# Two New Kinds of Interference Alignment Schemes for Cellular $K$-user MIMO
Downlink Networks

Jingfu Li, Wenjiang Feng, F. Richard Yu, and Weiheng Jiang Jingfu Li, Wenjiang
Feng and Weiheng Jiang are with the School of Microelectronics and
Communication Engineering, Chongqing University, Chongqing 400044, China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>).Fei
Richard Yu is with the Department of System and Computer Engineering, Carleton
University, Ottawa, ON K1S 5B6, Canada (e-mail: richard.yu@carleton.ca).

###### Abstract

It is known that interference alignment (IA) plays an important role in
improving the degree of freedom (DoF) of multi-input and multi-output (MIMO)
systems. However, most of the traditional IA schemes suffer from the high
computational complexity and require the global and instantaneous channel
state information (CSI), both of which make them difficult to be extended to
cellular MIMO systems. To handle these issues, two new interference alignment
schemes, i.e., the retrospective interference regeneration (RIR) scheme and
the beamforming based distributed retrospective interference alignment
(B-DRIA) scheme, are proposed for cellular $K$-user MIMO downlink networks.
For the RIR scheme, it adopts interference elimination algorithm to erase
redundant symbols in inter-cell interference (ICI) signals, and then uses
interference regeneration algorithm to avoid secondary interference. The RIR
scheme obtains greater DoF gain than the retrospective interference alignment
(RIA) scheme, but incurs performance degradation when the transceiver antennas
ratio (TAR) approaches 1. Therefore, the B-DRIA scheme is further proposed.
For the B-DRIA scheme, the cellular beamforming matrix is introduced to
eliminate the ICI, and meanwhile distributed retrospective interference
alignment algorithm is adopted to align inter-user interference (IUI). The
simulation results show that the B-DRIA scheme obtains larger DoF than the RIR
scheme locally. Specifically, when TAR approaches 1, two schemes obtain the
same DoF. While TAR approaches 2, the DoF of the B-DRIA scheme is superior
than the RIR scheme.

###### Index Terms:

Cellular K-user MIMO downlink networks, degrees of freedom, delayed CSIT,
retrospective interference regeneration, beamforming based distributed
retrospective interference alignment.

##  I Introduction

The interference alignment (IA) technique is an elegant way towards improving
the degree of freedom (DoF) of MIMO systems. With the help of IA technique,
the $K$-user time-varying interference channel almost surely has
${K\mathord{\left/{\vphantom{K2}}\right.\kern-1.2pt}2}$ DoF [1]. However, the
DoF gain caused by IA technique comes from the cost of perfect instantaneous
channel state information at the transmitter (CSIT) acquisition [2, 3, 4, 5],
which is a critical challenge, especially for distributed cellular networks
[6, 7, 8]. That is, in distributed networks, the CSI feedback from receivers
to transmitters experiences unavoidable delay, and meanwhile CSIT sharing
among multiple transmitters will take up lots of feedback resources and bring
burden to transmitters. To handle these issues, different kinds of approaches
have been proposed, such as the blind interference alignment (BIA), the space-
time interference alignment (STIA) and the retrospective interference
alignment (RIA).

Concerning the broadcast channel (BC), a novel approach is to take IA
technique without CSIT, which is the BIA [9]. Under certain heterogeneous
block fading models, the BIA makes use of channel characteristics to keep
channel gain constant in adjacent slots, and meanwhile alters the values of
signals to reduce the dimension of interference signals. However, since the
real channel fading model is somewhat between independent distributed fading
model and block fading model, the scheme still cannot work in practical
application. Therefore, for the $K$-user MISO channel systems with
reconfigurable antennas, a new blind interference alignment is presented in
[10, 11]. Owing to the circulant and non-circulant structures of the scheme,
the channels over different users keep correlated, which means that the inter-
user interference cancellation and inter-subblock interference cancellation
can be used to reduce the dimension of interference signals. Subsequently, the
BIA gets extended to other systems, i.e., the cellular interference systems
[12] and heterogeneous systems [13]. Nevertheless, the BIA still has the
defect that the complexity of precoding matrix increases exponentially with
the number of user antennas, which makes it hard to be used in MIMO systems.

To make IA technique practical and applicable in MIMO systems, the idea of
making use of the moderate delayed CSIT is taken into account. Specifically,
[14] firstly proposes STIA scheme for the $K$-user MISO BC. The core idea of
the STIA is that, in block fading model, the transmitter combines IA technique
with physical network coding (PNC) to eliminate interference in space-time
domain. Subsequent studies achieve extensions of the STIA to different
application scenarios [15, 16, 17]. In particular, [15] puts forward
distributed STIA (DSTIA) for $2\times 2$ user SISO $X$-channel (XC) and
$3\times 3$ user SISO interference channel (IC) which achieves
${4\mathord{\left/{\vphantom{43}}\right.\kern-1.2pt}3}$ and
${6\mathord{\left/{\vphantom{65}}\right.\kern-1.2pt}5}$ DoF respectively. [16]
introduces DSTIA into the $K$-user MISO IC and achieves $K-1$ DoF. For
$K$-user MIMO BC and $K$-user MIMO IC, [17] analyzes the trade-off strategy
between achievable DoF and the range of CSI delay. However, the DoF gain of
the STIA is restricted to the time span of the feedback delay, i.e., the time
span of the feedback delay cannot exceed the time span of the coherent time.
At the same time, owing to the inverse operation of the precoding matrix, the
scheme brings burden to the transceivers.

To handle the issue of feedback delay, the delayed CSIT is studied. The RIA is
firstly proposed [18] for MISO networks and gets extended to $K$-user MIMO
networks [19]. The core idea of the RIA is adopting the repetition coding,
i.e., at the transmitter, partial data is repeatedly transmitted $R$ times
during the slots, and then the received signals are jointly decoded at the
receivers’ side at last. Subsequently, the scheme is extended to the IC
scenarios, i.e., SISO IC [20], MISO IC [21], MIMO IC [22], MIMO BC [23, 24],
and meanwhile the achievable DoF gets further studied. However, with different
channels, the upper bounds of the DoF are quite different. On the one hand,
[25] explores the upper bounds of the DoF for 2-user MIMO BC and 3-user MIMO
BC, where the bounds are tight to the DoF obtained by RIA. On the other hand,
by taking RIA scheme, the obtained DoF is relaxed to the upper bounds in the
$K$-user MIMO IBC [26] and $2\times 2$-user SISO XC [27] for the sake of the
distributed transmitter loss. Therefore, the TDMA groups (TG) scheme and
3-user PSR scheme are introduced to maintain the DoF gain made by RIA, under
which the system can be adaptive with different kinds of system configurations
[28]. After making tradeoff among three schemes, unfortunately, the obtained
DoF is still not optimal in the cellular system, because it cannot properly
handle the issue of inter-cell interference (ICI) [29] by the middle way.

From the above illustration, we know that, when the number of user antennas is
large, the BIA suffers from high computational complexity so that its
application is not practical in MIMO systems. The STIA is adopted in MIMO
systems and gets DoF gain, but its performance is restricted to the time span
of the feedback delay. The RIA scheme solves the issue of feedback delay, but
the effect of DoF gain gets affected in cellular networks. Therefore, an
interference alignment scheme suitable for cellular networks is desired. In
this paper, two new interference alignment schemes, i.e., the retrospective
interference regeneration (RIR) scheme and the beamforming based distributed
retrospective interference alignment (B-DRIA) scheme, are proposed for K-user
cellular MIMO downlink systems. The main contribution of this paper is three-
fold as follows:

  * •

The paper proposes the RIR scheme which adopts interference elimination
algorithm to erase redundant symbols in inter-cell interference (ICI) signals,
and then uses interference regeneration algorithm to avoid secondary ICI
interference. The scheme achieves higher DoF gain than the RIA scheme, i.e.,
${{LM}\mathord{\left/{\vphantom{{LM}{(L+{1\mathord{\left/{\vphantom{1{\left\lfloor{{1\mathord{\left/{\vphantom{1{(\rho-1)}}}\right.\kern-1.2pt}{(\rho-1)}}}\right\rfloor}}}\right.\kern-1.2pt}{\left\lfloor{{1\mathord{\left/{\vphantom{1{(\rho-1)}}}\right.\kern-1.2pt}{(\rho-1)}}}\right\rfloor}})}}}\right.\kern-1.2pt}{(L+{1\mathord{\left/{\vphantom{1{\left\lfloor{{1\mathord{\left/{\vphantom{1{(\rho-1)}}}\right.\kern-1.2pt}{(\rho-1)}}}\right\rfloor}}}\right.\kern-1.2pt}{\left\lfloor{{1\mathord{\left/{\vphantom{1{(\rho-1)}}}\right.\kern-1.2pt}{(\rho-1)}}}\right\rfloor}})}}$
, but incurs performance degradation when transceiver antennas ratio
approaches 1.

  * •

To avoid the performance degradation of the RIR scheme under certain
circumstances, the paper further proposes the B-DRIA scheme which adopts the
cellular beamforming matrix to eliminate the ICI, and meanwhile utilizes
distributed retrospective interference alignment algorithm to align inter-user
interference (IUI). The scheme maintains the same performance improvement for
the DoF, i.e.,
${{L\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil
K(\bar{\varphi}{+}1)}\mathord{\left/{\vphantom{{L\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil
K(\bar{\varphi}{+}1)}{(L{+}\bar{\varphi}{+}}1)}}\right.\kern-1.2pt}{(L{+}\bar{\varphi}{+}1)}}$,
and in the meantime avoids performance degradation of the RIR scheme.

  * •

To further analyze the performance of the two proposed schemes effected by
transceiver antennas ratio, we perform numerical evaluations via simulations.
The simulation results show that, when the transceiver antennas ratio
approaches 2, both RIR scheme and B-DRIA scheme obtain greater DoF gain than
the RIA scheme, while as the transceiver antennas ratio approaches 1, B-DRIA
scheme achieves greater DoF gain than RIR scheme.

The rest of this paper is organized as follows. The cellular distributed
multi-user MIMO system is presented in Section II, which includes the
description of the network in time domain and angular domain, and meanwhile
the CSIT feedback model and the performance criteria are introduced. Then, two
novel interference alignment schemes, i.e., the RIR scheme and the B-DRIA
scheme, are proposed and the corresponding typical applications are given in
Section III and Section IV, respectively. The performances of the proposed
schemes are evaluated in Section V and we conclude at last.

##  II System Model

Consider cellular distributed $K$-user MIMO downlink network [30] as
illustrated in Fig. 1. Define the cells set as ${\cal
L}{=}\\{1,2,\cdots,L\\}$, and for each cell $i,\forall i\in{\cal L}$, the base
station is configured with $M$ antennas to provide service to the users in set
${{\textbf{K}}_{i}}{=}\left\\{{1,\cdots,K}\right\\}$. In addition, each served
user $k,\forall k\in{{\textbf{K}}_{i}}$ in the cell $i$ is equipped with $N$
antennas where $M>N$, which means that no user can decode its message without
additional information.

Figure 1: Cellular distributed multi-user MIMO network.

In the following, for the simplicity, we use $i,\forall i\in{\cal L}$ and
$j,\forall j\in{\cal L}$ to denote the cell which the transmitter and the
receiver belong to, respectively. Then, we further classify the cells into two
categories: if $j=i$, the cell $i$ is called target cell, otherwise denotes
interference cell. Assume that the network takes
${\textbf{T}}{=}\\{1,2,\cdots,T\\}$ slots to transmit signals and for each
slot $t,\forall t\in{\textbf{T}}$, the base station $i$ sends messages
${{\bf{S}}_{i}}[t]=\left[{{{\bf{S}}_{i,1}}[t]\cdots{{\bf{S}}_{i,k}}[t]}\right]$
to its served users. Due to the broadcast nature of the wireless
communication, on the one hand, the received signals of the user $k,\forall
k\in{\textbf{K}}$ are composed of two parts, i.e., the disred signals
${{\bf{S}}_{i}}[t]{=}\left[{{{\bf{S}}_{i,1}}[t]\cdots{{\bf{S}}_{i,k}}[t]}\right]$
from the cell $i$ and undesired signals
${{\bf{S}}_{-i}}[t]{=}\left[{{{\bf{S}}_{-i,1}}[t]\cdots{{\bf{S}}_{-i,k}}[t]}\right]$
from the interference cells $-i={\cal L}\backslash i$. On the other hand, the
desired signals is made up of desired symbols
${{\bf{S}}_{i,k}}[t]{=}\left[{{s_{k,1}}[t]\cdots{s_{k,n}}[t]}\right]$ where
$0\\!\\!<\\!n\\!\\!\leq\\!N$ and undesired symbols
${{\bf{S}}_{i,-k}}[t]{=}\left[{{s_{-k,1}}[t]\cdots{s_{-k,n}}[t]}\right]$ where
$-k={\textbf{K}}\backslash k$.

Herein, for the cellular distributed K-user MIMO downlink network, since all
users share the same channel resources, i.e., the IUI and the ICI coexist in
each slot. In order to implement our schemes to align IUI and ICI, the CSI
should be further differentiated. Specifically, the component of the CSI
should be expanded into two parts, the channel gain information and the angel
information. Thus, the system model should be described in time domain [31]
and angular domain [32], as that presented in the section II.A and II.B,
respectively. Meanwhile, based on the relationship between the coherent time
and the time delay, the types of the CSI should be classified into three
categories, the instantaneous CSI, the moderate delayed CSI and the delayed
CSI. In specific, the CSIT feedback model is presented in the section II.C
[33].

###  II-A Time Domain Model

To acquire the channel gain information of CSI, the system model mentioned
above should be described as the time domain model [31]. Taking the
transmission of one slot $t,\forall t\in{\textbf{T}}$ as example, the base
station $i$ transmits messages ${\boldsymbol{{S}}_{i,k}}[t]$ to the user $k$
in the cell $i$. Then the received signal of the user $k$ in cell $j,\forall
j\in{\cal L}$ can be written by

${\boldsymbol{{y}}_{j,k}}[t]{=}\sum\limits_{i=1}^{L}{\sum\limits_{k=1}^{K}{{\bf{H}}_{i}^{j,k}[t]{{\boldsymbol{{S}}}_{i,k}}[t]}}{+}\overline{\boldsymbol{{N}}}.$
(1)

where ${\bf{H}}_{i}^{j,k}[t]\in{\textbf{C}}{{}^{M\times N}}$ denotes the
channel matrix between the base station $i$ and the user $k$ of the cell $j$,
and it is subject to independent and identically distributed
$\left({i.i.d.}\right)$ according to ${\cal C}{\cal N}(0,1)$.
$\overline{\boldsymbol{N}}\in{\textbf{C}}{{}^{N\times 1}}$ denotes the
additive white Gaussian noise (AWGN) term at the user $k$ of the cell $j$.

If the transmitted messages ${{\boldsymbol{S}}_{i,k}}[t]$ is known to the user
$k$ in the cell $i$, the channel matrix ${\bf{H}}_{i}^{j,k}[t]$ can be gotten
in the form of the time domain, which denotes the channel gain information of
CSI.

###  II-B Angular Domain Model

Similarly, to acquire the angel information of CSI, the system model of the
network should be described in angular domain [32]. The application scenario
is the same as it is assumed in the section II.A. The received signals are
given by

${{\boldsymbol{y}}_{j,k}}[t]{=}\sum\limits_{i=1}^{L}{\sum\limits_{k=1}^{K}{{\bf{A}}_{i}^{j,k}[{\theta_{i}}]{{\boldsymbol{S}}_{i,k}}[t]}}{+}\overline{\boldsymbol{N}}.$
(2)

where ${\bf{A}}_{i}^{j,k}[{\theta_{i}}]\in{\textbf{C}}{{}^{N\times M}}$ is the
direction of arrival (DoA) matrix between the base station $i$ and the user
$k$ in the cell $j$ and meanwhile,
$\overline{\boldsymbol{N}}\in{\textbf{C}}{{}^{N\times 1}}$ denotes AWGN. By
analyzing the ingredient of ${\bf{A}}_{i}^{j,k}[{\theta_{i}}]$, it is composed
of $M$ direction vectors,

${\bf{A}}_{i}^{j,k}[{\theta_{i}}]{=}\left[{\begin{array}[]{*{20}{c}}{{{\boldsymbol{a}}}_{i,1}^{j,k}[{\theta_{i}}]}&\cdots&{{{\boldsymbol{a}}}_{i,M}^{j,k}[{\theta_{i}}]}\end{array}}\right],$
(3)

where the direction vector ${{\boldsymbol{a}}}_{i,m}^{j,k}[{\theta_{i}}]$ can
be regarded as the function of the DoA angular ${\theta_{i}}$. As for each
${{\boldsymbol{a}}}_{i,m}^{j,k}[{\theta_{i}}]$, it represents the direction
from the antenna $m\in[1,M]$ of the cell i to the user $k$ in the cell $j$ and
is written as

${\boldsymbol{a}}_{i,m}^{j,k}[{\theta_{i}}]={\left[{1,{e^{-j{\mu_{m}}}},\cdots,{e^{-j(N-1){\mu_{m}}}}}\right]^{T}}.$
(4)

Note that ${\mu_{m}}=({{2\pi d}\mathord{\left/{\vphantom{{2\pi
d}\lambda}}\right.\kern-1.2pt}\lambda})\sin{\theta_{i}}$, where $d$ is the
array spacing and $\lambda$ is the carrier wavelength. The DoA angular
${\theta_{i}}$ denotes the angel information of CSI.

