Low-Overhead Coordination in
Sub-28 Millimeter-Wave Networks
Hadi Ghauch

Taejoon Kim

Mikael Skoglund

Carlo Fischione

arXiv:1802.10473v1 [] 28 Feb 2018

School of Electrical Engrng Dept. of Electrical Engrng School of Electrical Engineering School of Electrical Engrng
Royal Inst. of Techn., KTH
and Computer Science
Royal Inst. of Techn., KTH
Royal Inst. of Techn., KTH
ghauch@kth.se
Univ of Kansas, Lawrence
skoglund@kth.se
carlofi@kth.se
taejoonkim@ku.edu

Abstract—In this paper, we present some contributions from
our recent investigation. We address the open issue of interference
coordination for sub-28 GHz millimeter-wave communication,
by proposing fast-converging coordination algorithms, for dense
multi-user multi-cell networks. We propose to optimize a lower
bound on the network sum-rate, after investigating its tightness.
The bound in question results in distributed optimization, requiring local information at each base station and user. We derive the
optimal solution to the transmit and receive filter updates, that
we dub non-homogeneous waterfilling, and show its convergence
to a stationary point of the bound. We also underline a builtin mechanism to turn-off data streams with low SINR, and
allocate power to high-SNR streams. This “stream control” is
a at the root of the fast-converging nature of the algorithm. Our
numerical result conclude that low-overhead coordination offers
large gains, for dense sub-28 GHz systems. These findings bear
direct relevance to the ongoing discussions around 5G New Radio.
Index Terms—Sub-28 GHz Millimeter-wave, low-overhead
coordination, Difference of Log and Trace (DLT), Nonhomogeneous Waterfilling, max-DLT.

I. I NTRODUCTION
To address the exponentially increasing demand in 5G systems, communications in the millimeter-wave (mmWave) band
are among the most promising candidates [1], due to the large
mmWave spectrum. While most investigations of mmWave
communcation have been focused on systems above 28 GHz,
in the current work, we study multi-user multi-cell coordination, in sub-28 GHz systems, e.g., X-band (8-12) GHz,
Ku-band (12-18) GHz, and 28 GHz in the Ka band. These
systems are characterized by a relatively large antenna spacing
(compared to systems beyond 60 GHz), thereby implying
that tens (rather than hundreds) of antennas can be fitted
on transmitters/receivers. Thus, the urge for hybrid analogdigital precoding is not stringent and fully digital precoding/combining is preferred. Moreover, propagation channels
are still dominated by Rayleigh/Rician components in non
line-of-sight environments [2]. As a result, conventional pilotbased channel estimation techniques are more efficient than
beam alignment/sounding [3].
The implication of highly directional wireless links at 60
GHz (and above) is that they are almost interference-free.
However, in sub-28 GHz systems, channels are less sparse
(in terms of eigenmodes), and beamforming has relatively

lower directivity than systems in the higher bands. This is
attributed to the presence of significant multi-path components,
in urban propagation (confirmed by narrowband/wideband
channel measurements in the 9.6 GHz, 11.4 GHz, and 28.8
GHz bands [4]).
Consequently, interference may still be a limiting factor in
these systems, especially when considering dense multi-cell
scenarios, where interference management and coordination
are still beneficial. Coordination in multi-user multi-cell networks generally refers to the exchanges of information among
base stations, to increase the network sum-rate. While these
aspects have been investigated at the MAC layer [5], they are
still essentially unaddressed at the physical layer. Indeed, the
benefits/costs of coordination in sub-28 GHz systems is still an
open problem, especially in the case of ultra dense networks
- believed to be pervasive in future networks [6]: In these
scenarios, ignoring interference for cell-edge users may be a
limiting factor on the sum-rate.
In the context of multi-user multi-cell networks, coordination is done using the framework of Forward-Backward (F-B)
iterations: this over-the-air training leverages local Channel
State Information (CSI) at each Base Station (BS) and user,
to iteratively optimize the filter at each BS/user, in a fully
distributed manner. This framework has been at the heart of
most distributed coordination algorithms, such as interference
leakage minimization [7], max-SINR [7], minimum meansquared error [8], and (weighted) sum-rate maximization [9].
Unfortunately, these conventional schemes suffer from extremely elevated overhead, as they require hundreds/thousands
of F-B iterations before convergence [10], where the latter
increases with the number of BSs, users and transmit/receive
antennas [10]. Moreover, mmWave systems will have a larger
number of BSs/cells per unit-area (due to their inherent
short range), and require a much larger number of BS/user
antennas (to mitigate pathloss through array gain), compared to
sub-6 GHz systems. Consequently, this shortcoming severely
impairs the applicability of conventional coordination, to the
systems in question. This limitation is reinforced by the lower
coherence time of mmWave channels. Despite bearing direct
relevance to conventional sub-6 GHz, this major limitation has
only been addressed in a few recent works [11]–[14], and
remains essentially unexplored.

