRs is obtained for a particular
choice of ⇢kl, it can also be achieved for any other choice of
⇢kl.
Proof: We omit the proof due to the lack of space andalso refer the reader to [13] for the proof.
From this point, we assume that ⇢kl = ⇢ for all k, l. Notethat allocation of base station transmit powers Pkl, in contrast,gives large SINR gain, as it is shown in Section IV.
5774
xr,ck
Signal Model - UL Data Transmission
�7
Nt � U � 1Nt � U � 1Nt � U � 1 UU U
Central station c
UNt � U � 1
Nt � U � 1
Nt � U � 1 U UCentral station b
xr,bu
Transmit signals from users in cell c is Base station b observes
!!
Base station b applies a linear detector to The linear detector is designed based on estimate of channels
�2
Wb 2 CNt⇥U
1 1
hbcu[n] :=R1/2bcugbcu[n]
1
�8
Channel Model with Spatial Corr.
UL channel vector from user u in cell c to base station b !!
: fast fading, uncorrelated WSS complex Gaussian
: deterministic Hermitian positive definite matrix
may include several effects Pathloss and shadowing
Spatial correlation due to inefficient antenna spacing
This model facilitates analysis of distributed antenna systems is a (block) diagonal matrix
(Block) diagonal entries correspond to cluster spatial correlation matrices
Spatial correlation matrices of clusters have different AoA/AoD and pathlosses
�8
�1⌘MRCr,bu =
|tr�bbu|2�2b
prtr �bbu +
P(c,k) 6=(b,u) trRbck�bbu +
Pc 6=b |tr�bcu|2
=AMRC
r,bu
BMRCr,bu + CMRC
r,bu +DMRCr,bu
u
�as
�as
The Wireless Networking and Communications Group
Rate Calculations
�10
General Uplink Achievable Rate
Treating as a SISO channel with known channel of Channel est. error & intf. are treated as uncorrelated noise=> worst case
Ergodic post-processing SINR for user u in cell b
!
!
!
!
Uplink ergodic achievable rate for user u in cell bchannel estimation error local noiseinterference
departure (AoD)), and a given angle spread (AoS) �as
. Note,however, that our approach does work with other more com-plex antenna and correlation models at the expense of compu-tational complexity. Moreover, we assume that the AoAs/AoDsare distributed according to a certain power azimuth spectrum(PAS). The PAS is modeled by the truncated Laplacian pdf,which is given by
P�(�) =
(�asp2�as
e
�|p2�/�as|
, if � 2 [�⇡,⇡]
0, otherwise,(3)
where � is the random variable describing the AoA/AoD withrespect to the mean angle �
0
, and �
as
=
⇣1� e
�p2⇡/�as
⌘�1
.Let �s be the angle separation between two adjacent antennaelements and ⇢ be the rind radius of the circular array. Define✓m(�
0
) := e
jk�⇢ cos(�0�m�s), for m = 0, · · · , Nt
� 1. Next,define the array response as
a(�0
) := [✓
0
(�
0
), ✓
1
(�
0
), · · · , ✓Nt�1
(�
0
)] . (4)
Define B(�
0
,�
as
) 2 CNt⇥Nt as the characteristic function ofthe Laplacian pdf in (3), where for m,n = 0, · · · , N
t
� 1
[B(�
0
,�
as
)]m,n
:=
�
1 +
�2as2
[k�⇢(sin(�0
�m�s)� sin(�
0
� n�s))]2
. (5)
A closed-form approximate expression of the spatial correla-tion matrix for a single channel tap with low �
as
between auser and a base station is given by [4].
R(�
0
,�
as
) =[a(�0
)a⇤(�0
)]⌦B(�
0
,�
as
). (6)
B. Pilot-based Channel Estimation Error
Base station b needs to estimate the channels between itselfand the users in the same cell based on uplink pilots to designthe linear receive filter Wb. We assume that all cell sharethe same set of U pair-wisely orthogonal pilot signals :=
[ T1
, · · · , TU ] 2 CU⇥⌧ , where ⌧ � U and ⇤
= IU . Let pp
be the average transmit power at each user during the trainingstage. Let Z
p,b 2 CNt⇥⌧ be spatially white additive Gaussiannoise vector at base station b during the training stage. Thereceived training signal at base station b is
Yp,b =
pp
p
⌧
CX
c=1
Hbc + Zp,b. (7)
Base station b correlates Yp,b with to obtain ˜Y
p,b =
Yp,b
⇤ 2 CNt⇥U . Specifically, base station b has the fol-lowing noisy observation of the channel to user u 2 Ub
˜yp,bu = hbbu|{z}
desired channel
+
X
c 6=b
hbcu
| {z }interference
+
1
pp
p
⌧
˜zp,bu
| {z }noise
, (8)
where ˜zp,bu ⇠ CN (0,�2
r
INt) is the post-processed noise atbase station b. The effect of these interference channels onchannel estimation error and then on system performance is
called pilot contamination [6]. Similarly, that of noise is callednoise contamination [6].
Base station b applies minimum mean squared error(MMSE) estimation to the right-hand side of (8) to estimatehbbu for u 2 Ub. Define for c, b 2 C and u 2 Ub
Qbu =
�
2
r
p
p
⌧
INt +
CX
c=1
Rbcu
!�1
(9)
�bcu =RbbuQbuRbcu. (10)
The MMSE estimate of hbbu is [7]
ˆhbbu =RbbuQbu
CX
c=1
hbcu +
1
pp
p
⌧
˜zp,bu
!, (11)
where ˆhbbu ⇠ CN (0,�bbu). To compute the channel esti-mates, central station c needs to know Rbbu and
PCc=1
Rbcu.Due to the orthogonality property of the MMSE estimation, theobserved channel can be decomposed as hbbu =
ˆhbbu +
˜hbbu,where ˜hbbu ⇠ CN (0,Rbbu � �bbu) is a channel estimationerror vector. Note that ˜hbbu is uncorrelated with ˆhbbu. Because˜hbbu and ˆhbbu are jointly Gaussian, they are statisticallyindependent. Define ˆHbb := [
ˆhbb1,ˆhbb2, · · · , ˆhbbU ].
