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Cyclic interference alignment for MIMO interference channels: A hybrid approach of MTLI and PSO
Applied Soft Computing 50 (2017) 158–165
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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Review Article
Cyclic interference alignment for MIMO interference channels: A hybrid approach of MTLI and PSO夽 Hoang-Yang Lu Department of Electrical Engineering, National Taiwan Ocean University, Keelung, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 1 September 2015 Received in revised form 10 November 2016 Accepted 10 November 2016 Available online 17 November 2016 Keywords: Interference alignment Particle swarm optimization Multiple-input-multiple-output Minimum total leakage interference
a b s t r a c t The paper presents an interference alignment (IA) scheme for wireless systems with multiple-inputmultiple-output (MIMO) interference channels. The scheme executes the sum rate optimization task in an iterative cycle. Specifically, in each cycle, the minimum total leakage interference (MTLI) method is used initially to find a tentative solution candidate. Next, the particle swarm optimization (PSO) method uses the tentative candidate to assist the joint search for feasible solutions that will maximize the total sum rate. Then, at the end of each cycle, the best solution candidate derived by PSO is passed to the next cycle to help the MTLI mechanism refine the candidate. The cyclic cooperation among the mechanisms of MTLI’s receivers, MTLI’s transmitters, and PSO allows the candidates to be refined cycle by cycle. The results of simulations and complexity analysis show that the proposed IA scheme not only reduces the computational complexity significantly, but also achieves a better performance gain than four existing approaches. © 2016 Elsevier B.V. All rights reserved.
Contents 1. 2. 3.
4.
5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 The proposed scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.1. Minimum total leakage interference, MTLI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.2. Steps of the proposed scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Simulations and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.1. Convergence of the proposed scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.2. Performance comparison with existing methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.3. Comparisons of computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
1. Introduction Because of the rapidly increasing demand for quality of service (QoS), current and future wireless communication systems must provide superior performance, as measured by high data rates and low power consumption [1–4]. To this end, several researchers have developed promising methods. Multiple-input-multiple-output (MIMO), which deploys multiple antennas at both the transmitter and receiver sides, is one such method [5,6]. It has been shown that
夽 This work was supported by Ministry of Science and Technology of R.O.C. under contract MOST 103-2221-E-019-014. http://dx.doi.org/10.1016/j.asoc.2016.11.017 1568-4946/© 2016 Elsevier B.V. All rights reserved.
MIMO can effectively enhance the performance without the allocation of extra frequency resources [7]. The significant performance enhancement of MIMO has generated a great deal of research interest [8–15]. In particular, a new wireless communication system, called the MIMO interference channel, has been investigated extensively [16,17]. On a MIMO interference channel, each transmitter sends its signal to its corresponding receiver. However, because of the radio propagation feature, the signal affects other receivers simultaneously and is referred to as interference. According to the literature, interference is the main factor that limits system performance [7]. Therefore, developing feasible methods to effectively alleviate the interference impairment of MIMO interference
H.-Y. Lu / Applied Soft Computing 50 (2017) 158–165
channels is an important aspect of satisfying the rapidly growing QoS demands of current and future wireless communications. Interference alignment (IA) is one of the feasible schemes that can effectively reduce the interference effect of MIMO interference channels [18]. In practice, IA uses MIMO mechanisms to align the unwanted interference sent by users to a small dimensional subspace at each receiver to help mitigate the interference. Furthermore, it has been shown that IA can achieve the maximum degree of freedom (i.e., spatial multiplexing gain) on MIMO interference channels [16,17]. The efficiency of IA to alleviate interference has motivated several researchers to develop IA algorithms to further enhance the performance of MIMO interference channels [10–15]. For example, Dahrouj et al. [10] proposed an IA method to find the downlink beamforming vectors. The objective is to either minimize the total weight of the transmitted power or maximize the per-antenna power across all base stations subject to the signal-to-interference-and-noise-ratio (SINR) constraints of remote users. As the uplink beamforming problem is easier to solve, the authors used the Lagrangian duality theory to transform it into an uplink problem. However, the duality method inherently limits the beamforming solutions of IA, such that they are only suitable for time-division duplex (TDD) systems. In [11], instead of maximizing the SINR, an iteratively weighted minimum mean square error (MMSE) IA method was proposed. The goal of the method is to design linear transceivers that maximize the system throughput of MIMO interference channels. Peters and Heath [12] proposed an iterative subspace method called minimum total leakage interference (MTLI), which tries to minimize the total leakage interference of MIMO interference channels. To further improve MTLI’s performance, the authors proposed two IA schemes called minimum interference plus noise leakage (MINL) and MMSE respectively [13]. The IA problems considered in [11–13] are all non-convex initially, so it is not easy to find joint solutions of IA concurrently. To solve the problem, [14–16] exploited the alternating minimization approach [14] to find one IA solution at a time, while ensuring that the other IA solutions remain unchanged. As a result, the above-mentioned schemes may incur some computational overhead and can lead to suboptimal solutions. To reduce the computational complexity, two low complexity IA algorithms were proposed in [15]; however, they only derive suboptimal solutions because they are still linear approaches. Recently, a computer intelligence approach called particle swarm optimization (PSO), which is based on observations of bird flight, was proposed in [19]. PSO has been used effectively in many engineering fields, including control [20,21], communications [9,22], and signal processing [23,24]. Because it only searches for parts of the solutions in a solution space, PSO can complete the solution search process quickly. Hence, in general, PSO only requires low computational complexity to find optimal or near-optimal solutions [19]. Furthermore, because of its inherent mechanism for searching joint solutions, PSO can solve nonlinear problems easily. It has also been shown that PSO usually performs better than other computational intelligence methods, such as the genetic algorithm (GA) [25], in some applications [19,26]. As a result, PSO has generated a great deal of research interest in recent years. This paper presents a novel IA scheme for MIMO interference channels. Specifically, a hybrid approach of MTLI and PSO is used to find joint IA solutions iteratively. The proposed scheme performs the sum rate optimization task in an iterative cycle. In each cycle, the MTLI method is used initially to find a tentative solution candidate. Then, PSO exploits the tentative candidate to help jointly search for feasible solutions in order to maximize the total sum rate. At the end of each cycle, the best solution candidate derived by PSO is passed to the next cycle to help the MTLI mechanism find a more feasible tentative candidate. Because of the cooperation between
159
the MTLI and the PSO, the candidates can be refined cycle by cycle. To evaluate the performance of the proposed IA scheme, intensive computer simulations and complexity analysis are conducted. The results show that the scheme outperforms some existing works and has lower computational complexity. The remainder of this paper is organized as follows. Section 2 describes the system model. In Section 3, the proposed hybrid MTLIPSO method is discussed for joint IA. In Section 4, the results of computer simulations and complexity analysis using the proposed scheme are compared to the results from four existing methods. Section 5 contains our concluding remarks.