communicate d data streams. Let L be the set of BSs, Kl the
set of users served by BS l ∈ L, and I the total set of users.
Moreover, we denote by user lj ∈ I the jth user (j ∈ Kl ), in
the lth cell (l ∈ L). The recovered signal for user j ∈ Kl , in
cell l ∈ L (denoted as user lj ∈ I hereafter),
s̃ lj = U †lj H l,lj V lj s lj
XX †
+
(1)
U lj H l,ik V ik s ik + U †lj n l , ∀ lj ∈ I
i∈L k∈K
i6=l
M×d

Fig. 1: L-cell MIMO Interfering Multiple-Access Channel
In this work, we design low-overhead distributed coordination algorithms, constrained to operate in a just a few FB iterations for increasing dimensions of the networks. In
the algorithm design, we further aim at a tenfold reduction in the communication overhead of conventional algorithms. We derive and optimize a lower bound the sum-rate
maximization problem in MIMO Interfering Multiple-Access
Channels (MIMO IMAC), that we dub Difference of Log
and trace (DLT). Unlike the sum-rate, when combined with
alternating optimization methods, the DLT expression results
in subproblems that are distributed (only requiring local CSI
at each BS and user) [15]. Despite their non-convexity, we
derive the optimal solution to each of these subproblems,
that we dub non-homogeneous waterfilling (a variation on the
classical waterfilling). This solution turns-off data streams with
low-SINR, and allocates power to streams with high SNR.
The built-in “stream-control” is key to achieving the tenfold
increase in convergence speed. Moreover, we show that the
devised algorithm converges to a locally optimal solution of
the DLT bound. It is revealed that the proposed fast-converging
algorithm offers large sum-rate gains, compared to many
standard and fast-converging benchmarks. Coordination is still
a vital aspect of these systems, and offers major performance
gains over uncoordinated transmission While the approach is
developed for the MIMO IMAC (i.e., uplink communication),
the methods/results are applicable to the downlink, and all its
special cases.
II. S YSTEM M ODEL
The proposed algorithms operate within the framework of F-B
iterations/training by exploiting the uplink (UL) and downlink
(DL) channel reciprocity. The proposed scheme is designed
to operate in the low-overhead regime, by restricting the
number of F-B iterations, T , to T ≤ 5 (a tenfold reduction
in communication overhead, over conventional coordination
algorithms). We assume that each BS and user posses local
CSI only, that is assumed to be perfect (i.e., no CSI errors).
We consider a multi-user multi-cell setting, with L
cells/BSs, serving K users each. In the MIMO IMAC case,
transmitters at users and receivers are BSs. Each transmitter
and receiver have M and N antennas, respectively, and

where 1V ik ∈ C
is the linear transmit filter for user ik ∈
I, U lj ∈ CN ×d is the linear receive filter for user lj ∈ I, and
H l,ik the N × M MIMO channel from user ik ∈ I, to BS l
assumed to be block-fading.2 s ik ∈ Cd is the transmit signal
of user ik ∈ I with unit power symbols, and n l is the AWGM
nln †l ] = σl2I N .
noise at receiver l, with E[n
We assume simple decoding, i.e., treating interference as
noise , without successive interference cancellation. Then, the
achievable rate of user lj ∈ I is given by,
U †lj Q lj U lj )−1 |, lj ∈ I (2)
U †lj R lj U lj )(U
rlj = log2 |II d + (U
where R lj and Q lj are the desired signal and interferenceplus-noise (I+N) covariance matrices for user j, at BS l,
respectively, and are given by,
R lj = H l,lj V lj V †lj H †l,lj , lj ∈ I
Q lj =

L X
K
X

H l,ik V ik V †ik H †l,ik + σl2I N − R lj , lj ∈ I.

