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Approaching the capacity of K-user MIMO interference channel with interference counteraction scheme
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Approaching the capacity of K-user MIMO interference channel with interference counteraction scheme Zhuo Li a,b,∗, Linzhong Song c, Heping Shi d a
College of Electronic and Communication Engineering, Tianjin Normal University, Tianjin, China Tianjin Key Laboratory of Wireless Mobile Communications and Power Transmission, Tianjin Normal University, Tianjin, China c School of Electronic Information Engineering, Tianjin University, Tianjin, China d School of Automotive and Transportation, Tianjin University of Technology and Education, Tianjin, China b
a r t i c l e
i n f o
Article history: Received 7 January 2016 Revised 5 February 2016 Accepted 14 February 2016 Available online xxx Keywords: Degrees of freedom (DoF) Imperfect channel estimation Interference alignment (IA) Interference counteraction (IC)
a b s t r a c t This paper study the general K-user MIMO interference channel with M antennas at each transmitter and N antennas at the corresponding receiver. Interference counteraction scheme is proposed to improve the entire achievable rate of such channel under the assumption that the global channel state information (CSI) is available to the receivers. Comparing with some of the existing interference alignment approaches, the proposed scheme performs better in terms of resisting the correlation of transmitters and accommodating N −1 ) interferences, if only the constraint on degrees of freedom (d ≤ min{N − 2, N (K−1 } ) is provided. The performance of interference counteraction scheme is also discussed for the situation with channel estimation errors. Simulation results show that the novel scheme is robust at low and medium SNR when CSI is imperfect. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Interference management has been a severe problem in wireless communication. During the past decades, many researches have been explored to improve the capacity for the K-user interference channel. Among them, joint Minimum Mean Square Error (MMSE) designs have been firstly proposed in [1,2]. Based on regarding interferences as noises, the iterative water-filling and interference avoidance methods have been studied in [3–5], in which all the transmitters tend to allocate their transmissions along those dimensions where the intended receivers undergoes the least interferences. In view of this, the two type of methods are considered to be selfish. Interference alignment scheme known as ”do not harm” approach has been maturely proposed in [6], where degrees of freedom (also referred to as multiplexing gain) is used to characterize
∗
Corresponding author. Tel.: +8613821465706. E-mail address: [email protected] (Z. Li).
the capacity performance of the K-user MIMO interference channel. By virtue of the interference alignment idea, many approaches have been developed to suppress the undesired signals from interference sources. In [7], the distributed interference alignment methods have been demonstrated by utilizing the reciprocity of wireless networks. The authors in [8] proposed the cooperative approach based on the MMSE principle which uses Karush Kuhn Tucker(KKT) conditions to find the suboptimal pre-coding matrices and combining matrices. This two type of matrices can be used to restrain the interferences. Besides, least squares approach and joint transceiver design for interference alignment have also been discussed in [9] and [10], as a development of the joint design schemes in [1,2]. Among the methods introduced above, the schemes for interference alignment have drawn great attention due to their outstanding capacity performance. In practice, the condition on degrees of freedom derived in [11–13] has to be satisfied to ensure the validity of interference alignment schemes. This condition is generally tight and it should be loosened to accommodate more users in K-user
http://dx.doi.org/10.1016/j.adhoc.2016.02.009 1570-8705/© 2016 Elsevier B.V. All rights reserved.
Please cite this article as: Z. Li et al., Approaching the capacity of K-user MIMO interference channel with interference counteraction scheme, Ad Hoc Networks (2016), http://dx.doi.org/10.1016/j.adhoc.2016.02.009
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achieving high rate defined in [6]. They manage to minimize the mean square error between the desired signal and the reconstructed signal by seeking pre-coding matrices and combining matrices Ui (i = 1, 2 . . . K ) belonging to CN × d , which are derived with constraints tr{UiUiH } = d and tr{ViViH } = d. This is also the entry point for the proposed scheme. 3. Interference counteraction scheme
Fig. 1. The K-user interference channel.
interference channel. In addition, interference alignment approaches have not been explored for such MIMO interference channel with coherent sources and imperfect CSI. In this letter, a new interference counteraction scheme will be introduced to deal with these circumstances. Notations: Scalars are denoted by lower-case or capital letters. Lower-case and capital letters in bold type are used to represent vectors and matrices, respectively. E(•), tr(•) and Re(•) are three functions used to get expected value, trace of a square matrix and real part separately. The superscript H is used to refer to Hermitian transpose and superscript ∗ means conjugate transformation. Identity matrices are uniformly denoted by I. 2. System model
K
3.1. Interference counteraction with perfect CSI Before proceeding, let x and x|xi refer to {x1 , x2 , . . . xK } and {x1 , . . . xi−1 , xi+1 , . . . xK } separately. In this subsection, perfect global CSI Hij ( j = 1, . . . K ) are accessible to the ith (i = 1, . . . K ) receiver. The objective function for the problem stated above can be formulated as (4) by only considering the case at Rei .
UiH yi − UiH HiiVi xi − ci (x|xi )2 ⎧ 2 ⎫ ⎬ ⎨ K = Ex UiH Hi jV j x j + UiH ni − ci (x|xi ) , ⎭ ⎩ j=i
Ji = Ex
Consider the general K-user interference channel depicted in Fig. 1, in which each transmitter has M antennas and the corresponding receiver has N antennas. Such channel is denoted by (M, N, K). We assume that transmitter Tri (i = 1, 2 . . . K ) sends d messages to its intended receiver Rei . Rei receives signals from both Tri and the other K − 1 transmitters which constitute interference sources. The received signals at Rei are formulated by (1).
yi = HiiVi xi +
Traditional IA approaches tend to restrain interferences in a direct way, which also reduces the power of the desired signals. Interferences can be removed indirectly and the desired signals will not suffer attenuation, if the information on interferences provided by mixed signals is utilized. Interference counteraction scheme under perfect CSI assumption is presented in subsection A based on this idea. The modified IC scheme is demonstrated in subsection B for the situation with imperfect CSI.
