arX
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201
1SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS 1
Energy and Spectral Efficiency of Very LargeMultiuser MIMO Systems
Hien Quoc Ngo, Erik G. Larsson, and Thomas L. Marzetta
Abstract
A multiplicity of autonomous terminals simultaneously transmits data streams to a compact array of antennas.
The array uses imperfect channel-state information derived from transmitted pilots to extract the individual data
streams. The power radiated by the terminals can be made inversely proportional to the square-root of the number
of base station antennas with no reduction in performance. In contrast if perfect channel-state information were
available the power could be made inversely proportional tothe number of antennas. Lower capacity bounds
for maximum-ratio combining (MRC), zero-forcing (ZF) and minimum mean-square error (MMSE) detection are
derived. A MRC receiver normally performs worse than ZF and MMSE. However as power levels are reduced,
the cross-talk introduced by the inferior maximum-ratio receiver eventually falls below the noise level and this
simple receiver becomes a viable option. The tradeoff between the energy efficiency (as measured in bits/J) and
spectral efficiency (as measured in bits/channel use/terminal) is quantified. It is shown that the use of moderately
large antenna arrays can improve the spectral and energy efficiency with orders of magnitude compared to a
single-antenna system.
Index Terms
Energy efficiency, spectral efficiency, multiuser MIMO, very large MIMO systems
I. INTRODUCTION
In multiuser multiple-input multiple-output (MU-MIMO) systems, a base station (BS) equipped with
multiple antennas serves a number of users. Such systems have attracted much attention for some time now
[2]. Conventionally, the communication between the BS and the users is performed by orthogonalizing
the channel so that the BS communicates with each user in separate time-frequency resources. This
is not optimal from an information-theoretic point of view,and higher rates can be achieved if the
BS communicates with several users in the same time-frequency resource [3], [4]. However, complex
techniques to mitigate inter-user interference must then be used, such as maximum-likelihood multiuser
detection on the uplink [5], or “dirty-paper coding” on the downlink [6], [7].
H. Q. Ngo and E. G. Larsson are with the Department of Electrical Engineering (ISY), Linkoping University, 581 83 Linkoping, Sweden(Email: [email protected]; [email protected]).
T. L. Marzetta is with Bell Laboratories, Alcatel-Lucent, 600 Moutain Avenue, Murray Hill, NJ 07974, USA (Email: [email protected]).
This work was supported in part by the Swedish Research Council (VR), the Swedish Foundation for Strategic Research (SSF), andELLIIT. E. Larsson is a Royal Swedish Academy of Sciences (KVA) Research Fellow supported by a grant from the Knut and AliceWallenberg Foundation.
Parts of this work were presented at the 2011 Allerton Conference on Communication, Control and Computing [1].
2 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
Recently, there has been a great deal of interest in MU-MIMO with very large antenna arraysat the
BS. Very large arrays can substantially reduce intracell interference with simple signal processing [8].
We refer to such systems as “very large MU-MIMO systems” here, and with very large we mean arrays
comprising say a hundred, or a few hundreds, of antennas, simultaneously serving tens of users. The
design and analysis of very large MU-MIMO systems is a fairlynew subject that is attracting substantial
interest [8]–[11]. The vision is that each individual antenna can have a small physical size, and be built
from inexpensive hardware. With a very large antenna array,things that were random before start to
look deterministic. As a consequence, the effect of small-scale fading can be averaged out. Furthermore,
when the number of BS antennas grows large, the random channel vectors between the users and the BS
become pairwisely orthogonal [10]. In the limit of an infinite number of antennas, with simple matched
filter processing at the base station, uncorrelated noise and intracell interference disappear completely [8].
Another important advantage of large MIMO systems is that they enable us to reduce the transmitted
power. On the uplink, reducing the transmit power of the terminals will drain their batteries slower.
On the downlink, much of the electrical power consumed by a BSis spent by power amplifiers and
associated circuits and cooling systems [12]. Hence reducing the emitted RF power would help in cutting
the electricity consumption of the BS.
This paper analyzes the potential for power savings on the uplink of very large MU-MIMO systems.
While it is well known that MIMO technology can offer improved power efficiency, owing to both array
gains and diversity effects [13], we are not aware of any workthat analyzes power efficiency of MU-
MIMO systems with receiver structures that are realistic for very large MIMO. We consider the power
efficiency of a single cell MU-MIMO system, and we focus on thecase where the users have a single
antenna each. The basic question that we address is how much the transmitted power of each user can be
reduced when increasing then number of antennas. The paper makes the following specific contributions:
• We show that, when the number of BS antennasM grows without bound, we can reduce the
transmitted power of each user proportionally to1/M if the BS has perfect channel state information
(CSI), and proportionally to1/√M if CSI is estimated from uplink pilot measurements. This holds
true even when using simple, linear receivers. We also derive closed-form expressions of lower bounds
on the uplink achievable rates for finite but largeM , for the cases of perfect and imperfect CSI,
assuming MRC, zero-forcing (ZF), and minimum mean-squarederror (MMSE) receivers, respectively.
See Section III.
• We study the tradeoff between spectral efficiency and energyefficiency. For imperfect CSI, in the low
transmit power regime, we can simultaneously increase the spectral-efficiency and energy-efficiency.
See Section IV.
3
Notation: The superscriptsT , ∗, andH stand for the transpose, conjugate, and conjugate-transpose,
respectively.[AAA]ij or AAAij denotes the (i, j)th entry of a matrixAAA, andIIIn is the n × n identity matrix.
The expectation operator and the Euclidean norm are denotedby E {·} and‖ · ‖, respectively. Finally, we
usezzz ∼ CN (0,ΣΣΣ) to denote a circularly symmetric complex Gaussian vectorzzz with covariance matrix
ΣΣΣ and zero mean.
II. SYSTEM MODEL AND PRELIMINARIES
A. MU-MIMO System Model
We consider the uplink of a MU-MIMO system. The system includes one BS equipped with an array
of M antennas that receive data fromK single-antenna users. The users transmit their data in the same
time-frequency resource. TheM × 1 received vector at the BS is
yyy =√puGGGxxx+ nnn (1)
whereGGG represents theM ×K channel matrix between the BS and theK users, i.e.,gmk , [GGG]mk is the
channel coefficient between themth antenna of the BS and thekth user;√puxxx is theK × 1 vector of
symbols simultaneously transmitted by theK users (the average transmitted power of each user ispu); and
nnn vector of additive white, zero-mean Gaussian noise. We takethe noise variance to be1, to minimize
notation, but without loss of generality. With this convention, pu has the interpretation of normalized
“transmit” SNR and is therefore dimensionless. The model (1) also applies to wideband channels handled
by OFDM over restricted intervals of frequency.
