GlobalSIP 2014: Massive MIMO Communications

Massive MIMO with Per-Antenna Power Constraint

Shuowen Zhang, Rui Zhang, and Teng Joon Lim
ECE Department, National University of Singapore
Email: {shuowen.zhang,elezhang,eleltj}@nus.edu.sg

Abstract—In this paper, we study the sum-rate maximization problem studied the optimal precoder under PAPC for the MISO BC, which
in a single-cell massive MIMO downlink system with K users. Unlike the was shown to have a more general structure than the conventional
conventional sum-power constraint (SPC) that limits the total average
channel pseudo-inverse based one under the SPC. The results in
power over all the transmit antennas, the more practical per-antenna
power constraint (PAPC) is considered. A precoding scheme based on [10] have been extended in [11] to the MIMO BC with general BD
the principle of equal gain transmission (EGT) is proposed to satisfy any (block diagonalization)-based precoding. The rate maximization with
given PAPC with low complexity. We show that as the number of transmit general LTCCs for the Gaussian MIMO BC-MAC was solved in [12]
antennas goes to infinity, the gap between the sum-rate achieved by our based on the uplink-downlink duality. The algorithms reported in
proposed scheme under PAPC and that under a single SPC achieved with
optimal maximum ratio transmission (MRT) approaches K log2 (4/π) the above prior works on solving the rate maximization problems in
in bits/sec/Hz under the condition of independent Rayleigh fading of MISO/MIMO BCs under SPC and/or PAPC require high complexity,
the user channels. The analytical results are also verified by numerical which becomes unaffordable as the number of transmit antennas
examples. Therefore, in massive MIMO systems, our proposed EGT- becomes very large as in massive MIMO systems. Therefore, new
based precoding scheme is near-optimal under PAPC with asymptotically
negligible capacity loss compared against the sum-rate upper bound by
approaches need to be adopted to tackle such difficulties by exploiting
the MRT-based precoding under the relaxed SPC, which justifies the use the unique characteristics of massive MIMO systems, such as the
of PAPC in practical massive MIMO systems. favorable asymptotic channel orthogonality. Furthermore, it is worth
pointing out that there was recently another line of research on
I. I NTRODUCTION investigating massive MIMO systems subject to the constant envelope
The potential gains in spectral and energy efficiency offered by (CE) constraint on the transmitted signals from each of the antennas,
massive MIMO, in which transmitters and/or receivers have a very which is even more stringent than the PAPC in limiting the dynamic
large number (in the hundreds) of antennas, are by now well under- range of transmitted signal power [13]–[15]. Generally speaking, low-
stood [1]–[4]. Under a so-called favorable channel condition, i.e., the complexity linear precoding design under the practical PAPC or CE
user channels become asymptotically orthogonal as the number of constraint in massive MIMO systems still remains open for further
transmit antennas increases, simple linear precoding schemes were investigation.
shown to be near-optimal [5]. However, it comes to our attention In this paper, we study the sum-rate maximization problem in a
that a vast body of existing literature on massive MIMO is based single-cell massive MIMO downlink system with K users, each with
on a single average sum-power constraint (SPC) at the multi-antenna a single antenna. Unlike the conventional SPC as considered in e.g.,
transmitter. Motivated by current designs where each antenna has [1], [4], [5], we assume that the more practical PAPC is employed
its own radio frequency (RF) chain and is thus limited by the at the transmitter. We propose a low-complexity linear precoding
dynamic range of its power amplifier (PA), along with the demand scheme which satisfies any given PAPC based on the principle
for inexpensive and power-efficient PAs due to the large number of equal gain transmission (EGT). It is shown that under users’
of antennas, the more stringent and hardware-friendly per-antenna independent Rayleigh channel distribution, the sum-rate achieved by
power constraint (PAPC) should be considered. Furthermore, in the proposed scheme under PAPC is only K log2 (4/π) bits/sec/Hz
certain massive MIMO applications using distributed antenna systems (bps/Hz) from the sum-rate achieved by optimal maximum ratio
(DASs), PAPC is more realistic than SPC since different remote transmission (MRT)-based precoding under a relaxed SPC as the
antenna units (RAUs) are not co-located and thus cannot share power number of transmit antennas goes to infinity. This result implies
with each other. that in massive MIMO systems, our proposed scheme performs near-
Massive MIMO precoding with PAPC is of course closely related optimally under the PAPC (although the optimal design still remains
to MIMO precoding with PAPC for an arbitrary (usually assumed unknown), and adopting the more stringent PAPC entails only a
small) number of antennas. There is a substantial body of work bounded sum-rate capacity loss as compared with the conventional
on this topic, as we briefly summarize here. It was proven in [6] SPC.
that the capacity region for the Gaussian MIMO broadcast channel
II. S YSTEM M ODEL
(BC) is achieved by dirty-paper coding (DPC) under any general
linear transmit covariance constraints (LTCCs). However, DPC is Consider a MISO downlink system with K single-antenna users
generally difficult to implement in practical systems due to its high in a single cell. The transmitter is equipped with M antennas, with
complexity. Consequently, linear precoding schemes based on MMSE M  K. We consider linear precoding at the transmitter in this paper
(minimum mean squared error) or ZF (zero forcing) criterion have and the precoding vector for user k is denoted by wk ∈ CM ×1 . Thus,
been proposed [7], [8] under a single SPC at the multi-antenna the transmitted signal is given by
transmitter. On the other hand, under the PAPC, [9] showed that the 
K

