This work was supported in part by the fundamental Research Funds forthe Central Universities under the Grant No.ZYGX2010J013. This paperis partially National Sci.& Tech. Major Project under the Grant No.2011ZX03004-002, 2011ZX03004-001.

New User Coexistence with Network Under Interference Alignment

Qing Li, Guangrong Yue, Zhigang Luo National Key Laboratory of Communication

University of Electronic Science and Technology of China Chengdu, Sichuan, China

liqing2601303005@163.com

Abstract—Recent results find numerical scheme for interference alignment (IA) to approach the Shannon capacity in interference network. In this paper, a constant multiple input multiple output interference channels where a group of user-pairs are communicating with each other through interference alignment while a group of new user-pairs desire to access the same channel. The primary users will not be affected when the new arrival users have enough transmit antennas that can null the interference from the new users at primary receiver. When a group of arrival users can not affect the primary user, we analyze the performance of sum rate while the new users use iterative algorithm to eliminate the interference from primary users. The iterative algorithm can make the new users network communicate without interference at the same time.

Keywords-iterative algorithms; interference channel; multiple input multiple output interference alignment

I. INTRODUCTION

The emergence new idea of interference alignment (IA) for multiple user wireless networks has shown that the capacity of wireless networks has much more potential [1]. Interference alignment can confine the interference to a subspace of the received signal space and another interference free subspace can be available for desired signal. Regardless of the number of interferers, every user is able to access one half of the spectrum free from interference other users. As an interference management technique, IA achieves the maximum degrees of freedom of the interference channel at high SNR [1].

Interference alignment schemes that are constructed over symbol extensions of time or frequency channel extensions, it is shown that by using long symbol extensions the degrees of freedom achieved per dimension approach arbitrarily close to the theoretical outrebound [2]. A lot of research on interference alignment focus on the areas such as how much degrees of freedom we can get[3], new schemes to achieve interference alignment[4][8][9] and the overhead for channel estimation[6][7]. If the channel estimation is imperfect, the IA will be affected. A MIMO-OFDM system has showed that IA can achieve much more degree of freedoms [10].

MIMO technology is emerging as the default choice for wireless networks [18]. The wireless industry is continuously pushing toward increasing the number of antennas per device. Simultaneously, there is a proliferation of wireless devices with diverse form factors. These range from large devices, like

desktops and laptops, to small devices, like temperature or light sensors. Future wireless networks will be populated by heterogeneous access points (AP) and clients supporting different numbers of antennas for two trends that a growth in the maximum number of antennas per device and an increase in device diversity.

The combination of interference alignment and MIMO technology will improve the capacity of the communication systems. A group of user-pairs can utilize multiple antennas to achieve IA through numerical algorithms in [2] and [5] . These users have aligned their interference and they desired to keep in the alignment state when a set of new users desire to access to the network [13][14]. The new users must not degrade the sum rate of users in existing network or the performance of each user, such as the secondary and primary user in cognitive radio. In this paper, a group of new user will access to the network, these users will not only have no effect on the ongoing transmission, but also utilize the same spectrum resources in their own network.

The remaining parts are organized as follows. In section II, we present the system model. In section III, we analyze the admission of new arriving users. In section IV, the numerical results of the simulation are presented. At last, section V summarizes the work we have done.

Vectors and matrices are set in bold-face letters.HA , TA , 1A� are conjugate transpose, transpose and inverse of

A respectively. ( )rank A And ( )tr A are the rank and the traceof A respectively. ,[ ]n nA is the element in row n and column of matrices A . NI is the N N� identity matrix

II. SYSTEM MODEL FOR MIMO INTERFERENCE CHANNELL

Consider the MIMO interference channel have K primaryusers and S new arrival users shown in Fig.1. Each user-pair has a transmitter with iM antennas and a desired receiver with iN antennas. Each transmitter has id datastreams for its desired receiver. Each datastream has a precoding matrix iV which is decoded by the desired receiver with a combining matrix iU .

