Yitian Chen,Shaoshuai Gao,Guofang Tu,Hao Qiu

School of Electronic,Electrical and Communication Engineering,the University of Chinese Academy of Sciences,Beijing 101408,China

Abstract: Massive multiple-input multiple-output(MIMO),a technique that can greatly increase spectral efficiency(SE)of cellular networks,has attracted significant interests in recent years.One of the major limitations of massive MIMO systems is pilot contamination,which will deteriorate the SE.The superimposed pilot-based scheme has been proved to be a viable method for pilot contamination reduction.However,it cannot break through another limitation of massive MIMO,i.e.,spatial correlation.In addition,it will also lead to interference between the pilot and user data since they are imposed together.In this paper,we try to tackle these two issues,which will be described as follows.Firstly,a column-wise asymptotically orthogonal matrix,named as pseudo-channel matrix,is developed by orthogonalization of received signal.To recover the information about the large-scale fading(LSF)coefficients,the pseudo-channel matrix is truncated according to the cardinality of adjacent users set(CAUS).By this means,spatial correlation can be mitigated effectively.Secondly,robust independent component analysis(RobustICA)is used to reduce the interference caused by user data,and as a result the system performance can be further improved.Numerical simulation results demonstrate the effectiveness of the proposed method.

Keywords: massive MIMO; pilot contamination;pseudo-channel matrix; spatial correlation; superimposed pilots.

I.INTRODUCTION

With the ever increasing number of users and data traffic,next generation wireless networks need to fulfill higher data rates and lower energy consumption.Massive multiple-input multiple-output (MIMO) systems have attracted tremendous interest from academia and industry in recent years for its ability to improve spectral efficiency (SE) and energy efficiency (EE)[1-3].Unlike traditional MIMO systems,massive MIMO systems require hundreds of antennas to be installed at the base station(BS),which can bring a lot of benefits.For instance,SE can be improved by means of spatial multiplexing,which allows multiple users to use the same time-frequency resource[4,5].Besides,a large number of antennas leads to improved EE[6,7].Moreover,simple linear processing turns out to be optimal when the number of BS antennas is sufficiently large[8].

The benefits described above are based on the crucial assumption that the BS has access to the accurate estimates of channel state information(CSI),which is usually estimated by using orthogonal pilot sequences.In order to save the required pilot sequences,Time Division Duplexing(TDD)is preferentially employed since only uplink pilot sequences need to be transmitted[6,9,10],while Frequency Division Duplexing(FDD)needs to transmit both uplink and downlink pilot sequences.However,the coherence interval is limited,which in turn limits the number of orthogonal pilot sequences for channel estimation.Therefore,some users may still share the same pilot sequence inevitably even with TDD if the number of users is large,which will result in serious coherent interference,called pilot contamination.

Pilot contamination has been widely investigated in recent years[11].According to whether spatial correlation is considered,the existing pilot contamination reduction methods are broadly classified into two categories,i.e.,uncorrelated and correlated approaches.The uncorrelated ones assume spatially uncorrelated channels,which are asymptotically orthogonal when the number of BS antennas becomes infinite.In[12],the authors showed that CSI can be estimated directly from the eigenvectors of the covariance matrix of the received signal when the number of BS antennas grows large.In[13],the authors proposed to project the received data onto the subspace of the received data through singular value decomposition (SVD).In[14],a semiblind channel estimation method was proposed,where the data obtained by robust independent component analysis (RobustICA) is used to estimate CSI.It should be noted that all the above three articles use time-multiplexed pilots (TP),i.e.,the pilots are time-multiplexed with data during the training phase.An alternative pilot structure to mitigate pilot contamination is superimposed pilots(SP),in which the entire coherent block can be utilized to transmit the pilot sequence,and as a result more users can be supported without incurring extra bandwidth.In[15],an iterative approach was proposed to reduce the interference that results from transmitting data alongside the pilots.In[16]and[17],a hybrid system of TP and SP was proposed to offer a higher downlink(DL)throughput.Different from the above schemes,the correlated approaches assume spatially correlated channels which are more practical in massive MIMO systems.In[18],pilot contamination can be completely eliminated by using minimum mean square error(MMSE)estimator if the desired and interfering users do not overlap in the angular domain.On this basis and according to the number of BS antennas and the location of the desired user,a desired angular region was characterized in[19],in which the interference can be eliminated effectively due to spatial separability.In[20],the authors proposed that the pilot contamination components can be identified by spatial signatures,and then the spatial filter can be constructed to eliminate pilot contamination.

