Ergodic interference alignment for MIMO interference system with variable channel

The Journal of China Universities of Posts and Telecommunications February 2015, 22(1): 11–16 www.sciencedirect.com/science/journal/10058885

http://jcupt.xsw.bupt.cn

Ergodic interference alignment for MIMO interference system with variable channel Ren Haiying, Liu Yuan’an, Liu Fang (

), Gao Jinchun, Liu Kaiming, Xie Gang

School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China

Abstract The authors pay focus on the K user multiple-input multiple-output (MIMO) Gaussian interference channel (IC) with M transmitting antennas and N receiving antennas, in which min ( M , N ) max ( M , N )≥ ( K − 1)  K ( K − 2 )  and K > 3 . The channel coefficients are variable, time varying or frequency selectively drawn from a continuous distribution. Based on ergodic interference alignment (IA), an achievable scheme was proposed to achieve a total of KMN ( M + N ) degrees of freedom (DoF). The ergodic IA scheme can reach the optimal DoF value with simply linear beamforming and finite symbols. Furthermore, the achievable rate of the ergodic IA scheme was derived at any signal-to-noise ratio (SNR). With numerical simulation, the performance of the proposed scheme is evaluated. Keywords

degrees of freedom, ergodic interference alignment, interference channel, multiple-input multiple-output

1 Introduction IA has become one of hotspots in wireless communications. When Cadambe and Jafar aligned an arbitrarily large number of interferences [1], the strength of IA as a general principle was established. Then, a variety of applications based on increasingly sophisticated forms emerges. The scenario of K user M × N MIMO IC is one of important applications. Although it plays an important role in the modern wireless communications, the research works are obviously inadequate. Hence, the K user M × N MIMO IC was mainly focused on. Each transmitter and each receiver is equipped with M and N antennas, respectively. One important measure for the performance of the K user MIMO IC is on the DoF, which is considered as the first-order approximation of sum capacity in the high SNR regime. The work of Refs. [2–3] only focused on a finite number of users, e.g., two or three users. In Table 1, we summarize the previous relevant work for theoretic maximal DoF of the general K user M × N MIMO IC. The Received date: 22-09-2014 Corresponding author: Liu Fang, E-mail: [email protected] DOI: 10.1016/S1005-8885(15)60619-7

work of Ref. [4] only focuses on the setting of M = N, and the result of Ref. [5] requires R is an integer. Additionally, the result of Ref. [6] is only suitable for the constant channel. The DoF upper bound of Ref. [7] was derived by using the notion of cooperation among subsets of users and side information. Once the cooperation and side information are not allowed, it becomes unknown whether the result can be valid is. Table 1 Information theoretic maximal DoF for the K user M × N MIMO IC Ref. [4]

DoF KM / 2

Condition M =N

[5]

 K min ( M , N ) ; K ≤R   R ; K>R  K min ( M , N ) R +1 

 max ( M , N )  R=   min ( M , N ) 

[6]

KMN M +N

K≥

M +N gcd ( M , N )

K≥M + N or

[7]

KMN M +N

[8]

 min ( M , N ) K −1 ≤  Kf ( K , M , N ) ; max ( M , N ) K ( K − 2 )    KMN ; min ( M , N ) ≥ K − 1 M + N max ( M , N ) K ( K − 2 ) 

N +1 < K < M + N , M N is an integer. M f ( K , M , N ) is

defined in Refs. [8–9].

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Hence, taking into account of all the work, it can be easily obtained that the optimal DoF upper bound of MIMO IC is the one of Ref. [8], where f ( K , M , N ) is defined in both Ref. [8] and Ref. [9]. For the optimal DoF upper bound of [8], when min ( M , N ) max ( M , N ) ≤ ( K − 1)  K ( K − 2 )  and K =

3 , Wang et al. in REF. [9] proposed achievable schemes using linear beamforming at the transmitters and zero forcing at the receivers. The scheme is suitable for both constant and time-varying channels. In contrast, when min ( M , N ) max ( M , N )≥ ( K − 1)  K ( K − 2 )  and K > 3, the achievable scheme in Ref. [6] can reach the optimal DoF by decomposing antennas at both transmitter and receiver sides and using the asymptotic IA scheme [4]. Nevertheless, the scheme can only be applied to the fixed channel coefficients. If the channel coefficients are time varying, there is only an achievable scheme for the case where M = N [4]. However, for the case of general M and N, there is no scheme to achieve the optimal DoF upper bound. In this article, we would like to solve above issue. Firstly, based on the ergodic IA Refs. [10–12], an linear IA scheme was proposed for the general K user M × N MIMO IC when the channel is variable, e.g., time vary or frequency selective. The scheme can reach the optimal DoF upper bound when min ( M , N ) max ( M , N )≥

