Full-Duplex MIMO Relay-Assisted Interference Alignment Algorithm in K-user Interference Channels

In this paper, we investigate the transceiver design schemes for the full-duplex multiple-input multiple-output relay-assisted K-user single-input multiple-output interference channels. Firstly, we propose an iterative optimized reference vector for IA (IORV-IA) algorithm in the perfect channel state information (CSI) scenario. The proposed IORV-IA algorithm not only achieves the alignment of interference signals at each receiver, but also iteratively optimizes the IA reference vector by orthogonalizing the directions of the interference signals and the desired signal. With the optimized IA reference vector, the relay processing matrix and the receiving filter vectors are designed to further improve the system performance. Considering that the relay cannot obtain perfect CSIs in practice due to many factors, and the performance of the IA scheme is very sensitive to this error. Furthermore, we propose a robust transceiver design scheme based on mean square error (MSE) in the imperfect CSI scenario, which minimizes the sum of MSEs in the worst case through iteration. The proposed algorithms are evaluated in terms of the average sum rate and bit error rate performance and the simulation results show the advantages of the proposed algorithms over existing centralized IA and centralized zero-forcing algorithms.

In principle, IA technology can be implemented in time, frequency or space dimensions. Due to the wide applications of multi-input and multi-output (MIMO) technique nowadays, more and more attention has been paid to the study of IA technology in space dimension, i.e., Spatial-domain IA, which is also termed as MIMO IA. In MIMO IA networks, all the undesired interference signals are constrained into the same pre-defined subspace with dimension smaller than the number of interferers at each receiver, and the desired signal can be retrieved by the interference suppression matrix which is orthogonal to the direction of interferences. Hence, the more dimensions can be used for the desired signals transmission.
The research work on MIMO IA mainly focuses on the design of IA algorithms and the achievable DoF brought by IA. For instance, in a K-user MIMO IC with uncoordinated interference, two rank minimization methods are proposed under perfect and imperfect channel state information (CSI) scenarios to enhance the performance of IA in [21]. In [7], the authors characterize the DOF of the K-user MIMO Gaussian IC and introduce a new upper-bound on the DoF. By introducing relay technology in MIMO networks can expand the coverage and enhance the link reliability, and is one of the key technologies of future wireless networks [34]. While the combination of relay strategy and IA algorithm can greatly improve the system performance. For instance, in a MIMO broadcast channel with K MIMO relays, the authors consider the more practical situation that the transmitter has outdated CSIs and propose a relay-aided IA algorithm in [17]. The DoF region that IA can achieve in this scenario is derived. Compared with non-relay-aided IA, the relay-aided IA obtains better performance in most of the outdated CSI scenarios. Further, the combination of IA and relay technology can also enhance system security. In the K-user MIMO IC where exists an external eavesdropper, the authors proposed an IA iteration algorithm in [29]. The proposed algorithm requires the receiver to transmit artificial noise and the amplify-and-forward (AF) relay cooperation, which improves the system performance while ensuring the security.
For relay assisted MIMO IC channels, AF relay can operate in half-duplex (HD) or full-duplex (FD) operating modes. In a K-user M × N IC with a HD AF relay [35], it is shown that KM/2 DoF can be achieved without the CSI at the transmitters (CSITs) when M = N . However, time extensions are still essential when M ≠ N . In [36], an opposite directional IA (ODIA) scheme for the K-user IC with a HD MIMO relay is proposed, which aligns the interference from the relay link along the opposite direction of the 1 3 interference from the direct link. The ODIA scheme requires only global CSI available at the relay, no receive filters are needed which results in a significant complexity reduction. By replacing HD relay by FD AF working mode, two schemes termed as centralized IA (CIA) and centralized zero-forcing (CZF) respectively are proposed in a K-user MIMO ICs [37]. Both of the CIA and CZF schemes require only the global CSI available at the relay, which can be realized by channel reciprocity and feedback from receivers. Moreover, CZF scheme can avoid the receive beamforming matrices, which results in a significant complexity reduction.
From the above illustration,it can be seen that most existing IA schemes only consider suppressing and eliminating interference, without taking the negative effect on the desired signal into account. In other words, these proposals only constrain all the interference signals into a certain interference space or align the interference signals to any random selected IA reference vector, and then eliminate the interference through receiving filter. Due to the randomness of interference space or IA reference vector, such operation will weaken the desired signal and ultimately have a negative effect on system performance. Therefore, novel IA schemes need to be designed to further improve the system performance.