###  II-C CSIT Feedback Model

Due to the unavoidable feedback delay from the receiver to the transmitter,
after making expansion of the component of CSI, the obtained CSIT is a delayed
version. It means that, at the transmitter side, the obtained CSIT cannot be
used to instruct the design of precoding matrix at the current slot.
Therefore, the types of the CSI should be further classified by CSIT feedback
model [33] which is summarized in Fig. 2.

Figure 2: CSIT feedback model.

We assume that the receiver can perfectly estimate the CSI and then send it
back to the corresponding transmitter through an error-free but delayed
feedback link. Therefore, the transmitter can continuously track the variation
of the channel matrix over different slots. To distinguish different versions
of CSIT, we define the channel feedback delay and the period of coherence time
as ${T_{fb}}$ and ${T_{c}}$, respectively. Then, the definition of the ratio
$\lambda$ is given by

$\lambda{=}{{{T_{fb}}}\mathord{\left/{\vphantom{{{T_{fb}}}{{T_{c}}}}}\right.\kern-1.2pt}{{T_{c}}}},$
(5)

which is naturally divided into three types and each type represents one kind
of CSI:

  * •

$\lambda=0$, the obtained CSI is an instantaneous CSI, that is the
transmitters acquire the CSI feedback from the receivers immediately.

  * •

$0<\lambda<1$, this situation should be subdivided into two subcases, i.e.,
the moment of getting CSI belongs to
$\left[{{T_{fb}},{T_{c}}-{T_{fb}}}\right]$ and
$\left[{{T_{c}}-{T_{fb}},{T_{c}}}\right]$. In the former interval, the
obtained CSI is equivalent to the instantaneous CSI, while in the later
interval, the obtained CSI represents the moderate delayed CSI, which means
that, after feedback delay, the transmitters can also make use of the CSI in
the rest time of the current slot.

  * •

$\lambda\geq 1$, the obtained CSI is a delayed version, that is the CSI is
absolutely outdated. Thus, the channel gain for the current slot is unknown
and the CSI is no longer limited by the feedback delay.

The type of CSI adopted in Section III and Section IV should be a
comprehensive combination of the latter two types, i.e., the moderate delayed
CSI and the delayed CSI.

###  II-D Sum Degrees of Freedom

For the considered system, a measurement criterion is desired to compare the
performance of the proposed schemes with existing IA schemes. Herein, the sum
degrees of freedom [34] is extended to the cellular distributed $K$-user MIMO
downlink network. Specifically, in each slot $t,\forall
t\\!\in\\!{\textbf{T}}$, the base stations simultaneously transmit independent
messages. Taking the base station $i,\forall i\in{\cal L}$ for example, the
messages ${{\bf{S}}_{i,1}}[t],\cdots,{{\bf{S}}_{i,k}}[t]$ are sent with the
rates of ${R_{i,1}},\cdots,{R_{i,k}}$ bits/s/Hz, respectively. For the user
$k,\forall k\in{\textbf{K}}$, its achievable rate is characterized by

${R_{i,k}}{=}{d_{i,k}}\log(1{+}SNR)+o\left({\log(SNR)}\right),$ (6)

where ${d_{i,k}}$ denotes the degree of freedom belonging to the user $k$ in
the cell $i$ and can be expressed as

${d_{i,k}}=\mathop{\lim}\limits_{SNR\to\infty}{{{R_{i,k}}}\mathord{\left/{\vphantom{{{R_{i,k}}}{{{\log}_{2}}(SNR)}}}\right.\kern-1.2pt}{{{\log}_{2}}(SNR)}}.$
(7)

The formula (7) measures the channel capacity of a single cell per user, i.e.,
the number of data streams that can be transmitted reliably by one user within
one slot. Then, we extend it to the case of cellar $K$-user, given by

$DoF=\mathop{\lim}\limits_{SNR\to\infty}\mathop{\max}\limits_{\boldsymbol{R}}\sum\limits_{i=1}^{L}{\sum\limits_{k=1}^{K}{{{{R_{i,k}}}\mathord{\left/{\vphantom{{{R_{i,k}}}{{{\log}_{2}}(SNR)}}}\right.\kern-1.2pt}{{{\log}_{2}}(SNR)}}}},$
(8)

where
${\boldsymbol{R}}{=}\left[{{R_{i,1}},\cdots,{R_{i,k}}}\right]\in{{\textbf{R}}^{K}}$
denotes the vector of achievable rates for the users in the cell $i$. Thus,
the DoF measures the channel capacity of the whole network.

##  III Retrospective Interference Regeneration Scheme

Although the RIA scheme achieves greater DoF than TDMA scheme [18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29], when the numbers of cells and users increase,
some space resources are wasted and the DoF gain will be decreased. To handle
this issue, we propose the retrospective interference regeneration (RIR)
scheme. The core idea of the scheme is making use of the interference signals
from interference cells to align ICI, i.e., when the receivers feed
interference signals back, the base stations eliminate redundant symbols in
the interference signals to extract desired symbols and meanwhile, provide
desired symbols to the target cell. In the following, we firstly illustrate
the details of the proposed scheme which includes three phases, i.e., signal
transmission, interference elimination and interference regeneration, and
interference retransmission. Then, the DoF of the RIR scheme is analyzed. To
make it understandable, we also present an example of the scheme at last.

###  III-A Signal Transmission

The ﬁrst phase spans ${\cal L}$ groups of
$\varphi=\left\lfloor{{N\mathord{\left/{\vphantom{N{\left({M-N}\right)}}}\right.\kern-1.2pt}{\left({M-N}\right)}}}\right\rfloor$
slots where $\left\lfloor{*}\right\rfloor$ denotes the round down operation.
In each slot $t\in\left\\{{1,\cdots,\varphi}\right\\}$ of period $i\in{\cal
L}$, the base station $i$ transmits signals while the other base stations keep
silent. The transmitted signal is characterized by

${\boldsymbol{S}}_{i}[(i-1)\varphi+t]={\left[{{s_{i,1}}[(i-1)\varphi+t],\cdots,{s_{i,M}}[(i-1)\varphi+t]}\right]^{T}}.$
(9)

Then the signals received by user $k$ of the cell $j$ are

${\boldsymbol{y}}_{j,k}[(i-1)\varphi+t]={\bf{H}}_{i}^{j,k}[(i-1)\varphi+t]{{\boldsymbol{S}}_{i}}[(i-1)\varphi+t],j\in{\cal
L}.$ (10)

As mentioned earlier, if $i\neq j$, the cell $i$ is the interference cell and
the corresponding received signals are ICI. If $i=j$ , the cell $i$ is the
target cell but, for the user $k$, only part of
${{\boldsymbol{S}}_{i}}[(i-1)\varphi+t]$ are desired symbols, i.e.,
$\left\\{{{s_{i,(k-1)\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil+1}}\left[{(i-1)\varphi+t}\right],\cdots,{s_{i,k\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil}}\left[{(i-1)\varphi+t}\right]}\right\\}$
and the left are the interference symbols. According to whether this user
belong to the target cell, the user feedback different information to the
transmitter, i.e., if $i=j$, the users feed the channel gain information back
to the target cell’s base station by delayed feedback links. While if $i\neq
j$, the users feed the ICI back to the interference cell’s base station, i.e.,
${\boldsymbol{y}}_{j,k}[(i-1)\varphi+t]$. This process is illustrated by Fig.
3.

Figure 3: Signal transmission in 1st phase.

As mentioned earlier, since $M>N$, no user can decode the messages with only
one-slot transmitted signals. To handle this issue, the additional
interference alignment process is developed to the system, i.e., interference
elimination and interference regeneration.

###  III-B Interference Elimination and Interference Regeneration

By analyzing the components of ICI, we find that, for the user $k$, part of
the information in ICI can be utilized, while the remainder is still the
interference. Therefore, to distinguish these two parts, the ICI can be
written as

$\displaystyle{{\textbf{L}}_{j}}[(i\\!-\\!1)\varphi\\!+\\!t]{=}$
$\displaystyle{{\textbf{L}}_{j}}\left({{s_{i,1}}[(i\\!-\\!1)\varphi\\!+\\!t],\\!\cdots\\!,{s_{i,M}}[(i\\!-\\!1)\varphi\\!+\\!t]}\right)$
$\displaystyle{=}$
$\displaystyle{\bf{H}}_{i}^{j,k}[(i-1)\varphi+t]{{\boldsymbol{S}}_{i}}[(i-1)\varphi+t],$
$\displaystyle{\cal L},j\neq i,k\in{\textbf{K}}$ $\displaystyle{=}$
$\displaystyle{\boldsymbol{L}}_{j}^{i,k}[(i-1)\varphi+t]{+}{\boldsymbol{I}}_{j}^{i,k}[(i-1)\varphi+t],$
(11)

where ${\boldsymbol{L}}_{j}^{i,k}[(i-1)\varphi+t]$ represents the useful part
and ${\boldsymbol{I}}_{j}^{i,k}[(i-1)\varphi+t]$ represents the rest, i.e.,
the interference. Then, we can define these two parts as partial desired
signals and partial interference signals as follows, respectively,

$\displaystyle{\boldsymbol{L}}_{j}^{i,k}[(i-1)\varphi+t]{=}$
$\displaystyle{\boldsymbol{L}}_{j}^{i,k}({s_{i,(k-1)\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil+1}}[(i-1)\varphi+t],$
$\displaystyle\cdots,{s_{i,k\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil}}[(i-1)\varphi+t])$
$\displaystyle{=}$
$\displaystyle{\bf{H}}_{i}^{j,k}[(i-1)\varphi+t]{{\boldsymbol{S}}_{i}}[(i-1)\varphi+t],$
$\displaystyle j\in{\cal L},j\neq i,$ (12)
$\displaystyle{\boldsymbol{I}}_{j}^{i,k}[(i-1)\varphi+t]{=}$
$\displaystyle{\boldsymbol{I}}_{j}^{i,k}({s_{i,(-k-1)\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil+1}}[(i-1)\varphi+t],$
$\displaystyle\cdots,{s_{i,-k\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil}}[(i-1)\varphi+t])$
$\displaystyle{=}$
$\displaystyle{\bf{H}}_{i}^{j,k}[(i-1)\varphi+t]{{\boldsymbol{S}}_{i}}[(i-1)\varphi+t],$
$\displaystyle j\in{\cal L},j\neq i,-k={\textbf{K}}\backslash k,$ (13)

For the interference cell, the interference space of
${\boldsymbol{L}}_{j}^{i,k}[(i-1)\varphi+t]$ is a subspace of
${{\textbf{L}}_{j}}[(i-1)\varphi+t]$. Hence, according to the feedback
interference signals, the interference cell’s base station can design a
precoding matrix ${{\bf{U}}_{j}}[(i-1)\varphi+t]$ to achieve partial
interference elimination, i.e.,

${{\bf{U}}_{j}}[(i-1)\varphi+t]{{\textbf{L}}_{j}}[(i-1)\varphi+t]={\boldsymbol{L}}_{j}^{i,k}[(i-1)\varphi+t].$
(14)

For (14), according to the extension theorem of linear subspace [36], the
precoding matrix ${{\bf{U}}_{j}}[t]$ must exist, and the solution of the
matrix can be obtained by cyclic-zero-padding precoding matrix [35].

After the interference elimination, the partial desired signal
${\boldsymbol{L}}_{j}^{i,k}[(i-1)\varphi+t]$ can be transmitted in the
interference retransmission phase. Unfortunately, the interference
retransmission will cause additional interference to other cells except the
target cell, as illustrated in Fig. 4.

Figure 4: Additional ICI in 2nd phase.

To handle this issue, the base station should design interference regeneration
matrices to satisfy the following condition

$\begin{array}[]{l}{{\bf{V}}_{p}}[(i-1)\varphi+t]{\boldsymbol{L}}_{p}^{i,k}[(i-1)\varphi+t]\\\
{=}{{\bf{V}}_{q}}[(i-1)\varphi+t]{\boldsymbol{L}}_{q}^{i,k}[(i-1)\varphi+t],\forall
p,q\in{\cal L}\backslash i,\end{array}$ (15)

where each interference regeneration matrix ${{\bf{V}}_{p}}[(i-1)\varphi+t]$
is an
$\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil\times\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil$
matrix that can align the interference signals into the same space. Since the
vectors ${\boldsymbol{L}}_{p}^{i,k}[(i-1)\varphi+t]$ and
${\boldsymbol{L}}_{q}^{i,k}[(i-1)\varphi+t]$ are irrelevant with same
dimension
$\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil$,
the precoding matrices ${{\bf{V}}_{p}}[(i-1)\varphi+t]$ and
${{\bf{V}}_{q}}[(i-1)\varphi+t]$ can be regarded as row transformation
matrices. Thus, from the property of equivalence matrix [36], the interference
regeneration matrices are always exist, and cyclic-zero-padding precoding [35]
can be used to obtain the particular solution. The details are summarized in
Fig. 5.

Figure 5: Interference regeneration in 2nd phase.

