On the Feasibility of Interference Alignment in
Compounded MIMO Broadcast Channels with
Antenna Correlation and Mixed User Classes
arXiv:1711.07175v1 [] 20 Nov 2017

Galymzhan Nauryzbayev, Member, IEEE, Emad Alsusa, Senior Member, IEEE,
and Mohamed Abdallah, Senior Member, IEEE

Abstract
This paper presents a generalized closed-form beamforming technique that can achieve the maximum degrees of
freedom in compounded multiple-input multiple-output (MIMO) broadcast channels with mixed classes of multipleantenna users. The contribution is firstly described within the context of a three-cell network and later extended
to the general multi-cell scenario where we also show how to determine the conditions required to align the
interference in a subspace that is orthogonal to the one reserved for the desired signals. This is then followed by
an analysis of the impact of antenna correlation for different channel state information acquisition models. The
proposed scheme is examined under both conventional and Large-scale MIMO systems. It will be shown that the
proposed technique enables networks with any combination of user classes to achieve superior performance even
under significant antenna correlation, particularly in the case of the Large-scale MIMO systems.
Index Terms
Antenna Correlation, Broadcast Channel (BC), Channel State Information (CSI), Degree of Freedom (DoF),
Interference Alignment (IA), Multiple-Input Multiple-Output (MIMO).

I. I NTRODUCTION

I

NTERFERENCE alignment (IA) is a potential capacity-achieving technique in interference-limited
networks, that was initially proposed in [1] to establish additional degrees of freedom (DoF) in a

single-input single-output (SISO) model, where it was shown that the optimal DoF is

K
2

in K-user time

varying interference channel (IC). The concept of IA (is concerned with) aligning all interfering signals
into a specific subspace at the receiver side in order to exclusively reserve (a linearly independent) subspace
for interference-free data transmission. In a relatively short time, the IA concept has attracted a significant

2

amount of the researcher attention and a number of algorithms have been proposed for multiple-input
multiple-output (MIMO) systems [2]-[21], [25]-[31], to name a few.
The feasibility conditions of IA for MIMO IC were analysed in [2] and [3]. In [4], the authors considered
the case of K-user M × N MIMO network. It was shown that the achievable total number of DoF equals
Kmin(M, N), if K ≤ R, and

RK
min(M, N),
R+1

antennas, respectively, and R =

max(M,N )
.
min(M,N )

if K > R; M and N are the number of transmit and receive

The authors in [5] investigated the X-channel network scenario

and obtained remarkably high DoF compared to the findings in [6]. The derived outcomes of two DoF
per user, [5], [7], prompted the work in [8], where the authors presented an IA scheme for the two-user
X-channel with three antennas at each node. In [9], the authors achieved a total DoF of

MN
M +N −1

for the

case of M × N time-varying X-channels. In [10], the authors proposed an IA scheme to estimate the
achievable DoF in cellular networks which is based on aligning the interference into a multi-dimensional
subspace at multiple non-intended base stations (BS). On the other hand, in [11], the authors provided a
precise expression of the spatial multiplexing gain for two mutually interfering MIMO broadcast channels
(BCs) using a linear transceiver. Moreover, for the same scenario, the authors in [12] proposed a novel
IA technique for jointly designing the transmit and receive beamforming (BF) vectors with a closed-form
non-iterative expression. It was shown both analytically and numerically that the proposed scheme achieves
the optimal DoF. Furthermore, the authors in [13] showed how the extension of the grouping method,
[12], can be outperformed in terms of capacity and BS complexity. On the other hand, the authors in
[14] considered a network scenario comprising all the aforementioned cases and proposed a closed-form
IA scheme that can mitigate such mixture of interference. However, due to the limited available physical
space, resulting in small separations between the antenna elements, each channel is affected by antenna
correlation which hinders further enhancements of the transmission rate.
A. Main Contributions
In contrast to [14] and [15], this paper considers the compounded MIMO BC network scenario when
the users have different number of antennas and require different numbers of data streams. The derived
algorithm can be regarded as a generalization that can be also used in the special cases of [14] and [15],
where it was assumed that all users demand an equal number of data streams. Thus, this paper focuses
on the feasibility of IA, achievable sum rate and DoF region under various scenarios of channel state
information (CSI) acquisition and spatial correlation. The efficient channel model is used to capture all
possible cases of CSI mismatch. Furthermore, the paper investigates the impact of spatial correlation

