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A signal subspace interference alignment scheme with sum rate maximization and altruistic-egoistic Bayesian gaming.

1. Introduction

The interference has become the major factor to impact the performance of multi-user wireless communication networks with growth of the modern systems. These effects are highlighted by both theoretical analysis [1] and experimental measurements [2][3]. The conventional interference management techniques used in practical wireless communication systems either consider interference signals as noise and orthogonalize the available channel resource. However, the former ignores the structure of interference signals and it is known to be effective only when the power of interference signals is low; while the latter leads to inefficient use of wireless channel resources. The multi-user detection schemes such as [18-20] can be used to eliminate interference and decode the desired signal in the cases where interference is strong [4], [5] or very strong [6]. However, the complexity of the multi-user detection schemes is unacceptable especially when the number of users in the interference channel exceeds three.

A. Previous Work

In a recent study, Cadambe and Jafar [7] have shown that a total K/2 degrees of freedom (DoF) is achievable from the perspective of information theory by deploying interference alignment scheme at the transmitters and zero-forcing at the receivers for K-user (K [greater than or equal to] 2) single-input single-output (SISO) frequency selective interference channels. Where the DoF is the one-order approximation of the sum rate at high SNR and it can be used to characterize the number of transmitted data streams free from interference in the channels. The key insight of the interference alignment is that the desired signal at the receivers can occupy 1/2 of the total available wireless channel resources at most in the case where suitable precoding matrices at the transmitter are designed to constrain interference subspace spanned by the interference signal.

A closed-form solution of interference alignment for transmitter precoding matrices is proposed in [7] for a 3-user interference channel. However, this solution assumes that the transmitters know global channel state information, which would become an increased overhead for system design. Moreover, the calculation of closed-form solution may become a difficult or impossible task in the cases where the number of uses is greater than three. Therefore, the application of interference alignment presented in [7] is limited to only small number of users. Later on, Gomadam, Cadambe and Jafar [10] proposed a distributed interference alignment algorithms based on iterative scheme. The scheme utilizes the reciprocity of wireless networks and cycle iteration to minimize the objective function of the interference leakage to achieve asymptotical interference alignment. The algorithm requires the transmitters to obtain only the local channel state information between communication pairs (the local channel state information of transmitter and its corresponding receiver), which implies that the interference alignment can be achieved with reduced channel feedback overheads. Moreover, asymptotical interference alignment can also be achieved in the interference channel with more than 3 users in this algorithm. However, the algorithm in [10] needs to exploit channel reciprocity to alternate between the forward and reverse channels that requires tight synchronization at both ends. The processing of alternation will produce significant overheads in a dynamic communication environment where the channel varies rapidly. Another approach on the interference alignment without the requirement of reciprocal channel has proposed in [11] that alternatively optimizes the precoder matrices at transmitters and the interference subspace at receivers. Their formulation is more conducive to mathematical and geometrical analysis. [11] uses alignment of signal subspace as the main optimization objective from the perspective of geometry, which is asymptotically optimal when the Signal to Noise Ratio (SNR) tends to infinity. However, only focusing on high SNR region and interference alignment of signal subspaces will result in linear transceivers with suboptimal performance at low to intermediate SNRs. In [15], the authors propose an approach to design the precoding vectors at each data stream in the framework of game theory that aims to provide a compromise between egoistic beamforming gain at the intended receiver; and altruistic alignment and cancellation of the interference created towards other receivers. In [16], the interference alignment scheme in frequency domain is applied to the practical heterogeneous cellular network with different cell sizes of cells and the satellite communication system including monobeam and multibeam satellites. In [17], a regularized transceiver designs is proposed to achieve a high sum-rate performance at overall SNR regime for 2-user and 3-user interference channels. Other interference alignment algorithms and interference alignment-inspired alogithms have been discussed in [21-25].