2. System model Consider a K-user MIMO interference channel as shown in Fig. 1, where each user utilizes M antennas to send the individual d symbol signals over the fading channel to the corresponding receiver with N antennas. For ease of description, the parameter setting of the MIMO interference channel considered is denoted as (M, N, d)K . Because of the radio propagation characteristic, each transmitted signal impinges on the intended receiver as well as the other K − 1 receivers. This means that each receiver will receive the signal from the corresponding transmitter and the interference from the other K − 1 transmitters. The system assumes that (1) all the channels are independent with complex Gaussian distributed coefficients, whose effect is identical to the multipath Rayleigh fading channel, commonly used in the literature [11,13,15,18]; (2) the users are all stationary, (i.e. they are not moving during the transmissions); and (3) the channel fading coefficients remain fixed in a symbol duration and vary independently symbol by symbol. In addition, for ease of derivation, the transmissions of the K users are assumed to be synchronized, using schemes like those in [7]. The transmitted signal of user k can be expressed as sk = Tk bk , where bk and Tk are user k’s d × 1 transmitted symbol vector and M × d precoder matrix respectively. It is assumed that each symbol of bk is independent and identically distributed such that E(bk bH k ) = Id . E(·), Id , and (·)H denote the expectation operation, d × d identity matrix, and Hermitian transpose operation respectively [27]. Furthermore, the transmission power of user k is also assumed to be subject to pk , i.e. ||Tk ||2F = pk , where ||A||F is the Frobenius norm of matrix A [27]. In addition, for ease of derivation, it is also assumed that the transmission power for each user is the same, i.e. p1 = p2 = · · · = pK . The signal sk is then transmitted through the transmit antennas over the fading channel to the intended receiver k. As a result, the base band equivalent signal received at the receiver k is [13]
rk = Hkk Tk bk +
K
Hki Ti bi + nk ,
1 ≤ k ≤ K,
(1)
i=1,i = / k
where Hki is the N × M channel matrix between transmitter i and receiver k; and nk is the additive white Gaussian noise (AWGN) vector with the complex normal distribution, CN(0, k2 IN ). For ease of analysis, all {Hki } are assumed to be full rank and mutually independent. It is also assumed that the variance of AWGN received at all receivers is the same, i.e. k2 = 2 , 1 ≤ k ≤ K. Note that the first two terms on the right-hand side of Eq. (1) represent, respectively, the desired signal and the interference from the other K − 1 users. A number of studies have shown that the system performance is limited by the interference [7]. To mitigate the interference effect, IA designs the precoders {Tk }, 1 ≤ k ≤ K to align the interference so that it lies in the small dimensional subspace of the receivers. Because of the alignment mechanisms, IA can improve the performance of MIMO interference channels
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Fig. 1. Block diagram of the MIMO interference channel.
effectively [18]. After determining the precoders, the instantaneous sum rate of the system can be calculated easily as follows [13]:
R(T1 , . . ., TK ) =
K
H log |IN + C−1 Hkk Tk TH k Hkk |, k
(2)
k=1
where
Ck =
K
H 2 Hki Ti TH i Hki + IN .
(3)
i=1,i = / k
Fig. 2. Block diagram of cyclic interference alignment.
denotes the inverse matrix of Ck . As the sum rate is one of C−1 k the most important measurements of system performance, a new IA method was proposed, which can concurrently find the joint precoders for the maximization of the sum rate in Eq. (2). However, the task is difficult because searching for the joint IA precoders in Eq. (2) is a nonlinear problem. Therefore, some researchers have simplified the problem or re-formulated it by using other metrics, such as MMSE. As a result, solutions of existing approaches are only suboptimal.
The goal of MTLI is to minimize the total leakage interference and then use each receiver’s decoder Dk to remove the receiver’s residual interference. Therefore, the problem for MTLI is formulated as follows [12]:
3. The proposed scheme
arg
3.1. Minimum total leakage interference, MTLI
min{Ti },
K {Dk } k=1
In this section, the proposed hybrid MTLI-PSO scheme is described, which is designed to determine the joint IA precoders for MIMO interference channels. Fig. 2 shows the structure of the scheme. The main function of the scheme is to search for joint IA solutions by using MTLI and PSO in an iterative cycle. In each iteration, MTLI is used to find an initial candidate, which is then provided to PSO for concurrently finding all the precoders in order to maximize the system sum rate in Eq. (2). At the end of each iteration, the best candidate found by the PSO mechanism is passed to the next cycle to assist MTLI to refine the candidate. Since the candidates are refined iteratively in a cyclic manner, the proposed scheme is thus named as cyclic interference alignment. In the following subsections, MTLI is briefly reviewed and then the steps of the proposed scheme are described in detail.
s.t.
TH i Ti =
DH k Dk = Id ,
pi I , d d
E||DH k
K
Hki Ti bi ||2F ,
(4)
i=1, i = / k
(5) (6)
where 1 ≤ i, k ≤ K. The constraint in Eq. (5) is used to ensure that the transmitted power is uniformly allocated across all transmit antennas of user i; while the constraint in Eq. (6) builds the orthonormal detectors for user k’s receiver. Because the precoders {Tk } and the decoders {Dk }, 1 ≤ k ≤ K are mutually dependent in Eq. (4), it is difficult to find a closed-form solution directly. To resolve the problem, Chae et al. used the alternating minimizing approach [14] to iteratively search for the precoders and the decoders [12]. According to
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the results in [12], the decoder solution for user k at the nth iteration is d
Dk (n) = {min e.v. (
K
Hki Ti (n − 1)Ti (n − 1)H HH ki )},
(7)
i=1,i = / k d where 1 ≤ i, k ≤ K; min e.v. (X) denotes the eigenvectors of matrix X whose corresponding eigenvalues are the d minimum ones; and (X) represents the normalization operation of matrix X’s column vectors for the constraint in Eq. (6). Then, with the solutions {Dk (n)} given in Eq. (7), the solution for the precoder of user k at the nth iteration is d Tk (n) = {min e.v. (
K
Furthermore, the new precoders {Tk (i)} of MTLI will be passed to the PSO mechanism to find better joint solutions for the precoders. Step 4: Evaluation and Update of the PSO: In the ith iteration, PSO first replaces the particle’s position with the lowest sum rate value by the new precoders {Tk (i)}, k = 1, . . ., K of MTLI found in Step 3. Next, the P particles’ positions {P1l (i)}, . . ., {PPl (i)}, l = 1, . . ., K are individually substituted into (2) to compute their corresponding sum rates. Then, the P sum rates of the particles are sorted, the position of the particle with the highest sum rate is chosen as the global optimum position for the precoders, denoted as Gl , l = 1, . . ., K. In addition, the local optimum position of the jth particle in the ith iteration is updated by j
H HH ik Di (n)Di (n) Hik )},
(8)
i=1,i = / k
where {X} denotes the normalization operation of matrix X’s column vectors to satisfy the constraint in Eq. (5) [27]. The alternating minimizing method [14] updates Eqs. (7) and (8) alternately to find the solutions iteratively until it reaches the prescribed number of iterations or a feasible MTLI result in Eq. (4). However, this alternating approach is a linear method that finds the precoders or decoders separately. As a result, MTLI’s performance deteriorates and the solutions are generally sub-optimal. 3.2. Steps of the proposed scheme Steps of the proposed scheme, shown in the block diagram in Fig. 2, are depicted in detail as follows.. Step 1: Initialization: First, for the initial decoders of MTLI, the proposed scheme chooses an arbitrary orthonormal basis for the precoders {Tk (0)}, 1 ≤ k ≤ K of the transmitters. The initial precoders {Tk (0)} will be used in Step 2 to compute Eq. (7) for the decoders {Dk (1)} in the first iteration. Ideally, with the help of the precoders {Tk } and the decoders {Dk }, the output signal of the decoders should be derived primarily from user k’s transmitted signal. This is because the interference is orthogonal to the subspace of user k’s decoder, so it will be removed completely at user k. For the PSO task, particles are created as the swarm’s initial population. The particles are the solution candidates for the precoders and will be refined cycle by cycle during the evolutional process of the proposed scheme. In addition, the following PSO feature vectors are assigned to each particle: the initial position, local optimal position, and velocity. The initial posij tion of the jth particle is denoted as {Pl (1)}, l = 1, . . ., K, j = 1, . . ., P whose elements are complex Gaussian random variables with disj tribution CN(0, 1). The column vectors of {Pl (1)}, l = 1, . . ., K, j = 1, . . ., P should be normalized in each iteration to satisfy the power constraint in (5). Particle j’s local optimal position and velocity are j j denoted as {Ll }, and {Vl (1)}, l = 1, . . ., K, j = 1, . . ., P, respectively. j
j
Moreover, for the initialization, let Ll = Pl (1) and the initial velocity j Vl (1)
= 0, l = 1, . . ., K, j = 1, . . ., P. vector of particle j be zero, i. e . Step 2: Update of MTLI’s decoders: In the ith iteration, i = 1, . . ., G, MTLI initially uses the PSO mechanism’s best candidate in the (i − 1)th iteration as the precoders. Next, with the help of the best precoder candidate, MTLI computes Eq. (7) to find better decoders {Dk (i)} for the receivers. The derived decoders are then passed to the MTLI precoder mechanism to find better precoders. Step 3: Update of MTLI’s precoders: In this step, MTLI uses the decoders {Dk (i)} found in Step 2 to compute the new precoders of Eq. (8). As the decoders derived in the ith iteration are usually better than those of the previous iterations, the new precoders {Tk (i)} may be better than those obtained earlier. Thus, the precoders can be refined iteration by iteration.