3

between the antenna elements which arises due to the physical limitations of the available space. With
this in mind, the model was manipulated to involve both the impact of spatial correlation and CSI
mismatch in the compounded MIMO BC when the users have various antenna configurations and require to
decode different numbers of data streams. Therefore, a comprehensive algorithm is proposed to define the
minimum required antenna configuration to achieve a given DoF. The proposed technique is demonstrated
within the contexts of both conventional and Large-scale MIMO (LS-MIMO) systems under a wide range
of channel conditions and spatial correlations. Finally, the complexity of this technique is studied and
compared to a well-known benchmark technique.
The results demonstrate the effectiveness of the proposed technique particularly under highly correlated
channels. It is shown that this technique is not only associated with relatively lower computational
requirement, but also can achieve the maximum DoF even when the multiplexed users belong to different
data classes.
II. S YSTEM M ODEL
The system model considered here presents a cellular network comprising of L cells serving multiple
users randomly distributed as shown in Fig. 1. In terms of the network coverage, the whole combined
area can be determined as non-overlapped and overlapped areas, [22], [23]. The overlapped area denotes
a space where BSs cover a common region. Hence, the number of BSs creating the overlapped area varies
from 2 to L. According to this number of BSs, we can further define totally and partially overlapped areas.
Any user located in the totally overlapped area experiences the worst scenario when high interference
degrades the network performance. Thus, a main focus of the work is the users located in the totally
overlapped area.
Under a dense network scenario, [24], it is feasible to assume that each BS aims to communicate with
a large number of users K. We can also assume that K ≤ K users reside in the totally overlapped area,
and that K is identical for all L cells.
Since a concept of the point-to-point MIMO model can be applied for designing the cellular network,
we consider the IC and X-channel models that are differentiable from each other by the message set ups,
[1], [25], e.g., IC is the scenario when each BS i serves only the corresponding user i while the X-channel
scenario is characterised by a message set when each BS i has different messages to all users. Therefore,
a cellular MIMO network can be determined as a compounded IC-X channel scenario when each cell
consists of cell-edge users experiencing a multi-source transmission from L BSs such that 1 < L < L,

4

BS1

BS2

A totally overlapped
area
A partially overlapped
area

BS3

Fig. 1. A three-cell MIMO network with randomly distributed users.

and L is thought to be the same for all users. Hence, the user of interest can classify the observed network
as a set of (L − L) IC and L X-channels, and then the corresponding interferences can be defined as ICI
and XCI, [14], [15].
Since we focus on multi-antenna users in the totally overlapped area, it is reasonable to assume that
the user of interest may wish to receive data from several BSs. In turn, those BSs might send different
numbers of data streams to the interested users. In this work, we consider the network scenario when BSs
serve mixed user classes that can be determined as the case when users with different numbers of antennas
require a different number of data streams to be decoded. Hence, we define the network configuration
as (Mi , Ni , K, L, di ), where Mi and Ni indicate the number of transmit and receive antennas per node in
cell i, respectively, K is the number of users per cell in the totally overlapped area, and di denotes the
number of data streams transmitted from BS i. For the sake of brevity, we assume that each BS serves
only one user in the totally overlapped area (K = 1), and the received signal at the user of interest can
be written as, [14],
ỹj =

UH
j

j
X

Hj,i Vi si + UH
j

i=j−L+1

[j]

where Uj ∈ CNj ×d

|

{z

desired + XCI signals

L
X

Hj,l Vl sl + ñj , ∀j ∈ L,

(1)

l=1,l6=i

}

|

{z

ICI

}

[j]

d ×1
denotes the receive suppression matrix of user j, and ñj = UH
is the
j nj ∈ C

effective zero-mean additive white Gaussian noise (AWGN) vector, with E ñj ñH
= σñ2 I. d[j] is the
j

number of data streams required to be decoded at user j. Hj,i ∈ CNj ×Mi is the channel matrix between

5

BS i and user j. As in [14], the transmit BF design can be proceeded as
[ICI]

Vi = Vi
[ICI]

[XCI]

× Vi

, ∀i ∈ L,

(2)

[XCI]

∈ CQi ×di are the BF matrices responsible for mitigating the ICI and

XCI terms in (1), respectively. Vi ∈ CMi ×di is the transmit BF matrix at BS i, with tr Vi ViH = 1. We
where Vi

∈ CMi ×Qi and Vi

assume that si ∈ Cd×1 is the data vector comprising of the symbols drawn from independent and identically

distributed (i.i.d.) Gaussian input signaling and chosen from a desired constellation, with E si sH
= I.
i

Then, all these conditions sufficiently meet the average power constraint at the BS.

With this in mind and due to the nature of broadcast transmissions, we can assume that the users within
one cell desire to receive the same data, thus the transmitted signal can be expressed as
si

=



c[i,i]T

c[i,i+1]T

···

c[i,l]T

···

c[i,i+L−1]T

T

,

∀i

∈

L, (3)

where c[i,l] ∈ Cd[i,l] ×1 and d[i,l] indicate the message vector and its corresponding number of data streams
transmitted from BS i to the user in cell l, respectively. More specifically, (3) implies that BS i dedicates the
first part of the data to its corresponding user i, while the second part of the data is transmitted to the user
belonging to the neighbouring cell (i+1), and so on. The superscript numeration l ∈ {i, i+1, . . . , i+L−1}
changes circularly such that1
l=



 L,

if modL (l) = 0,

(4)


 mod (l), otherwise.
L

For instance, for i = 4 and L = 4 (L = 3), we have modL ({i, i + 1, i + 2}) = mod4 ({4, 5, 6}) →
{4, 1, 2}.
To decode the required signal successfully in (1), the desired signal should be aligned into a subspace
at the receiver side such that both ICI and XCI interfering signals are aligned into the subspace that is
orthogonal to the desired signal. Therefore, the following conditions must be satisfied at user j
j
X

i=j−L+1
L
X


[j]
rank UH
j Hj,i Vi = d , ∀j ∈ L,

UH
j Hj,l Vl = 0, ∀j ∈ L, l 6= i,

l=1

1

It is worth to note that all notations with ∗ (·) should satisfy Eq. (4).