B. Contribution

In this paper, we propose a distributed interference alignment algorithm that provides a game-theoretic interpretation for both the interference signal subspace minimization and the sum rate maximization in K-user MIMO interference channels by directly building on the egoistic and altruistic game equilibria. Furthermore, a conductive mathematical and geometrical analysis and interpretation about the egoistic and altruistic cost function for interference channel in the frame of game theory is presented. To obtain optimal system performance, each transmitter will consider its role in the interference channel from the following two aspects. On one hand, each transmitter tries to maximize its data rate by transmitting along those signaling dimensions where the desired receiver anticipates the least interference. On the other hand, each transmitter primarily tries to minimize the interference to undesired receivers. From the perspective of game theory, the former considers the impact of the interference from the egoistic aspect and the latter considers the same problem from the altruistic aspect. Therefore, we can build a game theory framework to interpret the distributed interference alignment problem. It is assumed that each transmitter only knows the local channel state information between communication pairs (local channel state information of a transmitter and its corresponding receiver). A class of games suitable to the case of partial information based decision making, called Bayesian games, can be used in this scenario. The proposed algorithm is different from the most of the existing interference alignment schemes. The existing scheme only consider the power of the interference leakage as effective optimization objective and neglects sum rate maximization at low to intermediate SNR regions. The proposed scheme in this study is a two stage framework that is implemented by: firstly, defining the egoistic and altruistic objective functions, deriving analytically the game equilibrium and interpretation of interference alignment problem by using the mathematical and geometrical analysis; secondly adapting the obtained equilibrium solution to heuristic design of a practical beamforming technique based on the sum rate maximization scheme. The proposed approach combines the interference signal subspace minimization scheme and the sum rate maximization scheme of interference networks with a game-theoretic interpretation based on the egoistic and altruistic game equilibria. The feasibility of proposed algorithm is provided and a comparison of the DoF and sum rate performance with other interference alignment algorithms is studied using simulations. The simulation results show that the proposed scheme can effectively minimize the leakage interference in desired signal subspace at each receiver and provide a moderate average sum rate performance in comparison to several existing interference alignment schemes.

C. Organization and Notations

The rest of the paper is organized as follows. The system model under consideration is presented in next section. In section 3, the Bayesian game model of interference channel is presented using the mathematical and geometrical analysis and interpretation of interference alignment problem. Section 4 describes the game relationship in solving the sum rate maximization problem. In section 5, a calculation method is presented for the interference signal subspace at each receiver. Section 6 explains the convergence criterion and the concrete algorithm of proposed scheme. The simulation results and conclusion are presented in section 7 and section 8, respectively.

Boldface letters denote matrices or vectors, while upper and lower case letters denote scalars. [([]).sup.*] refers to the conjugate transpose of ([]). For a matrix B, [parallel]B[[parallel].sub.F] is the Frobenius norm of B and tr(B) is the trace of B. E[[]] denotes expectation operator. [[].sup.MxN] represents the set of all M x N matrices. [[].sub.j] represents interference subspace at receiver j. [partial derivative](f(W))/[partial derivative](W) represents the derivative for the function f(W).

2. System Model

A multi-user MIMO interference channel with K communication pairs is shown in Fig. 1. The signal of each transmitter can be received by all the receivers; however a given transmitter only intends to have its signal decoded by the targeted receiver. It is assumed that each transmitter is equipped with M transmitting antennas and each receiving is equipped with N receiver antennas. It is also assumed that only one data stream is transmitted by each user and [W.sub.k] [member of] [[].sup.Mx1], k [member of] {1,2, ..., K} is the transmission precoder for transmitter k with unit norm. The received signal at receiver k is given as:

[y.sub.k] = [H.sub.kk][W.sub.k][x.sub.k] + [K.summation over (j[not equal to]k)][H.sub.kj][W.sub.j][x.sub.j] + [Z.sub.k], k [member of] {1, 2, ..., K} (1)

where [H.sub.kj] [member of] [[].sup.NxM] isa frequency-flat fading channel between transmitter j and receiver k for k, j [member of] {1,2, ..., K}, where each entry is assumed as i.i.d complex Gaussian random variables with zero mean and unit variance CN (0,1). [Z.sub.k] [member of] [[].sup.Nx1] is a zero mean circularly symmetric additive white Gaussian noise vector (AWGN) at receiver k, and it has the covariance E[[Z.sub.k][Z.sup.*.sub.k]] = [[sigma].sup.2.sub.k][I.sub.N]. Here, the transmitted symbol [x.sub.k], k [member of] {1,2, ..., K} at the kth transmitter is generated with power budget [P.sub.k]; i.e., E[[x.sub.k] [x.sup.*.sub.k]] = [P.sub.k]. In equation (1), the first term [H.sub.kk][W.sub.k][x.sub.k] is the desired signal vector sent by the kth transmitter and the second term [K.summation over (j[not equal to]k)][H.sub.kj][W.sub.j][x.sub.j] represents the interference from other transmitters. Furthermore, we denote an orthonormal basis vector [V.sub.k] [member of] [[].sup.Nx1] as receive matrix at receiver k. The instantaneous rate of communication pair k can be obtained as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The instantaneous sum rate of the system is:

[bar.R] = [K.summation over (k=1)][R.sub.k] (3)

The discussion in the following section will be based on the channel model presented in Fig. 1.

