Partial Interference Alignment for Multi-Cell and Multi-User MIMO Downlink Transmission Wanfang Zhang, Yang Yu, ChengWang, Weidong Wang, Chaowei Wang Information & Electronics Technology Lab,Beijing University of Posts and Telecommunications Key Laboratory of Universal Wireless Communications, Ministry of Education Beijing University of Posts and Telecommunications, Beijing, China
[email protected] Abstract—As a promising technology to effectively mitigate interference and improve the performance of a wireless communication network, interference alignment (IA) has been attracted extensive concern. In order to solve the problem that perfect IA needs a large number of antennas and facing the cell selection, this paper proposes a partial interference alignment scheme based on user grouping for a network with multi-cell multi-user multiple-input and multiple-output (MIMO) under downlink transmission scenario. In the proposed scheme, the interference at different directions does not need to be aligned to the same subspace. Furthermore, each BS eliminates inter-user interference within its cell and the dominant interference to users in its neighboring cells. Analysis and simulation results show that the proposed approach outperforms the extension of grouping method in terms of antenna number and sum capacity. Keywords—MIMO; multi-cell; multi-user; downlink transmission; partial interference alignment; user grouping.
I.
INTRODUCTION
Interference is one of the key factors that limit the capacity of wireless networks, especially the throughput of cell-edge. Hence, an effective method that could mitigate interference is the goal which is always pursued by the experts in wireless field. Interference alignment, which was first considered in [1] for MIMO X channels, has been extensively researched. The work of [2] proved that for K-user time-varying interference channel, the total achievable degree of freedom(DoF) is K/2 through IA. Considering that the scheme in [2] needed the global channel state information (CSI), in [3], the authors provided two schemes using iterative algorithms to achieve IA with only local CSI at each node in spatial domain. The authors of [4]-[5] shown that IA could also be used to improve the capacity of a cellular network. A new version of uplink IA for cellular networks, namely subspace IA, was proposed in [4]. However, the scheme in [4] required extensive CSI to be exchanged between base-stations (BSs) of different cells, hence for a downlink cellular system, authors in [5] developed an IA scheme which could provide substantial gain and the scheme required feedback only within a cell. In the multi-cell multi-user MIMO system under downlink transmission, each BS supports multiple users within its cell. When the BS transmits signal to its serving users, there exist two kinds of interference, namely inter-user interference (IUI) and inter-cell interference (ICI). To mitigate both ICI and IUI, for a network with two-cell and two-user MIMO Gaussian
interference broadcast channels (MIMO-IFBC), authors in [6] proposed a novel IA technique in a closed-form expression without any iteration. Unlike the IA schemes in [2]-[5], the scheme in [6] firstly designed the receive beamforming matrices to make the interference aligned at the BS. However, when the number of cells is more than or equal to three, there is no solution for users’ beamforming matrix. To solve this problem, the authors of [7] and [8] gave an extension of the grouping method which consists of two parts: 1) All users of a cell and one of its neighboring cells are divided into a group, then in the group, users cooperatively design their own receive beamforming matrix to make ICI channels from the neighboring cell aligned to one subspace, which could be seen as perfectly aligned. 2) Each BS designs transmit beamforming matrices for its serving users to mitigate IUI, the aligned effective ICI and ICI from other neighboring cells. However, for users in different regions of the target cell, the strength of interference from different neighboring cells is quite different, it is difficult for all users of the target cell to choose a common neighboring cell to align its interference perfectly, which is ignored by the extension of the grouping method. What’s more, in [7] and [8], the large demand for antennas will lead to high complexity of BS. Considering these issues, in this paper, we propose a partial IA scheme based on user grouping for the multi-cell multi-user MIMO downlink transmission. Different from [7] and [8], in this paper: 1) We group the users in each cell, and each user group aligns partial ICI channels from one neighboring cell, thus the ICI channels of a neighboring cell are aligned to several subspaces, rather than one subspace as [7] and [8]. 2) We design transmit beamforming matrices to eliminate IUI and the strong ICI. 3) We eliminate the influence of cell number on the number of antennas. This paper is organized as follows. The system model is described in Section II. In Section III, we elaborate the proposed approach in the multi-cell multi-user MIMO scenario. Simulation results and analysis are given in Section IV. Finally, conclusions are drawn in Section V. Notations in the paper: lower case with normal font is scalars, bold upper and lower case letters stand for matrices and vectors, separately. I M denotes an M × M identity matrix, E [ i ] and det [i] indicate the expectation and determinant ∗
operator respectively. tr ( A ) , ( A ) , ( A ) , and ( A ) represent
This work is sponsored by the Research Fund of ZTE Corporation.