Hi jV j x j + ni ,
∀i ∈ {1 , 2 , . . . K }
(1)
j=i
in which ci (x|xi ) is a function independent of xi and changes with interferences. Interference counteraction will be realized from (4) if ci (x|xi ) is constructed elaborately. Here, the decomposition and reorganisation of the received signals are utilized to construct ci (x|xi ) which is given in (5). It is interpreted as the estimation or reconstruction of the interferences.
where (i) xi ∈ Cd denotes the complex messages (or signals) sent by Tri , (ii) Vi ∈ CM × d is the pre-coding matrix at Tri , (iii) ni ∈ CN refers to the additive white Gaussian noise at Rei , (iv) Hij ∈ CN × M presents the channel matrix from Trj to Rei . Its entries are supposed to follow independent and random distributions, take finite values and change slowly. The constraints on signal sources are presented in (2) and (3). Ri j is correlation matrix satisfying Ri j = RHji .
E {ni nH i } = I, E {xi xHj } = Ri j ,
E {x j nH i } = 0,
∀i, j ∈ {1, 2, . . . K }
∀i, j ∈ {1, 2, . . . K }
(2) (3)
Most traditional IA approaches have common features in recovering the desired signal xi from mixed signal yi or
(4)
ci ( x|xi ) =
N
UikH Pik
K
Hi j ( k )V j x j + ni ( k )
(5)
j=i
k=1
Hi j ( k ) presents the remaining part of Hij after the kth line is removed. Uik is (N − 1 ) × d auxiliary matrix. Pik denotes the orthogonal projection matrix of Hii ( k )Vi given by (6) under the condition N − d ≥ 2. H Pik = I − Sik Sik ,
k = 1, . . . N
(6)
where Sik is the (N − 1 ) × d unitary matrix satisfying the compact singular value decomposition in (7). ik and Dik are diagonal and unitary matrices, respectively.
Hii ( k )Vi = Sik ik Dik .
(7)
Generally, global optimum solutions are more difficult to derive for (4) with orthogonality constraints. Nevertheless, suboptimal Vi and Ui can be found by resorting to the following theorem.
Please cite this article as: Z. Li et al., Approaching the capacity of K-user MIMO interference channel with interference counteraction scheme, Ad Hoc Networks (2016), http://dx.doi.org/10.1016/j.adhoc.2016.02.009
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Z. Li et al. / Ad Hoc Networks 000 (2016) 1–6 N −1 ) Theorem 1. Provided that d ≤ min{ N (K−1 , N − 2} are satisfied, there are matrices Wik (k = 1, . . . N ) belonging to C (N−1 )×N that satisfy Uik = WikUi . The dominant part contributed by interferences in Ji can be eliminated based on Uik . The residual part is just the limited combinations of noises.
Proof. Decomposition optimization strategy is applied to complete the proof, since that the transmit power is often greater than that of noises. For (4), it can be restated as (8).
Ji = Ex tr (
UiH
× UiH
K
Hi jV j x j + ni ) − ci (x|xi )
j=i K
Hi jV j x j + ni
− ci ( x|xi )
H
= Jim + Jir
j=i
(8) where
Jim = tr UiH
×
K
Hi jV j R jl VlH HilH Ui +
j,l=i K
UiH Hi jV j
j=i
K
N
Jir = tr UiH Ui − 2Re UiH
R jl VlH
N
N
UikH Pik E
The optimization for Jir on Vi is very complicated due to the introduction of Pik . For simplicity, Vi is supposed to take the value which maximizes Hii Vi 2 and satisfies tr{ViViH } = d, if Hii is known to Tri . Otherwise, Vi takes arbitrary random matrix with full rank satisfying the constraint. With this precondition, the optimization problem are formulated as (13). The solution for Ui is given by (14).
minimize Jir
Hil ( k )H PikH Uik
, (9)
tr{UiUiH } = d.
N
{ni ( k )ni ( f ) }
PiHf Ui f
.
(10)
Jim occupies the major role in (8) and Jir consists of the combinations of noises according to (9) and (10). Jim are firstly minimized. Jir is then optimized based on this step. For Jim , the derivative about Uik is given in (11).
N K ∂ Jim = Pik Hi j ( k )V j R jl VlH Hil ( f )H PiHf Ui f ∂ Uik∗ f =1 j,l=i Hi j ( k )V j R jl VlH HilH Ui ,
N
(13)
E {ni ni ( k )H }PikH Wik
WikH Pik E ni ( k )ni ( f )
H
PiHf Wik ,
(14)
k, f =1
in which eigvd {•} refers to the matrix which consists of the eigenvectors corresponding to the d smaller eigenvalues of the matrix surrounded by brace. Taken together, the proof for Theorem 1 has been completed. We introduce the way of finding pre-coding and combining matrices for interference counteraction scheme through the proof. In contrast with the IA schemes in [7,8], the proposed scheme can avoid iterations.
E {ni ni ( k )H }PikH Uik
H
( ∀k = 1 , 2 , . . . N )
subject to Uik = WikUi ,
+
k, f =1
K
(12)
k=1
k=1
k=1
− Pik
f =1
∀l = 1, 2, . . . K, l = i
Ui = eigvd I − 2Re
UikH Pik
l=i
+
VlH Hil ( f )H PiHf Ui f = VlH HilH Ui ,
k, f =1
N
N
j,l=i
−2Re
on matrix theory and denoted by Wik Ui . By this point, the remaining work is to optimize Jir with respect to Ui and Vi .