In this paper, we investigate the transceiver design schemes for the FD MIMO relayassisted K-user single-input multiple-output (SIMO) ICs. Firstly, we propose an iterative optimized reference vector for IA (IORV-IA) algorithm in the perfect CSI scenario. The proposed IORV-IA scheme not only considers the achievement of IA at each receiver, but also reduces the negative impact of the aligned interference signals on the desired signal by iteratively optimizing the IA reference vector. Based on the optimized IA reference vector, the relay processing matrix and receiving filter vectors are designed to avoid the reduction of the desired signal when the receivers perform interference cancellation. We investigate the proposed IORV-IA algorithm in two scenarios where the number of receiving antennas is equal to two and more than two. The former scenario is termed as do the best while the latter is termed as partial, which will be introduced in Section III. In addition, the feasibility conditions of the proposed IORV-IA algorithm are also presented. Furthermore, we propose a robust transceiver design scheme based on mean square error (MSE) in the imperfect CSI scenario. The proposed robust MSE algorithm minimizes the sum of MSEs in the worst case through iteration, in order to further improve the system performance. Although iteration is needed for the proposed algorithms, both of the two algorithms converge soon after several iterations. Besides the computational complexity of the algorithms actually is not high. To the best of the authors' knowledge, this paper is the first time to focus on the iterative optimization of IA reference vector, sum rate and BER performance at low to medium SNR range in the FD MIMO relay-assisted K-user SIMO IC. The main contributions of this paper are summarized as follows.
• The feasibility conditions of the proposed IORV-IA scheme are derived, which shows that the number of relay antennas required by the proposed scheme is less than that of traditional CIA scheme under certain conditions. It is also showed that the minimum number of antennas required by the relay of the proposed IORV-IA is always much less than that of CZF scheme. • For the two IA schemes, the proposed IORV-IA is always much better than that of CIA scheme in terms of both average sum rate and BER performance. • For the proposed robust MSE algorithm, the average sum rate performance in the low to moderate SNR region is better than that of CZF scheme. Moreover, the BER performance in any SNR region is significantly better than that of CZF scheme.
The rest of this paper is organized as follows. Section II introduces the relay-assisted K-user IC model. Section III describes the proposed IORV-IA and robust MSE schemes. Section IV provides the simulation results of the proposed IORV-IA and robust MSE, CIA and CZF schemes and the conclusion is drawn in Section V. Notation: Throughout the paper, the superscript (⋅) T and (⋅) H denote transposition and Hermitian transposition, respectively. Matrices are set in bold-face uppercase letters whereas vectors are set in bold-face lowercase letters. C M×N is the space of complex M × N matrices. (⋅) indicates expectation. |x| means the magnitude of a complex number x. ‖x‖ and ‖X‖ denote Euclidean norm of vector x and the Frobenius norm of matrix X , respectively. The operator vec(⋅) stacks the elements of a matrix in one long column vector. ker(A) denotes the kernel of the matrix A . rank (A) and tr(A) represent the rank and trace of A , respectively. ∃ means there exists. ⊗ denotes Kronecker product. I M represents M × M identity matrix.

System Model
We consider a symmetric network in which K-transmitters communicate with their associated receiver through a FD MIMO relay as shown in Fig. 1 ( TX k and RX k represent the k-th transceiver pair). The MIMO Relay works in FD mode and employs AF protocol, which includes two reasons. On the one hand, FD communication can double the network capacity with respect to traditional HD by virtue of effective self-interference cancellation technology, because the available spectral resources can be fully utilized in time and frequency [38]. On the other hand, AF relay has lower hardware complexity due to not requiring decoding and demodulation information compared with DF protocol. Suppose that the direct communication between each transmitter and its corresponding receiver is not considered due to the strong path-loss and attenuation. Therefore, the relay needs not only to forward the signals, but also to assist in doing IA, so that each receiver can decode the desired signal without interference after processing by the receiving filter. Note that single, M and N(2 ≤ N < K) antennas are equipped at the transmitter, the relay and each receiver, respectively. Each transmitter transmits a single data symbol using the same carrier frequency at the same time. In other words, time division or frequency division multiple access schemes are not used. We denote by h k ∈ C M×1 the channel from the k-th transmitter to the relay and F k ∈ C N×M the channel from the relay to receiver k. Furthermore, we assume that all channel coefficients are independent and identically distributed (i.i.d) circularly symmetric complex Gaussian distributions with a zero mean and unit variance. Further, no beamforming technology is employed due to only one antenna at every transmitter. Hence, CSI at the transmitters is not required and receivers is required to acquire only their local CSI. In addition, it is assumed that the global CSI is available at the relay, which can be realized through channel reciprocity and receiver feedback mechanism [37]. The relay uses the obtained global CSI to design the processing matrix G , which is multiplied by the transmitted signals and then forwarded to the receivers. Then, at each receiver, the final received interference signals are aligned to the same direction (or subspace), which means IA is achieved.
