Adaptive Hybrid Beamforming with Massive
Phased Arrays in Macro-Cellular Networks

arXiv:1801.09029v2 [] 3 Feb 2018

Shahram Shahsavari† , S. Amir Hosseini , Chris Ng , and Elza Erkip†
† ECE Department of New York University, NYU Tandon School of Engineering, New York, USA
 Blue Danube Systems, Warren, New Jersey, USA
† {shahram.shahsavari, elza}@nyu.edu,  {amir.hosseini, chris.ng}@bluedanube.com
Abstract—Hybrid beamforming via large antenna arrays has
shown a great potential for increasing data rate in cellular
networks by delivering multiple data streams simultaneously. In
this paper, several beamforming design algorithms are proposed
based on the long-term channel information for macro-cellular
environments where the base station is equipped with a massive
phased array under per-antenna power constraint. Using an
adaptive scheme, beamforming vectors are updated whenever
the long-term channel information changes. First, the problem is
studied when the base station has a single RF chain (single-beam
scenario). Semi-definite relaxation (SDR) with randomization
is used to solve the problem. As a second approach, a lowcomplexity heuristic beam composition algorithm is proposed
which performs very close to the upper-bound obtained by SDR.
Next, the problem is studied for a generic number of RF chains
(multi-beam scenario) where the Gradient Projection method is
used to obtain local solutions. Numerical results reveal that using
massive antenna arrays with optimized beamforming vectors can
lead to 5X network throughput improvement over systems with
conventional antennas.

I. I NTRODUCTION
In light of the rapid development of fifth generation cellular
networks (5G), Massive MIMO has proven to improve the network performance significantly [1]. These systems comprise of
an array of many antenna elements. The user data is precoded
in the digital domain first and then, each of the digital streams
is converted to a radio frequency signal through a circuit
referred to as RF chain. Each signal is then transmitted by the
antenna element connected to that RF chain. This process is
best suited to a rich scattering propagation environment that
provides a large number of degrees of freedom. In macrocellular environment, however, these conditions often do not
hold. A more efficient alternative is the use of hybrid massive
MIMO systems in such scenarios [2].
In hybrid Massive MIMO systems, there are fewer RF
chains than antenna elements. This helps the overall system
to be much less power hungry and more cost effective, since
each RF chain consists of power consuming and expensive
elements such as A/D and D/A converters which do not
follow Moore’s law. However, these systems rely on accurate
channel estimation and typically are applied to TDD networks
to alleviate the estimation overhead [2]. On the other hand,
common deployment of LTE in North America is FDD based.
In this paper, we focus on a class of hybrid massive MIMO
systems where all antenna elements maintain RF coherency
[3]. This means that all antenna elements are closely spaced

and have matching phase and magnitude characteristics at the
operating frequency [4]. Using this technique, applicable also
in FDD with existing LTE protocols, the antenna system can
be used as a phased array and macro-cellular transmission is
achieved through hybrid beamforming (BF) [5].
In hybrid BF, each RF chain carries a stream of data and
is connected to each antenna element through a separate pair
of variable gain amplifier and phase shifter. By setting the
values of the amplifier and phase shifts (equivalently designing
BF vectors), multiple beams are generated, each carrying
one data stream over the air. Generating beams using phased
arrays generally requires channel information of all users. By
keeping the beam pattern constant over an extended period
of time, e.g., one hour, small scale channel variations can
be averaged out. Hence, the BF direction corresponds to a
dominant multipath component [6] which mainly depends on
the user location in macro-cellular environment due to the
primarily LOS channels. Whenever user location information
is updated, the system can adaptively switch to a different
beam pattern to constantly provide enhanced service to the
users. We refer to this technique as long-term adaptive BF.
The radiated power from an antenna array is constrained and
power constraints are chosen to limit the non-linear effects of
the amplifiers [7]. Generally, two types of power constraints
are considered in research problems: i) sum power constraint
(SPC) in which an upper-bound is considered for the total
power consumption of the array, and ii) per-antenna power
constraint (PAPC) in which an upper-bound is considered
for the power consumption of each antenna in the array [8],
[9]. Although it is more convenient to consider SPC for
research problems [10], [11], it is not applicable to practical
implementations, where each antenna element is equipped with
a separate power amplifier.
Generating adaptive beams that maximize the overall network throughput plays a significant role in exploiting the
benefits of hybrid BF in a cellular system. Any method that is
proposed should have a manageable complexity and operate
within the power constraints of the array. The goal of this paper
is to propose methods for long-term adaptive BF under PAPC
to maximize the average network rate using hybrid phased
arrays with arbitrary number of beams. First, we focus the
optimization on an individual cell where the interference from
other cells is treated as noise. We use well-known theoretical
and numerical techniques for finding the optimal beam pattern