The channel matrixGGG models independent fast fading, geometric attenuation, and log-normal shadow
fading. The coefficientgmk can be written as
gmk = hmk
√
βk, m = 1, 2, ...,M (2)
wherehmk is the fast fading coefficient from thekth user to themth antenna of the BS.√βk models the
geometric attenuation and shadow fading which is assumed tobe independent overm and to be constant
over many coherence time intervals and known a priori. This assumption is reasonable since the distances
between the users and the BS are much larger than the distancebetween the antennas, and the value of
βk changes very slowly with time. Then, we have
GGG =HHHDDD1/2 (3)
whereHHH is theM×K matrix of fast fading coefficients between theK users and the BS, i.e.,[HHH]mk = hmk,
andDDD is aK ×K diagonal matrix, where[DDD]kk = βk. Therefore, (1) can be written as
yyy =√puHHHDDD1/2xxx+ nnn. (4)
4 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
B. Review of Some Results on Very Long Random Vectors
Here we review some limit results for random vectors [14] that will be useful later on. Letppp ,
[p1 ... pn]T andqqq , [q1 ... qn]
T be n× 1 vectors whose elements are independent, identically distributed
(i.i.d.) random variables (RVs) withE {pi} = E {qi} = 0, E{
|pi|2}
= σ2p , and E
{
|qi|2}
= σ2q , i =
1, 2, ..., n. Assume further thatppp andqqq are independent. Then from the law of large numbers,
1
npppHppp
a.s.→ σ2p , and
1
npppHqqq
a.s.→ 0, asn → ∞. (5)
wherea.s.→ denotes the almost sure convergence. Also, from the Lindeberg-Levy central limit theorem,
1√npppHqqq
d→CN(
0, σ2pσ
2q
)
, asn → ∞ (6)
whered→ denotes convergence in distribution.
C. Favorable Propagation
Throughout the rest of the paper, we assume that the fast fading coefficicents, i.e., the elements ofHHH
are i.i.d. RVs with zero mean and unit variance. Then the conditions in (5)–(6) are satisfied withppp andqqq
being any two distinct columns ofGGG. In this case we have
GGGHGGG
M=DDD1/2HHH
HHHH
MDDD1/2 ≈DDD, M ≫ K
and we say that we havefavorable propagation. Clearly, if all fading coefficients are i.i.d. and zero
mean, we have favorable propagation. Recent channel measurements campaigns have shown that multiuser
MIMO systems with large antenna arrays have characteristics that approximate the favorable-propagation
assumption well [10], and therefore provide experimental justification for this assumption.
To understand why favorable propagation is desirable, considerM×K uplink (multiple-access) MIMO
channelHHH, whereM ≥ K, neglecting for now path loss and shadowing factors inDDD. This channel can
offer a sum-rate of
R =K∑
k=1
log2(
1 + puλ2k
)
(7)
where pu is the power spent per terminal and{λk}Kk=1 are the singular values ofHHH, see [13]. If the
channel matrix is normalized such that|Hij| ∼ 1 (where∼ means equality of the order of magnitude),
then∑K
k=1 λ2k = ‖HHH‖2 ≈ MK. Under this constraint the rateR is bounded as
log2 (1 +MKpu) ≤ R ≤ K · log2 (1 +Mpu) (8)
The lower bound (left inequality) is satisfied with equalityif λ21 = MK and λ2
2 = · · · = λ2K = 0
and corresponds to a rank-one (line-of-sight) channel. Theupper bound (right inequality) is achieved if
λ21 = · · · = λ2
K = M . This occurs if the columns ofHHH are mutually orthogonal and have the same norm,
which is the case when we have favorable propagation.
5
III. A CHIEVABLE RATE AND ASYMPTOTIC (M → ∞) POWER EFFICIENCY
By using a large antenna array, we can reduce the transmittedpower of the users asM grows large,
while maintaining a given, desired quality-of-service. Inthis section, we quantify this potential for power
decrease, and derive achievable rates of the uplink. Theoretically, the BS can use the maximum-likelihood
detector to obtain optimal performance. However, the complexity of this detector grows exponentially
with K. The interesting operating regime is when bothM andK are large, butM is still (much) larger
thanK, i.e.,1 ≪ K ≪ M . It is known that in this case, linear detectors (MRC, ZF and MMSE) perform
fairly well [8] and therefore we will restrict consideration to those detectors in this paper. We treat the
cases of perfect CSI (Section III-A) and estimated CSI (Section III-B).
A. Perfect Channel State Information
We first consider the case when the BS has perfect CSI, i.e. it knowsGGG. Let AAA be anM ×K linear
detector matrix which depends on the channelGGG. By using the linear detector, the received signal is
separated into streams by multiplying it withAAAH as follows
rrr = AAAHyyy. (9)
We consider three conventional linear detectors MRC, ZF, and MMSE, i.e.,
AAA =
GGG for MRC
GGG(
GGGHGGG)−1
for ZF
GGG(
GGGHGGG+ 1puIIIK
)−1
for MMSE
(10)
From (1) and (9), the received vector after using the linear detector is given by
rrr =√puAAA
HGGGxxx+AAAHnnn. (11)
Let rk andxk be thekth elements of theK × 1 vectorsrrr andxxx, respectively. Then,
rk =√puaaa
Hk GGGxxx+ aaaHk nnn =
√puaaa
Hk gggkxk +
√pu
K∑
i=1,i 6=k
aaaHk gggixi + aaaHk nnn (12)
whereaaak andgggk are thekth columns of the matricesAAA andGGG, respectively. For a fixed channel realization
GGG, the noise-plus-interference term is a random variable with zero mean and variancepu∑K
i=1,i 6=k |aaaHk gggi|2+‖aaak‖2. By modeling this term as additive Gaussian noise independent of xk we can obtain a lower bound
on the achievable rate. Assuming further that the channel isergodic so that each codeword spans over a
large (infinite) number of realizations of the fast-fading factor ofGGG, the ergodic achievable uplink rate of
the kth user is
RP,k = E
{
log2
(
1 +pu|aaaHk gggk|2
pu∑K
i=1,i 6=k |aaaHk gggi|2 + ‖aaak‖2
)}
(13)
6 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
To approach this capacity lower bound, the message has to be encoded over many realizations of all sources
of randomness that enter the model (noise and channel). In practice, assuming wideband operation, this
can be achieved by coding over the frequency domain, using, for example code OFDM.
Proposition 1: Assume that the BS has perfect CSI and that the transmit powerof each user is scaled
with M according topu = Eu
M, whereEu is fixed. Then,
RP,k → log2 (1 + βkEu) ,M → ∞. (14)
Proof: We give the proof for the case of an MRC receiver. With MRC,AAA = GGG so aaak = gggk. From
(13), the achievable uplink rate of thekth user is
Rmrc
P,k = E
{
log2
(
1 +pu‖gggk‖4
pu∑K
i=1,i 6=k |gggHk gggi|2 + ‖gggk‖2
)}
(15)
Substitutingpu = Eu
Minto (15), and using (5), we obtain (14). By using the law of large numbers, we
can arrive at the same result for the ZF and MMSE receivers. Note from (3) and (5) that whenM grows
large, 1MGGGHGGG tends toDDD, and hence the ZF and MMSE filters tend to the MRC.