978-1-4673-6187-3/13/$31.00 ©2013 IEEE
-1
T
the trace, inversion, transpose, and conjugate transpose of matrix A separately. span ( A ) and null ( A ) indicate the space spanned by the column vectors and orthonormal basis for the null space of matrix A respectively. II.
SYSTEM MODEL
The system model considered in this paper is a cellular network with L cells, each cell consists of K users and has one BS equipped with M ≥ 2 antennas, while each user equips N ≥ 2 antennas. An example for the case of L = 7 is illustrated in Fig. 1. In Fig. 1, BS 1 sends data to user 1 while introducing IUI within cell 1 and ICI to users in other 6 neighboring cells, and it is similar to BS 2 ~ BS 7. We assume each BS aims to convey d s data streams to its corresponding users, where d s ≤ min( M , N ) . We refer to the kth user in the
Fig. 1. System model for 7 cells
lth cell as user [k , l ] .Then, the received signal y [ k,l ] of the user [k , l ] could be expressed as L
K
m=1
j=1
∗
y [ k , l ] = ( R[ k ,l ] ) y[ k ,l ] ∗ = ( R[ k , l ] ) H l[ k ,l ]T[ k ,l ] s[ k ,l ]
y[ k , l ] = ∑ H[mk ,l ] ∑ x[ j , m ] + n[ k , l ] =H
[ k ,l ] l
[ k ,l ]
T
[ k ,l ]
s
+ ∑ H
desired signal L
K
j=1, j ≠ k
+ ( R[ k , l ] ) [ j ,l ] l
[ j ,l]
T
[ j ,l ]
s
(1)
m=1,m ≠ l, j =1
where
inter -cell interference
) is the signal intended
2 for user [k , l ] , and E ⎡⎢ x[ k , l ] ⎤⎥ ≤ P[ k , l ] ( P[ k , l ] is the transmitted ⎣ ⎦ power for user [k , l ] , and assumed the power is equally T
allocated to users in cell l ). s[ k , l ] = ⎡⎣⎢ s1[ k , l ] , s2[ k , l ] , ... , sd[ k , l ] ⎤⎦⎥ ∈
d s ×1
s
denotes the transmitted data vector for user [k , l ] , satisfying an average power constraint. T[ k , l ] = ⎡⎣⎢t1[ k , l ] , t[2k , l ] , ... , t[dk , l ] ⎤⎦⎥ ∈
M ×ds
s
M ×1
∈
R
[k , l ]
K
After multiplying the received signal by a receive beamforming matrix, which could decode the desired signals coming from its corresponding BS, the received signal for user [k , l ] is presented as (2)
N ×ds
s
indicates
the
receive beamforming matrix for user [ k , l ] , and n[ k , l ] = R[ k , l ]n[ k , l ] denotes the equivalent noise which is distributed according to CΝ (0,1) . As in [7] and [8], we define the DoF for our multi-cell multi-user network as the pre-log factor of the total capacity. This is one of the key metrics used for assessing the performance of a multiple antenna based system at high SNR regime, which is defined as d
[k , l ] i
H[mk, l ] ∈ N × M ( m, l ∈{1, 2,..., L} ) indicates the channel matrix from BS m to user [k , l ] and the entries of which are independent and identically distributed (i.i.d.) random variables, and each entry is generated according to CΝ (0,1) ( complex normal distribution with zero-mean and unitvariance). And it is assumed that each channel obeys a quasistationary and frequency flat fading. n[ k , l ] ∈ N ×1 is additive white Gaussian noise (AWGN) vector with zero mean and variance δ 2 per entry at user [k , l ] .