3. Bayesian game model of interference channel

The Bayesian game's definition of interference channel presented in [12] will be used to construct egoistic Bayesian altruistic Bayesian game cost functions. Particularly, two aspects of the problem will be considered here: 1) In absence of cooperation, each transmitter will "selfishly" choose its beamforming vector to maximize the transmission rate towards its desired receiver; 2) Each transmitter hopes that their transmitted signal causes the least interference to the desired signal at other undesired receivers. The former can be considered as the objective of egoistic Bayesian game cost function, while the latter can be mapped to the objective of altruistic Bayesian game cost function. The methods to construct these two functions are explained in the following subsections.

3.1 Altruistic Bayesian game cost function

Lemma 3.1: Given K-1 arbitrary p-dimensional subspaces [[].sub.j] with respective orthonormal bases [Q.sub.j] and a N x M matrix [H.sub.jk], the vector [F.sub.k] such that B = [H.sub.jk] [F.sub.k], [F.sub.k] [member of] [[].sup.Mx1], which minimizes the squared Euclidean distance from B to the subspaces, is equal to the eigenvector corresponding to the minimum eigenvalue of [K.summation over (j[not equal to]k)][H.sup.*.sub.jk](I - [Q.sub.j][Q.sup.*.sub.j])[H.sub.jk]

The proof of Lemma 1 can refer to the lemma 2 in [11] and is omitted to avoid repetition.

Remark: In interference channel, the impact of transmitter k over K - 1 undesired receivers can be embodied in the sum of the squared Euclidean distances from [H.sub.jk][W.sub.k] (j [not equal to] k, j [member of] {1,2, ... K}) to the K - 1 interference subspaces [[].sub.j] (j [not equal to] k) of undesired receivers. Fig. 2 presents an illustration of relationship between interference signal vectors [H.sub.jk][W.sub.k](j [not equal to] k) (red lines with solid arrows) from kth transmitting signal source and interference subspaces [[].sub.j] (j [not equal to] k) (planes with different colors) at K - 1 interference signal destinations (from 1st Rx to Kth RX). Where [W.sub.k] is precoding vector of signal source from transmitter k, [H.sub.jk] [W.sub.k] (j [not equal to] k) is direction vector of interference signal at undesired destination, and [[].sub.j] (j [epsilon] k) is interference subspace spanned by interference signals at undesired destination. It can be argued that the power leakage from interference subspace of K - 1 undesired receivers due to the transmitter j is driven proportionally with the sum of the squared Euclidean distances. When the sum of the squared Euclidean distance is zero, the interference that transmitter j produces aligns to the interference subspace at K - 1 different undesired receivers

According to the analysis provided in preceeding content, it can be argued that aligning the interference produced by transmitter k to the interference signal subspace [[].sub.j] (j [not equal to] k) at receiver j is an effective way to reduce the interference at undesired receiver j. This is exactly an altruistic action. So, the altruistic Bayesian game cost function can be constructed by calculating the sum of the squared Euclidean distances from interference signal [H.sub.jk][W.sub.k](j [not equal to] k, j [member of] {1,2, ... K}) produced by transmitter k to the K - 1 interference subspaces [[].sub.j] (j [not equal to] k) of undesired receivers:

[[phi].sub.k]([W.sub.k]) = -[K.summation over (j[not equal to]k)][parallel][H.sub.kj][W.sub.j] - [Q.sub.j][Q.sup.*.sub.j][H.sub.jk][W.sub.k][[parallel].sup.2.sub.F] (4)

To obtain the maximum benefit at transmitter k, the altruistic precoding vector [W.sup.Alt.sub.k] can be formulated by maximizing the expression (4):

[W.sup.Alt.sub.k] = arg max [[phi].sub.k]([W.sub.k]) (5)

Then the expression (5) can be rewritten as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

According to Lemma 3.1, we can obtain the solution of expression (6):

[W.sup.Alt.sub.k] = [V.sup.min] ([K.summation over (j[not equal to]k)][A.sub.jk] (7)

where

[A.sub.jk] = [H.sup.*.sub.jk] ([I.sub.N] - [Q.sup.j][Q.sup.*.sub.j])[H.sub.jk] (8)

[V.sup.min](A) represents the eigenvector corresponding to the minimum eigenvalue of A. The [A.sub.jk] is considered as the altruistic Bayesian game cost function between transmitter k and receiver j.

3.2 Egoistic Bayesian game cost function

Lemma 3.2: Given an arbitrary p-dimensional subspace [[].sub.k] with orthonormal base [Q.sub.k] and N x M vector [H.sub.kk], the vector [F.sub.k] such that B = [H.sub.kk][F.sub.k], [F.sub.k] [member of] [[].sub.Mx1], which maximizes the squared Euclidean distance from B to the subspace, is equal to the eigenvector corresponding to the maximum eigenvalues of [H.sup.*.sub.kk](I - [Q.sub.k][Q.sup.*.sub.k])[H.sub.kk].