In the aforementioned studies,some of them consider the spatial correlation and utilize some methods to reduce pilot contamination.However,its impairment effects are not taken into account.As a matter of fact,spatial channel correlation degrades the system SE[21,22].There are some research works that analyse the impact of spatial correlation in massive MIMO systems.Under the condition of favorable propagation,the ultimate performance limit can be achieved when the number of antennas grows infinitely.However,in[8],the authors showed that if the channel is spatially correlated,the signal to interference plus noise(SINR)is smaller than the ultimate performance limit.In[23],the authors showed that the unlimited capacity can be achieved only if the rank of the channel covariance matrix scales sublinearly with the number of BS antennas.In[24],the authors showed that increasing the number of antennas cannot effectively improve the system performance due to spatial correlation.Some literatures used interference alignment(IA) technique to reduce multi-cell interference such as[25]and[26].However,these papers assume that channel matrices of different users are independent.

In this paper,we propose a spatial correlation mitigation technique based on pseudo-channel matrix truncation.The pseudo-channel matrix,a columnwise asymptotically orthogonal matrix,is obtained by orthogonalization of the received signal through eigenvalue decomposition (EVD).It should be noted that pilot transmission and uplink data transmission are in the same phase for superimposed pilot scheme,which means that an unique pseudo-channel matrix can be obtained and therefore the CSI and the data information can be estimated accurately1.The large-scale fading (LSF) coefficients are suppressed in pseudochannel matrix,which will cause low SINR.To recover the information about LSF coefficients,the pseudo-channel matrix will be truncated according to the cardinality of adjacent users set (CAUS).As a result,spatial correlation can be mitigated.Moreover,since the pseudo-channel matrix is column-wise asymptotically orthogonal,the RobustICA is considered to reduce the interference caused by user data.More specifically,the contributions of our paper are as follows:

·Firstly,we show that for large number of BS antennas,the probability that the channel matrix of users (in a target cell and its adjacent cells) is a full rank matrix approaches to one.Therefore,it is possible to distinguish user data from the received signal.

·Secondly,a spatial correlation mitigation method is proposed.We demonstrate that the received signal can be converted into a column-wise asymptotically orthogonal matrix named pseudochannel matrix multiplied by the data matrix.However,the LSF coefficients are suppressed in the pseudo-channel matrix.Based on the fact that the values in each column in the matrix can be distinguished if their LSF coefficients differ greatly,the CAUS-based pseudo-channel matrix truncation is applied.By this means,the interference caused by spatial correlation can be mitigated effectively.

·Thirdly,combined with pseudo-channel matrix truncation,constrained RobustICA is employed to reduce the interference caused by uplink data symbols in superimposed pilot systems,and therefore the symbol error rate (SER) performance can be further improved.

The remaining of this paper is organized as follows.In Section II,the system model comprising the network model,the channel model and superimposed pilots is introduced.Besides,the concept of spatial filter and the problem of multipath channel are presented.In Section III,a novel method based on pseudo-channel matrix truncation is proposed to reduce the spatial correlation interference.In Section IV,constrained RobustICA is introduced to reduce the interference caused by uplink data and further improve the system performance.Section V demonstrates the performance of the proposed methods.Section VI draws conclusions.

Notation:Lower case and upper case boldface letters denote column vectors and matrices,respectively.The notations (·)∗,(·)T,(·)H,and (·)-1represent the conjugate,transpose,Hermitian transpose,and inverse,respectively.The notation INdenotes anN×Nidentity matrix,and‖·‖denotes the Euclidean norm of a vector.represents the expectation of any random variable,vector or matrix.(A)Ωis a matrix formed by setting the rows of A,which correspond to indexes not in the set Ω,to zero.(a)Ωis a vector formed by setting the elements of a,which correspond to indexes not in the set Ω,to zero.[A]:,nis a vector which contains then-th column element of A.represents a complex Gaussian distribution with meanμand covariance matrix Σ.