###  III-C Interference Retransmission

After interference elimination and interference regeneration, base stations
are capable to simultaneously transmit partial desired signals as

$\displaystyle{{\boldsymbol{X}}_{j}}[L\varphi+1]$
$\displaystyle=\\!\sum\nolimits_{i\in{\cal L}\backslash
j}\\!{{\boldsymbol{X}}_{j}^{i}[L\varphi+1]}$
$\displaystyle=\\!\sum\nolimits_{i\in{\cal L}\backslash
j}\\!\\!{\sum\limits_{t=\left({i-1}\right)\varphi+1}^{i\varphi}\\!{{{\bf{V}}_{j}}[t]{{\bf{U}}_{j}}[t]({\bf{H}}_{i}^{j,k}[t]{{\boldsymbol{S}}_{i}}[t])}}$
$\displaystyle=\sum\nolimits_{i\in{\cal L}\backslash
j}{\sum\limits_{t=\left({i-1}\right)\varphi+1}^{i\varphi}{{{\bf{V}}_{j}}[t]{\boldsymbol{L}}_{j}^{i,k}[t]}}.$
(16)

With the preprocessing of ${{\bf{V}}_{j}}[t]$ and ${{\bf{U}}_{j}}[t]$, for the
whole cells, the additional signals coming from the interference
retransmission phase are convert to the partial desired signals, which means
that the retransmitted process will not bring about ICI and IUI, as
illustrated in Fig. 6. Herein, for the user $k$ in the target cell $i$, the
received signals are written as,

${{\bf{y}}_{i,k}}[L\varphi+1]=\sum\nolimits_{j\in{\cal
L}}{{\bf{H}}_{j}^{i,k}[L\varphi+1]{{\bf{X}}_{j}}[L\varphi+1]},$ (17)

Figure 6: Interference retransmission in 3rd phase.

and the whole process can be summarized as the equation (18),

$\underbrace{\left[{\begin{array}[]{*{20}{c}}{{{\bf{y}}_{i,k}}[(i-1)\varphi+1]}\\\
{{{\bf{y}}_{i,k}}[(i-1)\varphi+2]}\\\ \vdots\\\
{{{\bf{y}}_{i,k}}[i\varphi]}\\\
{{{\bf{y}}_{i,k}}[L\varphi+1]}\end{array}}\right]}_{\bf{Y}}\\!{=}\\!\underbrace{\left[{\begin{array}[]{*{20}{c}}{{\bf{H}}_{i}^{i,k}[(i-1)\varphi+1]}&0&\cdots&0\\\
0&\ddots&\ddots&\vdots\\\ \vdots&\ddots&\ddots&0\\\
0&\cdots&0&{{\bf{H}}_{i}^{i,k}[i\varphi]}\\\ {\sum\limits_{j\in{\cal
L}}{{\bf{H}}_{j}^{i,k}[L\varphi+1]}{{\boldsymbol{\beta}}_{(i-1)\varphi+1}}}&{\sum\limits_{j\in{\cal
L}}{{\bf{H}}_{j}^{i,k}[L\varphi+1]{{\boldsymbol{\beta}}_{(i-1)\varphi+2}}}}&\cdots&{\sum\limits_{j\in{\cal
L}}{{\bf{H}}_{j}^{i,k}[L\varphi+1]{{\boldsymbol{\beta}}_{i\varphi}}}}\end{array}}\right]}_{\bf{H}}\underbrace{\left[{\begin{array}[]{*{20}{c}}{{{\bf{S}}_{i}}[(i-1)\varphi+1]}\\\
\vdots\\\ \vdots\\\ \vdots\\\
{{{\bf{S}}_{i}}[i\varphi]}\end{array}}\right]}_{\bf{S}}$ (18)

where ${{\boldsymbol{\beta}}_{t}}$ is the combination precoding of the
interference cells. The precoding matrix is composed of interference
elimination matrix, interference regeneration matrix and channel matrix in
slot $t$, denoted by

${{\boldsymbol{\beta}}_{t}}=\sum\nolimits_{j\in{\cal
L}{\backslash}i}{{{\bf{V}}_{j}}[t]{{\bf{U}}_{j}}[t]{\bf{H}}_{i}^{j,k}[t]}.$
(19)

For our proposed scheme, ${{\bf{V}}_{j}}[t]$ , ${{\bf{U}}_{j}}[t]$ and
${\bf{H}}_{j}^{i,k}[L\varphi+1]$ are non-singular matrix which means that they
don’t change the rank of ${\bf{H}}_{i}^{j,k}[t]$, and in the meantime, for any
two subchannel matrices, they either come from different time slots, e.g.,
${\bf{H}}_{i}^{i,k}[(i-1)\varphi+1]$ and ${\bf{H}}_{i}^{i,k}[i\varphi]$, or
come from different spaces, e.g., ${\bf{H}}_{i}^{i,k}[(i-1)\varphi+1]$ and
${\bf{H}}_{j}^{i,k}[(i-1)\varphi+1],j\neq i$, are independent. The former is
for the sake of the delayed CSIT and the latter is on account of the
irrelevance of the channels among different cells. Herein, the elements of
$\bf{H}$ are independent so that the matrix itself is a
$\left\lfloor{{N\mathord{\left/{\vphantom{N{(M-N}}}\right.\kern-1.2pt}{(M-N}})}\right\rfloor
M$ dimensional non-singular matrix. Therefore, at the user side, the Least
Square (LS) method [37] can be used to decode the messages of the whole slots.

###  III-D Degrees of Freedom

In this subsection, the DoF of the proposed RIR scheme is analyzed. In
specific, since the base stations spend $L\varphi{+}1$ incoherent slots to
transmit
$L\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil
K\varphi$ independent symbols in all. We assume that the transmission is
reliable, i.e., the value of signal-to-noise ratio approaches infinity.
Therefore, its achievable DoF is

$DoF{=}{{(L\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil
K\varphi)}\mathord{\left/{\vphantom{{(L\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil
K\varphi)}{(L\varphi{+}1)}}}\right.\kern-1.2pt}{(L\varphi{+}1)}}.$ (20)

To facilitate the analysis, we assume that $M$ can be divided by $K$. Then the
DoF can be simplified as

$DoF{{{=}LM}\mathord{\left/{\vphantom{{(L+{1\mathord{\left/{\vphantom{1\varphi}}\right.\kern-1.2pt}\varphi})}}}\right.\kern-1.2pt}{(L+{1\mathord{\left/{\vphantom{1\varphi}}\right.\kern-1.2pt}\varphi})}}.$
(21)

In practice, the number of cells $L$ is fixed and $\varphi$ is a function of
$M$ and $N$, thus the value of DoF mainly depends on the relationship between
$M$ and $N$. Define
$\rho={M\mathord{\left/{\vphantom{MN}}\right.\kern-1.2pt}N}$ as the
transceiver antennas ratio [18] and then, the DoF can be simplified as

$DoF{{{=}LM}\mathord{\left/{\vphantom{{{=}LM}{(L+{1\mathord{\left/{\vphantom{1{\left\lfloor{{1\mathord{\left/{\vphantom{1{(\rho-1)}}}\right.\kern-1.2pt}{(\rho-1)}}}\right\rfloor}}}\right.\kern-1.2pt}{\left\lfloor{{1\mathord{\left/{\vphantom{1{(\rho-1)}}}\right.\kern-1.2pt}{(\rho-1)}}}\right\rfloor}})}}}\right.\kern-1.2pt}{(L+{1\mathord{\left/{\vphantom{1{\left\lfloor{{1\mathord{\left/{\vphantom{1{(\rho-1)}}}\right.\kern-1.2pt}{(\rho-1)}}}\right\rfloor}}}\right.\kern-1.2pt}{\left\lfloor{{1\mathord{\left/{\vphantom{1{(\rho-1)}}}\right.\kern-1.2pt}{(\rho-1)}}}\right\rfloor}})}}.$
(22)

To analyze the formula (22), we can get the conclusion that, with the increase
of the ratio $\rho$, the DoF should be distinct. That is, when $M$ is not
divisible by $M-N$, the value of
$\varphi=\left\lfloor{{N\mathord{\left/{\vphantom{N{\left({M-N}\right)}}}\right.\kern-1.2pt}{\left({M-N}\right)}}}\right\rfloor$
will not change owing to the round down operation
$\left\lfloor{*}\right\rfloor$ and the DoF keeps constant. While if $M$ is
divisible by $M-N$, the value of the DoF will grow sharply. Therefore, the
value of $\varphi$ is divided into three cases, i.e., $\varphi>2$,
$\varphi=2$, $\varphi=1$, and the corresponding DoF of each situation is
further discussed in Section V.

###  III-E Typical Application

In this subsection, we take a typical application as example to demonstrate
the whole process of the RIR scheme. In particular, the configuration of the
network is set as $(L,M,K,N)=(3,4,2,3)$, i.e., the network is composed of
three cells and in each cell, the base station with four antennas provides
services to two users, where each user has three antennas. For this scenario,
the details about the application of the RIR scheme are as follows.

The ﬁrst phase spans ${\cal L}=3$ groups of $\varphi=3$ slots. From slot 1 to
slot 3 of the first group, the symbols sent by the base station 1 are as
follows

$\begin{array}[]{l}{{\boldsymbol{S}}_{1}}[1]=\left({\begin{array}[]{*{20}{c}}{{a_{1}}},{{a_{2}}},{{b_{1}}},{{b_{2}}}\end{array}}\right)^{T},\\\
{{\boldsymbol{S}}_{1}}[2]=\left({\begin{array}[]{*{20}{c}}{{a_{3}}},{{a_{4}}},{{b_{3}}},{{b_{4}}}\end{array}}\right)^{T},\\\
{{\boldsymbol{S}}_{1}}[3]=\left({\begin{array}[]{*{20}{c}}{{a_{5}}},{{a_{6}}},{{b_{5}}},{{b_{6}}}\end{array}}\right)^{T},\end{array}$
(23)

where the sets ${\bf{A}}=\left\\{{{a_{1}},\cdots,{a_{6}}}\right\\}$ and
${\bf{B}}=\left\\{{{b_{1}},\cdots,{b_{6}}}\right\\}$ are the desired symbols
for the users 1 and 2 in the target cell 1, respectively. Then, the received
signals at user $k\in\left\\{{1,2}\right\\}$ can be characterized as

$\begin{array}[]{l}{{\bf{y}}_{1,k}}[t]={\bf{H}}_{1}^{1,k}[t]{{\boldsymbol{S}}_{1}}[t]\\\
\;\;\;\;\;\;\;\quad{=}\left[{\begin{array}[]{*{20}{c}}{h_{1,1}^{k,1}[t]}&{h_{1,2}^{k,1}[t]}&{h_{1,3}^{k,1}[t]}&{h_{1,4}^{k,1}[t]}\\\
{h_{1,1}^{k,2}[t]}&{h_{1,2}^{k,2}[t]}&{h_{1,3}^{k,2}[t]}&{h_{1,4}^{k,2}[t]}\\\
{h_{1,1}^{k,3}[t]}&{h_{1,2}^{k,3}[t]}&{h_{1,3}^{k,3}[t]}&{h_{1,4}^{k,3}[t]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{2t-1}}}\\\
{{a_{2t}}}\\\ {{b_{2t-1}}}\\\ {{b_{2t}}}\end{array}}\right]\\\
\;\;\;\;\;\;\;\quad{=}{{\boldsymbol{L}}_{1}}[t]\left({{a_{2t-1}},{a_{2t}},{b_{2t-1}},{b_{2t}}}\right),t\in\\{1,2,3\\}.\end{array}$
(24)

Since $M>N$, no users can decode the messages within one slot. From (24), we
know that, in each slot, there are 4 independent values but we only have 3
equations. Thus, for this group, additional $\varphi\times(M-N)=3$ equations
are needed. The users feed the channel estimation ${\bf{H}}_{1}^{1,k}[t]$ back
to the base station 1. Similarly, in the second and third group, the
transmitted symbols are defined as (25) and (26) respectively,

$\begin{array}[]{l}{{\boldsymbol{S}}_{2}}[4]\\!=\\!\left({\begin{array}[]{*{20}{c}}{{c_{1}}},{{c_{2}}},{{d_{1}}},{{d_{2}}}\end{array}}\right)^{T},\\\
{{\boldsymbol{S}}_{2}}[5]\\!=\\!\left({\begin{array}[]{*{20}{c}}{{c_{3}}},{{c_{4}}},{{d_{3}}},{{d_{4}}}\end{array}}\right)^{T},\\!\\\
{{\boldsymbol{S}}_{2}}[6]\\!=\\!\left({\begin{array}[]{*{20}{c}}{{c_{5}}},{{c_{6}}},{{d_{5}}},{{d_{6}}}\end{array}}\right)^{T},\end{array}$
(25)
$\begin{array}[]{l}{{\boldsymbol{S}}_{3}}[7]\\!=\\!\left({\begin{array}[]{*{20}{c}}{{e_{1}}},{{e_{2}}},{{f_{1}}},{{f_{2}}}\end{array}}\right)^{T},\\\
{{\boldsymbol{S}}_{3}}[8]\\!=\\!\left({\begin{array}[]{*{20}{c}}{{e_{3}}},{{e_{4}}},{{f_{3}}},{{f_{4}}}\end{array}}\right)^{T},\\\
{{\boldsymbol{S}}_{3}}[9]\\!=\\!\left({\begin{array}[]{*{20}{c}}{{e_{5}}},{{e_{6}}},{{f_{5}}},{{f_{6}}}\end{array}}\right)^{T}.\end{array}$
(26)

During the second group and the third group of the first phase, the base
stations of the interference cells transmit signals to their served users in
turn, but at the same time, the users in cell 1 will receive ICI inevitably,
that is

$\begin{array}[]{l}{{\bf{y}}_{1,k}}[t]={\bf{H}}_{2}^{1,k}[t]{{\boldsymbol{S}}_{2}}[t]\\\
\;\;\;\;\;\;\;\;\;\;\,{=}\left[{\begin{array}[]{*{20}{c}}{h_{2,1}^{k,1}[t]}&{h_{2,2}^{k,1}[t]}&{h_{2,3}^{k,1}[t]}&{h_{2,4}^{k,1}[t]}\\\
{h_{2,1}^{k,2}[t]}&{h_{2,2}^{k,2}[t]}&{h_{2,3}^{k,2}[t]}&{h_{2,4}^{k,2}[t]}\\\
{h_{2,1}^{k,3}[t]}&{h_{2,2}^{k,3}[t]}&{h_{2,3}^{k,3}[t]}&{h_{2,4}^{k,3}[t]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{{c}_{2t-7}}}\\\
{{{c}_{2t-6}}}\\\ {{d_{2t-7}}}\\\ {{d_{2t-6}}}\end{array}}\right]\\\
\;\;\;\;\;\;\;\;\;\;{=}{{\textbf{L}}_{2}}\left({{{c}_{2t-7}},{{c}_{2t-6}},{d_{2t-7}},{d_{2t-6}}}\right),t\in\\{4,5,6\\},\end{array}$
(27)
$\begin{array}[]{l}{{\bf{y}}_{1,k}}[t]={\bf{H}}_{3}^{1,k}[t]{{\boldsymbol{S}}_{3}}[t]\\\
\;\;\;\;\;\;\;\;\;\,{=}\left[{\begin{array}[]{*{20}{c}}{h_{3,1}^{k,1}[t]}&{h_{3,2}^{k,1}[t]}&{h_{3,3}^{k,1}[t]}&{h_{3,4}^{k,1}[t]}\\\
{h_{3,1}^{k,2}[t]}&{h_{3,2}^{k,2}[t]}&{h_{3,3}^{k,2}[t]}&{h_{3,4}^{k,2}[t]}\\\
{h_{3,1}^{k,3}[t]}&{h_{3,2}^{k,3}[t]}&{h_{3,3}^{k,3}[t]}&{h_{3,4}^{k,3}[t]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{e_{2t-13}}}\\\
{{e_{2t{-12}}}}\\\ {{f_{2t-13}}}\\\ {{f_{2t{-}12}}}\end{array}}\right]\\\
\;\;\;\;\;\;\;\;\;{=}{{\textbf{L}}_{3}}\left(\\!{{e_{2t-13}},{e_{2t{-12}}},{f_{2t-13}},{f_{2t{-}12}}}\right)\\!,\\!t\\!\in\\!\\{7,8,9\\}\end{array}$
(28)

From (27), we know that
${{\textbf{L}}_{2}}\left({{{c}_{2t-7}},{{c}_{2t-6}}}\right),t\\!\in\\!\\{4,5,6\\}$
are the partical desired signals for the user 1 in cell 2 but the partical
interference signals for the user 2 in cell 2, and
${{\textbf{L}}_{2}}\left({{d_{2t-7}},{d_{2t-6}}}\right),t\\!\in\\!\\{4,5,6\\}$
is opposite to the former. In the meantime, the similar conclusion can be
gotten from (28). Therefore, to utilize partial desired signals, partial
interference signals should be eliminated with precoding matrix which is
desined at base station side. Therefore, the users feed interference signals
${{\textbf{L}}_{2}}\left({{{c}_{2t-7}},{{c}_{2t-6}},{d_{2t-7}},{d_{2t-6}}}\right),t\\!\in\\!\\{4,5,6\\}$
and
${{\textbf{L}}_{3}}\left({{e_{2t-13}},{e_{2t{-12}}},{f_{2t-13}},{f_{2t{-}12}}}\right),t\\!\in\\!\\{7,8,9\\}$
back to the base station 1 to achieve interference elimination by
${{\bf{U}}_{1}}[t]$.

In the second phase, as the analysis before,
$\left({{c}_{2t-7}},{{c}_{2t-6}}\right),t\\!\in\\!\\{4,5,6\\}$ and
$\left({{e_{2t-13}},{e_{2t{-}12}}}\right),t\\!\in\\!\\{7,8,9\\}$ are the
desidered symbols for the user 1 of cell 2 and the user 1 of cell 3,
respectively. After receiving feedback information, i.e.,
${\boldsymbol{y}}_{1,k}[t],t\\!\in\\!\\{4,5,6,7,8,9\\}$, the precoding matrix
${{\bf{U}}_{1}}[t],t\\!\in\\!\\{4,5,6\\}$ and
${{\bf{U}}_{1}}[t],t\\!\in\\!\\{7,8,9\\}$ are designed as (29) and (30),
respectively.