The proof of Lemma 3.2 is similar as lemma 3.1 therefor it is omitted to avoid repetition here.

Remark: To maximize the transmission rate of the desired signal from transmitter k, the optimal precoding vector at transmitter k should be designed to isolate the desired signal from the interference subspace of corresponding receiver k. Fig. 3 provides an illustration of the relationship between desired signal [H.sub.kk][W.sub.k] (red line with solid arrow) from the transmitter k and the interference signal subspace D k (red plane) at receiver k. We choose the squared Euclidean distance from [H.sub.kk][W.sub.k] to the interference subspace [[].sub.k] of receiver k as the measurement of the extent by which the desired signal is away from the interference signal subspace at receiver k. As mentioned earlier, the interference by which the desired signal suffers from the other undesired transmitters at receiver k decreased with an increase in squared Euclidean distance. When the desired signal is independent of the interference subspace spanned by interference from unintended transmitters, zero forcing approach can be used to recover the desired signal without any interference.

Now, the egoistic Bayesian game cost function can be constucted by calculating the squared Euclidean distance from desired signal [H.sub.kk][W.sub.k] over transmitter k to the interference subspaces [[].sub.k] of desired receiver k:

[[phi].sub.k]([W.sub.k]) = [parallel] [H.sub.kk][W.sub.k] - [Q.sub.k] [Q.sup.*.sub.k] [H.sub.kk][W.sub.k][[parallel].sup.2.sub.F] (9)

To obtain the maximum benefit at transmitter k, the egoistic precoding vector [W.sup.Ego.sub.k] can be formulated by maximizing the expression (9):

[W.sup.Ego.sub.k] = arg max [[phi].sub.k] ([W.sub.k]) (10)

Then the expression (10) can be rewritten as:

[W.sup.Ego.sub.k] = arg max [parallel] [H.sub.kk][W.sub.k] - [Q.sub.k][Q.sup.*.sub.k] [H.sub.kk][W.sub.k][[parallel].sup.2.sub.F] (11)

According to Lemma 3.2, we can obtain the solution of expression (11):

[W.sup.Ego.sub.k] = [V.sup.max]([E.sub.k]) (12)

where

[E.sub.k] = [H.sup.*.sub.kk] (I - [Q.sup.k][Q.sup.*.sub.k]) [H.sub.kk] (13)

[V.sup.max] (A) represents the eigenvector corresponding to the maximum eigenvalue of A. [E.sub.k] denotes the egoistic Bayesian game cost function between transmitter k and receiver k.

4. Game Relationship in Sum Rate

In this section, the precoding vector that maximizes sum rate for the multi-user MIMO interference channel will be solved using the approach of Lagrange multipliers. The relationship will be explored between the sum rate and Bayesian game cost functions mentioned in the preceding section. The description of sum rate for K-user MIMO interference channel in section 2 is given as:

[bar.R] = [K.summation over (k=1)] [R.sub.k] (14)

where [R.sub.k] = [log.sub.2] [1 + [[V.sup.*.sub.k][H.sub.kk][W.sub.kk][W.sup.*.sub.k][H.sup.*.sub.kk][V.sub.kk][P.sub.k]/[K.summation over (i[not equal to]k)][V.sup.*.sub.k] [H.sub.ki][W.sub.i][W.sup.*.sub.i][H.sup.*.sub.ki][V.sub.k][P.sub.i] + [[sigma].sup.2.sub.k]) and the definitions of parameters in this expression are explained in the system model in section 2. The constraint conditions can be employed, expression (14) and [W.sup.*.sub.k] [W.sub.k] = 1, to construct a Lagrange function for solving sum rate maximization problem:

larg([W.sub.k], [mu]) = [bar.R] - [[mu].sub.max] ([W.sup.*.sub.k][W.sub.k] - 1) (15)

where [[mu].sub.max] is the Lagrangian. The necessary condition for maximizing sum rate [bar.R] in the expression (14) is:

[partial derivative]larg ([W.sub.k], [mu])/[partial derivative][W.sup.*.sub.k] = 0 (16)

The above expressions can be derived to:

[partial derivative][R.sub.k]/[partial derivative][W.sup.*.sub.k] + [K.summation over (k=1)][[partial derivative][R.sub.j]/[partial derivative][W.sup.*.sub.k]] - [[mu].sub.max][W.sub.k] = 0 (17)

Substituting (2) into (17):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Simplifying the expression (18):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

According to the relationship between orthonormal base [Q.sub.k] for received interference signal subspace [[].sub.k] and linear receiver [V.sub.k] at receiver k, we can obtain:

[V.sub.*][V.sup.*.sub.k] = [I.sub.N] - [Q.sub.k][Q.sup.*.sub.k] (20)

Multiplying by (1/[P.sub.k])([K.summation over (i=1)] [V.sup.*.sub.k][H.sub.ki][W.sub.i] [W.sup.*.sub.k] [H.sup.*.sub.ki [V.sub.k][P.sub.i] + [[sigma].sup.2.sub.k]) on both sides of expression (19), the following expression can be constructed:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Substituting (8) and (13) into (21)

([E.sub.k] + [K.summation over (j[not equal to]k)][[lambda].sup.opt.sub.jk][A.sub.jk])[W.sub.k] = [[mu]'.sub.max][W.sub.k] (22)

According to expression (21):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

[[mu]'.sub.max] = [([K.summation over (i=1)] [V.sup.*.sub.k][H.sub.ki][W.sub.i][W.sup.*.sub.i][H.sup.*.sub.ki][V.sub.k][P.sub.i] + [[sigma].sup.2.sub.k])/[P.sub.k]][[mu].sub.max] (24)

In expression (22), [E.sub.k] and [A.sub.jk] correspond to the egoistic and the altruistic Bayesian game cost functions. Therefore, the principle of altruism and egoism in sum rate maximization is synthesized through a linear combination. According to the matrix theory, the precoding vector [W.sub.k] for transmitter k is just the dominant eigenvector of the linear combination [E.sub.k] + [K.summation over (j[not equal to]k)][[lambda].sup.opt.sub.jk][A.sub.jk], where [[lambda].sup.opt.sub.jk] plays a role of balancing factor between the egoistic and the altruistic Bayesian game cost functions. The parameter [[lambda].sup.opt.sub.jk] in expression (22) requires the global channel state information, which implies excessive training and a significant amount of feedback overhead in practical system. To eliminate this dependency on global state information, the parameter [[lambda].sup.opt.sub.jk] can be obtained through a suboptimal egoism-altruism balancing method mentioned in [12]. In this paper, the focus is on the feasibility of solution method for the parameter [[lambda].sup.opt.sub.jk] in practical wireless communication systems. Although a suboptimal egoism-altruism balancing method is demonstrated in [12], the parameter [[lambda].sup.opt.sub.jk] is still obtained through a complicated expression that requires a significant amount of channel state information and calculations. Therefore, an empirical estimation scheme for the estimation of parameter [[lambda].sup.opt.sub.jk] is presented in this study to improve our algorithm's application value. Section 7 explains the impact of the parameter [[lambda].sup.opt.sub.jk] on the algorithm across the different performance factors including the average sum-rate, the average number of iterations for our algorithm, the ratio of leakage interference in desired signal space.

5. Calculation of Interference Signal Subspace at Receivers

In the discussion in preceding section, it is assumed that the orthonormal base [Q.sub.k], k [member of] (1, 2, ..., K) for interference signal subspace [[].sub.k] at receiver k is known by transmitter k, thus an appropriate precoding vector [W.sub.k], k [member of] (1, 2, ..., K) can be designed to achieve interference alignment at each transmitter. In this section, the interference signal subspace of each receiver will be solved for fixed precoding vector [W.sub.k], k [member of] (1, 2, ..., K), assuring that the interference signal from undesired transmitters falls into the interference signal subspace as much as possible.

Let us assume that each transmitter knows its local Channel State Information (CSI) and local CSI of the corresponding receiver. The local CSI at each transmitter can be obtained by estimating the pilot signal sent from each receiver. Each transmitter can obtain the local CSI of the corresponding receiver through the feedback link from the corresponding receiver.

Lemma 5.1: Given K - 1 arbitrary vectors [H.sub.jk][W.sub.k] [member of] [[].sup.Nx1] (k [not equal to] j), the p- dimensional subspace [[].sub.j] with minimum overall Euclidean distance to all vectors [H.sub.jk][W.sub.k] (k [not equal to] j) has orthonormal basis [Q.sub.j], where the columns of [Q.sub.j] are the eigenvectors associated with the p largest eigenvalues of [K.summation over (k[not equal to]j)][H.sub.jk][W.sub.k][W.sup.*.sub.k][H.sup.*.sub.jk].