C. Uplink Sum-Rate Analysis
Given knowledge of ˆHbb, base station b designs the lineardetector Wb to detect xb. Specifically, we consider the linearmaximal ratio combining (MRC) and MMSE detectors. DefineˆRr,k as the sum covariance matrix of the channel estimation
error and interfering channels (from users in the other cell tobase station b 2 C), which is given by
ˆRb =E
2
4 ˜Hbb˜H⇤
bb +
X
c 6=b
GbcG⇤bc
3
5 (12)
=
CX
c=1
UX
u=1
Rbcu �UX
u=1
�bbu. (13)
The linear detector wbu to detect xr,bu for u 2 Ub is
wbu =
8<
:
ˆhbbu, MRC detector⇣ˆHbb
ˆH⇤bb +
ˆRb +�2r
prI˜Nt
⌘�1
ˆhbbu, MMSE detector.
Note that base station b needs to have the second-orderstatistics of the channels between itself and the users in thenetwork ˆRb for designing the MMSE detector. Moreover,wbu depends only on ˆHbb, the second-order statistics of otherinterference channels from the users in the other cells to basestation b, and the noise power at base station b.
Based on the worst-case uncorrelated additive noise [2],[8], a standard bound on the uplink ergodic achievable ratecorresponding to user u in cell b is
Rbu = E[log2
(1 + ⌘bu)], (14)
where ⌘bu is the associated signal-to-interference-plus-noiseratio (SINR) and is given in (15) on the top of the next page.
where ˆRb =
PCc=1
PUu=1
Rbcu �PU
u=1
�bbu. Note that fordesigning the MMSE detector, base station b needs to haveˆHbb, the second-order statistics of the channels between itselfand the users in the network ˆRb, and its noise power.
3) General Achievable Rate Analysis: Based on the worst-case uncorrelated additive noise [2], [7], a standard bound onthe uplink achievable rate corresponding to user u in cell b is
Rbu = E[log2
(1 + ⌘bu)], (14)
where ⌘bu is the associated signal-to-interference-plus-noiseratio (SINR) and is given in (15) on the top of the next page.
⌘bu =
|w⇤buˆhbbu|2
Ew⇤
bu
⇣hbbuh⇤
bbu +
P(c,v)
hbcvg⇤bcv +
�2r
⌘wbu
��� ˆHbb
�.
(15)
The exact computation of (14) is difficult since ⌘bu is a randomvariable with fading in the signal and interference terms, notto mention the receive filter.
4) Deterministic Equivalent Rate Analysis: To reduce thecomputational complexity, we use the deterministic equivalentanalysis to approximate the calculation of (14). The core ofthese calculations comes from the asymptotic results derivedin [2]. While these results are asymptotic in the sense thatthey are derived under an assumption that the number ofantennas grows large, simulations in [2] show that the fitbetween simulation and approximation is good, even for smallnumbers of antennas. The key ingredient in this analysis isthe deterministic equivalent SINR of user u in cell b, which isdenoted as ⌘bu. It follows from Theorem 3 in [2] that ⌘bu inthe case of the linear MRC detector is
⌘bu =
| tr�bbu|2�2b
prtr�bbu +
P(c,v)
trRbcv�bbu +
Pc 6=b
|tr�bcu|2. (16)
The reader is referred to Theorem 5 in [2] for ⌘bu in the caseof the linear MMSE detector to save space as its expression ismore complicated and hence its computation is more involved.
Remark 1: The deterministic equivalent SINR of user u incell b in (16) the full multiuser MIMO strategy for the DASin Section III-C simplifies to (17) on the top of the next page.
B. User-Grouping Multiuser MIMO for DAS
To reduce the amount of data sent over the backhaul links,we propose to divide the users in a cell evenly into K groups,each is served by a dedicated RRU in the same cell. Foranalysis, we assume RRU k in cell c serves users in cell c
with indices in Uck := {(k � 1)
˜
U + 1, · · · , k ˜U}. Note thatUc = Uc1 [ · · · [ UcK .
1) Signal Model: Denote the transmitted symbol vector bythe users served by RRU k in cell c as ˜xck 2 C ˜U⇥1. Let˜wbku 2 C ˜Nt⇥1, where | ˜wbku|2 = 1, be the linear detectorused by RRU k in cell b to detect xbu for u 2 Ubk. Thepost-processed signal corresponding to user u 2 Ubk served
by RRU k in cell b is
ybku =w⇤bkuhbkbuxbu| {z }
desired signal
+
X
(c,v) 6=(b,u)
w⇤buhbkcvxcv
| {z }interference
+
1
pp
r
zbku
| {z }noise
,
where zbku is the spatially filtered Gaussian noise for thedetection of xbu at RRU k in cell b.
2) Deterministic Equivalent Rate Analysis: Using the samederivation as in [2] for the full multiuser MIMO strategy, weobtain the deterministic equivalent SINR for user u served byRRU k in cell c as
⌘
(ug)
bku =
�
2
bkbu
1
˜Nt
h�2b
pr+
P(c,v)
�bkcv
i�bku +
Pc 6=b �
2
bkcu
. (18)
The achievable sum-rates of the users in cell b in this case is
R
(ug)
b,sum =
X
k2Kb
X
u2Ubk
log
2
(1 + ⌘
(ug)
bku ). (19)
3) A Greedy User Grouping Algorithm: Note that the rateanalysis in the section works for any method of dividingthe users in each cell into K groups. Moreover, ⌘
(ug)
bku isan increasing function of �bkbu because we can rewire (18)as ⌘
(ug)
bku (�bkbu) =
�A�
�2
bkbu +B�
�1
bkbu +D
��1, where A, B,and D are all positive and indepedent of �bkbu. Recall thatin the uncorrelated channel model, �bkcu acts as the gainof the channel from user u in cell c to RRU k in cell b.Although we can perform an exhaustive search for the globallyoptimal user group solution to maximize R
(ug)
b,sum, the requiredcomputational complexity may be high. Thus, we propose totake a greedy approach to user grouping. Let sbu 2 Kb bethe index of the RRU assigned to serve user u in cell b. Notethat Sb := {sbu : u 2 Ub} represents the result of the usergrouping algorithm. Let ¯Kb be the index set of the availableRRUs in cell b at a time in the process of the algorithm, i.e.,those having been assigned to serve less than ˜
U users at thattime. To initialize, randomly permute the indices of the usersand let ¯Ub denote the resulting user index set. Let µ(u) be theoriginal index of user u 2 ¯Ub. Set ¯Kb = Kb. Then startingwith the first user in ¯Ub in the first iteration, find the RRUamong those having the index in ¯Kb that maximizes �bkbµ(1).Let k
1
2 Kb be the index of the found RRU. Set sµ(1) = k
1
.Accounting for this user, if the found RRU has been assignedto serve ˜
U users then remove its index from ¯Kb. Continue forthe next user in ¯Ub in the next iteration until ¯Kb is empty. Thisalgorithm requires (U � 1) iterations.