(5)

(6)

6

where d[j] is the number of resolvable interference-free data streams at user j. Condition (5) ensures that
the user of interest can extract the desired signal coming from the channel links simultaneously carrying
both the desired and XCI signals, while (6) implies that the ICI signals are totally mitigated at the receiver
side.

A. Imperfect CSI
Since the CSI acquisition in practice is presented by imperfect estimates of the channel parameters, the
system performance is likely to be degraded. Similar to the same assumption as in [26], [27], [28], we
assume that the precoders and combiners are designed with the knowledge of CSI mismatch. Similar to
[29], [30], we exploit the following model to construct the CSI mismatch as
Ĥ = G + E,

(7)

where the actual channel realization G is thought to be independent of the channel measurement error
matrix E. Defining the nominal SNR as ρ =

P
2 ,
σn

we further regard E as a Gaussian matrix where each

entry is generated using i.i.d. random variables with zero mean and variance τ such that
vec (E) ∼ CN (0, τ I) with τ , βρ−α , β > 0, α ≥ 0,

(8)

where vec (·) indicates the vector operator and CN (µ, Σ) denotes the multivariate complex normal
distribution with mean vector µ and covariance matrix Σ.
In this model, the error variance can be used to capture a variety of CSI acquisition scenarios, i.e.,
dependent on the SNR (α 6= 0) or be independent of that (α = 0). In particular, perfect CSI scenario can
be obtained by setting τ = 0 (α → ∞). Two distinct cases of CSI acquisition, known as reciprocal and
CSI feedback channels, can be described by setting α = 0 and α = 1, respectively.
To facilitate further analysis, it is more appropriate to derive the statistical properties of G conditioned
on Ĥ. Since Ĥ = G + E, with G and E being statistically independent Gaussian variables, G and Ĥ
are jointly Gaussian. Therefore, after conditioning on Ĥ, [31], [32], the real channel realization can be
expressed as
G=

1
Ĥ + Ê,
1+τ

 

τ
where vec Ê ∼ CN 0, 1+τ
I is statistically independent of Ĥ.

(9)

7

B. Kronecker Product based Channel Modeling with Antenna Correlation
The downlink channel Hj,i in (1) is modeled as a correlated flat fading channel. Since the system model
in [14] presumes that both transmitter (Tx) and receiver (Rx) nodes are deployed with multiple antennas,
we assume that the fading is correlated at both sides.
The general model of the correlated channel is given as, [33], [34], [35],
vec (H) = R1/2 vec (G) ,

(10)

where G is the i.i.d. MIMO channel with either perfect CSI or CSI mismatch as in (9), and R is the
covariance matrix defined as
n

R , E vec (G) vec (G)

H

o

.

(11)

A popular MIMO channel simulation model is the Kronecker model, [35], [36], [37]. The Kronecker
model is limited because it does not take into account the coupling between the direction of departure
(DoD) at the transmitter and the direction of arrival (DoA) at the receiver, which is typical for MIMO
channels. Despite the limitation of ignoring the coupling between the DoD and DoA at the transmit and
receive ends, the Kronecker model is widely used in information theoretic capacity analysis and simulation
studies, [38], [39], [40], [41]. Though the Kronecker model is expected to be inaccurate for increasing
array size and angular resolution, it still finds use in large MIMO system studies because of its simplicity.
Therefore, the Kronecker model is found to be sufficient to analyse the effect of spatial correlation on
the network performances.
Since this model is based on the assumption of separability of the transmit and receive correlation
coefficients, the transmitter does not affect the spatial properties of the received signal. In such a case the
correlation matrix is given as
R = Rr ⊗ Rt ,

(12)

where the notation ⊗ stands for the Kronecker product operator, Rr ∈ CN ×N and Rt ∈ CM ×M are the
receive and transmit correlation matrices defined as
1 
E GGH ,
M
1 
Rt = E GH G .
N
Rr =

(13)
(14)

8

Then, the channel realization, with respect to imperfect CSI (9), can be modeled as, [42], [43],
H=
=
1/2

1/2

where Ẽ = Rr ÊRt
H̃ =

1/2
R1/2
r GRt

=

R1/2
r




1
1/2
Ĥ + Ê Rt
1+τ

1
1/2
1/2
R1/2 ĤRt + R1/2
= H̃ + Ẽ,
r ÊRt
1+τ r

∼ CN (0, Rr ⊗

τ
I
τ +1

(15)

⊗ Rt ) is the estimation error which is uncorrelated with

1/2
1/2
1
Rr ĤRt .
1+τ

With this in mind, we refer to [44], where, for the case of both-end correlation, the authors showed that
the Kronecker-based exponential and uniform models are empirically reasonable to apply at the transmitter
and receiver sides, respectively. Moreover, these simple single-parameter models allow one to investigate
the effects of both-end correlation on the achievable sum rate and DoF to achieve clearer insights in an
explicit way.
1) Exponential correlation coefficient model: The most common and easy model that accounts for
potential antenna correlation is the exponential model, [35], [37], which can be accordingly utilized at
the transmitter side as follows

Rt[m,n] =



 r |m−n| ,

if m ≥ n,

(16)


 (r † )|m−n| , if m < n,

where the subscript notation [m, n] indicates the matrix element located in the mth row and nth column.
(† ) denotes a complex conjugate and r = aejθ is the complex correlation coefficient with 0 ≤ a < 1. For
simplicity, we assume that r = a throughout the paper, unless it is restated.
2) Uniform correlation coefficient model: The uniform coefficient model, [45], is the worst case
scenario where the correlation coefficients are defined as


 r |m−n| , if m = n,
Rr[m,n] =

 r,
if m =
6 n,

(17)

where we assume that the coefficients of all neighboring subchannels are equal to those of the distant
channels.