Proposition 1 shows that with perfect CSI at the BS and a largeM , the performance of a MU-MIMO
system withM antennas at the BS and a transmit power per user ofEu/M is equal to the performance of
a SISO system with transmit powerEu, without any intra-cell interference and without any fast fading. In
other words, by using a large number of BS antennas, we can scale down the transmit power proportionally
to 1/M . At the same time we increase the spectral efficiencyK times by simultaneously servingK users
in the same time-frequency resource.
1) Maximum-Ratio Combining:For MRC, from (15), by the convexity oflog2(
1 + 1x
)
and using
Jensen’s inequality, we obtain the following lower bound onthe achievable rate:
Rmrc
P,k ≥ Rmrc
P,k , log2
1 +
(
E
{
pu∑K
i=1,i 6=k |gggHk gggi|2 + ‖gggk‖2pu‖gggk‖4
})−1
(16)
Proposition 2: With perfect CSI, Rayleigh fading, andM ≥ 2, the uplink achievable rate from thekth
user for MRC can be lower bounded as follows:
Rmrc
P,k = log2
(
1 +pu (M − 1)βk
pu∑K
i=1,i 6=k βi + 1
)
(17)
Proof: See Appendix B.
If pu = Eu/M , andM grows without bound, then from (17), we have
Rmrc
P,k = log2
(
1 +Eu
M(M − 1)βk
Eu
M
∑Ki=1,i 6=k βi + 1
)
→ log2 (1 + βkEu) , M → ∞ (18)
Equation (18) shows that the lower bound in (17) becomes equal to the exact limit in Proposition 1 as
M → ∞. This is so because whenM → ∞, things that were random before become deterministic and
hence, Jensen’s inequality will hold with equality.
7
2) Zero-Forcing Receiver:With ZF, AAAH =(
GGGHGGG)−1
GGGH , or AAAHGGG = IIIK . Therefore,
aaaHk gggi = δki (19)
whereδki = 1 when k = i and 0 otherwise. From (13) and (19), the achievable uplink rate for the kth
user is
Rzf
P,k = E
log2
1 +pu
[
(
GGGHGGG)−1]
kk
. (20)
By using Jensen’s inequality, we obtain the following lowerbound on the achievable rate:
Rzf
P,k ≥ Rzf
P,k = log2
1 +pu
E
{[
(
GGGHGGG)−1]
kk
}
(21)
Proposition 3: When using ZF, in Rayleigh fading, and provided thatM ≥ K + 1, the achievable
uplink rate for thekth user is lower bounded by
Rzf
P,k = log2 (1 + pu (M −K)βk) (22)
Proof: See Appendix C.
If pu = Eu/M , andM grows large, we have
Rzf
P,k = log2
(
1 +Eu
M(M −K)βk
)
→ log2 (1 + βkEu) , M → ∞ (23)
We can see again from (23) that the lower bound becomes exact at largeM .
3) Minimum Mean-Squared Error Receiver:For MMSE, the detector matrixAAA is
AAA =
(
GGGHGGG+1
puIIIK
)−1
GGGH = GGGH
(
GGGGGGH +1
puIIIM
)−1
(24)
Therefore, thekth column ofAAA is given by [15]
aaak =
(
GGGGGGH +1
puIIIM
)−1
gggk =ΛΛΛ−1
k gggkgggHk ΛΛΛ
−1k gggk + 1
(25)
whereΛΛΛk ,∑K
i=1,i 6=k gggigggHi + 1
puIIIM . Substituting (25) into (13), we obtain the uplink achievable rate for
the kth user as
Rmmse
P,k = E{
log2(
1 + gggHk ΛΛΛ−1k gggk
)} (a)= E
log2
1
1− gggHk
(
1puIIIM +GGGGGGH
)−1
gggk
= E
log2
1
1−[
GGGH(
1puIIIM +GGGGGGH
)−1
GGG
]
kk
(b)= E
log2
1[
(
IIIK + puGGGHGGG)−1]
kk
(26)
8 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
where(a) is obtained directly from (25), and(b) is obtained by using
GGGH
(
1
puIIIM +GGGGGGH
)−1
GGG =
(
1
puIIIK +GGGHGGG
)−1
GGGHGGG = IIIK −(
IIIK + puGGGHGGG)−1
.
By using Jensen’s inequality, we obtain the following lowerbound on the achievable uplink rate:
Rmmse
P,k ≥ Rmmse
P,k = log2
1 +1
E
{
1γk
}
(27)
where γk = 1[
(IIIK+puGGGHGGG)
−1]
kk
− 1. For Rayleigh fading, the exact distribution ofγk can be found in
[16]. This distribution is analytically intractable. To proceed, we approximate it with a distribution which
has an analytically tractable form. More specifically, the PDF of γk can be approximated by a Gamma
distribution as follows [17]:
pγk (γ) =γαk−1e−γ/θk
Γ (αk) θαk
k
(28)
where
αk =(M −K + 1 + (K − 1)µ)2
M −K + 1 + (K − 1) κ, θk =
M −K + 1 + (K − 1) κ
M −K + 1 + (K − 1)µpuβk (29)
whereµ andκ are determined by solving following equations:
µ =1
K − 1
K∑
i=1,i 6=k
1
Mpuβi
(
1− K−1M
+ K−1M
µ)
+ 1
κ
(
1 +K∑
i=1,i 6=k
puβi(
Mpuβi
(
1− K−1M
+ K−1M
µ)
+ 1)2
)
=K∑
i=1,i 6=k
puβiµ+ 1(
Mpuβi
(
1− K−1M
+ K−1M
µ)
+ 1)2 (30)
Using the approximate PDF ofγk given by (28), we have the following proposition.
Proposition 4: With perfect CSI, Rayleigh fading, and MMSE, the lower boundon the achievable rate
for the kth user can be approximated as
Rmmse
P,k = log2 (1 + (αk − 1) θk) . (31)
Proof: Substituting (28) into (27), and using the identity [18, eq.(3.326.2)], we obtain
Rmmse
P,k = log2
(
1 +Γ (αk)
Γ (αk − 1)θk
)
(32)
whereΓ (·) is the Gamma function. Then, usingΓ (x+ 1) = xΓ (x), we obtain the desired result (31).
Remark 1:From (13), the achievable rateRP,k can be rewritten as
RP,k=E
{
log2
(
1+|aaaHk gggk|2aaaHk ΛΛΛkaaak
)}
≤E
{
log2
(
1+‖aaaHk ΛΛΛ1/2
k ‖2‖ΛΛΛ−1/2k gggk‖2
aaaHk ΛΛΛkaaak
)}
= E{
log2(
1 + gggHk ΛΛΛ−1k gggk
)}
(33)
The inequality is obtained by using Cauchy-Schwarz’ inequality, which holds with equality whenaaak =
cΛΛΛ−1k gggk, for any c ∈ C. This corresponds to the MMSE detector (see (25)). This implies that the MMSE
detector is optimal in the sense that it maximizes the achievable rate given by (13).
