Dynamic Interference Steering in Heterogeneous
Cellular Networks
Zhao Li∗ , Canyu Shu∗ , Fengjuan Guo∗ , Kang G. Shin† , Jia Liu‡

∗

arXiv:1801.00145v1 [] 30 Dec 2017

State Key Laboratory of Integrated Service Networks, Xidian University
†
The University of Michigan, USA
‡
National Institute of Informatics, Japan
zli@xidian.edu.cn, cyshu112@126.com, fjguo@stu.xidian.edu.cn, kgshin@umich.edu, jliu@nii.ac.jp

Abstract—With the development of diverse wireless communication technologies, interference has become a key impediment
in network performance, thus making effective interference
management (IM) essential to accommodate a rapidly increasing
number of subscribers with diverse services. Although there have
been numerous IM schemes proposed thus far, none of them
are free of some form of cost. It is, therefore, important to
balance the benefit brought by and cost of each adopted IM
scheme by adapting its operating parameters to various network
deployments and dynamic channel conditions.
We propose a novel IM scheme, called dynamic interference
steering (DIS), by recognizing the fact that interference can
be not only suppressed or mitigated but also steered in a
particular direction. Specifically, DIS exploits both channel state
information (CSI) and the data contained in the interfering signal
to generate a signal that modifies the spatial feature of the
original interference to partially or fully cancel the interference
appearing at the victim receiver. By intelligently determining the
strength of the steering signal, DIS can steer the interference
in an optimal direction to balance the transmitter’s power used
for IS and the desired signal’s transmission. DIS is shown via
simulation to be able to make better use of the transmit power,
hence enhancing users’ spectral efficiency (SE) effectively.

I. I NTRODUCTION
Due to the broadcast nature of wireless communications,
concurrent transmissions from multiple source nodes may
cause interferences to each other, thus degrading subscribers’
data rate. Interference management (IM) is, therefore, crucial
to meet the ever-increasing demand of diverse users’ Qualityof-Service (QoS).
Interference alignment (IA) is a powerful means of controlling interference and has thus been under development
in recent years. By preprocessing signals at the transmitter,
multiple interfering signals are mapped into a certain signal
subspace, i.e., the overall interference space at the receiver is
minimized, leaving the remaining subspace interference-free
[1-2]. IA is shown to be able to achieve the informationtheoretic maximum DoF (Degree of Freedom) in some interference networks [3-4]. However, to achieve such a promising
gain, it is required to use either infinite symbol extensions
over time/frequency [4] or a large number of antennas at each
receiver [5], both which are not realistic. That is, IA emerges
as a promising IM scheme, but its applicability is severely
limited by the high DoF requirement.
Some researchers have attempted to circumvent the stringent
DoF requirement by proposing other IM schemes, such as

interference neutralization (IN). IN refers to the distributed
zero-forcing of interference when the interfering signal traverses multiple nodes before arriving at the undesired receivers/destinations. The basic idea of IN has been applied to
deterministic channels and interference networks [6-10], both
of which employ relays. IM was studied in the context of
a deterministic wireless interaction model [6-7]. In [6], IN
was proposed and an admissible rate region of the Gaussian
ZS and ZZ interference-relay networks was obtained, where
ZS and ZZ denote two special configurations of a two-stage
interference-relay network in which some of the cross-links are
weak. The authors of [7] further translated the exact capacity
region obtained in [6] into a universal characterization for the
Gaussian network. The key idea used in the above interference
networks with relays is to control the precoder at the relay
so that the sum of the channel gains of the newly-created
signal path via the relay and the direct path to the destination
becomes zero. A new scheme called aligned interference
neutralization was proposed in [8] by combining IA and IN.
It provides a way to align interference terms over each hop
so as to cancel them over the air at the last hop. However,
this conventional relay incurs a processing delay — while
the direct path does not — between a source-destination pair,
limiting the DoF gain in a wireless interference network. To
remedy this problem, an instantaneous relay (or relay-withoutdelay) was introduced in [9-10] to obtain a larger capacity than
the conventional relay, and a higher DoF gain was achieved
without requiring any memory at the relay. Although the
studies based on this instantaneous relay provide some useful
theoretical results, this type of relay is not practical. On
the other hand, IN can mitigate interference, but the power
overhead of generating neutralization signals also affects the
system’s performance. To the best of our knowledge, the power
overhead has not been considered in any of the existing studies
related to IN — either more recent interference neutralization [8-10] or the same ideas known via other names for
many years, such as distributed orthogonalization, distributed
zero-forcing, multiuser zero-forcing, and orthogonalize-andforward [11-12]. In practice, higher transmit power will be
used by IN when the interference is strong, thus making less
power available for the desired data transmission. Furthermore,
IN may not even be available for mobile terminals due to their
limited power budget.

By recognizing the fact that interference can be not only
neutralized but also steered in a particular direction, one
can “steer” interference, which we call interference steering
(IS) [13]. That is, a steering signal is generated to modify
the interference’s spatial feature so as to steer the original
interference in the direction orthogonal to that of the desired
signal perceived at the victim receiver. Note, however, that
IS simply steers the original interference in the direction
orthogonal to the desired signal, which we call Orthogonal-IS
(OIS) in the following discussion, regardless of the underlying
channel conditions, as well as the strength and spatial feature
of interference(s) and the intended transmission. Therefore, the
tradeoff between the benefit of IS (i.e., interference suppression) and its power cost was not considered there. Since the
more transmit power is spent on interference steering, the less
power for the desired signal’s transmission will be available,
one can naturally raise a question: “Is it always necessary to
steer interference in the direction orthogonal to the desired
signal?”
To answer the above question, we propose a new IM
scheme, called dynamic interference steering (DIS). With DIS,
the spatial feature of the steered interference at the intended
receiver is intelligently determined so as to achieve a balance
between the transmit power consumed by IS and the residual
interference due to the imperfect interference suppression, thus
improving the user’s SE.
The contributions of this paper are two-fold:
●

●

Proposal of a novel IM scheme called dynamic interference steering (DIS). By intelligently determining the
strength of steering signal, we balance the transmit power
used for IS and that for the desired signal’s transmission.
DIS can also subsume orthogonal-IS as a special case,
making it more general.
Extension of DIS to general cases where the number of
interferences from macro base station (MBS), the number
of desired signals from a pico base station (PBS) to its
intended pico user equipment (PUE), and the number of
PBSs and PUEs are all variable.