optimal MMSE-based downlink precoding problem can be solved by x= wk s k , (1)

converting it into an equivalent uplink problem via the celebrated k=1

uplink-downlink duality; the problem can be alternatively solved where sk denotes the information-bearing signal for user k. We
directly in the downlink by formulating it as a second-order cone assume sk ’s are independent and identically distributed (i.i.d.) circu-
programming (SOCP) [8]. For the case of ZF-based precoding, [10] larly symmetric complex Gaussian (CSCG) random variables (RVs)

978-1-4799-7088-9/14/$31.00 ©2014 IEEE 642

 GlobalSIP 2014: Massive MIMO Communications

with zero mean and unit variance, denoted by sk ∼ CN (0, 1), ∀k. our study for the PAPC case in Section IV, we obtain the following
Denote the channel vectors from the transmitter to the kth user lemma.
H√
as hH k = gk βk ∈ C1×M with hH k = [hk1 , hk2 , ..., hkM ] and Lemma 3.1: Given the MRT precoding in (8), the resulting SINR
gkH = [gk1 , gk2 , ..., gkM ], where gkH ’s are assumed to be i.i.d. for user k as M → ∞ is given by
random vectors representing Rayleigh fast fading, with elements gki ’s
SINRMRT ρλ β
modelled by independent CSCG RVs, i.e., gki ∼ CN (0, 1), i = lim k
=  k k . (9)
M →∞ M 1 + j=k ρλj βk
1, 2, ..., M , and βk specifies the long-term channel power gain for
user k, which depends on both shadowing and distance-dependent Proof: With i.i.d. RVs gki ∼ CN (0, 1), ∀k and ∀i, we have
attenuation. We assume that the channel vectors of all users are E[|hki |2 ] = βk E[|gki |2 ] = βk and E[hki hjl ] = 0, k = j or i = l.
perfectly known at the base station (BS). Thus, by law of large numbers, the received signal power for user k
For the kth user, the received signal is given by is given by
√ MRT 2
yk = ρhH k x + zk , k = 1, 2, ..., K, (2) SkMRT k wk
ρ|hH |
lim = lim
M →∞ M M →∞ M
where ρ is a normalization factor for the average signal-to-noise ratio  2
(10)
(SNR) and zk ∼ CN (0, 1) is the (normalized) Gaussian noise at ρλk M i=1 |hki |
= lim = ρλk βk .
the kth receiver. By treating the interference from all other users as M →∞ M
additional Gaussian noise, the signal-to-interference-plus-noise ratio The MUI power IkMRT is then given by
(SINR) for the kth user is given by 
lim IkMRT = lim ρ|hH MRT 2
k wj |
Sk ρ|hH w |2 M →∞ M →∞
SINRk = =  k kH , (3) M
j=k