9
B. Imperfect Channel State Information
In practice, the channel matrixGGG has to be estimated at the BS. The standard way of doing this isto
use uplink pilots. A part of the coherence interval of the channel is then used for the uplink training. Let
T be the length of the coherence interval and letτ be the number of symbols used for pilots. During
the training part of the coherence interval, all users simultaneously transmit pairwisely orthogonal pilot
sequences of lengthτ symbols. The pilot sequences used by theK users can be represented by aτ ×K
matrix√ppΦΦΦ (τ ≥ K), which satisfiesΦΦΦHΦΦΦ = IIIK , wherepp = τpu. Then, theM × τ received pilot
matrix at the BS is given by
YYY p =√ppGGGΦΦΦ
T +NNN (34)
whereNNN is anM × τ matrix of zero-mean, unit variance white Gaussian noise. The minimum-mean-
square-error (MMSE) estimate ofGGG givenYYY is
GGG =1
√ppYYY pΦΦΦ
∗DDD =
(
GGG+1
√ppWWW
)
DDD (35)
whereWWW , NNNΦΦΦ∗ is of dimensionM ×K, andDDD ,(
1ppDDD−1 + IIIK
)−1
. SinceΦΦΦHΦΦΦ = IIIK , WWW has i.i.d.
zero-mean complex Gaussian elements with unit variance.
Denote byEEE , GGG−GGG. Then, from (35), the elements of theith column ofEEE are RVs with zero means
and variances βi
ppβi+1. Furthermore, owing to the properties of MMSE estimation,EEE is independent ofGGG.
The received vector at the BS can be rewritten as
rrr = AAAH(√
puGGGxxx−√puEEExxx+ nnn
)
. (36)
Therefore, after using the linear detector, the received signal associated with thekth user is
rk =√puaaa
Hk GGGxxx−√
puaaaHk EEExxx+ aaaHk nnn =
√puaaa
Hk gggkxk +
√pu
K∑
i=1,i 6=k
aaaHk gggixi −√pu
K∑
i=1
aaaHk εεεixi + aaaHk nnn (37)
whereaaak, gggi, andεεεi are theith columns ofAAA, GGG, andEEE , respectively.
SinceGGG andEEE are independent,AAA andEEE are independent too. The BS treats the channel estimate as the
true channel, and the part including the last three terms of (37) is considered as interference and noise.
Therefore, an achievable rate of the uplink transmission from thekth user is given by
RIP,k = E
{
log2
(
1 +pu|aaaHk gggk|2
pu∑K
i=1,i 6=k |aaaHk gggi|2 + pu‖aaak‖2∑K
i=1βi
τpuβi+1+ ‖aaak‖2
)}
(38)
Intuitively, if we cut the transmitted power of each user, both the data signal and the pilot signal suffer
from the reduction in power. Since these signals are multiplied together at the receiver, we expect that
there will be a “squaring effect”. As a consequence, we cannot reduce power proportionally to1/M as
10 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
in the case of perfect CSI. The following proposition shows that it is possible to reduce the power (only)
proportionally to1/√M .
Proposition 5: Assume that the BS has imperfect CSI, obtained by MMSE estimation from uplink
pilots, and that the transmit power of each user ispu = Eu√M
, whereEu is fixed. Then,
RIP,k → log2(
1 + τβ2kE
2u
)
,M → ∞. (39)
Proof: For MRC, substitutingaaak = gggk into (38), we obtain the achievable uplink rate as
Rmrc
IP,k = E
{
log2
(
1 +pu‖gggk‖4
pu∑K
i=1,i 6=k |gggHk gggi|2 + pu‖gggk‖2∑K
i=1βi
τpuβi+1+ ‖gggk‖2
)}
(40)
Substitutingpu = Eu/√M into (40), and again using (5) with the fact that each elementof gggk is a RV
with zero mean and varianceppβ2i
ppβi+1, we obtain (39). We can obtain (39) for ZF and MMSE in a similar
way.
Proposition 5 implies that with imperfect CSI and a largeM , the performance of a MU-MIMO system
with an M-antenna array at the BS and with the transmit power per user set toEu/√M is equal to the
performance of an interference-free SISO link with transmit powerτβkE2u, without fast fading.
Remark 2:From the proof of Proposition 5, we see that if we cut the transmit power proportionally
to 1/Mα, whereα > 1/2, then the SINR of the uplink transmission from thekth user will go to zero as
M → ∞. This means that1/√M is the fastest rate at which we can cut the transmit power of each user
and still maintain a fixed rate.
Remark 3: In general, each user can use different transmit powers which depend on the geometric
attenuation and the shadow fading. This can be done by assuming that thekth user knowsβk and performs
power control. In this case, the reasoning leading to Proposition 5 can be extended to show that to achieve
the same rate as in a SISO system using transmit powerEu, we must choose the transmit power of the
kth user to be√
Eu
Mτβk.
Remark 4: It can be seen directly from (15) and (40) that the power-scaling laws still hold even for
the most unfavorable propagation (HHH has rank of one). However, for this case, the multiplexing gains do
not materialize since the intracell interference cannot becancelled whenM grows without bound.
We have considered asingle-cell MU-MIMO system. This simplifies the analysis, and it gives us
important insights into how power can be scaled with the number of antennas in very large MIMO
systems. A natural question is to what extent this power-scaling law still holds formulticell MU-MIMO
systems? Intuitively, when we reduce the transmit power of each user, the effect of interference from other
cells also reduces and hence, the signal-to-intercell-interference ratio will stay unchanged. Therefore we
will have the same power-scaling law as in the single-cell scenario. Appendix A explains this argument
in more detail.
11
1) Maximum-Ratio Combining:By following a similar line of reasoning as in the case of perfect CSI,
we can obtain lower bounds on the achievable rate.
Proposition 6: With imperfect CSI, Rayleigh fading, MRC processing, and for M ≥ 2, the achievable
uplink rate for thekth user is lower bounded by
Rmrc
IP,k = log2
(
1 +τp2u (M − 1)β2
k
pu (τpuβk + 1)∑K
i=1,i 6=k βi + (τ + 1) puβk + 1
)
(41)
By choosingpu = Eu/√M , we obtain
Rmrc
IP,k → log2(
1 + τβ2kE
2u
)
, M → ∞ (42)
Again, whenM → ∞, the asymptotic bound on the rate equals the exact limit obtained from Proposition 5.
2) ZF Receiver:For the ZF receiver, we haveaaaHk gggi = δki. From (38), we obtain the achievable uplink
rate for thekth user as
Rzf
IP,k = E
log2
1 +pu
(
∑Ki=1
puβi
τpuβi+1+ 1)
[
(
GGGHGGG)−1]
kk
. (43)
Following the same derivations as in Section III-A2 for the case of prefect CSI, we obtain the following
lower bound on the achievable uplink rate.