𝑠=1

𝑠=𝐿

Figure 1: Single cell scenario with L sections

as well as a theoretical upper bound for the solution. Then,
we propose a low complexity heuristic algorithm that performs
close to the obtained upper bound.
Notation: We use uppercase bold letters to denote matrices
and lowercase bold letters to denote column vectors. Xmm and
wm are the (m, m)th and mth element of matrix X and vector
w, respectively. (.)T , (.)H , Tr{.}, and ||.||F are transpose,
hermitian, trace and Frobenius norm operations, respectively.
[N ] denotes the set of integers from 1 to N .
II. PROBLEM STATEMENT
We consider the downlink of a single-cell scenario consisting of a BS with M antennas and L  M radio frequency
(RF) transceiver chains [2]. Since each RF chain can carry
a single data stream, the BS can serve L User Equipments
(UEs) simultaneously. As a result, the cell site is partitioned
into L sections (Fig. 1) and one UE per section is activated at
each time slot as will be explained later. We assume that user
equipments (UEs) are clustered in hotspots within the cell.
Let Hsi , (s, i) ∈ [L] × [Ks ] denote hotspot i from section s
which consists of a group of Nsi nearby UEs. We P
let Ks
be the number of hotspots in section s and K =
s Ks
denotes the total number of hotspots in the cell. The fraction
of UEs located at Hsi among the P
UEs in section s is defined
by αsi = Nsi /Ns , where Ns = i Nsi is the total number
n
of UEs in section s. Let Usi
, n ∈ [Nsi ], denote the nth UE
of hotspot Hsi .
We consider a macro-cellular environment in which the
channels are primarily LOS with the possibility of having
local scatterers around the UEs. We assume that only the longterm channel state information of UEs is available at the BS
and can be used to perform long-term BF. Furthermore, we
assume that the long-term channel vectors between the BS
and the users belonging
√ to a hotspot are the same due to their
proximity. Let gsi = βsi hsi , denote the long-term channel
vector between the BS and the UEs located at Hsi , where
βsi ∈ R+ and hsi ∈ CM denote the pathloss and spatial
signature between BS and Hsi , respectively. We consider the
Vandermonde model where hsi = [ejθsi , ej2θsi , . . . , ejM θsi ]T .
A use case of this channel model is when the users are located
in the far-field of a uniform linear array with M antennas
in a primarily line-of-sight environment [10]. In such cases
we have θsi = 2πd sin(ψsi )/λ, where d denotes the spacing
between successive elements, λ is the wavelength, and ψsi is
direction of Hsi relative to the BS. In order to model other

types of antenna arrays such as rectangular and circular arrays,
hsi can be changed accordingly. We note that this model
relates the long-term channel information to the location of the
hotspots. In [12], the validity of this model is demonstrated
using a variety of test-bed experiments.
Each RF chain is connected to each antenna element through
a separate pair of variable gain amplifier and phase shifter. We
model the corresponding gain and phase shift by a complex
coefficient. As a result, there are M complex coefficients
corresponding to each RF chain creating a BF vector. The
radiation pattern (or equivalently beam pattern) of the antenna
array corresponding to each RF chain can be modified by
controlling the corresponding BF vector [5]. We assume that
the BS uses BF vector ws ∈ CM , s ∈ [L] to generate a beam
pattern (or a beam in short) for serving UEs located in section
s. To reduce the complexity, these BF vectors are designed
based on the long-term channel information and are adaptively
modified when the long-term channel information changes,
i.e., when there is substantial change in the geographical
distribution of the hotspots. Moreover, we assume that the
UEs are scheduled within each section based on a round robin
n
n
scheduler. Let qsi
∈ C denote the signal to be sent to Usi
,
n
n 2
n∗
where E(ssi ) = 0 and E(|qsi | ) = 1. Also, let Usi∗ be the
scheduled UE in section s at √
a generic
time slot. Hence,
P
n∗
the BS transmit vector is x = P s∈[L] qsi
∗ ws , where P
denotes the average transmit power of the BS. Subsequently,
n∗
Usi
∗ receives signal
Xp
p
∗
∗
n
n∗ H
ysi
P βsi∗ qsi
P βs0 i∗ qsn0 i∗ hH
∗ =
∗ hsi∗ ws +
si∗ ws0 + v,
s0 6=s

where v ∼ CN (0, σ 2 ) is the noise. The first term corresponds
to the desired signal received from beam s and the second term
is the interference received from other L−1 beams. Therefore,
n
whenever Usi
is scheduled, the corresponding SINR is
SINRsi (W) =

wH Q w
P s siH s
,
1 + s0 6=s ws0 Qsi ws0

(1)