$\begin{array}[]{l}{{\bf{U}}_{1}}[t]{{\textbf{L}}_{2}}[t]={{\bf{U}}_{1}}[t]\left[{\begin{array}[]{*{20}{c}}{h_{2,1}^{k,1}[t]}&{h_{2,2}^{k,1}[t]}&{h_{2,3}^{k,1}[t]}&{h_{2,4}^{k,1}[t]}\\\
{h_{2,1}^{k,2}[t]}&{h_{2,2}^{k,2}[t]}&{h_{2,3}^{k,2}[t]}&{h_{2,4}^{k,2}[t]}\\\
{h_{2,1}^{k,3}[t]}&{h_{2,2}^{k,3}[t]}&{h_{2,3}^{k,3}[t]}&{h_{2,4}^{k,3}[t]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{{c}_{2t-7}}}\\\
{{{c}_{2t-6}}}\\\ {{d_{2t-7}}}\\\ {{d_{2t-6}}}\end{array}}\right]\\\
\qquad\qquad\quad\\!\\!={{\bf{U}}_{1}}[t]{\left[{\begin{array}[]{*{20}{c}}{h_{2,1}^{k,1}[t]}&{h_{2,1}^{k,2}[t]}&{h_{2,1}^{k,3}[t]}\\\
{h_{2,2}^{k,1}[t]}&{h_{2,2}^{k,2}[t]}&{h_{2,2}^{k,3}[t]}\end{array}}\right]^{T}}\left[{\begin{array}[]{*{20}{c}}{{{c}_{2t-7}}}\\\
{{{c}_{2t-6}}}\end{array}}\right]{+}{{\bf{U}}_{1}}[t]{\left[{\begin{array}[]{*{20}{c}}{h_{2,3}^{k,1}[t]}&{h_{2,3}^{k,2}[t]}&{h_{2,3}^{k,3}[t]}\\\
{h_{2,4}^{k,1}[t]}&{h_{2,4}^{k,2}[t]}&{h_{2,4}^{k,3}[t]}\end{array}}\right]^{T}}\left[{\begin{array}[]{*{20}{c}}{{d_{2t-7}}}\\\
{{d_{2t-6}}}\end{array}}\right]\\\
\qquad\qquad\quad\\!\\!={{\bf{U}}_{1}}[t]{\left[{\begin{array}[]{*{20}{c}}{h_{2,1}^{k,1}[t]}&{h_{2,1}^{k,2}[t]}&{h_{2,1}^{k,3}[t]}\\\
{h_{2,2}^{k,1}[t]}&{h_{2,2}^{k,2}[t]}&{h_{2,2}^{k,3}[t]}\end{array}}\right]^{T}}\left[{\begin{array}[]{*{20}{c}}{{{c}_{2t-7}}}\\\
{{{c}_{2t-6}}}\end{array}}\right]+{\bf{0}}\\\
\qquad\qquad\quad\\!\\!={\bf{L}}_{1}^{2,1}({{c}_{2t-7}},{{c}_{2t-6}})[t],t\in\\{4,5,6\\}\end{array}$
(29)

${{\bf{U}}_{1}}[t]{{}_{3}}[t]={\bf{L}}_{1}^{3,1}\left({{e_{2t-13}},{e_{2t{-}12}}}\right)[t],t\in\\{7,8,9\\}.$
(30)

Note that for the user 2 in the interference cells, the partial interference
signals ${\boldsymbol{L}}_{1}^{2,1}({{c}_{2t-7}},{{c}_{2t-6}})[t]$ and
${\boldsymbol{L}}_{1}^{3,1}\left({{e_{2t-13}},{e_{2t{-}12}}}\right)[t]$ can
also be used to eliminate IUI so that they can achieve equivalent functions
for the user 2. Then the interference regeneration matrix is used at the base
station 1 to avoid the additional interference in the third phases, i.e.

$\displaystyle{{\bf{V}}_{1}}[t]{\boldsymbol{L}}_{1}^{2,1}({{c}_{2t-7}},{{c}_{2t-6}})[t]$
$\displaystyle={{\bf{V}}_{3}}[t]{\boldsymbol{L}}_{3}^{2,1}({{c}_{2t-7}},{{c}_{2t-6}})[t],$
(31)
$\displaystyle{{\bf{V}}_{1}}[t]{\boldsymbol{L}}_{1}^{3,1}\left({{e_{2t-13}},{e_{2t{-}12}}}\right)[t]$
$\displaystyle={{\bf{V}}_{2}}[t]{\boldsymbol{L}}_{2}^{3,1}\left({{e_{2t-13}},{e_{2t{-}12}}}\right)[t].$
(32)

Herein, for the users in the cell 3, the additional partial interference
signals
${{\bf{V}}_{1}}[t]\boldsymbol{L}_{1}^{2,1}({{c}_{2t-7}},{{c}_{2t-6}})[t]$ is
convert to the partial interference signals, i.e.,
${{\bf{V}}_{3}}[t]{\boldsymbol{L}}_{3}^{2,1}({{c}_{2t-7}},{{c}_{2t-6}})[t]$
that the later one is known by the users in the cell 3. Meanwhile, for the
users in the cell 2, the signals are processed in the same way. Henceforth,
the additional interference caused by interference retransmission process can
be avoid.

In the third phase, all the base stations simutanously retransmit partial
desired signals as follows

$\displaystyle{{\bf{X}}_{1}}[10]$
$\displaystyle=\sum\limits_{t=4}^{6}{{{\bf{V}}_{1}}[t]}{\boldsymbol{L}}_{1}^{2,1}[t]{+}\sum\limits_{t=7}^{9}{{{\bf{V}}_{1}}[t]}{\boldsymbol{L}}_{1}^{3,1}[t],$
(33) $\displaystyle{{\bf{X}}_{2}}[10]$
$\displaystyle=\sum\limits_{t=1}^{3}{{{\bf{V}}_{2}}[t]}{\boldsymbol{L}}_{2}^{1,1}[t]{+}\sum\limits_{t=7}^{9}{{{\bf{V}}_{2}}[t]}{\boldsymbol{L}}_{2}^{3,1}[t],$
(34) $\displaystyle{{\bf{X}}_{3}}[10]$
$\displaystyle=\sum\limits_{t=1}^{3}{{{\bf{V}}_{3}}[t]}{\boldsymbol{L}}_{3}^{1,1}[t]{+}\sum\limits_{t=4}^{6}{{{\bf{V}}_{3}}[t]}{\boldsymbol{L}}_{3}^{2,1}[t].$
(35)

For the user $k$ in the cell 1, received signals can be written as

$\begin{array}[]{l}{{\bf{y}}_{1,k}}[10]={\bf{H}}_{1}^{[1,k]}[10]\sum\limits_{t=4}^{6}{{{\bf{V}}_{1}}[t]\boldsymbol{L}_{1}^{2,1}[t]\left({{c_{2t-7}},{c_{2t-6}}}\right)}\\\
+{\bf{H}}_{1}^{[1,k]}[10]\sum\limits_{t=7}^{9}{{{\bf{V}}_{1}}[t]{\boldsymbol{L}}_{1}^{3,1}[t]\left({{e_{2t-13}},{e_{2t-12}}}\right)}\\\
+{\bf{H}}_{2}^{[1,k]}[10]\sum\limits_{t=1}^{3}{{{\bf{V}}_{2}}[t]{\boldsymbol{L}}_{2}^{1,1}[t]\left({{a_{2t-1}},{a_{2t}}}\right)}\\\
+{\bf{H}}_{2}^{[1,k]}[10]\sum\limits_{t=7}^{9}{{{\bf{V}}_{2}}[t]{\boldsymbol{L}}_{2}^{3,1}[t]\left({{e_{2t-13}},{e_{2t-12}}}\right)}\\\
+{\bf{H}}_{3}^{[1,k]}[10]\sum\limits_{t=1}^{3}{{{\bf{V}}_{3}}[t]{\boldsymbol{L}}_{3}^{1,1}[t]\left({{a_{2t-1}},{a_{2t}}}\right)}\\\
+{\bf{H}}_{3}^{[1,k]}[10]\sum\limits_{t=4}^{6}{{{\bf{V}}_{3}}[t]{\boldsymbol{L}}_{3}^{2,1}[t]\left({{c_{2t-7}},{c_{2t-6}}}\right)}.\end{array}$
(36)

Owing to the existence of interference regeneration matrix, the partial
interference signals can be converted to the partial interference signals
known by the user $k$ in the cell 1. Thus, (36) can be simplified as

$\small\begin{array}[]{l}{{\bf{y}}^{i,k}}[10]\;{=}\left({{\bf{H}}_{2}^{[1,k]}[10]{+}{\bf{H}}_{3}^{[1,k]}[10]}\right)\sum\limits_{t=1}^{3}{{{\bf{V}}_{2}}[t]{\boldsymbol{L}}_{2}^{1,1}[t]\left({{a_{2t-1}},{a_{2t}}}\right){}}\\\
{+}\left({{\bf{H}}_{1}^{[1,k]}[10]{+}{\bf{H}}_{3}^{[1,k]}[10]}\right)\sum\limits_{t=4}^{6}{{{\bf{V}}_{1}}[t]{\boldsymbol{L}}_{1}^{2,1}[t]\left({{{c}_{2t-7}},{c_{2t{-6}}}}\right)}\\\
{+}\left({{\bf{H}}_{1}^{[1,k]}[10]{+}{\bf{H}}_{2}^{[1,k]}[10]}\right)\sum\limits_{t=7}^{9}{{{\bf{V}}_{1}}[t]{\boldsymbol{L}}_{1}^{3,1}[t]\left({{e_{2t-13}},{e_{2t{-12}}}}\right)}.\end{array}$
(37)

For (37), we know that
$({\bf{H}}_{2}^{[1,k]}[10]{+}{\bf{H}}_{3}^{[1,k]}[10])\sum\limits_{t=1}^{3}{{{\bf{V}}_{2}}[t]{\boldsymbol{L}}_{2}^{1,1}[t]}\\\
({a_{2t-1}},{a_{2t}})$ is the partial desired signals and the rest is the
known partial interference signals. Therefore, for the user 1 in the target
cell 1, the whole process can be summarized as the following equation (38)

$\left[{\begin{array}[]{*{20}{c}}{{{\boldsymbol{y}}_{1,1}}[1]},{{{\boldsymbol{y}}_{1,1}}[2]},{{{\boldsymbol{y}}_{1,1}}[3]},{{{\boldsymbol{y}}_{1,1}}[10]}\end{array}}\right]^{T}{\rm{=}}$

$\left[{\begin{array}[]{*{20}{c}}{h_{1,1}^{1,1}[1]}&{h_{1,2}^{1,1}[1]}&{h_{1,3}^{1,1}[1]}&{h_{1,4}^{1,1}[1]}&\cdots&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\\
{h_{1,1}^{1,2}[1]}&{h_{1,2}^{1,2}[1]}&{h_{1,3}^{1,2}[1]}&{h_{1,4}^{1,2}[1]}&\cdots&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\\
{h_{1,1}^{1,3}[1]}&{h_{1,2}^{1,3}[1]}&{h_{1,3}^{1,3}[1]}&{h_{1,4}^{1,3}[1]}&\cdots&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&\cdots&{h_{1,1}^{1,1}[3]}&{h_{1,2}^{1,1}[3]}&{h_{1,3}^{1,1}[3]}&{h_{1,4}^{1,1}[3]}\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&\cdots&{h_{1,1}^{1,2}[3]}&{h_{1,2}^{1,2}[3]}&{h_{1,3}^{1,2}[3]}&{h_{1,4}^{1,2}[3]}\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&\cdots&{h_{1,1}^{1,3}[3]}&{h_{1,2}^{1,3}[3]}&{h_{1,3}^{1,3}[3]}&{h_{1,4}^{1,3}[3]}\\\
{\sum\limits_{j=2,3}{h_{j,1}^{1,1}[10]}{\boldsymbol{\beta}_{1}}}&{\sum\limits_{j=2,3}{h_{j,2}^{1,1}[10]}{\boldsymbol{\beta}_{1}}}&{\sum\limits_{j=2,3}{h_{j,3}^{1,1}[10]}{\boldsymbol{\beta}_{1}}}&{\sum\limits_{j=2,3}{h_{j,4}^{1,1}[10]}{\boldsymbol{\beta}_{1}}}&\cdots&{\sum\limits_{j=2,3}{h_{j,1}^{1,1}[10]}{\boldsymbol{\beta}_{3}}}&{\sum\limits_{j=2,3}{h_{j,2}^{1,1}[10]}{\boldsymbol{\beta}_{3}}}&{\sum\limits_{j=2,3}{h_{j,3}^{1,1}[10]}{\boldsymbol{\beta}_{3}}}&{\sum\limits_{j=2,3}{h_{j,4}^{1,1}[10]}{\boldsymbol{\beta}_{3}}}\\\
{\sum\limits_{j=2,3}{h_{j,1}^{1,2}[10]}{\boldsymbol{\beta}_{1}}}&{\sum\limits_{j=2,3}{h_{j,2}^{1,2}[10]}{\boldsymbol{\beta}_{1}}}&{\sum\limits_{j=2,3}{h_{j,3}^{1,2}[10]}{\boldsymbol{\beta}_{1}}}&{\sum\limits_{j=2,3}{h_{j,4}^{1,2}[10]}{\boldsymbol{\beta}_{1}}}&\cdots&{\sum\limits_{j=2,3}{h_{j,1}^{1,2}[10]}{\boldsymbol{\beta}_{3}}}&{\sum\limits_{j=2,3}{h_{j,2}^{1,2}[10]}{\boldsymbol{\beta}_{3}}}&{\sum\limits_{j=2,3}{h_{j,3}^{1,2}[10]}{\boldsymbol{\beta}_{3}}}&{\sum\limits_{j=2,3}{h_{j,4}^{1,2}[10]}{\boldsymbol{\beta}_{3}}}\\\
{\sum\limits_{j=2,3}{h_{j,1}^{1,3}[10]}{\boldsymbol{\beta}_{1}}}&{\sum\limits_{j=2,3}{h_{j,2}^{1,3}[10]}{\boldsymbol{\beta}_{1}}}&{\sum\limits_{j=2,3}{h_{j,3}^{1,3}[10]}{\boldsymbol{\beta}_{1}}}&{\sum\limits_{j=2,3}{h_{j,4}^{1,3}[10]}{\boldsymbol{\beta}_{1}}}&\cdots&{\sum\limits_{j=2,3}{h_{j,1}^{1,3}[10]}{\boldsymbol{\beta}_{3}}}&{\sum\limits_{j=2,3}{h_{j,2}^{1,3}[10]}{\boldsymbol{\beta}_{3}}}&{\sum\limits_{j=2,3}{h_{j,3}^{1,3}[10]}{\boldsymbol{\beta}_{3}}}&{\sum\limits_{j=2,3}{h_{j,4}^{1,3}[10]}{\boldsymbol{\beta}_{3}}}\end{array}}\right]\\!\\!\\!\\!\left[{\begin{array}[]{*{20}{c}}{{a_{1}}}\\\
{{a_{2}}}\\\ {{b_{1}}}\\\ {{b_{2}}}\\\ \vdots\\\ {{a_{5}}}\\\ {{a_{6}}}\\\
{{b_{5}}}\\\ {{b_{6}}}\end{array}}\right]$ (38)

where
${{\boldsymbol{\beta}}_{t}}{=}{v_{2}}[t]{u_{2}}[t]h_{1}^{2,1}[t]+{v_{3}}[t]{u_{3}}[t]h_{1}^{3,1}[t]$.
It is obvious that the channel matrix is a $12\times 12$ non-singular matrix.
Herein, user 1 in the target cell 1 can decode its messages with delayed CSI.

From the above illustration, 36 independent symbols are transmitted by three
base stations over 10 coherent slots. Therefore, the DoF of the proposed RIR
scheme is 3.6. While under the same conditions, i.e., $(L,M,K,N)=(3,4,2,3)$,
the DoF for the TDMA scheme and RIA scheme are only 3 and 3.28, respectively.

##  IV Beamforming Based Distributed Retrospective Interference Alignment
Scheme

For the proposed RIR scheme, though it can achieve greater DoF than the RIA
scheme, the scheme has the problem of performance degradation as the
transceiver antennas ratio $\rho$ approaches 1. To handle this issue, a new
beamforming based distributed retrospective interference alignment (B-DRIA)
scheme is proposed. The core idea of this scheme is that the beamforming
technique is adopted to eliminate ICI and then, several additional slots are
taken for the base stations to simultaneously transmit desired signals.
Finally, we design distributed retrospective interference matrices to align
the IUI at the user side and decode the messages separately. In the following,
we firstly illustrate the details of the proposed scheme which includes three
phases, i.e., CSIT acquisition process, cellular beamforming and signal
transmission, and distributed retrospective interference alignment. Then, the
DoF of the B-DRIA scheme is analyzed. To make it understandable, we also
present an example of the scheme at last.