V. NUMERICAL RESULTS
In this section, we present numerical results on impacts ofspatial correlation and distributed antennas on uplink sum-rates of massive MIMO systems. Specifically, we simulatea network of 7 cells as illustrated in Fig. 3. The cellshave the same base station antenna configurations. Severalkey simulation parameters are provided in Table I. We areinterested in computing the average uplink achievable sum-rates of the users in the center cell. Note that in the system in
departure (AoD)), and a given angle spread (AoS) �as
. Note,however, that our approach does work with other more com-plex antenna and correlation models at the expense of compu-tational complexity. Moreover, we assume that the AoAs/AoDsare distributed according to a certain power azimuth spectrum(PAS). The PAS is modeled by the truncated Laplacian pdf,which is given by
P�(�) =
(�asp2�as
e
�|p2�/�as|
, if � 2 [�⇡,⇡]
0, otherwise,(3)
where � is the random variable describing the AoA/AoD withrespect to the mean angle �
0
, and �
as
=
⇣1� e
�p2⇡/�as
⌘�1
.Let �s be the angle separation between two adjacent antennaelements and ⇢ be the rind radius of the circular array. Define✓m(�
0
) := e
jk�⇢ cos(�0�m�s), for m = 0, · · · , Nt
� 1. Next,define the array response as
a(�0
) := [✓
0
(�
0
), ✓
1
(�
0
), · · · , ✓Nt�1
(�
0
)] . (4)
Define B(�
0
,�
as
) 2 CNt⇥Nt as the characteristic function ofthe Laplacian pdf in (3), where for m,n = 0, · · · , N
t
� 1
[B(�
0
,�
as
)]m,n
:=
�
1 +
�2as2
[k�⇢(sin(�0
�m�s)� sin(�
0
� n�s))]2
. (5)
A closed-form approximate expression of the spatial correla-tion matrix for a single channel tap with low �
as
between auser and a base station is given by [4].
R(�
0
,�
as
) =[a(�0
)a⇤(�0
)]⌦B(�
0
,�
as
). (6)
B. Pilot-based Channel Estimation Error
Base station b needs to estimate the channels between itselfand the users in the same cell based on uplink pilots to designthe linear receive filter Wb. We assume that all cell sharethe same set of U pair-wisely orthogonal pilot signals :=
[ T1
, · · · , TU ] 2 CU⇥⌧ , where ⌧ � U and ⇤
= IU . Let pp
be the average transmit power at each user during the trainingstage. Let Z
p,b 2 CNt⇥⌧ be spatially white additive Gaussiannoise vector at base station b during the training stage. Thereceived training signal at base station b is
Yp,b =
pp
p
⌧
CX
c=1
Hbc + Zp,b. (7)
Base station b correlates Yp,b with to obtain ˜Y
p,b =
Yp,b
⇤ 2 CNt⇥U . Specifically, base station b has the fol-lowing noisy observation of the channel to user u 2 Ub
˜yp,bu = hbbu|{z}
desired channel
+
X
c 6=b
hbcu
| {z }interference
+
1
pp
p
⌧
˜zp,bu
| {z }noise
, (8)
where ˜zp,bu ⇠ CN (0,�2
r
INt) is the post-processed noise atbase station b. The effect of these interference channels onchannel estimation error and then on system performance is
called pilot contamination [6]. Similarly, that of noise is callednoise contamination [6].
Base station b applies minimum mean squared error(MMSE) estimation to the right-hand side of (8) to estimatehbbu for u 2 Ub. Define for c, b 2 C and u 2 Ub
Qbu =
�
2
r
p
p
⌧
INt +
CX
c=1
Rbcu
!�1
(9)
�bcu =RbbuQbuRbcu. (10)
The MMSE estimate of hbbu is [7]
ˆhbbu =RbbuQbu
CX
c=1
hbcu +
1
pp
p
⌧
˜zp,bu
!, (11)
where ˆhbbu ⇠ CN (0,�bbu). To compute the channel esti-mates, central station c needs to know Rbbu and
PCc=1
Rbcu.Due to the orthogonality property of the MMSE estimation, theobserved channel can be decomposed as hbbu =
ˆhbbu +
˜hbbu,where ˜hbbu ⇠ CN (0,Rbbu � �bbu) is a channel estimationerror vector. Note that ˜hbbu is uncorrelated with ˆhbbu. Because˜hbbu and ˆhbbu are jointly Gaussian, they are statisticallyindependent. Define ˆHbb := [
ˆhbb1,ˆhbb2, · · · , ˆhbbU ].
C. Uplink Sum-Rate Analysis
Given knowledge of ˆHbb, base station b designs the lineardetector Wb to detect xb. Specifically, we consider the linearmaximal ratio combining (MRC) and MMSE detectors. DefineˆRr,k as the sum covariance matrix of the channel estimation
error and interfering channels (from users in the other cell tobase station b 2 C), which is given by
ˆRb =E
2
4 ˜Hbb˜H⇤
bb +
X
c 6=b
GbcG⇤bc
3
5 (12)
=
CX
c=1
UX
u=1
Rbcu �UX
u=1
�bbu. (13)
The linear detector wbu to detect xr,bu for u 2 Ub is
wbu =
8<
:
ˆhbbu, MRC detector⇣ˆHbb
ˆH⇤bb +
ˆRb +�2r
prI˜Nt
⌘�1
ˆhbbu, MMSE detector.
Note that base station b needs to have the second-orderstatistics of the channels between itself and the users in thenetwork ˆRb for designing the MMSE detector. Moreover,wbu depends only on ˆHbb, the second-order statistics of otherinterference channels from the users in the other cells to basestation b, and the noise power at base station b.
Based on the worst-case uncorrelated additive noise [2],[8], a standard bound on the uplink ergodic achievable ratecorresponding to user u in cell b is
Rbu = E[log2
(1 + ⌘bu)], (14)
where ⌘bu is the associated signal-to-interference-plus-noiseratio (SINR) and is given in (15) on the top of the next page.
T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Tran. Wireless Commun., vol. 9, no. 11, pp. 3590–3600, Nov. 2010. J. Jose, A. Ashikhmin, T. L. Marzetta, and S. Vishwanath, “Pilot contamination and precoding in multi-cell TDD systems,” IEEE Trans. Wireless Commun., vol. 10, no. 8, pp. 2640–2651, Aug. 2011.
