Spectrum-Efficient Topology Management of ... - IEEE Xplore

IEEE/CIC ICCC 2014 Symposium on Wireless Communications Systems

Spectrum-Efficient Topology Management of Asymmetric Interference Alignment Networks † §

Xinyu Zhang† , F. Richard Yu§ , Ying He† and Nan Zhao†

School of Information and Communication Engineering, Dalian University of Technology, Dalian, China Department of Systems and Computer Engineering, Carleton University, Ottawa, ON, K1S 5B6, Canada

Abstract—Interference alignment (IA) is a promising technique in wireless networks. However, existing works are mostly based on symmetric IA networks. To meet the requirements of practical applications, we consider asymmetric IA networks based on the various path-loss. In this paper, a spectrum-efficient topology management scheme is proposed for the asymmetric IA networks. In the scheme, for the user far away from others, solely adopting spatial multiplexing (SM) as a point-to-point subnetwork is more spectrum-efficient. On the other hand, for the others aggregating together, jointly comprising an IA subnetwork may be a better choice. We first present the criterion to decide which is more spectrum-efficient for the topology management scheme, i.e., IA or SM. Then, the topology management scheme is elaborated with the graph theory. In addition, the designs of the precoding and decoding matrices are presented in the IA and SM schemes, respectively. Simulation results show that the proposed topology management scheme is much more spectrum-efficient than the conventional IA scheme in the asymmetric multiuser network. Index Terms—Interference alignment, asymmetric multiuser networks, topology management, spectrum efficiency.

I. I NTRODUCTION Interference is a major factor that restricts the performance in modern wireless networks. Interference alignment (IA) is a novel method to solve the interference problem in multiuser multiple-input and multiple-output (MIMO) networks, which has been recently proposed by in [1]. However, the closed-form solutions of the IA problem were only found in several special cases [2]. Instead of obtaining the closed-form solutions, some iterative algorithms were proposed to design the precoding and decoding matrices directly in [3], which focused on making the interference leakage minimum. To guarantee the convergence of these iterative algorithms, the feasibility conditions of interference alignment should be satisfied [4]. Due to solving the IA problem effectively, these algorithms have been applied to many multiuser wireless communication systems. However, there are many problems that impede the application of IA, and some of our previous works have been concentrated on improving the practicability of IA [5], [6]. Nevertheless, all the works in [1]–[6] are based on the symmetric IA networks, i.e., both the distance between each other The corresponding author of the paper is Nan Zhao. This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 61201224, China Postdoctoral Science Foundation Special Funded Project under 2013T60282, and the Fundamental Research Funds for the Central Universities under DUT14QY44.

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and the power allocated to each transmitter are the same. However, in practical wireless networks, the requirements of the symmetry are redundant and rigorous due to the variety of the network topology. Hence, there have been growing interests in the asymmetric IA networks [7], [8]. In [7], the authors introduced the large-scale fading gain into the channel matrix. The distribution of users was no longer symmetric, hence the heterogeneous path loss and spatial correlations were taken into consideration to improve the performance. In [8], clustering methods for the multiuser asymmetric IA network were presented. However, those works just considered the IA scheme in asymmetric multiuser networks. As is known, in the point-topoint MIMO system, spatial multiplexing (SM) can achieve excellent performance, by which multiple data streams can be transmitted parallelly. However, the interference problem impedes the application of the SM scheme in the multiuser MIMO systems. Nevertheless, when a user is far away from all the others in an asymmetric multiuser network, due to the large path-loss, the user’s interference is sufficiently weak, and the SM scheme can be used by the user instead of joining the IA network with others. In this paper, we propose a spectrum-efficient topology management scheme in asymmetric IA networks. Through making use of the asymmetry based on the various path-loss, the criterion obtained by using the bisection search algorithm is derived to decide which is more spectrum-efficient for a certain user of the asymmetric IA networks, i.e., IA or SM. For convenience, the concepts and algorithms in the graph theory are applied to elaborate the topology management scheme of the networks. Simulation results show the effectiveness of the proposed topology management scheme. Notation: Id represents the d×d identity matrix. A† , det(A) and Tr(A) are the Hermitian transpose, the determinant and the trace of matrix A, respectively. a2 and A2 are the 2 -norm of vector a and matrix A, respectively. rank (H) represents the rank of matrix H. E(·) stands for expectation. CN (a, A) is the complex Gaussian distribution with mean a and covariance matrix A. II. S YSTEM M ODEL A. Topology of Asymmetric IA Networks Consider a K-pair MIMO interference wireless network where the j-th Tx-Rx pair is equipped with M [j] and N [j]