II.SYSTEM MODEL

2.1 Network Model

Here we consider an uplink scenario of a multi-cellular massive MIMO system consisting ofL ＞1 hexagonal cells.Each cell containsKsingle-antenna users and one BS equipped withM ≫Kantennas.The same frequency band is used for allLcells and the TDD mode is employed.The length of coherence interval is denoted byC,within which the effect of channels reduces to a multiplication of a complex-valued scalar gain.It can be divided intoCuandCd,which are the number of symbols in the uplink and downlink time slots,respectively.The location of userkin thej-th cell is denoted bywhile the location of the BS in thei-th cell is written as.

2.2 Channel Model

The uplink channel of userkfrom celljto BSiis denoted byFor ease of analysis,it is assumed that the antenna arrangement is an uniform linear array (ULA) and the antenna spacing is half the wavelength[19].Hence,the following multipath model can be obtained.

whereBis the arbitrary number of i.i.d.paths,βijkis the LSF coefficient which is written asfor a path-loss exponentηand a known constantαthat is dependent on the prescribed average cell-edge signal noise ratio(SNR)[19],bis the path index,is the random phase of theb-th path,andis the antenna steering vector shown as

The AoA distribution based on the“one-ring”model shown in Figure 1 is usually considered[27].The main parameters of the model include thex-axial distanceDijkbetween BSiand userkfrom cellj,the radiusrijkof the scatterer ring,the angleφijkbetween BS and user,and the angular spread Δijk.The AoA distribution is confined within[φijk −Δijk,φijk+can be represented as

Figure 1. Illustration of the“one-ring”model.

Figure 2. Illustration of matrix QH.The left matrix represents the matrix HP,in which the big red block represents the asymptotically orthogonal transformation of QA and the blue blocks represent one.The right matrix represents the matrix ,in which the row vectors of are asymptotically orthogonal to the rows of matrix ST.The white blocks represent zero.

For ease of analysis,it is assumed that all users have the same angular spread,i.e.,Δijk=Δ,∀i,j,k.

Remark 1.It should be noted that strong scatterers may cause large angular spreads,or the location of users may not be accurately known.These aspects can be incorporated into the system by allowing a larger radius rijk,which means a larger range of angle[19].

Remark 2.Although different users have different angular spreads,to simplify the mathematical presentation,the largest one among them will be chosen as the angular spread.

The channel vectors can be represented in the angular domain with inverse discrete Fourier transform(IDFT) when the ULA is considered.Let (hθ)ijk ∈be the angular domain representation of channel vector hijk.Then,the channel can be written as[20]whereis the discrete Fourier transform(DFT)matrix whose elements areDue to the compact deployment of antennas and the limitation of the number of scatterers,massive MIMO channels show a correlation in space and have fewer degrees of freedom than the number of antennas[28,29].This means that a substantial number of element values in the angular domain will be zero or close to zero.Letbe the index of the non-zero value of(hθ)ijk.When the number of antennas is sufficiently large,the relation betweenand the AoA of theb-th path can be written as

Therefore,the corresponding indexes of non-zero values can be found if the AoAs of each user are known.It is assumed that the BS knows the location of each user2so that the AoA distribution of each user can be calculated according to (3)[19].For the convenience of description,letbe the support set of (hθ)ijkwhich contains the indexes of non-zero values.And letrepresent its complementary set.

2.3 Superimposed Pilots

To avoid the adverse effects due to pilot contamination,the superimposed pilots based scheme will be considered.In the hexagonal shaped cells,each user can be assigned an unique orthogonal pilot if the total number of users in the target cell and its adjacent cells is less than the number of uplink symbols,i.e.,7K ＜Cu.In this case,the received signal at thei-th BS can be written as

The spatial filter can be operated by setting the rows ofwhich correspond to the indexes not in the set Ωijk,to zero,i.e.,Solving(7)yields

The third term in(8)represents the interference caused by the uplink data,i.e.,xmn.For the sake of clarity and without loss of generality,two indexesj,kare replaced with a single indexmthat lies in the range 0≤m ≤N −1,whereBesides,we consider the signal received by the BS in cell 0 unless otherwise specified so that the subscriptiof the BS can be ignored.To achieve lower receiver complexity,the matched filter(MF)detector,which is a linear detector and maximizes the gain of the desired signal,is employed for data estimation.By using(8),the detected output can be given by(9)

III.SPATIAL CORRELATION MITIGATION

In this section,we will propose a spatial correlation mitigation method based on pseudo-channel matrix truncation for massive MIMO systems.Rewrite the received signal given by(6)as

3.1 The Rank Of Channel Matrix

With a bounded support of AoAs,the rank of channel covariance matrix Cijksatisfies

Theorem 1.For a large number of BS antennas M,it is assumed that for each user,its paths do not overlap completely after arriving at the BS.ConsideringK users whose number of paths is B,the probability that the channel matrix is a full rank matrix can be represented as

where.