�11
Deterministic Equivalent Analysis
Post-processed uplink SINR for linear MRC detector [HoyEtAl13]yr,b
N (m)t
traditional interference
local noise
interference due to pilot contamination
J. Hoydis, S. T. Brink, and M. Debbah, “Massive MIMO in the UL/DL of cellular networks: How many antennas do we need?” IEEE J. Sel. Areas Commun., vol. 31, no. 2, pp. 160–171, Feb. 2013. K. T. Truong and R. W. Heath, Jr., “Effects of channel aging in massive MIMO systems,” J. Commun. Networks, vol. 15, no. 4, pp. 338-351, Aug. 2013.
log(1 + ⌘MRCr,bu )
Assumes MMSE estimation
The Wireless Networking and Communications Group
DAS and User Grouping
DAS without Spatial Correlation
�13
Nt � U � 1Nt � U � 1Nt � U � 1 UU U
Central station c
UNt � U � 1
Nt � U � 1
Nt � U � 1 U UCentral station b
Spatial correlation is not considered to focus on impact of DAS Each cell has K remote radio units (RRUs)
Nt and U are divisible by K, denote
: large-scale fading coefficient from user u in cell c to RRU k in cell b
Base stations (or RRUs) use linear MRC detector The same pilot-based channel estimation as before
Nt = Nt/K; U = U/K
�bkcu
UEU
RRUk
�bkcu
Full MU-MIMO Strategy for DAS
RRUs in each cell coordinate with each other for detection
Deterministic equivalent SINR of user u in cell b !
!
RRUs in cell b far from user u do not contribute much to its SINR
Small large-scale fading channel gain & large channel estimation error �14
UNt � U � 1U U
Central station b
RRUm
⌘bu =
|w⇤buˆhbbu|2
Ehw⇤
bu
⇣˜hbbu
˜h⇤bbu +
P(c,k) 6=(b,u) hbckg⇤
bck +
�2r
⌘wbu
��� ˆHbb
i. (15)
⌘cu =
8><
>:
tr |�bbu|2�2b
prtr�bbu+
P(c,k) 6=(b,u) trRbck�bbu+
Pc 6=b|tr�bcu|2
, MRC detector
�2bu�2r
Ntpr+
1Nt
tr�bbuT(1)b +
1Nt
P(c,k) 6=(b,u) µbcuk+
Pc 6=b |#bcu|2
, MMSE detector(16)
The exact computation of (14) is difficult since ⌘bu is a randomvariable with fading in the signal and interference terms, notto mention the receive filter. To reduce the computationalcomplexity, we use the deterministic equivalent analysis toapproximate the calculation of (14). The core of these cal-culations comes from the asymptotic results derived in [2]using deterministic equivalents, where each user has a differentspatial correlation matrix. While these results are asymptoticin the sense that they are derived under an assumption that thenumber of antennas grows large, simulations in [2] show thatthe fit between simulation and approximation is good, evenfor small numbers of antennas. The approximations are derivedunder some technical assumptions, that essentially we summa-rize as (i) the maximum eigenvalue of any spatial correlationmatrix is finite, (ii) all spatial correlation matrices have non-zero energy, and (iii) the intercell interference matrix includingchannel estimation errors are finite. From a practical perspec-tive, the assumptions are reasonable. From the perspective ofdoing the calculations, only (ii) is problematic. Essentially, onehas to remember not to use zero-valued correlation matricesin the expressions. The key ingredient in this analysis is thedeterministic equivalent SINR of user u in cell b, which we call⌘bu. It follows from Theorem 3 and Theorem 5 in [2] that ⌘buis given in (16) on the top of the next page. Note that ⌘bu is afunction of the covariance matrices Rbcu, the signal-to-noiseratio (SNR), and various quantities computed from them. Theexpressions of the quantities are provided in Appendix A. Theuplink ergordic achievable rate of user u in cell b given in (14)is approximated as Rbu ⇡ log
2
(1 + ⌘bu). The approximationerror grows smaller as N
t
and U grow large.
IV. NUMERICAL RESULTS
In this section, we present numerical results on impacts ofspatial correlation and distributed antennas on uplink sum-rates of massive MIMO systems. Specifically, we simulatea network of 7 cells as illustrated in Fig. 3. The cellshave the same base station antenna configurations. Severalkey simulation parameters are provided in Table I. We areinterested in computing the average uplink achievable sum-rates of the users in the center cell. Note that in the system inFig. 3, the antennas are deployed at the same location, in thiscase ideally at the center of each cell. We refer to this as the(centralized) 1-cluster antenna configuration. We also considertwo other antenna configurations. In one configuration, theantennas are divided evenly among three different sites. Inanother configuration, the antennas are divided evenly among
TABLE ISIMULATION PARAMETERS
Parameter Description
Inter-site distance 500 metersNumber of users per cell 12Path-loss model PL = 128.1 + 37.6 log
10
d
where d is the transmission dis-tance in kilometers
Penetration loss 20dBUser antennas omnidirectional with 0dBi gainUser dropping in a cell uniformly distributedShadowing model not consideredUser assignment each user is served by the base
station in the same cellBS antenna gain 10dBiBS total transmit power 46dBmFrequency carrier 2GHzBandwidth 10MHzThermal noise density -174dBm/HzBS noise figure 5dBUE noise figure 9dB
Fig. 3. The simulated 7-cell cellular network. The base station antennas arecollocated at the center of the cells and are illustrated by the red squares. Theusers of interest are located in the center cell.
eight different sites, one being at the center of the cell. Fig. 4illustrates the locations of the sites in a cell. Note that each siteaccommodates a cluster of antennas. We refer to the config-urations as the (decentralized) 3-cluster antenna configurationand the (distributed) 8-cluster antenna configuration.
First, we focus on impact of spatial correlation on theuplink sum-rates of massive MIMO systems. To do this, weconsider the centralized antenna configuration. We select the
�bmbu �bkbuRRUk
Nt � U � 1Nt � U � 1Nt � U � 1
Nt � U � 1Nt � U � 1
the linear detector wbu to detect xr,bu for u 2 Ub is
wbu =
8<
:
ˆhbbu, MRC detector⇣ˆHbb
ˆH⇤bb +
ˆRb +�2r
prI˜Nt
⌘�1
ˆhbbu, MMSE detector,
where ˆRb =
PCc=1
PUu=1
Rbcu �PU
u=1
�bbu. Note that fordesigning the MMSE detector, base station b needs to haveˆHbb, the second-order statistics of the channels between itselfand the users in the network ˆRb, and its noise power.