i=1 k=1

We let
Rik = H †i,ik U ik U †ik H i,ik , ik ∈ I
R̄
Qik =
Q̄

K
L X
X

Rik , ik ∈ I
H †l,ik U lj U †lj H l,ik + σ̄i2k I M − R̄

l=1 j=1

denote the desired and I+N covariance matrices of user ik , in
the backwark DL network (where σ̄i2k is the noise power at
receiver ik ∈ I). Moreover, we let L lj L †lj be the Cholesky
Qik .
Decomposition of Q lj , and K ik K †ik as that of Q̄
We aim
at
maximizing
the
sum-rate,
i.e.,

P
 max
lj ∈I rlj
U lj ,V
V lj }
(3)
(P ) {U
s. t. kU
V lj k2F = Pt , ∀ lj ∈ I
U lj k2F = Pr , kV
While distributed multi-user multi-cell optimization generally
entails a sum-power constraint on the users of a cell (e.g. WMMSE [9]), we adopt an equal power allocation among the
all users within a cell: The BS power is equally split among
all its UL (or DL) users, to simplify the presentation. Note
1 Notation: we use bold upper-case letters to denote matrices, and bold
A ) as its trace,
lower-case denote vectors. For a given matrix A , we define tr(A
A k2F as its Frobenius norm, |A
A| as its determinant, A † as its conjugate
kA
A † )−1 . In addition, A (i) denotes its ith column,
transpose, and A −† as (A
A] the ith eigenvalue
A i:j columns i to j, A (i,j) element (i, j) in A , λi [A
of a Hermitian matrix A (assuming the eigenvalues are sorted in decreasing
A ] denotes the d dominant eigenvectors of A . Furthermore,
order), and v1:d [A
A ≻ 0 (resp. A  0 ) implies that A is positive definite (resp. positive semidefinite). Finally, I n denotes the n × n identity matrix, {n} = {1, · · · , n},
and x+ = max{0, x}.
2 The model/results assume M, N and d for simplicity, and can be easily
extended to differ across users and BSs.

Algorithm 1 Maximal DLT (max-DLT)

that this does not affect the generality of the results.
III. P ROPOSED A PPROACH
Sum-rate maximization problems, such as (P ), are known
to be NP-hard [16]. Our proposed approach is based on a
tractable lower bound formulation that transfers the sum-rate
maximization, which is an originally coupled problem, into
separable subproblems.
A. Problem Formulation
We focus on the interference-limited regime (in a dense
deployment for instance), where we assume
U †lj Q lj U lj ] → ∞, ∀i ∈ {d}
(4)
λi [U
Proposition 1. Under the conditions in (4), the rlj in (2)
satisfies
U †lj Q lj U lj ) , rl(LB)
, (5)
rlj ≥ log2 |II d + U †lj R lj U lj | − tr(U
j
(LB)

where the gap ∆lj , rlj − rlj

is characterized by

U †lj Q lj U lj |
U †lj Q lj U lj ) − log2 |U
∆lj = tr(U
U †lj R lj U lj )−1 ]),
U †lj Q lj U lj )(U
+ O(tr[(U

∀lj ∈ I.

(6)

Proof: Refer to [15][Appendix B]
(LB)
Difference
In what follows, we shall dub the quantity rlj
of Log-Trace (DLT). The DLT becomes significant when it is
used as an alternative objective of the sum-rate objective in
(P ). Note that DLT is a lower bound on the sum-rate, RΣ ,
and can be written in the following ways:
X
(LB)
U †lj Q lj U lj ) (7)
RΣ
=
log2 |II d + U †lj R lj U lj | − tr(U
lj ∈I

=

X

R ik V ik | − tr(V
V †ik Q̄
Qik V ik ) (8)
log2 |II d + V †ik R̄

ik ∈I

The above expressions reveal that DLT makes both the receive
filters in (7) and the transmit filters in (8) decoupled, and facilitates the aimed distributed F-B implementation. We formulate
the maximal DLT (max-DLT) criterion as a surrogate objective
to the sum-rate
maximization in (P ),