Proof.Firstly,arbitrarily choose a user that occupiesBangles in the angular domain and then select another user.In order to ensure that the channel matrix is a full rank matrix,their support sets cannot be the same.Therefore,the second user can only selectofcombinations to ensure full rank,and the probability can be expressed asThek-th user must ensure that the support set is different from the previousk −1 users.Therefore,the number of combinations that thek-th user can select isand the corresponding probability isMultiplying the probabilities,we arrive at the desired result(13).

For largeM,the result of Theorem 1 is approximately equal to the actual full rank probability even if we only consider the case where the paths of each user do not overlap.For example,letM= 100,K=5,B= 2,the probability computed from (13) is 0.9980.The simulated probability is 0.9979,which is simulated over 105independent iterations.Please noted that it is closed to the actual probability when the number of iterations is large enough.

3.2 Pseudo-Channel Matrix

In this subsection,we present the basic concept of the pseudo-channel matrix,which is a column-wise asymptotically orthogonal matrix.It is assumed that the total number of users in the target cell and its adjacent cells is less than the number of antennas at BS and the number of uplink symbols,i.e.,N ＜MandN ＜Cu.This assumption usually holds because the number of BS antennas is far more than the number of users per cell and there are also enough uplink symbols in general[1,15].

According to(1),the channel matrix can be written aswheredenotes the diagonal matrix with LSF coefficients,denotes the small-scale fading matrix.By using SVD,Z can be rewritten aswhere U∈are unitary matrices with the columns containing the singular vectors ofandis a diagonal matrix containing the singular values.It should be noted that the elements of diagonal matrixandare all real numbers,thusThe following formula is considered and the proof is given in Appendix A.

Combining(14)and(15),we have

Figure 3. Illustration of pseudo-channel matrix truncation.The left part represents the intensity of pseudo-channel matrix Hp and the right part represents the intensity of the truncated matrix.The blue blocks represent one while the white blocks represent zero.

3.3 Pseudo-Channel Matrix Truncation

As we have mentioned above,the suppression of LSF coefficients matrix causes performance degradation.In order to solve this problem,a pseudo-channel matrix truncation method will be proposed in this section.This method is based on the fact that for each column in pseudo-channel matrix Hp,the values can be distinguished if the LSF coefficients differ greatly.

Theorem 2.In a two-dimensional case,letA =diag(β1,β2)and β1≫β2,Vis a two-dimensional unitary matrix.[u11,u21]T and[u12,u22]T are the eigenvectors of matrixAcorresponding to the larger eigenvalue and the smaller eigenvalue,respectively.The following holds true:Proof.See Appendix C.

This conclusion can be extended to an-dimensional case (see Appendix C) and the left part of Figure 3 shows this phenomenon.The darker the color,the larger the value and vice versa.Hence,the pseudochannel matrix Hpcan be truncated to the right part of Figure 3 to recover the information about the LSF coefficients.Now the problem is how to determine the optimal truncation length.To solve this issue,we will redefine the concept of adjacent users as follow.

Figure 4. SINR(dB) of users in the target cell with M =200,Cu =300,K =15,Δ=π/10.(a)The SINR(dB)of a user in the target cell vs.|Km|.(b)Cumulative distribution of the achieved SINR(dB)of users in the target cell.

Figure 5. Cumulative distribution of the normalized channel estimation error in dBs of users in the target cell with M =200,Cu =300,K =15,Δ=π/10,δLSF =0.1.

Definition 1.If user m and user n overlap in the angular domain,which means that their support sets intersect,i.e.,Ωm ∩Ωn /=⊘,and their LSF coefficients satisfy the following relation

where βm,βn represent the LSF coefficients of user m and user n respectively,δLSF ≤1represents the threshold,then user m and user n are called adjacent users.