9

III. T HE IA F EASIBILITY C ONDITIONS

AND

D O F R EGION A NALYSIS

As stated in [14], [15], the compounded MIMO BC scenario represents the network model where the
users located in the totally overlapped area experience a multi-source transmission by L BSs. In particular,
the authors in [14] focused on the scenario when each Tx-Rx pair has an identical antenna configuration
with an equal number of data streams transmitted to the users of interest. In contrast to [14], in this
work, we consider the network scenario when the BSs serve mixed user classes. Therefore, we exploit
a three-cell network scenario to analyze how the proposed IA scheme performs under this transmission
strategy. For the sake of simplicity, we assume that each cell consists of one BS serving a single user
residing in the totally overlapped area. Thus, each BS aims to send data to two users of interest (L = 2).
Accordingly, the transmitted signal in (3) can be rewritten as

si = c[i,i]T

c[i,i+1]T

T

, ∀i ∈ L,

(18)

where c[i,i] ∈ Cd[i,i] ×1 is the data vector transmitted from BS i to the corresponding user belonging the
same cell, while c[i,i+1] ∈ Cd[i,i+1] ×1 indicates the data transmitted to a desired user in the neighboring
cell.
In the following sections, we show how to define the antenna configuration for each Tx-Rx pair to
achieve a given DoF in the three-cell MIMO network (see Fig. 2) and determine the performance metrics
accounting for the impact of spatial correlation.

A. The Feasibility Conditions of Interference Alignment
Since we decouple the transmit BF matrix into two parts, we need to start with the design of the
part responsible for IC interference mitigation. For the sake of brevity, we refer to [14], where the given
scheme is also applicable for the case with a single user per cell.
[ICI]

After applying that scheme, we come up with the Vi

matrix with the dimension of Mi ×(Mi −Ni−1 ).

Hence, the matrix dimension leads to the following condition
(Mi − Ni−1 )+ ≥ di , ∀i ∈ L

(19)

to be satisfied for a successful transmission of all the desired data to the interested users, and (·)+ represents
a function that returns a non-negative value. di is the number of data streams transmitted from BS i and

10

M1

1

s1

s2

M2

s3

N2

1

y2

U2

V2
M3

1

y1

U1

V1
1

N1

1

N3

1

U3

V3

y3

Fig. 2. A system model with three-cell compounded MIMO BCs for K = 1 user per cell with various numbers of antennas at each Tx-Rx
pair.

defined with respect to (18) as
di =

i+1
X

d[i,j], ∀i ∈ L.

(20)

j=i

For simplicity of the following derivations, we introduce a new variable indicating the number of columns
[ICI]

in Vi

, as well as the number of non-occupied antennas at the transmitter side, and define
Qi = (Mi − Ni−1 )+ , ∀i ∈ L.

(21)

Finally, we restate the condition of the successive ICI cancellation given in (19) as


[ICI]
rank Vi
≥ Qi , ∀i ∈ L.

(22)

Since we have different classes of receivers and BSs that aim to transmit different numbers of data
streams to the users of interest, we need to define the number of data streams desired to be decoded at
user j as
d[j] =

X
∀i

d[i,j], ∀j ∈ L.

(23)

11

In general, the number of transmitted data streams from BS i is not equal to the number of data streams
desired to be received by the corresponding user i, di 6= d[i] ; nevertheless, the total number of streams at
the transmitter side always matches the one at the receiver side as
L=3
X

di =

i=1

L=3
X

d[j].

(24)

j=1

With this in mind, the received signal at the user of interest can be written as
ỹj =

UH
j

j
X

UH
j

i=j−1

|
=

{z

}

desired + XCI signals

j
X

[ICI] [XCI]
UH
Vi
si
j Hj,i Vi

|

[ICI]

}

+ ñj =

j
X

[XCI]

H̄j,i Vi

si + ñj

i=j−1


H̄j,i V[i,i]XCI
=
| {z }
i=j−1
where H̄j,i = UH
j Hj,i Vi

{z

ICI=0

i=j−1
j
X

L=3
X

✘✘✘
✘✘
✘
[ICI]
[XCI]
✘
H✘
Vl
Vl
sl + ñj
+
j,l✘
✘✘✘
✘
✘
✘ l=1,l6=i

[ICI] [XCI]
si
Hj,i Vi Vi

Qi ×d[i,i]

V
|


[i,i+1]XCI
{z

Qi ×d[i,i+1]

}




c[i,i]
c

[i,i+1]



 + ñj , ∀j ∈ L,

(25)

indicates the effective channel matrix. The under-braced terms represent the

Qi × d[m,n] matrices responsible for the XCI cancellation of the undesired data, c[m,n] . Then, the user
of interest no longer experiences ICI, but the interference arriving along the desired directions presented
by the c[j−1,j−1] and c[j,j+1] data vectors with d[j−1,j−1] and d[j,j+1] data streams, respectively. Hence, we
define the minimum and maximum numbers of the interfering data streams as

ki = min d[j−1,j−1], d[j,j+1] ,

ri = max d[j−1,j−1], d[j,j+1] ,

(26)
(27)

and the corresponding difference as
w i = ri − k i .

