An iterative intercell interference cancellation algorithm is introduced to improve the receiver performance of uplink transmission in multicell networks. At first, the uplink signal detection is performed independently in each cell according to minimum mean squared error (MMSE) criterion. Subsequently, the detection results are applied to reconstruct the transmit signals of different users and cancel their interference to neighboring cells. With the help of reconstruction results, the MMSE detection matrix of each cell is updated. The channel responses of both efficient and interference links are estimated with the help of pilots. The pilot allocation parameter is introduced to indicate the quality of channel estimation. The simulation results indicate that intercell interference can be greatly mitigated by the proposed algorithm with a moderate number of receiver antennas at the base station.
National Natural Science Foundation of China616010476167108061871050Huawei HIRP ProjectHIRPO201611031. Introduction
Multicell cooperative receiver processing [1, 2] has attracted much attention from academia in recent years. The efficient cooperation algorithm can efficiently cancel the intercell interference and improve the spectral efficiency. In [3, 4], the belief propagation method in multicell detection is proposed and the soft detection results are iteratively transferred in the cell-user factor graph. Furthermore, the two-dimensional (2D) intercell model is introduced in [5] and the Gaussian belief propagation algorithm is proposed to mitigate the intercell interference. Based on the 2D model proposed in [5], the algorithm in [6] proposes a kind of two-dimensional belief propagation equalizer to improve the uplink receiver performance. Recently, a novel measure call leakage capacity is introduced in [7] to optimize the multicell cooperative processing, which has distributive property. However, the detection performance at the receiver is not fully analyzed in [7].
In the existing research [3–6], the multicell detection is mainly based on the idea of belief propagation, and its complexity increases exponentially with the number of transmit antennas. When the number of transmit antennas becomes large, the complexity of belief propagation algorithm is very high. Therefore, this paper introduces an iterative multicell interference cancellation algorithm based on minimum mean square error (MMSE) criterion under multiantenna configuration conditions. It is noted that the idea of MMSE-based iterative processing has been widely used. For instance, in [8], the MMSE-based iterative algorithm is only applied for channel estimation, while signal detection is not included in the iterative processing. In addition, in [9], the iterative intercell interference is performed on the downlink transmission of MC-CDMA (Multicarrier Code Domain Multiple Access), while the multiantenna receiver is not applied at the receiver and the effect of channel estimation error is not fully considered.
In this paper, the iterative intercell interference cancellation in multicell uplink transmission is proposed. At first, channel estimation is performed with the help of uplink pilots. In order to suppress the pilot contamination [10, 11] in neighboring cells, the pilot sequences in different cells are orthogonal with each other. This scheme is called nonreused pilot configuration [12]. Based on the channel estimation results, the iterative intercell interference cancellation algorithm is proposed. With channel estimation results, the base station (BS) in each cell performs uplink signal detection individually and reconstructs its interference to other cells. Subsequently, the reconstruction uplink signals are exchanged between neighboring cells for further interference cancellation. In our algorithm, the interference cancellation can be iteratively executed to increase the accuracy of signal reconstruction in each cell. From the simulation results, it is shown that the proposed scheme achieves performance close to that of interference-free case with a moderate number of receiver antennas at the base station.
According to the above descriptions, our contributions mainly include the following two points. The first contribution is that the iterative detection algorithm based on minimum mean squared error (MMSE) criterion is introduced. The detection results are exploited to reconstruct the intercell interference, which is cancelled at the receiver in the uplink transmission. During iterative processing, the MMSE receiver matrix is updated based on the signal detection result in the previous iteration. According to the proposed iterative algorithm, the detection performance of the uplink transmission is improved. The second contribution is that the detection at the receiver is implemented based on the estimated channel responses. The channel estimation error is fully considered and its effect is verified in the simulations.