2
where Qsi = γsi hsi hH
si with γsi = P βsi /σ . The BF matrix is
M ×L
defined as W , [w1 , w2 , . . . , wL ] ∈ C
and has columns
corresponding to BF vectors
of
different
sections.
We note
P
that PAPC corresponds to s∈[L] |Wms |2 ≤ 1/M, ∀m. Hence
we define the feasible set as A = {W ∈ CM ×L | ∀m :
P
2
s∈[L] |Wms | ≤ 1/M }. The goal is to find BF matrix
W which maximizes a network utility function, denoted by
R(W) over the feasible set A. In this paper, we consider
average network rate as the network utility, i.e.,
X X
R(W) =
αsi log(1 + SINRsi ).
(2)
s∈[L] i∈[Ks ]

Hence, The problem can be formulated as follows.
ΠL : Wopt = argmax R(W)
W∈A

We note that the sub-index L in ΠL corresponds to the number
of the beams (equivalently number of RF chains). Although
there is no minimum utility constraint defined for individual

UEs in problem ΠL , sum-log maximization induces a type of
proportional fairness. It can be shown that problem ΠL is not
in a convex form [13, chapters 3, 4]. Therefore, finding the
globally optimal solution of this problem is difficult. In section
III, we study the single-beam (L = 1) problem Π1 to find local
solutions and an upper-bound to evaluate the performance. In
section IV, we provide an iterative algorithm to find a suboptimal solution of problem ΠL for arbitrary L.
In Sections III and IV, we will need to find the projection
of a general beamforming matrix W ∈ CM ×L on set A which
is defined as
PA (W) = argmin ||X − W||2F .

(3)

X∈A

We note that A is a closed convex set which leads to a unique
PA (W) for every W ∈ CM ×L , introduced by Lemma 1. The
proof is provided in Apeendix A.
Lemma 1. We have Ŵ = PA (W) if and only if for every
m ∈ {1, 2, . . . , M }
(
P
2
Ŵms = Wms
if
s |Wms | ≤ 1/M,
p P
P
2
Ŵms = Wms /( M s |Wms |2 ) if
s |Wms | > 1/M.
III. S INGLE -B EAM S CENARIO
In this section, we study problem Π1 where every UE is
served by a single-beam generated by BF vector w, i.e., there
is only one section in the cell (s = 1) and PAPC corresponds
to |wm |2 ≤ 1/M, ∀m. In this scenario, there is no interference
since one P
UE is scheduled per time slot. Hence we have
H
R(w) =
i∈[K] αi log(1 + w Qi w). Please note that we
drop index s in the single-beam scenario because s = 1. Next,
we derive an upper-bound for the optimal value of Π1 and
provide two different methods to obtain local solutions of this
problem. We will use the upper-bound as a benchmark in the
simulations to evaluate the effectiveness of the local solutions.
A. Semi-definite relaxation with randomization
Since wH Qi w is a complex scalar, we have wH Qi w =
H
(wH Qi w)T = γi hH
∈ CM ×M is a
i Xhi , where X = ww
rank-one positive semi-definite matrix. Using this transformation, semi-definite relaxation of problem Π1 is as follows.
Π1r :

Xopt
r

= argmax

K
X

X∈CM ×M i=1

αi log(1 + γi hH
i Xhi )

subject to: X ≥ 0, ∀m : Xmm ≤ 1/M
We remark that Π1 is equivalent to Π1r plus a non-convex
constraint Rank(X) = 1. Removing the rank-one constraint
enlarges the feasible set and makes it possible to find solutions
with higher objective value. Hence, the optimal objective value
of Π1r is an upper-bound for the optimal objective value
of Π1 . This is a well-known technique called ‘Semi-definite
Relaxation (SDR)’ [14]. Note that Π1r can be solved using
convex programming techniques [13]. After solving the convex
problem Π1r there are two possibilities:

1) Rank(Xopt
r ) = 1: in this case the upper-bound is tight
H
and we have Xopt
= wopt wopt , where wopt is the
r
solution of Π1 .
2) Rank(Xopt
r ) > 1: in this case, the upper-bound is not
tight and finding the global solution of Π1 is difficult.
However, there are a number of methods developed to
generate a reasonable BF vector w for problem Π1 by processing Xopt
[14]. For example, using eigenvalue decomr
position, we have Xopt
= VΛVH . Let v1 , v2 , . . . , vM
r
be the eigenvectors in descending order of eigenvalues.
One simple approach is to use the eigenvector corresponding to the maximum eigenvalue and form BF vector as
1
. It should be noted that normalization
wmev = √Mv||v
1 ||2
is necessary for feasibility. Although this simple method is
optimal when Rank(Xopt
r ) = 1, it is not the best strategy
when Rank(Xopt
r ) > 1. Using different ‘randomization’
techniques can lead to better solutions [14]. Let us define
wsdr = √Mb||b|| , where b = VΛ1/2 e with random vector
2
e ∈ CM . The elements of e are i.i.d. random variables
uniformly distributed on the unit circle in the complex
plane. Alternative distributions such as Gaussian distribution can also be adopted for e [15]. The randomization
method is to generate a number of BF vectors {wsdr }
and pick the one resulting in the highest objective value
of Π1 . Note that using e = [1, 0, 0, . . . , 0]T would lead to
wsdr = wmev . The number of random instances denoted
by Ntrial depends on the number of the hotspots, which is
discussed more in the numerical examples.
B. Single-beam sub-beam composition
In this section, we introduce a heuristic algorithm to find
a BF vector w for Π1 with a relatively good performance
compared to the upper-bound obtained by SDR. Suppose there
is only one hotspot in the network, say Hi . Using CauchySchwarz inequality,
we can show that the solution of Π1 is
√
w , hi / M . In this technique, which is referred to as
conjugate beamforming, the BS creates a narrow beam towards
the location of Hi [5, chapter 19]. We can generalize this
method to generate a beam pattern serving all the hotspots,
by summing up the individually optimal BF vectors, and
normalizing the result to satisfy
PK PAPC. Hence, the resulting
BF vector is wsbc , PA ( i=1 wi ), where PA (.) is given
by Lemma 1. We call this method single-beam sub-beam
composition (SB-SBC) due to the fact that we form a beam
pattern by adding up multiple sub-beams.
Adding up individually optimal BF vectors and projecting the result on the feasible set A will perturb each of
them. Therefore, wsbc would not exactly point towards all
the hotspots. To compensate for this disturbance, we use
another approach called single-beam phase optimized subbeam composition (SB-POSBC). In SB-POSBC, we add a
separate phase shift for each BF
wi in the summation,
Pvector
K
i.e., we define wposbc , PA ( i=1 ejφi wi ). By choosing a
set of appropriate phase shifts, wposbc leads to a beam pattern
which points to all the hotspots, hence, it leads to a better
network utility. Since it is not easy to find optimal phase

Hotspot
Antenna Element
SBC
POSBC
SDR-R

0
-30

30

-60

60

-90

-20

-10

0

90
10

Figure 2: Comparison between the beam patterns (in dB)
generated by SB-SBC and SB-POSBC for a uniform linear
antenna array with eight antennas (8-ULA) and four hotspots.

such as non-convex non-linear optimization. Although there
is no guarantee that these methods find the global optimum,
they converge to a local optimum of if some conditions hold;
we refer the reader to [16] for details. To find a local solution
for problem ΠL , we use an iterative numerical method called
‘Gradient Projection (GP)’. Although there are different types
of GP, we use one that includes two steps at each iteration: i)
taking a step in the gradient direction of the objective function
with a step-size satisfying a condition called Armijo Rule
(AR), and ii) projecting the new point on the feasible set.
Let W[k] be the BF matrix at iteration k. We define
W[k+1] = PA (W[k] + r[k] G[k] ),
[k]

[k]

[k]

(4)

[k]

shifts analytically, one approach is to try a number of randomly
chosen sets of phase shifts and pick the one which leads to the
highest objective value in Π1 . One can think of this random
trials as the counterpart of randomization technique described
in section III-A. Note that if ∀i : φi = 0 then wposbc = wsbc ,
hence, if the case of zero phase shifts is included in the set
of random phase shifts, we can ensure that SB-POSBC will
perform at least as good as SB-SBC. One important parameter
in SB-POSBC is the number of random trials of phase shift
sets denoted by Ntrial which will be studied in section V-A.
Figure 2 depicts a network with four hotspots. This figure
also depicts the beam patterns corresponding to BF vectors
wsbc , wposbc , and wsdr with Ntrial = 1000. We can observe
how phase shifts in SB-POSBC compensate for the perturbation caused by SB-SBC. Furthermore, we can also see that
SB-POSBC creates a similar beam to SDR with randomization
while its complexity is much lower.