(28)

To be more specific, we consider the received signal at user 1, which can be presented with respect to

12

(18) as




[1,1]
[3,3]
c
c
 [1,1]XCI



 + H̄1,3 V[3,3]XCI V[3,1]XCI 
 + ñ1
ỹ1 = H̄1,1 V
V[1,2]XCI 
[1,2]
[3,1]
c
c
= H̄1,1 V[1,1]XCI c[1,1] + H̄1,3 V[3,1]XCI c[3,1] + ñ1 + H̄1,1 V[1,2]XCI c[1,2] + H̄1,3 V[3,3]XCI c[3,3] .
|
|
{z
}
{z
}
desired signal

(29)

interference

Since we assume that user 1 is deployed with N1 antennas, this number has to be enough to allocate
the desired signal into a subspace separate from the interference, and can be then defined as
N1 ≥ d[1] + dI [1] ,

(30)

where dI [1] is the subspace spanning the interfering c[1,2] and c[3,3] vectors present at user 1 and can be
expressed as
d[3,3]

d[1,2]

I

[1]

=

X

[1,2] [1,2]
H̄1,1 vi ci

+

X

[3,3] [3,3]
ci .

H̄1,3 vi

(31)

i=1

i=1

Therefore, the number of receive antennas has to be enough to decode the received signal and consequently
needs to satisfy the following requirement
N1 ≥ d[1] + d[1,2] + d[3,3] ,
which is not always possible to provide at the receive side due to the physical space limitation.
We define the k1 and r1 variables given in (26)–(27) as

k1 = min d[1,2] , d[3,3] ,

r1 = max d[1,2] , d[3,3] .
Accordingly, the corresponding difference between d[1,2] and d[3,3] is given as
w 1 = r1 − k 1 ,
where w1 indicates the number of the interfering data streams that can not be aligned with the other
interfering signal, and, therefore, these w1 vectors need to be mitigated at the receiver.
To reduce the subspace spanning the I [1] , we make sure that these interfering signals span a one-

13

I

[1]

=
=

d1,2
X

[1,2] [1,2]
H̄1,1 vi ci

i=1
k1 
X

+

[3,3] [3,3]
ci

H̄1,3 vi

i=1

[1,2] [1,2]
H̄1,1 vi ci

i=1

|

d3,3
X

+

[3,3] [3,3]
H̄1,3 vi ci

{z


}

space dimension range = 1

+

kX
1 +w1

[1,2] [1,2]
cl

H̄1,1vl

(33)

l=k1 +1

|

{z

}

space dimension range = 0

dimensional space, and determine the k1 pairs of precoding vectors such that
[1,2]

H̄1,1 vi
[m,n]

where vi

[3,3]

= −H̄1,3 vi

, ∀i ∈ {1, 2, . . . , k1},

(32)

is the ith column of the V[m,n]XCI matrix from (29), and the XCI notation is omitted for
[1,2]

[3,3]

brevity. We randomly pick the v1,...,k1 and v1,...,k1 vectors to guarantee that they are linearly independent
with probability one. This can be presented as shown in (33), where we assumed that r1 = d[1,2] .
Since user 1 obtains the c[1,2] and c[3,3] data vectors with d[1,2] and d[3,3] data streams, respectively, for
d1,2 6= d[3,3] , we then need to define which effective channel matrix, H̄1,x , needs to be cancelled, where
x can be derived from
{x, y} =



 {1, 2}, if r1 = d[1,2] ,


 {3, 3}, if r1 = d[3,3] .
Therefore, we have the w1 interfering BF vectors,

[x,y]
V{k1 +1:k1 +w1 }

n
o
[x,y]
[x,y]
= vk1 +1 , . . . , vk1 +w1 , that are obtained

by finding the null space of the corresponding effective channel as follows

[x,y]
V{k1 +1:k1 +w1 } = null H̄1,x ,

(34)

[x,y]

where V{k1 +1:k1 +w1 } denotes the part of the BF matrix with a range from the (k1 +1)th up to the (k1 +w1 )th
column. This leads to the following condition to be satisfied
w1 ≤ (Q1 − N1 )+ or w1 ≤ (Q3 − N1 )+ .

(35)

As a result, the interference observed by user 1 can be mitigated as shown in (37), where we assume
d[1,2] > d[3,3] and the boxed term is redundant, unless d[1,2] < d[3,3] . According to (33), the interference
at the user of interest spans a one-dimensional space, thus we redefine the number of receive antennas

14



|

 v[1,2]
 1

 |
H̄1,1 H̄1,3 
|
|
{z
}
 [3,3]

N1 ×(Q1 +Q3 )
v1
|
|

···
|
|
[1,2]
[1,2]
· · · vk1
vk1 +1
|
···
|
···
|
|
[3,3]
· · · vk1
0
···
|
|

···
|
[1,2]
· · · vk1 +w1
···
|
···
|
···
0
···
|
{z

|
0
|
|
[3,3]

vk1 +1
|

···
|
···
0
···
|
···
|
[3,3]
· · · vk1 +w1
···
|







| ··· |

 =  0 · · · 0  (37)


| ··· |

|
{z
}
N1 ×max(d[1,2] ,d[3,3] )

}

(Q1 +Q3 )×(k1 +w1 )

given in (30) as
N1 = d[1] + 1.