Hi j ( k )V j R jl VlH Hil ( f )H PiHf Ui f K
3
∀k ∈ {1 , 2 , . . . N }.
j,l=i
(11) Jim is non-negative and it should be suppressed to approaching zero. The equations involving Uik (k = 1, 2, . . . N ) are constructed in (12) by this fact. Jim will be removed completely by combining (11) with (9), if (12) is satisfied. In order to validate (12) or solve the auxiliary matrix Uik , the K − 1 matrix equations are combined to form equivalent (K − 1 )d linear equations with N (N − 1 ) variables. Thus, Uik can be attained with the given condition based
3.2. Analysis with imperfect CSI Actually, CSI deviates from the real value due to estimation errors denoted by Heij (i, j = 1, . . . , K ). It is necessary to discuss the performance of IC scheme in this situation. Hij and Hi j are used to denote the real and estimated ma-
trices separately, that is, Hi j = Hi j − Hei j . In contrast with Hij , Heii usually changes faster and takes smaller values. Hence, it is inappropriate to treat them equally and (4) cannot be directly applied here. We can analyse the performance of the proposed scheme statistically, since that the statistics information on Heii is generally known to receivers. Necessary assumptions on Heij are presented in (15), in which σ is standard deviation. Pik is the orthogonal projection matrix of Hii ( k )Vi . Heii Vi is assumed to only affect ik and Dik in (7), that is, Pik =Pik . Similarly, the modified objective function is given by has the analogous expression with J by (16), where Jim im replacing Hij , Hi j ( k ) and Pik in (9) with Hi j , Hi j ( k ) and Pik , respectively. ci (x|xi ) is given in (17). See appendix for the detailed expression of Jir .
E {Heii } = 0,
2 E {H eii H eH ii } = σ I
(15)
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+ J , = Jim ir
c ( x|x i
i
N
)=
K
The same result about Uik and Vi can be achieved for (16) according to Theorem 1. The solution for Ui here is slightly different from that in (14). It is directly given by (18), where Wik plays the same role as Wik .
Ui = eigvd I − 2Re
N
E {ni ni ( k )
H H P ikWik
}
k=1
W ik P ik E {ni ( k )ni ( f )H }P i f Wik H
H
k, f =1
+
K σ2
N
N
W ik P ik F H
f k
k, f =1,k= f
H × W i f P i f F
N k H H − 2Re F (k )P ikWik . f
(Hi j ( k ) − Hei j ( k ))V j x j + ni ( k ) (17)
N
H
k=1
(18)
j=i
k=1
+
H
(16)
UikH Pik
× W ik P ik P ikWik +
Ji = E{x,Heii } {UiH yi − UiH Hii Vi xi − ci (x|xi )2 }
j=1
tr{
V j R j jV jH
} I+
N k=1
By (16), the interference terms caused by the estimation errors fall into the small quantity part. This is reasonable for in reality, errors are generally quite small. Otherwise, the normal communication is difficult to realize. Now that the detailed presentation for IC scheme has been finished, in the following part, the new scheme will be verified through simulation results. 4. Simulations and discussions In order to verify the performance of the proposed scheme, simulations are conducted from three aspects depicted in Figs. 2, 3, 4. All the simulations are based on the (5, 5, 4) interference channel with the same d used by each communication pair. During simulations, 300 iterations are used for the existing interference alignment approaches.
Fig. 2. Average rate per user for 4-user 5 × 5 interference channel with d = 2 and 3 under the perfect CSI assumption.
Fig. 3. Average rate per user for 4-user 5 × 5 interference channel with perfect CSI under the correlation assumption with d = 2.
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5
Fig. 4. Average rate per user achieved by the proposed scheme for 4-user 5 × 5 interference channel with different d under the imperfect CSI assumption.
Random complex symmetric matrices are utilized to imitate Ri j . As illustrated in Fig. 2, the proposed scheme has advantages over the two existing interference alignment approaches minimum interference-plus-noise-leakage (MinINL) and maximization signal-to-interference-plus-noise ratio (Max-SINR) proposed in [7,8], in terms of the rate per user achieved with same d and the capability of accommodating interferences. For the (5, 5, 4) interference channel, the number of sent messages (or the degrees of freeN −1 ) dom used) falls into the constraint d ≤ min{N − 2, N (K−1 }. In the statistical sense, the main interferences are removed thoroughly by interference counteraction scheme. The remainder is just a small quantity part consisting of the combinations of noises. Roughly speaking, it can be regarded as a constant with respect to the transmit power or Signal Noise Ratio (SNR). However, the interferences left in the desired signals still occupy a certain proportion after the two IA approaches are employed. It become evident with increasing SNR. This causes the decline in rate increasement at high SNR. For d = 3, the performance of existing approaches deteriorates, because the constraint in [11] derived using interference alignment scheme is broken. Fig. 3 depicts that the proposed scheme outperforms the two IA approaches against the correlation sources. IC scheme is not fettered by the correlation assumption if d ≤ N −1 ) } is satisfied. For the existing approaches, min{N − 2, N (K−1 the correlation will add more cross terms to their objective functions. These extra terms invalidate the two IA approaches at medium and high SNR. This also reveals the weakness of some conventional IA approaches in accommodating enough communication pairs. According to Fig. 4, the estimation errors impact the capacity performance discriminatively with respect to different SNR. The average rate per user basically remains unchanged at relative high SNR. Therefore, using the proposed scheme will not acquire performance improvement only by increasing SNR when SNR has already reached a specific value. The interferences caused by estimation errors are at the same level as noises at low and medium SNR, which contributes