⎞ ⎠
= ⎡⎣r1[ k , l ] , r2[ k , l ] ,...,rd[ k , l ] ⎤⎦ ∈
and
(with the unity norm constraint, i.e., t =1 ) represent the linear transmit beamforming matrix corresponding to s[ k , l ] and vector for symbol si[ k , l ] respectively. t
[k , l ] i
(2)
T[ j ,l ] s[ j ,l ]
j ,l ]
m=1,m ≠ l j=1
noise
M ×1
K
⎜ ∑ H[ ⎜ j=1, j ≠ k l ⎝
+ ∑ ∑ H[mk ,l ]T[ j , m ] s[ j , m ] ⎟⎟ + n[ k , l ]
inter -user interference
K
+ ∑ ∑ H[mk ,l ]Tm[ j , m ] s[ j , m ] + n[ k , l ]
where x[ k , l ] = T[ k , l ]s[ k , l ] ( x[ k , l ] ∈
L
∗⎛
lim
SNR →∞
C∑ ( SNR ) L K [ k , l ] =∑∑d log( SNR ) l =1 k =1
(3)
where C ∑ ( SNR ) denotes the sum capacity that can be L
K
achieved for a given SNR and C ∑ ( SNR ) = ∑ ∑ C[ k , l ] , C[ k , l ] is l =1 k =1
the capacity of user [k , l ] which can be written as (4) on the bottom of this page. d [ k , l ] is the number of data streams transmitted to user [k , l ] , and in this paper, we assumed that d [k , l ] = ds .
According to [7] and [8], in order to decode the desired signal efficiently, both the ICI and IUI should be aligned into the same interference subspace at the receiver and this subspace should be orthogonal to R[ k ,l ] , furthermore, the desired signal space should be linearly independent of the interference space. And the dimension of signal space for the desired signal should be at least d [ k , l ] . Hence the following conditions (5) should be satisfied for user [k , l ] .
* ∗ * ⎛ ⎡ ( R[k , l ] ) H[l k , l ]T[ k , l ] (T[k , l ] ) ( H[l k , l ] ) R[k , l ] ⎤⎥ ⎞⎟ C[ k , l ] = log 2 ⎜ det ⎢I + P[ k , l ] * ⎜ ⎢ ⎥⎟ δ 2 ( R[ k , l ] ) R [ k , l ] ⎣ ⎦⎠ ⎝
(4)
(R ) H (R ) H [ k ,l ] ∗ [ k ,l ] ∗
{
rank ( R
[ k ,l ] m
[ k ,l ] l [ k ,l ] ∗
T[ j , m ] = 0 ,∀m ≠ l, j ∈ {1,2,..., K }
T[ j ,l ] = 0 ,∀k ≠ j, k , j {1,2,..., K }
)H
[ k ,l ] l
T
[k , l ]
}= d
(5)
s
In the next section, we will explicitly give the proposed partial interference alignment scheme based on user grouping. III.
where (G , m) ∈ {( a, 2),(b,3),(c, 4),( d ,5),(e,6),( f ,7)} , and H[mk , G ,1] denotes the interference channel from cell m to user k in ICI group [G ,1] . H[(effective indicates the aligned effective ICI G ,1) , m ] channel from cell m to users in group [G ,1] , and the size of which is M × d s .