II. S YSTEM M ODEL
We consider the downlink1 transmission in heterogeneous
cellular networks (HCNs) composed of overlapping macro and
pico cells [14]. As shown in Fig. 1, macro and pico base
stations (MBSs and PBSs) are equipped with NT1 and NT0
antennas, whereas macro user equipment (MUE) and PUE are
equipped with NR1 and NR0 > 1 antennas, respectively. Since
mobile stations/devices are subject to severer restrictions in
cost and hardware, than a base station (BS), the BS is assumed
to have no less antennas than a UE, i.e., NTi ≥ NRi where
i = 0, 1. The radio range, d, of a picocell is known to be
300m or less, whereas the radius, D, of a macrocell is around
3000m [14]. Let x1 and x0 denote the transmit data vectors
from MBS and PBS to their serving subscribers, respectively.
E(∥x1 ∥2 ) = E(∥x0 ∥2 ) = 1 holds. For clarity of exposition, our
design begins with the assumption of beamforming (BF), i.e.,
only one data stream is sent from MBS to MUE (or from PBS
to PUE). Then, x1 and x0 become scalars x1 and x0 . We will
generalize this to multiple data streams sent from MBS and
PBS in Section IV. We use P1 and P0 to denote the transmit
power of MBS and PBS, respectively. Let H0 ∈ CNR0 ×NT0 and
H1 ∈ CNR1 ×NT1 be the channel matrices from MBS to MUE
and from PBS to PUE, respectively, whereas that from MBS
to PUE is denoted by H10 ∈ CNR0 ×NT1 . We adopt a spatially
uncorrelated Rayleigh flat fading channel model to model the
elements of the above matrices as independent and identically
distributed zero-mean unit-variance complex Gaussian random
variables. We assume that all users experience block fading,
i.e., channel parameters remain constant in a block consisting
of several successive time slots and vary randomly between
successive blocks. Each user can accurately estimate CSI
w.r.t. its intended and unintended Txs and feed it back to
the associated BS via a low-rate error-free link. We assume
reliable links for the delivery of CSI and signaling. The
delivery delay is negligible relative to the time scale on which
the channel state varies.
NT1
NT0

H10

H0

H1

x0

N R0

The rest of this paper is organized as follows. Section II describes the system model, while Section III details the dynamic
interference steering. Section IV presents the generalization of
DIS and Section V evaluates its performance and overhead.
Finally, Section VI concludes the paper.
Throughout this paper, we use the following notations.
The set of complex numbers is denoted as C, while vectors
and matrices are represented by bold lower-case and uppercase letters, respectively. Let XT , XH and X−1 denote the
transpose, Hermitian, and inverse of matrix X. ∥ ⋅ ∥ and ∣ ⋅ ∣
indicate the Euclidean norm and the absolute value. E(⋅)
denotes statistical expectation and ⟨a, b⟩ represents the inner
product of two vectors.

x1

N R1

Spatial feature of signals
perceived at PUE.

Fig. 1. System model.

As mobile data traffic has increased significantly in recent
years, network operators have strong preference of open access
to offload users’ traffic from heavily loaded macrocells to
other infrastructures such as picocells [14-15]. Following this
trend, we assume each PBS operates in an open mode, i.e.,
users in the coverage of a PBS are allowed to access it.
The transmission from MBS to MUE will interfere with the
1 Neither IS nor DIS is applicable for uplink transmission due to their
requirement of Tx’s cooperation.

intended transmission from PBS at PUE. Nevertheless, due
to the limited coverage of a picocell, PBS will not cause too
much interference to MUE, and is thus omitted in this paper.
As a result, the interference shown in Fig. 1 is asymmetric.
Since picocells are deployed to improve the capacity and
coverage of existing cellular systems, each picocell has subordinate features as compared to the macrocell, and hence
the macrocell transmission is given priority over the picocell’s
transmission. Specifically, MBS will not adjust its transmission
for pico-users. However, we assume that PBS can acquire the
information of x1 via inter-BS collaboration; this is easy to
achieve because PBS and MBS are deployed by the same
operator [16]. With such information, DIS can be implemented
to adjust the disturbance in a proper direction at PUE. Since
the transmission from MBS to MUE depends only on H1 and
is free from interference, we only focus on the pico-users’
transmission performance.
Although we take HCN as an example to design our scheme,
it should be noticed that other types of network as long as
they are featured as 1) collaboration between the interfering
Tx and victim Tx is available, and 2) the interference topology
is asymmetric, our scheme is applicable.
III. DYNAMIC I NTERFERENCE S TEERING
As mentioned earlier, by generating a duplicate of the
interference and sending it along with the desired signal,
the interference could be steered in the direction orthogonal
to the desired signal at the intended PUE with orthogonalIS. However, the tradeoff between the benefit of interference
steering and its power cost has not been considered before.
That is, under a transmit power constraint, the more power
consumed for IS, the less power will be available for the
intended signal’s transmission. To remedy this deficiency,
we propose a novel IM scheme called dynamic interference
steering (DIS). By intelligently determining the strength of
steering signal, the original interference is adjusted in an
appropriate direction. DIS balances the transmit power used
for generating the steering signal and that for the desired
signal’s transmission.
A. Signal Processing of DIS
As mentioned above, since the macrocell receives higher
priority than picocells, MBS will not adjust its transmission
for pico-users. In what follows, we use NTi = NRi ≥ 2 where
i = 0, 1 as an example, but our scheme can be easily extended
to the case of NTi ≥ NRi .
Due to path loss, the mixed signal received at PUE can be
expressed as:
√
√
r0 = P0 10−0.1L0 H0 p0 x0 + P1 10−0.1L10 H10 p1 x1 +n0 (1)
where the column vectors p0 and p1 represent the precoders
for data symbols x0 and x1 sent from PBS and MBS,
respectively. The first term on the right hand side (RHS)
of Eq. (1) is the desired signal, the second term denotes
the interference from MBS, and n0 represents for the additive white Gaussian noise (AWGN) with zero-mean and

variance σn2 . The path loss from MBS and PBS to a PUE
is modeled as L10 = 128.1 + 37.6 log10 (η10 /103 ) dB and
L0 = 38 + 30 log10 (η0 ) dB, respectively [17], where the
variable η(⋅) , measured in meters (m), is the distance from
the transmitter to the receiver.
The estimated signal at PUE after post-processing can be
written as r̃0 = f0H r0 where f0 denotes the receive filter. Recall
that the picocell operates in an open mode, and a MUE in the
area covered by PBS will become a PUE and then be served
by the PBS. The interference model shown in Fig. 1 has an
asymmetric feature in which only the interference from MBS
to PUE is considered. Moreover, since the macrocell is given
priority over the picocells, MBS will not adjust its transmission
for the PUEs, and hence transmit packets to MUE based only
on H1 . Here we adopt the singular value decomposition (SVD)
based BF transmission, but can also use other types of preand post-processing. Applying SVD to Hi (i = 0, 1), we get
(1)
(1)
Hi = Ui Σi ViH . We then employ pi = vi and fi = ui ,
(1)
(1)
where vi and ui are the first column vectors of the right
and left singular matrices (Vi and Ui ), respectively, both of
which correspond to the principal eigen-mode of Hi .
From Fig. 1 one can see that the strengths of desired signal
and interference at PUE depend on the network topology,
differences of transmit power at PBS and MBS, as well as
channel conditions. All of these factors affect the effectiveness
of IM. For clarity of presentation, we define P0e = P0 10−0.1L0 ,
P1e = P1 10−0.1L10 , where P0e and P1e indicate the transmit
power of PBS and MBS incorporated with the path loss perceived by PUE. With this definition, consideration of various
network topologies and transmit power differences can be
simplified to P0e and P1e . In what follows, we first present
the basic principle of orthogonal-IS (OIS), and then elaborate
on the design of dynamic-IS (DIS) where we provide the
existence and calculation of the optimal steering signal.
i QQ
sstIntIn
sst t