161
j
j
j
If R(L1 , . . ., LK ) < R(P1 (i), . . ., PK (i)) j
j
(9)
then Ll = Pl (i), l = 1, . . ., K, where j = 1, . . ., P and R( · ) is the sum rate function in (2). It is noted that error propagation may occur in the proposed scheme due to improper tentative candidates being created and delivered to the next step or the next iteration. To avoid error propagation, in step 4, only the worst particle position with the lowest sum rate value is replaced by the new precoders found via MTLI in Step 3. As a result, most of the better particles found in the previous iteration are still kept to help PSO refine the joint candidate results. In addition, as shown in Eq. (9), PSO updates the local optimum positions only when the new candidates are better. This update mechanism implies the results derived by PSO at least do not become worse, thus reducing the occurrence of error propagation. After the above evaluation of PSO, the local optimum positions of j the particles, {Ll }, l = 1, . . ., K, j = 1, . . ., P, and the global optimum positions, Gl , l = 1, . . ., K are determined. Next, PSO updates the process to refine the particles’ positions and corresponding velocities. For the update of particle j’s velocity in the ith iteration, j
j
j
j
Vl (i + 1) = ˛Vl (i) + ˇX1 (Ll − Pl (i)) j
+ X2 (Gl − Pl (i)), l = 1, . . ., K,
(10)
where j = 1, . . ., P; ˛ is the inertial weight; ˇ and are the movement weights of the local and the global optimum positions respectively; X1 and X2 are M × d matrices, whose entries are random variables uniformly distributed between 0 and 1; and is the Hadamard product operator [27]. Then the corresponding particle j’s positions for the next (i + 1)th iteration can be updated as follows: j
j
j
Pl (i + 1) = Pl (i) + Vl (i + 1)t, l = 1, . . ., K,
(11)
where j = 1, . . ., P; and t is the time duration, which is usually set at 1. Note that after the positions have been updated, they must be normalized to satisfy the power constraint in Eq. (5). Repeat/End: Steps 2 to 4 are repeated until the number of iterations is G, i.e. i = G. Finally, the global optimum position Gl , l = 1, . . ., K derived by the PSO mechanism is chosen as the solution of the precoders {Tk } for the IA problem. This means the PSO particle with the highest sum rate is the final precoder solution of the proposed IA scheme. 4. Simulations and discussion To evaluate the proposed scheme, computer simulations and complexity analysis are considered. For ease of presentation, the channels for all users are assumed to be complex Gaussian fading [11,13,15,18]. The channel coefficients remain fixed during symbol duration and vary independently symbol by symbol. In addition, it is also assumed that the channel state information is perfectly estimated at the corresponding receivers and shared with the other
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Fig. 3. Convergence of the proposed scheme.
transmitters and receivers in the system. For simplicity, the transmission power of each user is the same, denoted as p. Based on the setting in [13], the number of iterations for MTLI [12], MINL ¯ = 100. Fig. 3 shows the conver[13], and MMSE [13] is all set as G gence of the proposed scheme; and Figs. 4–7 show the scheme’s sum rate performance under different system settings. All the simulation results shown in these figures are obtained by averaging individual results over 256,000 independent trials.
Table 1 Parameters of the PSO mechanism in the proposed scheme. Parameter
Value
˛ ˇ, t Minimum velocity Maximum velocity The number of iterations, G
0.9 0.5 1.0 −3.0 3.0 8
4.1. Convergence of the proposed scheme First, simulations are conducted to assess the convergence performance of the proposed scheme. The parameter settings of PSO in the proposed scheme are listed in Table 1. Two scenarios with
system settings (M, N, d)K = (2, 2, 1)3 and (M, N, d)K = (4, 4, 2)3 are considered respectively. Fig. 3 shows the corresponding convergence performance with the sum rate versus the numbers of iterations. From the figure, we observe that when the transmission
Fig. 4. Sum rate performances of the proposed scheme and four existing methods, system setting: (2, 2, 1)3 .
H.-Y. Lu / Applied Soft Computing 50 (2017) 158–165
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Fig. 5. Sum rate performances of the proposed scheme and four existing methods, system setting: (4, 4, 2)3 .
power of each user, p = 24 dB in the two scenarios, the sum rate performance gain of the proposed scheme increases significantly from the 1st to the 7th iteration. In addition, the performance gain almost saturates after the 8th iteration. Hence, the proposed scheme tends to converge at the 8th iteration when p = 24 dB. Fig. 3 also shows that when p = 12 dB, the proposed scheme saturates at about the 14th iteration for (M, N, d)K = (2, 2, 1)3 and at about the 20th iteration for (M, N, d)K = (4, 4, 2)3 . The results imply that the proposed scheme requires more iterations for lower transmission power scenarios. Conducting more iterations would obviously increase the computational complexity. Setting the number of iterations, G, to be 8 in the following simulations achieves a satisfactory trade-off between computational complexity and sum rate performance.
4.2. Performance comparison with existing methods Next, the sum rate performance of the proposed scheme with four existing methods, namely, MTLI [12], MINL [13], MMSE [13], PSO [19] are compared. For fairness, the parameter settings of PSO and the proposed scheme’s PSO mechanism were the same, as shown in Table 1. In the first proper scenario, the system setting is (M, N, d)K = (2, 2, 1)3 . Based on the conclusion in [28], this setting satisfies the general feasibility condition, i. e . M + N − (K + 1)d ≥ 0. The population size of both PSO and the proposed scheme’s PSO mechanism is also set as 6. (i. e . P = 6). Fig. 4 shows the sum rate performance versus the transmission power of each user, p, for the proposed scheme and four existing methods. The results show
Fig. 6. Sum rate performances of the proposed scheme and four existing methods, system setting: (4, 4, 3)3 .
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Fig. 7. Sum rate performances of the proposed scheme and four existing methods, system setting: (3, 3, 3)3 .
that the proposed scheme significantly outperforms the other four methods. PSO yields the worst performance even though it executes the joint search for IA precoders iteratively. This is because PSO lacks a mechanism to create better initialization to improve the performance. MTLI, MINL, and MMSE do not perform as well as the proposed scheme because they find IA solutions in a linear manner. In the second proper scenario, the system setting (M, N, d)K = (4, 4, 2)3 is considered, which still satisfies the general feasibility condition [28]. For this setting, the population size P for both PSO and the proposed scheme’s PSO mechanism is set as 10. As shown in Fig. 5, all the five methods yield better sum rate gains than those in Fig. 4. This is because both the number of transmit and receive antennas increase from 2 to 4. Like the first proper scenario, the proposed scheme outperforms the other four methods when the transmission power p ≥ 3 dB; and PSO still yields the worst performance gain. To gain more insight into the proposed scheme’s performance, the improper scenario with the system setting (M, N, d)K = (4, 4, 3)3 is considered. First, based on the conclusion in [28], the setting does not satisfy the general feasibility condition, i.e. M + N − (K + 1)d ≤ 0. This means the signal space is too small to provide a feasible space for signal transmission. In this scenario, the population size P is set as 16 for both PSO and the proposed scheme. Compared to the other four methods, the proposed scheme still achieves a significant performance gain when the transmission power p > 5 dB, as shown in Fig. 6. Once again, PSO yields the worst performance because it does not have suitable mechanisms for initialization of the candidates. The performance gains of MINL and MTLI are almost the same and slightly better than the gain of PSO. MMSE performs better than PSO, MINL and MTLI, but its performance gain is not as good as that of the proposed scheme. Hence, in the improper scenario, only the proposed scheme is suitable for signal transmission. A second improper scenario with the system setting (M, N, d)K = (3, 3, 3)3 is also considered, and P is set as 12 for both PSO and the proposed scheme. As shown in Fig. 7, the performance gains of each of the five methods declines significantly under this setting. The performance gains of PSO, MINL, and MTLI are almost the same when the transmission power p > 10 dB. However,
the proposed scheme still yields the best performance gain when the transmission power p > 5 dB, thus achieving more robust performance gain than those of the other four methods in different scenarios.