3) General Achievable Rate Analysis: Based on the worst-case uncorrelated additive noise [2], [7], a standard bound onthe uplink achievable rate corresponding to user u in cell b is
Rbu = E[log2
(1 + ⌘bu)], (14)
where ⌘bu is the associated signal-to-interference-plus-noiseratio (SINR) and is given in (15) on the top of the next page.
⌘bu =
|w⇤buˆhbbu|2
Ew⇤
bu
⇣hbbuh⇤
bbu +
P(c,v)
hbcvg⇤bcv +
�2r
⌘wbu
��� ˆHbb
�.
(15)
The exact computation of (14) is difficult since ⌘bu is a randomvariable with fading in the signal and interference terms, notto mention the receive filter.
4) Deterministic Equivalent Rate Analysis: To reduce thecomputational complexity, we use the deterministic equivalentanalysis to approximate the calculation of (14). The core ofthese calculations comes from the asymptotic results derivedin [2]. While these results are asymptotic in the sense thatthey are derived under an assumption that the number ofantennas grows large, simulations in [2] show that the fitbetween simulation and approximation is good, even for smallnumbers of antennas. The key ingredient in this analysis isthe deterministic equivalent SINR of user u in cell b, which isdenoted as ⌘bu. It follows from Theorem 3 in [2] that ⌘bu inthe case of the linear MRC detector is
⌘bu =
| tr�bbu|2�2b
prtr�bbu +
P(c,v)
trRbcv�bbu +
Pc 6=b
|tr�bcu|2. (16)
The reader is referred to Theorem 5 in [2] for ⌘bu in the caseof the linear MMSE detector to save space as its expression ismore complicated and hence its computation is more involved.
Remark 1: The deterministic equivalent SINR of user u incell b in (16) the full multiuser MIMO strategy for the DASin Section III-C simplifies to (17) on the top of the next page.
B. User-Grouped Multiuser MIMO for DAS
Note from (17) that the RRUs corresponding to small large-scale fading channel gains (and hence low accurate channelestimates) to user u in cell b do not contribute much to therate of that user. Thus, it may be better if each user is servedby only its close-by RRU. In this section, we propose to dividethe users in a cell evenly into K groups, each is served bya dedicated RRU in the same cell. For analysis, we assume
RRU k in cell c serves users in cell c with indices in Uck :=
{(k � 1)
˜
U + 1, · · · , k ˜U}. Note that Uc = Uc1 [ · · · [ UcK .
1) Signal Model: Denote the transmitted symbol vector bythe users served by RRU k in cell c as ˜xck 2 C ˜U⇥1. Let˜wbku 2 C ˜Nt⇥1, where | ˜wbku|2 = 1, be the linear detectorused by RRU k in cell b to detect xbu for u 2 Ubk. Thepost-processed signal corresponding to user u 2 Ubk servedby RRU k in cell b is
ybku =w⇤bkuhbkbuxbu| {z }
desired signal
+
X
(c,v) 6=(b,u)
w⇤buhbkcvxcv
| {z }interference
+
1
pp
r
zbku
| {z }noise
,
where zbku is the spatially filtered Gaussian noise for thedetection of xbu at RRU k in cell b.
2) Deterministic Equivalent Rate Analysis: Using the samederivation as in [2] for the full multiuser MIMO strategy, weobtain the deterministic equivalent SINR for user u served byRRU k in cell c as
⌘
(ug)
bku =
�
2
bkbu
1
˜Nt
h�2b
pr+
P(c,v) 6=(b,u)
�bkcv
i�bku +
Pc 6=b
�
2
bkcu
. (18)
The achievable sum-rates of the users in cell b in this case is
R
(ug)
b,sum =
X
k2Kb
X
u2Ubk
log
2
(1 + ⌘
(ug)
bku ). (19)
3) A Greedy User Grouping Algorithm: Note that the rateanalysis in the section works for any method of dividingthe users in each cell into K groups. Moreover, ⌘
(ug)
bku isan increasing function of �bkbu because we can rewire (18)as ⌘
(ug)
bku (�bkbu) =
�A�
�2
bkbu +B�
�1
bkbu +D
��1, where A, B,and D are all positive and indepedent of �bkbu. Recall thatin the uncorrelated channel model, �bkcu acts as the gainof the channel from user u in cell c to RRU k in cell b.Although we can perform an exhaustive search for the globallyoptimal user group solution to maximize R
(ug)
b,sum, the requiredcomputational complexity may be high. Thus, we propose totake a greedy approach to user grouping. Let sbu 2 Kb bethe index of the RRU assigned to serve user u in cell b. Notethat Sb := {sbu : u 2 Ub} represents the result of the usergrouping algorithm. Let ¯Kb be the index set of the availableRRUs in cell b at a time in the process of the algorithm, i.e.,those having been assigned to serve less than ˜
U users at thattime. To initialize, randomly permute the indices of the usersand let ¯Ub denote the resulting user index set. Let µ(u) be theoriginal index of user u 2 ¯Ub. Set ¯Kb = Kb. Then startingwith the first user in ¯Ub in the first iteration, find the RRUamong those having the index in ¯Kb that maximizes �bkbµ(1).Let k
1
2 Kb be the index of the found RRU. Set sµ(1) = k
1
.Accounting for this user, if the found RRU has been assignedto serve ˜
U users then remove its index from ¯Kb. Continue forthe next user in ¯Ub in the next iteration until ¯Kb is empty. Thisalgorithm requires (U � 1) iterations.
User-Grouped Strategy for DAS
Each RRU serves an equal number of users ( users) in the cell Use its own linear MRC detector to detect signals from users it serves
SINR of user u served by RRU k in cell b for any user grouping !