where G[k] = [g1 , g2 , . . . , gL ] with gs , ∇ws R(W[k] ),
r[k] > 0. r[k] denotes the step-size at iteration k and PA (W)
is the projection of BF matrix W on the feasible set A which
is given by Lemma 1. We observe that the projection rule is
relatively simple and does not impose high implementation
complexity to the problem.
The step-size calculation rule directly affects the convergence of GP. Applying AR to problem ΠL , we have r[k] =
[k]
r̃β l where r̃ > 0 is a fixed scalar and l[k] is the smallest
non-negative integer satisfying R(W[k+1] ) − R(W[k] ) ≥
σ Re [ Tr{(W[k+1] − W[k] )H G[k] }] and W[k+1] is given
by (4). In order to find r[k] at iteration k, we start from
l[k] = 0 and increase l[k] one unit at a time until the above
condition is satisfied. 0 < σ < 1 and 0 < β < 1 are AR
parameters. In practice, σ is usually chosen close to zero, e.g.,
σ ∈ [10−5 , 10−1 ]. Also, β is usually chosen between 0.1 and
0.5 [16].

IV. M ULTI -B EAM S CENARIO

Lemma 2. Let {W[k] } be a sequence of BF matrices generated by gradient projection in (4) with step-size r[k] chosen
by the Armijo rule, described above. Then, every limit point
of {W[k] } is stationary.

In this section, we study problem ΠL for generic L. First
we present a heuristic similar to SB-SBC and SB-POSBC,
described in Section III-B, and then we introduce an iterative
algorithm to find a local solution of problem ΠL .
A. Multi-beam sub-beam composition
Similar to what is described in Section III-B, one can obtain
L BF vectors each of which generates a beam to cover a
section. To this end, we can consider each section and its
associated hotspots and use SB-SBC (or SB-POSBC) to find
a BF vector for that section. Furthermore, we assume that the
power is equally divided among the BF vectors. Hence, after
applying SB-SBC (or SB-POSBC) to find a BF vector√for
each section separately, we divide all the vectors by 1/ L.
We call this method MB-SBC (or MB-POSBC). We note that
this method does not consider inter-beam interference, because
each BF vector is obtained independently from the others.
B. Gradient projection
Numerical optimization methods can be used to find a
local solution of ΠL for arbitrary L. These methods are more
valuable when it is difficult to find a closed-form solution,

We refer the reader to [16, chapter 2] for the proof. To
implement GP, we need an initial point W[1] and a termination condition. We use MB-SBC described in Section
IV-A to generate an initial point for the numerical examples. For the termination condition, we define the error as
err[k+1] , ||W[k+1] − W[k] ||F and we stop after iteration k if
err[k+1] ≤ , where  is a predefined error threshold. Although
the numerical examples will show that GP converges fast
with AR, we specify a threshold for the number of iterations
denoted by Niter to avoid slow convergence.
Fig. 3 illustrates a network with two beams where sections
1 and 2 are the left and right half planes, respectively, and each
beam serves 2 hotspots. Fig. 3a shows the beams generated
by double-beam SBC described in Section IV-A. We observe
that the BS suffers from inter-beam interference in this case.
GP takes the solution of double-beam SBC as initial point and
iteratively updates the BF coefficients of each beam (Fig. 3b),
which greatly reduces the inter-beam interference.

0
-30

Hotspot
Antenna Element
Beam 1 (left)
Beam 2 (right)

30

-60

60

-90

-20

-10

90
10

0

(a) Double-beam SBC
0
-30

Hotspot
Antenna Element
Beam 1 (left)
Beam 2 (right)

30

-60

we consider 100 random network realizations. We run both
algorithms with Ntrial = 100 , 101 , 102 , 103 , 104 and find the
BF vectors wsdr and wposbc and the corresponding network
utilities in bps/Hz. We also obtain the Upper-Bound (UB) by
solving the relaxed problem Π1r . Table II lists the average
performance of these algorithms given two values for number
of hotspots, K. We observe that the larger Ntrial becomes,
the closer the performance gets to the UB, which in turn
slows down the pace of improvement. For larger K, however,
the performance keeps increasing with the number of trials,
which suggests that the number of trials should be proportional
to the number of hotspots. While both algorithms provide
performance close to the UB with large enough Ntrial , SDR-R
outperforms SB-POSBC in some cases. This improved performance comes at the cost of higher computational complexity.