During a certain time slot, the received signal at the relay is given by Where s k denotes the transmitted data symbol transmitted by the k-th transmitter with [ ‖ ‖ s k ‖ ‖ 2 ] = P . n r is the additive white Gaussian noise (AGWN) vector at the relay with zero mean and variance 2 r . Then, the relay forwards the received signals to all receivers after linear processing operation. The forwarded signal is described as Where x r is transmitted signal vector with transmitting power tr (x r x H r ) = P r at the relay. G = G r denotes the relay processing matrix with dimension M × M 1 . is the amplification coefficient satisfying the relay transmission power constraint, which is The received signal at the k-th receiver can be represented as Where the equivalent noise ñ k = F k Gn r + n k , n k is the AGWN vector at receiver k with zero mean and variance 2 . At the k-th receiver, a linear zero-forcing receive filter u k is applied to eliminate the interference signals coming from undesired transmitters, where u H k u k = 1 and u H k q k = 0 should be satisfied. Then the desired data symbol after postprocessing can be given as (2) x r = Gy r = G r y r .
Therefore, the achievable rate at receiver k is given by If interference is aligned into the null space of u k , then the following conditions must be satisfied: Where condition (7) assures that the received interference of receiver k is aligned in the direction of the reference vector, and condition (8) guarantees that the desired signal subspace has one dimension which can be automatically satisfied if there is no any special structure about the channel matrices.
If the IA conditions are satisfied, then the achievable rate at receiver k is simplified to

The Proposed IORV-IA and Robust MSE Algorithms
In this section, we first describe the approach of the proposed IORV-IA algorithm for FD MIMO relay assisted K-user SIMO ICs in the perfect CSI scenario, and the feasibility conditions of the proposed algorithm are presented. Then another robust transceiver design scheme based on MSE in the imperfect CSI scenario is proposed.
Note that when N = 2 , i.e., there are only two antennas at each receiver. In this case, all of the K − 1 interference signals at each receiver should be aligned to a predetermined reference vector. The proposed IORV-IA in this scenario is termed as do the best IA. When 2 < N < K , which means the available dimensions for all of the interference signals is N − 1 at every receiver. Therefore, it is not necessary to align all of the interference signals to the reference vector, only part of them need to be (5) selected for IA. The proposed IORV-IA in this scenario can be termed as partial IA.
Consequently, for the convenience of description, we will first introduce the proposed IORV-IA scheme in the scenario when N = 2 . Then the results can be extended to the case of N > 2.

Do the Best IA
According to (4) and denote Gh j = w j for j = 1, … , K . Then, the IA condition at the kth receiver for 1 ≤ k ≤ K can be given as where q k = F k k for an arbitrary vector k ∈ C M×1 . q k is referred to as the IA reference vector, which is random due to the arbitrarily selected k . Note that when the subscript in (10) satisfies What we need to do next is to work out w i s that meet the IA condition in (10), and then calculate G from Gh i = w i . It is noted that w k+j for j = 1, … , K − 1 in (10) can be regarded as the solution vectors of the system of non-homogeneous linear equations F k x = q k , and k is a particular solution. Then, based on the theory of solution of nonhomogeneous linear equations, w k+j can be expressed as a linear combination of k and S k . S k = ker(F k ) , the column vectors of which constitute a fundamental solution of homogeneous linear equation system F k x = 0 . Where 0 is a zero vector with the same dimension as q . Mathematically, ∃ a are known as the scalar and vector linear combination coefficients of w k+j at the k-th receiver.
In order to obtain the concrete expression of w i for i = 1, … , K , let's put all of the linear combinations of w i together. Note that w i participates in IA condition at all receivers except the i-th. Therefore, there is a linear combination of w i at each receiver except the i-th, and there are K − 1 equations of w i in total, as shown in the following.