(36)

Finally, if all the conditions above are satisfied, user 1 experiences the interference-free data transmission.
Similar to cell 1, we define the conditions satisfying the ability to maintain the proposed scheme for
the rest of cells.
Below we briefly explain how to derive the receive suppression matrices. The given example will be
generalized for user i. Due to the help of transmit beamforming matrices, we ensure that the interfering
signals coming from BSs j (∀j ∈ L = 3, j 6= i) are aligned along one shared space (for brevity, we
denote this effective interference as H̃i ). Having this, we define the space spanning the interference as
Ti = span{H̃H
i Ui },

(38)

where span{·} denotes the subspace spanned by the column vectors of a matrix. Next, we rewrite this
equation as
h
I Mi

i




− H̃H
i

Ti
Ui



 = Fi Wi = 0.

(39)

Since the size of the matrix Fi is Mi × Mi + Pi (Pi = max(dj , dk ) and dj and dk is the numbers of
data streams transmitted from BSs j and k, where i 6= {j, k}), the null space always exists because
Mi + Pi > Mi , ∀Pi > 0. For more details, the reader can refer to [13].
B. The DoF Region Analysis
With respect to the number of cells L, we define an upper bound on the DoF region. Thus, for the
compounded MIMO BC network scenario with L cells, we have ((L + 2)L + 1) constraints that determine
the upper limit of the DoF region.

15

While the set Dout describes an outer limit for all achievable d[i,j] on the compounded L−cell MIMO
BC channel, maximization of the sum of d[i,j] over Dout is a linear LP problem, where it is required to
explicitly estimate all extreme points of the feasible space, to calculate the objective values and, finally,
to eliminate all redundant limits.
Dout , maximize

L
X

∗ mod (i+(L−L))
L

i=1

X

d[i,j] ∈ RLL
+ ,

j=i

∗ mod (i+(L−L))
L

X

subject to di =

d[i,j] ≤ Qi ,

(40.1)

d[j,i] + 1 ≤ Ni ,

(40.2)

j=i

∗ mod (i+(L−1))
L

X
j=i

|

[k]

{z

d[i]

}

Di = d[j] ∪ Qi ≤ max(Nj , Qi ),

(40.3)

j ∈ {i, . . . ,∗ modL (i + (L − L))},
|
{z
}
k samples

∀i ∈ L, k ∈ {1, . . . , (L − L)},

where

Qt = wj + Nj , ∀j ∈ L,

(40.4)

t = m ← d[m,n] = rj in (27),

(40.5)

where t and ∪ indicate the first subscript index and the union operation, respectively. The case of wj = 0
implies that at this particular user the numbers of interfering streams are identical. Q indicates the variable
that captures uncertainty of which Q should be chosen according to (35).
Although at a glance it might seem that the DoF region above is not too different from the one
presented in [15], it is worth mentioning that [15] focused on the network case with an identical antenna
configuration leading to identical numbers of data streams causing the interference at the receiver side,
that is, in [15] the authors considered the users of one user class. Moreover, the algorithm in [15] can not
be applied for the compounded MIMO BC with mixed user classes. Thus, considering different classes
of users entails more conditions related to the XCI mitigation shown by the second term in (33), which
spans a zero-dimensional space.
The set Dout provides the conditions that define the outer limit for all the attainable d[j,i] under the
compounded MIMO BC by maximizing a weighted sum of d[j,i] which is a linear programming problem.
At the same time, this set of conditions is suitable either to analyse the achievable DoF region under a

16

given network antenna configuration or to calculate the minimum number of antennas at each node in
order to obtain the required DoF. Subsequently, we provide the algorithm that allows us to compute the
minimum required number of antennas at each Tx-Rx pair with maximum DoF.
Algorithm 1 Defining the Antenna Configuration
Require: 
1: inputs d[i,j] , ∀i, j ∈ L
2: Qi and Nj from (40.1)–(40.2)
Ensure:
3: for j = 1 : L do
4:
if (40.3) is not valid then
[k]
5:
update Qi ← Di
6:
end if
7:
return Qi .
8: end for
9: for j = 1 : L do
10:
calculate wj , according to (28).
11:
find the value of t with respect to (40.5).
12:
if t = j then
13:
Qt := Qj
14:
if Qj > Qj then
15:
update Qj ← Qj
16:
end if
17:
else
Qt := Qj−1
18:
19:
if Qj−1 > Qj−1 then
20:
update Qj−1 ← Qj−1
21:
end if
22:
end if
23:
return Qj−1 or Qj (if updated only).
24:
This derived value will be used in the next iteration
25: end for
26: utilize Qi and Nj , ∀i, j ∈ L, to calculate the numbers of transmit antennas as in (21).