60

K

-90

-20

-10

0

90
10

Ntrial
100

101

102

103

104

4

SB − POSBC
SDR − R
UB

5.039
5.252
5.783

5.505
5.584
5.783

5.602
5.609
5.783

5.627
5.611
5.783

5.637
5.611
5.783

16

SB − POSBC
SDR − R
UB

3.617
4.123
4.715

4.220
4.521
4.715

4.403
4.572
4.715

4.498
4.585
4.715

4.556
4.587
4.715

(b) Gradient projection

Figure 3: Beam patterns generated by double-beam SBC and
gradient projection in a sample network with 2 beams, four
hotspots, and a uniform linear array with eight antennas.

Method

Table II: Average network utility in (bps/Hz)
B. Computational complexity and performance

V. N UMERICAL EXAMPLES
In this section, we provide numerical examples to evaluate
and compare the performance of the proposed methods. We
simulate the downlink of a three dimensional network with a
BS consisting of a 4 × 12 uniform rectangular array serving
a 120◦ sector of a cell. Table I lists the network parameters.
Hotspots are distributed uniformly at random in a ring around
the BS with inner and outer radii of 300 m and 577 m,
respectively. We use CVX package [17] to solve the convex
problem Π1r . We also use  = 10−4 and Niter = 104 for GP.
Parameter

Value

Scenario
Cell radius
Bandwidth
Noise spectral density
BS transmit power (P )
Number of hotspots (K)
Pathloss in dB (β −1 )

single-beam, double-beam
577 m
20 MHz
−174 dBm/Hz
20 dBm
4, 8, 16
128.1 + 37.6 log10 (d in km)

Table I: Simulation parameters
A. Effect of number of trials on the performance
In this section, we consider the single-beam scenario described in Section III. We focus on SDR with randomization
(SDR-R) and SB-POSBC described in Sections III-A and
III-B, respectively. In both of these algorithms there are Ntrial
random trials. To evaluate the performance of these algorithms,

In this part, we compare the performance and numerical
complexity of different algorithms devised for the single-beam
scenario (L = 1). Table III lists the average throughput (in
bps/Hz) and average time spent on a typical desktop computer
(3.1 GHz Core i5 CPU, 16 GB RAM) to find the BF vector
using GP, SB-POSBC, and SDR-R algorithms. The reported
values are average values over 100 random network realizations. It is assumed that Ntrial = 103 for SDR-R and SBPOSBC. While GP and SB-POSBC have sub-second runtime,
SDR-R has a much higher complexity. This is because a
complex convex optimization problem has to be solved in
the first step of SDR-R, whereas GP and SB-POSBC only
require simple mathematical operations. We also observe that
the performance of these algorithms are very close. Overall, we
can conclude that GP and SB-POSBC are superior to SDRR since they achieve similar performance with much lower
computational complexity. The results also reveal that network
utility decreases for each of the algorithms when the number
of hotspots increases. This is the cost of having a single beam
pattern. In fact, given a fixed antenna array aperture, it is more
difficult to provide good BF gain for larger number of hotspots
with a single beam.
C. Performance evaluation
In this section, we consider single-beam (L = 1) and
double-beam (L = 2) scenarios. In order to compare the performance of the algorithms described in Sections III and IV,

K

Method

Run Time

Network Utility

4

GP
SB − POSBC
SDR − R

0.024
0.081
9.268

5.541
5.627
5.611

16

GP
SB − POSBC
SDR − R

0.040
0.293
21.026

4.958
4.498
4.585

Table III: Average run time (in seconds) and utility in (bps/Hz)
1

Empirical CDF

0.8
0.6
0.4

Single-beam GP
Single-beam POSBC
Single-beam SDR-R
Single-beam Upper-bound
Double-beam SBC
Double-beam GP

0.2
0
4

6

8

10

12

designed to optimize beam patterns in macro-cellular networks
to enable effective antenna deployment. Fig. 5 illustrates
the map of a sample cellular network in Danville, VA in
three different deployment scenarios. Fig. 5a represents the
case where all cells are equipped with conventional passive
antennas, whereas the other two figures showcase the deployment of BeamCraftTM 500, an active antenna array designed
and manufactured by Blue Danube Systems. The white dots
show the distribution of demand inside the network and the
illuminated patterns illustrate the SINR at each point. Fig. 5b
shows the single-beam scenario where the beams are optimized
using the GP algorithm. Fig. 5c illustrates the same result for
the double-beam scenario. It can be seen that double-beam
active antenna arrays with optimal beam patterns can offer
close to 5X throughput improvement over current systems with
conventional antennas.
A PPENDIX A
P ROOF OF L EMMA 1
Based on the definition of optimal projection in (3), we have

