Multicell multiuser massive MIMO channel estimation and MPSK signal block detection applying twodimensional compressed sensing
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Abstract
For the uplink multicell massive multipleinput multipleoutput (MIMO) block fading systems, a twodimensional smoothed l_{0} channel estimation method (2DSL0CE) with the aid of virtual channel representation is firstly exploited in this paper, which can jointly estimate the desired multiuser channels of the target cell and the interference links from neighbor cells without inducing pilot contamination. Then, a 2DSL0 signal detection method (2DSL0SD) with the aid of sparse decomposing and the modified 2D sl_{0} recovery algorithm is proposed, which can jointly decode Mary phaseshift keying (MPSK) signal block for whole desired users. Moreover, an improved 2DSL0SD is also proposed to remove multiuser interference of neighbor cells in high SNR scenario. Simulation results show that the 2DSL0CE method can remove performance floor induced by pilot contamination and need less pilot overhead than the conventional least square (LS) method. When detecting QPSK signal blocks at 12 dB SNR, the 2DSL0SD method with perfect channel state information (CSI) can obtain 10^{−2} BER. Moreover, in the case of 8PSK signals, the 2DSL0SD joining with the 2DSL0CE can obtain 10^{−2} BER at 20 dB SNR.
Keywords
Massive MIMO Sparse channel estimation Data block detection Twodimensional smoothed l_{0} (2DSL0)Abbreviations
 2DSL0
Twodimensional smoothed l_{0}
 2DSL0CE
Twodimensional smoothed l_{0}norm channel estimation
 2DSL0SD
Twodimensional smoothed l_{0}norm signal detection
 2DSL0SDIC
Twodimensional smoothed l_{0}norm signal detection with interference cancel
 BER
Bit error rate
 BS
Base station
 CS
Compressed sensing
 CSI
Channel state information
 FDD
Frequency division duplexing
 JSDM
Joint spatial division and multiplexing
 LS
Least square
 MIMO
Multipleinput multipleoutput
 ML
Maximum likelihood
 MMSE
Minimal mean square error
 MPSK
Mary phaseshift keying
 NMSE
Normalized mean square error
 QPSK
Quadrature phaseshift keying
 SNR
Signaltonoise ratio
 TDD
Time division duplexing
 ULA
Uniform linear array
1 Introduction
The high energy and spectrum efficiency of massive multipleinput multipleoutput (MIMO) systems heavily build on the premise that the base stations (BS) obtain channel state information (CSI) with reasonable quality, which is generally estimated via pilot sequences [1]. However, in the uplink massive MIMO systems, the pilot overhead demanded should be proportional to the number of users and would be prohibitively large as the number of users increase. In the uplink multicell massive MIMO, this results in pilot contamination as the same pilot sequences have to be reused by neighbor cells to serve a large number of users [2]. Moreover, the pilot contamination is a major limiting factor to system performance [3]. Hence, the massive MIMO urgently needs efficient channel estimation scheme without producing pilot contamination and requiring too much pilot overhead. Based on the estimated CSI, the signals received at base stations are typically detected through linear methods with low complexity, such as zeroforcing [4, 5, 6] and matched filter [7, 8]. However, the performances of linear detector are typically far inferior to the optimal maximum likelihood (ML) detector whose computational complexity exponentially scales up with the signal constellation size and the number of antennas [9]. Thus, the development of computationally efficient and reliable detector for massive MIMO also needs to be thoroughly addressed [10].
In the past few years, several types of schemes have been exploited to mitigate or reduce the impact of pilot contamination in multicell massive MIMO systems. (1) Semiblind or blind approaches, such as [11, 12, 13, 14]—the eigenvalue decompositionbased method with a short training sequence was proposed in [11]. Hu et al. [12] proposed a semiblind method without requiring the statistical information of channels. Another lowcomplexity semiblind approach was proposed in [13], which the received signal are firstly projected onto the subspace with minimal interference, then alternatively refined the channel estimation and detected the data symbols. Applying the theory of large random matrices, [14] proposed a blind pilot decontamination with subspace projection. (2) Optimization design of nonorthogonal pilot signals, such as [15, 16, 17, 18]—when training slots are not large enough to construct the orthogonal pilot signals, [15] exploits a pilot design criterion and shows that the line packing on a complex Grassmannian manifold is the optimization scheme, which is based on minimal mean square error (MMSE) estimator. A generalized Welchbound equalitybased pilot signal design method is proposed in [16], which has low correlation coefficients and ensures the network to satisfy the requirement of user capacity. For a given pilot length, [17] proposes an alternating minimizationbased pilot design algorithm. (3) The precodingbased approaches, such as [19, 20, 21]—a MMSEbased precoding is exploited in [19] to alleviate the impact of pilot contamination. A pilot contamination mitigation method along with zeroforcing precoding is proposed in [20], which can generate orthogonal pilot signals across neighboring cells through multiplying the ZadoffChu sequences elementwise with a specific orthogonal variable spreading factor code.