___________________________________ 978-1-4673-2101-3/12/$31.00 ©2012 IEEE

H11

H22

HK3

H21

HK1

H12

HK2

H 13

H23

Z1

Z2

ZK

User1

User2

UserK

H11

H22

HS3

H21

HS1

H12

HS2

H 13

H23

Z1

Z2

ZS

User1

User2

UserS

HSK

Primary network

New arrival users

Figure 1. MIMO interference channel with K primary users and S new access users

The signal received at receiver i can be expressed as

[ ] [ ] [ ] [ ] [ ]K S

ii i ij j ii i j

j iY H V X H V X Z

�

�

� � �� (1)

where [ ]iZ is additive white Gaussian noise(AWGN) at the receivers with each term distributed as ����� 0N ��� ���

����� �� ��� � �� ����� ���� ��� � �� Hi iX X �� P ��

[ ]ijH ��� �� �������� �� � ������ � ��� ����� � j �� ����!�� i �� ����� ��� �� ���� �������� I �"#���������������� �����������$����� �� � "

In primary user network, IA algorithms are used by receiver for canceling interferences from undesired transmitters.� The feasibility of IA [12] is to design the precoding and combining matrices satisfied following:

i

Hi i dV V I� (2)

i

Hi i dU U I� (3)

� [ ]

[ ]

( ), 1,...,

0i j

H iii i i

H iji i d d

rank U H V di j K

U H V i j�

�� � � � � ��� (4)

The IA algorithms need global CSI to achieve the goal that the subspace of desired id interference-free streams and the

i iN d� dimensional subspace where the interference signal are confined.

We assume global channel state information (CSI) of primary network is available to the primary users and they will not know any information of new users. The new users have

their own global CSI and the CSI of links that each new transmitter to all primary receivers.

In primary network, if IA is achieved and the receiver used the zero forcing equalizer, then each user has no interference.

[ ] 1 [ ] 1 [ ][ ] [ ]H ii H ii H ii i i i i i iU H V Y X U H V U Z� �� � (5)

And the total achievable sum rate of primary users is[11]

[ ] [ ] 11 1 0 ,

/log(1 )

[ ( ) ]

idKP isum H ii H H ii

i n i i i i n n

P dRN V H U U H V �

� �

� ��� (6)

III. USER ARRIVAL IN PRIMARY NETWORK The users in primary network do not want to change the

state that they have achieved IA and have no necessary to sense the new arrival, so they will not cooperate with new users to achieve the goal that all users communicate concurrently. In the presence of the new users, the sum rate of the primary network is

[ ] [ ] 11 1 0 ,

/log(1 )

[[ ( ) ] ]

idKP isum H ii H H ii

i n i i i i n n

P dR

N V H U U H V L�� �

� ���� (7)

where 1

K S H H Hi ik k k ik ik K

o k

PL U H V V H UN d

�

� �� � .

If 0L � , then the new users will not degrade the sum rate of the primary network. Interference nulling is a MIMO technique that transmitter creates a null at the antenna of receiver. For primary receiver i , the interference from new users can be confined to its interference subspace by utilizing the interference nulling at new transmitter. Therefore, the precoding matrix of the new user k should satisfy

0kk k R dH V �� (8)

where 1 1[( ) ,..., ( ) ]H H H H Hk k K KkH U H U H�

1

Kii

R N�

� � .

The new users will not affect the primary network if they transmit their own kd streams and have enough transmit

antennas 1

K

k i ki

M d d�

� �� [14].

The method desires to null their own interference using their own multiple antennas. Based on interference null, it provides the required minimum number of antennas at each new user and enables the nodes of new users to communicate without affecting the performance of the primary users. In this paper we mainly focus on optimizing the performance of the new users.

For mathematical tractability and to simplify the statement, we consider the condition that all primary users have same transmit antennas M and same receive antennas N . Each user transmits d streams. The new user pair has sM transmit antennas and sN receive antennas. Each new user transmits

sd streams.