Proposition 7: With ZF processing using imperfect CSI, Rayleigh fading, and for M ≥ K + 1, the
achievable uplink rate for thekth user is bounded as
Rzf
IP,k = log2
(
1 +τp2u (M −K) β2
k
(τpuβk + 1)∑K
i=1puβi
τpuβi+1+ τpuβk + 1
)
. (44)
Similarly, with pu = Eu/√M , whenM → ∞, the achievable uplink rate and its lower bound tend to
the ones for MRC (see (42)), i.e.,
Rzf
IP,k → log2(
1 + τβ2kE
2u
)
, M → ∞ (45)
which equals the rate value obtained from Proposition 5.
3) MMSE Receiver:With imperfect CSI, the received vector at the BS can be rewritten as
yyy =√puGGGxxx−√
puEEExxx+ nnn (46)
Therefore, for the MMSE receiver, thekth column ofAAA is given by
aaak =
(
GGGGGGH+
1
puCov (−√
puEEExxx+ nnn)
)−1
gggk =ΛΛΛ
−1
k gggk
gggHk ΛΛΛ−1
k gggk + 1(47)
12 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
whereCov (aaa) denotes the covariance matrix of a random vectoraaa, and
ΛΛΛk ,K∑
i=1,i 6=k
gggigggHi +
(
K∑
i=1
βi
τpuβi + 1+
1
pu
)
IIIM (48)
Similarly to in Remark 1, by using Cauchy-Schwarz’ inequality, we can show that the MMSE receiver
given by (47) is the optimal detector in the sense that it maximizes the rate given by (38).
Substituting (47) into (38), we get the achievable uplink rate for thekth user with MMSE receivers as
Rmmse
P,k = E
{
log2
(
1 + gggHk ΛΛΛ−1
k gggk
)}
= E
log2
1[
(
IIIK+(
∑Ki=1
βi
τpuβi+1+ 1
pu
)−1
GGGHGGG
)−1]
kk
. (49)
Again, using an approximate distribution for the SINR, we can obtain a lower bound on the achievable
uplink rate in closed form.
Proposition 8: With imperfect CSI and Rayleigh fading, the achievable ratefor thekth user with MMSE
processing is approximately lower bounded as follows:
Rmmse
IP,k = log2
(
1 + (αk − 1) θk
)
(50)
where
αk =(M −K + 1 + (K − 1) µ)2
M −K + 1 + (K − 1) κ, θk =
M −K + 1 + (K − 1) κ
M −K + 1 + (K − 1) µωβk (51)
whereω ,(
∑Ki=1
βi
τpuβi+1+ 1
pu
)−1
, βk , τpuβ2k
τpuβk+1, µ and κ are obtained by using following equations:
µ =1
K − 1
K∑
i=1,i 6=k
1
Mωβi
(
1− K−1M
+ K−1M
µ)
+ 1
κ
1 +
K∑
i=1,i 6=k
ωβi(
Mωβi
(
1− K−1M
+ K−1M
µ)
+ 1)2
=
K∑
i=1,i 6=k
ωβiµ+ 1(
Mωβi
(
1− K−1M
+ K−1M
µ)
+ 1)2 (52)
Table I summarizes the lower bounds on the achievable rates for linear receivers derived in this section,
distinguishing between the cases of perfect and imperfect CSI, respectively.
IV. ENERGY-EFFICIENCY AND SPECTRAL-EFFICIENCY TRADEOFF
The energy-efficiency (in bits/Joule) of a system is defined as the spectral-efficiency (sum-rate in
bits/channel use) divided by the transmit power expended (in Joules/channel use). Increasing the spectral
efficiency is associated with increasing the power and hence, with decreasing the energy-efficiency.
Therefore, there is a fundamental tradeoff between the energy efficiency and the spectral efficiency. In
13
this section, we study this tradeoff for the uplink of MU-MIMO systems using linear receivers at the base
station.
We define the spectral efficiency for perfect and imperfect CSI, respectively, as follows
RAP =
K∑
k=1
RAP,k, andRA
IP =T − τ
T
K∑
k=1
RAIP,k (53)
whereA ∈ {mrc, zf, mmse} corresponds to MRC, ZF and MMSE, andT is the coherence interval in
symbols. The energy-efficiency for perfect and imperfect CSI is defined as
ηAP =1
puRA
P , andηAIP =1
puRA
IP (54)
For analytical tractability, we ignore the effect of the large-scale fading here, i.e., we setDDD = IIIK . Also,
we only consider MRC and ZF receivers.1
For the case of perfect CSI, it is straightforward to show from (17), (22), and (54) that when the spectral
efficiency increases, the energy efficiency decreases. For imperfect CSI, this is not always so, as we shall
see next. In what follows, we focus on the case of imperfect CSI since this is the case of interest in
practice.
A. Maximum-Ratio Combining
From (41), the spectral efficiency and energy efficiency withMRC processing are given by
Rmrc
IP =T − τ
TK log2
(
1 +τ (M − 1) p2u
τ (K − 1) p2u + (K + τ) pu + 1
)
, andηmrcIP =1
puRmrc
IP (55)
We have
limpu→0
ηmrcIP = limpu→0
1
puRmrc
IP
(a)= lim
pu→0
T − τ
TK
(log2 e) τ (M − 1) puτ (K − 1) p2u + (K + τ) pu + 1
= 0 (56)
and
limpu→∞
ηmrcIP = limpu→∞
1
puRmrc
IP = 0 (57)
where(a) is obtained by using the fact thatlimx→0ln(1+x)
x= 1.
Equations (56) and (57) imply that for lowpu, the energy efficiency increases whenpu increases, and
for high pu the energy efficiency decreases whenpu increases. Since∂Rmrc
IP
∂pu> 0, ∀pu > 0, Rmrc
IP is a
monotonically increasing function ofpu. Therefore, at lowpu (and hence at low spectral efficiency), the
energy efficiency increases as the spectral efficiency increases and vice versa at highpu. The reason is
that, the spectral efficiency suffers from a “square effect”when the received data signal is multiplied with
1 WhenM is large, the performance of the MMSE receiver is very close to that of the ZF receiver (see Section V). Therefore, the insights
on energy versus spectral efficiency obtained from studyingthe performance of ZF can be used to draw conclusions about MMSE as well.
14 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
the received pilots. Hence, atpu ≪ 1, the spectral-efficiency behaves as∼ p2u. As a consequence, the
energy efficiency (which is defined as the spectral efficiencydivided bypu) increases linearly withpu. In
more detail, expanding the rate in a Taylor series forpu ≪ 1, we obtain
Rmrc
IP ≈ Rmrc
IP |pu=0 +∂Rmrc
IP
∂pu
∣
∣
∣
∣
pu=0
pu +1
2
∂2Rmrc
IP
∂p2u
∣
∣
∣
∣
pu=0
p2u =T − τ
TK log2 (e) τ (M − 1) p2u (58)
This gives the following relation between the spectral efficiency and energy efficiency atpu ≪ 1:
ηmrcIP =
√
T − τ
TK log2 (e) τ (M − 1)Rmrc
IP (59)
We can see that, whenpu ≪ 1, by doubling the spectral efficiency, or by doublingM , we can increase
the energy efficiency by1.5 dB.