4.3. Comparisons of computational complexity Next, computational complexity in terms of the number of complex multiplications and additions (CMAs) required by the five methods is analyzed. A CMA contains 4 real multiplications and 2 real additions. First, in each iteration, the proposed scheme calculates each user’s eigenvectors according to (7). Computing K H T (n − 1)Ti (n − 1)H HH ki and eigenvalue decompositions / k ki i i=1,i = of M × M matrix requires approximately (K − 1)(NMd + M2 d) and 1 3 M CMAs respectively. Thus, in each iteration, the total compu3 tational complexity of (7) for K users is K((K − 1)(NMd + M 2 d) + 1 3 M ) CMAs. The proposed scheme also requires K((K − 1)(NMd + 3 N 2 d) + 13 N 3 ) CMAs to compute (8) in each iteration for K users. When computing each solution candidate’s sum rate based on (2) in each iteration, our scheme requires (K − 1)(NMd + M2 d), 13 N 3 ,
K
H T (n − 1)Ti (n − 1)H HH and 13 N 3 CMAs to compute ki , the / k ki i i=1,i = inversion of N × N matrix, and the determinate of N × N matrix respectively. The total computational complexity for K users in each iteration is P( 23 N 3 + (K − 1)(NMd + M 2 d)) CMAs; and the update of the PSO mechanism based on (10) is 5P Md CMAs. Therefore, the total computational complexity of the proposed scheme in each iteration is P( 23 N 3 + (K − 1)(NMd + M 2 d) + 5Md) + K((K − 1)(2NMd + N 2 d + M 2 d) + 13 N 3 + 13 M 3 ) CMAs. Similar to the PSO mechanism of the proposed scheme, the PSO method requires a total of P( 23 N 3 + (K − 1)(NMd + M 2 d) + 5Md) CMAs for each iteration. The computational complexity of MTLI and the MINL in each iteration is almost the same as that of the MTLI part of the proposed scheme, which is K((K − 1)(2NMd + N 2 d + M 2 d) + 13 N 3 + 1 3 M ) CMAs. Meanwhile, MMSE needs approximately K(K(2NMd + 3 N 2 d + M 2 d) + 23 M 3 + 23 N 3 + 2NMd) CMAs. Table 2 shows the analytic expressions of the computational complexity for the five compared methods.
H.-Y. Lu / Applied Soft Computing 50 (2017) 158–165 Table 2 Expressions of the computational complexity.
References
Methods
Complex multiplications and additions (CMAs)
Proposed scheme
GP( 23 N 3
PSO MTLI Min. INL MMSE
165
+ (K − 1)(NMd + M d) + 5Md) + GK((K − 1) (2NMd + N 2 d + M 2 d) + 13 N 3 + 31 M 3 ) 2 3 GP( 3 N + (K − 1)(NMd + M 2 d) + 5Md) ¯ GK((K − 1)(2NMd + N 2 d + M 2 d) + 13 N 3 + 13 M 3 ) ¯ GK((K − 1)(2NMd + N 2 d + M 2 d) + 13 N 3 + 13 M 3 ) ¯ GK(((2NMd + N 2 d + M 2 d) + 23 N 3 + 23 N 3 + 2NMd) 2
Table 3 Complexity comparison in terms of the number of complex multiplications and additions (CMAs). Methods
(2, 2, 1)3 G=8, P=6 ¯ = 100 G
(4, 4, 2)3 G=8, P=10 ¯ = 100 G
(4, 4, 3)3 G=8, P=16 ¯ = 100 G
(3, 3, 3)3 G=8, P=12 ¯ = 100 G
Proposed scheme PSO MTLI Min. INL MMSE
2400 1504 11,200 11,200 18,400
24,021 16,853 89,600 89,600 147,200
47,957 37,717 128,000 128,000 214,400
22,032 16,416 70,200 70,200 118,800
To gain further insight into the computational complexity, some practical values are substituted into the expressions in Table 2 to compare the real number of CMAs needed by the five schemes. The results, listed in Table 3, show that the PSO method needs the smallest number of CMAs, and the proposed scheme requires fewer CMAs than the other three linear methods. However, our scheme achieves the highest sum rate performance gain among the five methods. Note that in fact, the complexities of the MTLI method and the proposed scheme are significantly different, mainly due to their required numbers of iteration. Specifically in Table 3, the ¯ for the MTLI method is 100, while the numnumber of iterations G ber of iterations G for the proposed scheme is only 8. Hence, the proposed scheme requires conducting MTLI only 8 times, as compared to 100 times for the MTLI method. As a result, the complexity of the proposed scheme is lower than that of the MTLI method. 5. Conclusions The proposed interference alignment (IA) scheme for wireless systems with multiple-input-multiple-output (MIMO) interference channels executes the sum rate optimization task in an iterative cycle. In each cycle, the MTLI mechanism is used initially to find a tentative solution candidate. Next, the PSO mechanism uses the candidate to assist in the joint search for feasible solutions to maximize the total sum rate. Then, at the end of each cycle, the best candidate derived by the PSO mechanism is passed to the next cycle to help the MTLI mechanism refine the candidate. Because of the cyclic cooperation among the mechanisms of MTLI’s decoders, MTLI’s precoders, and the PSO mechanism, the candidates can be refined cycle by cycle. The results of simulations and complexity analysis show that the proposed scheme not only reduces the computational complexity significantly, but also achieves better performance gains than four existing approaches.
[1] D. Gesbert, S. Hanly, H. Huang, S.S. Shitz, O. Simeone, W. Yu, Multi-cell MIMO cooperative networks: a new look at interference, IEEE J. Sel. Areas Commun. 28 (9) (Dec 2010) 1380–1407. [2] R. Rusek, Scaling up MIMO: opportunities and challenges with very large arrays, IEEE Trans. Signal Proc. 30 (1) (2013) 40–46. [3] C. Rowell, S. Han, Z. Xu, G. Li, Z. Pan, Toward green and soft: a 5G perspective, IEEE Commun. Mag. (Feb 2014) 66–73. [4] K.G. Andrews, S. Buzzi, W. Choi, S.V. Hanly, A. Lozano, A.C.K. Soong, J.C. Zhang, What will be 5G Be, IEEE J. Sel. Areas Commun. 32 (6) (June 2014) 1065–1082. [5] M. Jiang, L. Hanzo, Multiuser MIMO-OFDM for next-generation wireless systems, Proc. IEEE 95 (7) (2007). [6] Y. Kim, H. Ji, J. Lee, Y.-H. Nam, B.L. Ng, I. Tzanidis, Y. Li, J. (Charlie) Zhang, Full dimension MIMO (FD-MIMO) – the next evolution of MIMO in LTE systems, IEEE Wire. Commun. (2014) 26–33. [7] Y.S. Cho, J. Kim, W.Y. Yang, C.-G. Kang, MIMO-OFDM Wireless Communications With MATLAB, Johns Wiley & Sons Pvt Ltd, 2010. [8] Z. Zhao, M. Peng, Z. Ding, W. Wang, H.-H. Chen, Denoise-and-forward network coding for two-way relay MIMO systems, IEEE Trans. Vehi. Technol. 63 (2) (2014) 775–788. [9] H.-Y. Lu, Particle swarm optimization assisted joint transmit/receive antenna combining for multiple relays in cooperative MIMO systems, Appl. Soft Comput. 12 (7) (2012) 1865–1874. [10] H. Dahrouj, W. Yu, Coordinated beamforming for the multicell multi-antenna wireless systems, IEEE Trans. Wire. Commun. 9 (5) (2010) 1748–1795. [11] Q. Shi, M. Razaviyayn, Z.-Q. Luo, C. He, An iteratively weighted MMSE approach to distributed sum-utility maximization for a MIMO interferencing broadcast channel, IEEE Trans. Signal Proc. 59 (9) (2011) 4331–4340. [12] S.W. Peters, R.W. Heath Jr., Interference alignment via alternating minimization, IEEE Acoustic Speech Signal Proc. (2009) 2445–2448. [13] S.W. Peters, R.W. Heath Jr., Cooperative algorithms for MIMO interference channels, IEEE Trans. Vehi. Technol. 60 (1) (Jan 2011) 206–218. [14] I. Csisizár, G. Tusnády, Information geometry and alternating minimization procedures, Stat. Decis. 1 (1984) 205–237. [15] C. Sun, Y. Yang, Y. Yuan, Low complexity interference alignment algorithms for desired signal power maximization problem of MIMO channels, EURASIP J. Adv. Signal Proc. (2012), 13 pages. [16] M. Maddah-Ali, A. Motahari, A. Khandani, Communication over MIMO X channels: interference alignment, decomposition, and performance analysis, IEEE Trans. Inf. Theory 54 (8) (Aug 2008) 3457–3470. [17] V.R. Cadabe, S.A. Jafar, Interference alignment and degrees of freedom of the K-user interference channel, IEEE Trans. Inf. Theory 54 (8) (Aug 2008) 3425–3441. [18] E.L. Ayach, S.W. Peters, R.W. Heath, The practical challenges of interference alignment, IEEE Wire. Commun. (2013) 35–42. [19] J. Kennedy, R.C. Eberhart, Y. Shi, Swarm Intelligence, Morgan Kaufmann Publishers, San Francisco, 2001. [20] Y. Song, Z. Chen, Z. Yuan, New chaotic PSO-based neural network predictive control for nonlinear process, IEEE Trans. Neural Netw. 18 (2) (Mar 2007) 595–601. [21] J.O. Pedroa, M. Dangora, O.A. Dahunsia, M.M. Ali, Intelligent feedback linearization control of nonlinear electrohydraulic suspension systems using particle swarm optimization, Appl. Soft Comput. 24 (7) (2014) 50–62. [22] R.V. Kulkarni, A. Föster, G.K. Venayagamoorthy, Computional inteligence in wireless sensor networks: a survey, IEEE Commun. Surv. Tutor. Trans. 13 (1) (2011) 68–96, Fir. Qua. [23] M. Zubair, M.A.S. Chouchry, A. Naveed, I.M. Qureshi, Joint channel and data estimation using particle swarm optimization, IEICE Trans. Commun E91-B (9) (2008) 3033–3036. [24] H.-H. Tsai, B.-M. Chang, S.-H. Liou, Rotation-invariant texture image retrieval using particle swarm optimization and support vector regression, Appl. Soft Comput. 17 (2014) 127–139. [25] J.H. Holland, Genetic algorithms, Sci. Am. (1992). [26] H. Liu, Ji Li, A particle swarm optimization-based multiuser detection for receive-diversity-aided STBC systems, IEEE Signal Proc. Lett. (2008) 29–32. [27] G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, 1996. [28] C.M. Yetis, T. Gou, S.A. Jafar, A.H. Kayran, On feasibility of interference alignment in MIMO interference networks, IEEE Trans. Signal Proc. 58 (9) (Sep 2010) 4771–4782.
