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PAPER

Generic Iterative Downlink Interference Alignment∗ Won-Yong SHIN†a) , Member and Jangho YOON††b) , Nonmember

SUMMARY In this paper, we introduce a promising iterative interference alignment (IA) strategy for multiple-input multiple-output (MIMO) multi-cell downlink networks, which utilizes the channel reciprocity between uplink/downlink channels. We intelligently combine iterative beamforming and downlink IA issues to design an iterative multiuser MIMO IA algorithm. The proposed scheme uses two cascaded beamforming matrices to construct a precoder at each base station (BS), which not only efficiently reduce the effect of inter-cell interference from other-cell BSs, referred to as leakage of interference, but also perfectly eliminate intra-cell interference among spatial streams in the same cell. The transmit and receive beamforming matrices are iteratively updated until convergence. Numerical results indicate that our IA scheme exhibits higher sum-rates than those of the conventional iterative IA schemes. Note that our iterative IA scheme operates with local channel state information, no time/frequency expansion, and even relatively a small number of mobile stations (MSs), unlike opportunistic IA which requires a great number of MSs. key words: beamforming, interference alignment (IA), iterative algorithm, leakage of interference, multi-cell downlink network

1.

Introduction

Interference management has been recognized as a crucial problem in communication systems where multiple users share the same resources. There has been a great deal of research on characterizing the asymptotic capacity of interference channels using the simple notion of degrees-offreedom, also known as multiplexing gain. Recently, interference alignment (IA) was proposed by fundamentally solving the interference problem when there are two communication pairs [1]. It was shown in [2] that the IA scheme can achieve the optimal degrees-of-freedom, equal to K/2, in the K-user interference channel with time-varying channel coefficients. Following up this success for IA, the underlying idea of [2] has been widely applied to various wireless network environments: multiple-input multiple-output (MIMO) interference networks [3]–[5], X networks [6]–[8], and cellular networks [9]–[16]. Some conventional IA schemes [2], [4] operate based on global channel state information (CSI) including CSI of Manuscript received August 6, 2014. Manuscript revised December 22, 2014. † The author is with the Department of Computer Science and Engineering, Dankook University, Yongin 448-701, Republic of Korea. †† The author is with the Department of Electrical Engineering, KAIST, Daejeon 305-701, Republic of Korea. ∗ The present research was conducted by the research fund by Dankook University in 2014. a) E-mail: [email protected] b) E-mail: [email protected] DOI: 10.1587/transcom.E98.B.834

other communication links. Furthermore, a huge number of dimensions based on time/frequency expansion are required to achieve the optimal degrees-of-freedom [2], [4], [6]–[9]. These constraints need to be relaxed in order to apply IA to more practical systems. In [5], a distributed IA scheme was introduced for the K-user MIMO interference channel with time-invariant coefficients, where a number of iterations are performed until designed transmit/receive beamforming vectors converge prior to data transmission. As another approach, in [14]–[16], an opportunistic IA protocol was introduced for wireless multi-cell uplink networks with time-invariant coefficients, where user scheduling is incorporated into the classical IA framework by opportunistically selecting mobile stations (MSs) in each cell in the sense that inter-cell interference is aligned at a pre-defined interference space. Assuming the channel reciprocity between forword/backward channels, the schemes in [5], [14]–[17] require only local CSI at each node that can be acquired from all received channel links via pilot signaling, thus resulting in easier implementation than the original scheme [2]. In this paper, we introduce a new iterative IA framework as a promising interference mitigating technique for MIMO multi-cell downlink networks, also known as the interfering broadcast channel (IBC) [10], [18], with timeinvariant channel coefficients (or equivalently slow fading channel coefficients). As in other iterative IA schemes [5], [10], our scheme is assumed to operate in the time-division duplexing (TDD) mode, where the channel reciprocity between uplink/downlink channels can be utilized to design transceivers. The proposed IA jointly takes into account iterative beamforming and downlink IA issues to design an iterative multiuser MIMO IA algorithm. In particular, inspired by the precoder design in [10], we use two cascaded beamforming matrices to construct a precoder at each base station (BS). The first transmit beamforming matrix is designed in the sense of minimizing the total amount of generating inter-cell interference, termed leakage of interference (LIF), only using local CSI. On the other hand, to design the second transmit beamforming matrix, we conduct a linear zero-forcing (ZF) filtering which completely eliminates intra-cell interference among MSs (i.e., spatial streams) in the same cell. The receive beamforming is also designed at each MS in the sense of minimizing the total amount of LIF to other BSs in a distributed fashion. Based on the TDD operation, the transmit and receive beamforming matrices (i.e., precoder and postcoder) are iteratively updated until convergence. Numerical results indicate that our IA scheme