###  IV-A CSIT Acquisition Process

The first phase takes $L$ slots and in which the base stations take turns to
send messages to its served users. Since $M>N$, these desired messages cannot
be decoded within only one slot and additional transmissions are required.
Therefore, the users do not complete decoding in the current slot but feedback
the estimated CSI to the base stations. The CSI contains both the channel gain
estimation and the DoA estimation where the former part is used for
retrospective interference alignment and the latter part is taken in cellular
beamforming. Taking the cell 1 for example, the process of the first phase is
illustrated as Fig. 7.

Figure 7: CSIT Acquisition process in 1st phase.

To make it universal applicable, assume that in the slot $i\in{\cal L}$, the
base station $i\in{\cal L}$ transmits messages as

${\boldsymbol{S}}_{i}[i]=\left[{{s_{i,1}}[i],\cdots{s_{i,\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil}}[i],\cdots,{s_{i,K\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil}}[i]}\right].$
(39)

Then for user $k$ in the target cell $j$, its received signals can be
expressed in time domain and angular domain as follows

$\displaystyle{\boldsymbol{y}}_{j,k}[i]={\bf{H}}_{i}^{j,k}[t]{{\boldsymbol{S}}_{i}}[t],i\in{\cal
L},j=i,$ (40)
$\displaystyle{\boldsymbol{y}}_{j,k}[t]={\bf{A}}_{i}^{j,k}[t]{{\boldsymbol{S}}_{i}}[t],i\in{\cal
L},j=i.$ (41)

Based on the received signals, the LS algorithm [37] and the multiple signal
classification algorithm algorithm (MUSIC) [38] can be used at the user side
to estimate the channel gain and DoA, respectively. With channel estimation,
the feedback information from user $k$ is extended into two parts, i.e., the
channel gain information and the angel information, and its form can be
characterized by

$CSIT{\\!=\\!}\left\\{{\left({{\bf{H}}_{i}^{i,1}[i]\\!,\\!{\theta_{1}}[i]}\right)\\!,\\!\\!\cdots\\!\\!,\\!\left({{\bf{H}}_{i}^{i,k}[i]\\!,\\!{\theta_{k}}[i]}\right)}\right\\}\\!,\\!i\in{\cal
L},$ (42)

where $\left({{\bf{H}}_{i}^{i,k}[i],{\theta_{k}}[i]}\right)$ denotes the
channel matrix from base station $i$ to user $k$ in the target cell $i$ and
the corresponding DoA of that user.

###  IV-B Cellular Beamforming and Signal Transmission

Following the CSIT acquisition process, we can further perform the process of
cellular beamforming and signal transmission. While this phase spans
$\bar{\varphi}=\left\lfloor{{{2N-M}\mathord{\left/{\vphantom{{2N-M}{M-N}}}\right.\kern-1.2pt}{M-N}}}\right\rfloor$
slots, the base stations simultaneously transmit independent messages to their
served users. However, the desired signals received at the user sides are
contaminated by both ICI and IUI. Therefore, additional operations are needed
to eliminate interference signals.

In order to cancel the ICI, beamforming is adopted at the base stations.
Define the slots set of this phase as
$\Psi{=}\left\\{{L+1,\cdots,L{+}\bar{\varphi}}\right\\}$ and the beamforming
matrices are

$\displaystyle{\bf{W}}_{i}^{j,k}[t]{=}{{\bf{0}}_{M\times M}},j\in{\cal
L}\backslash i,k\in{\textbf{K}},t\in\Psi,$ (43)
$\displaystyle{\bf{W}}_{i}^{j,k}[t]{=}{{\bf{I}}_{M\times
M}},i=j,k\in{\textbf{K}},t\in\Psi.$ (44)

where ${{\bf{0}}_{M\times M}}$ is a zero matrix and ${{\bf{I}}_{M\times M}}$
is an $M$-dimensional identity matrix. Then, for one slot $t,\forall
t\in\Psi$, the signals sent by base station $i$ to user $k$ in the target cell
$j$ is

$\begin{array}[]{l}{\boldsymbol{X}}_{i}^{j,k}[t]={\bf{W}}_{i}^{j,k}[t]{\boldsymbol{S}}_{i}[t]\\\
\qquad\quad\,={{\bf{I}}_{M\times
M}}{\left[{{s_{i,1}}[t],\cdots,{s_{i,M}}[t]}\right]^{T}}\\\
\qquad\quad\,={\left[{{s_{i,1}}[t],\cdots,{s_{i,M}}[t]}\right]^{T}}.\end{array}$
(45)

After performing the beamforming with ${\bf{W}}_{i}^{j,k}$, the ICI is
eliminated which means that all the users will not receive signals from the
base stations of the interference cells. This process is shown in Fig. 8.

Figure 8: Signals transmission with beamforming in 2nd phase.

Thus, the received signals of the user $k$ in the cell $j$ can be expressed as

${\boldsymbol{y}}_{j,k}[t]={\bf{H}}_{i}^{i,k}[t]{\boldsymbol{X}}_{i}^{j,k}[t].$
(46)

Due to the presence of the IUI, no users can decode the received message in
one slot. By analyzing the components of the received signals, we find that,
for user $k$, the IUI contained desired information for the other users, i.e.,
the received signals of the user $k$ can be written as

$\begin{array}[]{l}{\boldsymbol{y}}_{j,k}[t]={\bf{H}}_{i}^{i,k}[t]{{\boldsymbol{S}}_{i}}[t]+{\bf{H}}_{i}^{i,-k}[t]{{\boldsymbol{S}}_{i}}[t]\\\
\qquad\quad\\!=\boldsymbol{L}\left({{s_{i,(k-1)\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil+1}}[L+t],\cdots,{s_{i,k\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil}}[L+t]}\right)\\\
\qquad\quad\\!+\boldsymbol{I}\left({{s_{i,(k-1)\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil+1}}[L+t],\cdots,{s_{i,k\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil}}[L+t]}\right)\\\
\qquad\quad\\!=\boldsymbol{L}_{i}^{i,k}[t]{+}\boldsymbol{I}_{i}^{i,k}[t],-k={\textbf{K}}\backslash
k,\end{array}$ (47)

where $\boldsymbol{L}_{i}^{i,k}[t]$ represents the desired signals and
$\boldsymbol{I}_{i}^{i,k}[t]$ represents IUI. Therefore, through
$L{+}\bar{\varphi}$ slots, there are $\bar{\varphi}M+M$ messages needed to be
decoded for each user and we only have $\bar{\varphi}N+N$ equations. In order
to decode these messages, some extra information is desired, i.e., to
formulate $N$ additional equations of each user.

###  IV-C Distributed Retrospective Interference Alignment

Based on the first phase and the second phase, especially the CSI estimation
and feedback, each base station can obtain the knowledge of the CSI and the
IUI feedback, i.e., $\boldsymbol{I}_{i}^{i,k}[t]$. Therefore, the distributed
retrospective interference alignment is developed to make use of the
combination of the IUI from the previous slots. Part of which can be extracted
by the users to supplement $N$ additional equations. In specific, the designed
vector from the base station in the cell $i$ is shown as follows

${{\boldsymbol{S}}_{i}}[L\\!+\\!\bar{\varphi}{\\!+\\!}1]{=}\left[{\begin{array}[]{*{20}{c}}{\sum\limits_{k=1}^{K}{\left({I_{i}^{i,k}[i]\\!+\\!I_{i}^{i,k}[L\\!+\\!1]\\!+\\!\cdots\\!+\\!I_{i}^{i,k}[L\\!+\\!\bar{\varphi}]}\right)}}\\\
0\\\ \vdots\\\ 0\end{array}}\right],$ (48)

where
${\sum\limits_{k=1}^{K}{\left({\boldsymbol{I}_{i}^{i,k}[i]\\!+\\!\boldsymbol{I}_{i}^{i,k}[L\\!+\\!1]\\!+\\!\cdots\\!+\\!\boldsymbol{I}_{i}^{i,k}[L\\!+\\!\bar{\varphi}]}\right)}}$
is a $N$-dimensional vector composed of the whole past slots’ IUI. By the pre-
mentioned beamforming process and the distributed interference alignment
scheme, the base stations are capable of transmitting the vector
${{\boldsymbol{S}}_{i}}[L\\!+\\!\bar{\varphi}{\\!+\\!}1]$ without introducing
new ICI or IUI, as shown in Fig. 9.

Figure 9: Regenerated interference retransmission in 3rd phase.

Then, the received signals of the user $k$ in the cell $j$ can be expressed as

$\begin{array}[]{l}{\boldsymbol{y}}_{j,k}[L+\bar{\varphi}{+}1]\\\
={\bf{H}}_{i}^{j,k}[L+\bar{\varphi}{+}1]{\bf{W}}_{i}^{j,k}[L+\bar{\varphi}{+}1]{{\boldsymbol{S}}_{i}}[L+\bar{\varphi}{+}1].\end{array}$
(49)

From (49), the desired signals can be extracted from the combination of the
IUI by the interference elimination process

$\begin{array}[]{l}{\boldsymbol{y}}_{j,k}[L+\bar{\varphi}{+}1]-{\bf{H}}_{i}^{j,k}[L+\bar{\varphi}{+}1]\sum\limits_{t=1}^{L+\bar{\varphi}}{\boldsymbol{I}_{i}^{i,-k}[t]}\\\
={\bf{H}}_{i}^{j,k}[L+\bar{\varphi}{+}1](\boldsymbol{I}_{i}^{i,k}[i]+\sum\limits_{t=L+1}^{L+\bar{\varphi}}{\boldsymbol{I}_{i}^{i,k}[t]}),\end{array}$
(50)

where
$\boldsymbol{I}_{i}^{i,k}[i]+\sum\limits_{t=L+1}^{L+\bar{\varphi}}{\boldsymbol{I}_{i}^{i,k}[t]}$
is the supplementary of the $N$ additional equations. Herein, the whole
process of the proposed B-DRIA scheme can be summarized in (51),

$\underbrace{\left[{\begin{array}[]{*{20}{c}}{{\bf{y}}_{j,k}[i]}\\\
{{\bf{y}}_{j,k}[L+1]}\\\ \vdots\\\ {{\bf{y}}_{j,k}[L+\bar{\varphi}]}\\\
{{\bf{y}}_{j,k}[L+\bar{\varphi}+1]}\end{array}}\right]}_{\bf{Y}}{=}\underbrace{\left[{\begin{array}[]{*{20}{c}}{{\bf{H}}_{i}^{i,k}[i]}&0&\cdots&0\\\
0&{{\bf{H}}_{i}^{i,k}[L+1]}&\cdots&\vdots\\\ \vdots&\ddots&\ddots&0\\\
0&\cdots&0&{{\bf{H}}_{i}^{i,k}[L+\bar{\varphi}]}\\\
{{\bf{H}}_{i}^{i,k}[L+\bar{\varphi}{+}1]{{\bf{\beta}}_{i}}}&{{\bf{H}}_{i}^{i,k}[L+\bar{\varphi}{+}1]{{\bf{\beta}}_{L+1}}}&\cdots&{{\bf{H}}_{i}^{i,k}[L+\bar{\varphi}{+}1]{{\bf{\beta}}_{L+\bar{\varphi}}}}\end{array}}\right]}_{\bf{H}}\underbrace{\left[{\begin{array}[]{*{20}{c}}{{{\bf{S}}_{i}}[i]}\\\
{{{\bf{S}}_{i}}[L+1]}\\\ \vdots\\\
{{{\bf{S}}_{i}}[L+\bar{\varphi}]}\end{array}}\right]}_{\bf{S}}$ (51)

where ${{\boldsymbol{\beta}}_{t}}$ is the precoding matrix defined as follows

${{\boldsymbol{\beta}}_{t}}={\bf{W}}_{i}^{j,k}[t]\sum\nolimits_{k={\textbf{K}}\backslash
k}{{\bf{H}}_{i}^{i,k}[t]}.$ (52)

Since the delayed CSIT is adopted in the transmission process, which means
that, for any two different slots, the time span exceeds the coherence time,
the channels are independent with each other over different slots, e.g.,
${\bf{H}}_{i}^{i,k}[L+1]$ and ${\bf{H}}_{i}^{i,k}[L+\bar{\varphi}]$. In
addition, in each slot, the channel matrix is non-singular, which means that
each submatrix has a full rank. Herein, the matrix $\bf{H}$ is a
$\left\lfloor{{N\mathord{\left/{\vphantom{N{M-N}}}\right.\kern-1.2pt}{M-N}}}\right\rfloor
M$ dimensional non-singular matrix, and the LS method [37] can be used to
decode the desired messages.

###  IV-D Degrees of Freedom

Following the above illustration, we analyze the DoF of the proposed B-DRIA
scheme. Since the base stations spend $L{+}\bar{\varphi}{+}1$ incoherent slots
and transmit
$L\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil
K(\bar{\varphi}{+}1)$ independent symbols, we assume that the transmission is
reliable, i.e., the value of signal-to-noise ratio (SNR) is approximated to
infinity. Hence the DoF of the B-DRIA scheme can be calculated as

$DoF{=}{{L\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil
K(\bar{\varphi}{+}1)}\mathord{\left/{\vphantom{{L\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil
K(\bar{\varphi}{+}1)}{(L{+}\bar{\varphi}{+}1)}}}\right.\kern-1.2pt}{(L{+}\bar{\varphi}{+}1)}}.$
(53)

To facilitate the analysis, we assume that $M$ is divisible by $K$. Then, the
above equation can be simplified as

$\begin{array}[]{l}DoF{{{=}LM}\mathord{\left/{\vphantom{{{=}LM}{(1+{L\mathord{\left/{\vphantom{L{(\bar{\varphi}{+}1)}}}\right.\kern-1.2pt}{(\bar{\varphi}{+}1)}})}}}\right.\kern-1.2pt}{(1+{L\mathord{\left/{\vphantom{L{(\bar{\varphi}{+}1)}}}\right.\kern-1.2pt}{(\bar{\varphi}{+}1)}})}}.\end{array}$
(54)

As mentioned earlier, the number of cells $L$ is constant and the signal
transmission slots in the second phase $\bar{\varphi}$ is a function of $M$
and $N$. Hence, the value of DoF mainly depends on transceiver $\rho$ and $M$,
and the DoF is rewritten as

$DoF{{{=}LM}\mathord{\left/{\vphantom{{{=}LM}{(1+{L\mathord{\left/{\vphantom{L{\left\lfloor{{3\mathord{\left/{\vphantom{3{(\rho-1)}}}\right.\kern-1.2pt}{(\rho-1)}}}\right\rfloor}}}\right.\kern-1.2pt}{\left\lfloor{{3\mathord{\left/{\vphantom{3{(\rho-1)}}}\right.\kern-1.2pt}{(\rho-1)}}}\right\rfloor}})}}}\right.\kern-1.2pt}{(1+{L\mathord{\left/{\vphantom{L{\left\lfloor{{3\mathord{\left/{\vphantom{3{(\rho-1)}}}\right.\kern-1.2pt}{(\rho-1)}}}\right\rfloor}}}\right.\kern-1.2pt}{\left\lfloor{{3\mathord{\left/{\vphantom{3{(\rho-1)}}}\right.\kern-1.2pt}{(\rho-1)}}}\right\rfloor}})}}.$
(55)

With the increase of $\rho$, the valued of the DoF can be separately discussed
under two cases/conditions. That is, when $M$ is not divisible by $M-N$, the
value of
$\bar{\varphi}=\left\lfloor{{{2N-M}\mathord{\left/{\vphantom{{2N-M}{M-N}}}\right.\kern-1.2pt}{M-N}}}\right\rfloor$
will not change owing to the round down operation
$\left\lfloor{*}\right\rfloor$ and the DoF keeps constant. While if $M$ is
divisible by $M-N$, the value of the DoF will grow sharply. Therefore, the
value of $\bar{\varphi}$ is divided into three cases, i.e., $\bar{\varphi}>1$,
$\bar{\varphi}=1$, $\bar{\varphi}=0$, and the corresponding DoF gains of each
situation are further discussed in Section V.