1 + Ik 1 + j=k ρ|hk wj |2  ρλj 2
|hki |2 + hji h∗ki hkl h∗jl )
i=1 (|hji | l=i
= lim (11)
where Sk and Ik denote the received signal power and multi-user M →∞ M βj
j=k
interference (MUI) power at the kth user, respectively. The sum-rate 
achievable by all users is thus given by (in bps/Hz) = ρλj βk .
j=k

K
R= log2 (1 + SINRk ). (4) With (10) and (11) substituted into (3), the proof of Lemma 3.1 is
k=1 completed.
Our aim is to maximize the sum-rate given in (4) subject to PAPC From (4) and (9), the asymptotic sum-rate with MRT-based pre-
at the transmitter, expressed as follows: coding as M → ∞ is given by
 
1 K
E[|xi |2 ] ≤ , i = 1, 2, ..., M, (5) R MRT
→ K log2 M + log2 
ρλk βk
. (12)
M 1 + j=k ρλj βk
K 1
k=1
which can be alternatively expressed as [ k=1 wk wkH ]i,i ≤ M , ∀i,
where [·]i,j denotes the (i, j)-th element of a matrix. For comparison, From (12), it follows that the sum-rate achievable with MRT-based
the conventional SPC is given by scheme scales asymptotically with K log2 M as M → ∞, which is
consistent with [1]. The second term in (12) does not depend on
E[x2 ] ≤ 1, (6) M and can be further maximized by solving the following power
K 2 allocation problem:
which is equivalent to k=1 wk  ≤ 1. Notice that any transmit
signals satisfying the PAPC given in (5) will satisfy the SPC in (6),  
K
λk βk
but not necessarily vice versa. max log2 
{λk } 1 + j=k ρλj βk
k=1
III. S UM -R ATE M AXIMIZATION UNDER C ONVENTIONAL
K (13)
S UM -P OWER C ONSTRAINT s.t. λk ≤ 1,
In this section, we revisit the sum-rate maximization problem k=1
subject to a single SPC given in (6), which is expressed explicitly as λk > 0, k = 1, 2, ..., K,

K which is equivalent to:
max log2 (1 + SINRk ) 
{wk }
k=1 
K
1+ j=k ρλj βk
(7) min
K {λk } λk βk
s.t. wk 2 ≤ 1. k=1

k=1 K (14)
s.t. λk ≤ 1,
It has been shown in [1] that under the i.i.d. Rayleigh fading k=1
condition, the user channels become asymptotically orthogonal as the λk > 0, k = 1, 2, ..., K.
number of transmit antennas, M , goes to infinity, and in this case
the well-known maximum ratio transmission (MRT)-based precoding Notice that λk > 0, ∀k is needed to achieve the sum-rate scaling
serves as the asymptotically optimal scheme, i.e., K log2 M in (12). Problem (14) can be shown to be a geometric
√ programming (GP) [16], which can be solved by existing software,
λk hk
wkMRT = , (8) e.g., CVX [17].
hk  Remark 3.1: For the special single-user case with K = 1, we omit
λk denotes the power allocation for user k, with λk ≥ 0, ∀k,
where the user index for brevity. In this case, it is well-known that MRT
h
and K k=1 λk ≤ 1. In order to re-confirm this result and also motivate
w = h is the optimal precoder under the SPC, even with finite M .

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From (12), the asymptotic capacity as M → ∞ for the single-user is more stringent than SPC, the sum-rate capacity under SPC is an
case is given by upper bound for the maximum sum-rate under PAPC. Therefore,
MRT our proposed EGT-based scheme is near-optimal within the class of
CSU → log2 M + log2 (ρβ). (15) precoders under PAPC, with nearly negligible loss that does not scale
IV. S UM -R ATE M AXIMIZATION UNDER P ER - ANTENNA P OWER with M as M increases. Furthermore, since the sum-rate achieved
C ONSTRAINT by our proposed scheme is a lower bound on the sum-rate capacity
under PAPC, it follows that the capacity loss due to PAPC compared
Now, consider the sum-rate maximization problem subject to the
with SPC is upper-bounded by K log2 ( π4 ), although the sum-rate
PAPC given in (5):
capacity under PAPC still remains open for massive MIMO systems.