In the following, lower and upper bold face letter denote the vector and matrix, respectively. For the matrix X, XT, XH, and X+ denote its transpose, Hermitian, and pseudoinverse, respectively. In addition, vec(X) denotes a vector whose element is taken column-wise from matrix X. The N×N identity matrix is denoted by IN. The operator E(·) denotes the statistical expectation. For a complex scalar x, x∗ denotes its complex conjugate. The operator ⊗ denotes the Kronecker product.
2. System Model
In this paper, our focus is on uplink signal transmission in multicell scenario. We assume that there is only one base station per cell and the base station has multiple antennas. Since each cell has only one base station, the cell index p also denotes its corresponding base station. Furthermore, we assume that base stations of multicells can exchange detection results through backhaul link during iterative signal detection. In the following, we assume that the base station of each cell has the same number of antennas and each cell has the same number of users. Consequently, the base station antenna number of each cell is denoted by M and the user number in each cell is denoted by K. Moreover, the overall cell number in the uplink transmission is denoted by L. The above scenario description is reflected in Figure 1 with L=19.
A multicell network with L=19 cells. The red dots denote BSs, and the blue crosses denote users.
In the multicell scenario shown in Figure 1, the uplink transmission in the pth cell is affected by the users in the neighboring cells. For clearness, when describing the uplink signal model in the pth cell, the neighboring cell index is denoted by q, q≠p. In addition, there are multiple users in each cell. Consequently, the intercell channel response from user i in cell q to BS p is given by hq,ip=(Rq,ip)1/2wq,ip, where Rq,ip≜E(hq,ip(hq,ip)H) denotes receiver correlation corresponding to hq,ip, and wq,ip~CN(0,IM) is a Rayleigh fading channel coefficient vector. With K users in the qth cell, Hp,q=[hq,1p,…,hq,Kp] is introduced to denote the channel coefficient matrix from users in the qth cell to BS p.
In the following analysis, the block fading assumption is applied, in which the channel response maintains constant during one coherent interval. In addition, the pilot sequence is inserted into the beginning of each coherent interval and the duration of the pilot transmission is called the training phase.
In order to describe the pilot configuration, the pilot allocation parameter F, is introduced to indicate pilot reuse in the L-cell scenario shown in Figure 1. For example, with F=1/L, this means that the pilot sequences of different cells are not the same. When F>1/L, this means that at least two cells share the same pilot sequence set.
During the training phase, sp,k∈Cτ×1 denotes the pilot sequence of user k in cell p, in which τ denotes length of pilot sequence. In the following, the pilot sequence length τ is set equal to K/F. At the same time, the overall possible pilot sequence number is set equal to K/F. With Chu sequences [13] applied in this paper, the above pilot length and number setting ensures orthogonal property between different pilot sequences. During pilot sequence allocation, we assume that K users in the p cell can choose K different pilot sequences. Therefore, the channel estimations of K users in the same cell have no interference with each other. The pilot contamination only comes from intercell pilot reuse, which depends on the pilot allocation parameter F. For example, with L=2 and F=1, the above setting means that K users in cell 1 have K different pilot sequences. At the same time, K users in cell 2 share the same K pilot sequences with cell 1. Consequently, the pilot contamination between cell 1 and cell 2 exists.
In this paper, the transmit power of each pilot sequence is proportional to its length τ, so we have sp,kHsp,k=τ,1≤p≤L,1≤k≤K. During the training phase, the received signals corresponding to the pilots at BS p are packed into the following matrix formulation.(1)Yp=∑k=1Khp,kpsp,kT+∑q=1,q≠pL∑j=1Khq,jpsq,jT+Zpwhere Yp∈CM×τ denotes the received signal matrix at BS p during training phase, hq,jp∈CM×1 denotes the channel response from user j of cell q to BS p, and Zp is the additive white Gaussian noise matrix at BS p whose elements have zero mean and σ2 variance.
According to the above analysis, when the pilot allocation parameter F>1/L, the second term in expression (1) introduces pilot contamination. As F≤1/L, the pilot sequences of users in different cells are orthogonal with each other, and the effect of pilot contamination is eliminated.