(LB)
 max RΣ
V lj ,U
U lj }
{V
(PLB )
(9)
s. t. kU
V lj k2F = Pt , ∀lj ∈ I.
U lj k2F = Pr , kV
It should be noted that although DLT allows distributed F-B
implementation, the problem in (PLB ) is not directly solvable
since it is non-convex due to the coupling between the transmit
and receive filters, and the quadratic equality constraints.
B. Proposed Algorithm
We underline that a coupled optimization like (PLB ) can
ideally be handled by a Block Coordinate Descent (BCD)
approach. This means if the superscript (n) is adopted to
denote the iteration number, problem (PLB ) is decomposed
into a sequence of subproblems that are solved via F-B
iterations as below


(n)
(LB)
V
U
U (n+1)
}
},
{V
{U
}
,
argmax
(J1) {U
R
lj
Σ
lj
lj
U lj }
{U



(LB)
(n+1)
V (n+1)
V
U
}
,
argmax
}
, (10)
(J2) {V
},
{V
R
{U
lj
lj
Σ
lj
V lj }
{V

for t = 1, 2, ..., T do
// forward network optimization: receive filter update
Estimate R lj , Q lj , and compute L lj , ∀lj
−†
L−1
U lj ← L −†
lj R lj L lj ]Σlj , ∀lj
lj v1:d [L
// backwark network optimization: transmit filter update
R ik , Q̄
Qik , and compute K ik , ∀ik
Estimate R̄
K −1
R ik K −†
Λik , ∀ik
V ik ← K −†
v
ik R̄
ik 1:d [K
ik ]Λ
end for

for n = 1, 2, 3.... As seen from (7), at each iteration n, given
V (n)
the fixed {V
lj }, problem (J1) is decouples in the receive
U lj }, yielding
filters {U

min tr(U
U †lj Qlj U lj ) − log2 |II d + U †lj Rlj U lj |
U lj
(J1)
(11)
s. t. kU
U lj k2F = Pr .
Likewise, if the receive filters are fixed, problem (J2) decouples, as
 seen from (8), in the transmit filters, resulting in
min tr(V
V †ik Q̄
Qik V ik ) − log2 |II d + V †ik R̄
Rik V ik |
V ik
(12)
(J2)
s. t. kV
V ik k2F = Pt .
As aforementioned, the feasible sets of (11) and (12) are nonconvex. Nevertheless, we show in the result below, that their
globally optimal solutions can still be found.
Lemma 1. Non-homogeneous Waterfilling.
Consider the following problem,
(
X ) , tr(X
X †QX )−log2 |II d +X
X †RX |
min f (X
X ∈Cn×r

X k2F = ζ.
s. t. kX

(13)

where Q ≻ 0 and R  0 , r < n. Let Q , LL † be the
Cholesky factorization of Q , and M , L −1RL −† , M  0 ,
M ]}ri=1 , Ψ , v1:r [M
M ],
and define the following, {αi , λi [M
†
†
−1
r
L L ) Ψ (i) }i=1 . Then the optimal solution for
{βi , Ψ (i) (L
(13) is,
X ⋆ = L −†Ψ Σ⋆ ,
(14)
⋆
where Σ (diagonal) is the optimal power allocation. Moreover, optimal powerrallocation in Σ⋆ is,
+

(15)
1/(1 + µ⋆ βi ) − 1/αi , ∀i,
Σ⋆(i,i) =
P
r
where µ⋆ is the unique root to g(µ) , i=1 βi 1/(1 + µβi)−
+
1/αi − ζ, on the interval [−1/(maxi βi ), ∞], and g(µ) is
monotonically decreasing on that interval.
Proof: Refer to [15][Appendix C]
With Lemma 1, the optimal transmit and receive filter
updates are formulated as below
−†
⋆
L −1
U ⋆lj = L −†
lj R lj L lj ], ∀ lj ,
lj Ψ lj Σlj , Ψ lj , v1:d [L
⋆
K −1
R ik K −†
V ⋆ik = K −†
ik R̄
ik Θ ik Λ ik , Θ ik , v1:d [K
ik ], ∀ ik , (16)
⋆
⋆
where Σlj and Λ ik are the optimal power allocation, given
in Lemma 1. Denoting by T the predefined number of F-B
iterations, the max-DLT algorithm is in Algorithm 1.