Some significant efforts have been made to reduce the pilot overhead for massive MIMO systems, which can be divided into two broad categories. (1) Lowrank channel covariance matrices based methods, such as [22, 23, 24]—the finite scattering environment and small angular spread result in high correlation of different paths between the user and the BS [25, 26, 27, 28, 29] and lowrank channel covariance matrix. Through exploiting the correlation characteristic of channel vectors, the joint spatial division and multiplexing (JSDM) was proposed in [23] which significantly reduced the overhead of downlink training and uplink feedback for frequency division duplexing (FDD) massive MIMO systems. When the number of pilot signals is no less than the rank of channel covariance matrix and the noise interference disappear, [24] proves that the MMSE estimator can recovery channel vectors exactly. (2) Compressed channel sensing method—exploiting the channel sparsity and applying the compressed sensing (CS) to reduce the overhead of CSI feedback has been investigated in [30, 31, 32]. A spare channel estimation method applying Gaussianmixture Bayesian learning has been proposed in [33] to estimate the whole channel parameters including the desired and interference links, which can mitigate pilot contamination and reduce pilot overhead, but every time, the approach just can estimate the channel response at one beam.
with A∈R^{60×200}, B∈R^{30×100}, x=vec(X), and y=vec(Y), which results in Φ=B⊗A with dimensions 1800×20,000. The signs of vec() and \(\bigotimes \) denote vectorization of a matrix and Kronecker product, respectively.

We propose a channel estimation method named as 2DSL0CE applying the twodimensional smoothed l_{0}norm compressed sensing recovery algorithm [37, 38], which are able to jointly estimate multiuser CSI. Using virtual channel representation, the 2DSL0CE formulates the joint channel estimation problem, comprising both the target and interference channels, as a 2D sparse signal reconstruction problem in CS, which not only can mitigate the pilot contamination but also can significantly reduce pilot overhead.

We propose a signal detection method named as 2DSL0SD using our modified 2DSL0 algorithm, which can decode multiuser Mary phaseshift keying (MPSK) signal block. Applying sparse decomposition, the 2DSL0SD models the detection problem of multiuser signal block as a 2D sparse signal reconstruction problem whose elements are binaries {0,1}. Moreover, in the high SNR scenario, through exploiting the estimated CSI of interference links, an improved 2DSL0SD is also proposed to remove the decoding bottleneck induced by interference from neighboring cells.
The remaining paper is organized as follows. The system model and the least square (LS) channel estimation methods are described in Section 2. Section 3 models the multicell multiuser channel estimation problem as a twodimensional compressed sensing problem and describes the proposed channel estimation algorithm. Section 4 models the signal decoding problem of multiuser as a 2D sparse signal recovery problem and presents the steps of the proposed 2DSL0SD method. Section 5 gives and analyzes the numerical results. Section 6 draws a conclusion of the whole paper.
Notations: diag(x) represents a diagonal matrix with diagonal elements being the vector x. Superscripts T and † denote the transpose and pseudoinverse, respectively. CN(0,1) denotes complex Gaussian variables with zero mean and unit variance. \(\bigotimes \) denotes Kronecker product. Vectors and matrices are denoted by boldface lowercase and uppercase letters, respectively.
2 System model
with H_{ji}=[h_{ji1},⋯,h_{jiK}], G_{ji}=[g_{ji1},⋯,g_{jiK}] and \(\mathbf {D}_{ji}=\text {diag}\left (\sqrt {\beta _{ji1}},\cdots,\sqrt {\beta _{jiK}}\right)\).
where P_{tr} denotes the training signaltonoise ratio (SNR), the rows of \(\mathbf {X}_{i}^{tr}\in C^{K\times T_{tr}}\) are the pilot sequences of ith cell, and \(\mathbf {N}_{j}^{tr}\) is the noise with i.i.d. elements distributed as CN(0,1).
where the second term denotes pilot contamination resulting in the same orthogonal pilots reused by adjacent cells.
3 2DSL0CE channel estimation method
The direction θ_{m} is related to the physical angle ϕ_{m}∈[−π/2,π/2] as θ_{m}=dsin(ϕ_{m})/λ with λ being the carrier wavelength and d being the antenna spacing [39]. We uniformly sample the principal θ period to set the virtual spatial angles, i.e., θ_{m}=m/M, and resulting in an M×M unitary discrete Fourier transform matrix A_{R}. The element \(\mathbf {G}_{ji,mk}^{v}\) of M×K matrix \(\mathbf {G}_{ji}^{v}\) represents the coupling gain from the kth terminal to the mth virtual receive angle θ_{m}. Therefore, the element will be zero when there is no corresponding coupling, and the \(\mathbf {G}_{ji}^{v}\) will be a sparse matrix when the number of nonzero elements is much smaller than that of the total elements.
where b∈R^{M×1} is a sparse vector, and the parameter σ determines the quality of the approximation and how smooth the function F_{σ}(b). Consequently, the minimum l_{0}norm solution can be found by maximizing F_{σ}(b). In Algorithm 1, steps 2–9 gradually decrease the value of σ and maximize the objective function for each value of σ.