After pilot transmission, the data symbols of users are transmitted. In the following, xp,k denotes the data symbol of user k in the pth cell. With intercell interference, the received signal yp∈CM×1 at the base station of the pth cell is given by(2)yp=∑k=1Khp,kpxp,k+∑q=1,q≠pL∑j=1Khq,jpxq,j+zp=Hp,pxp+∑q=1,q≠pLHp,qxq+zpwhere the second equation transforms the signal model into the matrix formulation, xp=[xp,1,…,xp,K]T, and zp~CN(0,σ2IM) is an additive white Gaussian noise (AWGN) vector.
Based on the above two expressions, the following discussion is mainly focused on the iterative intercell interference cancellation algorithm with pilot-aided channel estimation.
3. Channel Estimation and Intercell Interference Cancellation
At first, the channel estimation is realized based on the pilot sequences. The estimation is realized according to minimizing mean squared error (MMSE) criterion. Afterwards, the iterative intercell interference cancellation algorithm is introduced.
3.1. Channel Estimation
With the pilot allocation parameter F>1/L, more than one cell has the same pilot sequence set. In addition, we assume that the user number in each cell is equal to K. Consequently, the set Fp,k={(q,k)|sq,k=sp,k,q≠p} is introduced to denote user index set which has the same pilot sequence to user k in cell p.
In channel estimation, the channel response is obtained based on the received signal and the pilot sequence. In order to better represent the process of channel estimation, the received signal matrix Yp in (1) needs to be rewritten in the vector formulation y~p=vec(Yp), whose dimensionality is equal to M×τ. In order to accommodate the received signal y~p, the pilot sequence sp,k is extended to the following matrix expression Sp,k=sp,k⊗IM, which has M×τ rows and M columns. Therefore, the received pilot signal model shown in (1) can be rewritten as follows:(3)y~p=Sp,khp,kp+∑i=1,i≠kKSp,ihp,ip+∑q=1,q≠pL∑j=1KSq,jhq,jp+z~p, where z~p=vecZp.
The pilot setting described in Section 2 makes the extended pilot matrix Sp,k has the following property:(4)Sp,kHSp,k=τIM,Sp,iHSp,k=0M,k≠i,Sq,kHSp,k=τIM,q,k∈Fp,k,Sq,jHSp,k=0M,q≠p,j≠k.
According to the above property, by left multiplying Sp,kH, we have(5)bp,k=1τSp,kHy~p=hp,kp+∑q,k∈Fp,khq,kp+wp,kwhere ∑q,k∈Fp,khq,kp denotes the intercell pilot contamination and wp,k=(1/τ)Sp,kHzp, which obeys complex Gaussian distribution CN(0,σ2/τIM).
Based on MMSE criterion [14], the estimation result of hp,kp is given by(6)h^p,kp=Rp,kpRp,kp+∑q,k∈Fp,kRq,kp+σ2τIM-1bp,k. In addition, the intercell channel response hq,kp,q≠p, can be also obtained by the process similar to expression (5) and (6).
In the following, we define Δhp,kp≜hp,kp-h^p,kp as channel estimation error of hp,kp. Consequently, the covariance matrix of the above error can be given by(7)Cp,kp=Rp,kp-Rp,kpRp,kp+∑q,k∈Fp,kRq,kp+σ2τIM-1Rp,kp Moreover, for the matrix Hp,p, ΔHp,p=[Δhp,1p,⋯,Δhp,Kp] denotes its channel estimation error. And the covariance of ΔHp,p is given by(8)Cp,p=∑k=1KCp,kp Similarly, the intercell channel estimation error ΔHp,q and its covariance Cp,q can be also achieved.
In the next subsection, the iterative intercell interference cancellation algorithm is implemented based on the above channel estimation results.