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antennas, respectively, and d[j] data streams are transmitted to the j-th Rx from its corresponding Tx. Different from the previous system models in [1]–[4], all the Tx-Rx pairs are randomly distributed in a certain area, and the distance between any two pairs is no longer equal. Hence, denote the large-scale fading gain from the i-th Tx to the j-th Rx as −α ρ[ji] = r[ji] [9], which is scaled by the distance r[ji] with the path-loss exponent α. The small-scale fading matrix from [j] [i] the i-th Tx to the j-th Rx can be defined as H[ji] ∈ CN ×M , and the transmitted power of each transmitter is denoted as Pt . Thus in the asymmetric IA network, the output at the j-th receiver is given by y

[j]

K = ρ[ji] U[j]† H[ji] V[i] x[i] + U[j]† z[j] ,

(1)

[j] where z[j] ∈ CN ×1 ∼ CN 0, σ 2 IN [j] is the white Gaussian noise at the j-th Rx. V[i] is the precoding matrix of the i-th Tx with V[i]† V[i] = Id[i] . U[j] is the decoding matrix of the j-th Rx with U[j]† U[j] = Id[j] . x[i] ∼ CN 0, dP[i]t Id[i] represents the transmitted symbols from the i-th Tx to its corresponding Rx with a power constraint Pt . Assume that each entity of H[ji] is independent and identically distributed (i.i.d.) with the complex Gaussian distribution CN (0, 1), and the channel state information (CSI) can be perfectly obtained at all the transceivers. In the conventional IA scheme, to eliminate the interferences from other pairs, the following conditions should be met as

(3)

[jj]

where H = U[j]† H[jj] V[j] is the effective channel matrix from the j-th Tx to its corresponding Rx, and the power of residual interference at the j-th Rx I [j] can be expressed as K i=1,i=j

Pt [ji]

[j]† [ji] [i] 2 ρ H V

U

. d[i]

(5)

Although the asymmetric IA network is considered, the configuration and parameter of each pair is assumed to be the same in this paper, i.e., M [j] = M, N [j] = N, d[j] = d. The feasibility condition of IA should be satisfied as [4] M + N ≥ d · (K + 1).

where m = min (M, N ) represents the maximum of K

Pt [ji] [j]† [ji] [i] 2 U H V rank H[jj] , I [j] = is the d ρ effective noise power by treating the interferences from other IA subnetworks as white gaussian noise, W[j] is the Wishart matrix, and it can be expressed as [jj] [jj]† N CIA .

(9)

As expressed in (5), the residual interference is sufficiently weak, and in order to handle it more conveniently, we can ignore the impact of the residual interference on the capacity. Hence, the following expression is adopted instead of (4) as

ρ[jj] Pt [jj] [jj]† [j] CIA = log2 det Id[j] + [j] 2 H H . (10) d σ Due to the number of the interferers decreasing, the expression of the interference in (7) shall be modulated slightly. Assume that the variation of the network topology is mainly caused by the random distribution of the users’ locations, other than the fluctuation of the channel matrix. When the relative location between each other is fixed, the long-term statistical property of the channels is invariable, and the network

628


15

Transmission Rate (bits/s/hz)

SM IA r=5 10

r=4 r=3

5

r=2 r=1

0

0

5

10

15 20 25 Transmit SNR (dB)