The following provides the total number of DoF by explicitly solving the LP problem for the three-cell
compounded MIMO BC case.
P roposition 1 :
η , max
Dout

L=3
X
i=1

∗ mod (i+1)
L

X

d[i,j]

j=i

= min {Q1 + Q2 + Q3 , N1 + N2 + N3 − 3, max(N1 , Q1 ) + max(N3 , Q2 ),
max(N1 , Q3 ) + max(N2 , Q2 ), max(N2 , Q1 ) + max(N3 , Q3 ),

(41)

17

max(N1 , Q1 ) + max(N1 , Q3 ) + max(N3 , Q2 ) + max(N2 , Q2 )
,
2
max(N1 , Q3 ) + max(N2 , Q1 ) + max(N3 , Q3 ) + max(N2 , Q2 )
,
2
max(N1 , Q1 ) + max(N2 , Q1 ) + max(N3 , Q2 ) + max(N3 , Q3 )
,
2
max(N1 , Q1 ) + max(N1 , Q3 ) + Q2 + N2 + N3
− 1,
2
max(N1 , Q1 ) + max(N2 , Q1 ) + Q2 + Q3 + N3 − 1
,
2
max(N1 , Q1 ) + max(N3 , Q2 ) + Q1 + Q2 + Q3
,
2
max(N1 , Q1 ) + max(N3 , Q2 ) + N1 + N2 + N3 − 1
− 1,
2
max(N1 , Q3 ) + max(N2 , Q2 ) + Q1 + Q2 + Q3
,
2
max(N1 , Q3 ) + max(N2 , Q2 ) + N1 + N2 + N3 − 1
− 1,
2
max(N1 , Q3 ) + max(N3 , Q3 ) + Q1 + Q2 + N2 − 1
,
2
max(N2 , Q1 ) + max(N2 , Q2 ) + Q3 + N1 + N3
− 1,
2
max(N2 , Q1 ) + max(N3 , Q3 ) + Q1 + Q2 + Q3
,
2
max(N2 , Q1 ) + max(N3 , Q3 ) + N1 + N2 + N3 − 1
− 1,
2
max(N2 , Q2 ) + max(N3 , Q2 ) + Q1 + Q3 + N1 − 1
,
2
max(N3 , Q2 ) + max(N3 , Q3 ) + Q1 + N1 + N2
− 1,
2
max(N1 , Q1 ) + max(N1 , Q3 ) + max(N2 , Q1 )
3

max(N2 , Q2 ) + max(N3 , Q2 ) + max(N3 , Q3 )
.
+
3
Outline of the Proof : Proposition 1 can be verified by solving the dual problem by linear programming

max d[1,1] + d[1,2] + d[2,2] + d[2,3] + d[3,3] + d[3,1] .
Since all the extreme points of the feasible space can be directly evaluated, we compute the objective
value at these points and eliminate the limits that can be regarded redundant. Using the fundamental
theorem of LP, [46], [47], we find the solution. For the sake of brevity, the derivation details are omitted.
It is worth noting that all the terms given in Proposition 1 are essential because any of them is valid
for a certain antenna configuration.

18

We define the achievable DoF for our multi-cell network as the pre-log factor of the sum rate, [4], [48].
This is one of the key metrics used for assessing the performance of a multiple antenna based system in
the high SNR region defined as
L

X
IP (SNR)
η = lim
=
d[j] ,
SN R→∞ log2 (SNR)
j=1
where IP (SNR) denotes the sum rate that can be achieved at a given SNR defined as IP (SNR) =

(42)
L
P

Ij ,

j=1

where Ij and d[j] are the data rate and the number of the successfully decoded data streams at user j,
respectively.
Therefore, the received signal at user j, with respect to the considered channel model in (15), can be
rewritten as
ỹj =

UH
j

= UH
j

L=3
X

i=1
L=3
X

i=1,

|
=

Hj,iVi si + UH
j Hj,j+1 Vj+1 sj+1 + ñj
|
{z
}
i6=j+1
ICI
{z
}

desired signal + XCI
L=3
X

UH
j

i=1,

|
=

Hj,i Vi si + ñj





H̃j,i + Ẽj,i Vi si + UH
H̃
+
Ẽ
j,j+1
j,j+1 Vj+1 sj+1 + ñj
j
|
{z
}
i6=j+1
{z
}
ICI
desired signal + XCI

j

UH
j

X

H̃j,i V

[i,j] [i,j]

c

i=j−1

|
+

{z

}

desired signal

UH
j
|

j
X

+

UH
j

|

L=3
X

Ẽj,i Vi si

i=1

{z

CSI mismatch

}

✭✭✭

H
✭✭
U✭
V✭
H̃j,i V[i,l{l6=j}]c[i,l{l6=j}] + ✭
✭✭
j,j+1
j+1 sj+1 + ñj , ∀j ∈ L.
j H̃
|
{z
}
i=j−1
ICI = 0
{z
}

(43)

XCI

With this in mind, we determine the data rate achievable at the user of interest as shown in (44), where
Jj in (45) indicates the interference terms in (43).
IV. S IMULATION R ESULTS
In this section, we present our results using Monte Carlo simulations with 105 trials to investigate the
impact of antenna correlation and CSI mismatch on the network performances. The simulations assume a
Gaussian modulation and frequency flat fading which is designed according to (9). The users are distributed