For simplicity, the proposed iterative algorithm is firstly given in two-cell scenario. In each cell, there is only one user. The two-cell signal model is given by(9)y1=H1,1x1+H1,2x2+z1y2=H2,2x2+H2,1x1+z2The iterative structure with two cells is given in Figure 2. When iteration number is equal to t, we define v2,1(t) and v1,2(t) as the estimation result of the interferences H2,1x1 and H1,2x2, respectively. In addition, the residual interference at iteration t is given as follows:(10)Δv2,1t≜H2,1x1-v2,1t,Δv1,2t≜H1,2x2-v1,2t,and their covariance matrices are given by (11)W2,1t≜EΔv2,1tΔv2,1tH,W1,2t≜EΔv1,2tΔv1,2tH.When no cancellation happens with t=0, we have v2,1(0)=v1,2(0)=0 and(12)W2,10=EH2,1x1x1HH2,1H=bH^2,1H^2,1H+C2,1W1,20=EH1,2x2x2HH1,2H=cH^1,2H^1,2H+C1,2where C2,1 and C1,2 denote the channel estimation error covariance based on the analysis shown in the above section.
The iterative intercell interference scheme for a two-cell network.
For t=1, the following expectations is required to calculate the linear MMSE (LMMSE) estimators v2,1(1) of H2,1x1 and v1,2(1) of H1,2x2. They are given by(13)EH2,1x1y1H=H^2,1H^1,1EH1,2x2y2H=H^1,2H^2,2and(14)Ey1y1H=H^1,1H^1,1H+C1,1+W1,20+σ2IM≜U10Ey2y2H=H^2,2H^2,2H+C2,2+W2,10+σ2IM≜U20
Based on (12)-(14) and LMMSE estimation in [14], the signal detection and intercell interference construction can be denoted by the following expressions:(15)v2,11=H^2,1H^1,1U10-1y1v1,21=H^1,2H^2,2U20-1y2and the covariance matrices of estimation errors are given by(16)W2,11=W2,10-H^2,1H^1,1HU10-1H^1,1H^2,1H,W1,21=W1,20-H^1,2H^2,2HU20-1H^2,2H^1,2H.
After v2,1(1) and v1,2(1) are exchanged between the two BSs, they are subtracted from the received signal. So we have(17)y11≜y1-v1,21=H1,1x1+Δv1,21+n1,y21≜y2-v2,11=H2,2x2+Δv2,11+n2.
When the iterative processing converges, y1(1) and y2(1) are used to generate the final detection result of x1 and x2. Otherwise, they are used to calculate the estimator v2,1(2) and v1,2(2) for t=2 based on expression (13)-(15).
Furthermore, for the general multicell networks, we define vq,p(t) to be the interference estimation result from cell p to BS q in the tth iteration. At first, we define vq,p(0)=0. The estimation error at iteration t is denoted by Δvq,p(t)≜Hq,pxp-vq,p(t). Similar to (10), we define the covariance matrix of Δvq,p(t),(18)Wq,pt=EΔvq,ptΔvq,ptH Similar to that in (12), we have(19)Wq,p0=EHq,pxpxpHHq,pH=H^q,pH^q,pH+Cq,pBefore estimating the interference to other cells at one iteration, each BS subtracts from its received signal the interference estimated by other cells. Similar to (17), the signal after subtraction at BS p of iteration t is defined as yp(t), which is denoted by(20)ypt=yp-∑l≠pLvp,lt=Hp,pxp+∑l≠pLΔvp,lt+zpIt is noted that interference Hq,pxp from cell p to BS q is independent of estimation error Δvp,l(t),∀l≠p. To obtain the LMMSE estimator vq,p(t+1) of Hq,pxp using yp(t) at iteration t+1, we need the following expressions:(21)EHq,pxpyptH=H^q,pH^p,pHand Up(t)≜E(yp(t)(yp(t))H) which can be written as(22)Upt=H^p,pH^p,pH+Cp,p+∑l≠pLWp,lt+σ2IMApplying (19), (20), and (22), we obtain the LMMSE estimator of Hq,pxp at iteration t+1 as(23)vq,pt+1=H^q,pH^p,pHUpt-1yptand the covariance matrix of the estimation error(24)Wq,pt+1=Wq,p0-H^q,pH^p,pHUpt-1H^p,pH^q,pHIn addition, the LMMSE estimator of xp at iteration t can be expressed as(25)x^pt=H^p,pHUpt-1ypt
3.3. Distributed Implementation
In practice, interferences from nonneighboring cells are much smaller than those from neighboring cells. For the sake of complexity, we assume that each BS only interferes its neighboring BSs. Generally speaking, for any BS p, the set Ωp contains its neighboring BSs. Consequently, at BS p, the covariance matrices Up(t) and Wq,p(t+1) shown in (22) and (24) are generated from H^q,p and Wp,q(t) with q∈Ωp. Because of the block fading assumption, Up(t) needs to be built only once for one coherent interval in the tth iteration.