LetKm ⊂{0,...,N −1}be the set that contains the adjacent users of usermand itself.Then the cardinality of the setKm,which is represented as|Km|,can be used as the truncation length.The operation of truncation can be done by constructing an auxiliary matrix Gm.TheNth-order leading principle minor of G is an unitary matrix and the remaining values are zero.By left multiplying G,we can obtainwhereIt should be noted that the complexity of computing the truncation lengths for a target cell is aboutO(7K2).Compared with the computation cost of data and channel estimation,this computation cost can be ignored.

Remark 3.The smaller the value of δLSF,the more users are selected as adjacent users,which means that the value of|Km|is large and the differences between LSF coefficients cannot be recovered fully.Increasing the value of δLSF reduces the number of adjacent users,but at the same time the column orthogonality ofwill be destroyed and the system performance also deteriorates.Therefore,it is necessary to make a trade-off between the number of adjacent users and the column orthogonality ofto find an optimal δLSF.For the case of two users,we demonstrate that δLSF=0.1can be a good compromising value and the explanation is provided in Appendix D.In the case of more than two users,the analytical solution of optimal δLSF cannot be obtained.In Section V,simulation results show that the system will perform well when δLSF= 0.1is exploited in the case of more than two users.Hence,δLSF= 0.1can be regarded as a compromising value for any number of users.

3.4 Analysis of SINR

Then,MF detector mentioned in (9) is performed to estimate data by using(21),

Now,the estimated data can be used to estimate the CSI of userm,

If we restrict the parametersρmandλmsuch thatρm=ρ,λm=λ,∀m,the approximate SINR of usermat the output of MF detector is given in(24)and is derived in Appendix B.

IV.CONSTRAINED ROBUSTICA

Due to the mixture of pilots and user data,the channel estimation of the superimposed pilot scheme suffers from co-interference from user data[17].In fact,if the data of interest can be extracted from(20)directly,the interference of user data will be eliminated.This method can be simply described as blind source separation (BSS).Independent component analysis (ICA)is the most natural tool for BSS in the case of instantaneous linear mixtures[30].Compared to many ICA algorithms,RobustICA has a lot of advantages.Firstly,it can process both real-valued and complex-valued sources without any modification.Secondly,it does not require prewhitening,which make it more suitable than whitening-based techniques for spatially correlated sources[30].As we have discussed in Section III,the truncation operation will destroy the column orthogonality of the pseudo-channel matrix.Thus,RobustICA can provide better performance than other ICA algorithms in our system.To incorporate more assumptions and prior information,constrained RobustICA is applied in this paper,which is a systematic and flexible method[31].

4.1 Brief Introduction of Constrained RobustICA

For(20),an extracting vector w is first sought so that the estimatemaximizes the kurtosis,which is defined as

In order to obtain a more accurate result,some constraints are imposed on the extracting vector.The following constrained maximization problem is formulated.

whereγmis a threshold.We can obtain

whereζis defined as the summation of the last two terms in(27).Then we have

Using the method of Lagrange multipliers,the optimal solution of (26) can be obtained by solving the following unconstrained optimization problem

After normalizing w,λis updated as

The globally optimal step sizeμoptcan be obtained by finding the roots ofL(w,λ)which can be expressed as polynomials of the step sizeμ.The calculation details ofμoptcan refer to[14].

The above search procedure will be ceased when

or the maximum iteration number is reached.w(j)denotes the values of w obtained after thej-th iteration andεis a small constant.

After the solution of (26) is obtained,the source from them-th user can be extracted as

Note that for the ICA method,there exist phase and scale ambiguities in the extracted source becauseis a complex vector and w is normalized.Therefore,the pilot sequence is used to correct the ambiguities.

