Degrees of Freedom on the K-User MIMO Interference ... - IEEE Xplore

Degrees of Freedom on the K -User MIMO Interference Channel with Constant Channel Coefficients for Downlink Communications Namyoon Lee, Dohyung Park, Young-Doo Kim Communication Lab Samsung Advanced Institute of Technology Samsung Electronics Co., LTD. Yongin-si, Korea Email: [email protected], [email protected], and [email protected] Abstract— In this paper, we study degrees of freedom for the K-User multiple-input multiple-output (MIMO)-interference channel (IFC) with constant channel coefficients. In this channel, we investigate how many total number of transmit antennas, M1 + M2 + . . . + MK , are required in minimum to achieve di = 1, ∀i degrees of freedom when all receivers have N = 2 antennas, which is a downlink communication scenario. To answer this question, we propose a new interference alignment scheme based on intersection subspace property of the vector space. The proposed interference alignment scheme can be easily generalized regardless of the number of users. In addition, we investigate degrees of freedom for the partially connected MIMOIFC where some arbitrary interfering links are disconnected due to the large path loss or deep fades. In this channel model, we examine how these disconnected links are considered on designing the beamforming vectors for interference alignment.

I. I NTRODUCTION Many researchers have been devoted to seeking for the capacity of the Gaussian interference channel (IFC) over the past three decades. Even though the exact capacity of the IFC is not known in general, several inner bounds and outer bounds of the Gaussian IFC capacity have been found [1][7]. Most of the previous studies, however, are focused on the two-users interference channel because the generalization to more than three-users is too complicated. Recently, Cadambe and Jafar have studied the capacity characterization of the K-User interference channel using the degrees of freedom approach, which shows us the capacity scaling behavior at the high signal-to-noise ratio (SNR) regime. The authors in [8] have shown that the capacity of the K-User time/frequencyvarying interference channel was characterized as C(SN R) = K 2 log(SN R) + o (log(SN R)) using the idea of interference alignment, originally presented in [9] and [10]. This is an even more surprising capacity result than prior forecasts using cakecutting approach like time division multiple access (TDMA). Although the capacity of the interference channel can be increased with number of users (K), two crucial assumptions, i.e., infinitely many signal dimensions and time-varying channels, are required for obtaining this capacity result. In practical communication scenarios, however, infinitely many signal dimensions are unavailable due to limited resources (frequency slots, time slots, and number of antennas). From a practical point of view, accordingly, we are interested in the capacity characterization of a multiple-input-

multiple-output (MIMO)-IFC with constant channel coefficients which exploits finite spatial signal dimensions by multiple antennas not frequency/time slots. There have been a number of studies related to the MIMO-IFC with constant channel coefficients. In [11], the authors show that when K = 2, the degrees of freedom is characterized as min{M1 + M2 , N1 + N2 , max{M1 , N2 }, max{M2 , N1 }} where Mi and Ni are the number of antennas at the ith transmitter and receiver, respectively. In [8] it is shown that 3M 2 degrees of freedom are achievable for K = 3 under the assumption that all transmitters and receivers have equal even number of antennas i.e., M = Mi = Ni (When M is odd, twotime symbol extension is required to achieve this degrees of freedom). To generalize the more than three user case (K ≥ 4) RM degrees of the authors in [12] show that RM + R2 +2R−1 freedom can be achieved for the R+2-User MIMO-IFC where each transmitter and receiver has M and RM (R = 2, 3, . . . ,) antennas, respectively. Theses results are some special cases of the K-User MIMO-IFC. In previous studies, the degrees of freedom were investigated in the perspective of the maximum number of independent streams for given transmitter and receiver antenna configurations. In this paper, however, we change the view of the degrees of freedom such as how many total number of transmit antennas, M1 + M2 + . . . + MK , are needed in minimum to achieve d1 + d2 + . . . + dK degrees of freedom when all receiver have N = 2 antennas. Using this point of view, we provide an inner bound of the degree of freedom for the K-User MIMO-IFC. To derive the inner bound of the K-User MIMO-IFC with constant channel coefficients, we propose a new interference alignment scheme based on intersection subspace property of the vector space. The proposed interference alignment scheme can be easily generalized regardless of the number of users. In addition, we consider the partially connected interference channel where some interfering links are disconnected. In practical cellular scenarios, if the number of K increase, the fully connected interference channel is invalid. Because this channel model cannot contain the nature of wireless link such that the signals coming far from sources with respect to the receiver provides negligible interference due to a large path loss. From this investigation, we show that how the partially connected link conditions affect the design of interference alignment scheme.