3.4. Complexity Analysis
According to the above analysis, for interference cancellation of L cells, the main complexity comes from expression (22)-(25). With known Up(t), x^pt is firstly calculated according to expression (25) and its complexity is M2+MK. Afterwards, with Up(t) and x^pt, expressions (23) and (24) are achieved, whose complexity is equal to MK and M2K+3M3, respectively. Finally, according to expression (22), the updating of Up(t+1) in the (t+1)th iteration has a complexity of (L+1)M3.
With L cells, the overall complexity in one iteration is given by(26)LM2+2MK+M2K+L+4M3.
4. Numerical Results
In this section, the performance of the proposed iterative interference cancellation algorithm is given. Users in each cell are randomly distributed. The intracell receiver correlation matrix is given by Rp,ip=IM. At the same time, the intercell receiver correlation matrix is given by Rq,ip=βq,ipIM,q≠p. The parameter βq,ip is defined as βq,ip=dq,ip/dp,ip-4 similar to that in [15], where dp,ip is the distance between the user i of cell p to BS p and dq,ip is the distance between the user with the same index i of cell q to BS p. According to the above analysis, the parameter βq,ip reflects the strength of intercell interference. In addition, the above setting makes the received signals of different antennas be independent with each other. The transmit modulation symbols of the users are 4QAM symbols. The length of pilot sequence is given by τ=K/F, in which the pilot allocation parameter F is configurable. Unless specifically declared, the MMSE channel estimation results are applied in the proposed iterative algorithm. The distributive implementation indicates that intercell interference cancellation only performs between neighboring BSs.
In Figure 3, the bit error rate (BER) performance of the proposed algorithm with L≤4 is given. The simulation parameters in Figure 3 are given in Table 1.
The simulation parameter in Figure 3.
BS Antenna Number M
40
User Number Per Cell K
30
Cell Number L
1,2,3,4
Iteration Number t
≤2
Bit error rate (BER) performance of the proposed algorithm with L≤4. In addition, the receiver antenna number M is equal to 40 and the user number K in each cell is equal to 30. The index t denotes the number of iterations.
From Figure 3, it can be seen that the pilot allocation parameter F is equal to 1 and 1/2 respectively, when L=1. In this case, there is no intercell interference. When F=1/2, the length of pilot sequence is longer than that with F=1. Consequently, the channel estimation performance with F=1/2 is better. The simulation result shown in Figure 3 indicates the BER performance advantage of F=1/2 over the case with F=1 when L=1.
In addition, the performance curve with F=1/2 and L=1 can be seen as the lower bound of the case with L=2. From Figure 3, it can be seen that after 1 iteration, the performance curve with F=1/2 and L=2 can approach the lower bound with F=1/2 and L=1. This means that the proposed algorithm can efficiently cancel the intercell interference when L=2. In addition, when L=3 and L=4, the intercell interference becomes large. After 2 iterations, the performance loss of L=3 and L=4 is about 1 dB and 1.3 dB compared with the lower bound with L=1 and F=1/2 when BER approaches 10-3.