K
Remark 4.1: In the single-user case under PAPC, it is shown in
max log2 (1 + SINRk ) [18] that EGT achieves the capacity for the MISO channel, with
{wk }
k=1
K  (16) any finite number of antennas. From (20), the asymptotic capacity as
 1 M → ∞ for the single-user channel is given by
s.t. wk wkH ≤ , i = 1, 2, ..., M.
k=1
M π
i,i EGT
CSU → log2 M + log2 (ρβ). (22)
Due to the high complexity of applying existing methods [8], [9] 4
to solve the above problem with very large M , we propose a low- Comparing (22) with (15) under SPC, the asymptotic capacity gap
complexity precoding scheme based on single-user EGT. For the as M → ∞ is log2 ( π4 ), which is consistent to the multi-user result
single-user MISO channel case, suppose the channel vector is given in (21) with K = 1.
by hH = [|h1 |ejθ1 , |h2 |ejθ2 , ..., |hM |ejθM ]; then the EGT assigns a
phase correlation and equal power on each antenna:
V. N UMERICAL R ESULTS
1 −jθ1 −jθ2
wEGT = [e ,e , ..., e−jθM ]T . (17)
M Consider K = 10 users in a single hexagonal cell with radius
1600m. For simplicity, we assume the distance between each user and
Based on this result, for the multi-user case, by denoting hH
k =
the transmitter is uniformly distributed between 100m and 1600m.
[|hk1 |ejθk1 , |hk2 |ejθk2 , ..., |hkM |ejθkM ], our proposed EGT-based
Assume βk = K0 ( ddk0 )α without shadowing, where d0 = 10m
precoding is
is a reference distance, dk > d0 represents the distance from the
λk −jθk1 −jθk2 transmitter to the kth user, α = 3.8 is the path loss coefficient, and
wkEGT = [e ,e , ..., e−jθkM ]T , k = 1, 2, ..., K,
M the constant K0 = ( 4πd λ
0
)2 where λ denotes the wavelength at a
(18) carrier frequency of fc = 900MHz. The transmit power is assumed
where λk > 0 denotes
 the power allocation  for user
K
k. Since it can be to be P = 1 watt or 30dBm and the average receiver noise power is
K EGT EGT H k=1 λk
easily verified that k=1 w k w k = M
, the PAPC set to be −94dBm over a signal bandwidth of B = 10MHz, which
i,i
in problem (16) is satisfied by the proposed precoder design with yields an average downlink SNR at the user located at 500m from
K
k=1 λk ≤ 1.
the transmitter as 7.91dB.
Lemma 4.1: With our proposed EGT-based precoding in (18), the Fig. 1 shows the sum-rate achieved by our proposed EGT-based
resulting SINR for user k as M → ∞ is given by precoding under PAPC versus the number of transmit antennas, M , as
compared with the sum-rate achieved by MRT-based precoding under
SINREGT π
4
ρλk βk
lim k
=  . (19) SPC. Both analytical results based on (12) and (20) and simulation
M →∞ M 1 + j=k ρλj βk results by averaging over 103 random channel realizations for each
Proof: Please refer to Appendix A. given value of M are plotted, which are observed to match closely
From (19), the asymptotic sum-rate with EGT-based precoding as for both EGT and MRT-based precoding. Note that in this example,
M → ∞ is given by we assume equal-power allocation for the users in both precoding
1
  schemes, i.e., λk = K , ∀k. Fig. 2 shows the sum-rate gap between
π  K
ρλk βk
R EGT
→ K log2 M + log2  . (20) the two schemes. It is observed that the sum-rate gap by simulation
4
k=1
1 + j=k ρλj βk converges to the constant K log2 ( π4 ) as M grows, which is expected
from Theorem 4.1.
Theorem 4.1: The gap between the sum-rate achieved by the EGT-
based precoding in (18) under PAPC in (5) and that by the MRT-
based precoding in (8) under SPC in (6) converges to a constant as VI. C ONCLUSIONS
M → ∞:
  In this paper, a low-complexity EGT-based precoding scheme
4
lim (RMRT − REGT ) = K log2 . (21) is proposed to meet the practical PAPC in a single-cell massive
M →∞ π MIMO downlink system with K users. Both analytical and numerical
Proof: Comparing (20) with (12), we see that both sum-rate results show that the gap between the sum-rate achieved by our
expressions have the identical second-term given the same power proposed scheme under PAPC and that achieved by MRT-based
allocation {λk }. Hence, as M → ∞, the asymptotic sum-rate gap in precoding under SPC converges to a constant K log2 (4/π) in bps/Hz
(21) follows. as M → ∞ under i.i.d. Rayleigh fading channels. Our results indicate
Theorem 4.1 shows that our proposed EGT-based precoding that for massive MIMO systems, the practical PAPC endures nearly
scheme under PAPC achieves the same scaling order with the sum- negligible rate loss from the ideal SPC as the number of transmit
rate capacity by MRT-based precoding under the relaxed SPC, within antennas increases, and the simple EGT-based precoding is already
a constant gap of K log2 ( π4 ) that is independent of M . Since PAPC capacity near-optimal.