OQ
stQt

i Qi Q

ii
stst
iiInIn ss

(a) An illustration of OIS.

i i
iR

OO

In In
iR i i s s

(b) An illustration of DIS.

Fig. 2. Illustrations of OIS and DIS.

With OIS, PBS acquires interference information, including
data and CSI, from MBS via inter-BS collaboration and
by PUE’s estimation and feedback, respectively. PBS then
generates a duplicate of the interference and sends it along
with the desired signal. The former is used for interference
steering at PUE, whereas the latter carries the payload. The
received signal at PUE then becomes:
√
√
e
r0 = P0e − POIS
H0 p0 x0 + P1e H10 p1 x1
√
(2)
e
+ POIS
H0 pOIS x1 + n0
e
where POIS
= POIS 10−0.1L0 . POIS represents the power
overhead of OIS, and pOIS is the precoder for the steering

signal. We first define the directions of the desired signal and
the original interference combined√with the √steering signal
P e H10 p1 + P e H0 pOIS
H0 p0
as ds = ∥H
and di+st = ∥√P1e H p +√POIS
,
e
H0 pOIS ∥
0 p0 ∥
10 1
1
OIS
respectively. Then, the original interference should be steered
in the direction orthogonal to the desired signal by letting
⟨ds , di+st ⟩ = 0. As shown in Fig. 2(a), both the disturbance
(i) and the steering signal (st ), can be decomposed into an
in-phase component and a quadrature component, denoted by
the superscripts In and Q, respectively, w.r.t. the intended
Q
transmission s, i.e., i = iIn + iQ and st = sIn
t + st . When
In
In
st = −i , OIS is realized. Furthermore, since the length of a
vector indicates the signal’s strength, OIS with the minimum
Q
power overhead is achieved when st = sIn
t , i.e., st = 0. Hence,
Q
in order to reduce√power cost, we let st = 0. It can be easily
seen that iIn = P1e PH10 p1 where P = ds (dTs ds )−1 dTs
denotes the projection√matrix. To implement
√ OIS, the steering
e
H0 pOIS = − P1e PH10 p1 . This
signal should satisfy POIS
equation can be decomposed into H0 pOIS = −αPH10 p1 and
e
POIS
= βP1e where αβ = 1, so that we can get pOIS =
−1
−αH0 PH10 p1 . Note that ∥pOIS ∥ = 1 is not guaranteed, i.e.,
pOIS affects the power cost of OIS.
When NTi > NRi (i = 0, 1), the inverse of H0 should be replaced by its Moore-Penrose pseudo-inverse. The mechanism
can then be generalized. In addition, when the interference is
too strong, P0 may not be sufficient for OIS, in such a case,
we can simply switch to the non-interference management
(non-IM) mode, e.g., matched filtering (MF) at the victim
receiver, or other IM schemes with less or no transmit power
consumption such as zero-forcing reception.
(1)
By adopting f0 = u0 as the receive filter, the SE of PUE
with OIS can be computed as:
⎧
(1) ⎫
e
⎪
(P0e − POIS
)[λ0 ]2 ⎪
⎪
⎪
⎬,
(3)
cOIS
=
log
⎨
1
+
2
0
⎪
⎪
σn2
⎪
⎪
⎭
⎩
(1)
where λ0 is the largest singular value of H0 , indicating the
amplitude gain of the principal spatial sub-channel.
From Eq. (3), we can see that although the original interference is steered into the orthogonal direction of the desired
signal and the disturbance to the intended transmission is
completely eliminated, it accompanies a transmit power loss,
e
POIS
, degrading the received desired signal strength. One
can then raise a question: “is the orthogonal-IS always necessary/worthwhile?” To answer this question, we may adjust
both the direction and strength of the steering signal adaptively
to implement dynamic IS. Note, however, that in order to
minimize the transmit power overhead, the steering signal
should be opposite to the spatial feature of the desired signal.
Thus, only the strength of steering signal needs to be adjusted.
In what follows, we generalize the OIS to DIS by introducing
a coefficient ρ ∈ (0, 1] called the steering factor, representing
the portion of in-phase component of the disturbance w.r.t.
the desired signal to be mitigated. When ρ = 1, orthogonalIS is realized, while DIS approaches non-IM as ρ → 0.
Without ambiguity, we adopt st to represent√the dynamic
steering signal. Then, we have st = −ρiIn = −ρ P1e PH10 p1 .

As illustrated by Fig. 2(b), when ρ < 1, interference i is
steered into a direction not orthogonal to ds , i.e., iIn is not
completed eliminated. Then, the steered interference becomes
i + st = (1 − ρ)iIn + iQ , whose projection on ds is non-zero,
i.e., provided that ρ < 1, a residual interference, expressed as
iR = iIn + st = (1 − ρ)iIn , exists.
Similarly to the discussion about OIS, to implement DIS,
the following equation should hold:
√
√
e
(4)
PDIS
H0 pDIS = −ρ P1e PH10 p1 .
For clarity of exposition, we normalize the precoder so
that the direction and strength requirements for DIS could
be decoupled from each other. Then, the expression of DIS
implementation is given as:
−1
pDIS = −H−1
0 PH10 p1 /∥H0 PH10 p1 ∥
,
(5)
e
2
PDIS
= ρ2 P1e ∥H−1
PH
p
10 1 ∥
0
e
where pDIS is the precoder for steering signal and PDIS
denotes the power overhead for DIS at PBS, i.e., PDIS ,
incorporated with path loss 10−0.1L0 .
The received signal at PUE with DIS is then
√
√
e
H0 p0 x0 + P1e (1−ρ)PH10 p1 x1 +n0 , (6)
r0 = P0e − PDIS