 K ( K − 2 )  and K > 3 . In addition, the simplicity of coding scheme allows us to derive the closed form of achievable rate expression at any SNR. By numerical simulation, the performance of the ergodic IA scheme is analyzed. This article was organized as follows. In Sect. 2, the system model was introduced. In Sect. 3, the achievable scheme based on ergodic IA was designed. In this section, firstly, the K user multiple-input single-output (MISO) IC, which is the IC equipped with multiple transmitting antennas and single receiving antenna, was considered. Then, the scenario was extended to K user MIMO IC. In Sect. 4, the achievable rate was derived, and the numerical simulation was presented. The conclusion was given in Sect. 5.

receiver is equipped with M antennas and N antennas, respectively. We assume the channel coefficients are time varying or frequency selective, and follow a continuous distribution. As we know, the coherence time is related to the mobility and the coherence bandwidth is related to multipath delay spread. In other words, there are two settings that can satisfy our assumption. One is that the user is movable, the other is that the system is in a multipath environment. Of course, if the system not only owns the movable users but also is in a multipath environment, then such scenario satisfies the assumption naturally. Furthermore, suppose perfect knowledge of all the channel coefficients is available to all transmitters and receivers. Let S denotes the set of the all quantizing channels coefficients and the elements in S are always enough for requirement.

( K − 1)

2 System model Consider the K user MIMO IC composed of K transmitter-receiver pairs. Each transmitter and each

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Fig. 1

System model

The received signal of the ith user, i = 1, 2,..., K , is K

∑

yi = H i ,iVi xi +

H i , jV j x j + ni

(1)

j =1, j ≠ i K

where

H i ,iVi xi

is desired signal,

∑

H i , jV j x j

is

j =1, j ≠ i

interference,

ni

(AWGN). H i , j ∈ C

is additive white Gaussian noise N ×M

denotes the channel from the jth

transmitter to the ith receiver, j = 1, 2,..., K . When i = j ,

H i , j are the direct channels, otherwise H i , j are cross channels,

Vi ∈ C M × M

is

the

beamforming

matrix,

T

1 2 M xi =  xi[ ] xi[ ] ⋯ xi[ ]  is a M × 1 vector of symbols from the kth transmitter, ni obeys CN ( 0， I N ) and is i.i.d

over all the time. The transmitting power of each transmitter is P. The DoF shows the number of signals that can be transmitted in the system at the same time. The DoF

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definition used in this work employed the standard sense of Ref. [1], as shown below. C (ρ) D = lim (2) ρ →+∞ lb ρ where C ( ρ ) is the sum capacity of the system, ρ is

3.2

SNR. And the DoF upper bound is taken as the maximum DoF value. Since C ( ρ ) is considered as the maximum

supersymbol structure for the M × N MIMO IC is identical with the one of the ( M N ) × 1 MISO IC, and is trivial.

sum capacity in this paper, the definition for the DoF also holds for the DoF upper bound.

We mainly focus on the case where M | N ≠ 0 . In this

3 Ergodic IA for K user MIMO IC In this section, firstly, based on ergodic IA, the achievable scheme for the scenario of K user MISO IC was presented. Then, the generalization from the MISO IC to MIMO IC was analyzed, and the ergodic IA scheme for the general case of K user MIMO IC was designed. 3.1

K user MISO IC

K user MIMO IC

In this subsection, the general case of K user M×N MIMO IC was considered. The key is still the supersymbol. We define M | N = mod ( M , N ) . If M | N = 0 , the

work, the supersymbol for the M×N MIMO IC is based on the one of the M ′ ×1 MISO IC, where M ′ =  M N  + 1 . Hence, we first considere the generalization from the M ′ × 1 MISO IC to the M × N MIMO IC. To realize the generalization from the MISO IC to the MIMO IC, the two operations, expansion and repeat, are needed. The symbols after expansion are different from each other, while the ones after repeating are identical with each other. An analogy between the supersymbol of M × N MIMO IC and M ′ × 1 MISO IC is shown in Fig. 3(a), where M * = M | N .