Discussions
Based on the generalized eigenvalue analysis, we have that
L−1RL −† ]}ri=1 are also the eigenvalues of Q −1R .
{αi , λi [L
This means {αi } can be viewed as a (quasi)-SINR measure
of each data stream. The proposed method in (15) allocates
no power to streams that have low-SINR, since Σ⋆(i,i) tends to
zero as αi → 0. Moreover, as seen from (15), {βi } models the
price of activating each of the streams, mimicking the original
waterfilling principle. The difference however is that (14) fills
the power level based on the SINR and cost for the stream
activation, namely the non-homogeneous waterfilling solution.
This readily enables the algorithm to not allocate power to
some low SINR streams. Finally, since the global optimizer is
found at each iteration in Algorithm 1, we can conclude that
(n)
(LB)
(n)
V lj }) in (PLB ) is monotonically decreasing
U lj }, {V
RΣ ({U
with n, and converges to a stationary point of the DLT bound.
While the ‘stream-control’ greatly speeds up the convergence,
it evidently raises fairness issues, as some users/streams with
low-SINR, may not get served. This can be remedied by
introducing user weights in (P ), with minor modifications in
the problem/solutions.
IV. P RACTICAL A SPECTS
A. Comparisons
Our approach is applicable to other communication scenarios
such as the MIMO Interfering Broadcast Channel (MIMO
IBC), the MIMO Interference Channel (MIMO IFC). We
benchmark our algorithms against widely adopted ones,
o max-SINR [7] in the MIMO IMAC / MIMO IFC / MIMO
IBC
o MMSE and Weighted-MMSE [9], [17] in the MIMO IFC
/ MIMO IBC
o Uncoordinated (Eigen-beamforming): each transmit
(resp. receive) filter uses right (resp. let) singular
eigenvectors of the desired channel
We also include relevant fast-converging algorithms,
o CCP-WMMSE [12]: an accelerated version of WMMSE
algorithm for the MIMO IMAC
o IWU [14]: a fast-convergent leakage minimization algorithm for the MIMO IFC
o AIMS: our previously proposed generalization of maxSINR [14], for MIMO IMAC / MIMO IFC / MIMO IBC
Algorithms such as IWU and CCP-WMMSE use so-called
turbo iterations, where I inner-loop iterations are performed
within each F-B iteration. Unlike IWU where the turbo iterations are done at the BS/user (i.e., offline), these iterations are
carried over-the-air for CCP-WMMSE.
B. Communication Overhead
The operation of the proposed scheme hinges on each transmitter and receiver’s having knowledge of effective channels,
for the desired and interfering links. We note that investigating
different mechanisms for the distributed acquisition of CSI is
outside the scope of the current work (we refer the reader
to [18]). However, we have outlined a simple mechanism that
goes hand-in-hand with F-B iterations, in Fig. I. We recall that
T F-B iterations are carried out.

Pilots

Estim. cov.
matrices
at receiver

Optimize
receive
filter

Pilots

Estim. cov.
matrices
at transmitter

Optimize
transmit
filter

Data

TABLE I: Basic structure of Forward-Backward Iteration

It becomes clear that each F-B iteration has an associated
communication overhead. While total overhead comprises of
bidirectional transmission of pilots, synchronization, frequency
offset calibration, etc, it is dominated by the pilot overhead, if the case of cellular coordination [19]. Thus, we can
safely approximate the communication overhead by the total
number of pilot symbols, for channel estimation, after T FB iterations. In conventional coordination, it is typical to
assume T = 100 ∼ 1000 until convergence, even for small
systems [10]. Moreover, this number increases with more BSs,
cells and transmit/receive antennas, all of which are prevalent
in sub-28 GHz systems. This limitation is compounded by
the naturally lower coherence time of mmWave channels, thus
further restricting the possible number of F-B iteration (before
the channel changes). Indeed, simple calculations reveal that
conventional algorithms would fail in these systems, as the
overhead would destroy the sum-rate gains from coordination.
Thus, we aggressively limit the number of F-B iteration to
T ≤ 5, thereby resulting in a drastic tenfold reduction in the
communication overhead.
For simplicity, we additionally assume that the minimal
number of orthogonal pilots is used, i.e. d pilot symbols for
each UL/DL effective channel, resulting in a total of KLd
orthogonal pilots for each UL/DL phase. The total overhead
for max-DLT, in the number of channel uses (c.u.), is given by,
Ωprop = T (KLd
| {z }) = 2T KLd
| {z } + KLd
UL

DL

The overhead is the same for schemes such as max-SINR,
IWU and MMSE. Similar calculations can be made to estimate
the overhead of CCP-WMMSE and WMMSE (in c.u.),
Ωccp-wmmse = T [( KLM
− 1) + I ×( KLN
| {z } ) ×(L
| {z } )]
| {z } |{z}
UL chann.
estim

CSI
sharing

Ωw-mmse = T (KLd
| {z } + KLd
| {z })
| {z } + KLM
UL

weights

turbo

cov. mat
upd.