4 2DSL0SD signal detection method
where \(\mathbf {Y}_{j}\in \mathcal {C}^{M\times N}\) is the received data, \(\sqrt {P_{\text {data}}}\) denotes the uplink SNR, X_{i} denotes the transmitted data matrix of the ith cell whose element is selected from a finite alphabet constellation defined as {s_{1},⋯,s_{Q}} with Q being the finite alphabet cardinal, N_{j} is the noise with elements distributed as CN(01), and W_{j} represents the noise plus interference faced by the received data of jth BS.
where \(\mathbf {B}_{s}=\mathbf {I}_{K} \bigotimes \mathbf {s}\) is a block diagonal matrix of size K×KQ, and the nth column of the matrix E_{i} is [(e_{i,1n})^{T},⋯,(e_{i,Kn})^{T}]^{T}.
with \(\mathbf {A}_{j}=\sqrt {P_{\text {data}}}\mathbf {H}_{jj}\mathbf {B}_{s}\). The detection problem of signal block has been modelled as a 2D sparse binary {0,1} reconstruction problem in CS, then based on Y_{j}, A_{j}, and I_{N}, the signal block X_{j} can be detected using the modify 2DSL0 algorithm which suits to reconstruct 2D sparse binary {0,1} signal. The process of 2DSL0SD is summarized in Algorithm 2. Because the elements of E_{j} needed to be recovered are 0 or 1, but the elements recovered by the original 2DSL0 algorithm are not exactly 0 or 1, step 10 of Algorithm 2 is added to find which element of \(\hat {\mathbf {e}}_{j,mn}\) maybe 1 with highest probability and reset such element to 1 and others to 0.
Now, the decoding problem without multiuser interference has also been modelled as a 2D sparse {0,1} signal reconstruction problem, which has the same form as that of (14), and can be solved through the processes of Algorithm 2 whose A_{j} needs to be replaced by A. Hereafter, the improved 2DSL0SD with interference cancel is named as 2DSL0SDIC. Comparing (16) with (14), it can be observed that the recovery object of the 2DSL0SDIC is E, including not only the E_{j} of the objective cell but also the E_{i} of the L−1 neighbor cells (i=1,⋯L,i≠j), and the noise is the only interference source.
It is worth noting that the proposed 2DSL0SD is only suitable to decode constant modulus signal, such as MPSK. Since there is only one element of e_{i,mn} in (12) equaling to 1 and the others are zeros, Algorithm 2 needs a ruler to reset the values of the estimated \(\hat {\mathbf {e}}_{i,mn}\). In step 10 of Algorithm 2, the ruler is that the element of \(\hat {\mathbf {e}}_{j,mn}\) with the largest real part is viewed as such element whose value is 1 with the highest probability. Such ruler requires that the elements of s in (12) have the same modulus.
5 Numerical results and discussion
System and algorithm parameter values
Parameters  Values  Parameters  Values 

Number of BS antennas M  256  Standard deviation σ_{shadow}  8 dB 
Number of cells L  7  Threshold value σ_{min}  10^{−4} 
Cell radius R  1000 m  Descent factor ρ  0.6 
Cellhole radius r_{h}  50 m  Step size μ  2 
Path loss exponent α  3.5  Number of iterations P  3 
Number of users in each cell K  20  Number of simulations N_{MC}  1000 
6 Conclusions
This paper has investigated the two challenging problems for block fading massive MIMO systems. The one is to exploit efficient uplink channel estimation method that requires acceptable pilot overhead and can mitigate pilot contamination. The other one is to jointly decode multiuser data block. The joint multiuser channel estimation and data block detection problems have both been modelled and solved as a 2D sparse signal reconstruction problem in the CS framework. More specifically, through using a linear virtual channel representation for ULA, the 2DSL0CE compressed channel sensing method is proposed, which needs less pilot overhead than LS method, and can jointly estimate the desired and interference uplinks. Through sparse representation in a finite alphabet set for each transmitted data symbol, the 2DSL0SD data decoding method is proposed which can simultaneously decode a MPSK data block for multiuser. Simulation results demonstrate that joint the 2DSL0CE with 2DSL0SD to estimate channel and decode MPSK data for multiuser massive MIMO is a feasible scheme.
Notes
Funding
This work was partly supported by the National Natural Science Foundation of China (Grant Nos. 61601005, 61801114), Natural Science Foundation of Anhui Province (Grant Nos. 1808085MF164, 1608085QF138), Key Projects of the Outstanding Young Talents Program in the Universities of Anhui Province (Grant No. gxyqZD2016027), the Natural Science Foundation of Jiangsu Province (Grant No. BK20170688) and Doctoral Scientific Research Foundation of Anhui Normal University (Grant Nos. 2014bsqdjj38, 2018XJJ40).
Availability of data and materials
Mostly, I got the writing material from different journals as presented in the references. A MATLAB tool has been used to simulate my concept.
Authors’ contributions
XY conceived and designed the methods. XY performed the experiments and wrote the paper. AZ analyzed the simulation data. GZ and XG gave valuable suggestions on the structure of the paper. LY revised the original manuscript. All authors read and agreed the manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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