!
is an increasing function of large-scale fading channel gain
�15
Central station b
RRUm
⌘bu =
|w⇤buˆhbbu|2
Ehw⇤
bu
⇣˜hbbu
˜h⇤bbu +
P(c,k) 6=(b,u) hbckg⇤
bck +
�2r
⌘wbu
��� ˆHbb
i. (15)
⌘cu =
8><
>:
tr |�bbu|2�2b
prtr�bbu+
P(c,k) 6=(b,u) trRbck�bbu+
Pc 6=b|tr�bcu|2
, MRC detector
�2bu�2r
Ntpr+
1Nt
tr�bbuT(1)b +
1Nt
P(c,k) 6=(b,u) µbcuk+
Pc 6=b |#bcu|2
, MMSE detector(16)
The exact computation of (14) is difficult since ⌘bu is a randomvariable with fading in the signal and interference terms, notto mention the receive filter. To reduce the computationalcomplexity, we use the deterministic equivalent analysis toapproximate the calculation of (14). The core of these cal-culations comes from the asymptotic results derived in [2]using deterministic equivalents, where each user has a differentspatial correlation matrix. While these results are asymptoticin the sense that they are derived under an assumption that thenumber of antennas grows large, simulations in [2] show thatthe fit between simulation and approximation is good, evenfor small numbers of antennas. The approximations are derivedunder some technical assumptions, that essentially we summa-rize as (i) the maximum eigenvalue of any spatial correlationmatrix is finite, (ii) all spatial correlation matrices have non-zero energy, and (iii) the intercell interference matrix includingchannel estimation errors are finite. From a practical perspec-tive, the assumptions are reasonable. From the perspective ofdoing the calculations, only (ii) is problematic. Essentially, onehas to remember not to use zero-valued correlation matricesin the expressions. The key ingredient in this analysis is thedeterministic equivalent SINR of user u in cell b, which we call⌘bu. It follows from Theorem 3 and Theorem 5 in [2] that ⌘buis given in (16) on the top of the next page. Note that ⌘bu is afunction of the covariance matrices Rbcu, the signal-to-noiseratio (SNR), and various quantities computed from them. Theexpressions of the quantities are provided in Appendix A. Theuplink ergordic achievable rate of user u in cell b given in (14)is approximated as Rbu ⇡ log
2
(1 + ⌘bu). The approximationerror grows smaller as N
t
and U grow large.
IV. NUMERICAL RESULTS
In this section, we present numerical results on impacts ofspatial correlation and distributed antennas on uplink sum-rates of massive MIMO systems. Specifically, we simulatea network of 7 cells as illustrated in Fig. 3. The cellshave the same base station antenna configurations. Severalkey simulation parameters are provided in Table I. We areinterested in computing the average uplink achievable sum-rates of the users in the center cell. Note that in the system inFig. 3, the antennas are deployed at the same location, in thiscase ideally at the center of each cell. We refer to this as the(centralized) 1-cluster antenna configuration. We also considertwo other antenna configurations. In one configuration, theantennas are divided evenly among three different sites. Inanother configuration, the antennas are divided evenly among
TABLE ISIMULATION PARAMETERS
Parameter Description
Inter-site distance 500 metersNumber of users per cell 12Path-loss model PL = 128.1 + 37.6 log
10
d
where d is the transmission dis-tance in kilometers
Penetration loss 20dBUser antennas omnidirectional with 0dBi gainUser dropping in a cell uniformly distributedShadowing model not consideredUser assignment each user is served by the base
station in the same cellBS antenna gain 10dBiBS total transmit power 46dBmFrequency carrier 2GHzBandwidth 10MHzThermal noise density -174dBm/HzBS noise figure 5dBUE noise figure 9dB
Fig. 3. The simulated 7-cell cellular network. The base station antennas arecollocated at the center of the cells and are illustrated by the red squares. Theusers of interest are located in the center cell.
eight different sites, one being at the center of the cell. Fig. 4illustrates the locations of the sites in a cell. Note that each siteaccommodates a cluster of antennas. We refer to the config-urations as the (decentralized) 3-cluster antenna configurationand the (distributed) 8-cluster antenna configuration.
First, we focus on impact of spatial correlation on theuplink sum-rates of massive MIMO systems. To do this, weconsider the centralized antenna configuration. We select the
RRUk UNt � U � 1U U Nt � U � 1Nt � U � 1
Nt � U � 1Nt � U � 1
Nt � U � 1
U
�bkbu⌘
(full)
bu =
⇣PKk=1
�
2
bkbu��1
bku
⌘2
�2b
pr˜Nt
⇣PKk=1
�
2
bkbu��1
bku
⌘+
1
˜Nt
PKk=1
⇣ P(c,v) 6=(b,u)
�bkcv
⌘�
2
bkbu��1
bku +
Pc 6=b
⇣PKk=1
�bkbu�bkcu��1
bku
⌘2
. (17)
TABLE ISIMULATION PARAMETERS
Parameter DescriptionInter-site distance 500 metersNumber of users per cell 12Path-loss model PL = 128.1 + 37.6 log
10
d
where d is the transmission dis-tance in kilometers
Penetration loss 20dBUser antennas omnidirectional with 0dBi gainUser dropping in a cell uniformly distributedShadowing model not consideredUser assignment each user is served by the base
station in the same cellBS antenna gain 10dBiBS total transmit power 46dBmFrequency carrier 2GHzBandwidth 10MHzThermal noise density -174dBm/HzBS noise figure 5dBUE noise figure 9dB
V. NUMERICAL RESULTS
In this section, we present numerical results on impacts ofspatial correlation and distributed antennas on uplink sum-rates of massive MIMO systems. Specifically, we simulatea network of 7 cells as illustrated in Fig. 3. The cellshave the same base station antenna configurations. Severalkey simulation parameters are provided in Table I. We areinterested in computing the average uplink achievable sum-rates of the users in the center cell. Note that in the system inFig. 3, the antennas are deployed at the same location, in thiscase ideally at the center of each cell. We refer to this as the(centralized) 1-cluster antenna configuration. We also considertwo other antenna configurations. In one configuration, theantennas are divided evenly among three different sites. Inanother configuration, the antennas are divided evenly amongeight different sites, one being at the center of the cell. Fig. 4illustrates the locations of the sites in a cell. Note that each siteaccommodates a cluster of antennas. We refer to the config-urations as the (decentralized) 3-cluster antenna configurationand the (distributed) 8-cluster antenna configuration.
First, we focus on impact of spatial correlation on theuplink sum-rates of massive MIMO systems. To do this, weconsider the centralized antenna configuration. We select theuncorrelated channel model with Rbcu = INt for all b, c 2 Cand u 2 Ub as the baseline for performance comparison.
Fig. 3. The simulated 7-cell cellular network. The base station antennas arecollocated at the center of the cells and are illustrated by the red squares. Theusers of interest are located in the center cell.
a) 3-cluster antenna configuration b) 8-cluster antenna configuration
Fig. 4. Illustration of multiple-cluster antenna configurations in a cell wherethe base station antennas are divided evenly among a number of antenna sitesin the cell area. The antenna sites are illustrated by the red squares.