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

Under above considerations, the contributions of this paper can be summarized as follows: • An inner bound of the capacity for K-User MIMO-IFC with constant channel coefficients can Kbe characterized as K log(SN R) + o(log(SN R)) if i=1 Mi ≥ K(K − 2)N , where Ni = N = 2 is the number of receive antennas. • An inner bound of the capacity for for the K-User partially connected MIMO-IFC with constant channel coefficients can be characterized as K log(SN R) + K o(log(SN R)) if i=1 Mi ≥ {K(K − 2) − L}N , where Ni = N = 2 is the number of receive antennas and L is the number of disconnected channel links out of total K(K − 1) interference links on the network. The remainder of this paper is organized as follows. Section II describes the signal model used in the current study. The inner bound of K-User MIMO-IFC is derived using interference alignment scheme based on intersection subspace approach in Section III. In Section IV, we also provide an inner bound of the degrees of freedom for the K-User partially connected MIMO-IFC. In Section V, computer simulation results are presented to demonstrate the performance of proposed technique. Section VI contains the conclusions. II. S YSTEM MODEL In this section, we describe the system model for the KUser MIMO-IFC with constant channel coefficients as shown in Fig. 1 (K=4). The system of the the K-User MIMO-IFC consists of K transmitters with Mi , i = 1, 2, . . . , K, antennas and K receivers with N = 2 antennas. Each transmitter wants to convey independent messages to its corresponding receiver. The received signal at the ith receiver is represented as Y[i] =

K

H[i,j] X[j] + N[i] ,

(1)

j=1

where N[i] is the N × 1 additive white Gaussian noise (AWGN) vector at the ith receiver, X[i] is the Mi × 1 transmit vector at the ith transmitter, and H[i,j] is the N × Mi channel matrix from the jth transmitter to the ith receiver. The channel matrices H[i,j] for ∀i, j ∈ {1, 2, . . . , K} are generated so that each entry of the matrix is independently and identically distributed (i.i.d.) according to CN (0, 1). This implies that all channel matrices have full rank almost surely, i.e., rank H[i,j] = min{Mi , N }. Throughout this paper, we assume that perfect channel state information (CSI) is available at all transmitters and receivers. A. Degrees of freedom By using multiple antennas at both ends of the wireless links, channel capacity can be increased due to multiple signal space dimensions. The degrees of freedom which is the prelog factor of the capacity (d) is one of the main metrics for evaluating the performance of the signaling in multiple antenna systems, which is defined as d

C(SN R) , SN R→∞ log(SN R) lim

(2)

Fig. 1.

MIMO-IFC when K=4, Mi , and N =2.

where C(SN R) denotes the sum capacity at signal-to-noise ratio (SN R). III. A N INNER BOUND ON THE DEGREES OF FREEDOM Theorem 1: For the K-User MIMO Gaussian interference channel with non-zero constant channel coefficients where kth transmitter has KMk > 1 antennas and all receivers have N = 2 antennas, if k=1 Mk ≥ K(K −2)N , then KN 2 = K degrees of freedom can be achieved. A. Proof for Theorem 1 with K = 4 In this subsection, we show the case of K = 4 because the extension of the proof of the Theorem 1 to arbitrary K is straightforward based on the case of K = 4. Proof : The achievability of the Theorem 1 is provided using intersection subspace based interference alignment scheme as described below. We show that each user can achieve 4di = 1, 1 ≤ i ≤ 4, degree of freedom and thus, total i=1 di = 4 degrees of freedom can be achieved in this network. Transmitter i conveys message Wi to Receiver i using di = 1 independently encoded stream along vectors v[i] , i.e, X[i] = v[i] xi ,

i = 1, 2, . . . , 4

where v[i] is the Mi × 1 dimensional beamforming vector. Then, the received signal at the jthe receiver is Y[j] =