{

where the second term on the RHS of Eq. (6) indicates the
residual interference, √
which is the in-phase component of the
original interference
P1e H10 p1 x1 combined with the steer√ e
ing signal ρ PDIS H0 pDIS x1 w.r.t. the desired transmission.
(1)
By employing f0 = u0 as the receive filter, the achievable
SE of PUE employing DIS can then be calculated as:
⎧
(1) ⎫
e
⎪
(P0e − PDIS
)[λ0 ]2 ⎪
⎪
⎪
cDIS
=
log
⎨
1
+
⎬,
(7)
2
0
⎪
⎪
σn2 + IR
⎪
⎪
⎩
⎭
where IR = P1e ∥f0H (1 − ρ)PH10 p1 ∥2 denotes the strength of
residual interference after post-processing at PUE.
Based on the above discussion, it can be easily seen that
IR = 0 when ρ = 1, i.e., DIS becomes OIS. So, DIS includes
OIS as a special case, making it more general.
B. Optimization of Steering Factor ρ
In what follows, we will discuss the existence of the
optimal ρ, denoted by ρ∗ , with which PUE’s SE can be
maximized with limited P0 . Based on the Shannon’s equation,
we can instead optimize the signal-to-interference-plus-noise
ratio (SINR) of PUE, denoted by ϕ0 .
Substituting Eq. (5) into Eq. (7), we can obtain ϕ0 as:
(1)

ϕ0 =

2
2
(P0e − ρ2 P1e ∥H−1
0 PH10 p1 ∥ ) [λ0 ]

(1 − ρ)2 P1e ∥f0H PH10 p1 ∥2 + σn2
(1)

(1)

P e [λ ]2 − ρ2 P1e ∥g∥2 [λ0 ]2
= 20 e 0 2
ρ P1 ∣χ∣ − (2ρ − 1)P1e ∣χ∣2 + σn2

,

(8)

H
where g = H−1
0 PH10 p1 and χ = f0 PH10 p1 .
Eq. (8) can be simplified as:

ϕ0 =

A − ρ2 B
C − ρD + ρ2 E

(9)

(1)

(1)

where A = P0e [λ0 ]2 , B = P1e ∥g∥2 [λ0 ]2 , C = P1e ∣χ∣2 + σn2 ,
D = 2P1e ∣χ∣2 and E = P1e ∣χ∣2 . Note that all of these coefficients
are positive.
Next, we elaborate on the existence of ρ∗ under the P0
constraint, with which ϕ0 is maximized. By substituting g into
Eq. (5), we can see that when P0e > P1e ∥g∥2 , PBS has enough
power to steer the interference into the direction orthogonal
to the desired signal, i.e., OIS is achievable. Otherwise, when
P0e ≤ P1e ∥g∥2 , the maximum of ρ, denoted by √
ρmax is limited

Then, we can get:
1
1
P0e
P0e
)+∣ −
∣
ρ∗+ > ( +
e
2
2 2P1 ∥g∥
2 2P1e ∥g∥2
.
P0e
⎧
⎪
≤ 12
⎪ 1,
2P1e ∥g∥2
e
= ⎨ P0e
P0
1
⎪
⎪
⎩ P1e ∥g∥2 , 2P1e ∥g∥2 > 2

P0e
should not be less
2P1e ∥g∥2
e
P
P0e
1
∗
0
than or equal to 12 . However, when 2P e ∥g∥
2 > 2 , ρ+ > 2P e ∥g∥2
1
1
is equivalent to√
ρ∗+ > 1. As a result, ρ∗+ ∉ (0, ρmax ] where
P0e
∗
ρmax = min (1, P e ∥g∥
2 ), i.e., ρ+ is not acceptable.
1
√
(BC+AE)− ∆
∗
, since ABD2 > 0, BC +
As for
BD
√ ρ− =
AE > ∆ holds, thus justifying ρ∗− > 0. Then, we prove
ρ∗− < ρmax as follows.
First, we define a function g(C) =
√

Note that ρ∗+ ≤ 1, and hence

Pe

0
is the
by the PBS’s transmit power. In such a case,
P1e ∥g∥2
maximum portion of the in-phase component of the original
interference w.r.t. the desired signal that can be mitigated with
P0 . Based on the above√discussion, we have ρ∗ ∈ (0, ρmax ]

where ρmax = min (1,

P0e
).
P1e ∥g∥2
∗

In what follows, we first

prove the solvability of ρ , and then show the quality of the
resulting solution(s).
By computing the derivative of ϕ0 to ρ and setting it to 0,
we get:
BD 2
ρ − (BC + AE)ρ + AD
2
2
= 0.
(10)
1
2 E)2
(C
−
Dρ
+
ρ
2

(BC + AE)√ − ∆. Since the derivative of g(C) to C,
√
< 0, g(C) is a monotonically
g ′ (C) = B ∆−B(BC+AE)
∆
decreasing function of variable C. Let C ′ = PTe1 ∣χ∣2 , then
C = σn2 +P1e ∣χ∣2 > C ′ , thus leading to g(C) < g(C ′ ). Similarly
to the derivations of Eqs. (13)–(15), we get:

Since the denominator cannot be 0, we only need to solve
AD
BD 2
ρ − (BC + AE)ρ +
=0
(11)
2
2
which is a quadratic equation with one unknown. Let’s define
2
2
∆
√ = (BC + AE) −√ABD . Since ∆ = (BC√+ AE +
ABD)(BC + AE − ABD), and BC + AE +
√ ABD is
positive, we only need to show that BC + AE − ABD > 0.
The proof is given below by Eq. (12).
Based on the √
above discussion, we can obtain two solutions
(BC+AE)± ∆
∗
ρ± =
. We then need to verify the qualification
BD
of the two solutions ρ∗± . We first investigate
the feasibility of
√
∆
the larger solution ρ∗+ = (BC+AE)+
.
The
first term of ρ∗+
BD
can be rewritten as:
(1)

1

1
P0e
= +
2 2P1e ∥g∥2

0

1

The second term of ρ∗+ is:
√
√
∆
(BC + AE)2 A
=
−
BD
B 2 D2
B
¿
2
Á 1
P0e
P0e .
À( +
>Á
)
−
e
e
2 2P1 ∥g∥2
P1 ∥g∥2
e
1
P0
=∣ −
∣
2 2P1e ∥g∥2
BC + AE −

√

ρ∗−

⎧ 1,
g(C) g(C ′ ) ⎪
⎪
=
<
= ⎨ P0e
⎪ e 2,
BD
BD
⎪
⎩ P1 ∥g∥

P0e
2P1e ∥g∥2
P0e
2P1e ∥g∥2

≥
<

1
2
1
2

.