B. Zero-Forcing Receiver
From (44), the spectral efficiency and energy efficiency for ZF are given by
Rzf
IP =T − τ
TK log2
(
1 +τ (M −K) p2u(K + τ) pu + 1
)
, andηzfIP =1
puRmrc
IP (60)
Similarly to in the analysis of MRC, we can show that at low transmit powerpu, the energy efficiency
increases when the spectral efficiency increases. In the low-pu regime, we obtain the following Taylor
series expansion
Rzf
IP ≈ T − τ
TK log2 (e) τ (M −K) p2u, for pu ≪ 1 (61)
Therefore,
ηzfIP =
√
T − τ
TK log2 (e) τ (M −K)Rzf
IP (62)
Again, atpu ≪ 1, by doublingM or Rzf
IP, we can increase the energy efficiency by1.5 dB.
V. NUMERICAL RESULTS
We consider a hexagonal cell with a radius (from center to vertex) of 1000 meters. The users are located
uniformly at random in the cell and we assume that no user is closer to the BS thanrh = 100 meters.
The large-scale fading is modelled viaβk = zk/(rk/rh)ν , wherezk is a log-normal random variable with
standard deviationσshadow, rk is the distance between thekth user and the BS, andν is the path loss
exponent. For all examples, we chooseσshadow = 8 dB, andν = 3.8.
We assume that the transmitted data are modulated with OFDM.Here, we choose the parameters that
resemble those of LTE standard: the OFDM symbol durationTs = 71.4µs, and the useful symbol duration
Tu = 66.7µs. Therefore, the guard intervalTg = Ts − Tu = 4.7µs. We choose the channel coherence time
to be Tc = 1 ms. Then,T = Tc
Ts
Tu
Tg= 196, where Tc
Ts= 14 is the number of OFDM symbols in1 ms
coherence interval, andTu
Tg= 14 corresponds to the “frequency smoothness interval” [8]. For all examples,
we computed the spectral efficiency using (53).
15
A. Power-Scaling Law
We first conduct an experiment to validate the tightness of our proposed capacity bounds. Fig. 1 shows
the simulated spectral efficiency and the proposed analytical bounds for MRC, ZF, and MMSE receivers
with perfect and imperfect CSI atpu = 10 dB. In this example there areK = 10 users. For CSI estimation
from uplink pilots, we choose pilot sequences of lengthτ = K. (This is the smallest amount of training
that can be used.) Clearly, all bounds are very tight, especially at largeM . Therefore, in the following,
we will use these bounds for all numerical work.
We next illustrate the power scaling laws. Fig. 2 shows the spectral efficiency on the uplink versus
the number of BS antennas forpu = Eu/M andpu = Eu/√M with perfect and imperfect receiver CSI,
and with MRC, ZF, and MMSE processing, respectively. Here, we chooseEu = 20 dB. At this SNR,
the spectral efficiency is in the order of 10–30 bits/s/Hz, corresponding to a spectral efficiency per user
of 1–3 bits/s/Hz. These operating points are reasonable from a practical point of view. For example, 64-
QAM with a rate-1/2 channel code would correspond to 3 bits/s/Hz. (Figure 3, see below, shows results
at lower SNR.) As expected, withpu = Eu/M , whenM increases, the spectral efficiency approaches a
constant value for the case of perfect CSI, but decreases to0 for the case of imperfect CSI. However, with
pu = Eu/√M , for the case of perfect CSI the spectral efficiency grows without bound (logarithmically
fast withM) whenM → ∞ and with imperfect CSI, the spectral efficiency converges toa nonzero limit
asM → ∞. These results confirm that we can scale down the transmittedpower of each user asEu/M
for the perfect CSI case, and asEu/√M for the imperfect CSI case whenM is large.
Typically ZF is better than MRC at high SNR, and vice versa at low SNR [13]. MMSE always performs
the best across the entire SNR range (see Remark 1). When comparing MRC and ZF in Fig. 2, we see
that here, when the transmitted power is proportional to1/√M , the power is not low enough to make
MRC perform as well as ZF. But when the transmitted power is proportional to1/M , MRC performs
almost as well as ZF for largeM . Furthermore, as we can see from the figure, MMSE is always better
than MRC or ZF, and its performance is very close to ZF.
In Fig. 3, we consider the same setting as in Fig. 2, but we chooseEu = 5 dB. This figure provides
the same insights as Fig. 2. The gap between the performance of MRC and that of ZF (or MMSE) is
reduced compared with Fig. 2. This is so because the relativeeffect of crosstalk interference as compared
to the thermal noise is larger here than in Fig. 2.
We next show the transmit power per user that is needed to reach a fixed spectral efficiency. Fig. 4
shows the normalized power (pu) required to achieve1 bit/channel use per user as a function ofM . As
predicted by the analysis, by doublingM , we can cut back the power by approximately 3 dB and 1.5
dB for the cases of perfect and imperfect CSI, respectively.WhenM is large (M/K ' 6), the difference
16 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
in performance between MRC and ZF (or MMSE) is less than1 dB and3 dB for the cases of perfect
and imperfect CSI, respectively. This difference increases when we increase the target spectral efficiency.
Fig. 5 shows the normalized power required for2 bit/s/Hz per user. Here, the crosstalk interference is
more significant (relative to the thermal noise) and hence the ZF and MMSE receivers perform relatively
better.
B. Energy Efficiency versus Spectral Efficiency Tradeoff
We next examine the tradeoff between energy efficiency and spectral efficiency in more detail. Here,
we ignore the effect of large-scale fading, i.e., we setDDD = IIIK . We normalize the energy efficiency against
a reference mode corresponding to a single-antenna BS serving one single-antenna user with a transmit
power ofpu = 10 dB. For this reference mode, the spectral efficiencies and energy efficiencies for MRC,
ZF, and MMSE are equal, and given by (from (40) and (53))
R0IP =
T − τ
TE
{
log2
(
1 +τp2u|z|2
1 + pu (1 + τ)
)}
, η0IP = R0IP/pu
wherez is a Gaussian RV with zero mean and unit variance. For the reference mode, the spectral-efficiency
is obtained by choosing the duration of the uplink pilot sequenceτ to maximizeR0IP. Numerically we
find thatR0IP = 2.65 bits/s/Hz andη0IP = 0.265 bits/J.