4

H[j,i] X[i] + N[j] .

i=1

In order for the each receiver to decode its di = 1 desired signal stream by zero forcing interference cancelation, the signal dimension occupied by all interference signals at each receiver has to be less than or equal to N − di = 2 − 1 = 1. Note that there are three interference vectors at Receiver i. Therefore, we should verify that whether three interference signal vectors can be aligned within the 1 dimensional signal space for interference at each receiver. To prove this, we propose a novel interference alignment scheme in the perspective of intersection signal subspace. Firstly, consider Receiver 1. Because Receiver 1 has N − d1 = 1 dimensional signal space for containing the interference signal vectors, the interference vectors, i.e., H[1,2] v[2] , H[1,3] v[3] , and H[1,4] v[4] should be aligned within


1 dimensional signal space spanned by u[1] . In similar manner, all other receivers should also contain the interference vectors within 1 dimensional signal space spanned by u[2] , u[3] , and u[4] , respectively. These interference align conditions for each receiver are represented as span(u[1] ) = span H[1,2] v[2] , H[1,3] v[3] , H[1,4] v[4] span(u[2] ) = span H[2,1] v[1] , H[2,3] v[3] , H[2,4] v[4] span(u[3] ) = span H[3,1] v[1] , H[3,2] v[2] , H[3,4] v[4] span(u[4] ) = span H[4,1] v[1] , H[4,2] v[2] , H[4,3] v[3] .(3) Without loss of generality, the conditions in (3) are rewritten as 1 [1,2] [2] 1 [1,3] [3] 1 [1,4] [4] H v = H v = H v u[1] = α1 α2 α3 1 [2,1] [1] 1 [2,3] [3] 1 [2,4] [4] u[2] = H v = H v = H v α4 α5 α6 1 [3,1] [1] 1 [3,2] [2] 1 [3,4] [4] u[3] = H v = H v = H v α7 α7 α9 1 1 1 u[4] = H[4,1] v[1]= H[4,2] v[2]= H[4,3] v[3] , (4) α10 α11 α12 where αi , 1 ≤ i ≤ 12, are constant values which represents relative magnitude of vectors. From the lemma 1 in the appendix, we find the intersection subspace where all interfering signal vectors are aligned at each receiver. At Receiver 1, the intersection subspace spanned by u[1] and beamforming vectors v[2] , v[3] , and v[4] satisfy the following equation ⎡ ⎤ ⎡ ⎤ u[1] [1,2] 0 0 α1 IN −H ⎢ v[2] ⎥ ⎥ ⎣ α2 IN (5) 0 −H[1,3] 0 ⎦⎢ ⎣ v[3] ⎦ = 0, [1,4] 0 0 −H α3 IN [4] v (K−1)N ×(N +M2 +M3 +M4 )

and in similar manner, intersection subspaces for alignment can be found at Receiver 2, Receiver 3, and Receiver 4, respectively as follows : Receiver 2: ⎡ ⎤ ⎤ u[2] ⎡ [2,1] 0 0 α4 IN −H ⎢ v[1] ⎥ ⎥ ⎣ α5 IN (6) 0 −H[2,3] 0 ⎦⎢ ⎣ v[3] ⎦ = 0. [2,4] 0 0 −H α6 IN [4] v (K−1)N ×(N +M1 +M3 +M4 )

Receiver 3 : ⎡ ⎤ ⎤ u[3] ⎡ [3,1] 0 0 α7 IN −H ⎢ v[1] ⎥ ⎥ ⎣ α8 IN 0 −H[3,2] 0 ⎦⎢ ⎣ v[2] ⎦ = 0. [3,4] 0 0 −H α9 IN [4] v

(7)

(K−1)N ×(N +M1 +M2 +M4 )

Receiver 4 : ⎡ ⎤ ⎤ u[4] ⎡ α10 IN −H[4,1] 0 0 ⎢ v[1] ⎥ ⎥ ⎣ α11 IN 0 −H[4,2] 0 ⎦⎢ ⎣ v[2] ⎦ = 0. [4,3] 0 0 −H α12 IN [3] v (K−1)N ×(N +M1 +M2 +M3 )