(16)

Pe

0
Eq. (16) is equivalent to ρ∗− < ρ2max = min (1, P e ∥g∥
2 ). Since
1
2
2
∗
0 < ρmax ≤ 1, ρmax ≤ ρmax holds, thus proving ρ− < ρmax .
Finally, we prove that ρ∗− could achieve the maximum ϕ0 .
ρ2 −(BC+AE)ρ+ AD
is
Since it can be proved that h(ρ) = BD
2
2
a monotonically decreasing function of variable ρ ∈ (0, ρmax ],
when 0 < ρ < ρ∗− , we get h(ρ) > h(ρ∗ ) = 0. Similarly, when
ρ∗− < ρ < ρmax , h(ρ) < 0 can be derived. Thus, ρ∗− corresponds
to the maximum ϕ0 . The optimal steering factor is calculated
as ρ∗ = ρ∗− .

IV. G ENERALIZATION OF DIS

(1)

BC + AE P1e ∥g∥2 [λ0 ]2 P1e ∣χ∣2 + P0e [λ0 ]2 P1e ∣χ∣2
>
(1)
BD
2P e ∥g∥2 [λ ]2 P e ∣χ∣2

(15)

A. Generalized Number of Interferences
.

So far, we have assumed that the MBS sends a single data
stream to MUE, i.e., only one interference is imposed on the
PUE. When multiple desired signals are sent from a MBS,
the proposed DIS can be extended as follows. Since picocells
are deployed within the coverage of a macrocell, interferences
from the other MBSs are negligible. For clarity of presentation,
Fig. 3 shows a two-interference situation as an example, where
i1 and i2 are the interferences. Only one desired signal is
considered.
As shown in this figure, each interference can be decomposed into an in-phase component and a quadrature component
w.r.t. the desired signal. Then, DIS can be applied to each

(13)

(14)

√
(1)
(1)
(1)
(1)
ABD = P1e ∥g∥2 [λ0 ]2 (σ02 + P1e ∣χ∣2 ) + P0e [λ0 ]2 P1e ∣χ∣2 − 2 P0e [λ0 ]2 P1e ∥g∥2 [λ0 ]2 P1e ∣χ∣2
(1)

(1)

1

3

(1)

> (P1e )2 ∥g∥2 [λ0 ]2 ∣χ∣2 + P0e P1e [λ0 ]2 ∣χ∣2 − 2(P0e ) 2 (P1e ) 2 [λ0 ]2 ∥g∥∣χ∣2
(1)

1

1

2

= P1e [λ0 ]2 ∣χ∣2 [(P1e ) 2 ∥g∥ − (P0e ) 2 ] ≥ 0

.

(12)

i1Q
i2Q

st ,2

st ,1

For simplicity, we use the function f (ρ1 , ρ2 ) to denote cDIS
0
under N = 2 in the following description. The above algorithm
can be extended to the case of N > 2. Due to space limitation,
we do not elaborate on this any further in this paper.

i1
i2

O

i1In

i2In s

Fig. 3. Generalization of the number of interferences.

interference separately, following the processing as described
in Section III.
For N > 1 interferences, the received signal at the victim
PUE with DIS can be expressed as:
¿
N
N √
Á
ÀP e − ∑ P e
r =Á
H p x + ∑ Pe H p x
0

0

DIS,n

n=1
N

+∑

√

0 0 0

1,n

10 1,n 1,n

n=1

,

e
PDIS,n
H0 pDIS,n x1,n + n0

n=1

(17)
where x1 = [x1,1 , ⋯, x1,n , ⋯, x1,N ] is the transmit data vector
of MBS. The transmission of x1,n causes the interference term
√
e
e H p
P1,n
10 1,n x1,n to the PUE, in which P1,n is the transmit
power for x1,n , i.e., P1,n , incorporated with the path loss
e
e
10−0.1L10 . p1,n denotes the precoder for x1,n . ∑N
n=1 P1,n = P1
holds. PBS generates a duplicate of this interference with the
e
power overhead PDIS,n where PDIS,n
= PDIS,n 10−0.1L0 and
the precoder pDIS,n so as to adjust the interference to an
appropriate direction at the victim PUE.
Similarly to the derivation of Eq. (5), we can obtain the DIS
design for the n-th interfering component as:
−1
p
= −H−1
0 PH10 p1,n /∥H0 PH10 p1,n ∥
{ DIS,n
,
e
2 e
−1
PDIS,n = ρn P1,n ∥H0 PH10 p1,n ∥2

Algorithm 1
1: Take the derivative of f (ρ1 , ρ2 ) to ρ1 and ρ2 respectively,
1 ,ρ2 )
to obtain fρ′ n (ρ1 , ρ2 ) = ∂f (ρ
where n = 1, 2.
∂ρn
2: Compute the stationary point (ρ̃1 , ρ̃2 ) of f (ρ1 , ρ2 ) by
solving the equation fρ′ n (ρ1 , ρ2 ) = 0. We define set Φ
consisting of (ρ̃1 , ρ̃2 ).
3: Calculate the second-order derivative of f (ρ1 , ρ2 ) at
the stationary point (ρ̃1 , ρ̃2 ), i.e., fρ′′1 ,ρ2 (ρ̃1 , ρ̃2 ) =
∂f (ρ1 ,ρ2 )
∣
. For clarity of exposition, we de∂ρ1 ∂ρ2 ρ1 =ρ̃1 ,ρ2 =ρ̃2
fine A = fρ′′1 ,ρ2 (ρ̃1 , ρ̃2 ). Similarly, we define variables
B = fρ′′1 ,ρ1 (ρ̃1 , ρ̃2 ) and C = fρ′′2 ,ρ2 (ρ̃1 , ρ̃2 ).
4: Check whether the stationary point is an extreme point or
not. If A2 − BC < 0 and B < 0, (ρ̃1 , ρ̃2 ) is an extreme
point; otherwise not. We can obtain the set of extreme
points and the value of f (ρ1 , ρ2 ) at each extreme point
correspondingly. We define the extreme value set as Ω.
5: Since both ρ1 and ρ2 range from 0 to 1, i.e., ρn ∈ (0, 1],
the maximum value of f (ρ1 , ρ2 ) may exist at the boundary
points. We define set F = f (ρ1 , ρ2 ) ∣ρn ∈{0,1},n=1,2 .
∗
6: Determine the optimal (ρ∗
1 , ρ2 ) outputting the maximum
value of f (ρ1 , ρ2 ) ∣ρ1 =ρ∗1 ,ρ2 =ρ∗2 by searching all the elements in sets Ω and F.