Fig. 5 presents the simulation results of uplink achievable sum-rates of the users in the center cell for the MRC and MMSEdetectors for both the spatially correlated channel model andthe uncorrelated channel model. We notice that when thesimple MRC detector is deployed on the uplink, the averagesum-rate performance under the uncorrelated channel model isslightly better than that under the spatially correlated channelmodel. Nevertheless, when the MMSE detector is deployedon the uplink, the average sum-rate performance under theuncorrelated channel model is significantly worse than thatunder the spatially correlated channel model. Thus, impactof spatial correlation on uplink sum-rates of massive MIMOsystems depend on the deployed reception strategies.
Next, we investigates the impacts of antenna distributionson the uplink sum-rates of massive MIMO systems. Fig. 6presents the simulation results on Uplink achievable sum-ratesfor different antenna distributions and different detectors forthe spatially correlated channel model. We notice that theMMSE detectors outperform significantly the MRC detectorfor all antenna distributions. We also notice that increasing thenumber of base station antenna clusters per cell improves theaverage achievable sum-rates per cell on the uplink. We canmake similar observations for the uncorrelated channel model,however, those results are omitted to save space.
Circular Array
scattering cluster
user 1
user 2
AoA (angle-of-arrival)AoD (angle-of-departure)
⇢
0
s
Fig. 2. The spatial correlation model under consideration. Each user isassociated with a single scattering cluster.
choose to use a circular array (as illustrated in Fig. 2) becausethey are commercially deployed in other systems that exploitreciprocity, i.e. TD-SCDMA in China, and it is possibleto compute the spatial correlation matrices efficiently usingalgorithms developed in prior work [4].
We consider mainly the uniformly circular array (UCA)configuration and assume that the antenna array configurationhas a uniformly chosen angle-of-arrival (AoA) (or angle-of-departure (AoD)), and a given angle spread (AoS) �
as
. Note,however, that our approach does work with other more com-plex antenna and correlation models at the expense of compu-tational complexity. Moreover, we assume that the AoAs/AoDsare distributed according to a certain power azimuth spectrum(PAS). The PAS is modeled by the truncated Laplacian pdf,which is given by
P�(�) =
(�asp2�as
e
�|p2�/�as|
, if � 2 [�⇡,⇡]
0, otherwise,(6)
where � is the random variable describing the AoA/AoD withrespect to the mean angle �
0
, and �
as
=
⇣1� e
�p2⇡/�as
⌘�1
.Let �s be the angle separation between two adjacent antennaelements and ⇢ be the rind radius of the circular array. Define✓m(�
0
) := e
jk�⇢ cos(�0�m�s), for m = 0, · · · , Nt
� 1. Next,define the array response as
a(�0
) := [✓
0
(�
0
), ✓
1
(�
0
), · · · , ✓Nt�1
(�
0
)] . (7)
Define B(�
0
,�
as
) 2 CNt⇥Nt as the characteristic function ofthe Laplacian pdf in (6), where for m,n = 0, · · · , N
t
� 1
[B(�
0
,�
as
)]m,n
:=
�
1 +
�2as2
[k�⇢(sin(�0
�m�s)� sin(�
0
� n�s))]2
. (8)
A closed-form approximate expression of the spatial correla-tion matrix for a single channel tap with low �
as
between auser and a base station is given by [4].
R(�
0
,�
as
) =[a(�0
)a⇤(�0
)]⌦B(�
0
,�
as
). (9)
C. Distributed Antenna Systems
To investigate the impact of distributing base station anten-nas, we consider a distributed antenna system (DAS) withoutcorrelation where the base stations use the simple linearmaximal ratio combining (MRC) detector. Define �bkcu as thelarge-scale fading coefficient of the channel from user u in cell
c to antenna cluster k in cell b, for k = 1, · · · ,K. As a result,Rbcu can be modeled as a diagonal matrix with the diagonalentries of �bmcu for m = 1, · · · , N
t
, where �bmcu = �bkcu
and m 2 [(k � 1) ⇤ ˜
N
t
+ 1, k
˜
N
t
]. For notational convenience,for k = 1, · · · ,K, define
�bku =
�
2
r
p
p
⌧
+
CX
c=1
�bkcu. (10)
In this case, Qbu is a diagonal matrix with the followingdiagonal entries for m = 1, · · · , N
t
qbmu =�
�1
bku, (11)
with m 2 [(k � 1) ⇤ ˜
N
t
+ 1, k
˜
N
t
]. As a result, �bcu is also adiagonal matrix with the following diagonal entries
�bmcu =�bmbuqbum�bmcu (12)=�bkbu�bkcu�
�1
bku, (13)
with m 2 [(k � 1) ⇤ ˜
N
t
+ 1, k
˜
N
t
].
IV. UPLINK PERFORMANCE ANALYSIS
In the paper, we focus on the uplink, however, the counter-part results can be obtained in a similar, but more involved,manner for the downlink. Let p
r
be the average transmit powerat each user during the uplink data transmission stage, assumedto be the same for each user. Let xcu be the transmitted symbolsent by user u in cell c 2 C on the uplink, where E[|xcu|2] = 1.The transmitted symbols sent by the users in the network aremutually independent. We consider two different multiuseruplink reception strategies: i) full multiuser MIMO and ii)user-grouped multiuser MIMO for DAS.
A. Full Multiuser MIMO
In the full multiuser MIMO strategy, the RRUs in a cellcoordinate with each other to receive and then detect thesignals sent from the users in the same cell.
1) Signal Model: Define xc := [xc1, xc2, · · · , xcU ]T 2
CU⇥1 as the transmitted symbol vector by the users in cell c.Let wbu 2 CNt⇥1, where |wbu|2 = 1, be the linear detectorused by base station b to detect xbu for u 2 Ub. The post-processed signal corresponding to user u 2 Ub is
ybu = w⇤buhbbuxbu| {z }
desired signal
+
X
(c,k) 6=(b,u)
w⇤buhbckxck
| {z }interference
+
1
pp
r
zbu
| {z }noise
,
where zbu is the spatially filtered Gaussian noise for thedetection of xbu at base station b.
2) Linear Detector Design: Given knowledge of ˆHbb, basestation b designs the linear detector Wb to detect xb. We con-sider only the linear MRC and MMSE detectors. Specifically,the linear detector wbu to detect x
r,bu for u 2 Ub is
wbu =
8<
:
ˆhbbu, MRC detector⇣ˆHbb
ˆH⇤bb +
ˆRb +�2r
prI˜Nt
⌘�1
ˆhbbu, MMSE detector,
Greedy User Grouping Alg. for DASInitialization
Randomly permute the user indices to obtain
Initial set of available RRUs is
Iteration
Among the available RRUs, find the RRU that maximizes the large-scale fading channel gain from itself to user in the same cell (i.e. cell b)
!!