30

35

40

Fig. 1. The transmission rate of the j-th pair versus the transmit SNR for M = N = 3, d = 1, α = 3. The variable r (km) represent the distance between the i-th pair and the j-th pair. The distance from the transmitter to corresponding receiver is set to 1 km.

topology is unchanged. Therefore, we use the mean capacity over time to obtain the decision instead of the instantaneous capacity (9) as [j] [j] E CSM > E CIA . (11) Fig. 1 presents the transmission rate of the j-th pair versus the transmit SNR. As we expected, at a specific transmit signal to noise ratio (SNR), as the distance r between the two pairs increasing, the interference at the j-th Rx becomes weaker, and the performance of the SM scheme is becoming better than the IA scheme for the j-th pair gradually. However, the results just dwell on the qualitative description about the decision criterion. Thus, a quantitative description shall also be presented. Through analyzing (11), we can find that the left hand side of inequation depends on the transmit SNR and the distance r, however, the right hand side only depends on the transmit SNR. Generally, the variation of the transmit SNR is trivial in a network, while the change of the relative distance r between each other may be drastic due to the move of the pairs. Hence, to solve the inequation (11) conveniently, we first fix the transmit SNR, and then find the minimum distance D(SN R) to satisfy the inequation at the transmit SNR. Due to the difficulty to obtain the closed-form solutions of (11), the bisection search method is leveraged, and the Algorithm 1 is represented as follows. B. Reinterpretation of Asymmetric IA Networks In the graph theory, a graph is denoted as G =< V, E >, where V is a nonempty set of vertices, E is a set of edges. When all the edges are undirected, the graph is called an undirected graph. We can reinterpret the asymmetric IA network as a weighted undirected graph. Each vertex represents a Tx-Rx pair. The weight of each edge depends on the distance between the two pairs connected by the edge. When the distance is larger than the minimum distance D(SN R) calculated by Algorithm 1, the weight w is set to 0. Otherwise, w is set to 1. In the graph theory, the weighted undirected graph can

Algorithm 1 Solution to the inequation (11) 1: Set solution interval D(SN R) ∈ [a, b], precision ξ, transmit SNR γ. 2: repeat 3: Calculate the midpoint of the interval c. 4: Calculate the mean capacity of the IA scheme [j] E[CIA (γ)]. 5: Calculate the mean capacity of the SM scheme [j] E[CSM (γ, c)]. [j] [j] 6: Compare E[CSM (γ, c)] with E[CIA (γ)]: [j] [j] 7: if E[CSM (γ, c)] = E[CIA (γ)] then 8: The solution D(SN R) = c, break. [j] [j] 9: else if E[CSM (γ, c)] > E[CIA (γ)] then 10: b = c. 11: else 12: a = c. 13: end if 14: until |b − a| < ξ 15: The solution D(SN R) = a or D(SN R) = b.

be also expressed by the adjacency matrix as ⎡ w11 w12 · · · w1K ⎢ w21 w22 · · · w2K ⎢ W =⎢ . .. .. .. ⎣ .. . . . wK1 wK2 · · · wKK

⎤ ⎥ ⎥ ⎥, ⎦

(12)

where wij ∈ {0, 1}, i = j, i, j ∈ {1, · · · , K} is the weight of the edge connecting the i-th and the j-th vertexes, and the diagonal element wii = 0, i ∈ {1, · · · , K}. Take the network shown in Fig. 2 as an example, the adjacency matrix of the graph can be expressed as ⎤ ⎡ 0 1 1 1 0 ⎢ 1 0 1 1 0 ⎥ ⎥ ⎢ ⎥ (13) W =⎢ ⎢ 1 1 0 1 0 ⎥. ⎣ 1 1 1 0 0 ⎦ 0 0 0 0 0 From the adjacency matrix, the network topology can be shown more clearly. Therefore, the algorithm in the graph theory can be applied to analyze the network structure. Then, based on the network structure, the spectrum-efficient topology management of the network can be established. IV. T OPOLOGY M ANAGEMENT S CHEME AND THE D ESIGN OF THE P RECODING AND D ECODING M ATRICES A. Topology Management Scheme Due to the one-to-one correspondence between the graph and the network, the topology management problem can be transformed into a weighted undirected graph partition problem. In consideration of the connection of a graph, G may be comprised of several independent subgraphs, i.e., the whole network can be divided into several subnetworks. To describe the problem conveniently, some definitions shall be introduced as follows.