Contents lists available at ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Review Article
Cyclic interference alignment for MIMO interference channels: A hybrid approach of MTLI and PSO夽 Hoang-Yang Lu Department of Electrical Engineering, National Taiwan Ocean University, Keelung, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 1 September 2015 Received in revised form 10 November 2016 Accepted 10 November 2016 Available online 17 November 2016 Keywords: Interference alignment Particle swarm optimization Multiple-input-multiple-output Minimum total leakage interference
a b s t r a c t The paper presents an interference alignment (IA) scheme for wireless systems with multiple-inputmultiple-output (MIMO) interference channels. The scheme executes the sum rate optimization task in an iterative cycle. Specifically, in each cycle, the minimum total leakage interference (MTLI) method is used initially to find a tentative solution candidate. Next, the particle swarm optimization (PSO) method uses the tentative candidate to assist the joint search for feasible solutions that will maximize the total sum rate. Then, at the end of each cycle, the best solution candidate derived by PSO is passed to the next cycle to help the MTLI mechanism refine the candidate. The cyclic cooperation among the mechanisms of MTLI’s receivers, MTLI’s transmitters, and PSO allows the candidates to be refined cycle by cycle. The results of simulations and complexity analysis show that the proposed IA scheme not only reduces the computational complexity significantly, but also achieves a better performance gain than four existing approaches. © 2016 Elsevier B.V. All rights reserved.
Contents 1. 2. 3.
4.
5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 The proposed scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.1. Minimum total leakage interference, MTLI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.2. Steps of the proposed scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Simulations and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.1. Convergence of the proposed scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.2. Performance comparison with existing methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.3. Comparisons of computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
1. Introduction Because of the rapidly increasing demand for quality of service (QoS), current and future wireless communication systems must provide superior performance, as measured by high data rates and low power consumption [1–4]. To this end, several researchers have developed promising methods. Multiple-input-multiple-output (MIMO), which deploys multiple antennas at both the transmitter and receiver sides, is one such method [5,6]. It has been shown that
夽 This work was supported by Ministry of Science and Technology of R.O.C. under contract MOST 103-2221-E-019-014. http://dx.doi.org/10.1016/j.asoc.2016.11.017 1568-4946/© 2016 Elsevier B.V. All rights reserved.
MIMO can effectively enhance the performance without the allocation of extra frequency resources [7]. The significant performance enhancement of MIMO has generated a great deal of research interest [8–15]. In particular, a new wireless communication system, called the MIMO interference channel, has been investigated extensively [16,17]. On a MIMO interference channel, each transmitter sends its signal to its corresponding receiver. However, because of the radio propagation feature, the signal affects other receivers simultaneously and is referred to as interference. According to the literature, interference is the main factor that limits system performance [7]. Therefore, developing feasible methods to effectively alleviate the interference impairment of MIMO interference
H.-Y. Lu / Applied Soft Computing 50 (2017) 158–165
channels is an important aspect of satisfying the rapidly growing QoS demands of current and future wireless communications. Interference alignment (IA) is one of the feasible schemes that can effectively reduce the interference effect of MIMO interference channels [18]. In practice, IA uses MIMO mechanisms to align the unwanted interference sent by users to a small dimensional subspace at each receiver to help mitigate the interference. Furthermore, it has been shown that IA can achieve the maximum degree of freedom (i.e., spatial multiplexing gain) on MIMO interference channels [16,17]. The efficiency of IA to alleviate interference has motivated several researchers to develop IA algorithms to further enhance the performance of MIMO interference channels [10–15]. For example, Dahrouj et al. [10] proposed an IA method to find the downlink beamforming vectors. The objective is to either minimize the total weight of the transmitted power or maximize the per-antenna power across all base stations subject to the signal-to-interference-and-noise-ratio (SINR) constraints of remote users. As the uplink beamforming problem is easier to solve, the authors used the Lagrangian duality theory to transform it into an uplink problem. However, the duality method inherently limits the beamforming solutions of IA, such that they are only suitable for time-division duplex (TDD) systems. In [11], instead of maximizing the SINR, an iteratively weighted minimum mean square error (MMSE) IA method was proposed. The goal of the method is to design linear transceivers that maximize the system throughput of MIMO interference channels. Peters and Heath [12] proposed an iterative subspace method called minimum total leakage interference (MTLI), which tries to minimize the total leakage interference of MIMO interference channels. To further improve MTLI’s performance, the authors proposed two IA schemes called minimum interference plus noise leakage (MINL) and MMSE respectively [13]. The IA problems considered in [11–13] are all non-convex initially, so it is not easy to find joint solutions of IA concurrently. To solve the problem, [14–16] exploited the alternating minimization approach [14] to find one IA solution at a time, while ensuring that the other IA solutions remain unchanged. As a result, the above-mentioned schemes may incur some computational overhead and can lead to suboptimal solutions. To reduce the computational complexity, two low complexity IA algorithms were proposed in [15]; however, they only derive suboptimal solutions because they are still linear approaches. Recently, a computer intelligence approach called particle swarm optimization (PSO), which is based on observations of bird flight, was proposed in [19]. PSO has been used effectively in many engineering fields, including control [20,21], communications [9,22], and signal processing [23,24]. Because it only searches for parts of the solutions in a solution space, PSO can complete the solution search process quickly. Hence, in general, PSO only requires low computational complexity to find optimal or near-optimal solutions [19]. Furthermore, because of its inherent mechanism for searching joint solutions, PSO can solve nonlinear problems easily. It has also been shown that PSO usually performs better than other computational intelligence methods, such as the genetic algorithm (GA) [25], in some applications [19,26]. As a result, PSO has generated a great deal of research interest in recent years. This paper presents a novel IA scheme for MIMO interference channels. Specifically, a hybrid approach of MTLI and PSO is used to find joint IA solutions iteratively. The proposed scheme performs the sum rate optimization task in an iterative cycle. In each cycle, the MTLI method is used initially to find a tentative solution candidate. Then, PSO exploits the tentative candidate to help jointly search for feasible solutions in order to maximize the total sum rate. At the end of each cycle, the best solution candidate derived by PSO is passed to the next cycle to help the MTLI mechanism find a more feasible tentative candidate. Because of the cooperation between
159
the MTLI and the PSO, the candidates can be refined cycle by cycle. To evaluate the performance of the proposed IA scheme, intensive computer simulations and complexity analysis are conducted. The results show that the scheme outperforms some existing works and has lower computational complexity. The remainder of this paper is organized as follows. Section 2 describes the system model. In Section 3, the proposed hybrid MTLIPSO method is discussed for joint IA. In Section 4, the results of computer simulations and complexity analysis using the proposed scheme are compared to the results from four existing methods. Section 5 contains our concluding remarks.