DL

where I denotes the number of turbo iterations. These simple
calculation reveal that the overhead for W-MMSE and CCPWMMSE is significantly higher than that of max-DLT. Furthermore, the turbo iteration in CCP-WMMSE (outlined in
Sec IV-A) is carried over-the-air, and thus induces a massively
higher overhead, compared to other schemes. We include the
overhead of these algorithms in the simulation results.
C. Complexity
We can approximate the computational complexity of maxDLT, by noticing that it is dominated by the complexity of
the Cholesky Decomposition of the I+N covariance matrix,
O(N 3 ), and that of Eigenvalue Decomposition of M , O(M 3 ),
Cprop = O((M + N )3 ) .
One can verify that the above also holds for max-SINR, IWU,
MMSE, and WMMSE. Unlike other methods, the acceleration

Sum−rate (bps/Hz)

50
40

280

max−DLT (Ω=48)
AIMS (Ω=48)
MMSE [Peters] (Ω=48)
max−SINR (Ω=48)
IWU − RR (Ω=48)
WMMSE (T=200, Ω=4800)
WMMSE (T=4, Ω=96)

260
240

Sum−rate (bps/Hz)

60

30
20

220
200
180
160
140
max−DLT (Ω=288T)
AIMS (Ω=288T)
Uncoordinated (Ω=864)

120
100

10
0
0

80
1

5

10

15

20
25
1/ σ2 (dB)
vs 1/σ 2 , for L

30

35

40

Sum−rate (bps/Hz)

35

I = 2, Ω=48T

I = 1, Ω=32T

30

4

T

5

6

7

8

350
300

Sum−rate (bps/Hz)

I = 50, Ω=816T

3

Fig. 4: Ergodic sum-rate vs T , for dense uplink (N = 8, M =
4, d = 2, L = 9, K = 8).

Fig. 2: Ergodic sum-rate
= 3, K = 1, M =
N = 4,Sum−rate
d = 2, Tfor=M4=(MIMO
IFC)
4, N = 4, L
= 2, K = 2, d = 2, it = 8, SNR=20
40

2

250
200
150
max−DLT (Ω=144T)
100

WMMSE (Ω=288T)
Uncoordinated (1296)

50

25

0
1

20

10
1

2

3

4

T

5

6

7

3

4

T

5

6

7

8

Fig. 5: Ergodic sum-rate vs T for dense downlink (M =
16, N = 4, d = 1, L = 9, K = 8)

max−DLT (Ω=16T)
AIMS (Ω=16T)
max−SINR (Ω=16T)
CCP−WMMSE

15

2

2) Multi-user Multi-cell MIMO uplink
8

Fig. 3: Ergodic sum-rate vs T , for L = 2, K = 2, M = 4, N =
4, d = 2
does not require gradient/Hessian, and thus comes at a negligible added computational cost, compared to conventional
algorithms. However, each turbo iteration for CCP-WMMSE
involves running a series of semidefinite programs (using
interior point solvers), which render the algorithm very costly.
V. N UMERICAL R ESULTS
A. Performance in Sub-6 GHz systems
We start with presenting results for conventional multi-cell
multi-user MIMO, to illustrate desired features of max-DLT.
We refer the reader to [15] for a detailed discussion of the
simulation setup.
1) Single-user Multi-cell MIMO Uplink
We start with a widely used coordination test case, a MIMO
IFC with L = 3, K = 1, M = N = 4, d = 2 where the
set T = 4 for all algorithms. We include W-MMSE results
for T = 4, and T = 200 F-B iterations (as an upper bound).
Fig 2 reveals that while max-DLT and W-MMSE (with T = 4)
have similar performance in the low- and medium-SNR range,
this gap increases sharply as the SNR increases. This is in
spite of two-fold increase in communication overhead for
WMMSE. Moreover, the proposed scheme yields better sumrate performance than all benchmarks, with this gap becoming
significant in the high-SNR: as the following results will
show, the gap increases further with more users, antennas, and
BS/cells, under a low number of F-B iterations.