4.2 The Whole Procedure of the Proposed Method

Algorithm 1.The whole procedure of the proposed method.Input:Received Signal Y,pilot matrix P,threshold δLSF Output:Channel estimate ˆHm,user data ˆXm 1:Arrange the user in the decreasing order of their LSF coefficients and set m=1;2:Compute the support set of user m through (4),i.e.,Ωm;3:Compute the IDFT of received Signal Y through(6),and then employ spatial filter to obtain(¯Y)Ωm;4:Obtain eigenvectors Qm by computing the EVD of1 MCu(¯Y)HΩm(¯Y)Ωm through(15);5:According to δLSF,compute adjacent users set of user m through(19),i.e.,Km;6:Construct the auxiliary matrix Gm,the rank of which is|Km|;7:Calculate the new received signalimages/BZ_147_2067_1354_2111_1400.pngimages/BZ_147_2111_1343_2136_1388.pngYm =√CuGmQm;8:Solving(26)to obtain the extracting vector w;9:Extract the m-th source as in(33);10:Estimate the channel hm through(23);11:Update the received signal as(34);12:Set m=m+1.If m ＜=K,go to step 2,otherwise,end algorithm;

For largeM,since the SINRs of the users are proportional to the users’ LSF coefficients,the users are arranged in the decreasing order of their LSF coefficients,i.e.,β1＞β2＞···＞βN3.We first estimate the user with the largest LSF coefficients,then remove the user from the original received signal and estimate the next user.After estimating them-th user,the received signal used to estimate the remaining users can be expressed as

where Y(m)represents the received signal used to estimate them-th user and Y(1)= Y.In summary,the whole procedure of the proposed method is summarized in Algorithm 1.The proposed method mainly includes two parts,i.e.,pseudo-channel matrix truncation(step 2 to step 7)and RobustICA(step 8 to step 11).The complexity of these two parts are aboutfor largeM.The parameterJrepresents the number of RobustICA iterations,which is usually smaller than 10[14].Thus the complexity of the proposed method is about(2J+ 3)KMCu) for largeM.Compared with the spatial filtering method whose complexity is aboutO(KM2Cu),the complexity of the proposed method is increased since it is implemented on the basis of spatial filter.For example,letM=300,Cu=100,K=5,J=10,the simulation result shows that the computation cost of the proposed method is 49 percent more than that of spatial filter.However,the benefit is that the SE and SER performance can be improved.

V.SIMULATION RESULTS

In this section,we present numerical results to evaluate the proposed method.

5.1 Simulation Scenario

We consider hexagonal shaped cells and simulations are performed with a multi-cell system withL= 49 cells[32].The simulation parameters are almost identical to those employed in[18]and[19],as shown in Table 2.We keep these parameters fixed for the simulations unless otherwise specified.Moreover,we restrict the parametersρmandλmsuch thatρm=ρ,λm=λ,∀mand set the values ofρandλasFor a cell radiusrcell,αis set toα[dB]=γSNR+10ηlog10(rcell)+10 log10(σ2),whereγSNRis the cell-edge SNR in dB andσ2is the receiver noise power variance.Monte Carlo simulation results are averaged over 104independent trials.Two scenarios are considered:

Table 1. The most commonly used notation.

Table 2. Simulation parameters.

(a)Scenario 1:The users are uniformly distributed in hexagonal cells.

(b)Scenario 2:The users are on a circle with a radius of 800m.This scenario is used to demonstrate the edge effects.

The performance of the proposed scheme is compared with several estimation schemes.The schemes are as follows.

·TP:Time-multiplexed pilots scheme in scenario

1.This is a conventional pilot structure.

·SP:Superimposed pilots scheme in scenario 1.This scheme benefits from a higher value ofCuand can avoid the adverse effects due to pilot contamination[15].

·SPS:Superimposed pilots with spatial filter scheme in scenario 1.Spatial filter can be constructed to filter out some interferences while maintain most energy of the desired signals,and therefore users,especially cell-edge users,can achieve better performance[20].

·Proposed-NOICA(S1):Proposed method without RobustICA in scenario 1.

·Proposed-ICA(S1):Proposed method with RobustICA in scenario 1.

·Proposed-NOICA(S2):Proposed method without RobustICA in scenario 2.

·Proposed-ICA(S2):Proposed method with RobustICA in scenario 2.