2. System model Consider a K-user MIMO interference channel as shown in Fig. 1, where each user utilizes M antennas to send the individual d symbol signals over the fading channel to the corresponding receiver with N antennas. For ease of description, the parameter setting of the MIMO interference channel considered is denoted as (M, N, d)K . Because of the radio propagation characteristic, each transmitted signal impinges on the intended receiver as well as the other K − 1 receivers. This means that each receiver will receive the signal from the corresponding transmitter and the interference from the other K − 1 transmitters. The system assumes that (1) all the channels are independent with complex Gaussian distributed coefficients, whose effect is identical to the multipath Rayleigh fading channel, commonly used in the literature [11,13,15,18]; (2) the users are all stationary, (i.e. they are not moving during the transmissions); and (3) the channel fading coefficients remain fixed in a symbol duration and vary independently symbol by symbol. In addition, for ease of derivation, the transmissions of the K users are assumed to be synchronized, using schemes like those in [7]. The transmitted signal of user k can be expressed as sk = Tk bk , where bk and Tk are user k’s d × 1 transmitted symbol vector and M × d precoder matrix respectively. It is assumed that each symbol of bk is independent and identically distributed such that E(bk bH k ) = Id . E(·), Id , and (·)H denote the expectation operation, d × d identity matrix, and Hermitian transpose operation respectively [27]. Furthermore, the transmission power of user k is also assumed to be subject to pk , i.e. ||Tk ||2F = pk , where ||A||F is the Frobenius norm of matrix A [27]. In addition, for ease of derivation, it is also assumed that the transmission power for each user is the same, i.e. p1 = p2 = · · · = pK . The signal sk is then transmitted through the transmit antennas over the fading channel to the intended receiver k. As a result, the base band equivalent signal received at the receiver k is [13]
rk = Hkk Tk bk +
K
Hki Ti bi + nk ,
1 ≤ k ≤ K,
(1)
i=1,i = / k
where Hki is the N × M channel matrix between transmitter i and receiver k; and nk is the additive white Gaussian noise (AWGN) vector with the complex normal distribution, CN(0, k2 IN ). For ease of analysis, all {Hki } are assumed to be full rank and mutually independent. It is also assumed that the variance of AWGN received at all receivers is the same, i.e. k2 = 2 , 1 ≤ k ≤ K. Note that the first two terms on the right-hand side of Eq. (1) represent, respectively, the desired signal and the interference from the other K − 1 users. A number of studies have shown that the system performance is limited by the interference [7]. To mitigate the interference effect, IA designs the precoders {Tk }, 1 ≤ k ≤ K to align the interference so that it lies in the small dimensional subspace of the receivers. Because of the alignment mechanisms, IA can improve the performance of MIMO interference channels
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Fig. 1. Block diagram of the MIMO interference channel.
effectively [18]. After determining the precoders, the instantaneous sum rate of the system can be calculated easily as follows [13]:
R(T1 , . . ., TK ) =
K
H log |IN + C−1 Hkk Tk TH k Hkk |, k
(2)
k=1
where
Ck =
K
H 2 Hki Ti TH i Hki + IN .
(3)
i=1,i = / k
Fig. 2. Block diagram of cyclic interference alignment.
denotes the inverse matrix of Ck . As the sum rate is one of C−1 k the most important measurements of system performance, a new IA method was proposed, which can concurrently find the joint precoders for the maximization of the sum rate in Eq. (2). However, the task is difficult because searching for the joint IA precoders in Eq. (2) is a nonlinear problem. Therefore, some researchers have simplified the problem or re-formulated it by using other metrics, such as MMSE. As a result, solutions of existing approaches are only suboptimal.
The goal of MTLI is to minimize the total leakage interference and then use each receiver’s decoder Dk to remove the receiver’s residual interference. Therefore, the problem for MTLI is formulated as follows [12]:
3. The proposed scheme
arg
3.1. Minimum total leakage interference, MTLI
min{Ti },
K {Dk } k=1
In this section, the proposed hybrid MTLI-PSO scheme is described, which is designed to determine the joint IA precoders for MIMO interference channels. Fig. 2 shows the structure of the scheme. The main function of the scheme is to search for joint IA solutions by using MTLI and PSO in an iterative cycle. In each iteration, MTLI is used to find an initial candidate, which is then provided to PSO for concurrently finding all the precoders in order to maximize the system sum rate in Eq. (2). At the end of each iteration, the best candidate found by the PSO mechanism is passed to the next cycle to assist MTLI to refine the candidate. Since the candidates are refined iteratively in a cyclic manner, the proposed scheme is thus named as cyclic interference alignment. In the following subsections, MTLI is briefly reviewed and then the steps of the proposed scheme are described in detail.
s.t.