Proof: The proof of Lemma 5.1 is relegated to the Appendix

Remark: At receiver j, the interference suffering from K - 1 unintended transmitters can be embodied in the sum of the squared Euclidean distance from K - 1 matrices [H.sub.jk][W.sub.k] (k [not equal to] j) to the interference signal subspace [[].sub.j], j [not equal to] k. Fig. 4 provides an illustration of relationship between interference signal vectors [H.sub.jk][W.sub.k] (k [not equal to] j) (colorized lines with solid arrows) from K - 1 undesired transmitting signal sources (from 1st Tx to Kth TX) and interference subspaces [[].sub.j] (j [not equal to] k) (planes with different colors) at receiver j. It is argued that the smaller the sum of the squared Euclidean distance is, the less power of the interference that K - 1 undesired transmitters produce at receiver j leaks out from interference subspace of the jth receiver. When the sum of the squared Euclidean distance is equal to zero, the interference from K - 1 unintended transmitters aligns to interference subspaces [[].sub.j] at receiver j. Fig. 3 illustrates the relationship between K - 1 interference vectors [H.sub.jk][W.sub.k] (k [not equal to] j) and interference signal subspace at receiver j. According to the Lemma 5.1, the optimal orthonormal base [Q.sub.k], k [member of] (1, 2, ..., K) for interference signal subspace [[].sub.k] at receiver k can be obtained by expression (30) in the Appendix.

6. Convergence criterion and algorithm

At receiver j, the total leakage-interference caused by K - 1 undesired transmitters (k [not equal to] j) is given by:

[IL.sub.j] = [V.sup.*.sub.j][O.sub.j][V.sub.j] (31)

where

[O.sub.j] = [K.summation over (k=1,k[not equal to]j)] [P.sub.k][H.sub.jk][W.sub.k][W.sup.*.sub.k][H.sup.*.sub.jk] (32)

and [V.sub.j] is linear receiver at receiver j.

The metric representing average interference ratio in desired signal space is defined as:

[MeanR.sub.IL] = [1/K][K.summation over (j=1)][[O.sub.j]/[IL.sub.j]] (33)

The value of [MeanR.sub.IL] is reduced by each iteration of the algorithm Since [MeanR.sub.IL] is bounded below a threshold value, this implies that the algorithm must converge. According to the convergence criterion, the proposed algorithm can be described as follows:

Step 1: start with K arbitrary M x 1 precoding vectors [W.sub.k](k = 1, 2, ..., K) at transmitters and guarantee [W.sub.k][W.sup.*.sub.k] = I(k = 1, 2, ..., K) .

Begin iteration.

Step 2: calculate the interference signal subspace according to (30) and obtain the linear receiver according to (20) for each receiver.

Step 3: calculate the altruistic Bayesian game cost function [A.sub.jk](j [not equal to] k, j = 1, 2, ..., K) between the transmitter k and each undesired receiver j according to (8) and the egoistic Bayesian game cost function [E.sub.k] between the transmitter k and the receiver k according to (13).

Step 4: calculate the precoding vectors [W.sub.k](k = 1, 2, ..., K) at transmitters according to (22), the rearrangement of the equation is given as:

[W.sub.k] = [V.sup.min] ([E.sub.k] + [K.summation over (j[not equal to]k)] [[lambda].sup.opt.sub.jk] [A.sub.jk]) (34)

where [[lambda].sup.opt.sub.jk] is determined by statistical channel information.

Step 5: go to step 2.

Step 6: continue until parameter [MeanR.sub.IL] converges.

7. Simulation result

In this section, the numerical evaluation performance of the proposed algorithm is presented considering two major factor: 1) average sum-rate; 2) average ratio of leakage interference in desired signal space. The algorithm realization for 3-user interference channel has been considered with 2 antennas at each node. The transmission power budgets are set to 1 for all the transmitters and noise power [[sigma].sup.2.sub.k] is set equally for each transmitter, where [[sigma].sup.2.sub.k] = 1/([10.sup.SNR/10]). An uncorrelated fading channel model is used with channel coefficients generated from the complex Gaussian distribution CN(0,1). Each simulation result is averaged over 100 random channel realizations and the permitted maximum number of iterations for the proposed algorithm is 2000. In these simulations, an average of the instantaneous sum-rate (we call it average sum-rate in simulations) defined in formula (3) is chosen and average ratio of leakage interference to desired signal space defined in formula (33) is selected as the performance metrics.

According to Section 4, the parameter [[lambda].sup.opt.sub.jk] in (23) has been defined as a balancing factor between the egoistic Bayesian game cost function and the altruistic Bayesian game cost function. The parameter [[lambda].sup.opt.sub.jk] used in this study is given as:

[[lambda].sup.opt.sub.jk] = [lambda] * ([10.sup.SNR/10]), (j, k) [member of] (1,2, ..., K) (35)

The performance of the proposed algorithm is evaluated with different values of parameter [lambda] in (35) to obtain the optimal balancing factor [[lambda].sup.opt.sub.jk].