c 2015 The Institute of Electronics, Information and Communication Engineers Copyright 

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exhibits higher sum-rates than those of the conventional iterative IA schemes [5], [10] for the multiuser network model. The proposed iterative IA is generic in the sense that one can apply our scheme to cellular networks with arbitrary numbers of cells and transmit/receive antennas. We also remark that our scheme fully utilizes the property of multicell downlink networks, unlike prior work [10], and operates even with only a small number of MSs, unlike opportunistic IA [14]–[17] which requires a great number of MSs. The rest of this paper is organized as follows. Section 2 describes the system and channel models. In Sect. 3, our proposed iterative downlink IA algorithm is described. In Sect. 4, numerical evaluation is provided. Finally, we summarize the paper with some concluding remarks in Sect. 5. Throughout this paper, the superscript † denotes the conjugate transpose of a matrix (or a vector). C is the field of complex numbers, In is the identity matrix of size n × n, and  ·  denotes L2 -norm of a vector. 2.

System and Channel Models

We consider the MIMO K-cell IBC as one of practical multicell downlink networks, where each cell consists of a BS equipped with M antennas and N MSs, each with L antennas. The number of simultaneously active MSs in a cell to receive downlink signals is denoted by S (≤ M). Then, it is assumed that S MSs are randomly selected out of N MSs in each cell and each selected MS receives a single spatial stream. Under the model, each MS in a cell is interested only in traffic demands of the BS in its cell. To take into account nontrivial cases, we assume that L < (K − 1)S + 1 and M < KS , because, otherwise, all inter-cell interference can be completely canceled at the receivers. The channel matrix from the kth BS to the nth MS in the ith cell is denoted by Hk[i,n] ∈ CL×M , where i, k ∈ {1, · · · , K} and n ∈ {1, · · · , N}. Each element of Hk[i,n] is assumed to be independent and identically distributed (i.i.d.) and to have zero mean and unit variance. In addition, frequency-flat block fading (i.e., quasi-static fading) model is assumed, which means that channel coefficients are constant during one transmission block and change to new independent values for every transmission block. The TDD operation plays an important role to perform our iterative beamformer updates since, otherwise (i.e., under other duplexing methods such as frequency-division duplexing), the forward channel Hk[i,n] cannot be the same as the backward channel. Owing to the channel reciprocity of TDD systems, the nth MS in the ith cell can estimate the channels Hk[i,n] for k = 1, · · · , K, using a pilot signaling sent from all the BSs, i.e., local CSI at each node is available. Figure 1 illustrates an example of the MIMO IBC model, where K = 3, M = 4, S = 2, L = 2, and N = 2. The detailed description in the figure will be shown in the next section. According to the above channel model, the received signal vector y[i,n] ∈ CL×1 at the nth MS in the ith cell, before being postprocessed by the receive beamforming matrix, is

given by y[i,n] =

K 

Hk[i,n] xk + z[i,n] ,

(1)

k=1

where xk ∈ C M×1 is the transmit signal vector of the kth BS, preprocessed by the two cascaded beamforming matrices. The received signal y[i,n] is corrupted by the independent and identically distributed (i.i.d.) and circularly symmetric complex additive white Gaussian noise (AWGN) vector z[i,n] ∈ CL×1 whose elements have zero-mean and variance N0 . It is assumed that each receiver (i.e., each MS) uses a single-user decoder such that interference from other cells is treated as noise. 3.