THE PROPOSED PARTIAL IA SCHEME
To maximize the sum rate performance of the MIMOIFBC, the transmitter and the receive beamforming matrices are usually designed by applying an iterative optimization algorithm as in [3]. However, the iterative scheme requires a considerable number of iterations and the channel state should keep unchanged during iteration. As mentioned before，for the grouping method in [7] and [8], for users in different regions of the target cell, the strength of interference from different neighboring cells is quite different, it is difficult for all users of the target cell to choose a common interfering cell to align its interference perfectly. Meaning time, the required minimum number of transmit antennas M and receive antennas N are [ K ( L − 1) +1]d s ( K is the total number of users in a cell) and [(K − 1)( L − 1) +1]d s separately, which are rapidly increased with the product of the users’ and cells’ number and result high complexity of the BS and users.
Fig. 2. The user grouping scheme for 7 cells
Step 2: Obtaining the effective ICI channel vectors
In this section, we elaborate our partial IA scheme based on user grouping. In our scheme, we avoid the problem of cell selection through user grouping, then by eliminating IUI and dominant ICI simultaneously without any iterative and partial interference alignment, we remove the influence of the cell number on the number of antennas. We take cell 1 for example and present the scheme using the following threestep.
To obtain the effective ICI channels of cell 1 to users in other six neighboring cells, the users in group [G , m] ( (G , m) ∈ {( d , 2),(e,3),( f ,4),( a,5),(b,6),(c,7)} ) should jointly design their receive beamforming matrices respectively according to (7)
Step 1: Grouping the users and designing the receive beamforming matrices.
ICI where H[(effective G , m ) ,1] is the aligned effective ICI channel by users in group [G , m] . By solving (7) we can obtain six aligned effective ICI channels from cell 1 to six user groups in its neighboring cells.
First of all, we group users in each cell. Let us explain the grouping procedure in detail using the example as in Fig.2. All users in each cell are divided into six groups according to their areas, and we use [G , m] represents user group G in cell m , G ∈ {a,b,c,d,e, f } and m ∈ {1,2 ,3,...,7} . To users in group [ a,1] , [b,1] and [ d , 2] , the dominant interference is from cell 2, cell 3 and cell 1 respectively, etc. We assumed that the number of users in each group is the same and equal to K' , therefore the total number K of the users in a cell is K = 6 K' . Then, we design the receive beamforming matrices. Taking cell 1 for example, users in each group jointly design receive beamforming matrix R[ k ,G ,1] (matrix for user k in [G ,1] ), in our scheme, users in different user groups do not need to collaborate. And the designing of receive matrices in different user groups should satisfy the following principle of (6): span {H
[1, G ,1] m
R
} = ⋅⋅⋅ = span{H R } =H
[1,G ,1]
= ⋅⋅⋅ = span{H
[ K , G ,1] m
[ k , G ,1] m
[ K , G ,1]
R
effective ICI [( G ,1) , m ]
}
[ k ,G ,1]
(6)
span{H1[1,G , m ]R [1,G , m ]} = ⋅ ⋅ ⋅ = span {H1[ k ,G , m ]R[ k ,G , m ]} ICI = ⋅ ⋅⋅ = span{H1[ K , G , m ]R [ K , G , m ]} =H[(effective G , m ) ,1]
(7)
Since (6) and (7) have the same structure, we analyze them using the same method. As in [6], [7] and [8], we can obtain the intersection subspace and receive beamforming matrices satisfying the conditions (6) and (7) above by solving the matrix equation (8): ⎡I − ( H[1, G ,l ] )* ... 0 ⎤ ... 0 m ⎢ M ⎥ [ k ,G,l ] * ⎢I M ⎥ 0 ... 0 − (Hm ) ⎢ ⎥ . . ⎢ ⎥ . . ⎢ ⎥ . . ⎢ ⎥ [ K' , G , l ] * 0 ... 0 ... − ( H m ) ⎦⎥ ⎣⎢ I M
ICI ⎡H[(effective G , l ), m ] ⎤ ⎢ ⎥ ⎢ R [1,...G ,l] ⎥ = FX = 0 (8) ⎢ [ k ,G ,l] ⎥ ⎢ R ⎥ ... ⎥ ⎢ [ K' , G ,l] ⎣⎢ R ⎦⎥
Since the size of matrix F is ( K' ∗ M ) × ( M + K' ∗ N ) , from the perspective of the linear algebra, the condition for existing null space with at least d s dimensions is
⎛⎧ ⎜⎪ ∗ ⎪ ICI ICI ICI ICI ICI ICI ⎡( R[t ( t ≠ k ),1] )∗ ∗ H[ t (t ≠ k ),1] ⎤ T[ k ,1] ⊂ null ⎜ ⎨H[(effective H[(effective H[(effective H[(effective H[(effective H[(effective d ,2),1] e ,3),1] f ,4),1] a ,5),1] b ,6),1] c ,7),1] 1 ⎢⎣ ⎥⎦ ⎜ ⎜⎜ ⎪ aligned effective ICI channels ⎪ effective IUI channels ⎩ ⎝