B. Generalized Number of Desired Data Streams
(18)

where P represents the projection matrix depending only on
the spatial feature of the desired transmission, with which
we can calculate the in-phase component of the interference
caused by the transmission of x1,n w.r.t. the intended signal.
ρn is the steering factor for the steering signal carrying x1,n .
One should note that when there are multiple interferences,
it is difficult to determine the optimal steering factors for all
the interfering components. However, we can allocate a power
budget P0,n , satisfying ∑N
n=1 P0,n < P0 , to each interference,
and then by applying DIS to each disturbance under its power
budget constraint, a vector of n sub-optimal steering factors is
achieved. P0,n can be assigned with the same value or based
on the strength of interferences. The achievable SE of the PUE
can then be calculated as:
(1) 2 ⎫
⎧
e
⎪
(P0e − ∑N
⎪
⎪
n=1 PDIS,n ) [λ0 ] ⎪
DIS
⎬
(19)
c0
= log2 ⎨1 +
N
2
⎪
⎪
σn + ∑n=1 IR,n
⎪
⎪
⎩
⎭
where IR,n = P1e ∥f0H (1 − ρn )PH10 p1,n ∥2 (n = 1, ⋯, N ) is
the strength of the n-th residual interference to the intended
(1)
transmission. f0 = u0 is the receive filter for data x0 .
To further elaborate on the extension of the proposed
scheme, we provide below an algorithm for N = 2, with which
the optimal ρn can be determined, maximizing the system SE.

We now generalize the number of desired signals, denoted
by M , sent from PBS to its PUE. For clarity of exposition,
we take M = 2 and the number of interferences N = 1 as an
example as shown in Fig. 4. However, this discussion can be
readily extended to more general parameter settings. As can be
seen from the figure, the interference i forms a plane with each
of the desired signals sm where m = 1, 2. The projection of i
onto sm is denoted by iIn
m (in-phase component), whereas the
quadrature component is iQ
m . By applying DIS to each (i, sm )
pair, a set of steering signals can be determined.
i1Q

i

i2Q

st ,1
st ,2

O

i2In

i1In

s2

s1

Fig. 4. Generalization of the number of desired signals.

Since multiple data streams are sent from PBS to PUE via
mutually orthogonal eigenmodes/subchannels, and the steering
signal is opposite to the spatial feature of the desired transmission it intends to protect, an arbitrary steering signal, say st,m ,
is orthogonal to any of the other desired signals sm′ where

m′ ≠ m. Hence, no additional interference will be created by
the steering signal st,m .
Based on the above discussion, the number of steering
signals is equal to that of the desired data streams M . Thus,
the mixed signal received at PUE can be expressed as:
M √
√
e
e
− PDIS,m
H0 p0,m x0,m + P1e H10 p1 x1
r0 = ∑ P0,m
m=1
M √

+ ∑

e
PDIS,m
H0 pDIS,m x1 + n0

m=1

(20)
where x0 = [x0,1 , ⋯, x0,m , ⋯, x0,M ] is the transmit data vector
e
of PBS. P0,m
denotes the transmit power budget for x0,m ,
i.e., P0,m , incorporated with the path loss 10−0.1L0 . PDIS,m
is the power cost for steering the interference away from
e
the m-th desired signal, and PDIS,m
= PDIS,m 100.1L0 . p0,m
and pDIS,m represent the precoders for x0,m and its steering
signal, respectively. Similarly to the derivation of Eqs. (5) and
(18), the DIS solution for multi-desired-signal situation can
be readily obtained. In such a case, the achievable SE of PUE
can be expressed as:
(m)
⎧
e
e
M
⎪
⎪
) [λ0 ]2 ⎫
(P0,m
− PDIS,m
⎪
⎪
⎬ . (21)
cDIS
=
log
⎨
1
+
∑
2
0
2
⎪
⎪
σn + IR,m
⎪
⎪
m=1
⎭
⎩
(m)
λ0 denotes the amplitude gain of the m-th desired transmission from PBS to PUE. The residual interference IR,m =
(m)
H
P1e ∥f0,m
(1 − ρm )Pm H10 p1 ∥2 where f0,m = u0
is the
receive filter for data x0,m . The projection matrix Pm =
H p
ds,m (dTs,m ds,m )−1 dTs,m where ds,m = ∥H00 p0,m
.
0,m ∥
e
It should be noted that the optimal PDIS,m , or equivalently
e
e
e
ρm is dependent on P0,m
where ∑M
m=1 P0,m = P0 holds. Thus,
different power allocations will yield different DIS solutions.
For example, P0 can be equally allocated to the M intended
data transmissions, or in terms of the quality of subchannels
and/or the strength of interference imposed on each desired
signal. Then, suboptimal performance w.r.t. the transmission
from PBS to PUE is achieved. How to jointly determine the
optimal P0,m and PDIS,m is our future work.
C. Generalized Number of PBSs and PUEs
We discuss the generalization of the number of PBSs deployed in the coverage of a macrocell and the number of PUEs
served by each PBS. As mentioned before, PBSs are installed
by the network operator. Inter-picocell interference could
therefore be effectively avoided by the operator’s planned
deployment or resource allocation. Even when inter-picocell
interference exists, our scheme can be directly applied by
treating the interfering PBS as the MBS in this paper, and DIS
is implemented at the PBS associated with the victim PUE.
It should be noted that the proposed DIS is applicable to the
scenario with asymmetric interferences. Otherwise, concurrent
data transmissions should be scheduled or other schemes
should be adopted to address the interference problem, which
is beyond the scope of this paper. As for the multi-PUE case,
each PUE can be assigned an exclusive channel so as to
avoid co-channel interference, which is consistent with various