Iterative Downlink IA Algorithm

In this section, we describe the overall procedure of our proposed generic iterative downlink IA algorithm for the MIMO IBC model, and then prove that our algorithm converges. 3.1 Algorithm Description The overall procedure of our iterative downlink IA scheme is described as follows: • Step 1 (Initialization): First, each BS randomly/ arbitrarily selects S home-cell MSs among N MSs. The precoding matrix at BS k ∈ {1, · · · , K} is composed of the product of an interference suppression matrix and a user-specific beamforming matrix. In this step, we focus on the initial design of the precoding matrix. To employ a multiuser MIMO, supporting multiple MSs per cell simultaneously, BS k ∈ {1, · · · , K} independently generates an S -dimensional arbitrary precoding matrix Wk ∈ C M×S such that W†k Wk = IS , which is precisely given by   Wk = w[k,1] , · · · , w[k,S ] , where w[k,s] ∈ C M×1 is the sth column vector of Wk . Each MS can estimate the effective channels Hk[i,n] Wk using a pilot signaling broadcasted by all the BSs. • Step 2 (Receive Beamforming Update): In this step, we explain how to decide a receive beamforming vector at each MS. Let u[i,n] ∈ CL×1 denote the unitnorm weight vector at the nth MS in the ith cell, i.e., u[i,n] 2 = 1. Then, from the notion of Hk[i,n] and Wk , the nth MS in the ith cell can compute the quantity of received interference from the kth BS while using its receive beamforming vector u[i,n] , which is given by  2 Ik[i,n] = u†[i,n] Hk[i,n] Wk  , (2) where i ∈ {1, · · · , K}, n ∈ {1, · · · , N}, and k ∈ {1, · · · , K} \ i (= {1, · · · , i − 1, i + 1, · · · , K}. Using (2), the LIF metric at the nth MS in the ith cell, denoted

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Fig. 1 Iterative MU-MIMO Algorithm. The solid and dashed lines indicate the desired signal and interference links, respectively. The MIMO IBC model with K = 3, M = 4, S = 2, L = 2, and N = 2 is considered.

by I [i,n] , is defined as the sum of received interference power from other-cell BSs. That is, we have I [i,n] =

K 

Ik[i,n] .

(3)

k=1 ki

Then, for given Wk , it is apparent that each MS designs its receive beamforming vector ui,n such that u[i,n] = arg min u˜ i,n

K   u˜ † H[i,n] W 2 . k i,n k

(4)

K 

Qi(u) =

K  S  k=1 ki

†

Hk[i,n] Wk W†k Hk[i,n] .

k=1 ki

Then, the optimal receive vector at each MS is given by the eigenvector that corresponds to the minimum eigenvalue of the covariance matrix. That is, it follows that  (d)  u[i,n] = νL×1 Q[i,n] , (5) where νA×B (X) ∈ CA×B denotes an orthonormal matrix whose column vectors correspond to the minimum B eigenvalues of the Hermitian matrix X. As illustrated in Fig. 1, each MS needs to broadcast its updated receive vector ui,n (i.e., the interference suppression vector) to all the BSs. • Step 3 (Transmit Beamforming Update): Similarly as

†

H[k,s] u[k,s] u†[k,s] H[k,s] . i i

s=1

Then, for given u[k,n] , each BS finds its transmit beamforming matrix Wi ∈ C M×S such that

k=1 ki

To find the optimal receive vector in (4) in terms of minimizing the LIF, we first compute the inter-cell in(d) terference covariance matrix Q[i,n] , which is given by (d) Q[i,n] =

in Step 2, each BS also finds a subspace for the transmit beamforming, which minimizes the amount of generating inter-cell interference to other-cell MSs. Let Qi(u) denote the inter-cell interference covariance matrix, measured by the ith BS, which is given by