The transmitter of new users needs at least sKd d� antennas in order not to degrade the performance of the

primary users. This means that if s sM Kd d� � , the new users can utilize the extra antennas resource not only satisfy no degrade on performance of primary user but also improve the performance of their own link.

%����&��� �� ' � �� ��'� � ���'���� !� ���� ���

kH ������'� � �� ����� ���� �� �� �������� ��� kT �� �����

s sM Kd d� � , so kT ��� sM Kd� �� ������� ��������� � �

k k kV T G� �!� �����linear������� ������� �������������

kT ���������������� kG �������� �������"

#����������want� ���(���)� ��������� �'�!�����

1

1 0

log det( )K S

new H Hsum kk k k k k

k K s

PR I H V V H LN d

��

� �

� �� (9)

Where1,

0

( )k

K S H Hk N ki i i kii i k

i

PL I H VV HN d

�

� �� �� ��� ��

interference���������� ��� ����� ��"��

*�+,-.��their����$� ��� � ���� ���)�/�������!��'�

1

1

,..., 1 0

arg max log det( )

. . 0 1,...,K K S

K SH H H

kk k k k k kk kG G k K s

k k k

PI H T G G T H LN d

s t H T G k K K S� �

��

� �

�

� � � �

� (10)

When a single new user access to the network, the optimum precoder of new user solving (10) can be found in close-form and is given by the sd most significant eigenvectors of

1H Hk kk k kk kT H L H T� [14].

If there are two new users, in this case, we have

1 0 0

KH H H H

k ki i i ki kq q q kqi i s

P PL I H VV H H V V HN d N d�

� � �� (11)

where , { 1, 2}q k K K � � and q k� . A simple approach is that ignoring the third term in (11) when interference from primary users dominate the interference term and the optimization of (10) is same to the case of a single new user[14].

When new users have as many as primary users, ignoring the interference from other new users is not optimization. We have designed precoder kT at transmitter of new user which will not affect the performance of primary users in the network. From the original of interference alignment, we desire to create desired signal subspace and interference subspace for interference from primary users and other new users at receiver of new user. So we have to design precoding filter kG at the transmitter of new users and combining filter kU at the receiver of new users using iterative algorithm.

The objectives of the algorithm minimize the power in the leakage interference at each receiver in new user network. The leakage interference is the interference power remaining in the received signal after the receive interference combining filter is applied. The goal is to achieve interference alignment by progressively reducing the leakage interference. If the leakage interference converges to almost zero, then interference alignment is feasible.

Mathematically, the global leakage interference is represented by

2

1 1,[ || || ( )]

S SH Hk kl l l F k k k

k l l k l

PF U H T G tr U Q Ud� � �

� �� � (12)

where1

KH H

k ki i i kii i

PQ H VV Hd�

� � is interference covariance

matrix considering of interference from primary network and noise at the receiver of new user. So objective is to minimize F through choosing lG and kU , where H

l lG G I� , Hk kU U I� .

Since 2|| || ( ) ( )H HFA tr AA tr A A� � , (12) can be written as

1 1,

[ ( ) ( )]S S

H H H H Hk kl l l l l kl k k k k

k l l k l

P tr U H T G G T H U tr U Q Ud� � �

�� � (13)

To minimize their total leakage interference, we utilize the approach alternating minimization [16][17].

Step : Start with arbitrary precoding filter lG in new arrival network, receiver k chooses its interference combining filter kU to minimize the leakage interference from all undesired transmitters when all lG are fixed. The received signal subspace that contains the least interference is the space spanned by the eigenvectors corresponding to the sd smallest eigenvalues of the H H H

kl l l l l kl kH T G G T H Q� .