TH i Ti =
DH k Dk = Id ,
pi I , d d
E||DH k
K
Hki Ti bi ||2F ,
(4)
i=1, i = / k
(5) (6)
where 1 ≤ i, k ≤ K. The constraint in Eq. (5) is used to ensure that the transmitted power is uniformly allocated across all transmit antennas of user i; while the constraint in Eq. (6) builds the orthonormal detectors for user k’s receiver. Because the precoders {Tk } and the decoders {Dk }, 1 ≤ k ≤ K are mutually dependent in Eq. (4), it is difficult to find a closed-form solution directly. To resolve the problem, Chae et al. used the alternating minimizing approach [14] to iteratively search for the precoders and the decoders [12]. According to
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the results in [12], the decoder solution for user k at the nth iteration is d
Dk (n) = {min e.v. (
K
Hki Ti (n − 1)Ti (n − 1)H HH ki )},
(7)
i=1,i = / k d where 1 ≤ i, k ≤ K; min e.v. (X) denotes the eigenvectors of matrix X whose corresponding eigenvalues are the d minimum ones; and (X) represents the normalization operation of matrix X’s column vectors for the constraint in Eq. (6). Then, with the solutions {Dk (n)} given in Eq. (7), the solution for the precoder of user k at the nth iteration is d Tk (n) = {min e.v. (
K
Furthermore, the new precoders {Tk (i)} of MTLI will be passed to the PSO mechanism to find better joint solutions for the precoders. Step 4: Evaluation and Update of the PSO: In the ith iteration, PSO first replaces the particle’s position with the lowest sum rate value by the new precoders {Tk (i)}, k = 1, . . ., K of MTLI found in Step 3. Next, the P particles’ positions {P1l (i)}, . . ., {PPl (i)}, l = 1, . . ., K are individually substituted into (2) to compute their corresponding sum rates. Then, the P sum rates of the particles are sorted, the position of the particle with the highest sum rate is chosen as the global optimum position for the precoders, denoted as Gl , l = 1, . . ., K. In addition, the local optimum position of the jth particle in the ith iteration is updated by j
H HH ik Di (n)Di (n) Hik )},
(8)
i=1,i = / k
where {X} denotes the normalization operation of matrix X’s column vectors to satisfy the constraint in Eq. (5) [27]. The alternating minimizing method [14] updates Eqs. (7) and (8) alternately to find the solutions iteratively until it reaches the prescribed number of iterations or a feasible MTLI result in Eq. (4). However, this alternating approach is a linear method that finds the precoders or decoders separately. As a result, MTLI’s performance deteriorates and the solutions are generally sub-optimal. 3.2. Steps of the proposed scheme Steps of the proposed scheme, shown in the block diagram in Fig. 2, are depicted in detail as follows.. Step 1: Initialization: First, for the initial decoders of MTLI, the proposed scheme chooses an arbitrary orthonormal basis for the precoders {Tk (0)}, 1 ≤ k ≤ K of the transmitters. The initial precoders {Tk (0)} will be used in Step 2 to compute Eq. (7) for the decoders {Dk (1)} in the first iteration. Ideally, with the help of the precoders {Tk } and the decoders {Dk }, the output signal of the decoders should be derived primarily from user k’s transmitted signal. This is because the interference is orthogonal to the subspace of user k’s decoder, so it will be removed completely at user k. For the PSO task, particles are created as the swarm’s initial population. The particles are the solution candidates for the precoders and will be refined cycle by cycle during the evolutional process of the proposed scheme. In addition, the following PSO feature vectors are assigned to each particle: the initial position, local optimal position, and velocity. The initial posij tion of the jth particle is denoted as {Pl (1)}, l = 1, . . ., K, j = 1, . . ., P whose elements are complex Gaussian random variables with disj tribution CN(0, 1). The column vectors of {Pl (1)}, l = 1, . . ., K, j = 1, . . ., P should be normalized in each iteration to satisfy the power constraint in (5). Particle j’s local optimal position and velocity are j j denoted as {Ll }, and {Vl (1)}, l = 1, . . ., K, j = 1, . . ., P, respectively. j
j
Moreover, for the initialization, let Ll = Pl (1) and the initial velocity j Vl (1)
= 0, l = 1, . . ., K, j = 1, . . ., P. vector of particle j be zero, i. e . Step 2: Update of MTLI’s decoders: In the ith iteration, i = 1, . . ., G, MTLI initially uses the PSO mechanism’s best candidate in the (i − 1)th iteration as the precoders. Next, with the help of the best precoder candidate, MTLI computes Eq. (7) to find better decoders {Dk (i)} for the receivers. The derived decoders are then passed to the MTLI precoder mechanism to find better precoders. Step 3: Update of MTLI’s precoders: In this step, MTLI uses the decoders {Dk (i)} found in Step 2 to compute the new precoders of Eq. (8). As the decoders derived in the ith iteration are usually better than those of the previous iterations, the new precoders {Tk (i)} may be better than those obtained earlier. Thus, the precoders can be refined iteration by iteration.
161
j
j
j
If R(L1 , . . ., LK ) < R(P1 (i), . . ., PK (i)) j
j
(9)
then Ll = Pl (i), l = 1, . . ., K, where j = 1, . . ., P and R( · ) is the sum rate function in (2). It is noted that error propagation may occur in the proposed scheme due to improper tentative candidates being created and delivered to the next step or the next iteration. To avoid error propagation, in step 4, only the worst particle position with the lowest sum rate value is replaced by the new precoders found via MTLI in Step 3. As a result, most of the better particles found in the previous iteration are still kept to help PSO refine the joint candidate results. In addition, as shown in Eq. (9), PSO updates the local optimum positions only when the new candidates are better. This update mechanism implies the results derived by PSO at least do not become worse, thus reducing the occurrence of error propagation. After the above evaluation of PSO, the local optimum positions of j the particles, {Ll }, l = 1, . . ., K, j = 1, . . ., P, and the global optimum positions, Gl , l = 1, . . ., K are determined. Next, PSO updates the process to refine the particles’ positions and corresponding velocities. For the update of particle j’s velocity in the ith iteration, j
j
j
j
Vl (i + 1) = ˛Vl (i) + ˇX1 (Ll − Pl (i)) j
+ X2 (Gl − Pl (i)), l = 1, . . ., K,
(10)
where j = 1, . . ., P; ˛ is the inertial weight; ˇ and are the movement weights of the local and the global optimum positions respectively; X1 and X2 are M × d matrices, whose entries are random variables uniformly distributed between 0 and 1; and is the Hadamard product operator [27]. Then the corresponding particle j’s positions for the next (i + 1)th iteration can be updated as follows: j
j
j
Pl (i + 1) = Pl (i) + Vl (i + 1)t, l = 1, . . ., K,
(11)
where j = 1, . . ., P; and t is the time duration, which is usually set at 1. Note that after the positions have been updated, they must be normalized to satisfy the power constraint in Eq. (5). Repeat/End: Steps 2 to 4 are repeated until the number of iterations is G, i.e. i = G. Finally, the global optimum position Gl , l = 1, . . ., K derived by the PSO mechanism is chosen as the solution of the precoders {Tk } for the IA problem. This means the PSO particle with the highest sum rate is the final precoder solution of the proposed IA scheme. 4. Simulations and discussion To evaluate the proposed scheme, computer simulations and complexity analysis are considered. For ease of presentation, the channels for all users are assumed to be complex Gaussian fading [11,13,15,18]. The channel coefficients remain fixed during symbol duration and vary independently symbol by symbol. In addition, it is also assumed that the channel state information is perfectly estimated at the corresponding receivers and shared with the other
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Fig. 3. Convergence of the proposed scheme.
transmitters and receivers in the system. For simplicity, the transmission power of each user is the same, denoted as p. Based on the setting in [13], the number of iterations for MTLI [12], MINL ¯ = 100. Fig. 3 shows the conver[13], and MMSE [13] is all set as G gence of the proposed scheme; and Figs. 4–7 show the scheme’s sum rate performance under different system settings. All the simulation results shown in these figures are obtained by averaging individual results over 256,000 independent trials.
Table 1 Parameters of the PSO mechanism in the proposed scheme. Parameter
Value
˛ ˇ, t Minimum velocity Maximum velocity The number of iterations, G
0.9 0.5 1.0 −3.0 3.0 8
4.1. Convergence of the proposed scheme First, simulations are conducted to assess the convergence performance of the proposed scheme. The parameter settings of PSO in the proposed scheme are listed in Table 1. Two scenarios with
system settings (M, N, d)K = (2, 2, 1)3 and (M, N, d)K = (4, 4, 2)3 are considered respectively. Fig. 3 shows the corresponding convergence performance with the sum rate versus the numbers of iterations. From the figure, we observe that when the transmission
Fig. 4. Sum rate performances of the proposed scheme and four existing methods, system setting: (2, 2, 1)3 .
H.-Y. Lu / Applied Soft Computing 50 (2017) 158–165
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Fig. 5. Sum rate performances of the proposed scheme and four existing methods, system setting: (4, 4, 2)3 .
power of each user, p = 24 dB in the two scenarios, the sum rate performance gain of the proposed scheme increases significantly from the 1st to the 7th iteration. In addition, the performance gain almost saturates after the 8th iteration. Hence, the proposed scheme tends to converge at the 8th iteration when p = 24 dB. Fig. 3 also shows that when p = 12 dB, the proposed scheme saturates at about the 14th iteration for (M, N, d)K = (2, 2, 1)3 and at about the 20th iteration for (M, N, d)K = (4, 4, 2)3 . The results imply that the proposed scheme requires more iterations for lower transmission power scenarios. Conducting more iterations would obviously increase the computational complexity. Setting the number of iterations, G, to be 8 in the following simulations achieves a satisfactory trade-off between computational complexity and sum rate performance.