types of wireless communication systems, such as WLANs. In
summary, with an appropriate system design, the proposed DIS
can be applied to the system with multiple PBSs and PUEs.
V. E VALUATION
We evaluate the performance of the proposed mechanism
using MATLAB. We set d = 300m, D = 3000m, P0 = 23dBm
and P1 = 46dBm [14]. The path loss is set to L10 =
128.1+37.6 log10 (η10 /103 ) dB and L0 = 38+30 log10 (η0 ) dB
where η0 ≤ d and η10 ≤ D. Since L0 and L10 are dependent on
the network topology, P0e ranges from −89dBm to 23dBm,
whereas P1e varies between −100dBm to 46dBm. For clarity
of presentation, we adopt γ̄ = 10lg(γ) where γ = P1e /σn2 . We
also define ξ = P0e /P1e . Then, based on the above parameter
settings, ξ ∈ [−135, 123] dB. Note, however, that we obtained
this result for extreme boundary situations, making its range
too wide to be useful. Without specifications, the simulation
is done under NT0 = NT1 = NR0 = 2 antenna configuration. However, same conclusion can be drawn with various
parameter settings. In practice, a PBS should not be deployed
close to MBS and mobile users may select an access point
based on the strength of reference signals from multiple access
points. Considering this practice, we set ξ ∈ [0.1, 100] in our
simulation. There are M desired signals and N interferences.
In the following simulation, when the power overhead of an
IM scheme exceeds P0 at the victim Tx, we simply switch
to non-IM mode, i.e., matched filtering (MF) is employed by
(m)
letting f0,m = u0 , while the interference remains unchanged.
Fig. 5 shows two samples of the relationship between ρ
and PUE’s achievable SE. The interference shown in Fig. 5(a)
is relatively weak, and hence the transmit power of PBS is
sufficient for OIS, i.e., ρmax can be as large as 1. In Fig. 5(b),
since the interference is strong, and hence, when ρ > ρmax
where ρmax < 1, there won’t be enough power for PBS to
realize OIS. In such a case, we simply switch off IS and
adopt non-IM. In both figures, the optimal ρ, denoted by
ρ∗ , is computed as in Section III, which corresponds to the
maximum SE. We can conclude from Fig. 5 that in order
to better utilize the transmit power for both IS and data
transmission, it is necessary to intelligently determine the
appropriate strength of the steering signal, or equivalently, the
direction into which the interference is steered.
Fig. 6 plots the PUE’s average SE versus ρ for different
ξ. The average ρ∗ , marked by pentagram, which corresponds
to the PUE’s maximum SE, grows as ξ increases. When ξ
gets too high, a large portion of interference is preferred to be
mitigated. As shown in the figure, given ξ = 100, the average
ρ∗ is approximately 0.9. In addition, since the strength of
the desired signal relative to the interference grows with an
increase of ξ, the PUE’s SE performance improves with ρ.
Fig. 7 shows the average ρ∗ versus γ̄ under M = N = 1 and
different ξ. As the figure shows, with fixed ξ, the average ρ∗
grows with an increase of γ̄. This is because the interference
gets stronger with increasing γ̄, and hence, to achieve the
maximum SE, ρ∗ should increase, i.e., more interference

3
2.5
2
0.2

0.4

1
0.95
0.9
0.85

0.80.80

0.6

ρ

-1

4

[2,4,2]
[3,4,2]
[4,4,2]
[5,4,2]
[6,4,2]
Maximum cDIS
0

1.5
0

1

3.8
3.6
3.4
ρ*
3.2
3
2.8

2.6
0.4 0 0.6
ρ

10.2

5

[4,2,2]
[4,3,2]
[4,4,2]
[4,5,2]
[4,6,2]
Maximum cDIS
0

-1

0.8

(a) NT0 is variable.

ρmax=1

0.2
0.8

4.5

-1

0.6

ρ

Spectral efficiency (bit ⋅ s ⋅ Hz )

3.5

0.4

-1

-1

4

Spectral efficiency (bit ⋅ s ⋅ Hz )

4.5

-1

5

0.2

Spectral efficiency (bit ⋅ s ⋅ Hz )

2.6
0

Spectral efficiency (bit ⋅ s-1 ⋅ Hz-1)

S

2.8

10.4

ρ

0.6

0.8

4
3.5
3

2.5
0

1

[4,4,2]
[4,4,3]
[4,4,4]
Maximum cDIS
0

0.2

0.4

ρ

0.6

0.8

(c) NR0 is variable.

(b) NT1 is variable.

0.95
0.9
0.85
0.8
0

ρmax=1
0.2

0.4

0.6

ρ

0.8

-1

1.5

-1

ρ*

Spectral efficiency (bit ⋅ s ⋅ Hz )

1

Spectral efficiency (bit ⋅ s-1 ⋅ Hz-1)

Spectral efficiency (bit ⋅ s-1 ⋅ Hz-1)

5
Fig. 8. SE of PUE vs. ρ under
γ̄ = 5dB, ξ = 1, M = N = 1, and different antenna settings.
[2,4,2]

1

0.5

1

0
0

0.2

4.5 ρ* [3,4,2]
[4,4,2]
[5,4,2]
4
[6,4,2]
Maximum cDIS
0
3.5
ρ=0
3
2.5

0.4 2

ρmax
0.6

ρ

0.8

1.5

0
0.2
(b) Strong interference.

(a) Weak interference.

1
0.4

4
-1

Spectral efficiency (bit ⋅ s ⋅ Hz )

1

-1

Spectral efficiency (bit ⋅ s-1 ⋅ Hz-1)

Fig. 5. SE of PUE vs. ρ under γ̄ = 0dB, M = N = 1, and ξ = 1.
1.5

0.5

0
0

0.2

ρ*

Maximum cDIS
0

3.5

ρ=0

3
2.5

ρmax
20.4

ρ

0.6

0.8

1.5
1
0

1
ξ=0.3,0.5,0.8,1,5,10,20,50,100

0.2

0.4

ρ

0.6

0.8

1

express the antenna configuration. In Fig. 8(a), NT1 and NR0
are fixed, and NT0 varies from 2 to 6. Since the transmit array
gain of the desired signal grows with an increase of NT0 ,
meaning that the desired signal becomes stronger relative to
the interference, and more interference can be eliminated by
DIS with the same power overhead as NT0 grows, both the
average ρ∗ and the achievable SE improve as NT0 increases.
In Fig. 8(b), NT0 and NR0 are fixed while NT1 ranges from 2
0.8
toρ 6.0.6Although
NT11 varies, MBS causes random interferences
to PUE as the PUE adopts f0 to decode x0 regardless of
the interference channel H10 . Hence, both ρ∗ and the PUE’s
average SE under different NT1 remain similar. In Fig. 8(c),
NT0 and NT1 are fixed while NR0 ranges from 2 to 4.
With such antenna settings, since NT0 and NT1 are fixed,
the processing gain with the transmit antenna array doesn’t
change for the desired signal or the interference. However, as
NR0 increases, the receive gain for the intended signal grows
as the filter vector f0 , an NR ×1 vector designed to match H0 .
As a result, the desired signal, relative to the interference, after
the receive filtering becomes stronger with an increase of NR0 ,
thus enhancing the PUE’s SE.

Fig. 6. SE of PUE vs. ρ under γ̄ = 5dB, M = N = 1, and different ξ.

0.9
0.8

Average ρ*

1

0.7

( P > P0 )
Prob
Probability

imposing onto the desired signal should be mitigated. Given
the same γ̄, the average ρ∗ grows with an increase of ξ, which
is consistent with Fig. 6.

0.6
0.5
0.4
0.3

0.9

0.2

0.8

0.1

0.7

0
0

ξ=0.1,0.5,1,10,20,50,100

0.6

[1 1 DIS]
[1 1 OIS]
[1 1 IN]
[2 1 DIS]
[2 1 OIS]
[2 1 IN]
[1 0.5 DIS]
[1 0.5 OIS]
[1 0.5 IN]

5

10
γ .(dB
(dB))

15

20

Fig. 9. P rob(PM > P0 ) vs. γ̄ under M = 1, and different ξ and N .