Wi = arg min ˜i W

= ν M×S

K  S  2   † [k,n] ˜   u[k,n] Hi W i k=1 ki

n=1

 (u)  Qi ,

(6)

which is also shown in Fig. 1. As illustrated in Fig. 1, after the computation of the transmit beamforming matrix, each BS broadcasts its updated transmit beamforming matrix to all the other-cell MSs. • Step 4 (Iteration): We iteratively perform Steps 2 and 3, which correspond to the update of receive and transmit beamforming matrices, respectively, until convergence. • Step 5 (Cascaded Transmit Beamforming): Besides the design of the transmit inter-cell interference suppression matrix Wi , we find the user-specific beamforming matrix coming from the ZF filtering, which enables to completely eliminate intra-cell interference during the decoding process at each MS. In consequence, the cascaded transmit beamforming matrix Vi ∈ C M×S is given by Vi = Wi Pi ,

(7)

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where Wi is shown in (6) and Pi ∈ CS ×S is given by   Pi = p[i,1] · · · p[i,S ] , which indicates the user-specific beamforming matrix of the ith transmit beamforming matrix Vi such that u†[i,a] Hi[i,a] Wi p[i,b]

=0

desired signal

+

K  S  k=1 ki

1. Start with an S -dimensional arbitrary precoding matrix at each BS.   Wi = w[i,1] , w[i,2] , · · · , w[i,S ] ∈ C M×S , W†i Wi = IS 2. Begin iteration. 3. Compute the interference covariance matrix at each MS.

(8)

for any a, b ∈ {1, 2, · · · , S } and a  b. Here, p[i,s] ∈ S ×1 is C 2 the sth column vector of Pi with unit-norm, i.e., p[i,s]  = 1 for i ∈ {1, · · · , K} and s ∈ {1, · · · , S }. Note that, even if using the user-specific beamforming matrix Pi at each BS may lead to a change to the inter-cell interference level at each MS in other cells, it plays a vital role in greatly reducing the total amount of potential interference. • Step 6 (Downlink data transmission): Using the cascaded transmit beamforming matrix in Step 5, the received signal vector y[i,n] at the nth MS in the ith cell can be rewritten as y[i,n] = Hi[i,n] Wi p[i,n] s[i,n] +  

Algorithm 1: Iterative Downlink IA

S 

H[i,n] i Wi p[i,s] s[i,s]

s=1 sn

Q(d) = [i,n]

k=1 ki

4. Generate the interference suppression vector at each MS.   u[i,n] = νL×1 Q(d) [i,n] 5. Each MS broadcasts the updated interference suppression vector u[i,n] to all the BSs. 6. Each BS computes the inter-cell interference covariance matrix based on the updated vectors. Q(u) i =

k=1 ki

9. Continue until convergence.   10. Find Vi = Wi Pi = Wi p[i,1] · · · p[i,S ] such that each of which column vector can be spanned by Wi and

Hk[i,n] Wk p[k,s] s[k,s] +z[i,n] ,

u†[i,a] Hi[i,a] Wi p[i,b] = 0 for any a, b ∈ {1, 2, · · · , S } and a  b.

=

=

k=1 ki

K  S  K  i=1 n=1

Ik[i,n] †

u†[i,n] Hk[i,n] Wk W†k Hk[i,n] u[i,n] ,

(9)

k=1 ki

s=1

The above iterative procedure is summarized in Algorithm 1, and its pictorial representation is shown in Fig. 1. 3.2 Convergence Analysis We now show that our iterative algorithm converges. The proof technique essentially follows the same line as [5, Section V], and thus we provide a brief sketch of the proof. We start from letting Itotal denote the total LIF by adding I [i,n] in (3) up for all i ∈ {1, · · · , K} and n ∈ {1, · · · , S }, which is given by

i=1 n=1

K  S  K  i=1 n=1

u†[i,n] Hk[i,n] Wk p[k,s] s[k,s] + u†[i,n] z[i,n] .

Here, it is seen that intra-cell interference is completed cancelled due to the user-specific beamforming matrix Pi that behaves as the linear zero-forcing beamformer as shown in (8).