Assign the found RRU (i.e. RRU ) to serve user
If the found RRU has been assigned to serve users (including user ), then remove the index of the found RRU from
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Kb = {1, 2, · · · ,K}Ub = {u1, u2, · · · , uU}
n = 1, 2, · · · , U � 1
un
un
U unKb
k⇤n = arg max
k2Kb
�bkbun
k⇤n
Kb = Kb \ k⇤n
�17
Simulation Setup
Hexagonal cells with antennas per base station sector
users dropped randomly in each sector
Base station (BS)
user
Rbcu =E[hbcu[n]h⇤bcu[n]]
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Simulation Parameters
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Parameters Description
Number of sectors per cell 1
Number of users per cell 12
Inter-site distance 500m
Pathloss model PLNLOS = 128.1 + 37.6 log10(d), where d > 0.035km is the trans. distance
Penetration loss 20dB
Antenna array configuration at users 1 antenna omni with 0dBi gain
Channel estimation method MMSE
Angle spread 10 degrees
User dropping Uniformly distributed within a cell
Shadowing model Not considered
User assignment Each user is served by the BS in the same cell
BS antenna gain 10dBi
BS antenna spacing 0.5, 1, 1.5, and 2 wavelengths
BS total transmit power 40 watts or 46dBm
Thermal noise density -174dBm/Hz
BS noise figure (UL) 5dB
MS noise figure (DL) 9dB
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Antenna Clustering Configurations
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Collocated or 01-cluster
Distributed with 03 clusters
Distributed with 08 clustersNote: distributed antennas are equally-spaced on a ring of 2/3 of cell radius
BS antenna cluster
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Spatial Correlation Model
No standardized multi-user spatial correlation model Use single cluster model, with randomly located cluster
Employ low complexity model for ULA and circular array
Parameters Random AoA or AoD for each user
Laplacian power azimuth spectrum, angle spread of 10 degrees
Note: Spatial correlation is independent of mobility in our model
�20A. Forenza, D. J. Love, and R. W. Heath, Jr., ``Simplified Spatial Correlation Models for Clustered MIMO Channels with Different Array Configurations,''IEEE Trans. on Veh. Tech., vol. 56, no. 4, part 2, pp. 1924-1934, July 2007.
Circular Array
scattering cluster
user 1
user 2
AoA (angle-of-arrival)AoD (angle-of-departure) U
Yp,b =ppp
✓ CX
c=1
Hbc
◆ +Np,b,
U (m)
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Impacts of Spatial Correlation
Impacts of spatial correlation depend on reception strategies MRC: uncorrelated channel model is preferred
MMSE: spatial correlated channel model is preferred
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24 48 7210
20
30
40
50
60
70
Number of antennas at a base station
8SOLQN�DYHUDJH�DFKLHYDEOH�VXP
ïUDWHV�>ESV�+]@ Uncorrelated: MRC
Spatially correlated: MRCUncorrelated: MMSESpatially correlated: MMSE
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Effects of Antenna Distributions
Distributing antennas over cell areas bring considerable gains
Saturation is not observed at not-so-large numbers of antennas
�22
24 48 7210
20
30
40
50
60
70
Number of antennas at a base station
8SOLQN�DYHUDJH�DFKLHYDEOH�VXP
ïUDWHV�>ESV�+]@
01 cluster, MMSE03 clusters, MMSE08 clusters, MMSE01 cluster, MRC03 clusters, MRC08 clusters, MRC
Effects of User Grouping in DAS
Greedy user grouping outperforms full MU-MIMO strategy
Random user grouping is much worse than the other strategies
�23
72 96 1205
10
15
20
25
30
35
40
Number of antennas at a base station
8SOLQN�DYHUDJH�DFKLHYDEOH�VXP
ïUDWHV�>ESV�+]@
*UHHG\�XVHU�JURXSLQJ����LQLWLDOL]DWLRQV*UHHG\�XVHU�JURXSLQJ����LQLWLDOL]DWLRQ)XOO�08ï0,02�VWUDWHJ\Random user grouping
Distributed with 03 clusters
12 users in each cell are evenly divided into 3 groups, Each group is served by a dedicated cluster
ConclusionsMassive MIMO and spatial correlation
Impact depends on deployed detector
Massive MIMO and distributed antennas
Provide high performance versus centralized solutions
Remote radio unit selection offers high gain
Further investigation
Connections to power control?
Performance with non MMSE estimation?
Asymptotic performance trends?
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References[LarEtAl13] E. G. Larsson, F. Tufvesson, O. Edfors, and T. L. Marzetta, “Massive MIMO for next generation wireless systems”, to appear in IEEE Commun. Mag., 2013.
[RusEtAl13] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors and F. Tufvesson, “Scaling up MIMO: Opportunities and challenges with very large arrays,” IEEE Signal Processing Mag., vol. 30, no. 1, pp. 40–60, Jan. 2013.
[HoyEtAl13] J. Hoydis, S. T. Brink, and M. Debbah, “Massive MIMO in the UL/DL of cellular networks: How many antennas do we need?” IEEE J. Sel. Areas Commun., vol. 31, no. 2, pp. 160–171, Feb. 2013.
[TruHea13] K. T. Truong and R. W. Heath, Jr., “Effects of channel aging in massive MIMO systems,” J. Commun. Networks, vol. 15, no. 4, pp. 338-351, Aug. 2013.
[GaoEtAl11]X. Gao, O. Edfors, F. Rusek, and F. Tufvesson, “Linear pre-coding performance in measured very-large MIMO channels,” in Proceedings of IEEE Veh. Tech. Conf., Sep. 2011, pp. 1–5.
[ForEtAL07] A. Forenza, D. J. Love, and R. W. Heath, Jr., “Simplified spatial correlation models for clustered MIMO channels with different array configurations,” IEEE Trans. Veh. Tech., vol. 56, no. 4, pp. 1924–1934, Jul. 2007.
[3GPPLTE] 3GPP TR 36.814, “Further advancements for E-UTRA physical layer aspects,” Mar. 2010.
[Mar10] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Tran. Wireless Commun., vol. 9, no. 11, pp. 3590–3600, Nov. 2010.
[Ver98] S. Verdu, Multiuser Detection. Cambridge University Press, 1998.
[JosEtAl11] J. Jose, A. Ashikhmin, T. L. Marzetta, and S. Vishwanath, “Pilot contamination and precoding in multi-cell TDD systems,” IEEE Trans. Wireless Commun., vol. 10, no. 8, pp. 2640–2651, Aug. 2011.
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