In Fig. 5 and Fig. 6, different markers are used to present each of the cases of [lambda] = 0.01, 0.1, 1, 10, 100. Fig. 5 shows a plot of the average sum-rate values versus the SNRs for the 3-user interference channel with 2 antennas at each node in response to different values of parameters [lambda]. Fig. 6 illustrates the variation of average ratio of leakage interference in desired signal space under different values of parameter [lambda] as the values of SNR increase in the same scenario as the Fig. 5. It can be observed in Fig. 5 that the proposed algorithm provides almost the best performance on the average sum-rate when [lambda] = 10. From the Fig. 6, It can be noticed that the least average of leakage interference in desired signal space is obtained in the case of [lambda] = 100 ; and the performance gap between [lambda] = 100 and [lambda] = 10 is very small. Combining the results from these two simulations, the value 10 * ([10.sup.SNR/10]) is selected for [[lambda].sup.opt.sub.jk] as the optimal balancing factor for 3-user interference channel with 2 antennas at each node.

In the coming simulations, the performance of the proposed algorithm using the selected balancing factor is compared with the following algorithms on beamformer design for multiuser interference channel:

* Reciprocal IA-CJ08 [10]

* Alternating-Minimization [11]

* MMSE [13]

* Max SINR IA-CJ08 [10]

* WMMSE [14]

In Fig. 7, the performance on the average sum-rate for 3-user interference is measured with 2 antennas at each node, where the results from all of the above mentioned algorithms are averaged over 100 random channel realizations while all other configurations are same as in the simulation in Fig. 5. Further, it must also be noted that the number of spatial dimensions (i.e., antennas) is equal to two for all the algorithms. The result shows that the proposed algorithm produced almost similar sum-rate performance in comparison to Reciprocal IA-CJ08 and Max SINR IA-CJ08 at high SNR, which implies that the proposed scheme can achieve same total DoF as achieved by these interference alignment schemes with perfect CSI. The sum-rate advantage of the algorithms such as Reciprocal IA-CJ08, Max SINR IA-CJ08 and the proposed algorithm over others such as MMSE and WMMSE at high SNR is due to the interference alignment in the vector space. However, the Reciprocal IA-CJ08 and Max SINR IA-CJ08 require the reciprocity of channel and perfect CSI which indicate that both algorithms have limited application scenario in realistic system. Although the proposed algorithm does not produce best performance on the average sum-rate, it takes advantage of reduced external constrains over other two algorithms, hence it becomes a better alternative in practical applications. From the graph in Fig. 7, it can be seen that MMSE and WMMSE have better performance in low SNR that indicates a suboptimal "selfish" approach where a transmitter ignores the interference it causes and aims simply to maximize desired signal rate. These algorithms have their own advantages in low SNR's scenarios.

Fig. 8 illustrates the covergence behaviors of the proposed algorithm, Reciprocal IA-CJ08 algorithm [10], MMSE algorithm [13], Max SINR IA-CJ08 algorithm [10] and WMMSE algorithm [14] for the case where SNR = 30 (dB). The graph shows that the proposed algorithm and Reciprocal IA-CJ08 algorithm have better performance and converge in about 23 steps, where fast convergence means low algorithm overload.

Fig. 9 presents the performance of different algorithms on average ratio of leakage interference in desired signal space. It can be observed that the Reciprocal IA-CJ08 algorithm has significantly better performance compared to other algorithms. This is due to the fact that the Reciprocal IA-CJ08 aims to maximize the achievable DoF by minimizing the leakage interference in desired signal space. Other algorithms that address the maximization of the SINR or Sum-Rate have shown worse performance on leakage interference. By analyzing the structures of these algorithms such as MMSE and WMMSE, it is noticed that these algorithms always egoistically maximize the rate for data streams of desired signal by increasing the power of transmitted and these does not altruistically suppress the interference to other undesired receivers. From the plot in Fig. 7, it can be seen that the performance of the proposed algorithm is very close to the optimal Reciprocal IA-CJ08 algorithm almost at all the SNRs. The simulation results indicate that the proposed algorithm try to maximize the sum-rate of the 3-user interference channel while keeping the leakage interference in the desired signal space to minimum.

8. Conclusion

In this paper, a numerical algorithm to achieve spatial interference alignment for K-user interference channel has been presented that is based on the framework of game theory. The proposed algorithm accounts for the impact of interference management scheme on both DoF and sum rate performance for K-user interference channel. The achievable DoF of the interference channel and the average sum rate of systems have been optimized to provide a better alternate to existing interference alignment schemes. While the signal subspace analytical approaches can only solve the minimization problem of the leakage interference power from the perspective of DoF, and the conventional optimization approaches are always used to deal with the sum rate maximization problem of system. The main contribution of this work is the design of a numerical interference alignment algorithm aiming to maximize sum rate with signal subspace analytical method and game theory. Furthermore, in the proposed algorithm only requires each transmitter only require the information about the channel between the corresponding communication pairs to obtain the better performance, which implies that this algorithm can be used in a distributed manner in realistic systems. In the future work, we will consider how we could utilize the other interference management approaches to obtain a medium performance if each transmitter only knows the local channel state information.