Itotal =

s=1

  Wi = ν M×S Q(u) i

u†[i,n] Hi[i,n] Wi p[i,n] s[i,n]

S K  

†

H[k,s] u[k,s] u†[k,s] Hi[k,s] i

7. Generate the S -dimensional orthonormal matrix that minimizes the inter-cell interference at each BS.

where s[i,n] is the transmit symbol for the nth MS in the ith cell. After using the receive beamforming in Step 2, the received signal y˜ [i,n] at the nth MS in the ith cell is finally given by

k=1 ki

K  S 

8. Each BS broadcasts the updated matrix Wi to all the other-cell MSs.

inter-cell interference

+

†

intra-cell interference

s=1

K  S 

Hk[i,n] Wk W†k H[i,n] k

 

 

y˜ [i,n] =

K 

I [i,n]

where the last equality holds due to (2). We now show that each step in the algorithm reduces the total LIF Itotal , thereby eventually approaching zero. We first focus on how the design of the receive beamforming at each MS affects Itotal . From (9), it follows that min

u[i,n] ,∀n∈{1,··· ,S },i∈{1,··· ,K}

= =

Itotal K  S 

min

u[i,n] ,∀n∈{1,··· ,S },i∈{1,··· ,K} K  S  i=1 n=1

I [i,n]

i=1 n=1

minI [i,n] u[i,n]

⎡ ⎤ ⎥⎥⎥ K  S ⎢ K ⎢⎢⎢   † ⎥ † [i,n] † [i,n] ⎢ ⎢ = u[i,n] Hk Wk Wk Hk u[i,n] ⎥⎥⎥⎥⎥ , ⎢⎢⎢min u[i,n] ⎣ ⎦ k=1 i=1 n=1 ki

which indicates  that, given the values of Wk =  w[k,1] , · · · , w[k,S ] (k ∈ {1, · · · , K}), using (5) (or equiva-

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lently, (4)) in Step 2 minimizes the value of Itotal over all possible choices of u[i,n] (i ∈ {1, · · · , K} and n ∈ {1, · · · , S }). Let us turn to the transmit beamforming case by using the channel reciprocity between up/downlink channels. In a similar fashion, when we denote the sum of generating interference power to other-cell MSs from the kth BS by Ik , we have min

Wk ,∀k∈{1,··· ,K}

= =

min

Itotal

Wk ,∀k∈{1,··· ,K}

K 

Ik

k=1

K  min Ik . k=1

Wk

Thus, it is easy to see that, given the values of u[i,n] , using (6) in Step 3 minimizes the value of Itotal over all possible choices of Wk (k ∈ {1, · · · , K}). Since the value of Itotal is monotonically decreasing after every iteration, convergence of the algorithm is guaranteed. Note that the algorithm reduces the total LIF for every iteration and thus is guaranteed to converge while minimizing it. However, the convergence to a global minimum is not guaranteed since the optimization problem (i.e., the transmit/receive beamforming design problem) is nonconvex. In the next section, the convergence for the proposed algorithm will be shown using computer simulations. 4.

Numerical Evaluation

In this section, we perform computer simulations to verify the performance of the proposed iterative downlink IA algorithm for practical multi-cell downlink MIMO networks. To consider a more practical environment, suppose that the channel matrix between the kth BS and the nth MS in the ith√cell consists of the large-scale path-loss component 0 < βik ≤ 1 and the small-scale complex fading component Hk[i,n] , where i, k ∈ {1, · · · , K} and n ∈ {1, · · · , N}. In this case, the received signal vector y[i,n] in (1) can be rewritten as y[i,n] =

K   βik Hk[i,n] xk + z[i,n] . k=1

In our all simulations, for simplicity, it is assumed that βik = 1 if i = k and βik = 0.5 otherwise for i, k ∈ {1, · · · , K}. This is because, when i = k, the large-scale term βik corresponds to the intra-cell received signal strength, which is much stronger than the signal strength from the other-cell BS. The small-scale fading terms Hk[i,n] are generated 1×105 times for each system parameter. Similarly as in [5], the average amount of the total LIF, Itotal , is first evaluated as the number of iterations, denoted by Nit , increases. In Fig. 2, the log-log plot of Itotal versus Nit is shown as Nit increases.† In Fig. 2, when the parameter S † Even if it seems unrealistic to have a great number of iterations in practice, the range for parameter Nit is taken into account to precisely see some trends of curves varying with Nit .