Step : This step is identical to the first step, but performed in objective global function that is rewritten as

1 1,

[ ( ) ( )]S S

H H H H Hl l kl k k kl l l k k k

k l l k l

PF tr G T H U U H T G tr U Q Ud� � �

� �� � (14)

at this step, transmitter l chooses its precoding filter to minimize the leakage interference from itself at each receiver when all kU is fixed. We only focus on the terms which are related to precoding filter lG , so we ignore the second term in (14) and utilize the same solution in step . Then lG is the eigenvectors corresponding to the sM Kd� smallest eigenvalues of the H H H

l kl k k kl lT H U U H T . Note that from the beginning of the iterative algorithm, all lG start with arbitrary precoding filter. When step is finished ,then the algorithm returns to Step . The iterations continue in this manner until the algorithm converges.

This solution tries to align the interference from other users in the network and primary users outside of the coordinated portion of the network. The algorithm can update the receive subspaces to be the ones that interference caused by the fixed precoding filter is confined to interference subspace. At this step, the interference spaces have minimum sum squared Euclidean distance to the interference caused by the fixed precoding filter. At each transmitter l , the algorithm finds the precoder lG such that the interference caused by itself has minimum squared Euclidean distance to the interference subspace of undesired receiver. At each iteration, the objective function will converge.

We do not utilize the iterative algorithm when there is only one new user-pair. The global leakage interference is simplified as

( )Hk k kF tr U Q U� (15)

Then kU is the eigenvectors corresponding to the

sd smallest eigenvalues of the kQ and the precoding filter lG is arbitrary

IV. SIMULATION RESULTS Consider the model of 3 primary users and S new arrival

users. Each user has only one data stream to its corresponding receiver, and all channel coefficients are i.i.d. zero mean unit variance circularly symmetric complex Gaussian. For simplicity, BPSK is used for modulation. Each transmitter and receiver of primary user is equipped with 2 transmit antennas to send data streams. Each transmitter of new users is equipped with 5 antennas. Its selection of antennas of transmitter is for mathematical tractability in the sense that the IA system of primary network is not affected by new arrival users.

If 1S � , we assume that the new transmitter and receiver both are equipped with 5 antennas. We compare the total achievable sum rate of the new user between our scheme and the method given by [14] in fig. 2. In the low SNR regime, the algorithm in [14] is better than our algorithm, but it will be worse in the high SNR regime. The purpose of our algorithm is eliminating the interference from other users which is different from the purpose of the algorithm in [14] that mainly optimizes the performance of the capacity. If the SNR is low, then the interference is low. The algorithm in [14] will be optimal. If the SNR is high, the interference is the main factor that degrades the performance, so eliminating the interference is 5

/ sec/bit Hz better than optimizing the capacity directly.

If there are 3 new users, we focus on the performance of total achievable sum rate of the primary network and the network of new arrival users. We compare our scheme to the result of ignoring the terms in (11) when there are 3 new arrival users. The transmitter of new user has enough antennas means that it will have extra degree of freedom to optimize the transmit vector when the transmitter of new user have no degrade on the performance of the primary network.

Figure 2. The difference of capacity in our algorithm and method in [14] for one new user

Fig.3 shows that our algorithm can achieve the same performance of DOF as if all users had performed IA together. In the low SNR regime, the method that ignoring the interference terms is optimal [15], but the performance of the method will be worse in the high SNR regime. The performance difference between iterative algorithm and ignoring method to increase as more new arrival users access the network. The iterative algorithm used in new arrival network can achieve better performance no matter how many new arrival users.

Figure 3. Compare the performance of sum capacity between our algorithm

and the approach that ignoring the interference term in (11)

V. CONCLUSION In this paper, we consider the performance of the new

arrival users that will not affect an established network of primary users with IA. Using the knowledge of the interference subspaces at the primary receiver, the new users can mitigate the performance degradation of primary networks by interference nulling. The performance of new users will be better if using the iterative algorithms. The new arrival users perform their own IA, and the primary network also performs IA at the same time. Every user, include primary users and new users, is able to communicate to its corresponding receiver like only one user accessing the channel.

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