Now we aggregate interference alignment conditions for every receiver in (5), (6), (7), and (8) into a unified system equation to jointly construct beamforming vectors, which is written as ⎤ ⎡ [1,2] α1 IN

⎢ α2 IN ⎢ α3 IN ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0 0

0 0 0 0 −H 0 0 0 0 0 0 0 −H[1,3] 0 0 0 0 0 0 0 −H[1,4] α4 IN 0 0 −H[2,1] 0 0 0 α5 IN 0 0 0 0 −H[2,3] 0 α6 IN 0 0 0 0 0 −H[2,4] 0 α7 IN 0 −H[3,1] 0 0 0 0 α8 IN 0 0 −H[3,2] 0 0 0 α9 IN 0 0 0 0 −H[3,4] 0 0 α10 IN −H[4,1] 0 0 0 0 0 α11 IN 0 −H[4,2] 0 0 0 0 α12 IN 0 0 −H[4,3] 0

4

12N ×(4N +

= F(αi )y = 0,

Mi )

(9)

4

where F(αi ) is the 24 × (8 + i=1 Mi ) dimensional unified system matrix with unknown variables αi , 1 ≤ i ≤ 12. By solving unified system equation in (9) above, we can obtain the beamforming vectors for interference alignment. To solve this equation, if we set α1 = · · · = α12 = 1, i.e., fixed value is assumed, the system equation becomes linear. In this case, this homogenous system equation can be easily solved with non-trivial solutions by finding right null space of the F(αi = 1) if the sum of the cooperative transmit antennas 4 k=1 Mk ≥ K(K − 2)N + 1 = 17, because this condition ensures that the right null space of the unified system matrix F(αi = 1) exists with probability one. The main objective, however, is to seek the closed form solution of the equation (9) with the minimum sum of the transmitters’ antennas. Thus by designing αi with u[k] and v[k] jointly, the solution for interference alignment can be acquired with the smaller number of the sum of transmitters’ antennas in a closed form. By using the fact that nonlinear equations for interference alignment in (9) have the special structure that the only αi s cause nonlinearity, a novel method to solve the system equation in (9) is provided. The key concept of the proposing method is to utilize the auxiliary variables, q. This auxiliary variables make possible to solve the nonlinear equation in (9) by solving linearized system equation under the some condition. The linearized system equation with auxiliary variables is represented as ⎤ ⎡ [1,2] IN 0 0 0 IN IN IN 0 0 0 0 0 0 0 0

⎢ 0 ⎢ IN ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0

0 0 0 0 0 0 IN IN IN 0 0 0

= Qx = 0, (8)

i=1

⎥ ⎥⎡ u[1] ⎤ ⎥ [2] ⎥ u ⎥⎢ u[3] ⎥ ⎥⎢ [4] ⎥ ⎥⎢ u[1] ⎥ ⎥⎢ v ⎥ ⎥⎣ v[2] ⎦ ⎥ [3] ⎥ v ⎥ v[4] ⎦

0 0 −H 0 0 0 0 0 −H[1,3] 0 0 0 0 0 −H[1,4] 0 −H[2,1] 0 0 0 0 0 0 −H[2,3] 0 0 0 0 0 −H[2,4] 0 −H[3,1] 0 0 0 0 0 −H[3,2] 0 0 0 0 0 0 −H[3,4] IN −H[4,1] 0 0 0 IN 0 −H[4,2] 0 0 IN 0 0 −H[4,3] 0

24×(24+2)

0 I ⎥⎡ 0 ⎥ u[1] ⎥ [2] 0⎥⎢u [3] 0⎥⎢u ⎥ ⎢ [4] 0 ⎥ ⎢ u[1] v 0 ⎥ ⎢ [2] ⎥⎢v 0 ⎥ ⎣ [3] v 0 ⎥ v[4] ⎥ q 0⎦ 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(10)

where we assume that α1 = α3 = . . . = α12 = 1, (i.e., design the only α2 ), q = [q1 q2 ] is a 2 × 1 dimensional vector containing two auxiliary variables. If we assume that