###  IV-E Typical Application

For the comparison purpose, the condition used herein is the same as that
adopted for the RIR scheme, that is, $(L,M,K,N)=(3,4,2,3)$, which means that
the network is composed of 3 cells and in each cell, the base station with 4
antennas provides services to 2 users, where each user has 3 antennas. For
this scenario, the B-DRIA scheme is adopted and the whole process is
illustrated as follows.

The ﬁrst phase takes 3 slots, in which the base stations take turns to send
messages. In each slot, the symbols sent by the base stations are shown as
follows

$\begin{array}[]{l}{{\boldsymbol{S}}_{1}}[1]=\left({\begin{array}[]{*{20}{c}}{{a_{1}}},{{a_{2}}},{{b_{1}}},{{b_{2}}}\end{array}}\right)^{T},\\\
{{\boldsymbol{S}}_{2}}[2]=\left({\begin{array}[]{*{20}{c}}{{c_{1}}},{{c_{2}}},{{d_{1}}},{{d_{2}}}\end{array}}\right)^{T},\\\
{{\boldsymbol{S}}_{3}}[3]=\left({\begin{array}[]{*{20}{c}}{{e_{1}}},{{e_{2}}},{{f_{1}}},{{f_{2}}}\end{array}}\right)^{T}.\end{array}$
(56)

Take the user $k$ in the target cell 1 for example, in the slot 1, the
received signals can be expressed as

$\begin{array}[]{l}{\boldsymbol{y}}_{1,k}[1]\\!=\\!{\bf{H}}_{1}^{1,k}[1]{{\boldsymbol{S}}_{1}}[1]\\\
\qquad\quad\\!\\!=\\!\left[{\begin{array}[]{*{20}{c}}{h_{1,1}^{k,1}[1]}&{h_{1,2}^{k,1}[1]}&{h_{1,3}^{k,1}[1]}&{h_{1,4}^{k,1}[1]}\\\
{h_{1,1}^{k,2}[1]}&{h_{1,2}^{k,2}[1]}&{h_{1,3}^{k,2}[1]}&{h_{1,4}^{k,2}[1]}\\\
{h_{1,1}^{k,3}[1]}&{h_{1,2}^{k,3}[1]}&{h_{1,3}^{k,3}[1]}&{h_{1,4}^{k,3}[1]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{1}}}\\\
{{a_{2}}}\\\ {{b_{1}}}\\\ {{b_{2}}}\end{array}}\right]\end{array},$ (57)
$\begin{array}[]{l}{\boldsymbol{y}}_{1,k}[1]\\!=\\!{\bf{A}}_{1}^{1,k}[1]{{\boldsymbol{S}}_{1}}[1]\\\
\qquad\quad\\!\\!=\\!\left[{\begin{array}[]{*{20}{c}}1&1&1&1\\\
{{e^{-j{\mu_{1}}}}}&{{e^{-j{\mu_{2}}}}}&{{e^{-j{\mu_{3}}}}}&{{e^{-j{\mu_{4}}}}}\\\
{{e^{-j2{\mu_{1}}}}}&{{e^{-j2{\mu_{2}}}}}&{{e^{-j2{\mu_{3}}}}}&{{e^{-j2{\mu_{4}}}}}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{1}}}\\\
{{a_{2}}}\\\ {{b_{1}}}\\\ {{b_{2}}}\end{array}}\right]\end{array}.$ (58)

The former is the time domain expression where the channel gain matrix is
estimated by the LS algorithm[37], and the latter is the angular domain
expression where the DoA matrix is obtained by the MUSIC algorithm [38].
Therefore, the feedback estimation contains two parts, i.e., the channel gain
information ${\bf{H}}_{1}^{1,k}[1]$ and the angel information
${\theta_{1}}[1]$. Since $M>N$, user $k$ cannot decode the messages
${{\bf{S}}_{1}}[1]$ within one slot and meanwhile, the estimated CSI is fed
back to the base station 1. Similarly, the user $k$ of cell 2 receive the
signals in slot 2 and the user $k$ of cell 3 receive signals in slot 3 as

$\begin{array}[]{l}{\bf{y}}_{2,k}[2]={\bf{H}}_{2}^{2,k}[2]{{\boldsymbol{S}}_{2}}[2]\\\
\;\;\;\;\;\;\;\;=\left[{\begin{array}[]{*{20}{c}}{h_{2,1}^{k,1}[2]}&{h_{2,2}^{k,1}[2]}&{h_{2,3}^{k,1}[2]}&{h_{2,4}^{k,1}[2]}\\\
{h_{2,1}^{k,2}[2]}&{h_{2,2}^{k,2}[2]}&{h_{2,3}^{k,2}[2]}&{h_{2,4}^{k,2}[2]}\\\
{h_{2,1}^{k,3}[2]}&{h_{2,2}^{k,3}[2]}&{h_{2,3}^{k,3}[2]}&{h_{2,4}^{k,3}[2]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{c_{1}}}\\\
{{c_{2}}}\\\ {{d_{1}}}\\\ {{d_{2}}}\end{array}}\right]\end{array},$ (59)
$\begin{array}[]{l}{\bf{y}}_{3,k}[t]={\bf{H}}_{3}^{3,k}[t]{{\boldsymbol{S}}_{3}}[t]\\\
\;\;\;\;\;\;\;\;{=}\left[{\begin{array}[]{*{20}{c}}{h_{3,1}^{k,1}[3]}&{h_{3,2}^{k,1}[3]}&{h_{3,3}^{k,1}[3]}&{h_{3,4}^{k,1}[3]}\\\
{h_{3,1}^{k,2}[3]}&{h_{3,2}^{k,2}[3]}&{h_{3,3}^{k,2}[3]}&{h_{3,4}^{k,2}[3]}\\\
{h_{3,1}^{k,3}[3]}&{h_{3,2}^{k,3}[3]}&{h_{3,3}^{k,3}[3]}&{h_{3,4}^{k,3}[3]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{e_{1}}}\\\
{{e_{2}}}\\\ {{f_{1}}}\\\ {{f_{2}}}\end{array}}\right],\end{array}$ (60)

By channel estimation, both channel gain information ${\bf{H}}_{2}^{2,k}[2]$
and angel information ${\theta_{2}}[2]$ are obtained and fed back to the base
station 2, and it is the same for ${\bf{H}}_{3}^{3,k}[3]$ and
${\theta_{3}}[3]$.

In the second phase, after obtaining CSI, the beamforming is used by the base
station to eliminate the ICI. Taking base station 1 for example, beamforming
matrices are written as

$\begin{array}[]{l}{\bf{W}}_{1}^{j,k}[t]{=}\left[{\begin{array}[]{*{20}{c}}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}}\right]\\\ {=}{{\sl\bf{0}}_{4\times
4}},j\in{\cal L}\backslash 1,t\in\Psi,\end{array}$ (61)
$\begin{array}[]{l}{\bf{W}}_{1}^{1,k}[t]{=}\left[{\begin{array}[]{*{20}{c}}1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{array}}\right]\\\ {=}{{\bf{I}}_{4\times
4}},t\in\Psi,\end{array}$ (62)

where $\Psi$ denotes the slots set of the second phase. For base station 1,
the transmitted signals are changed to

$\begin{array}[]{l}{{\bf{X}}_{1}}[t]={{\bf{W}}_{1}}[t]{{\boldsymbol{S}}_{1}}[t]\\\
={\left[{\begin{array}[]{*{20}{c}}{{{\bf{I}}_{4\times 4}}}&{{{\bf{0}}_{4\times
4}}}&{{{\bf{0}}_{4\times
4}}}\end{array}}\right]^{T}}{\left[{\begin{array}[]{*{20}{c}}{{a_{1}}}&{{a_{2}}}&{{b_{1}}}&{{b_{2}}}\end{array}}\right]^{T}}.\end{array}$
(63)

After the process of the beamforming, the base stations are able to
simultaneously transmit signals without introducing additional ICI. However,
the unsolvable issue which caused by the condition $M>N$ is still exist. While
from (59) and (60), it is not difficult to infer that, for the case $M=4$ and
$N=3$, only 1 ($=M-N$) additional equation is desired in each slot. In order
to make full use of the time resources, 2 more slots are taken to transmit
signals, i.e., $\Psi{=}\left\\{{4,5}\right\\}$. Herein, for user $k$ in cell
1, in slot 4 and slot 5, the received signals can be written as follows,
respectively,

$\small\begin{array}[]{l}{\bf{y}}_{1,k}[4]\\\
={\bf{H}}_{1}^{1,k}[4]{\bf{W}}_{1}^{1,k}[4]{{\boldsymbol{S}}_{1}}[4]\\\
=\left[{\begin{array}[]{*{20}{c}}{h_{1,1}^{k,1}[4]}&{h_{1,2}^{k,1}[4]}&{h_{1,3}^{k,1}[4]}&{h_{1,4}^{k,1}[4]}\\\
{h_{1,1}^{k,2}[4]}&{h_{1,2}^{k,2}[4]}&{h_{1,3}^{k,2}[4]}&{h_{1,4}^{k,2}[4]}\\\
{h_{1,1}^{k,3}[4]}&{h_{1,2}^{k,3}[4]}&{h_{1,3}^{k,3}[4]}&{h_{1,4}^{k,3}[4]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{3}}}\\\
{{a_{4}}}\\\ {{b_{3}}}\\\ {{b_{4}}}\end{array}}\right]\\\
=\left[{\begin{array}[]{*{20}{c}}{h_{1,1}^{k,1}[4]}&{h_{1,2}^{k,1}[4]}\\\
{h_{1,1}^{k,2}[4]}&{h_{1,2}^{k,2}[4]}\\\
{h_{1,1}^{k,3}[4]}&{h_{1,2}^{k,3}[4]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{3}}}\\\
{{a_{4}}}\end{array}}\right]\vspace{0.5ex}\\\
+\left[{\begin{array}[]{*{20}{c}}{h_{1,3}^{k,1}[4]}&{h_{1,4}^{k,1}[4]}\\\
{h_{1,3}^{k,2}[4]}&{h_{1,4}^{k,2}[4]}\\\
{h_{1,3}^{k,3}[4]}&{h_{1,4}^{k,3}[4]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{b_{3}}}\\\
{{b_{4}}}\end{array}}\right]\\\
={\bf{L}}_{1}^{1,k}[4]{+}{\bf{I}}_{1}^{1,k}[4],\end{array}$ (64)
$\small\begin{array}[]{l}{\bf{y}}_{1,k}[5]\\\
={\bf{H}}_{1}^{1,k}[5]{\bf{W}}_{1}^{1,k}[5]{{\boldsymbol{S}}_{1}}[5]\\\
=\left[{\begin{array}[]{*{20}{c}}{h_{1,1}^{k,1}[5]}&{h_{1,2}^{k,1}[5]}&{h_{1,3}^{k,1}[5]}&{h_{1,4}^{k,1}[5]}\\\
{h_{1,1}^{k,2}[5]}&{h_{1,2}^{k,2}[5]}&{h_{1,3}^{k,2}[5]}&{h_{1,4}^{k,2}[5]}\\\
{h_{1,1}^{k,3}[5]}&{h_{1,2}^{k,3}[5]}&{h_{1,3}^{k,3}[5]}&{h_{1,4}^{k,3}[5]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{5}}}\\\
{{a_{6}}}\\\ {{b_{5}}}\\\ {{b_{6}}}\\\ \end{array}}\right]\\\
=\left[{\begin{array}[]{*{20}{c}}{h_{1,1}^{k,1}[5]}&{h_{1,2}^{k,1}[5]}\\\
{h_{1,1}^{k,2}[5]}&{h_{1,2}^{k,2}[5]}\\\
{h_{1,1}^{k,3}[5]}&{h_{1,2}^{k,3}[5]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{5}}}\\\
{{a_{6}}}\\\ \end{array}}\right]\vspace{0.5ex}\\\
+\left[{\begin{array}[]{*{20}{c}}{h_{1,3}^{k,1}[5]}&{h_{1,4}^{k,1}[5]}\\\
{h_{1,3}^{k,2}[5]}&{h_{1,4}^{k,2}[5]}\\\
{h_{1,3}^{k,3}[5]}&{h_{1,4}^{k,3}[5]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{b_{5}}}\\\
{{b_{6}}}\\\ \end{array}}\right]\\\
={\boldsymbol{L}}_{1}^{1,k}[5]{+}{\boldsymbol{I}}_{1}^{1,k}[5].\end{array}$
(65)

Note that $\boldsymbol{L}_{i}^{i,k}[t]$ represents the desired signals and
$\boldsymbol{I}_{i}^{i,k}[t]$ represents the IUI and which means that, if
$k=1$, the former is the desired signals while the later is the IUI, that is

${\boldsymbol{L}}_{1}^{1,1}[5]{=}\left[{\begin{array}[]{*{20}{c}}{h_{1,1}^{1,1}[5]}&{h_{1,2}^{1,1}[5]}\\\
{h_{1,1}^{1,2}[5]}&{h_{1,2}^{1,2}[5]}\\\
{h_{1,1}^{1,3}[5]}&{h_{1,2}^{1,3}[5]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{5}}}\\\
{{a_{6}}}\end{array}}\right],$ (66)
$\vspace{-0.5em}{\boldsymbol{I}}_{1}^{1,1}[5]{=}\left[{\begin{array}[]{*{20}{c}}{h_{1,3}^{1,1}[5]}&{h_{1,4}^{1,1}[5]}\\\
{h_{1,3}^{1,2}[5]}&{h_{1,4}^{1,2}[5]}\\\
{h_{1,3}^{1,3}[5]}&{h_{1,4}^{1,3}[5]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{b_{5}}}\\\
{{b_{6}}}\end{array}}\right],$ (67)

and this is the same for the case $k=2$. Now, the signals from different cells
are independent, thus they are able to align the interference separately and
the DRIA algorithm is adopted.

In the third phase, with the knowledge of delayed CSIT, base station 1
regenerates interference signals of previous slots as

$\begin{array}[]{l}{{\boldsymbol{I}}_{1}^{1,1}}[1]+{{\boldsymbol{I}}_{1}^{1,2}}[1]\\\
\\!=\\!\left[{\begin{array}[]{*{20}{c}}{h_{1,3}^{1,1}[1]}&{h_{1,4}^{1,1}[1]}\\\
{h_{1,3}^{1,2}[1]}&{h_{1,4}^{1,2}[1]}\\\
{h_{1,3}^{1,3}[1]}&{h_{1,4}^{1,3}[1]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{b_{1}}}\\\
{{b_{2}}}\end{array}}\right]\\!+\\!\left[{\begin{array}[]{*{20}{c}}{h_{1,3}^{2,1}[1]}&{h_{1,4}^{2,1}[1]}\\\
{h_{1,3}^{2,2}[1]}&{h_{1,4}^{2,2}[1]}\\\
{h_{1,3}^{2,3}[1]}&{h_{1,4}^{2,3}[1]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{1}}}\\\
{{a_{2}}}\end{array}}\right]\\\
\\!=\\!\left[{\begin{array}[]{*{20}{c}}{{{\boldsymbol{h}}_{1,3}^{2}}[1]}&{{{\boldsymbol{h}}_{1,4}^{2}}[1]}&{{{\boldsymbol{h}}_{1,3}^{1}}[1]}&{{{\boldsymbol{h}}_{1,4}^{1}[1]}}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{1}}}\\\
{{a_{2}}}\\\ {{b_{1}}}\\\ {{b_{2}}}\end{array}}\right],\end{array}$ (68)
$\begin{array}[]{l}{{\boldsymbol{I}}_{1}^{1,1}}[4]+{{\boldsymbol{I}}_{1}^{1,2}}[4]\\\
\\!=\\!\left[{\begin{array}[]{*{20}{c}}{h_{1,3}^{1,1}[4]}&{h_{1,4}^{1,1}[4]}\\\
{h_{1,3}^{1,2}[4]}&{h_{1,4}^{1,2}[4]}\\\
{h_{1,3}^{1,3}[4]}&{h_{1,4}^{1,3}[4]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{b_{3}}}\\\
{{b_{4}}}\end{array}}\right]\\!+\\!\left[{\begin{array}[]{*{20}{c}}{h_{1,3}^{2,1}[4]}&{h_{1,4}^{2,1}[4]}\\\
{h_{1,3}^{2,2}[4]}&{h_{1,4}^{2,2}[4]}\\\
{h_{1,3}^{2,3}[4]}&{h_{1,4}^{2,3}[4]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{3}}}\\\
{{a_{4}}}\end{array}}\right]\\\
\\!=\\!\left[{\begin{array}[]{*{20}{c}}{{{\boldsymbol{h}}_{1,3}^{2}}[4]}&{{{\boldsymbol{h}}_{1,4}^{2}}[4]}&{{{\boldsymbol{h}}_{1,3}^{1}}[4]}&{{{\boldsymbol{h}}_{1,4}^{1}}[4]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{3}}}\\\
{{a_{4}}}\\\ {{b_{3}}}\\\ {{b_{4}}}\end{array}}\right],\end{array}$ (69)
$\vspace{-0.5em}\begin{array}[]{l}\boldsymbol{I_{1}^{1,1}}[5]+\boldsymbol{I_{1}^{1,2}}[5]\\\
\\!=\\!\left[{\begin{array}[]{*{20}{c}}{h_{1,3}^{1,1}[5]}&{h_{1,4}^{1,1}[5]}\\\
{h_{1,3}^{1,2}[5]}&{h_{1,4}^{1,2}[5]}\\\
{h_{1,3}^{1,3}[5]}&{h_{1,4}^{1,3}[5]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{b_{5}}}\\\
{{b_{6}}}\end{array}}\right]\\!+\\!\left[{\begin{array}[]{*{20}{c}}{h_{1,3}^{2,1}[5]}&{h_{1,4}^{2,1}[5]}\\\
{h_{1,3}^{2,2}[5]}&{h_{1,4}^{2,2}[5]}\\\
{h_{1,3}^{2,3}[5]}&{h_{1,4}^{2,3}[5]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{5}}}\\\
{{a_{6}}}\end{array}}\right]\\\
\\!=\\!\left[{\begin{array}[]{*{20}{c}}{{{\boldsymbol{h}}_{1,3}^{2}}[5]}&{{{\boldsymbol{h}}_{1,4}^{2}}[5]}&{{{\boldsymbol{h}}_{1,3}^{1}}[5]}&{{{\boldsymbol{h}}_{1,4}^{1}}[5]}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{a_{5}}}\\\
{{a_{6}}}\\\ {{b_{5}}}\\\ {{b_{6}}}\end{array}}\right],\end{array}$ (70)

where ${\boldsymbol{h}}_{i,M}^{j}[t]$ denotes the channel vector from antenna
$M$ of the base station $i$ to user $k$ in the t-th slot and it has the form
as below