Fig. 6 shows the relative energy efficiency versus the the spectral efficiency for MRC and ZF. The
relative energy efficiency is obtained by normalizing the energy efficiency byη0IP and it is therefore
dimensionless. The dotted and dashed lines show the performances for the cases ofM = 1, K = 1 and
M = 100, K = 1, respectively. Each point on the curves is obtained by choosing the transmit powerpu
and pilot sequence lengthτ to maximize the energy efficiency for a given spectral efficiency. The solid
lines show the performance for the cases ofM = 50, and100. Each point on these curves is computed by
jointly choosingK, τ , andpu to maximize the energy-efficiency subject a fixed spectral-efficiency, i.e.,
arg maxpu,K,τ
ηAIP
s.t.RA
IP = const., K ≤ τ ≤ T
We first consider a single-user system withK = 1. We compare the performance of the casesM = 1
andM = 100. SinceK = 1 the performances of MRC and ZF are equal. With the same power used
as in the reference mode, i.e.,pu = 10 dB, using100 antennas can increase the spectral efficiency and
the energy efficiency by factors of4 and3, respectively. Reducing the transmit power by a factor of100,
from 10 dB to −10 dB yields a100-fold improvement in energy efficiency compared with that ofthe
reference mode with no reduction in spectral-efficiency.
We next consider a multiuser system (K > 1). Here the transmit powerpu, the number of usersK to be
served simultaneously, and the duration of the uplink pilotsequencesτ are chosen optimally for fixedM .
17
We considerM = 50 and100. Here the system performance improves very significantly compared to the
single-user case. For example, with MRC, atpu = 0 dB, compared with the case ofM = 1, K = 1, the
spectral-efficiency increases by factors of50 and 80, while the energy-efficiency increases by factors of
55 and75 for M = 50 andM = 100, respectively. As discussed in Section IV, at low spectral efficiency,
the energy efficiency increases when the spectral efficiencyincreases. Furthermore, we can see that at
high spectral efficiency, ZF outperforms MRC. This is due to the fact that the MRC receiver is limited by
the intracell interference, which is significant at high spectral efficiency. As a consequence, whenpu is
increased, the spectral efficiency of MRC approaches a constant value, while the energy efficiency goes
to zero (see (57)).
The corresponding optimum values ofK and τ as functions of the spectral efficiency forM = 100
are shown in Fig. 7. For MRC, the optimal number of users and uplink pilots are the same (this means
that the minimal possible length of training sequences are used). For ZF, more of the coherence interval
is used for training. Generally, at low transmit power and therefore at low spectral efficiency, we spend
more time on training than on payload data transmission. At high power (high spectral efficiency and low
energy efficiency), we can serve around55 users, andK = τ for both MRC and ZF.
VI. CONCLUSION
We studied the uplink transmission of data fromK terminals to an array ofM antennas at the BS,
with respect to energy efficiency and throughput. We found that energy efficiency is qualitatively different
depending on whether the receiver has perfect CSI or whetherit has only imperfect CSI derived from
uplink pilots. With perfect CSI the radiated power of the terminals can be made inversely proportional
to M while maintaining spatial multiplexing gains, while for imperfect CSI the power can only be made
inversely proportional to the square-root ofM . We also compared three receiver structures: MRC, ZF, and
MMSE. Among these detectors, the MMSE is always the best one,and its performance is very close to
that of ZF. Except for the case of imperfect channel knowledge and a small number of antennas, MMSE
and ZF outperform MRC. However, reducing the transmit powernarrows the performance gap between
MRC and ZF (or MMSE).
Furthermore, we investigated the tradeoff between spectral efficiency and energy efficiency. We showed
that, at low spectral efficiency, improving spectral efficiency also can improve the energy efficiency.
Taken together, a large excess of antennas offers high spectral efficiency, high energy efficiency, with
low-complexity processing.
18 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
APPENDIX
A. Power-Scaling Law for Multicell MU-MIMO Systems
We show that the power-scaling laws for multicell systems are the same as for single-cell systems. We
will use the MRC detector for our analysis. A similar analysis can be performed for the ZF and MMSE
detectors.
Consider the uplink of a multicell MU-MIMO system withL cells sharing the same frequency band.
Each cell includes one BS equipped withM antennas andK single-antenna users. TheM × 1 received
vector at thelth BS is given by
yyyl =√pu
L∑
i=1
GGGlixxxi +nnnl (63)
where√puxxxi is theK × 1 transmitted vector ofK users in theith cell; nnnl is an AWGN vector,nnnl ∼
CN (0, IIIM); andGGGli is theM ×K channel matrix between thelth BS and theK users in theith cell.
The channel matrixGGGli can be represented as
GGGli =HHH liDDD1/2li (64)
whereHHH li is the fast fading matrix between thelth BS and theK users in theith cell whose elements
have zero mean and unit variance; andDDDli is a K ×K diagonal matrix, where[DDDli]kk = βlik, with βlik
represents the large-scale fading between thekth user in thei cell and thelth BS.
1) Perfect CSI:With perfect CSI, the received signal at thelth BS after using MRC is given by
rrrl = GGGHll yyyl =
√puGGG
HllGGGllxxxl +
√pu
L∑
i=1,i 6=l
GGGHllGGGlixxxi +GGGH
ll nnnl (65)
With pu = Eu
M, then (65) can be rewritten as
1√M
rrrl =√
EuGGGH
llGGGll
Mxxxl +
√pu
L∑
i=1,i 6=l
GGGHllGGGli
Mxxxi +
1√M
GGGHll nnnl (66)
From (5)–(6), it follows that whenM grows large, the interference from other cells disappears.More
precisely,
1√M
rrrl →√
EuDDDllxxxl +DDD1/2ll nnnl (67)
wherennnl ∼ CN (0, III). Therefore, the SINR of the uplink transmission from thekth user in thelth cell
converges to a constant value whenM grows large, i.e.,
SINRPl,k → βllkEu, asM → ∞ (68)
This means that the power scaling law derived for single-cell systems is valid in multi-cell systems too.
19
2) Imperfect CSI:For this case, the channel estimate from the uplink pilots iscontaminated by inter-
ference from other cells. Following a similar derivation asin the case of perfect CSI, withpu = Eu/√M ,
the asymptotic SINR of the uplink transmission from thekth user in thelth cell is
SINRIPl,k →
τβ2llkE
2u
τ∑L
i 6=l β2likE
2u + 1
, asM → ∞. (69)
We can see that the1/√M power-scaling law still holds. Furthermore, transmissionfrom users in other
cells constitutes residual interference. The reason is that the pilot reuse implies pilot-contamination-induced
inter-cell interference which grows withM at the same rate as the desired signal.
B. Proof of Proposition 2
From (16), we have
Rmrc
P,k = log2
1 +
(
E
{
pu∑K
i=1,i 6=k |gi|2 + 1
pu‖gggk‖2
})−1
(70)
wheregi ,gggHkgggi
‖gggk‖. Conditioned ongggk, gi is a Gaussian RV with zero mean and varianceβi which does not
depend ongggk. Therefore,gi is Gaussian distributed and independent ofgggk, gi ∼ CN (0, βi). Then,
E
{
pu∑K
i=1,i 6=k |gi|2 + 1
pu‖gggk‖2
}
=
(
pu
K∑
i=1,i 6=k
E{
|gi|2}
+ 1
)
E
{
1
pu‖gggk‖2}
=
(
pu
K∑
i=1,i 6=k
βi + 1
)
E
{
1
pu‖gggk‖2}
(71)
Using the identity [19]
E{
tr
(
WWW−1)}
=m
n−m(72)
whereWWW ∼ Wm (n,IIIn) is anm×m central complex Wishart matrix withn (n > m) degrees of freedom,
we obtain
E
{
1
pu‖gggk‖2}
=1
pu (M − 1)βk, for M ≥ 2 (73)
Substituting (73) into (71), we arrive at the desired result(17).