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|g | |g |
70 Substituting Xi with ki M l=i kl
in Lemma A.1, we have
M  
60
MRT analytical
EGT analytical 1  |gki | l=i |gkl |
MRT simulation
EGT simulation 2
var
50 M i=1
M
⎧ ⎡ ⎤2 ⎛ ⎡ ⎤⎞2 ⎫
Sum-rate (bps/Hz)

40 ⎨ M  
M  ⎬
1
= 4 E⎣ |gki | |gkl |⎦ − ⎝E ⎣ |gki | |gkl |⎦⎠ .
30 M ⎩ ⎭
i=1 l=i i=1 l=i
20 (24)
It then follows that for Rayleigh distributed RV |gki |’s,
 
10

(E[ M 2  π 2
i=1 |gki | l=i |gkl |]) 4
0
1 2 3 lim = (E[|g ki |]) = , (25)
10 10
Number of transmit antennas, M
10
M →∞ M4 4
  2
E[( M i=1 |gki | l=i |gkl |) ]
Fig. 1. Sum-rate of EGT-based versus MRT-based precoding lim
M →∞ M 4
 
M (M − 1)E[(|gki1 | l1 =i1 |gkl1 |)(|gki2 | l2 =i2 |gkl2 |)]
l1 =i2 l2 =i1
= lim
M →∞ M4
(E[|gki |])4 M (M − 1)(M − 2)(M − 3)  π 2
5 = lim = .
rate gap simulation M →∞ M4 4
4.5 K log2 ( π4 ) (26)
4 Combining (25) and (26), we have
M  
3.5
1  |gki | l=i |gkl |  π 2  π 2
Sum-rate gap (bps/Hz)

3 lim var = − = 0.
M →∞ M 2
i=1
M 4 4
2.5
(27)
2
Hence, 

|g |the |g condition in Lemma A.1 is satisfied. Since
kl |
1.5
E ki M l=i
= π4 , the following result holds for any
1 positive :
# EGT  $
0.5
10
1
10
2
10
3
S π 
Number of transmit antennas, M
lim Prob  k − ρλk βk  <  = 1. (28)
M →∞ M 4
Fig. 2. Sum-rate gap between EGT-based and MRT-based precoding Next, for the MUI power at the kth user, we have
 ρλj 
M 
M
hki h∗ji hjl h∗kl
lim IkEGT = lim
A PPENDIX M →∞ M →∞ M i=1 |hji ||hjl |
j=k l=1
(29)
 ρλj βk 
M 
= lim |gki |2 = ρλj βk .
A. Proof of Lemma 4.1 M →∞ M i=1
j=k j=k

First, it is worth noting that the expected value of a Rayleigh By combining (28) and (29), the proof of Lemma 4.1 is thus
completed.
distributed RV R with probability density function (PDF) fR (r) =
−r 2 π 1
r
σ2
e 2σ 2 is σ
2
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