${\boldsymbol{h}}_{i,M}^{j}[t]{=}{\left[{\begin{array}[]{*{20}{c}}{h_{i,M}^{j,1}[t]}&{h_{i,M}^{j,2}[t]}&{h_{i,M}^{j,3}[t]}\end{array}}\right]^{T}}.$
(71)

Then the transmitted symbols can be characterized as

$\ \begin{array}[]{l}{{\boldsymbol{S}}_{1}}[6]{\\!=\\!}\\\
\left[{\begin{array}[]{*{20}{c}}{({\boldsymbol{I}}_{1}^{1,1}[1]\\!+\\!{\boldsymbol{I}}_{1}^{1,2}[1])\\!+\\!({\boldsymbol{I}}_{1}^{1,1}[4]\\!+\\!{\boldsymbol{I}}_{1}^{1,2}[4])\\!+\\!({\boldsymbol{I}}_{1}^{1,1}[5]\\!+\\!{\boldsymbol{I}}_{1}^{1,2}[5])}\\\
{{0_{1\times 1}}}\end{array}}\right].\end{array}$ (72)

For user 1 in the target cell 1, the received signals are

$\begin{array}[]{l}{{\boldsymbol{y}}_{1,1}}[6]={\bf{H}}_{1}^{1,1}[6]{\bf{W}}_{1}^{1,1}[6]{{\boldsymbol{S}}_{1}}[6]\\\
\quad\quad\quad=\left[{\begin{array}[]{*{20}{c}}{h_{1,1}^{1,1}[6]}&{h_{1,2}^{1,1}[6]}&{h_{1,3}^{1,1}[6]}\\\
{h_{1,1}^{1,2}[6]}&{h_{1,2}^{1,2}[6]}&{h_{1,3}^{1,2}[6]}\\\
{h_{1,1}^{1,3}[6]}&{h_{1,2}^{1,3}[6]}&{h_{1,3}^{1,3}[6]}\end{array}}\right]{{\boldsymbol{S}}_{1}}[6]\\\
\quad\quad\quad=\left[{\begin{array}[]{*{20}{c}}{\boldsymbol{h}_{1}^{1,1}[6]}\\\
{\boldsymbol{h}_{1}^{1,2}[6]}\\\
{\boldsymbol{h}_{1}^{1,3}[6]}\end{array}}\right]{{\boldsymbol{S}}_{1}}[6],\end{array}$
(73)

where ${\boldsymbol{h}}_{i}^{k,N}[t]$ denotes the channel vector which is
consist of the channel gains from the whole antennas of the base station $i$
to the $N$th antenna of the user $k$, and the vector is given by

${\boldsymbol{h}}_{i}^{k,N}[t]{=}\left[{\begin{array}[]{*{20}{c}}{h_{i,1}^{k,N}[t]}&{h_{i,2}^{k,N}[t]}&{h_{i,3}^{k,N}[t]}\end{array}}\right].$
(74)

Now, we have $(\bar{\varphi}+1)\times(M-N)=3$ additional equations and for
user 1 in the target cell 1, the whole process can be summarized by the
equation (75).

$\begin{array}[]{l}\left[{\begin{array}[]{*{20}{c}}{{{\bf{y}}_{1,1}}[1]}\\\
{{{\bf{y}}_{1,1}}[4]}\\\ {{{\bf{y}}_{1,1}}[5]}\\\
{{{\bf{y}}_{1,1}}[6]}\end{array}}\right]{\rm{=}}\left[{\begin{array}[]{*{20}{c}}{{{\bf{A}}_{3\times
4}}}&{{{\bf{0}}_{3\times 4}}}&{{{\bf{0}}_{3\times 4}}}\\\ {{{\bf{0}}_{3\times
4}}}&{{{\bf{B}}_{3\times 4}}}&{{{\bf{0}}_{3\times 4}}}\\\ {{{\bf{0}}_{3\times
4}}}&{{{\bf{0}}_{3\times 4}}}&{{{\bf{C}}_{3\times 4}}}\\\ {{{\bf{D}}_{3\times
4}}}&{{{\bf{E}}_{3\times 4}}}&{{{\bf{F}}_{3\times
4}}}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{{{\bf{X}}_{4\times
1}}}\\\ {{{\bf{Y}}_{4\times 1}}}\\\ {{{\bf{Z}}_{4\times
1}}}\end{array}}\right]\quad where\\\ {\rm{}}{{\bf{A}}_{3\times
4}}=\left[{\begin{array}[]{*{20}{c}}{h_{1,1}^{1,1}[1]}&{h_{1,2}^{1,1}[1]}&{h_{1,3}^{1,1}[1]}&{h_{1,4}^{1,1}[1]}\\\
{h_{1,1}^{1,2}[1]}&{h_{1,2}^{1,2}[1]}&{h_{1,3}^{1,2}[1]}&{h_{1,4}^{1,2}[1]}\\\
{h_{1,1}^{1,3}[1]}&{h_{1,2}^{1,3}[1]}&{h_{1,3}^{1,3}[1]}&{h_{1,4}^{1,3}[1]}\end{array}}\right],{{\bf{B}}_{3\times
4}}=\left[{\begin{array}[]{*{20}{c}}{h_{1,1}^{1,1}[4]}&{h_{1,2}^{1,1}[4]}&{h_{1,3}^{1,1}[4]}&{h_{1,4}^{1,1}[4]}\\\
{h_{1,1}^{1,2}[4]}&{h_{1,2}^{1,2}[4]}&{h_{1,3}^{1,2}[4]}&{h_{1,4}^{1,2}[4]}\\\
{h_{1,1}^{1,3}[4]}&{h_{1,2}^{1,3}[4]}&{h_{1,3}^{1,3}[4]}&{h_{1,4}^{1,3}[4]}\end{array}}\right],\\\
{{\bf{C}}_{3\times
4}}=\left[{\begin{array}[]{*{20}{c}}{h_{1,1}^{1,1}[5]}&{h_{1,2}^{1,1}[5]}&{h_{1,3}^{1,1}[5]}&{h_{1,4}^{1,1}[5]}\\\
{h_{1,1}^{1,2}[5]}&{h_{1,2}^{1,2}[5]}&{h_{1,3}^{1,2}[5]}&{h_{1,4}^{1,2}[5]}\\\
{h_{1,1}^{1,3}[5]}&{h_{1,2}^{1,3}[5]}&{h_{1,3}^{1,3}[5]}&{h_{1,4}^{1,3}[5]}\end{array}}\right],{\rm{}}{{\bf{X}}_{4\times
1}}=\left[{\begin{array}[]{*{20}{c}}{{a_{1}}}\\\ {{a_{2}}}\\\ {{b_{1}}}\\\
{{b_{2}}}\end{array}}\right],{{\bf{Y}}_{4\times
1}}=\left[{\begin{array}[]{*{20}{c}}{{a_{3}}}\\\ {{a_{4}}}\\\ {{b_{3}}}\\\
{{b_{4}}}\end{array}}\right],{{\bf{Z}}_{4\times
1}}=\left[{\begin{array}[]{*{20}{c}}{{a_{5}}}\\\ {{a_{6}}}\\\ {{b_{5}}}\\\
{{b_{6}}}\end{array}}\right],\\\ {\rm{}}{{\bf{D}}_{3\times
4}}=\left[{\begin{array}[]{*{20}{c}}{h_{1,3}^{2}[1]h_{1}^{1,1}[6]}&{h_{1,4}^{2}[1]h_{1}^{1,1}[6]}&{h_{1,3}^{1}[1]h_{1}^{1,1}[6]}&{h_{1,4}^{1}[1]h_{1}^{1,1}[6]}\\\
{h_{1,3}^{2}[1]h_{1}^{1,2}[6]}&{h_{1,4}^{2}[1]h_{1}^{1,2}[6]}&{h_{1,3}^{1}[1]h_{1}^{1,2}[6]}&{h_{1,4}^{1}[1]h_{1}^{1,2}[6]}\\\
{h_{1,3}^{2}[1]h_{1}^{1,3}[6]}&{h_{1,4}^{2}[1]h_{1}^{1,3}[6]}&{h_{1,3}^{1}[1]h_{1}^{1,3}[6]}&{h_{1,4}^{1}[1]h_{1}^{1,3}[6]}\end{array}}\right],\\\
{\rm{}}{{\bf{E}}_{3\times
4}}=\left[{\begin{array}[]{*{20}{c}}{h_{1,3}^{2}[4]h_{1}^{1,1}[6]}&{h_{1,4}^{2}[4]h_{1}^{1,1}[6]}&{h_{1,3}^{1}[4]h_{1}^{1,1}[6]}&{h_{1,4}^{1}[4]h_{1}^{1,1}[6]}\\\
{h_{1,3}^{2}[4]h_{1}^{1,2}[6]}&{h_{1,4}^{2}[4]h_{1}^{1,2}[6]}&{h_{1,3}^{1}[4]h_{1}^{1,2}[6]}&{h_{1,4}^{1}[4]h_{1}^{1,2}[6]}\\\
{h_{1,3}^{2}[4]h_{1}^{1,3}[6]}&{h_{1,4}^{2}[4]h_{1}^{1,3}[6]}&{h_{1,3}^{1}[4]h_{1}^{1,3}[6]}&{h_{1,4}^{1}[4]h_{1}^{1,3}[6]}\end{array}}\right],\\\
{\rm{}}{{\bf{F}}_{3\times
4}}=\left[{\begin{array}[]{*{20}{c}}{h_{1,3}^{2}[5]h_{1}^{1,1}[6]}&{h_{1,4}^{2}[5]h_{1}^{1,1}[6]}&{h_{1,3}^{1}[5]h_{1}^{1,1}[6]}&{h_{1,4}^{1}[5]h_{1}^{1,1}[6]}\\\
{h_{1,3}^{2}[5]h_{1}^{1,2}[6]}&{h_{1,4}^{2}[5]h_{1}^{1,2}[6]}&{h_{1,3}^{1}[5]h_{1}^{1,2}[6]}&{h_{1,4}^{1}[5]h_{1}^{1,2}[6]}\\\
{h_{1,3}^{2}[5]h_{1}^{1,3}[6]}&{h_{1,4}^{2}[5]h_{1}^{1,3}[6]}&{h_{1,3}^{1}[5]h_{1}^{1,3}[6]}&{h_{1,4}^{1}[5]h_{1}^{1,3}[6]}\end{array}}\right].\end{array}$
(75)

It is obvious that the channel matrix $\bf{H}$ is a $12\times 12$ non-singular
matrix and 36 independent symbols can be decoded over 6 slots. Thus, the DoF
of the B-DRIA is 6, which is better than both the RIA scheme and the RIR
scheme.

##  V Numerical Results

In this section, numerical results are presented to characterize the
performance of the proposed two IA schemes. In specific, to analyze the
performance of the proposed schemes, two benchmark schemes are introduced:

  * •

The TDMA scheme [39], that is at each slot, only one base station is selected
to transmit signals while the other base stations keep silent so that the
negative effect of the ICI is cancelled. However, owing to the pattern of
single base station transmission, both space resources and time resources are
wasted inevitably. Hence, the obtained DoF is relaxed to the outer bound [40].

  * •

The RIA scheme [41], that is in the previous $t$ slots, only one base station
is selected to transmit signals while the other base stations keep silent, and
in the last slot, all the base stations make use of the combination of IUI in
the previous slots to achieve simultaneous transmission. Therefore, the RIA
scheme gets DoF gain while the improvement is still limited. The main reason
is that the scheme only improves the space and time resources’ utilization in
the last slot.

Herein, we develop the RIR scheme and B-DRIA scheme to further improve the
utilization of time and space resources. As mentioned in Section III and
Section IV, in the specific scenario, both RIR scheme and B-DRIA scheme
achieve DoF gain better than RIR scheme, but the degree of the improvement of
the DoF mainly depends on the relationship between transceiver antennas ratio
$\rho$ and transmitter antennas $M$, whereas the relationship is nonlinear
correlated. Therefore, the relationship should be further discussed in two
cases:

  * •

Fix the value of $\rho$ and change the value of $M$, and for this case, the
interval of $\rho$ is from 1 to 2. In specific, owing to the assumption that
no users can decode the messages with only one-slot transmitted signals, the
lower bound of $\rho$ should be greater than 1. Meanwhile, in order to make
comparison with the RIA scheme, the upper bound of $\rho$ should be no more
than 2. Therefore, we fix the value of $\rho$ at 1, 3/2 or 2 and change the
value of $N$ from 5 to 25. The effects of $M$ on the DoF are shown in Fig. 10,
Fig. 11 and Fig. 12.

  * •

Fix the value of $M$ and change the value of $N$, and for this case, when $N$
increases and $M$ can be divided by $M-N$, the DoF grows sharply, otherwise
the value of DoF keeps constant. Therefore, we fix the value of $M$ at 72 to
make it divisible by as many integers as possible, and then change the value
of $N$ from 36 to 72. The effects of $\rho$ on the DoF are shown in Fig. 13
and Fig. 14.

###  V-A Performance Analysis with Fixed $\rho$

At first, we analyze how does the DoF is affected by the parameter $M$ under
the condition that the parameter $\rho$ is fixed, i.e.,
$\rho={3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}$. In specific, the
configuration of the network is set to
$[L,M,K,N=(2,\left\lfloor{{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}N}\right\rfloor,3,N)$,
and meanwhile the value of the $N$ is varying from 5 to 25. Under this
configuration, for the RIR scheme, $\varphi=2$ slots per group are used for
signal transmission and at the same time, for the B-DRIA scheme,
$\bar{\varphi}=1$ slot is used for signal transmission, respectively. From the
result shown in Fig. 10, we observe that, for all the value of the $M$, B-DRIA
scheme always has the best DoF performance, then it is the RIR scheme, and the
RIA scheme obtains the least DoF performance. This phenomenon can be explained
as that, on the one hand, the RIR scheme can aligns both ICI and IUI of which
the RIA scheme only can align the later one, so that the RIR scheme gets
higher DoF gain than RIA scheme. On the other hand, the B-DRIA scheme aligns
both IUI and ICI, and meanwhile takes less slots to achieve IA, hence it keeps
the highest DoF gain. In addition, we can note that, as the number of user
antennas reaches 25, compared with the TDMA scheme, the DoF gain of the RIA
scheme, RIR scheme and B-DRIA scheme are 11.54%, 24.8% and 56%, respectively.
Furthermore, it is worth noting that some points of the DoF keep constant
locally when $N$ increases. This comes from the fact that, when the base
station sends the same amount of information
$\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil$
to each user, if $M$ cannot be divided to $K$, the base station does not send
redundant symbols
$[{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}-\left\lceil{{M\mathord{\left/{\vphantom{MK}}\right.\kern-1.2pt}K}}\right\rceil$
separately. This measurement leads to no increase of the DoF and we call this
operation as round down operation loss.