( M + K' ∗ N ) − ( K' ∗ M ) ≥ d s
(9)
Thus the relationship between transmit antenna number M , receive antenna number N and data stream d s satisfies N ≥ [( K' − 1) ∗ M + d s ] / K'
(10)
After the two steps, the ICI channels from BS 1 to users in its six adjacent cells are aligned to six different subspaces corresponding to six different user groups, and to users in cell 1, each group aligns dominant interference from its correspond neighboring cell. Thus partial interference of each cell is aligned by one user group. While for the grouping method in [7], all users in a cell align interference from one of its neighboring cell to the same subspace.
Step 3: Designing the transmit beamforming matrices to mitigating IUI and dominant ICI Since the ICI channels from cell 1 are aligned by six user groups corresponding to its six neighboring cells, BS 1 can treats K' different interference vectors corresponding to users in group [G , m] as a single ICI channel vector, thus from the view of BS 1, there are six ICI channel vectors corresponding to its six neighboring cells. In our scheme, BS 1 only eliminates dominant interference to users in its neighboring cells, while the residual interference can be seen as noise. Therefore, if the transmit beamforming matrix T[ k ,1] for user k in cell 1 is designed as (11) which is presented on the top of this page, i.e., T[ k ,1] is orthogonal to the space spanned by effective ICI channels and IUI, BS 1 can send data steams to its users without introducing any IUI and also can eliminate strong interference to users in its neighboring cells. Now, we discuss the required minimum number of transmit antennas M and receive antennas N . The condition for existence of T[ k ,1] is that the matrix B in null ( i ) of (11) should have a null space with at least d s dimension. Since the size of B is M × [( K + 5) ∗ d s ] , according to (9), the required minimum number of antennas M is ( K + 6)d s , we substitute this result to (10) and obtain the minimum N is ⎡⎢( K − 5 / K' ) ∗ d s ⎤⎥ ( K' = K / 6 ). Compared to the minimum number of antennas M and N of the extension of grouping method in [7] and [8], it is obvious that in our scheme, the number of antennas has nothing to do with the number of cells. The analysis of the other cells is the same to cell 1 when L > 7 . IV.
DISCUSSIONS AND SIMULATION RESULTS
∗
⎫ ⎪⎪ ⎬ ⎪ ⎭⎪
⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠
(11)
A. Discussions In this subsection, we compare the proposed partial IA scheme with the extension of the grouping method in terms of DoF and antenna number. And we use the system configuration [ L, K , d s ] denotes there are L cells and K users in each cell, and each user wants d s data streams. For the case [7,12,1] , the minimum number M and number N are 18 and 10 respectively for the proposed partial IA scheme. While for grouping method in [7], the minimum number M and number N are 73 and 67 respectively. Meaning time, the obtained total DoF is 84 for both the two schemes. That is, at the same system configuration, our scheme needs fewer antennas. Therefore, the proposed scheme has an advantage in the efficient resource usage in terms of antenna number. As for feedback overhead, the extension of the grouping method only needs one user in interfered cell to feed back the effective ICI channel, i.e., ( R[ k ,l ] ) H[mk ,l ] ∈ *
d s ×M
, because
ICI channels are perfectly aligned. However, for our partial IA scheme, in a cell, there requires six users to feed back six different effective ICI channels to its corresponding strong interfering cell. Compared to [7], the feedback overhead of our scheme is increased, but compared to conventional schemes with global CSI, feedback overhead of our scheme is still significantly reduced. B. Simulation Results Through simulation results, we assess the achievable sum capacity performance of the proposed IA scheme and compare it with the extension of the grouping method in [7] and [8]. And also, we make a comparison between the numbers of transmit and receive antennas that two schemes require. Throughout the simulations, the noise variance of all receive antennas at a user is assumed to be the same and equal to unity, and the transmitted power P[ k , l ] of each user is equal.