From the last equation to the second in (11), each equation subtracts the previous one to obtain the following (12). (12) can be further rewritten as is defined as the combination coefficient matrix of w i and is given by According to (11), C i has a total of K − 1 columns. It should be noted that columns i and By combining the equations in (13), we can further obtain (15) (at the top of the next page).
Note that 0 and O in the short and fat matrix on the left side of (15) are zero vector and zero matrix of the same dimension as any i and S i , respectively. Besides ⃗ C i = vec(C i ). Denote the short and fat matrix in (15) as F i , then (15) can be expressed as ] . If (15) has solutions, i.e., C i exists. Then according to (11), w i can be combined linearly by any column of C i .
From (15), we can get the following feasibility conditions of the proposed IORV-IA scheme.
Proposition 1 For the relay-assisted K-use SIMO IC, when N = 2 , the number of antennas at the relay should satisfy M > K − 1.
We omit the proof for it is trivial.
Although we can seek out w i s by the above method and further obtain G , so as to achieve IA at each receiver. However, these efforts are far from enough from the perspective of desired signals. Because the IA reference vector chosen at the beginning is random, the power of the desired signal is likely to be weakened when the receiver performs interference cancellation using a linear filter, which has a negative impact on the system performance. In order to reduce this effect, the IA reference vector needs to be optimized, this is what the proposed algorithm takes into account.
Specifically, the IA reference vector is iteratively optimized according to the following process. Firstly, the IA reference vector can be updated by orthogonalizing it with the desired signal vector. Then, the updated w i s can be obtained through relevant calculation. Finally, the updated G can be obtained from the updated w i s. Mathematically, let Bring i s and S i s into (15), updated ⃗ C is obtained, through which w i s can be combined. After several iterations, the optimal w i s can be obtained. Finally, based on Gh i = w i , we can get G h 1 h 2 ⋯ h K = w 1 w 2 ⋯ w K . Denote W K = w 1 w 2 ⋯ w K ,the close-form solution of G is presented as 3.2 Partial IA As described above, when 2 < N < K , we only need to select K − N + 1 of the interference signals to align to the predetermined reference vector. Note that there are still N − 2 dimensions left at each receiver, which are just used for the remaining N − 2 interference signals. Therefore, similar to IA condition in (10), the following IA condition for the k-th receiver for 1 ≤ k ≤ K should be satisfied When the subscript in (17) What we shall do next is similar to that in the do the best IA subsection except for one difference, which is the number of linear combinations of w i that we finally obtain in the process of solving w i . In partial IA scenario, the number is K − N + 1 , while in do the best IA case, the number is K − 1 . Specifically speaking, there are K − N + 1 equations of w i and w i can be obtained similarly to solving (11). In addition, one more thing to remember is that we still need to update w i iteratively.
In partial IA scenario, the linear zero-forcing receiver u k not only needs to zero-force the aligned interference signals, but also has to eliminate the misaligned interference signals. In other words, u H k [q k , F k w k+(K−N+2) , ⋯ , F k w k+(K−1) ] = 0 should be satisfied.

Proposition 2
For the relay-assisted K-user SIMO IC, when 2 < N < K , the number of antennas at the relay should satisfy M > (K − N + 1) (N − 1).
Note that proposition 2 is still true when N = 2 , so do the best IA can be regarded as a special case of partial IA. Consider do the best IA and partial IA scenarios together, the following proposition can be obtained.

Proposition 3 For the relay-assisted K-user SIMO IC, when 2 ≤ N < K+1
2 , the minimum number of antennas required at the relay for the proposed iterative ORV-IA scheme is less than that of CIA scheme.
In order to facilitate the comparison with traditional CIA and CZF, the IA feasibility conditions of the two schemes are provided as follows [37].

Proposition 5 For the CZF scheme, the number of antennas at the relay should satisfy M > N(K − 1).
By comparing Proposition 4 with Proposition 2, the conclusion of Proposition 3 can be obtained easily. Moreover, it can be noted from Proposition 3 that when the number of users is large and the number of user antennas is small, which is consistent with most of the practical application scenarios, the proposed IORV-IA scheme has a great advantage in the number of relay antennas. For example, when K = 10, N = 2 , the minimum number of antennas required at the relay of CIA scheme is M = 17 , which is much more than the minimum required M = 10 for the proposed IORV-IA scheme.