Figure 10: DoF versus $N$ with fixed
$\rho={3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}$ of the schemes.

Then we further analyze how does the DoF is affected by the parameter $M$
under two special values of $\rho$, i.e., $\rho=1$ and $\rho=2$. For the
former, we set the value of $\rho$ approaching to the critical point 2 and the
other parameters used herein are the same as that used for the Fig. 10. Under
this configuration, for the RIR scheme, $\varphi=1$ slots per group are used
for signal transmission, and for the B-DRIA scheme, $\bar{\varphi}=0$ slot is
used for signal transmission, in other words, B-DRIA scheme does not implement
the second phase. It can be witnessed from the Fig. 11 that, the proposed two
schemes achieve 36% DoF gain than the benchmark schemes and meanwhile, both
RIR scheme and B-DRIA scheme get the same DoF. The former confirms that the
proposed schemes have a better effect on the improvement of DoF, and later is
lies in the absence of the second phase so that the spatial gain is lost. It
is also worth noting that the RIA scheme has the same performance as the TDMA
scheme. It can be explained as that RIA scheme transmits signals in the first
slot and then transmits interference in the second slots where the whole
process can be equivalent to the TDMA scheme with two slots. Notice that the
third reason can also be used to explain why does the B-DRIA scheme not get
the DoF gain from the past interference.

Figure 11: DoF versus $N$ with fixed $\rho=2$ of the schemes.

Finally, we set the value of $\rho$ approaching to the critical point 1 and
the other parameters used herein are the same as that used for the Fig. 10.
Under this configuration, for the RIR scheme and B-DRIA scheme, more than 2
slots per group and 1 slot per group are used for signal transmission,
respectively. The Fig. 12 shows that the RIA scheme obtains 19.5% DoF gain
than the TDMA scheme, and meanwhile the performance of the two proposed
schemes is polarized, i.e., the B-DRIA scheme acquires 100% DoF gain than TDMA
scheme while the performance of RIR scheme is degraded which is worse than RIA
scheme. The former result can be explained as that, for the B-DRIA scheme,
with the multi-slots’ simultaneous transmission, the space resources have been
fully utilized and the obtained DoF is tight to the upper bound [40]. The
later result is mainly because that, when approaches 1, too many slots of each
group are taken. In specific, when the number of user antennas reaches 25,
there are $\bar{\varphi}{{=}}26$ slots should be used for the single cell
transmission which leads to serious waste of the time and space resources.

It is worth noting that, although the performance of the B-DRIA scheme is
completely superior to the RIR scheme, in practice, the application scenario
of the former is restricted. In specific, for the sake of DoA estimation, the
location of the users should be stationary or moves within the current cell,
otherwise the estimation should be reacquired which leads to the poor mobility
of the network. Therefore, for the selection of the two proposed schemes, the
choice should be made according to specific application scenario.

Figure 12: DoF versus N with fixed $\rho=1$ of the schemes.

###  V-B Performance Analysis with Fixed M

We analyze how does the DoF is affected by the parameter $\rho$ under the
condition that the parameter $M$ is fixed, i.e., $M=72$. In specific, the
configuration of the network is set to $L,M,K,N=(2,72,3,N)$, and meanwhile the
value of the $N$ ranges from 36 to the 72. Aiming to this configuration the
parameter goes through three intervals, i.e.,
${3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}<\rho\leq 2$,
${4\mathord{\left/{\vphantom{43}}\right.\kern-1.2pt}3}<\rho\leq{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}$
and $1<\rho\leq{4\mathord{\left/{\vphantom{43}}\right.\kern-1.2pt}3}$ where
the corresponding intervals measured by the parameter $N$ is $36\leq N<48$
$48\leq N<54$ and $54\leq N<72$, respectively.

From the result shown in the Fig. 13, we find that, for the RIR scheme, the
DoF keeps constant in each interval, while increasing sharply when the value
of $N$ exceeds one critical point. This phenomenon is caused by round down
operation loss mentioned before. Meanwhile, when the value of $N$ reaches 36,
48 and 54, the obtained DoF gains are 20%, 14.3% and 10.3%, respectively. It
indicates that, with the increase of $N$, the improvement made by RIR scheme
becomes weaken. In another word, the DoF gain is inversely related to the
critical points of $\rho$. Herein, the local optimum of DoF gain is obtained
at critical points of $\rho$, and the global optimum of DoF gain is acquired
when $\rho$ approaches 2.

Figure 13: DoF versus $N$ with fixed $M=72$ of RIR scheme.

From the result shown in the Fig. 14, we discover that, for the B-DRIA scheme,
the DoF keeps constant in each interval, while increasing sharply when the
value of $N$ exceeds one critical point. The reason for this phenomenon is the
same as that for the RIR scheme. Meanwhile, when the value of $N$ reaches 36,
48, and 54, the obtained DoF gains are 33.3%, 50% and 60%, respectively. That
is to say, the DoF gain is positively related to the critical points of
$\rho$. Herein, the local optimum of DoF gain is obtained at the critical
points of $\rho$, and the global optimum of DoF gain is acquired when $\rho$
approaches 1.

Figure 14: DoF versus $\\!N\\!$ with fixed $M\\!=\\!72$ of B-DRIA scheme.

##  VI Conclusion

In this paper, two new IA schemes were proposed for cellular $K$-user MIMO
downlink networks, i.e., the RIR scheme and the B-DRIA scheme. For the former,
the redundant symbols of the interference signals are eliminated with delayed
CSIT and meanwhile, the desired symbols are provided to the target cell’s
users without causing secondary interference. The analysis indicates that it
achieves better DoF gain than the RIA scheme but, when the transceiver
antennas ratio $\rho$ approaches 1, the performance improvement becomes
weakened. Therefore, we further proposed the B-DRIA scheme, in which the
beamforming technique is adopted to eliminate ICI and meanwhile, the
distributed retrospective interference alignment is developed to use the past
interference signals. Simulation results show that the B-DRIA scheme obtains
larger DoF than the RIR scheme locally. Specifically, when $\rho$ approaches
1, two schemes obtain the same DoF. While as $\rho$ approaches 2, the DoF of
the B-DRIA scheme is superior than the later.

## References

  * [1] V. R. Cadambe and S. A. Jafar, “Interference alignment and degrees of freedom of the $k$-user interference channel,” _IEEE Transactions on Information Theory_ , vol. 54, no. 8, pp. 3425–3441, 2008.
  * [2] C. Suh and D. Tse, “Interference alignment for cellular networks,” in _46th Annual Allerton Conference on Communication, Control, and Computing_ , 2008, pp. 1037–1044.
  * [3] W. Shin, N. Lee, J. Lim, C. Shin, and K. Jang, “On the design of interference alignment scheme for two-cell mimo interfering broadcast channels,” _IEEE Transactions on Wireless Communications_ , vol. 10, no. 2, pp. 437–442, 2011.
  * [4] K. Lee, “Uplink interference alignment for two-cell mimo interference channels,” _IEEE Transactions on Vehicular Technology_ , vol. 62, no. 4, pp. 1861–1865, 2013.
  * [5] G. Sridharan and W. Yu, “Degrees of freedom of mimo cellular networks: Two-cell three-user-per-cell case,” in _Proc. IEEE Global Communications Conference (GLOBECOM)_ , 2013, pp. 1968–1973.
  * [6] D. J. Love, R. W. Heath, V. K. N. Lau, D. Gesbert, B. D. Rao, and M. Andrews, “An overview of limited feedback in wireless communication systems,” _IEEE Journal on Selected Areas in Communications_ , vol. 26, no. 8, pp. 1341–1365, 2008.
  * [7] C. Suh, M. Ho, and D. N. C. Tse, “Downlink interference alignment,” _IEEE Transactions on Communications_ , vol. 59, no. 9, pp. 2616–2626, 2011\.
  * [8] B. Zhuang, R. A. Berry, and M. L. Honig, “Interference alignment in mimo cellular networks,” in _Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2011, pp. 3356–3359.
  * [9] S. A. Jafar, “Blind interference alignment,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 6, no. 3, pp. 216–227, 2012.
  * [10] T. Gou, C. Wang, and S. A. Jafar, “Aiming perfectly in the dark-blind interference alignment through staggered antenna switching,” _IEEE Transactions on Signal Processing_ , vol. 59, no. 6, pp. 2734–2744, 2011.
  * [11] C. Wang, H. C. Papadopoulos, S. A. Ramprashad, and G. Caire, “Improved blind interference alignment in a cellular environment using power allocation and cell-based clusters,” in _Proc. IEEE International Conference on Communications (ICC)_ , 2011, pp. 1–6.
  * [12] Y. Lu and W. Zhang, “Blind interference alignment in the k-user miso interference channel,” in _Proc. IEEE Global Communications Conference (GLOBECOM)_ , 2013, pp. 3464–3469.
  * [13] Q. F. Zhou and Q. T. Zhang, “On blind interference alignment over homogeneous block fading channels,” _IEEE Communications Letters_ , vol. 16, no. 9, pp. 1432–1435, 2012.
  * [14] N. Lee and R. W. Heath, “Space-time interference alignment and degree-of-freedom regions for the miso broadcast channel with periodic csi feedback,” _IEEE Transactions on Information Theory_ , vol. 60, no. 1, pp. 515–528, 2014.
  * [15] N. Lee, R. Tandon, and R. W. Heath, “Distributed space–time interference alignment with moderately delayed csit,” _IEEE Transactions on Wireless Communications_ , vol. 14, no. 2, pp. 1048–1059, 2015\.
  * [16] N. Lee and R. W. Heath, “Csi feedback delay and degrees of freedom gain trade-off for the miso interference channel,” in _Proc. Conference Record of the Forty Sixth Asilomar Conference on Signals, Systems and Computers (ASILOMAR)_ , 2012, pp. 1851–1854.
  * [17] Y. Lejosne, D. Slock, and Y. Yuan-Wu, “Space time interference alignment scheme for the mimo bc and ic with delayed csit and finite coherence time,” in _Proc. IEEE International Conference on Acoustics, Speech and Signal Processing_ , 2013, pp. 4868–4872.
  * [18] M. A. Maddah-Ali and D. Tse, “Completely stale transmitter channel state information is still very useful,” _IEEE Transactions on Information Theory_ , vol. 58, no. 7, pp. 4418–4431, 2012.
  * [19] D. T. H. Kao and A. S. Avestimehr, “Linear degrees of freedom of the mimo x-channel with delayed csit,” in _Proc. IEEE International Symposium on Information Theory_ , 2014, pp. 366–370.
  * [20] Y. Lejosne, D. Slock, and Y. Yuan-Wu, “Achieving full sum dof in the siso interference channel with feedback delay,” _IEEE Communications Letters_ , vol. 18, no. 7, pp. 1202–1205, 2014.
  * [21] M. Torrellas, A. Agustin, and J. Vidal, “On the degrees of freedom of the k-user miso interference channel with imperfect delayed csit,” in _Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2014, pp. 1155–1159.
  * [22] C. Hao and B. Clerckx, “Achievable sum dof of the ${K}$ -user mimo interference channel with delayed csit,” _IEEE Transactions on Communications_ , vol. 64, no. 10, pp. 4165–4180, 2016.
  * [23] W. Shin and J. Lee, “Retrospective interference alignment for the two-cell mimo interfering multiple access channel,” _IEEE Transactions on Wireless Communications_ , vol. 14, no. 7, pp. 3937–3947, 2015.
  * [24] M. Torrellas, A. Agustin, and J. Vidal, “Retrospective interference alignment for the mimo interference broadcast channel,” in _Proc. IEEE International Symposium on Information Theory (ISIT)_ , 2015, pp. 1492–1496.
  * [25] C. S. Vaze and M. K. Varanasi, “The degrees of freedom region and interference alignment for the mimo interference channel with delayed csit,” _IEEE Transactions on Information Theory_ , vol. 58, no. 7, pp. 4396–4417, 2012.
  * [26] ——, “The degree-of-freedom regions of mimo broadcast, interference, and cognitive radio channels with no csit,” _IEEE Transactions on Information Theory_ , vol. 58, no. 8, pp. 5354–5374, 2012.
  * [27] R. Tandon, S. Mohajer, H. V. Poor, and S. Shamai, “Degrees of freedom region of the mimo interference channel with output feedback and delayed csit,” _IEEE Transactions on Information Theory_ , vol. 59, no. 3, pp. 1444–1457, 2013.
  * [28] M. Torrellas, A. Agustin, and J. Vidal, “Achievable dof-delay trade-offs for the $k$ -user mimo interference channel with delayed csit,” _IEEE Transactions on Information Theory_ , vol. 62, no. 12, pp. 7030–7055, 2016.
  * [29] H. Kwon, S. Ko, H. Seo, and B. G. Lee, “Inter-cell interference management for next-generation wireless communication systems,” _Journal of Communications and Networks_ , vol. 10, no. 3, pp. 258–267, 2008\.
  * [30] K. Wang, H. Li, F. R. Yu, W. Wei, and L. Suo, “Interference alignment in virtualized heterogeneous cellular networks with imperfect channel state information,” _IEEE Transactions on Vehicular Technology_ , vol. 66, no. 2, pp. 1519–1532, 2017.
  * [31] M. Rahmati and V. T. Vakili, “A transmitter model for performance improvement and interference mitigation of multi-user stbc-cdma system in correlated miso channels,” in _Proc. Third UKSim European Symposium on Computer Modeling and Simulation_ , 2009, pp. 569–573.
  * [32] R. Schmidt, “Multiple emitter location and signal parameter estimation,” _IEEE Transactions on Antennas and Propagation_ , vol. 34, no. 3, pp. 276–280, 1986.
  * [33] N. Lee, R. Tandon, and R. W. Heath, “Distributed space–time interference alignment with moderately delayed csit,” _IEEE Transactions on Wireless Communications_ , vol. 14, no. 2, pp. 1048–1059, 2015\.
  * [34] N. Zhao, F. R. Yu, M. Jin, Q. Yan, and V. C. M. Leung, “Interference alignment and its applications: A survey, research issues, and challenges,” _IEEE Communications Surveys Tutorials_ , vol. 18, no. 3, pp. 1779–1803, 2016\.
  * [35] T. Ying, W. Feng, and G. Liu, “Space-time interference alignment: Dof of two-user mimo x channel with alternating csit,” _IEEE Communications Letters_ , vol. 21, no. 5, pp. 1167–1170, 2017.
  * [36] A. Waldron, M. Serone, and J. Plefka, “Matrix theory s matrix,” _Phys.rev.lett_ , vol. 81, no. 14, pp. 2866–2869, 1998.
  * [37] B. Efron, “Discussion of least square regression, least angle regression,” _The Annals of Stats_ , vol. 32, no. 2, pp. 465–469, 2004.
  * [38] F. Yan, M. Jin, and X. Qiao, “Low-complexity doa estimation based on compressed music and its performance analysis,” _IEEE Transactions on Signal Processing_ , vol. 61, no. 8, pp. 1915–1930, 2013.
  * [39] F. U. Hai-Yang, W. Wei, and J. Zhuo-Lin, “Ameliorated cdma/tdma scheme in 3g mobile communication,” _Journal of Chongqing University of Posts and Telecommunications(Natural ence)_ , 2006.
  * [40] T. Gou and S. A. Jafar, “Degrees of freedom of the k user mimo interference channel,” in _Proc. 42nd Asilomar Conference on Signals, Systems and Computers_ , 2008, pp. 126–130.
  * [41] W. Shin and J. Lee, “Retrospective interference alignment for the two-cell mimo interfering multiple access channel,” _IEEE Transactions on Wireless Communications_ , vol. 14, no. 7, pp. 3937–3947, 2015.