4

i=1 Mi = 16, the Q becomes a 24×26 dimensional matrix. Then we decompose Q as Σ24×24 024×2 H H (11) A 1 A0 , Q=U 02×24 02×2

where U and Σ denotes the left singular vector matrix and a diagonal matrix which is composed of singular values, respectively. A1 and A0 are the right singular matrices consisted of the singular vectors corresponding to non-zero singular values and zero singular values, respectively. Note that left null space of Q consists of two column vectors, x[1] and x[2] , i.e., A0 = x[1] x[2] . Let the solution as the form of linear combination of these two vectors as x[sol] = λx[1] + x[2] , (12) T [i] [i] [i] , where x[i] = for i = 1, 2. x1 x2 · · · x26 Since this solution should solve (10) as well as the unified system equation in (9) simultaneously, we should determine the coefficients λ in order to satisfy the following condition,

for performing interference alignment which provides scalability when beamforming vectors are constructed. 3: The total number of transmitter’s antennas, Remark 4 M = K(K − 2)N , is not minimum for interference k k=1 alignment. This implies that we can achieve the same degrees of freedom with the reduced the sum of total transmit antennas by properly designing unknown variables, 4 αi in other manners. Actually, it has been observed that k=1 Mk = 12 is sufficient for interference alignment for K = 4 and N = 2 case by solving (9) using numerical approach such as the Newton’s method. Inaddition, the authors in [13] and [14] 4 also investigate that k=1 Mk = 12 is sufficient to achieve 4 degrees of freedom when K = 4, N = 2, dk = 1, and Mk = 3. The solving method of (9) with reduced unknown variables, (i.e., the reduced total sum of transmit antennas), will be explored through future works. IV. D EGREES OF FREEDOM ON THE PARTIALLY CONNECTED MIMO-IFC

In previous section, our studies are focused on the investigation of the degrees of freedom for the fully connected + + = ⇔ − = 0. (13) K-User MIMO-IFC with constant channel coefficients. In α2 = + + practical scenarios, however, the fully connected interference From the equation (13), the solution for the λ∗ is obtained as channel is not valid. Because this channel model cannot √ contain the nature of wireless link such that the signals −b ± b2 − 4ac ∗ , (14) coming far from sources with respect to the receiver provides λ = 2a negligible interference due to a large path loss. Thus, in [1] [1] [1] [1] [1] [2] [1] [2] [1] [2] where a = x2 x25 −x1 x26 , b = x25 x2 +x2 x25 −x26 x1 − this section, we investigate degrees of freedom on the K[1] [2] [2] [2] [2] [2] x1 x26 , and c = x2 x25 − x1 x26 . Using the λ∗ , we get the User partially connected MIMO-IFC with constant channel coefficients where arbitrary interfering links are disconnected. beamforming vectors for interference alignment as The degrees of freedom of the partially connected interference [sol] v[1] = x8+1:8+M1 channel have originally been studied in [15]. It is shown that [sol] is achievable for K ≥ 4 when the multiplexing gain KM [2] 2 v = x8+M +1:8+ 2 M 1 i i=1 number of interfering links are no more than 2 for every [sol] receiver. The result, however, has limitations on that each v[3] = x8+ 2 M +1:8+ 3 M i i i=1 i=1 receiver allow only two interference signals and the interfering [sol] v[4] = x8+ 3 M +1:8+ 4 M . (15) link should have some cyclic pattern. These limitations are i i i=1 i=1 Therefore, the beamforming vectors for interference alignment not applicable for general partially connected interference 4 can be constructed in closed form if k=1 Mk = K(K − channel. In similar manner with the previous section, our objective is 2)N = 16. to find out how many total transmit antennas are necessary to Now, we check the decodability of X[k] at each receiver. achieve the K degrees of freedom when arbitrary number of To show this, we need to prove that u[k] and H[k,k] v[k] are interfering links are disconnected in an arbitrary pattern. The linearly independent so that each receiver decode its message main result for this question is explained by the following using zero forcing. Since direct channel matrix H[k,k] does not theorem. depend on the interference alignment conditions, H[k,k] v[k] Theorem 2 : For the K-User partially connected MIMOwhich is linearly transformed by H[k,k] is independent of IFC with constant channel coefficients where kth transmitter KN [k] u with probability one. Consequently, 2 = 4 degrees has Mk > 1 antennas and all receivers have N = 2 of freedom is achieved. K antennas, if k=1 Mk ≥ {(K − 2)K − L}N , then KN 2 =K Remark 1: When K = 3 and N = Mk = 2 (1 ≤ k ≤ 3), degrees of freedom can be achieved where L is the number we can obtain the same degrees of freedom as the result in [8]. of disconnected interference links out of total K(K − 1) While the interference alignment scheme in [8] could be used interference links. in the only K = 3 case for MIMO-IFC with constant channel coefficients, the proposed interference alignment scheme can A. Proof for Theorem 2 with K = 4 and L = 2 be applied regardless of K. We show that if the number of disconnected interfering links L = 2, d1 + d2 + d3 + d4 = 4 is achieved by using Remark 2: The proposed interference alignment scheme 4 only depends on the sum of the total transmitter’s antennas. k=1 Mk ≥ {(4 − 2)4 − 2}2 = 12 total transmit antennas. This indicates that each transmitter can have different antennas In this example, we assume that two links from Transmitter [sol] x25 [sol] x1