Appendix

The problem is formulated by minimizing the sum of the squared Euclidean distances between all vectors [H.sub.jk][W.sub.k](k [not equal to] j) and their orthogonal projections onto [[].sub.j]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

where [Q.sub.j] is an orthonormal basis of [[].sub.j]. According to matrix theory, square of Frobenius norm has the following property:

[parallel]B[[parallel].sup.2.sub.F] = ([n.summation over (i=1)] [m.summation over (j=1)] [[absolute value of [b.sub.ij]].sup.2]) = Tr ([B.sup.*] B) (26)

Expression (25) can be extended as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

In expression (27), the first term of Tr(x) is constant and only the second term of Tr(x) is related to [Q.sup.opt.sub.j], it can lead to the following expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

Due to Tr(AB) = Tr(BA) in linear algebra, expression (28) can be rewritten as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

According to the matrix theory, the solution of [Q.sup.opt.sub.j] for expression (29) is equal to the p dominant eigenvectors of [K.summation over (k[not equal to]j)] [([H.sub.jk][W.sub.k]).sup.*] [H.sub.jk][W.sub.k], so the columns of [Q.sup.opt.sub.j] can be given by:

[Q.sup.opt.sub.j] = [V.sup.max.sub.p] ([K.summation over (k[not equal to]j)][H.sub.jk][W.sub.k][([H.sub.jk][W.sub.k]).sup.*]) (30)

where [V.sup.max.sub.p](A) represents the sets of eigenvectors corresponding to the p maximum eigenvalues of A.

This work was supported in part by the Sino-Korea Joint Research Project under grant 2012DFG12250, the Key Project of National Natural Science Foundation of China under grant 61231007, the National Natural Science Foundation of China under grant 61201168, the National High Technology Research and Development Program of China (863 Program) under grant 2012AA121604, the International Science and Technology Cooperation Program of China under grant 2012DFG12010, the National Science and Technology Major Project of the Ministry of Science and Technology of China under grant 2011ZX03003-001-04.

http://dx.doi.org/10.3837/tiis.2014.06.007

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Shixin Peng (1), Yingzhuang Liu (1), Hua Chen (2) and Zhengmin Kong (3)

(1) Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074--China

[e-mail: psx6050@163.com, liuyz@mail.hust.edu.cn]

(2) College of Mathematics and Computer Science, Wuhan Textile University, Wuhan, Hubei 430073--China

[e-mail: 562054584@qq.com]

(3) Department of Automation, Wuhan University, Wuhan, Hubei 430072--China

[e-mail: zmkong@whu.edu.com]

* Corresponding author: Shixin Peng

Received January 2, 2014; revised March 13, 2014; accepted May 15, 2014; published June 27, 2014

Shixin Peng is currently working towards the Ph.D.degree in communication engineering from Huazhong University Science & Technology, Wuhan. His research interests interference management scheme in MIMO interference channel, capacity analysis in multiuser communication system, 4G wireless communication system, signal processing.

Yingzhuang Liu received his Ph.D. degree in communications and information system in 2000 from Huazhong University of Science and Technology. He was the post doctor at University of Paris-Sud 11, France, from 2000 to 2001. He was the head of Chinese subgroup in the ALICE collaboration in European Organization for Nuclear Research (CREN) and the member of the PHENIX collaboration in Brookhaven National Laboratory (BNL). He is the member of China Broadband Wireless IP Standard Group and the expert of communications and intelligent networks for China National Ministry of Science and Technology. Now he is the Professor at Huazhong University of Science and Technology and Wuhan National Laboratory for Optoelectronics in China.

Chen Hua is a associate professor in Wuhan Textile University. Her main research field is Network Communications, include optimization theory and technology, etc. From 2005 to now, her presided more than 3 Hubei natural science fund projects and more than 2 Hubei youth fund projects, published more than 15 papers in the field of network communications.

Zhengmin Kong received the B.Eng. (in 2003) and Ph.D (in 2011) degree in the Department of Electronics and Information Engineering from Huazhong University of Science and Technology (HUST), Wuhan, China. From 2005 to 2011, he was with the Wuhan National Laboratory for Optoelectronics (WNLO) as a member of research staff and was involved in beyond-3G (B3G) and UWB system design. He is currently a lecturer in the Department of Automation, Wuhan University. His current research interests include interference management, UWB wireless communication, signal processing.
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