Fig. 2 The total LIF Itotal with respect to the number of iterations, Nit , for some S . The system with M = 6, L = 5, and K = 3 is considered.

varies, the case with M = 6, L = 5, and K = 3 is considered, where S denotes the number of simultaneously active MSs per cell. It is shown that, as S varies from 5 to 3, the total LIF Itotal decreases due to less interferers, which is rather obvious. The result, illustrated in Fig. 2, also indicates that Itotal tends to monotonically decrease with Nit , which means that convergence of the proposed algorithm is guaranteed. It is further seen how many iterations are required to guarantee that the total LIF is less than an arbitrarily small constant for given system parameters M, L, K, and S . Next, as illustrated in Fig. 3, the achievable sum-rates of the proposed IA scheme are evaluated according to the received signal-to-noise ratios (SNRs) (in dB scale) for some S and are compared to those of the following two conventional iterative IA schemes: Gomadam-Cadambe-Jafar (GCJ) scheme [5] and Suh-Tse (ST) scheme [10]. • In the GCJ scheme, the single-user MIMO strategy is used, where S spatial streams are served to randomly selected one MS among N MSs in each cell. The GCJ scheme iteratively updates transmit/receive beamforming matrices in terms of minimizing the LIF, whereas, unlike our structure, its precoder does not use cascaded beamforming matrices since the single-user MIMO is assumed. • In the ST scheme, the integration of IA with opportunistic scheduling is used to mitigate dominant intercell interference while the two cascaded beamforming matrices including the ZF filtering are generated at the BS to eliminate intra-cell interference. In the ST scheme, an iterative matched filter receiver is introduced to maximize the received signal-to-interferenceand-noise ratio at each MS, which is turned out to be efficient for a large range of practical SNR regimes. In the proposed and GCJ schemes, the round robin scheduler, providing resources cyclically to N MSs without taking into account channel conditions, is used. On the other hand, in the ST scheme, an opportunistic scheduler is employed to choose S MSs out of N MSs at a time such that the sum-rates are maximized. Our scheme based on round robin schedul-

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Fig. 3 The achievable sum-rates and upper bound with respect to the received SNR. The system with M = 6, L = 5, K = 3, N = 20, and Nit = 15 is considered.

ing, which provides the best fairness, leads to a lower bound on the sum-rates, compared to the case using another scheduler balancing throughput and fairness. To see the fundamental limit of the MIMO K-cell IBC model, we also show an upper bound that comes from a genie-aided removal of all the inter-cell interference. When we use the singular-value decomposition (SVD)-based beamforming, we can obtain K parallel MIMO systems. Simulation environments are given

Fig. 4 The achievable sum-rates according to various systems parameters M, L, K, and S . The system with N = 20, Nit = 15, and SNR=20 dB is considered.

by M = 6, L = 5, K = 3, N = 20, S = 3, 4, 5, and Nit = 15. It is shown that our iterative IA scheme significantly outperforms the conventional ones beyond a certain SNR level, which is in a low SNR regime. More precisely, as depicted in Fig. 3, it is seen that, in the low SNR regime, the ST scheme outperforms the other two schemes since it

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5.