14

Ŵ = argmin d(X, W),

Network Utility (bps/Hz)

Figure 4: Empirical CDF of network utility

we consider 4000 random network realizations and calculate
the network utility corresponding to each algorithm for each
realization. For the single-beam scenario, the upper-bound of
the network utility is obtained for each realization by solving
problem Π1r . It is assumed that Ntrial = 103 for SDRR and SB-POSBC. Fig. 4 illustrates the empirical CDF of
network utility corresponding to each algorithm for K = 8
hotspots. We observe that SDR-R outperforms SB-POSBC
and GP in the single-beam scenario. Moreover, SB-POSBC
performs very close to SDR-R. Having two beams will double
the number of transmissions compared to the single-beam
scenario which can potentially lead to significant network utility improvement if the interference due to multi-user activity
(i.e. inter-beam interference) is managed appropriately. Since
double-beam GP considers interference, it leads to almost 2X
improvement in network utility compared to the single-beam
algorithms. On the other hand, the performance of doublebeam SBC is remarkably inferior to double-beam GP, due to
lack of interference management.

where, d(X, W) , ||X − W||2F is convex in X. Besides, set
A is also convex, hence (5) is a convex optimization problem.
Let us assume that for the coefficient representing antenna
element m and beam s, Xms = Zms ejΦms with Xms ≥ 0
and Wms = Rms ejΨms with Rms ≥ 0. We can reformulate
(5) as follows
(Ẑ, Φ̂) = argmin
{Zms ,Φms }

X X 


2
2
Rms
+ Zms
− 2Rms Zms cos(Ψms − Φms ) ,

m∈[M ] s∈[L]

subject to: ∀m :

X

2
Zms
≤ 1/M, ∀m, s : Zms ≥ 0.

s∈[L]

Since ∀m, s : Zms , Rms ≥ 0, we have ∀m, s : Φ̂ms =
Ψms . Then,
is reduced to minimize
P
P the objective function
2
m∈[M ]
s∈[L] (Rms − Zms ) . Furthermore, since all terms
inside the sum are non-negative, the above problem can be
broken into M separate problems that can be solved independently for each antenna element index m. Hence, we can drop
the antenna element index and rewrite the problem as follows:
X
2
min
(Rs − Zs )
{Zs }

VI. C ONCLUDING R EMARKS
We have studied the hybrid BF problem for a single macrocell scenario where the BS is equipped with a massive phased
array. Long-term channel information is used to design the BF
vectors, where they are updated when there is a substantial
change in long-term channel information. Several algorithms
with different complexities have been proposed for designing
BF vectors in different scenarios.
Based on the multi-cell generalization of the proposed
algorithms, a commercial software has been developed by
Blue Danube Systems called BeamPlannerTM . The software is

(5)

X∈A

s∈[L]

subject to:

X

Zs2 ≤ 1/M,

∀s : Zs ≥ 0.

s∈[L]

It can be easily verified that the above problem is convex,
therefore, the Karush-Kuhn-Tucker (KKT) conditions will
result in the optimal solution [13, chapter 5]. First, we generate
the Lagrangian as follows:
L(Z, λ, µ) =
 X
 X
X
2
(Rs − Zs ) + µ
Zs2 − 1/M −
λs Zs .
s∈[L]

s∈[L]

s∈[L]

(a) Network with passive anten- (b) Network with optimized sin- (c) Network with optimized dounas. Average throughput is 53.0 gle beam phased arrays. Average ble beam phased arrays. Average
Mbps/km2 .
throughput is 89.8 Mbps/km2 .
throughput is 237.0 Mbps/km2 .

Figure 5: Sample cellular network located in Danville, VA optimized using BeamPlannerTM software.

KKT conditions to be checked are as follows:
∂L
= 0 ∀s ∈ [L],
∂Zs
Zs ≥ 0
∀s ∈ [L],

R EFERENCES
(6)
(7)

∀s ∈ [L],

(8)

µ, λs ≥ 0 ∀s ∈ [L],
X
Zs2 ≤ 1/M,

(9)

λs Zs = 0

(10)

s∈[L]

 X

µ
Zs2 − 1/M = 0.