to the satisfactory performance in consistent with that in the perfect CSI situation. 5. Conclusion A robust interference counteraction scheme is proposed in this letter to deal with the interferences in K-user MIMO interference channel. The novel scheme are confirmed to be valid in promoting the achievable rate. Besides, the proposed scheme will not be affected by the independence assumption on the desired signals and interferences, if specific constraints can be satisfied. The simulation results have also supported this point. According to the derivations and simulation results, there are some situations, in which the counteraction strategy is a better choice comparing with the direct suppression strategy used in prevailing IA approaches. Acknowledgments This research was supported by Research Fund for the Doctoral Programme of Tianjin Normal University (52XB1504). Appendix A. The expression of Jir Based on the preceding assumptions, Jir is given by
Jir = tr UiH Ui − 2Re UiH
N
E {ni ni ( k )
H
}P H U ik
ik
k=1
+
N
UikH P ik E {ni ( k )ni ( f )H }P i f Ui f } H
k, f =1
+EHei
K
tr{UiH H ei jV j R j jV jH H eH i j Ui
j=1
+
N
UikH P ik H ei j ( k )V j R j jV jH H ei j ( f )H P i f Ui f H
k, f =1
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N
−2Re UiH H ei jV j R j jV jH
= tr
− 2Re
H ei j ( k )P Uik
k=1
UiH Ui
H
UiH
N
E {ni ni ( k )
H
}P H U ik
ik
k=1
+
N
UikH P ik E {ni ( k )ni ( f )H }P i f Ui f } H
k, f =1
+
+
K σ2
t r{
N
V j R j jV jH
}t r UiH Ui +
H
UikH P ik F (kf )(UiHf P i f F ( kf ))H
k, f =1,k= f
−2Re
(UikH P ik P ikUik )
k=1
j=1 N
N
N
UiH F (k )P ikUik H
(19)
k=1
where F(k) and F (k) separately denote the remaining parts of (N − 1 ) × (N − 1 ) and N × N identity matrices after f their kth columns are removed. k has the following expression,
= f k
f −1 f min{k, f }
f − k ≥ 2, k − f ≥ 2, |k − f | = 1.
(20)
References [1] A.J. Tenenbaum, R.S. Adve, Joint multiuser transmit-receive optimization using linear processing, in: Proceedings of the IEEE International Conference on Communication (ICC), 1, 2004, pp. 558–592. [2] J. Zhang, Y. Wu, S. Zhou, J. Wang, Joint linear transmitter and receiver design for the downlink of multiuser MIMO systems, IEEE Commun. Lett. 9 (11) (2005) 991–993. [3] C. Rose, S. Ulukus, R. Yates, Wireless systems and interference avoidance, in: Proceedings of the IEEE Transactions on Wireless Communication, 1, 2002, pp. 415–418. [4] O. Popescu, C. Rose, D.C. Popescu, Signal space partitioning versus simultaneous water filling for mutually interfering systems, in: Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM), 5, 2004, pp. 3128–3132. [5] W. Yu, W. Rhee, S. Boyd, J.M. Cioff, Iterative water-filling for gaussian vector multiple-access channels, IEEE Trans. Inf. Theory 5 (1) (2005) 145–152. [6] V.R. Cadambe, S.A. Jafar, Interference alignment and degrees of freedom region for the k-user interference channel, IEEE Trans. Inf. Theory 54 (8) (2008) 3425–3441.
[7] K. Gomadam, V.R. Cadambe, S.A. Jafar, Approaching the capacity of wireless networks through distributed interference alignment, Proc 2008 IEEE GLOBACOM1–6. [8] S.W. Peters, R.W. Heath, Cooperative algorithms for MIMO interference channels, IEEE Trans. Veh. Technol. 60 (1) (2011) 206–218. [9] H. Yu, Y. Sung, Least squares approach to joint beam design for interference alignment in multiuser multi-input multi-output interference channels, IEEE Trans. Signal Process. 58 (9) (2010) 4960–4966. [10] M. Razaviyayn, M. Sanjabi, Z. Luo, Linear transceiver design for interference alignment:complexity and computation, IEEE Trans. Inf. Theory 58 (9) (2012) 2896–2910. [11] T. Gou, S.A. Jafar, Degrees of freedom of the k-user m × n MIMO interference channel, IEEE Trans. Inf. Theory 56 (12) (2010) 6040–6057. [12] G. Sridharan, W. Yu, Degrees of freedom of MIMO cellular networks: decomposition and linear beamforming design, IEEE Trans. Inf. Theory 61 (6) (2015) 3339–3364. [13] O. Gonzalez, C. Beltran, I. Santamaria, On the number of interference alignment solutions for the k-user MIMO channel with constant coefficients, IEEE Trans. Inf. Theory 61 (12) (2015) 6028–6048. Zhuo Li ([email protected]) is a lecturer with Collage of Electronic and Communication Engineering at Tianjin Normal University. He received his B.S. degree in computer science from Tianjin Normal University in 2007 and received M.S. degrees in computer science from East China Normal University in 2010. After that he joined Tianjin University and received his Ph.D. degrees in circuit and system in 2015. His research interests include Computer Networks, Future Internet Architecture, Named Data Networking and Wireless Communications. Linzhong Song ([email protected]) received his B.S. degree from Anhui University in 2012 and received M.S. degrees in circuits and systems from Tianjin University in 2015. His research interests include Computer Networks and Wireless Communications.
Heping Shi ([email protected]) received the B.S. and M.S. degrees from Tianjin University of Technology and education, Tianjin, China, in 2009 and 2012, respectively. He received M.S. degrees in circuits and systems from Tianjin University in 2015. He is currently a Lecturer in Tianjin University of Technology and education. His research interests include in the area of direction-of-arrival (DOA) and array signal processing.