Fig. 5 The achievable sum-rates with respect to the number of per-cell MSs, N. The system with M = 6, L = 5, K = 3, S = 3, Nit = 15, and SNR=20 dB is considered.

can obtain the power gain with the use of matched filtering. However, in the high SNR regime, much higher IA gains can be achieved by using the proposed iterative algorithm, which uses beamformers designed in the sense of fully mitigating interference. It is also seen that, as S increases (especially for S = 5), the sum-rates of the proposed IA scheme get degraded due to the existence of more interferers, but are still higher than those of the other two schemes in the high SNR. In addition, it is seen that, for S = 3, the upper bound and the sum-rates of the proposed scheme have almost the same slope, but as S increases, the performance gap between the two curves becomes large since interference is not completely aligned along with the proposed IA scheme. To better see how each of other parameters such as M, L, and K affects the sum-rate performance of the proposed IA scheme, the achievable sum-rates are evaluated according to M, L, and K in Figs. 4(a), 4(b), and 4(c), respectively. In this case, simulation environments are given by S = 3, 4, 5, N = 20, Nit = 15, and SNR=20 dB. It is obvious that the sum-rates of the proposed scheme get higher with increasing M or L, whereas they get degraded with increasing K. Interestingly, it is also seen that the optimal number of activated MSs per cell, S , in terms of maximizing the sum-rates depends heavily on the system parameters M and L. In Fig. 5, the achievable sum-rates are evaluated according to N where simulation environments are given by M = 6, L = 5, K = 3, S = 3, Nit = 15, and SNR=20 dB. Based on the result, the following observation is made. Our scheme and the GCJ scheme using round robin scheduling randomly select S and 1 per-cell MSs, respectively, among N MSs, and thus there is no performance gain as N increases. The sum-rates of the ST scheme, however, gets slightly improved owing to the multiuser diversity gain with increasing N since the opportunistic scheduler is employed, while our scheme still outperforms the ST scheme even for relatively large N (up to N = 100).

Conclusion

We introduced the generic iterative IA algorithm which intelligently combines iterative transmit/receive beamforming and downlink IA framework for multi-cell downlink MIMO networks. More precisely, the proposed IA scheme was constructed in the sense that the effect of inter-cell interference is significantly mitigated through the iterative update of beamforming matrices and intra-cell interference is completely eliminated. Hence, by fully utilizing the property of cellular downlink in designing IA, it turns out that our IA scheme outperforms the existing iterative IA schemes in terms of sum-rates. We finally remark that our scheme operates with a relatively small number of iterations, local CSI, no time/frequency expansion, and a small number of MSs, thereby resulting in an easier implementation. References [1] 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, no.8, pp.3457–3470, Aug. 2008. [2] V.R. Cadambe and S.A. Jafar, “Interference alignment and degrees of freedom of the K-user interference channel,” IEEE Trans. Inf. Theory, vol.54, no.8, pp.3425–3441, Aug. 2008. [3] C.M. Yetis, T. Gou, S.A. Jafar, and A.H. Kayran, “On feasibility of interference alignment in MIMO interference netweorks,” IEEE Trans. Signal Process., vol.58, no.9, pp.4771–4782, Sept. 2010. [4] T. Gou and S.A. Jafar, “Degrees of freedom of the K user M × N MIMO interference channel,” IEEE Trans. Inf. Theory, vol.56, no.12, pp.6040–6057, Dec. 2010. [5] K. Gomadam, V.R. Cadambe, and S.A. Jafar, “A distributed numerical approach to interference alignment and applications to wireless interference networks,” IEEE Trans. Inf. Theory, vol.57, no.6, pp.3309–3322, June 2011. [6] C. Huang, V. Cadambe, and S.A. Jafar, “Interference alignment and the generalized degrees of freedom of the X channel,” IEEE Trans. Inf. Theory, vol.58, no.8, pp.5130–5150, Aug. 2012. [7] V.R. Cadambe and S.A. Jafar, “Interference alignment and the degrees of freedom of wireless X networks,” IEEE Trans. Inf. Theory, vol.55, no.9, pp.3893–3908, Sept. 2009. [8] B. Nourani, S.A. Motahari, and A.K. Khandani, “Relay-aided interference alignment for the quasi-static X channel,” Proc. IEEE Int. Symp. Inf. Theory, pp.1764–1768, Seoul, Korea, June/July 2009. [9] C. Suh and D. Tse, “Interference alignment for cellular networks,” Proc. 46th Allerton Conf. Commun., Control, and Comput., pp.1037–1044, Urbana-Champaign, IL, Sept. 2008. [10] C. Suh and D.N.C. Tse, “Downlink interference alignment,” IEEE Trans. Commun., vol.59, no.9, pp.2616–2626, Sept. 2011. [11] W. Shin, N. Lee, J.-B. Lim, C. Shin, and K. Jang, “On the design of interference alignment scheme for two-cell MIMO interfering broadcast channels,” IEEE Trans. Wireless Commun., vol.10, no.2, pp.437–442, Feb. 2011. [12] Y. Ma, J. Li, and R. Chen, “On the achievability of interference alignment for three-cell constant cellular interfering networks,” IEEE Commun. Lett., vol.16, no.9, pp.1384–1387, Sept. 2012. [13] A.S. Motahari, S.O. Gharan, M.A. Maddah-Ali, and A.K. Khandani, “Real interference alignment: Exploiting the potential of single antenna systems,” IEEE Trans. Inf. Theory, vol.60, no.8, pp.4799– 4810, Aug. 2014. [14] B.C. Jung and W.-Y. Shin, “Opportunistic interference alignment for interference-limited cellular TDD uplink,” IEEE Commun. Lett.,