[sol] x26 [sol] x2

[1] λx25 [1] λx1

[2] x25 [2] x1

[1] λx26 [1] λx2

[2] x26 [2] x2


Partially connected MIMO-IFC when K=4,L=2, and N =2.

Fig. 2.

1 to Receiver 4 and from Transmitter 4 to Receiver 1 are disconnected as depicted in Fig. 2. The achievable scheme is provided using intersection subspace based interference alignment scheme. The transmitter’s beamforming vectors are picked so that interference alignment conditions for the partially connected interference channel are satisfied. The interference alignment conditions for Receiver 1 is that the dimension occupied by two interference signal vectors from Transmitter 2 and 3 should be not more than 1 (Transmitter 4 does not cause the interference to Receiver 1). In similar manner, all interference alignment conditions for every receiver are represented as 1 [1,2] [2] 1 [1,3] [3] H v = H v u[1] = β1 β2 1 [2,1] [1] 1 [2,3] [3] 1 [2,4] [4] u[2] = H v = H v = H v β3 β4 β5 1 [3,1] [1] 1 [3,2] [2] 1 [3,4] [4] u[3] = H v = H v = H v β6 β7 β8 1 [4,2] [2] 1 [4,3] [3] u[4] = H v = H v . (16) β9 β10 As in the same manner of Section III, we can make a unified system equation for interference alignment which is written as ⎡ β1 IN 0 ⎤ 0 0 0 −H[12] 0 0 0 0 0 0 0 −H[13] 0 β2 IN ⎤ ⎡ ⎢ 0 β3 IN 0 ⎥ u[1] 0 −H[21] 0 0 0 ⎢ ⎥ u[2] 0 0 0 −H[23] 0 ⎢ 0 β4 IN 0 ⎥⎢ u[3] ⎥ ⎢ 0 β5 IN 0 ⎢ [4] ⎥ 0 0 0 0 −H[24] ⎥ ⎥ ⎢ ⎥⎢ u[1] 0 β6 IN 0 −H[31] 0 0 0 ⎢ 0 ⎥⎢ v ⎥ ⎢ 0 ⎥ ⎣ [2] ⎦ 0 0 −H[32] 0 0 0 β7 IN ⎢ ⎥ v 0 0 0 0 −H[34] ⎦ v[3] 0 β8 IN ⎣ 0 [4] 0 0

0 0

0 0

= W(βi )y = 0,

β9 IN β10 IN

0 0

−H[42] 0 0 −H[43]

0 0

Fig. 3.

Simulation environment where Mk = 2, N = 2, and L = 4.