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Contents lists available at ScienceDirect
Ad Hoc Networks journal homepage: www.elsevier.com/locate/adhoc
Approaching the capacity of K-user MIMO interference channel with interference counteraction scheme Zhuo Li a,b,∗, Linzhong Song c, Heping Shi d a
College of Electronic and Communication Engineering, Tianjin Normal University, Tianjin, China Tianjin Key Laboratory of Wireless Mobile Communications and Power Transmission, Tianjin Normal University, Tianjin, China c School of Electronic Information Engineering, Tianjin University, Tianjin, China d School of Automotive and Transportation, Tianjin University of Technology and Education, Tianjin, China b
a r t i c l e
i n f o
Article history: Received 7 January 2016 Revised 5 February 2016 Accepted 14 February 2016 Available online xxx Keywords: Degrees of freedom (DoF) Imperfect channel estimation Interference alignment (IA) Interference counteraction (IC)
a b s t r a c t This paper study the general K-user MIMO interference channel with M antennas at each transmitter and N antennas at the corresponding receiver. Interference counteraction scheme is proposed to improve the entire achievable rate of such channel under the assumption that the global channel state information (CSI) is available to the receivers. Comparing with some of the existing interference alignment approaches, the proposed scheme performs better in terms of resisting the correlation of transmitters and accommodating N −1 ) interferences, if only the constraint on degrees of freedom (d ≤ min{N − 2, N (K−1 } ) is provided. The performance of interference counteraction scheme is also discussed for the situation with channel estimation errors. Simulation results show that the novel scheme is robust at low and medium SNR when CSI is imperfect. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Interference management has been a severe problem in wireless communication. During the past decades, many researches have been explored to improve the capacity for the K-user interference channel. Among them, joint Minimum Mean Square Error (MMSE) designs have been firstly proposed in [1,2]. Based on regarding interferences as noises, the iterative water-filling and interference avoidance methods have been studied in [3–5], in which all the transmitters tend to allocate their transmissions along those dimensions where the intended receivers undergoes the least interferences. In view of this, the two type of methods are considered to be selfish. Interference alignment scheme known as ”do not harm” approach has been maturely proposed in [6], where degrees of freedom (also referred to as multiplexing gain) is used to characterize
∗
Corresponding author. Tel.: +8613821465706. E-mail address: [email protected] (Z. Li).
the capacity performance of the K-user MIMO interference channel. By virtue of the interference alignment idea, many approaches have been developed to suppress the undesired signals from interference sources. In [7], the distributed interference alignment methods have been demonstrated by utilizing the reciprocity of wireless networks. The authors in [8] proposed the cooperative approach based on the MMSE principle which uses Karush Kuhn Tucker(KKT) conditions to find the suboptimal pre-coding matrices and combining matrices. This two type of matrices can be used to restrain the interferences. Besides, least squares approach and joint transceiver design for interference alignment have also been discussed in [9] and [10], as a development of the joint design schemes in [1,2]. Among the methods introduced above, the schemes for interference alignment have drawn great attention due to their outstanding capacity performance. In practice, the condition on degrees of freedom derived in [11–13] has to be satisfied to ensure the validity of interference alignment schemes. This condition is generally tight and it should be loosened to accommodate more users in K-user
http://dx.doi.org/10.1016/j.adhoc.2016.02.009 1570-8705/© 2016 Elsevier B.V. All rights reserved.
Please cite this article as: Z. Li et al., Approaching the capacity of K-user MIMO interference channel with interference counteraction scheme, Ad Hoc Networks (2016), http://dx.doi.org/10.1016/j.adhoc.2016.02.009
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achieving high rate defined in [6]. They manage to minimize the mean square error between the desired signal and the reconstructed signal by seeking pre-coding matrices and combining matrices Ui (i = 1, 2 . . . K ) belonging to CN × d , which are derived with constraints tr{UiUiH } = d and tr{ViViH } = d. This is also the entry point for the proposed scheme. 3. Interference counteraction scheme
Fig. 1. The K-user interference channel.
interference channel. In addition, interference alignment approaches have not been explored for such MIMO interference channel with coherent sources and imperfect CSI. In this letter, a new interference counteraction scheme will be introduced to deal with these circumstances. Notations: Scalars are denoted by lower-case or capital letters. Lower-case and capital letters in bold type are used to represent vectors and matrices, respectively. E(•), tr(•) and Re(•) are three functions used to get expected value, trace of a square matrix and real part separately. The superscript H is used to refer to Hermitian transpose and superscript ∗ means conjugate transformation. Identity matrices are uniformly denoted by I. 2. System model
K
3.1. Interference counteraction with perfect CSI Before proceeding, let x and x|xi refer to {x1 , x2 , . . . xK } and {x1 , . . . xi−1 , xi+1 , . . . xK } separately. In this subsection, perfect global CSI Hij ( j = 1, . . . K ) are accessible to the ith (i = 1, . . . K ) receiver. The objective function for the problem stated above can be formulated as (4) by only considering the case at Rei .
UiH yi − UiH HiiVi xi − ci (x|xi )2 ⎧ 2 ⎫ ⎬ ⎨ K = Ex UiH Hi jV j x j + UiH ni − ci (x|xi ) , ⎭ ⎩ j=i
Ji = Ex
Consider the general K-user interference channel depicted in Fig. 1, in which each transmitter has M antennas and the corresponding receiver has N antennas. Such channel is denoted by (M, N, K). We assume that transmitter Tri (i = 1, 2 . . . K ) sends d messages to its intended receiver Rei . Rei receives signals from both Tri and the other K − 1 transmitters which constitute interference sources. The received signals at Rei are formulated by (1).
yi = HiiVi xi +
Traditional IA approaches tend to restrain interferences in a direct way, which also reduces the power of the desired signals. Interferences can be removed indirectly and the desired signals will not suffer attenuation, if the information on interferences provided by mixed signals is utilized. Interference counteraction scheme under perfect CSI assumption is presented in subsection A based on this idea. The modified IC scheme is demonstrated in subsection B for the situation with imperfect CSI.