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vol.15, no.2, pp.148–150, Feb. 2011. [15] B.C. Jung, D. Park, and W.-Y. Shin, “Opportunistic interference mitigation achieves optimal degrees-of-freedom in wireless multi-cell uplink networks,” IEEE Trans. Commun., vol.60, no.7, pp.1935– 1944, July 2012. [16] H.J. Yang, W.-Y. Shin, B.C. Jung, and A. Paulraj, “Opportunistic interference alignment for MIMO interfering multiple-access channels,” IEEE Trans. Wireless Commun., vol.12, no.5, pp.2180–2192, May 2013. [17] H.J. Yang, B.C. Jung, W.-Y. Shin, and A. Paulraj, “Codebook-based opportunistic interference alignment,” IEEE Trans. Signal Process., vol.62, no.11, pp.2922–2937, June 2014. [18] W.-Y. Shin and B.C. Jung, “Network coordinated opportunistic beamforming in downlink cellular networks,” IEICE Trans. Commun., vol.E95-B, no.4, pp.1393–1396, April 2012.

Won-Yong Shin received the B.S. degree in electrical engineering from Yonsei University, Seoul, Korea, in 2002. He received the M.S. and the Ph.D. degrees in electrical engineering and computer science from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2004 and 2008, respectively. From February 2008 to April 2008, he was a Visiting Scholar in the School of Engineering and Applied Sciences, Harvard University, Cambridge, MA. From September 2008 to April 2009, he was with the Brain Korea Institute and CHiPS at KAIST as a Postdoctoral Fellow. From August 2008 to April 2009, he was with the Lumicomm, Inc., Daejeon, Korea, as a Visiting Researcher. In May 2009, he joined Harvard University as a Postdoctoral Fellow and was promoted to a Research Associate in October 2011. Since March 2012, he has been with the Division of Mobile Systems Engineering, College of International Studies and the Department of Computer Science and Engineering, Dankook University, Yongin, Korea, where he is currently an Assistant Professor. His research interests are in the areas of information theory, communications, signal processing, mobile computing, and their applications to multiuser networking issues. Dr. Shin has served as an Associate Editor for the IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS, COMMUNICATIONS, COMPUTER SCIENCES, for the IEIE TRANSACTIONS ON SMART PROCESSING & COMPUTING, and for the JOURNAL OF KOREA INFORMATION AND COMMUNICATIONS SOCIETY. He has also served as an Organizing Committee for the 2015 IEEE Information Theory Workshop.

Jangho Yoon received the B.S., M.S., and Ph.D. degrees in electrical engineering and computer science from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2006, 2008, and 2015, respectively. His research interests include in the areas of wireless communications, interference management, signal processing, and their applications to multiuser networking issues.