Note that as long as the initial assumption 2 ≤ N < K is satisfied, the minimum number of antennas required at the relay in the proposed IORV-IA scheme is always less than that of CZF scheme.
For the convenience of interpretation on the IA conditions of do the best IA and partial IA scenarios, we define a Latin square L of order K as follows.
It is a standard Latin square, i.e., the first row and the first column are arranged in natural order. We assume L particular implications: the K numbers in each row represent the transmit signal indexes from K transmitters while received at each receiver, so the elements in the i-th row of L indicates K signals indexes received by receiver i. For example, l 21 = 2 means the signal transmitted by the second transmitter and received at the second receiver, l 32 = 4 indicates the signal transmitted by the fourth transmitter and received at the third receiver. It is easy to know that the elements in the first column of L are indexes of the desired signals for each transmitter, columns 2 to K are indexes of the interference signals received by each receiver.
Based on L , it is convenient for us to understand the IA conditions in (10) and (17). In do the best IA case, the second to K-th columns of L are the signal indexes which need to be aligned at each receiver. While in partial IA scenario, the signal indexes for IA are the elements of column 2 to K − N + 2 . Moreover, it is necessary to point out that the second to K columns of L in do the best IA and the second to K − N + 2 columns of L in partial IA, every number from 1 to K happens to appear K − 1 and K − N + 1 times, respectively. This implies that there are K − 1 and K − N + 1 linear combination equations for each w i in do the best IA and partial IA scenarios, respectively.
For 2 ≤ N < K , we summarize the proposed IORV-IA scheme as Algorithm 1.

Robust MSE Scheme with Imperfect CSI
It is noted that IA technology is carried out at the MIMO relay for the proposed IORV-IA algorithm, so the relay needs to have all CSIs from transmitters to relay and from relay to receivers. To this end, the relay first estimates the transmitter-relay channel by using pilot symbols. Secondly, the relay also has to obtain the relay-receiver channel information from the receiver through the feedback channel. In the former case, the estimations of transmitterrelay may deviate from the actual channels whereas in the latter case, the feedback CSI is not perfect due to many factors such as channel estimation error, quantization error and feedback error or delay. Therefore, the relay cannot obtain perfect CSIs, and the performance of the IA scheme is very sensitive to this error [39]. In view of this, in this subsection, we propose a robust iterative scheme in the imperfect CSI scenario. The proposed MSE scheme aims to minimize the total MSE (sum-MSE) through the joint design of processing matrix at the relay and interference suppression vectors at receivers. The channel under imperfect CSI can be expressed as Where ĥ k and F k represent the estimated channel vector and matrix of transmitter k to relay and relay to receiver k, respectively. Assume channel error vector Δh k and error matrix ΔF k obey the Gaussian distribution with zero mean, and the norms are limited by the known limits, i.e Note that both the relay processing matrix and the receiver interference suppression vector are based on the estimated channel vector and matrix, then the estimate symbol ŝ k of s k at the kth receiver can be expressed as Then the SINR at receiver k can be given as The achievable rate of receiver k can be given as In order to minimize the sum-MSE of users in the worst case, we will joint design relay processing matrix and interference suppression vector under the condition of meeting the transmission power requirement of the relay. The MSE of the k-th user is defined as Substitute (19) and (20) into (23), we have (24).
Therefore, the optimization problem can be expressed as Denote MSE wc k as the MSE in the worst case of the kth receiver, so we have (29).
Therefore, it is easy to get Then the relay power limitation condition in (25) can be further rewritten as Consequently, the optimization problem of (25) can be further rewritten as Note that the sum of MSE wc k in (32) is a convex function for relay processing matrix or receiving filter vector, but it is not jointly convex for all quantities. Therefore, there is no standard convex optimization method to find the optimal solution. However, we first give the optimal û k ,Ĝ , and then based on the optimality condition, an iterative algorithm is used to find the effective optimal solution [40].
The Lagrange dual objective function can be constructed as in (33), where is the Lagrange multiplier related to relay transmission power limitation. According to the Karush-Kuhn-Tucker (KKT) conditions, we can obtain (34)- (38).