Moving on to a larger setup with L = 2, K = 2, M = 4, N =
4, d = 2 (MIMO IMAC), we benchmark max-DLT against
the fast-converging CCP-WMMSE (Sec. IV-A), by varying
the number of turbo iterations I, for CCP-WMMSE. Fig. 3
clearly exhibits the fast converging nature of max-DLT, that
achieves 95% of its final performance, after just 2 iterations. In
the low overhead regime (for T ≤ 5), max-DLT outperforms
CCP-WMMSE (for I = 1), although the overhead of the
latter is twice that of former. While additional turbo iterations
improve slightly the CCP-WMMSE performance, the overhead
increases linearly with I, e.g., the overhead of CCP-WMMSE
with I = 2 is threefold that of max-DLT. Achieving the
nominal performance of CCP-WMMSE relies on convergence
of the turbo iteration, which implies a (possibly arbitrary) large
number of turbo iteration. This results in (potentially) ordersof-magnitude higher overhead/complexity (e.g. CCP-WMMSE
with I = 50). Despite its fast-converging nature, CCPWMMSE is clearly ill-suited for the systems studied here.
B. Performance in Dense mmWave Deployments
Following recent measurements in the 28 GHz band [20], we
consider a dense urban mmWave setting. The full parametrization is detailed in [15][Sec. VI-C].
1) Dense Multi-user Multi-cell uplink
We consider a dense UL system with L = 9, K = 8, N =
8, M = 4, d = 2, where the average SNR (across users) is
set to 19 dB. Fig. 4 reveals that max-DLT offers significantly
better sum-rate, than all benchmarks. Interestingly, max-DLT
(with T = 3) provides a threefold increase in sum-rate with
respect to the uncoordinated scheme, however, with a similar
overhead.

2) Dense Multi-user Multi-cell downlink
We next consider DL scenario with L = 9, K = 8, M =
16, N = 4, d = 1, while setting the average SNR to 21dB, and
following the above simulation method. The fast-converging
nature of max-DLT is embodied in Fig. 5, where most the
performance is delivered in just 2 F-B iterations: this is due
to inherent stream control feature, that allows poor quality
streams to be shut down, thus converging quickly to a good
sum-rate. Note that max-DLT assumes equal power allocation
for users in each cell. In contrast, WMMSE performs power
allocation for users in each cell, as part of the algorithm.
Despite this unfavorable setup for max-DLT, we observe a
large sum-rate gain compared to WMMSE, while resulting
in a 50% decrease in overhead. Evidently, the sum-rate for
WMMSE will exceed that of max-DLT, as T increases (with
a huge overhead).
C. Discussions
We note the significant gap between the proposed scheme
and the benchmarks, may be attributed to the fast-converging
nature of the max-DLT, which is in turn due to the inherent
stream-control mechanism of the non homogeneous waterfilling solution. Moreover, the drastically limited number of F-B
iterations limits the performance of conventional algorithms,
due to significant levels of residual interference. As seen
in Figs. 4 and 5, that uncoordinated transmission performs
extremely poorly: max-DLT provides a threefold sum-rate
improvement over uncoordinated scheme, with a similar communication overhead. This provides a clear answer that lowoverhead coordination is a crucial, to achieving huge sum-rate
improvements in a dense multi-cell 28 GHz mmWave system.
This also implies that the same conclusions hold for sub-28
GHz systems, which are naturally more sensitive to inference.
VI. C ONCLUSIONS
We have proposed a low-overhead algorithm for coordination,
in dense multi-cell sub-28 GHz systems. The DLT bound - a
lower bound on the sum-rate, was derived and its tightness was
investigated. Moreover, we have proposed a distributed optimization algorithm (max-DLT), and showed its convergence to
a stationary point of the DLT bound. The non-homogeneous
waterfilling was derived as a solution to the optimal BS/user
filter update, and its ability to turn-off low-SINR streams was
underlined. We have tied this to the fast-convergence of the
algorithm, thus enabling a tenfold reduction in communication
overhead (over conventional coordination). Our numerical results have showed that low-overhead coordination offers huge
gains, in dense sub-28 GHz systems.
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