(11)

s∈[L]

From (6), we can conclude:
λs = 2(Zs − Rs ) + 2µZs .

(12)

2
By applying (11), we know
P that either µ = 0 or s∈L Zs =
1/M . If µ = 0 (hence s∈[L] Zs2 ≤ 1/M ), then from (12)
we obtain λs = 2(Zs − Rs ). According to (8), we either have
λs = 0 or Zs = 0. If Zs = 0, we will have λs < 0 which
contradicts (9). Otherwise if λs = 0, we have Zs = Rs .
POn the2 other hand, if we assume µ > 0, then we have
s∈[L] Zs = 1/M . Again, from (8), if λs > 0, then Zs = 0
and using (12) we have λs = −2Rs < 0 which is a
contradiction. If we set λs = 0, we have Zs = Rs /(1 + µ)
according to (12), which results in the following equalities:
X
X
Zs2 = 1/M,
(Rs /(1 + µ))2 = 1/M.

P

s∈[L]

s∈[L]

q P
L
With a simple substitution we obtain µ = M s=1 Rs2 − 1
q P
which yields Zs = Rs / M s∈[L] Rs2 . It should be noted
that µ > 0 and hence,
PRs > Zs . Therefore, this case represents
the cases in which s∈[L] Rs2 ≥ 1/M .
Consequently, the solution of the problem is as follows:
(
P
2
Rms
if
s∈[L] Rms < 1/M,
q
P
P
Ẑms =
2
2
if
Rms / M s∈[L] Rms
s∈[L] Rms ≥ 1/M.
for every m ∈ [M ] which concludes the proof.



[1] T. L. Marzetta et al., Fundamentals of Massive MIMO. Cambridge
University Press, 2016.
[2] A. F. Molisch et al., “Hybrid beamforming for massive mimo: A survey,”
IEEE Communications Magazine, vol. 55, no. 9, pp. 134–141, 2017.
[3] M. Banu, “HDAAS: An efficient massive MIMO technology,” 4th
Brooklyn 5G Summit, April 2017.
[4] Y. T. Lo and S. Lee, Antenna Handbook: theory, applications, and
design. Springer Science & Business Media, 2013.
[5] S. J. Orfanidis, Electromagnetic waves and antennas. Rutgers University New Brunswick, NJ, 2002.
[6] Q. Li et al., “MIMO techniques in WiMAX and LTE: a feature
overview,” IEEE Communications magazine, vol. 48, no. 5, 2010.
[7] J. Joung et al., “A survey on power-amplifier-centric techniques for
spectrum-and energy-efficient wireless communications,” IEEE Communications Surveys & Tutorials, vol. 17, no. 1, pp. 315–333, 2015.
[8] W. Yu and T. Lan, “Transmitter optimization for the multi-antenna
downlink with per-antenna power constraints,” IEEE Transactions on
Signal Processing, vol. 55, no. 6, pp. 2646–2660, 2007.
[9] C. T. Ng and H. Huang, “Linear precoding in cooperative MIMO cellular
networks with limited coordination clusters,” IEEE Journal on Selected
Areas in Communications, vol. 28, no. 9, pp. 1446–1454, 2010.
[10] E. Karipidis, N. D. Sidiropoulos, and Z. Q. Luo, “Far-field multicast
beamforming for uniform linear antenna arrays,” IEEE Transactions on
Signal Processing, vol. 55, no. 10, pp. 4916–4927, Oct 2007.
[11] A. B. Gershman et al., “Convex optimization-based beamforming,” IEEE
Signal Processing Magazine, vol. 27, no. 3, pp. 62–75, 2010.
[12] C. Zhang and R. C. Qiu, “Massive MIMO testbed - implementation and
initial results in system model validation,” vol. abs/1501.00035, 2015.
[13] S. Boyd and L. Vandenberghe, Convex optimization.
Cambridge
university press, 2004.
[14] Z.-Q. Luo et al., “Semidefinite relaxation of quadratic optimization
problems,” IEEE Signal Processing Magazine, vol. 27, no. 3, pp. 20–34,
2010.
[15] N. D. Sidiropoulos, T. N. Davidson, and Z.-Q. Luo, “Transmit beamforming for physical-layer multicasting,” IEEE Transactions on Signal
Processing, vol. 54, no. 6, pp. 2239–2251, 2006.
[16] D. P. Bertsekas, Nonlinear programming. Athena scientific Belmont,
2016.
[17] M. Grant, S. Boyd, and Y. Ye, “CVX: Matlab software for disciplined
convex programming,” 2008.