For K user M × 1 MISO IC, in order to achieve KM ( M + 1) DoF, M + 1 time slots were employed. Furthermore, the corresponding M + 1 channels constitute a supersymbol. Within the supersymbol, the direct channels change at every time slot, while the cross channels remain fixed at all time slots. The super symbol of user 1 is shown in Fig. 2, where each block denotes one time slot. The beamforming matrix of each symbol is an M × M identity matrix. Hence, the size of the entire beamforming matrix is ( M + 1) M × M .

Fig. 2

(a) Supersymbol

The super symbol for user 1 of M × 1 MISO IC

User 1 performs the operations of Eq. (3). Since the rank of the coefficient matrix is M, user 1 can resolve M desired signals, each corresponding to one DoF. Furthermore, there are M+1 time slots in the supersymbol. Hence, M ( M + 1) (normalized) DoF can be obtained by user 1. Systematically, KM ( M + 1) DoF can be achieved for the K user M × 1 MISO IC.   y1 (1) − y1 ( 2 )   H1,1 (1) − H1,1 ( 2 )       y1 (1) − y1 ( 3)   H1,1 (1) − H1,1 ( 3)  x1  = ⋮ ⋮      y1 (1) − y1 ( M + 1)   H1,1 (1) − H1,1 ( M + 1) 

rank = M

(b) Beamforming matrix Fig. 3 The supersymbol and beamforming matrix for M × N MIMO IC

(3) The first and second symbols from the supersymbol of the M ′ ×1 MISO IC are chosen to expand to M | N and N symbols, respectively. In contrast, the remaining symbols in the supersymbol of the M ′ ×1 MISO IC are

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chosen to repeat N times. The symbols of MIMO IC originated from each symbol of MISO IC are defined as one sub-block. Please note that the cross channels in the supersymbol of MIMO IC still remain constant and are omitted in Fig. 3(a). The corresponding beamforming matrix for the supersymbol of M × N MIMO IC is shown in the Fig. 3(b). The sizes of both I and 0 are M×M. Note that the beamforming matrix of the MISO IC corresponds to only one block column. However, the matrix of the MIMO IC corresponds to N block columns. Now, we resolve the desired signals for MIMO IC with the supersymbol and beamforming matrix of Fig. 3. For user k, each received signal from the first sub-block subtracts all the received signals from the second *

sub-blocks, as shown in Eq. (4), where A ∈ CM N × MN is the coefficient matrix. xk ( m ) , m = 1, 2,..., N , denotes the signals of mth block column.  yk (1) − yk ( M * + 1) − ⋯ − yk ( M * + N )   xk (1)     x (2)  * *  yk (2) − yk ( M + 1) − ⋯ − yk ( M + N )  k  (4)   =A    ⋮   ⋮     * * *  xk ( N )   yk ( M ) − yk ( M + 1) − ⋯ − yk ( M + N )  where  a11 a12 ⋯ a1N  a a22 ⋯ a2 N  A =  21  ⋮ ⋮ ⋮     aM 1 aM 1 ⋯ aMN 

a11 = H k , k (1) − H k , k ( M * + 1)

a12 = H k , k (1) − H k , k ( M + 2 ) a1N = H k , k (1) − H k , k ( M + N ) a21 = H k , k ( 2 ) − H k , k ( M * + 1) a22 = H k , k ( 2 ) − H k , k ( M * + 2 ) a2 N = H k , k ( 2 ) − H k , k ( M * + N ) aM 1 = H k , k ( M * ) − H k , k ( M * + 1)

Similarly, the receiving signals of the second and the sub-blocks, t = 2,3,..., M ′ − 1 , perform the

( t + 2 ) th

same operations in turn, as shown in Eq. (6). The coefficient matrix Bt is a N 2 × MN matrix as well.

 yk ( M * + 1) − yk ( M * + (t − 1) N + 1)   xk (1)      * *  yk ( M + 2 ) − yk ( M + (t − 1) N + 2 )   xk ( 2 )    = Bt   (6) ⋮   ⋮     * *  xk ( N )   yk ( M + N ) − yk ( M + tN )  where Bt = diag  H k , k ( M * + 1) − H k , k ( M * + N + t ) , H k , k ( M * + 2 ) − H k , k ( M * + N + t ) ,..., H k , k ( M * + N ) − H k ,k ( M * + N + t )  After all the operations are performed, the received signal can be presented as Eq. (7), where C is the entire coefficient matrix. Obviously, rank C = MN is got. Hence, the receiver 1 can resolve MN desired signals. Since the length of the supersymbol in Fig. 3(a) is M + N , MN ( M + N ) DoF can be obtained for user 1.

