IEEE INTERNET OF THINGS JOURNAL, VOL. 5, NO. 3, JUNE 2018 1957
Degrees of Freedom of the Circular Multirelay
MIMO Interference Channel in IoT Networks
Wenjing Lv, Rui Wang , Member, IEEE,JunWu , Senior Member, IEEE, Jun Xu, Ping Li, and Jianwu Dou
Abstract—In this paper, we study the degrees of freedom (DoF)
of a new network information flow model named the circular
multirelay multiple-input multiple-output interference channel
(CMMI). In this model, there are two clusters and each of them
contains three users. Each user equipped with M antennas in
one cluster intends to deliver data streams to another user in the
same cluster in a circular one-way transmission via the common
distributed KN-antenna relay nodes. The CMMI network model
can be considered as a basic component to construct the compli-
cated Internet of Things networks. By assuming linear processing
at the users and the relays, we show that the original analy-
sis of DoF comes down in finding solutions of some nonlinear
matrix equations with rank constraints. Toward this end, by using
linear precoding and post-processing techniques, we propose
two different approaches to solve the nonlinear matrix equa-
tions based on different antenna configurations. We show that a
DoF of max{min{M,(
√
6K/12)},min{(M/3), (KN/2)}} is achiev-
able for ∀(M/N) ∈ (0, +∞). In addition, to assess the optimal
DoF, the cut-set approach is used for deriving the DoF upper
bound by innovatively separating certain users to form two-pair
two-way relay channels. We show that the DoF of CMMI is
upper bounded by max{min{M,(KN/3)}, min{(2M/3), (KN/2)}}.
By combining the achievable DoF and the upper bound, we
finally show that the optimal DoF of CMMI can be achieved
for (M/N)∈[0,(
√
6K/12)]∪[(3K/2), +∞), ∀K ≥ 1.
Index Terms—Degrees of freedom (DoFs), Internet of Things
(IoT), multiple-input multiple-output (MIMO), signal align-
ment, the circular multirelay multiple-input multiple-output
interference channel (CMMI).
I. INTRODUCTION
T
HE INTERNET of Things (IoT) has been considered as a
novel paradigm that provides the ability for humans and
computers to learn and interact from billions of things. The
IoT offers numerous exciting potentials and opportunities for
us to build a smart world [1]–[4]. In general, the IoT networks
are very complicated networks, which are constructed by many
Manuscript received October 15, 2017; revised December 13, 2017;
accepted March 5, 2018. Date of publication March 20, 2018; date of cur-
rent version June 8, 2018. This work was supported in part by the National
Natural Science Foundation of China under Grant 61571329, Grant 61771345,
Grant 6140144, and Grant 61390513, in part by the National Science and
Technology Major Project under Grant 2016ZX03001015, and in part by the
National Science and Technology Support Plan under Grant 2012BAH15F03.
(Corresponding author: Rui Wang.)
W. Lv, R. Wang, and J. Wu are with the College of Electronics and
Information Engineering, Tongji University, Shanghai 201804, China (e-mail:
lvwenjing@tongji.edu.cn; ruiwang@tongji.edu.cn; wujun@tongji.edu.cn).
J. Xu, P. Li, and J. Dou are with the ZTE Corporation,
Shenzhen 518057, China (e-mail: xu.jun2@zte.com.cn; li.ping6@zte.com.cn;
dou.jianwu@zte.com.cn).
Digital Object Identifier 10.1109/JIOT.2018.2817580
different basic network protocol components due to numerus
different applications.
Relay technique has also been introduced in IoT networks
to assist a long-distance transmission [5], [6]. However, the
development of traditional relay technology is limited by the
low spectral efficiency due to the half-duplex constraint. To
overcome this disadvantage (especially in a scenario of IoT
transmissions), physical-layer network coding (PNC) has been
introduced and played a significant role in enhancing spectral
efficiency by allowing a relay node to decode and forward
message combinations rather than individual messages. PNC,
as an advanced relaying technique [7], was proposed initially
in two-way relay channel and then extended to the communi-
cations over multiway relay channels (mRCs), where multiple
users exchange data with the help of the relay. Several mRC
models have been studied. Particularly, Sezgin et al. [8] and
Chen and Yener [9] studied mRCs in which the users are
grouped into pairs and the two users in each pair exchange
information with each other. Later, Gunduz et al. [10] stud-
ied clustered mRCs, in which the users in the network
are grouped into clusters and each user in a cluster wants
to exchange information with the other users. Approximate
capacities of these mRC models were studied in [10] and [11],
while the exact capacity characterizations are still open top-
ics. Also, these initial works on mRC are limited to the single
antenna setup, i.e., each node in the network is equipped with
one antenna.
Multiple-input multiple-output (MIMO) techniques have
also been incorporated into PNC-aided relay networks to
achieve spatial multiplexing [12]. Lee et al. [13] and
Ding et al. [14] have studied the degrees of freedom (DoFs)
of the MIMO mRCs. Particularly, Lee et al. [13] studied the
optimal DoF of the MIMO Y channel with an arbitrary num-
ber of users and antenna setups but with a single relay. It
has been shown that the number of relay antennas is the
performance bottleneck of the mRC especially when the num-
ber of users is large. To overcome this shortage, multiple relays
were proposed to assist the data transmission over mRC. For
example, Ding et al. [14] investigated the optimal DoF of the
symmetric multirelay MIMO Y channel which has an arbitrary
number of relays rather than only a single relay. In order to
obtain the achievable DoF of this model, they also proposed
an uplink-downlink asymmetric design. As a further genera-
tion, Tian and Yener [15] and Yuan [16] studied MIMO mRCs
with clustered data exchange.
It is worth noting that the existing work on the MIMO mRCs
only involves several traditional data exchange models, lots of
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