Remark 4: When K = 4, L = 4, Mk = N = 2, and cyclic pattern of interference signals exists, we can reach the same result of multiplexing gain in [15]. However, the proposed interference alignment scheme is applicable regardless of the pattern of disconnected links and number of disconnected links L. Remark 5: By concerning the effective interferences on the network, we can reduce the number of the sum of transmitter’s K antennas k=1 Mk = N (K(K−2)−L) to achieve K degrees of freedom. This implies that the interference alignment scheme based on the cognition of effective interference links is fairly attractive. V. N UMERICAL R ESULTS

In this section, numerical results were provided to demonstrate the performance of the proposed interference alignment scheme. Fig. 3 depicts the simulation environment where each base station (BS) with Mk = 2 transmit antennas intends to transmit a single data stream to its serving mobile station (MS) with N = 2 antennas. In this environment, it was assumed that each MS suffers from the interferences coming two adjacent BSs, i.e., partially connected MIMO-IFC for L = 4. Throughout the simulations, transmit power of all the transmitters and noise variance of all receive antennas were 2 2 assumed to be the same, i.e., PBSi = P and σM Si = σn . Moreover, it is assumed that all perfect CSI are collected at the central controller through X2 interface without feedback delay so that the interference alignment beamforming vectors can be constructed. We define an achievable rate at the kth MS supported by kth BS is written by 2 v P w[k]H H[k,k] v[k] Rk = log2 1 + 2 . (18) K (17) σn 2 + j=1,j=k P w[k]H H[k,j] v[j]

where W(βi ) is the system matrix for designing beamforming vectors with the size ofN {K(K − 1) − L} × K K KN + k=1 Mk = 20 × (8 + k=1 Mk ). Note that the number of rows of W(βi ) are reduced as much as LN = 4 compared with those of F(αi ). From the solving technique used in previous K section, equation (17) can be solved if the condition , i=4 Mi ≥ {K(K − 2) − L}N = 12, is satisfied. We omit the proof of decodability of each data stream because it is analogous to the previous section.

where w[k] is the receive combining vector of the kth MS to eliminate the interference, i.e., w[k]H u[k] = 0 ,∀k. The achievable sum rate results are shown in Fig. 4. The proposed interference alignment method exhibits superior performance compared to the conventional TDMA with ZF scheme in which BS1 and BS2 transmit a single data streams to it’s corresponding MSs for the first time slot and BS3 and BS4 transmit a single data stream to it’s corresponding MSs for the second time slot, respectively.


+ K dimensional intersection exists k=1 Mk − (K − 1)N subspace denoted by ∩K R(A ) with probability one where k k=1 R(·) stands for the column space of a matrix of a matrix. proof : Consider a N × 1 vector q which lies in R(A1 ) ∩ R(A2 ) ∩ . . . ∩ R(AK ). Then, it is sure that there exist qk ∈ CMk ×1 such that

40

achievable sum rate (bits/sec/hz)

35

Proposed IA scheme TDMA with ZF scheme

D.O.F=4

30 25

q = A 1 q1 = A 2 q2 = . . . = A k qK .

20 15

The equation (A1) can be rewritten in a matrix form as ⎡ q ⎤ ⎡ IN −A1 0 ··· 0 ⎤ q1 IN 0 −A2 ··· 0 ⎢ q2 ⎥ ⎦ ⎣ . (A2) ⎣ . ⎦ = Mx = 0. .. 0 . . . −AK−1 0 ..

D.O.F=2

10 5 0

IN

0

5

10

15 SNR

20

25

Fig. 4.

Achievable sum rate performance comparison.

(A1)

30

We can see that the sum rate of the proposed interference alignment scheme grew linearly with the slope of 4 while the sum rate of the TDMA with ZF scheme was linearly increased with the slop of 2. The performance improvement of the proposed interference alignment scheme mainly came from the efficient utilization of the signal space and it led the result of the increased total degrees of freedom. Although the proposed interference alignment scheme shows an attractive gain in achievable sum rate, the assumption of the perfect CSI at the central controller seems to be a challenge because feedback overhead for obtaining perfect CSI is quite burdensome in practical communication system. Thus, codebook based signaling scheme should be considered on the limited feedback circumstance. In addition, we notice that the proposed interference alignment scheme does not contain the noise effect; an optimal signaling based on the sum rates maximization criterion considering noise effect remains as future works. VI. C ONCLUSION The degrees of freedom of the K-User MIMO-IFC with constant channel coefficients have been studied. To derive an inner bound of degrees of freedom, we proposed a novel interference alignment based on intersection subspace property of the vector space. Using the proposed scheme, it is shown that the capacity of the K-User MIMO-IFC with constant channel coefficients is characterized as K log(SN R) + o(log(SN R)) K M ≥ K(K − 2)N . In addition, we investigated if i i=1 degrees of freedom for the partially connected MIMO-IFC. In thischannel model, we achieved K degrees of freedom K with i=1 Mi ≥ {K(K − 2) − L}N total number of transmit antennas. It is shown that the proposed interference alignment scheme outperforms a conventional method in terms of degrees of freedom. A PPENDIX Lemma 1 : If Ak are N × Mk random matrices whose entries are drawn i.i.d. complex Gaussian CN (0, 1), there