KMN ( M + N )

DoF

can

also

be

(7)

C = [ A, B1 ,..., BM ′−1 ]

Τ

aM 2 = H k , k ( M * ) − H k , k ( M * + 2 )

4 Achievable rate

aMN = H k , k ( M * ) − H k , k ( M * + N ) Then, the receiving signals of the second and third coefficient matrix.

H k , k ( M * + N + 1) ,..., H k , k ( M * + N ) − H k , k ( M * + N + 1) 

obtained for the K user M × N MIMO IC.  xk (1)     x (2)  yk′ = C  k  ⋮    xk ( N )  where

*

sub-blocks perform Eq. (5), where B1 ∈ C N

 yk ( M * + 1) − yk ( M * + N + 1)   xk (1)       yk ( M * + 2 ) − yk ( M * + N + 2 )   xk ( 2 )  (5)   =B1   ⋮   ⋮     * *  xk ( N )   yk ( M + N ) − yk ( M + 2 N )  where B1 = diag  H k ,k ( M * + 1) − H k , k ( M * + N + 1) , H k , k ( M * + 2 ) −

Systematically,

*

2015

2

× MN

is the

Although KMN ( M + N ) DoF can be reached for the K user M × N MIMO IC, the noise will increase because of interference subtractions. Hence, in this section, we derive the achievable rate, which can evaluate the ergodic IA

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scheme proposed in Sect. 3. From Eq. (4), it can be seen that each received signal of the first sub-block should subtract all the symbols of the second sub-block. There are N symbols in the second sub-block. As a result, the noise is N + 1 times stronger in these operations. Normalizing the noise results in a scaling of a factor 1 N + 1 of the channels. In other words, the coefficient matrix A should be multiplied by a factor of 1 N + 1 . Similarly, for the coefficient matrix from other operations, e.g., Β1 from Eq. (5), should be multiplied by

1

2 . Therefore, after eliminating the interference and

(b) Rate comparison against different receiving antennas

normalizing the noise, the received signal in one supersymbol of user k is  xk (1)    x ( 2)  ɶyk = H k  k (8)  + zɶ ⋮    xk ( N )  where Τ

1 1  1  Hk =  A, B1 ,..., BM ′−1  2 2  N +1  zɶ obeys CN ( 0， I ) . We suppose the power allocated to each data is equal. Since there are M + N time slots in the one supersymbol, the achievable rate per time slot for user k is       P(M + N ) 1 Rk = Ε l b det  I + H k H k †    M +N   M   MN  M | N +   + 1    N        (9) The achievable sum rate for the K user M × N MIMO IC is simulated in Fig. 4.

(a) Rate comparison against different transmitting antennas

(c) Rate comparison against different number of users Fig. 4 The achievable sum rate for the K user MIMO IC

Except for the SNR, there are three system parameters, which are the transmitting antenna M, the receiving antenna N and the number of users K. The simulation results show that when two of the three variables are fixed, the sum rate increases with the remaining one. In addition, as mentioned in Sect. 2, we assume the channel coefficients are variable, i.e., time varying or frequency selective. Hence, the effects on the achievable rate that brought by the two different channel assumptions are discussed. Before analyzing the effects, let us recall the channel quantization procedure, which is identical with that in Ref. [12]. Please see Fig. 5 for an illustration, supposing the number of rings is γ , the number of segments per ring is η and all the quantized cells have the same probability in the whole complex plane. When the channels are time varying, the setting is the one that the users are movable. Then, the channel quantization is performed in the dimension of time. In contrast, when the channels are frequency selective, the setting is in a multipath environment, so that the channel quantization is performed in the dimension of frequency. No matter whether the quantization is performed in time dimension or in

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frequency dimension, the quantitative results in Fig. 5 will not be affected. In other words, no matter the channel coefficients are time varying or frequency selective, the ergodic IA schemes for the two settings are identical. Naturally, the achievable rates for the two settings are the same, too.