C. Proof of Proposition 3
From (3), we have
E
{[
(
GGGHGGG)−1]
kk
}
=1
βk
E
{[
(
HHHHHHH)−1]
kk
}
=1
Kβk
E
{
tr
[
(
HHHHHHH)−1]}
(a)=
1
(M −K) βk, for M ≥ K + 1 (74)
where(a) is obtained by using (72). Using (74), we get (22).
20 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
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21
TABLE I
LOWER BOUNDS ON THE ACHIEVABLE RATES OF THE UPLINK TRANSMISSION FOR THEkTH USER.
Perfect CSI Imperfect CSI
MRC log2
(
1 + pu(M−1)βk
pu∑
Ki=1,i6=k
βi+1
)
log2
(
1 +τp2u(M−1)β2
k
pu(τpuβk+1)∑
Ki=1,i6=k
βi+(τ+1)puβk+1
)
ZF log2 (1 + pu (M −K) βk) log2
(
1 +τp2u(M−K)β2
k
(τpuβk+1)∑
Ki=1
puβiτpuβi+1
+τpuβk+1
)
MMSE log2 (1 + (αk − 1) θk) log2
(
1 + (αk − 1) θk)
22 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
50 100 150 200 250 300 350 400 450 5000.0
10.0
20.0
30.0
40.0
50.0
Bounds Simulation
Number of Base Station Antennas (M)
Perfect CSI
MRC, ZF, MMSE
50 100 150 200 250 300 350 400 450 5000.0
10.0
20.0
30.0
Bounds Simulation
Sp
ect
ral-
Effi
cien
cy (
bits
/s/H
z)
Number of Base Station Antennas (M)
MRC, ZF, MMSE
Imperfect CSI
Fig. 1. Lower bounds and numerically evaluated values of thespectral efficiency for different numbers of BS antennas forMRC, ZF, and
MMSE with perfect and imperfect CSI. In this example there are K = 10 users, the coherence intervalT = 196, the transmit power per
terminal ispu = 10 dB, and the propagation channel parameters wereσshadow = 8 dB, andν = 3.8.
23
50 100 150 200 250 300 350 400 450 5000.0
10.0
20.0
30.0
40.0
Perfect CSI, MRC Imperfect CSI, MRC Perfect CSI, ZF Imperfect CSI, ZF Perfect CSI, MMSE Imperfect CSI, MMSE
� �
p E M=
� �
p E M=
Sp
ectr
al-E
ffici
ency
(b
its/s
/Hz)
Number of Base Station Antennas (M)
Eu = 20 dB
Fig. 2. Spectral efficiency versus the number of BS antennasM for MRC, ZF, and MMSE processing at the receiver, with perfect CSI and
with imperfect CSI (obtained from uplink pilots). In this exampleK = 10 users are served simultaneously, the reference transmit power is
Eu = 20 dB, and the propagation parameters wereσshadow = 8 dB andν = 3.8.
24 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
50 100 150 200 250 300 350 400 450 5000.0
2.0
4.0
6.0
8.0
10.0
Perfect CSI, MRC Imperfect CSI, MRC Perfect CSI, ZF Imperfect CSI, ZF Perfect CSI, MMSE Imperfect CSI, MMSE
� �
p E M=
� �
p E M=
Spe
ctra
l-Effi
cien
cy (
bits
/s/H
z)
Number of Base Station Antennas (M)
Eu = 5 dB
Fig. 3. Same as Figure 2, but withEu = 5 dB.
25
50 100 150 200 250 300 350 400 450 500-9.0
-6.0
-3.0
0.0
3.0
6.0
9.0
12.0
15.0
18.0 MRC ZF MMSE
Perfect CSI
Req
uire
d P
ow
er, N
orm
aliz
ed (
dB
)
Number of Base Station Antennas (M)
Imperfect CSI
1 bit/s/Hz
Fig. 4. Transmit power required to achieve1 bit/channel use per user for MRC, ZF, and MMSE processing, with perfect and imperfect CSI,
as a function of the numberM of BS antennas. The number of users is fixed toK = 10, and the propagation parameters areσshadow = 8
dB andν = 3.8.
26 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
50 100 150 200 250 300 350 400 450 500-3.0
0.0
3.0
6.0
9.0
12.0
15.0
18.0
21.0
24.0
27.0
30.0
MRC ZF MMSE
Perfect CSI
Req
uire
d P
ow
er, N
orm
aliz
ed (
dB
)
Number of Base Station Antennas (M)
Imperfect CSI
2 bits/s/Hz
Fig. 5. Same as Figure 4 but for a target spectral efficiency of2 bits/channel use per user.
27
0 10 20 30 40 50 60 70 80 9010
-1
100
101
102
103
104
K=1, M=1
MRC
20 dB
10 dB
0 dB
-10 dB
-20 dB
M=50
Rel
ativ
e E
ner
gy-
Effi
cie
ncy
(bits
/J)/(
bits
/J)
Spectral-Efficiency (bits/s/Hz)
Reference Mode
K=1, M=100
M=100
ZF
Fig. 6. Energy efficiency (normalized with respect to the reference mode) versus spectral efficiency for MRC and ZF receiver processing
with imperfect CSI. The reference mode corresponds toK = 1,M = 1 (single antenna, single user), and a transmit power ofpu = 10
dB. The coherence interval isT = 200 symbols. For the dashed curves (marked withK = 1), the transmit powerpu and the fraction of
the coherence intervalτ/T spent on training was optimized in order to maximize the energy efficiency for a fixed spectral efficiency. For
the green and red curves (marked MRC and ZF; shown forM = 50 andM = 100 antennas, respectively), the number of usersK was
optimized jointly withpu and τ/T to maximize the energy efficiency for given spectral efficiency. Any operating point on the curves can
be obtained by appropriately selectingpu and optimizing with respect toK and τ/T . The number marked next to the× marks on each
curve is the powerpu spent by the transmitter.
28 SUBMITTED TO THE IEEE TRANSACTIONS ON COMMUNICATIONS
0 10 20 30 40 50 600
20
40
60
80
100
120
140
number of users
Spectral-Efficiency (bits/s/Hz)
number of uplink pilots
ZF
MRC
M=100
Fig. 7. Optimal number of usersK and number of symbolsτ spent on training, out of a total ofT = 200 symbols per coherence interval,
for the curves in Fig. 6 corresponding toM = 100 antennas.
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