0.5
0.4
0.3
0

5

10

(dB))
γ .(dB

15

20

Fig. 7. Average ρ∗ vs. γ̄ under M = N = 1, and different ξ.

Fig. 8 plots PUE’s average SE along with ρ under different
antenna settings. We use a general form [NT0 NT1 NR0 ] to

Fig. 9 shows the probabilities that the power overhead of
DIS, OIS and IN are greater than P0 , i.e., P rob(PM > P0 )
where PM denotes the power cost at PBS with IM scheme
M. We use a general form [N ξ M] to denote the parameter
settings for different mechanisms, where N is the number of
interferences. ξ = P0e /P1e . P1e is the total power of the interferer, i.e., P1 , incorporated with path loss. P rob(PM > P0 ) is

1

Spectral efficiency (bit ⋅ s-1 ⋅ Hz-1)

Prob(P
> 0P))
( PM>P
Prob

0.8
0.6
0.4
[2,1,IN]
[1,1,IN]
[2,1,OIS]
[1,1,OIS]

0.2
0
0

0.5
1
1.5
Power normalized
by P0
P

2

7
6
5
4

8

DIS
OIS
IS w/ ρ = 0.7
IS w/ ρ = 0.5
IS w/ ρ = 0.3
ZF
ZFBF
IN

Spectral efficiency (bit ⋅ s-1 ⋅ Hz-1)

8

1

3
2
1
0
0

5

10

γ (dB)
.(dB)

15

20

7
6
5

[1,2,DIS]
[1,2,OIS]
[1,2,IN]
[2,1,DIS]
[2,1,OIS]
[2,1,IN]

4
3
2
1
0
0

5

10
γ (dB)

15

20

Fig. 10. P rob(PM > P̄ ) vs. P̄ under ξ = 1, M = 1, Fig. 11. SE of PUE vs. γ̄ with various IM schemes under Fig. 12. SE of PUE vs. γ̄ with various IM schemes under
M = N = 1, and ξ = 1.
ξ = 1 and different M and N .
and different N .

shown to increase as ξ decreases since a small ξ results in a
strong interference, incurring higher PM . P rob(PIN > P0 ) is
notably higher than P rob(POIS > P0 ) and P rob(PDIS > P0 ),
and P rob(PDIS > P0 ) is less than P rob(POIS > P0 ). This is
because IN consumes more power than DIS and OIS. For DIS,
the power overhead increases as γ̄ grows and approaches OIS
when γ̄ becomes too large. One may note in Fig. 9 that with
fixed ξ and P1e , P rob(PIN > P0 ) with N = 2 interferences
is higher than that with a single interference. However, as for
DIS and OIS, a larger N produces lower P rob(PM > P0 ).
This phenomenon can be explained by the results illustrated
in Fig. 10.
Fig. 10 plots the distribution of P̄M for an arbitrary γ̄
where P̄M and P̄ represents for PM and the power value
P normalized by P0 . Since P rob(P̄DIS > P̄ ) varies with γ̄
whereas P rob(P̄IN > P̄ ) and P rob(P̄OIS > P̄ ) do not, for
simplicity, we only study IN and OIS. As shown in the figure,
P rob(P̄IN > P̄ ) with N = 2 is no less than that with N = 1.
As for OIS, when P̄ < 0.5, P rob(P̄OIS > P̄ ) with N = 2 is
larger than that with N = 1. However, as P̄ grows larger than
0.5, 2 interferences incur statistically less power overhead.
When P̄ = 1, P rob(P̄M > P̄ ) with different schemes shown
in Fig. 10 is consistent with the results given in Fig. 9.
Fig. 11 shows the PUE’s average SE with different IM
schemes. Besides IN, OIS and DIS, zero-forcing (ZF) reception and zero-forcing beamforming (ZFBF) are also simulated.
With ZF reception, a receive filter being orthogonal to the
unintended signal is adopted so as to nullify the interference
at PUE, but an attenuation w.r.t. the desired signal results.
As for ZFBF, we let PBS adjust its beam so that the desired
signal is orthogonal to the interference at the intended receiver.
It should be noted that for either IN, IS with fixed ρ, OIS or
DIS, when the power overhead at PBS exceeds P0 , i.e., the IM
scheme is unavailable, we simply switch to ZF reception. As
shown in the figure, DIS yields the best SE performance. When
γ̄ is low, noise is the dominant factor affecting the PUE’s
SE. Therefore, IS with fixed ρ (ρ < 1) yields similar SE to
OIS. Moreover, although DIS can achieve the highest SE, its
benefit is limited in the low γ̄ region. As γ̄ grows large, ρ∗
increases accordingly, and hence IS with large ρ exceeds that
with small ρ in SE. In addition, OIS gradually outperforms
those IS schemes with fixed ρ as γ̄ increases. Moreover, by

intellectually determining ρ∗ , the advantage of DIS becomes
more pronounced with an increase of γ̄. Although IN yields
more power overhead than IS with fixed ρ, DIS and OIS,
with the help of ZF, IN yields slightly higher SE than ZF
reception. As for ZFBF, more desired signal power loss results
as compared to ZF, thus yielding inferior SE performance.
Fig. 12 plots the PUE’s average SE with various mechanisms under different numbers of interferences and desired
signals. We use a general form [M N M] to denote the
parameter settings, where M represents the number of desired
signals, N is the number of interferences, and M denotes the
IM schemes. When M > 1, equal power allocation is adopted,
e
i.e., P0,m
(m = 1, ⋯, M ) in Eq. (21) is P0e /M . As shown in the
figure, OIS achieves the highest SE among the three schemes,
whereas IN yields the lowest SE. Since ρ∗ approaches 1 as γ̄
increases, with the same M and N , DIS becomes OIS when γ̄
grows too large. Given fixed ξ, IN yields better SE when there
are 2 interfering signals than the single interference case. As
for DIS and OIS, SE with 2 interferences is lower than that
with one disturbance. This is consistent with the results shown
in Figs. 9 and 10.
VI. C ONCLUSION
In this paper, we proposed a new interference management scheme, called Dynamic Interference Steering (DIS), for
heterogeneous cellular networks. By intelligently determining
the strength of the steering signal, the original interference
is steered into an appropriate direction. DIS can balance the
transmit power used for generating the steering signal and that
for the desired signal’s transmission. Our in-depth simulation
results show that the proposed scheme makes better use of the
transmit power, and enhances users’ spectral efficiency.
ACKNOWLEDGMENT
This work was supported in part by NSFC (61173135, U14
05255); the 111 Project (B08038); the Fundamental Research
Funds for the Central Universities (JB171503). It was also
supported in part by the US National Science Foundation
under Grant 1317411.
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