4.2. Performance comparison with existing methods Next, the sum rate performance of the proposed scheme with four existing methods, namely, MTLI [12], MINL [13], MMSE [13], PSO [19] are compared. For fairness, the parameter settings of PSO and the proposed scheme’s PSO mechanism were the same, as shown in Table 1. In the first proper scenario, the system setting is (M, N, d)K = (2, 2, 1)3 . Based on the conclusion in [28], this setting satisfies the general feasibility condition, i. e . M + N − (K + 1)d ≥ 0. The population size of both PSO and the proposed scheme’s PSO mechanism is also set as 6. (i. e . P = 6). Fig. 4 shows the sum rate performance versus the transmission power of each user, p, for the proposed scheme and four existing methods. The results show
Fig. 6. Sum rate performances of the proposed scheme and four existing methods, system setting: (4, 4, 3)3 .
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Fig. 7. Sum rate performances of the proposed scheme and four existing methods, system setting: (3, 3, 3)3 .
that the proposed scheme significantly outperforms the other four methods. PSO yields the worst performance even though it executes the joint search for IA precoders iteratively. This is because PSO lacks a mechanism to create better initialization to improve the performance. MTLI, MINL, and MMSE do not perform as well as the proposed scheme because they find IA solutions in a linear manner. In the second proper scenario, the system setting (M, N, d)K = (4, 4, 2)3 is considered, which still satisfies the general feasibility condition [28]. For this setting, the population size P for both PSO and the proposed scheme’s PSO mechanism is set as 10. As shown in Fig. 5, all the five methods yield better sum rate gains than those in Fig. 4. This is because both the number of transmit and receive antennas increase from 2 to 4. Like the first proper scenario, the proposed scheme outperforms the other four methods when the transmission power p ≥ 3 dB; and PSO still yields the worst performance gain. To gain more insight into the proposed scheme’s performance, the improper scenario with the system setting (M, N, d)K = (4, 4, 3)3 is considered. First, based on the conclusion in [28], the setting does not satisfy the general feasibility condition, i.e. M + N − (K + 1)d ≤ 0. This means the signal space is too small to provide a feasible space for signal transmission. In this scenario, the population size P is set as 16 for both PSO and the proposed scheme. Compared to the other four methods, the proposed scheme still achieves a significant performance gain when the transmission power p > 5 dB, as shown in Fig. 6. Once again, PSO yields the worst performance because it does not have suitable mechanisms for initialization of the candidates. The performance gains of MINL and MTLI are almost the same and slightly better than the gain of PSO. MMSE performs better than PSO, MINL and MTLI, but its performance gain is not as good as that of the proposed scheme. Hence, in the improper scenario, only the proposed scheme is suitable for signal transmission. A second improper scenario with the system setting (M, N, d)K = (3, 3, 3)3 is also considered, and P is set as 12 for both PSO and the proposed scheme. As shown in Fig. 7, the performance gains of each of the five methods declines significantly under this setting. The performance gains of PSO, MINL, and MTLI are almost the same when the transmission power p > 10 dB. However,
the proposed scheme still yields the best performance gain when the transmission power p > 5 dB, thus achieving more robust performance gain than those of the other four methods in different scenarios.
4.3. Comparisons of computational complexity Next, computational complexity in terms of the number of complex multiplications and additions (CMAs) required by the five methods is analyzed. A CMA contains 4 real multiplications and 2 real additions. First, in each iteration, the proposed scheme calculates each user’s eigenvectors according to (7). Computing K H T (n − 1)Ti (n − 1)H HH ki and eigenvalue decompositions / k ki i i=1,i = of M × M matrix requires approximately (K − 1)(NMd + M2 d) and 1 3 M CMAs respectively. Thus, in each iteration, the total compu3 tational complexity of (7) for K users is K((K − 1)(NMd + M 2 d) + 1 3 M ) CMAs. The proposed scheme also requires K((K − 1)(NMd + 3 N 2 d) + 13 N 3 ) CMAs to compute (8) in each iteration for K users. When computing each solution candidate’s sum rate based on (2) in each iteration, our scheme requires (K − 1)(NMd + M2 d), 13 N 3 ,
K
H T (n − 1)Ti (n − 1)H HH and 13 N 3 CMAs to compute ki , the / k ki i i=1,i = inversion of N × N matrix, and the determinate of N × N matrix respectively. The total computational complexity for K users in each iteration is P( 23 N 3 + (K − 1)(NMd + M 2 d)) CMAs; and the update of the PSO mechanism based on (10) is 5P Md CMAs. Therefore, the total computational complexity of the proposed scheme in each iteration is P( 23 N 3 + (K − 1)(NMd + M 2 d) + 5Md) + K((K − 1)(2NMd + N 2 d + M 2 d) + 13 N 3 + 13 M 3 ) CMAs. Similar to the PSO mechanism of the proposed scheme, the PSO method requires a total of P( 23 N 3 + (K − 1)(NMd + M 2 d) + 5Md) CMAs for each iteration. The computational complexity of MTLI and the MINL in each iteration is almost the same as that of the MTLI part of the proposed scheme, which is K((K − 1)(2NMd + N 2 d + M 2 d) + 13 N 3 + 1 3 M ) CMAs. Meanwhile, MMSE needs approximately K(K(2NMd + 3 N 2 d + M 2 d) + 23 M 3 + 23 N 3 + 2NMd) CMAs. Table 2 shows the analytic expressions of the computational complexity for the five compared methods.
H.-Y. Lu / Applied Soft Computing 50 (2017) 158–165 Table 2 Expressions of the computational complexity.
References
Methods
Complex multiplications and additions (CMAs)
Proposed scheme
GP( 23 N 3
PSO MTLI Min. INL MMSE
165
+ (K − 1)(NMd + M d) + 5Md) + GK((K − 1) (2NMd + N 2 d + M 2 d) + 13 N 3 + 31 M 3 ) 2 3 GP( 3 N + (K − 1)(NMd + M 2 d) + 5Md) ¯ GK((K − 1)(2NMd + N 2 d + M 2 d) + 13 N 3 + 13 M 3 ) ¯ GK((K − 1)(2NMd + N 2 d + M 2 d) + 13 N 3 + 13 M 3 ) ¯ GK(((2NMd + N 2 d + M 2 d) + 23 N 3 + 23 N 3 + 2NMd) 2
Table 3 Complexity comparison in terms of the number of complex multiplications and additions (CMAs). Methods
(2, 2, 1)3 G=8, P=6 ¯ = 100 G
(4, 4, 2)3 G=8, P=10 ¯ = 100 G
(4, 4, 3)3 G=8, P=16 ¯ = 100 G
(3, 3, 3)3 G=8, P=12 ¯ = 100 G
Proposed scheme PSO MTLI Min. INL MMSE
2400 1504 11,200 11,200 18,400
24,021 16,853 89,600 89,600 147,200
47,957 37,717 128,000 128,000 214,400
22,032 16,416 70,200 70,200 118,800
To gain further insight into the computational complexity, some practical values are substituted into the expressions in Table 2 to compare the real number of CMAs needed by the five schemes. The results, listed in Table 3, show that the PSO method needs the smallest number of CMAs, and the proposed scheme requires fewer CMAs than the other three linear methods. However, our scheme achieves the highest sum rate performance gain among the five methods. Note that in fact, the complexities of the MTLI method and the proposed scheme are significantly different, mainly due to their required numbers of iteration. Specifically in Table 3, the ¯ for the MTLI method is 100, while the numnumber of iterations G ber of iterations G for the proposed scheme is only 8. Hence, the proposed scheme requires conducting MTLI only 8 times, as compared to 100 times for the MTLI method. As a result, the complexity of the proposed scheme is lower than that of the MTLI method. 5. Conclusions The proposed interference alignment (IA) scheme for wireless systems with multiple-input-multiple-output (MIMO) interference channels executes the sum rate optimization task in an iterative cycle. In each cycle, the MTLI mechanism is used initially to find a tentative solution candidate. Next, the PSO mechanism uses the candidate to assist in the joint search for feasible solutions to maximize the total sum rate. Then, at the end of each cycle, the best candidate derived by the PSO mechanism is passed to the next cycle to help the MTLI mechanism refine the candidate. Because of the cyclic cooperation among the mechanisms of MTLI’s decoders, MTLI’s precoders, and the PSO mechanism, the candidates can be refined cycle by cycle. The results of simulations and complexity analysis show that the proposed scheme not only reduces the computational complexity significantly, but also achieves better performance gains than four existing approaches.
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