0

···

0

−AK

qK

From equation (A2), we can find the dimension of R(A1 ) ∩ R(A2 ) ∩ . . . ∩ R(AK ) by equivalently calculating the nullity of M. Note that matrix M K has rank(M)=min{KN, k=1 Mk + N } with high probability because all entries of Ak ∀k are i.i.d. complex Gaussian random variable. As a result, by using the rank-nullity theorem of linear algebra, K we conclude that Null(M) = KN − min{KN, k=1 Mk + N } = + K with probability one where k=1 Mk − (K − 1)N (x)+ = max(x, 0). R EFERENCES [1] T.S. Han and K. Kobayashi, “A new achievable rate region for the interference channel,” IEEE Trans. Inf. Theory, vol. 27, pp. 49-60, Jan. 1981. [2] H. Sato, “The capacity of the Gaussian interference channel under strong interference,” IEEE Trans. Inf. Theory, vol. 27, pp. 786-788, Nov. 1981. [3] A.B. Carleial, “Outer bounds on the capacity of interference channels,” IEEE Trans. Inf. Theory, vol. 29, pp. 602-606, July 1983. [4] M.H.M. Costa, “On the Gaussian interference channel,” IEEE Trans. Inf. Theory, vol. 31, pp. 607-615, Sept. 1985. [5] G. Kramer, “Outer bounds on the capacity of Gaussian interference channels,” IEEE Trans. Inf. Theory, vol. 50, pp. 581-586, Mar. 2004. [6] R. Etkin, D. Tse and H. Wang, “Gaussian interference channel capacity to within one bit,” IEEE Trans. Inf. Theory, vol. 54, pp. 5534-5562, Dec. 2008. [7] X. Shang, G. Kramer, and B. Chen, “New outer bounds on the capacity region of Gaussian interference channels,” Proc. IEEE Int. Symp. on Inform. Theory (ISIT), Toronto, Canada, July 2008. [8] V. R. Cadambe and S. Jafar, “Interference alignment and degrees of freedom of the K-User interference channel,” IEEE Trans. Inf. Theory, vol. 54, pp. 3425-3441, Aug. 2008. [9] M.A. Maddah-Ali, A.S. Motahari, and A.K. Khandani, “Communication over MIMO X channels: interference alignment, decomposition, and Performance analysis,” IEEE Trans. Inf. Theory, vol. 54, pp. 3457-3470, Aug. 2008. [10] S. Jafar and S. Shamai, “Degrees of freedom region for the MIMO X channel,” IEEE Trans. Inf. Theory, vol. 54, pp. 151-170, Jan. 2008. [11] S. Jafar, M. Fakhereddin “Degrees of freedom for the MIMO interference channel” IEEE Trans. Inf. Theory, vol. 53, No. 7, pp. 2637-2642, July 2007. [12] T. Gou, S. Jafar, “Degrees of freedom of the K-User M × N MIMO interference Channel,” e-print, arXiv:0809.0099 [13] K. S. Gomadam, V. R. Cadambe, S. A. Jafar, “Approaching the capacity of wireless networks through distributed interference alignment,” e-print, arXiv:0803.3816. [14] C. M. Yetis, S. A. Jafar, and A. H. Kayran, “Feasibility conditions for interference alignment,” e-print, arXiv:0904.4526. [15] S. W. Choi and S-Y. Chung, “On the multiplexing gain of K-User partially connected interference channel,” e-print, arXiv:0806.4737v1.