In order to facilitate the effect of channel estimation, the simulation results with different pilot allocation parameters in L=3 scenario are given in Figure 4. The pilot allocation parameter F is equal to 1, 1/2, and 1/4, respectively. When F becomes larger, there exist more severe channel estimation errors. With F=1/4, the proposed algorithm can achieve better performance that with F=1 and F=1/2. In addition, the results in Figure 4 show that, after 2 iterations, the BER performance convergence can be achieved when F is equal to 1, 1/2, and 1/4.
Bit error rate (BER) performance of the proposed algorithm with L=3. The pilot allocation parameter F is equal to 1,1/2, and 1/4, respectively. In addition, the receiver antenna number M is equal to 40 and the user number K in each cell is equal to 30. The index t denotes the number of iterations.
In Figure 5, the scenario with L=7 and L=19 is considered. The corresponding parameters are given in Table 2. For comparison, the performance curve of the intercell interference-free case with L=1 and F=1/7 is given. It can be seen that, with F=1/7, the BER performance is better than that with F=1 because lower parameter F indicates longer pilot sequences and more reliable channel estimation results. In addition, it can be seen that, compared with the case with perfect channel estimation, the performance loss of F=1/7 is smaller than 1 dB when BER reaches 10-3. In the following, the curve with L=1 and F=1/7 can be seen as the lower bound of the proposed iterative algorithm with L=7. Figure 5 shows that, after 2 iterations, the performance curve with L=7 and F=1/7 has about 1.4dB loss compared with the above lower bound.
The simulation parameter in Figure 5.
BS Antenna Number M
20
User Number Per Cell K
10
Cell Number L
1,7,19
Iteration Number t
≤2
Bit error rate (BER) performance of the proposed algorithm with L=7 and L=19. In addition, the receiver antenna number M is equal to 20 and the user number K is equal to 10. The index t denotes the number of iterations.
In addition, it can be seen from Figure 1 that the distribution of cells presents a hexagonal structure. This means that there are at most 6 neighbors for each cell. In addition, we assume that the intercell interference mostly comes from neighboring cells. Therefore, the pilot allocation parameter F is set equal to 1/7 with L=19. Under this setting, the pilot contamination only comes from nonneighboring cells. Because of large path loss, the effect of nonneighboring cells can be ignored. The simulation results with L=19 and F=1/7 support the above analysis. It can be seen that, with 2 iterations, the performance curve with L=19 and F=1/7 approaches that with L=7 and F=1/7. This indicates that the interference of nonneighboring cells has very little effect on the proposed iterative algorithm.
Furthermore, the convergence property of the proposed algorithm is given in Figure 6. When L=3, the BER performance can achieve convergence after 2 iterations, with F equal to 1, 1/2, and 1/4, respectively. The performance gain is very marginal with more than 2 iterations. When F is equal to 1/4, the channel estimation performance is the same for the cases with L=3 and L=4. The simulation results shows that the performance with L=3 and F=1/4 is better than that with L=4 and F=1/4 since fewer neighboring cells introduces less intercell interference. At the same time, it is noted that the case with L=4 and F=1/4 can also converge after 2 iterations.
Bit error rate (BER) convergence of the proposed algorithm. In addition, the receiver antenna number M is equal to 40 and the user number K is equal to 30.
5. Conclusions
In this paper, an iterative intercell interference cancellation algorithm is proposed. The principle of this algorithm is based on minimizing mean squared error (MMSE) criterion. In addition, the concrete expressions of the iterative cancellation are derived. From the numerical results, it can be seen that intercell interference can be mitigated efficiently with a moderate number of antennas.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by National Natural Science Foundation of China (61601047, 61671080, and 61871050) and Huawei HIRP Project HIRPO20161103.
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