Hi jV j x j + ni ,
∀i ∈ {1 , 2 , . . . K }
(1)
j=i
in which ci (x|xi ) is a function independent of xi and changes with interferences. Interference counteraction will be realized from (4) if ci (x|xi ) is constructed elaborately. Here, the decomposition and reorganisation of the received signals are utilized to construct ci (x|xi ) which is given in (5). It is interpreted as the estimation or reconstruction of the interferences.
where (i) xi ∈ Cd denotes the complex messages (or signals) sent by Tri , (ii) Vi ∈ CM × d is the pre-coding matrix at Tri , (iii) ni ∈ CN refers to the additive white Gaussian noise at Rei , (iv) Hij ∈ CN × M presents the channel matrix from Trj to Rei . Its entries are supposed to follow independent and random distributions, take finite values and change slowly. The constraints on signal sources are presented in (2) and (3). Ri j is correlation matrix satisfying Ri j = RHji .
E {ni nH i } = I, E {xi xHj } = Ri j ,
E {x j nH i } = 0,
∀i, j ∈ {1, 2, . . . K }
∀i, j ∈ {1, 2, . . . K }
(2) (3)
Most traditional IA approaches have common features in recovering the desired signal xi from mixed signal yi or
(4)
ci ( x|xi ) =
N
UikH Pik
K
Hi j ( k )V j x j + ni ( k )
(5)
j=i
k=1
Hi j ( k ) presents the remaining part of Hij after the kth line is removed. Uik is (N − 1 ) × d auxiliary matrix. Pik denotes the orthogonal projection matrix of Hii ( k )Vi given by (6) under the condition N − d ≥ 2. H Pik = I − Sik Sik ,
k = 1, . . . N
(6)
where Sik is the (N − 1 ) × d unitary matrix satisfying the compact singular value decomposition in (7). ik and Dik are diagonal and unitary matrices, respectively.
Hii ( k )Vi = Sik ik Dik .
(7)
Generally, global optimum solutions are more difficult to derive for (4) with orthogonality constraints. Nevertheless, suboptimal Vi and Ui can be found by resorting to the following theorem.
Please cite this article as: Z. Li et al., Approaching the capacity of K-user MIMO interference channel with interference counteraction scheme, Ad Hoc Networks (2016), http://dx.doi.org/10.1016/j.adhoc.2016.02.009
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Z. Li et al. / Ad Hoc Networks 000 (2016) 1–6 N −1 ) Theorem 1. Provided that d ≤ min{ N (K−1 , N − 2} are satisfied, there are matrices Wik (k = 1, . . . N ) belonging to C (N−1 )×N that satisfy Uik = WikUi . The dominant part contributed by interferences in Ji can be eliminated based on Uik . The residual part is just the limited combinations of noises.
Proof. Decomposition optimization strategy is applied to complete the proof, since that the transmit power is often greater than that of noises. For (4), it can be restated as (8).
Ji = Ex tr (
UiH
× UiH
K
Hi jV j x j + ni ) − ci (x|xi )
j=i K
Hi jV j x j + ni
− ci ( x|xi )
H
= Jim + Jir
j=i
(8) where
Jim = tr UiH
×
K
Hi jV j R jl VlH HilH Ui +
j,l=i K
UiH Hi jV j
j=i
K
N
Jir = tr UiH Ui − 2Re UiH
R jl VlH
N
N
UikH Pik E
The optimization for Jir on Vi is very complicated due to the introduction of Pik . For simplicity, Vi is supposed to take the value which maximizes Hii Vi 2 and satisfies tr{ViViH } = d, if Hii is known to Tri . Otherwise, Vi takes arbitrary random matrix with full rank satisfying the constraint. With this precondition, the optimization problem are formulated as (13). The solution for Ui is given by (14).
minimize Jir
Hil ( k )H PikH Uik
, (9)
tr{UiUiH } = d.
N
{ni ( k )ni ( f ) }
PiHf Ui f
.
(10)
Jim occupies the major role in (8) and Jir consists of the combinations of noises according to (9) and (10). Jim are firstly minimized. Jir is then optimized based on this step. For Jim , the derivative about Uik is given in (11).
N K ∂ Jim = Pik Hi j ( k )V j R jl VlH Hil ( f )H PiHf Ui f ∂ Uik∗ f =1 j,l=i Hi j ( k )V j R jl VlH HilH Ui ,
N
(13)
E {ni ni ( k )H }PikH Wik
WikH Pik E ni ( k )ni ( f )
H
PiHf Wik ,
(14)
k, f =1
in which eigvd {•} refers to the matrix which consists of the eigenvectors corresponding to the d smaller eigenvalues of the matrix surrounded by brace. Taken together, the proof for Theorem 1 has been completed. We introduce the way of finding pre-coding and combining matrices for interference counteraction scheme through the proof. In contrast with the IA schemes in [7,8], the proposed scheme can avoid iterations.
E {ni ni ( k )H }PikH Uik
H
( ∀k = 1 , 2 , . . . N )
subject to Uik = WikUi ,
+
k, f =1
K
(12)
k=1
k=1
k=1
− Pik
f =1
∀l = 1, 2, . . . K, l = i
Ui = eigvd I − 2Re
UikH Pik
l=i
+
VlH Hil ( f )H PiHf Ui f = VlH HilH Ui ,
k, f =1
N
N
j,l=i
−2Re
on matrix theory and denoted by Wik Ui . By this point, the remaining work is to optimize Jir with respect to Ui and Vi .