5.2 Performance of SINR and NMSE of Channel Estimation

Figure 4a shows the SINR(dB)of an arbitrary user in the target cell.The blue curve represents the simulated curve and the red curve represents the theoretical curve which is computed from(24).When the value of|Km|is less than 8,the red curve performs better than the simulation results.This is because the column vectors ofare not completely orthogonal.When|Km|becomes larger,the main interference is caused by the adjacent users and the blue curve is close to the red one.Figure 4b shows the cumulative distribution of the achieved SINR of users in the target cell.We can observe that the proposed method is obviously superior to other existing methods.For example,at the 90%likely SINR(dB)point(where the vertical axis is 0.1),the Proposed-ICA(S1) can achieve 11 dB gain in comparison to the SPS.Besides,it can also be observed that the proposed method applied to scenario 2 only performs a little worse than that applied to scenario 1,which means that it has good performance for cell-edge users.Moreover,the proposed method with ICA performs better than the method without ICA because the former reduce the interference caused by uplink data.

Figure 6.SER of users in the target cell for different number of antennas at BS with Cu = 300,K = 15, Δ = π/10,δLSF =0.1.

Figure 7. SER of users in the target cell for different celledge SNR with M =200,Cu =300,K =15,Δ=π/10.

In Figure 5,we present the CDF of the normalized mean square error (NMSE) of channel estimation of users in the target cell.The performance of SP scheme is better than that of TP scheme because different users can be assigned different pilots.SPS scheme only considers the users who overlap with the desired users in space and filters out the others.This operation can improve the SER performance.Unlike the previous three methods which only use pilots to estimate CSI,the proposed method uses both user data and pilots to estimate channel.This will suppress the effect of correlation between data and pilots and therefore better performance can be achieved.For example,at the 90%likely NMSE point,the Proposed-ICA(S1)performs 2 dB better than the SPS.

Figure 8. SER of users in the target cell for different LSF threshold with M =200,Cu =300,K =15,Δ=π/10.

Figure 9. SER of users in the target cell for different modulation scheme with M =200,Cu =300,Δ=π/10.

5.3 Impact of System Parameters

In Figure 6,the SERs versus the number of BS antennasMis illustrated.It can be seen from the figures that the SER performance will get better when increasingM.It is mainly due to the fact that the orthogonality ofwill be enhanced in this case.Moreover,it is obvious that the proposed method performs better than others.ForM=200 in Figure 6,the SER of the Proposed-ICA(S1)is 1%of that of the SPS.From Figure 7,we can notice that the SNR is no longer the major factor affecting the system performance when the cell-edge SNR is higher than 0 dB.The performance improvement of the proposed method at-20 dB is not obvious compared with the other three schemes.This is because the noise will destroy the column orthogonality of the pseudo-channel matrix when the SNR is very low.

In Figure 8,the SER is shown versusδLSF.WhenδLSFis less than 0.1,SER becomes higher asδLSFdecreases because the number of adjacent users increases.WhenδLSFis higher than 0.1,SER becomes higher asδLSFincreases because the orthogonality ofare destroyed.The method with ICA performs worse than the method without ICA whenδLSF=0.3.This is because the performance of RobustICA relies on the orthogonality of the column vectors.Figure 9 shows the SER performance of the proposed method under different modulation.The SER performance will decrease when the order of modulation increases.It also implies that for higher-order modulation,the number of supported users must be reduced to maintain the best performance.

5.4 Effect of Location Error

The position estimated error can be incorporated into the system by allowing a larger radius,which means a larger angular spread.Hence,the increment of angular spreaddΔ can be used to represent location error.Consider a user whose parameters are set asDijk= 500,φijk=π/2 anddΔ =π/10.We keep the position of this user fixed and other parameters are shown in Table 2.Three cases are considered,i.e.,exact location knowledge(C1),only the desired user has location error (C2) and all users have location errors(C3).Table 3 shows the SER comparison of the proposed method and the SPS in these cases.From Table 3,the proposed method in C3 increases the SER by 307%and 62%relative to the proposed method in C1 whenrespectively.This means that location error will degrade system performance.However,it can be seen that the SER of the proposed method in C3 is 0.2%and 1.9%of that of the SPS in C1 whenrespectively.In other words,the proposed method with location error still performs better than the SPS with exact location.

Table 3. The SER comparison of the proposed method and the SPS under location error.