Simulation Results
In this section, the performance of the proposed IORV-IA and robust MSE algorithm will be evaluated through simulations in the perfect CSI and imperfect CSI scenarios, respectively. As a comparison, we also give the corresponding results of the traditional CIA and  CZF schemes. In this evaluation, we focus on the average achievable sum rate as well as BER performance, and we do not use any coding schemes, only QPSK modulator and demodulator are employed. The results shown below are the averages over 1000 and 10000 independent trials for average achievable sum rate and BER, respectively. Without loss of generality, set P r = P, 2 = 2 r , SNR=P∕ 2 . For i = 1, … , K , set h,i = F,i = = 0.05 , 0.1 or 0.2. It is noted that K, N, M and n here represent the number of users (or the number of transceivers), the number of receiver antennas, the number of relay antennas and the number of iterations of related algorithm, respectively. Figures 2 and 3 show the average sum rate performance of the proposed IORV-IA, CIA and CZF schemes in the perfect CSI scenario with N = 2 and N = 3 , respectively. Note that for the proposed IORV-IA scheme, the two figures correspond to the cases of do the best IA and partial IA. In Fig. 2, when K = 3 and N = 2 , the minimum number of antennas required at the relay for the proposed IORV-IA and CIA is the same, i.e., M = 3 , while for CZF scheme, the number is 5. As can be seen from Fig. 2, under the same configuration (K, N, M) = (3, 2, 3) , the sum rate performance of the proposed IORV-IA scheme is better than that of CIA scheme, and it can provide performance gain without too many iterations (the number of iterations n increases from 1 to 5). When (K, N, M) = (3, 2, 5) , the curve of the proposed IORV-IA scheme after 5 iterations tends to be consistent with that of CZF scheme, and the performance of the two schemes is better than that of CIA scheme. It is necessary to note that when N = 2 and M = 5 , for the proposed IORV-IA scheme, the maximum number of users supported to transmit simultaneously is K = 5 . While for CIA and CZF, the number is 4 and 3, respectively. It can be seen from Fig. 2 that, under the same antenna configuration, the maximum achievable sum rate of the proposed IORV-IA scheme far exceeds that of traditional CIA and CZF schemes.
When K = 4 and N = 3 , the minimum number of antennas required at the relay for the proposed IORV-IA, CIA and CZF schemes are 5, 5 and 10, respectively. It can be seen from Fig. 3 that for the two IA schemes with the same configuration, the sum rate performance of the proposed IORV-IA is always significantly better than that of CIA scheme. Although more antennas are needed at the relay for CZF scheme, the desired signal can fully occupy the antenna dimension at each receiver and thus can achieve better performance due to the diversity gain [37]. However, when N = 3 and M = 10 , the maximum number of users that can be transmitted simultaneously for the proposed IORV-IA and CIA schemes is 6, while for CZF, the number is 4. The proposed IORV-IA obviously outperforms CZF with the increase of SNR in this situation. In addition, it should be reminded that the partial IA is adopted by the proposed IORV-IA in this scenario. Although the reference vector is iteratively optimized, the misaligned interference signal at the receiver does not participate in the optimization process. As a result, the desired signal is very likely to be weakened to a certain extent when the receiver performs post-processing, which has a negative impact on the system performance. Therefore, it can be observed from the figure that the performance gain brought by the iteration of the proposed IORV-IA is not obvious, and increasing the number of relay antennas does not significantly improve the sum rate performance. Figure 4 depicts the BER performance of the proposed IORV-IA, CIA and CZF schemes in the perfect CSI scenario. It can be seen from the figure that, when (K, N, M) = (3, 2, 3) , the BER performance of the proposed IORV-IA scheme is much better than that of CIA scheme, and the performance gain brought by iteration is significant. After five iterations, 1 3 the performance of the proposed scheme is also better than that of CZF scheme. Remember that the minimum number of relay antennas required by CZF is 5, which is more than that of the proposed IORV-IA scheme. It is necessary to point out that, when K = 4 and N = 3 , the minimum number of antennas required at the relay for the proposed IORV-IA and CIA  is 5 and 4, respectively. For fair comparison, we set the number is 5. It can be seen that the BER performance of the proposed IORV-IA scheme is still better than that of CIA. The performance gain from iteration is small, which is also due to employing partial IA. It is worth noting that the BER performance of CIA scheme under two different  configurations is almost the same, which means that partial IA of CIA has little impact on BER performance. Figure 5 illustrates the average sum rate performance of different schemes in the imperfect CSI scenario with (K, N, M) = (3, 2, 5) . It is necessary to note that