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Generally, the MAX-SINR algorithm [3] is more spectrumefficient than the MinIL algotithm. Nevertheless its computational complexity is higher than that of the MinIL algorithm. 2) SM Subnetwork: For the SM subnetwork, we extend the statements of matrices V and U of IA to it. The design of two matrices of the SM scheme is easier than those in the IA scheme, because there are no need to obtain them iteratively. V and U can be calculated just through the singular value decomposition (SVD) of the channel matrix H ∈ CN ×M H = UΣV† , Fig. 2. Reinterpret the asymmetric IA network as a graph. Each vertex is a Tx-Rx pair. The blue and red solid lines indicate that the distance between two pairs is smaller or larger than the minimum distance D(SN R), respectively.

Definition 1. Isolated node/SM pair: When the weight of all the edges connected to the i-th vertex is 0, the i-th vertex is called Isolated node, also called SM pair. Definition 2. Nonisolated node/IA pair: When the weight of any one edge connected to the i-th vertex is 1, the i-th vertex is called Nonisolated node, also called IA pair. Definition 3. Fusion: If a vertex (pair) is included in two different subgraphs (subnetworks), the two subgraphs would be fused into one subgraph. The operation would stop when no more subgraphs can be fused. Finally, all the subgraphs are independent and disjoint with each other. Based on the adjacency matrix of the graph and the three definitions above, the depth first search (DFS) algorithm [11] can be applied to solve the graph partition problem. Hence, the topology management scheme can be summarized as 1) Apply the DFS algorithm to traverse the whole graph. The depth first spanning trees of the graph can be obtained. 2) Each tree represents an independent subgraph (subnetwork). All the trees jointly comprise a depth first spanning forest, which represents the whole graph (network). 3) Count the number num of nodes in each tree, num ≥ 1. 4) If num = 1, the subnetwork represented by the tree is call SM subnetwork, and the pair in the subnetwork would be deemed as a SM pair. Otherwise, the subnetwork is called IA subnetwork, and each pair in the subnetwork would be treated as an IA pair. B. The Design of the Precoding and Decoding Matrices 1) IA Subnetwork: Similar to the conventional IA, the precoding and decoding matrices V and U can be obtained by the MinIL algorithm to each IA subnetwork, respectively. The MinIL algorithm focuses on eliminating interference in each subnetwork, and ignores the interferences from other subnetworks. However, due to caring about the signaling spaces, but ignoring the receiver power level, the MinIL IA algorithm can’t obtain the optimal spectrum efficiency.

(14)

where Σ is a rectangular and √ √ diagonal matrix, √ whose diagonal elements ( λ1 ≥ λ2 ≥ · · · ≥ λm ) are the singular values of the channel matrix H. The right and left singular vector matrices, i.e., U ∈ CN ×N , UU† = IN and V ∈ CM ×M , VV† = IM are denoted as the decoding and precoding matrices, respectively. In Section II, to obtain the decision criterion conveniently, we assume that the power allocated to each antenna is equal in the SM scheme. However, when the CSIT is available, the water-filling PA algorithm can be applied to improve the spectrum efficiency of the SM subnetwork. The throughput of the SM subnetwork by the water-filling PA algorithm can be expressed as [jj] [j] [j] max CSM = log2 det Im + σ2ρ+I [j] W1 , (15) Subject to Tr(P) ≤ Pt , where [j]

W1 =

H[jj] PH[jj]† H[jj]† PH[jj]

N