SPS

Proposed-NOICA(S1)

VI.CONCLUSIONS

In this paper,two issues in massive MIMO systems are considered.Firstly,we have proposed a method based on pseudo-channel matrix truncation for mitigating the impairment effect of spatial correlation.The key idea is to replace the channel matrix with a columnwise asymptotically orthogonal matrix,called pseudochannel matrix.To recover the LSF coefficients that are suppressed by EVD,the CAUS-based pseudochannel matrix truncation is applied.We demonstrate that proper truncation length can greatly improve the system performance.Secondly,RobustICA is introduced to improve the performance of the superimposed pilot-based system by reducing the interference due to uplink user data.The proposed method does not need cell cooperation and considers the practical channels which are spatially correlated.Moreover,this method performs well when the number of BS antennas is limited and more users can be supported.Simulation results show that the proposed method performs better than the existing methods in terms of SER,SINR and NMSE of channel estimation.Specifically,compared to the existing methods,the proposed method withM= 200 can achieve 11 dB and 2 dB improvement in SINR and NMSE of channel estimation respectively and 99%reduction in SER.

ACKNOWLEDGEMENT

This work was supported in part by National Key R&D Program of China (2017YFB0403604),in part by the Fundamental Research Funds for the Central Universities (No.292021000242),in part by the National Natural Science Foundation of China(Grant No.61571416,61072045,61032006).

NOTES

1It does not mean that the proposed method is not suitable for TP.As a matter of fact,an unique pseudo-channel matrix can be obtained after the pilot transmission phase and the uplink data transmission are both completed.However,in such case,we have to design a pilot allocation scheme to mitigate pilot contamination.This is out of the scope of this paper.Furthermore,SP performs better than TP in channel estimate.Hence,only SP is considered in this paper.

2In communication systems,the user location information can be obtained by applying some positioning technologies,such as global positioning system(GPS)and global navigation satellite system(GNSS)[33,34].The discussion of location error has been shown in Remark 1 and its effect will be demonstrated in Section V.

3It is assumed that the BS has access to exact values of LSF coefficients[15].This assumption is reasonable because for large M,the LSF coefficients can be computed by averaging the power of the channel coefficients.

APPENDIX

A.PROOF OF(14)

According to(11),we have

The approximation errors between (14) and (35) are proportional toandIfM ≫1 andCu ≫1,then the second and third terms in (35) can be neglected.Similarly,Ifσ2≪Cu,then the effect of noise can also be neglected becauseHence,we arrive at the desired result(14).

B.PROOF OF(33)

Rewrite the estimation data in(22)as

where we define the estimation error of the channel as

The first term in (36) is the signal and the last term is the noise components.As we discussed in Section III,matrixis asymptotically orthogonal when the LSF coefficients differ greatly,i.e.,

The notations from f1to f4are the interference components.The average power of interference can be written as

Hence,the interference power can be obtained as

Using (40) and ignoring the noise in (36) which is small whenσ2≪Cu,the SINR can be obtain as(41).

C.PROOF OF THEOREM 2

In a two-dimensional case,we define

The eigenvectors of matrix A,i.e.,u=[u1,u2]T,satisfy with the following expression

whereλrepresents the eigenvalue and can be calculated through(44).

For the larger eigenvalue,sinceandλmax ＞0,we have

Hence,we have|u1| ≫|u2|.For the smaller eigenvalue,we can obtain|u1| ≪|u2|.This conclusion can be extended to an-dimensional case.For aNdimensional case,it is assumed that the LSF coefficients ofkusers are much greater than those of the restN −kusers,i.e.,β1≫β2.Equation(44)can be written as(47).

Similarly,a2,a3anda4can be defined.Now,we add the firstkequations in(47),and then add the restN−kequations.Two new equations can be obtained and can be represented as

Hence,for the larger eigenvalue,we have

In other words,|u1| ≫|uk+1|.It also means thatFor the smaller eigenvalue,the opposite conclusion can be obtained.

D.JUSTIFICATION TO THE COMPROMISING VALUE OF δLSF

Use the same symbolic definitions as Theorem 2.For the larger eigenvalue,we have

Letβ1=αβ2,whereis the ratio ofβ1toβ2.Then we have

In general,the value ofc22can be ignored when the value ofαis large andc11＞|c12|.It should be noted that the functionis a monotone decreasing function.Considering a worst-case wherec11=|c12|and letα= 10,we havewhich means that more than 90% of the energy of the eigenvector corresponding to the larger eigenvalue concentrates on its first index,i.e.,u1.If we increase the value ofα,the energy will not increase significantly and the number of adjacent users will increase.Hence,we considerα= 10,namelyδLSF= 0.1,as the compromising value.