Hi j ( k )V j R jl VlH Hil ( f )H PiHf Ui f K
3
∀k ∈ {1 , 2 , . . . N }.
j,l=i
(11) Jim is non-negative and it should be suppressed to approaching zero. The equations involving Uik (k = 1, 2, . . . N ) are constructed in (12) by this fact. Jim will be removed completely by combining (11) with (9), if (12) is satisfied. In order to validate (12) or solve the auxiliary matrix Uik , the K − 1 matrix equations are combined to form equivalent (K − 1 )d linear equations with N (N − 1 ) variables. Thus, Uik can be attained with the given condition based
3.2. Analysis with imperfect CSI Actually, CSI deviates from the real value due to estimation errors denoted by Heij (i, j = 1, . . . , K ). It is necessary to discuss the performance of IC scheme in this situation. Hij and Hi j are used to denote the real and estimated ma-
trices separately, that is, Hi j = Hi j − Hei j . In contrast with Hij , Heii usually changes faster and takes smaller values. Hence, it is inappropriate to treat them equally and (4) cannot be directly applied here. We can analyse the performance of the proposed scheme statistically, since that the statistics information on Heii is generally known to receivers. Necessary assumptions on Heij are presented in (15), in which σ is standard deviation. Pik is the orthogonal projection matrix of Hii ( k )Vi . Heii Vi is assumed to only affect ik and Dik in (7), that is, Pik =Pik . Similarly, the modified objective function is given by has the analogous expression with J by (16), where Jim im replacing Hij , Hi j ( k ) and Pik in (9) with Hi j , Hi j ( k ) and Pik , respectively. ci (x|xi ) is given in (17). See appendix for the detailed expression of Jir .
E {Heii } = 0,
2 E {H eii H eH ii } = σ I
(15)
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+ J , = Jim ir
c ( x|x i
i
N
)=
K
The same result about Uik and Vi can be achieved for (16) according to Theorem 1. The solution for Ui here is slightly different from that in (14). It is directly given by (18), where Wik plays the same role as Wik .
Ui = eigvd I − 2Re
N
E {ni ni ( k )
H H P ikWik
}
k=1
W ik P ik E {ni ( k )ni ( f )H }P i f Wik H
H
k, f =1
+
K σ2
N
N
W ik P ik F H
f k
k, f =1,k= f
H × W i f P i f F
N k H H − 2Re F (k )P ikWik . f
(Hi j ( k ) − Hei j ( k ))V j x j + ni ( k ) (17)
N
H
k=1
(18)
j=i
k=1
+
H
(16)
UikH Pik
× W ik P ik P ikWik +
Ji = E{x,Heii } {UiH yi − UiH Hii Vi xi − ci (x|xi )2 }
j=1
tr{
V j R j jV jH
} I+
N k=1
By (16), the interference terms caused by the estimation errors fall into the small quantity part. This is reasonable for in reality, errors are generally quite small. Otherwise, the normal communication is difficult to realize. Now that the detailed presentation for IC scheme has been finished, in the following part, the new scheme will be verified through simulation results. 4. Simulations and discussions In order to verify the performance of the proposed scheme, simulations are conducted from three aspects depicted in Figs. 2, 3, 4. All the simulations are based on the (5, 5, 4) interference channel with the same d used by each communication pair. During simulations, 300 iterations are used for the existing interference alignment approaches.
Fig. 2. Average rate per user for 4-user 5 × 5 interference channel with d = 2 and 3 under the perfect CSI assumption.
Fig. 3. Average rate per user for 4-user 5 × 5 interference channel with perfect CSI under the correlation assumption with d = 2.
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5
Fig. 4. Average rate per user achieved by the proposed scheme for 4-user 5 × 5 interference channel with different d under the imperfect CSI assumption.
Random complex symmetric matrices are utilized to imitate Ri j . As illustrated in Fig. 2, the proposed scheme has advantages over the two existing interference alignment approaches minimum interference-plus-noise-leakage (MinINL) and maximization signal-to-interference-plus-noise ratio (Max-SINR) proposed in [7,8], in terms of the rate per user achieved with same d and the capability of accommodating interferences. For the (5, 5, 4) interference channel, the number of sent messages (or the degrees of freeN −1 ) dom used) falls into the constraint d ≤ min{N − 2, N (K−1 }. In the statistical sense, the main interferences are removed thoroughly by interference counteraction scheme. The remainder is just a small quantity part consisting of the combinations of noises. Roughly speaking, it can be regarded as a constant with respect to the transmit power or Signal Noise Ratio (SNR). However, the interferences left in the desired signals still occupy a certain proportion after the two IA approaches are employed. It become evident with increasing SNR. This causes the decline in rate increasement at high SNR. For d = 3, the performance of existing approaches deteriorates, because the constraint in [11] derived using interference alignment scheme is broken. Fig. 3 depicts that the proposed scheme outperforms the two IA approaches against the correlation sources. IC scheme is not fettered by the correlation assumption if d ≤ N −1 ) } is satisfied. For the existing approaches, min{N − 2, N (K−1 the correlation will add more cross terms to their objective functions. These extra terms invalidate the two IA approaches at medium and high SNR. This also reveals the weakness of some conventional IA approaches in accommodating enough communication pairs. According to Fig. 4, the estimation errors impact the capacity performance discriminatively with respect to different SNR. The average rate per user basically remains unchanged at relative high SNR. Therefore, using the proposed scheme will not acquire performance improvement only by increasing SNR when SNR has already reached a specific value. The interferences caused by estimation errors are at the same level as noises at low and medium SNR, which contributes