Fig.3 depicts the minimum number of transmit and receive antennas versus the total number of users in per cell of the proposed partial IA scheme and compares it with the extension of the grouping method ，and we assumed that L = 7 and each user wants one data stream. As can be seen from the figure, compared to the extension of the grouping method, in our scheme, no matter the demand of transmit antenna number or that of receive antenna number, it is significantly reduced. Therefore, the proposed partial IA scheme outperforms the extension of the grouping method in terms of antenna number. Hence, our scheme could decreases the complexity of BS and user.
V.
200
Transmit antenna number M of the proposed scheme Transmit antenna number M of grouping method Receive antenna number N of the proposed scheme Receive antenna number N of grouping method
The minimum number of antennas
180 160 140 120 100 80 60 40 20 0
10
15
20
25
30
The number of total users in per cell
Fig. 3. The minimum number of antennas versus the total number of users in per cell ( d s = 1 and L = 7 ).
We then consider the sum capacity versus SNR of the proposed partial IA scheme and compares it with the extension of the grouping method at the same transmit antenna setting, and we also give the sum capacity of two schemes at different number of antenna setting. Fig. 4 illustrates the sum capacity versus SNR of the two schemes at different antenna setting when L = 7 . As presented in Fig. 4, at the same antenna setting with 18 transmit antennas, the proposed partial IA in this paper outperforms the extension of the grouping method in terms of sum capacity, because for the same number of transmit antennas, our scheme could support more users in each cell. And it also can be seen from the figure, the proposed partial IA, even with 18 transmit antennas, outperforms the extension of the grouping method with 24 transmit antennas. The main reason is that in our scheme, the number of transmit antennas has nothing to do with the cell number, when the number of transmit antennas is equal to 18, BS of each cell could support 12 users, while for the extension of the grouping method, the maximum number of users in each cell is 3 even when the number of transmit antennas is 24. Although our scheme provides suboptimal gain because we just eliminate strong ICI, the sum capacity is still improved, the reason is that at the same antenna configuration, in our scheme, the BS of each cell could support more users. 65
Proposed partial IA scheme(M=18) Extension of the grouping method(M=24) Extension of the grouping method(M=18)
60
Ahievable sum capacity (bit/sec/Hz)
55 50 45 40 35 30 25 20 15 10 5 0
0
2
4
6
8
10
12
14
16
18
20
SNR (dB)
Fig. 4. The achievable sum capacity for the proposed IA scheme and comparison to the extension of the grouping method.
CONCLUSION
In this paper, for the multi-cell and multi-user MIMO downlink transmission, we proposed a partial interference alignment scheme jointly designing the transmitting and receiving beamforming matrices. Compared to conventional schemes in literature, this proposed scheme could eliminate the influence of the cell number on the number of antennas, and it is more suitable for a large cellular network. The simulation results demonstrated the performance of the algorithms for various SNR values, and also validates the superiority of the proposed approach in terms of sum capacity and the number of antennas. However, in this paper, we just study IA with perfect CSI, and grouping users according to their location . The impact of the imperfect CSI and a more rational grouping method should be also investigated, which is our future work. REFERENCES [1]
[2]
[3]
[4]
[5] [6]
[7]
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