19

as it was described in the system model. To make a fair comparison, the total transmit power at BS is
constrained to unity irrespective of the number of transmit antennas. To create different classes of users
with various numbers of antennas, we assume that BS i transmits the i data streams to the corresponding
user i, and only one data stream to the collateral user of interest. Therefore, we have three receivers
deployed with three, four and five antennas, respectively.
According to [51], we examine three correlation regimes, namely, the low, medium and high correlations.
The low correlation mode indicates the scenario with no correlation with sufficient spacing (≥ λ/2)
between the antennas. According to (15), the medium and high correlation regimes at the transmitter side
can be modeled by r = 0.3 and r = 0.9, respectively; however, the same modes at the receiver side can
be presented by r = 0.9.
In Fig. 3, we evaluate the cumulative distribution function (CDF) of the achievable data rate for every
user in the assumed system model. The data rates are calculated using the proposed scheme under different
antenna correlation modes. For the sake of clarity, we consider the average data rate achievable by the
network. The observation point is 30 dB. We deploy transmitters with ten, thirty and fifty antennas to
estimate the potential benefit attainable from the LS-MIMO scenario. As we can see, for the case of low
correlation, the probability of attaining a higher data rate increases as the number of antennas at the BS
goes up. Regarding the medium correlation case, we observe severe degradation in the achievable data
rate, and the various numbers of transmit antennas do not seem to have much difference; however, a
different antenna deployment still matters as it is shown in the inset figure (see Fig. 3, where lines from

Fig. 3. CDF curves with low (black), medium (red) and high (blue) correlations for different numbers of transmit antennas at 30 dB for the
case of perfect CSI (when α → ∞).

20

L=3
X

L=3
X





IΣ =
Ij =
log2 det I +

j=1
j=1
where Jj =

j
X

i=j−1

j
P

i=j−1



[i,j] [i,j]H H
UH
V
H̃j,iUj 
j H̃j,i V

Jj +

σñ2 I

[i,l{l6=j}] [i,l{l6=j}]H H
UH
V
H̃j,i Uj +
j H̃j,i V

L=3
X


,


H H
UH
j Ẽj,k Vk Vk Ẽj,k Uj .

(44)

(45)

k=1

Fig. 4. CDF curves with low (black), medium (red) and high (blue) correlations for different CSI mismatch scenarios with M = 50
antennas per BS at 30 dB.

top to bottom refer to the cases when transmitter is equipped with 10, 30 and 50 antennas, respectively).
Finally, we consider the high correlation mode presented in blue lines with which produces a significant
loss in the achievable data rate. For M = 10, M = 30 and M = 50 antennas deployed at the transmitter
side, the average data rate of 8.8, 11.4 and 12.75 bits/s/Hz are achievable with a probability of 90%. It
is worth to note that the deployment of more antennas allow us to overcome the high correlation, and
accordingly, the system can be treated as though it is experiencing medium correlation. With this in mind,
in the next simulation, we consider only the network case where all the BSs are equipped with fifty
antennas.
Next, we want to investigate the combined effect of CSI mismatch and antenna correlation on the
achievable data rates. The observation point is 30 dB. As shown in Fig. 4, the case of (α = 1.5, β = 15)
performs worse than the other two scenarios of the CSI acquisition. The CSI mismatch cases given by
(α = 0.75, β = 10) and (α = 0, β = 0.05) act in a similar way under the medium and high correlation
regimes; however, for low correlation, the network with (α = 0, β = 0.05) slightly outperforms the one

21

modeled by (α = 0.75, β = 10). This result leads to the realization that the SNR-dependent and SNRindependent CSI acquisition scenarios, (α = 0, β = 0.05) and (α = 0.75, β = 10), do not differ from each
other in the cases of medium and high antenna correlations.
It is worth mentioning that the provided results in Fig. 4 and 5 also provide a representative insight
into how many DoFs can be achieved under the considered network scenario. Eq. (42) can be modified
to calculate the average number of DoFs as follows
DoF =

I(SNR)
,
log2 (SNR)

(46)

when the averaged sum rate values are scaled by the SNR value (the observation point is chosen to be
30 dB).

Fig. 5. The achievable data rate as a function of the correlation coefficient and number of transmit antennas.

Finally, we provide a 3D plot in Fig. 5 presenting the data rate achievable at 30 dB as a function of the
number of transmit antennas and correlation coefficient. The correlation at the transmitter side is assumed
to be high and modeled by using the exponential model as in (16) (r = 0.9). Accordingly, the antenna
correlation at the receiver side is given as in (17) and simulated within the range [0, 1]. As we can see, the
achievable data rate increases as the number of transmit antennas goes up, while the data rate decreases
as the correlation coefficient increases.
V. C ONCLUSION
In this work, we considered the compounded MIMO BC network scenario and proposed a generalized
transmit BF design that accounts for different user classes. We analysed the feasibility conditions of IA

22

and presented the DoF region analysis which were then utilized in design of an algorithm to define the
minimum antenna configuration to achieve a required number of data streams in the network. Moreover, we
investigated the impact of spatial antenna correlation under various CSI acquisition scenarios. Finally, the
proposed scheme was examined in traditional and LS-MIMO systems under different channel scenarios.
It was shown that the performance obtained for the latter case indicates that deploying more antennas
makes it possible to overcome the impact of high correlation by careful manipulation of the antenna array.
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