to the satisfactory performance in consistent with that in the perfect CSI situation. 5. Conclusion A robust interference counteraction scheme is proposed in this letter to deal with the interferences in K-user MIMO interference channel. The novel scheme are confirmed to be valid in promoting the achievable rate. Besides, the proposed scheme will not be affected by the independence assumption on the desired signals and interferences, if specific constraints can be satisfied. The simulation results have also supported this point. According to the derivations and simulation results, there are some situations, in which the counteraction strategy is a better choice comparing with the direct suppression strategy used in prevailing IA approaches. Acknowledgments This research was supported by Research Fund for the Doctoral Programme of Tianjin Normal University (52XB1504). Appendix A. The expression of Jir Based on the preceding assumptions, Jir is given by
Jir = tr UiH Ui − 2Re UiH
N
E {ni ni ( k )
H
}P H U ik
ik
k=1
+
N
UikH P ik E {ni ( k )ni ( f )H }P i f Ui f } H
k, f =1
+EHei
K
tr{UiH H ei jV j R j jV jH H eH i j Ui
j=1
+
N
UikH P ik H ei j ( k )V j R j jV jH H ei j ( f )H P i f Ui f H
k, f =1
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N
−2Re UiH H ei jV j R j jV jH
= tr
− 2Re
H ei j ( k )P Uik
k=1
UiH Ui
H
UiH
N
E {ni ni ( k )
H
}P H U ik
ik
k=1
+
N
UikH P ik E {ni ( k )ni ( f )H }P i f Ui f } H
k, f =1
+
+
K σ2
t r{
N
V j R j jV jH
}t r UiH Ui +
H
UikH P ik F (kf )(UiHf P i f F ( kf ))H
k, f =1,k= f
−2Re
(UikH P ik P ikUik )
k=1
j=1 N
N
N
UiH F (k )P ikUik H
(19)
k=1
where F(k) and F (k) separately denote the remaining parts of (N − 1 ) × (N − 1 ) and N × N identity matrices after f their kth columns are removed. k has the following expression,
= f k
f −1 f min{k, f }
f − k ≥ 2, k − f ≥ 2, |k − f | = 1.
(20)
References [1] A.J. Tenenbaum, R.S. Adve, Joint multiuser transmit-receive optimization using linear processing, in: Proceedings of the IEEE International Conference on Communication (ICC), 1, 2004, pp. 558–592. [2] J. Zhang, Y. Wu, S. Zhou, J. Wang, Joint linear transmitter and receiver design for the downlink of multiuser MIMO systems, IEEE Commun. Lett. 9 (11) (2005) 991–993. [3] C. Rose, S. Ulukus, R. Yates, Wireless systems and interference avoidance, in: Proceedings of the IEEE Transactions on Wireless Communication, 1, 2002, pp. 415–418. [4] O. Popescu, C. Rose, D.C. Popescu, Signal space partitioning versus simultaneous water filling for mutually interfering systems, in: Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM), 5, 2004, pp. 3128–3132. [5] W. Yu, W. Rhee, S. Boyd, J.M. Cioff, Iterative water-filling for gaussian vector multiple-access channels, IEEE Trans. Inf. Theory 5 (1) (2005) 145–152. [6] V.R. Cadambe, S.A. Jafar, Interference alignment and degrees of freedom region for the k-user interference channel, IEEE Trans. Inf. Theory 54 (8) (2008) 3425–3441.
[7] K. Gomadam, V.R. Cadambe, S.A. Jafar, Approaching the capacity of wireless networks through distributed interference alignment, Proc 2008 IEEE GLOBACOM1–6. [8] S.W. Peters, R.W. Heath, Cooperative algorithms for MIMO interference channels, IEEE Trans. Veh. Technol. 60 (1) (2011) 206–218. [9] H. Yu, Y. Sung, Least squares approach to joint beam design for interference alignment in multiuser multi-input multi-output interference channels, IEEE Trans. Signal Process. 58 (9) (2010) 4960–4966. [10] M. Razaviyayn, M. Sanjabi, Z. Luo, Linear transceiver design for interference alignment:complexity and computation, IEEE Trans. Inf. Theory 58 (9) (2012) 2896–2910. [11] T. Gou, S.A. Jafar, Degrees of freedom of the k-user m × n MIMO interference channel, IEEE Trans. Inf. Theory 56 (12) (2010) 6040–6057. [12] G. Sridharan, W. Yu, Degrees of freedom of MIMO cellular networks: decomposition and linear beamforming design, IEEE Trans. Inf. Theory 61 (6) (2015) 3339–3364. [13] O. Gonzalez, C. Beltran, I. Santamaria, On the number of interference alignment solutions for the k-user MIMO channel with constant coefficients, IEEE Trans. Inf. Theory 61 (12) (2015) 6028–6048. Zhuo Li ([email protected]) is a lecturer with Collage of Electronic and Communication Engineering at Tianjin Normal University. He received his B.S. degree in computer science from Tianjin Normal University in 2007 and received M.S. degrees in computer science from East China Normal University in 2010. After that he joined Tianjin University and received his Ph.D. degrees in circuit and system in 2015. His research interests include Computer Networks, Future Internet Architecture, Named Data Networking and Wireless Communications. Linzhong Song ([email protected]) received his B.S. degree from Anhui University in 2012 and received M.S. degrees in circuits and systems from Tianjin University in 2015. His research interests include Computer Networks and Wireless Communications.
Heping Shi ([email protected]) received the B.S. and M.S. degrees from Tianjin University of Technology and education, Tianjin, China, in 2009 and 2012, respectively. He received M.S. degrees in circuits and systems from Tianjin University in 2015. He is currently a Lecturer in Tianjin University of Technology and education. His research interests include in the area of direction-of-arrival (DOA) and array signal processing.
Please cite this article as: Z. Li et al., Approaching the capacity of K-user MIMO interference channel with interference counteraction scheme, Ad Hoc Networks (2016), http://dx.doi.org/10.1016/j.adhoc.2016.02.009