when K = 3 and N = 2 , the minimum number of antennas required at the relay for the proposed IORV-IA, CIA and CZF are 3, 3 and 5, respectively. For fair comparison, set M = 5 . It can be seen from the figure that imperfect CSI leads to system performance degradation. Greater channel error would render poorer system performance. Moreover, the sum rate curves of the proposed robust MSE and the proposed IORV-IA coincide with each other in the high SNR region, both of which exhibit better performance than that of CIA. In addition, the CZF scheme can achieve better performance through diversity gain, which is consistent with the description in Fig. 3. However, the proposed robust MSE scheme outperforms CZF scheme in the low to moderate SNR region. Figures 6 and 7 compare the BER performance of different schemes in imperfect CSI scenario with (K, N, M) = (3, 2, 5) and (K, N, M) = (4, 3, 5) , respectively. Firstly, it can be seen from both figures that, imperfect CSI degrades the BER performance of the system. It can be observed in Fig. 6 that the imperfect CSIs bring forth irreducible error floors even at moderate SNRs with = 0.2 for the four schemes. Moreover, the BER performance of the proposed two schemes is significantly better than that of CIA scheme, and the performance of the proposed robust MSE scheme also greatly outperforms that of CZF scheme, while the proposed IORV-IA scheme is no longer better than that of CZF scheme as shown in Fig. 4, indicating that the proposed IORV-IA scheme is more sensitive to channel error. Thankfully, compared with the proposed IORV-IA, the proposed robust MSE scheme further improves the BER performance of the system. In Fig. 7, the BER performance of the proposed two schemes IORV-IA and robust MSE is still better than that of CIA. Moreover, the performance of the proposed robust MSE scheme is better than that of the proposed IORV-IA, and the iteration brings about more obvious performance gain. It is worth noting that the BER performance of CIA scheme hardly changes when = 0.05 becomes 0.1. Figure 8 is presented to show the provable convergence of the proposed IORV-IA in the perfect CSI scenario and the robust MSE algorithm in the imperfect CSI scenario with different SNRs when K = 3, N = 2 and M = 3 . For the proposed robust MSE scheme, is set to 0.1. As can be seen from the figure, the average sum rate of the proposed two schemes grows as n increases, and is saturated after a few iterations. It can also be seen from the figure that imperfect CSI leads to obvious degradation of system performance. Figure 9 shows the relationship between the minimum number of antennas required at the relay (denoted as M min ) and the number of users K, and the relationship between M min and user antennas N for the proposed IORV-IA, CIA and CZF, respectively. We set N = 2 for sub-figure (a) and K = 31 for sub-figure (b). It can be seen first from sub-figure (a) that M min of the three schemes increases linearly with the increase of K. CIA and CZF have the same growth rate, which is nearly twice of the proposed IORV-IA schemes. In addition, as shown in sub-figure (b), M min of CZF increases sharply with the increase of N when K = 31 is fixed. However, for the proposed IORV-IA and CIA, M min and N show parabola relationship with opening downward. That is to say, M min increases first and then decreases with the increase of N, and the intersection point is K+1 2 = 16 . Moreover, when 2 ≤ N < 16 , M min of the proposed IORV-IA scheme is less than that of CIA scheme. But when 16 < N < 31 , the result is the opposite.

Conclusion
In this paper, by introducing a FD MIMO AF relay to assist transmission in a K-user SIMO IC, we first proposed a new IA algorithm termed IORV-IA in the perfect CSI scenario. Different from the traditional IA schemes, the proposed IORV-IA scheme considers not (a) (b) Fig. 9 The minimum number of antennas required at the relay versus K and N, respectively only the achievement of IA at each receiver, but also the negative impact of the aligned interference signals on the desired signal. Based on the iteratively optimized IA reference vector, the relay processing matrix and receiving filter vectors are designed to avoid the reduction of the desired signal when the receiver executes interference cancellation. Moreover, considering that the relay can not obtain perfect CSIs in practice due to many factors, another robust MSE iterative algorithm is proposed in the imperfect CSI scenario to further improve the system performance especially BER performance. Compared with the conventional CIA and CZF schemes, the proposed schemes have great advantages in the minimum number of antennas required at the relay, average sum rate performance and BER performance. Several directions naturally arise for future work. How to improve the system performance under partial IA scenario can be further studied and discussed. The joint design of transceivers and relay processing matrix in MIMO relay-assisted MIMO ICs, and the relationship between the number of users and the number of user antennas and relay antennas are also worthy of consideration and research.