2015

Acknowledgment The authors would like to thank the National Science and Technology Major Project (2012ZX03003001-004), the National Natural Science Foundation of China (61401042, 61327806), Beijing Key Laboratory of Work Safety Intelligent Monitoring (Beijing University of Posts and Telecommunications).

References

Fig. 5

Quantization of channel coefficients

5 Conclusions The article mainly focus on the K user M×N IC where min ( M , N ) max ( M , N )≥ ( K − 1)  K ( K − 2 )  and

K > 3 . The channel coefficients are variable. For this setting, we provide an achievable scheme to achieve a total of KMN ( M + N ) DoF, which is the optimal DoF upper bound. The achievable scheme is based on ergodic IA. The ergodic IA scheme can reach the DoF value with simply linear beamforming and finite symbols. Furthermore, the simplicity of the coding scheme allows us derive the closed form of achievable rate at any SNR. By numerical simulation, the performance of the ergodic IA scheme is evaluated. Finally, note that the channel coefficients in this work are time varying or frequency selective. If the channel coefficients are both time varying and frequency selective, the ergodic IA scheme is still suitable, and the quantization for the channels can be selected between the time and frequency dimensions. Whether the increase of the selective dimensions can improve the performance of ergodic IA scheme, is an interesting research in our future work.

1. Jafar S A, Fakhereddin M J. Degrees of freedom for the MIMO interference channel. IEEE Transactions on Information Theory, 2007, 53(7): 2637−2642 2. Wang C W, Gou T G, Jafar S A. Subspace alignment chains and the degrees of freedom of the three-user MIMO interference channel. IEEE Transactions on Information Theory, 2014, 60(5): 2432−2479 3. Cadambe V R, Jafar S A. Interference alignment and the degrees of freedom of the K user interference channel. IEEE Transactions on Information Theory, 2008, 54(8): 3425−3441 4. Jafar S A. On asymptotic interference alignment: Plenary talk. Proceedings of the 2010 International Conference on Signal Processing and Communications (SPCOM’10), Jul 18−21, 2010, Bangalore, India. Piscataway, NJ, USA: IEEE, 2010: 5p 5. Gou T G, Jafar S A. Degrees of freedom of the K user M×N MIMO interference channel. IEEE Transactions on Information Theory, 2010, 56(12): 6040−6057 6. Ghasemi A, Motahari A S, Khandani A K. Interference alignment for the K user MIMO interference channel. Proceedings of the 2010 IEEE International Symposium on Information Theory (ISIT’10), Jun 13−18, 2010, Austin, TX, USA. Piscataway, NJ, USA: IEEE, 2010: 360−364 7. Mohapatra P, Murthy C R. Outer bounds on the sum rate of the K-user MIMO Gaussian interference channel. IEEE Transactions on Communications, 2013, 61(1): 176−186 8. Wang C W, Sun H, Jafar S A. Genie chains and the degrees of freedom of the K-user MIMO interference channel. Proceedings of the 2012 IEEE International Symposium on Information Theory (ISIT’12), Jul 1−6, 2012, Cambridge, MA, USA. Piscataway, NJ, USA: IEEE, 2012: 2476−2480 9. Wang C W, Gou T G, Jafar S A. Subspace alignment chains and the degrees of freedom of the three-user MIMO interference channel. Proceedings of the 2012 IEEE International Symposium on Information Theory (ISIT'12), Jul 1−6, 2012, Cambridge, MA, USA. Piscataway, NJ, USA: IEEE, 2012: 2471−2475 10. Nazer B, Gastpar M, Jafar S A, et al. Ergodic interference alignment. Proceedings of the 2009 IEEE International Symposium on Information Theory (ISIT’09), Jun 28−Jul 3, Seoul, Republic of Korea. Piscataway, NJ, USA: IEEE, 2009: 1769−1773 11. Nazer B, Gastpa M, Jafar S A, et al. Ergodic interference alignment. IEEE Transactions on Information Theory, 2012, 58(10): 6355−6371 12. Geng C, Jafar S A. On optimal Ergodic interference alignment. Proceedings of the IEEE Global Communications Conference (GLOBECOM’12), Dec 3−7, 2012, Anaheim, CA, USA. Piscataway, NJ, USA: IEEE, 2012: 2311−2316

(Editor: Zhang Kexin)