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Lowcomplexity sparseaware multiuser detection for largescale MIMO systems
EURASIP Journal on Wireless Communications and Networking volume 2021, Article number: 58 (2021)
Abstract
Sparseaware (SA) detectors have attracted a lot attention due to its significant performance and lowcomplexity, in particular for largescale multipleinput multipleoutput (MIMO) systems. Similar to the conventional multiuser detectors, the nonlinear or compressive sensing based SA detectors provide the better performance but are not appropriate for the overdetermined multiuser MIMO systems in sense of power and time consumption. The linear SA detector provides a more elegant tradeoff between performance and complexity compared to the nonlinear ones. However, the major limitation of the linear SA detector is that, as the zeroforcing or minimum mean square error detector, it was derived by relaxing the finitealphabet constraints, and therefore its performance is still suboptimal. In this paper, we propose a novel SA detector, named singledimensional searchbased SA (SDSBSA) detector, for overdetermined uplink MIMO systems. The proposed SDSBSA detector adheres to the finitealphabet constraints so that it outperforms the conventional linear SA detector, in particular, in high SNR regime. Meanwhile, the proposed detector follows a singledimensional search manner, so it has a very low computational complexity which is feasible for lightware Internet of Thing devices for ultrareliable lowlatency communication. Numerical results show that the the proposed SDSBSA detector provides a relatively better tradeoff between the performance and complexity compared with several existing detectors.
Introduction
Largescale MIMO system which uses over tens antennas at both transmitter and receiver, promises to offer high data rates and has been identified as one of key techniques in modern wireless communications [1, 2]. Meanwhile, MIMO is also an emerging technology for supporting communications between enormous devices in IoT environments [3]. However, the multiantenna interference brings the fundamental limiting characteristic for MIMO systems [5]. Therefore, for downlink multiuser MIMO systems, transmission technologies such as quadrature spatial modulation (QSM) [4] can be considered for reducing the interference for largescale MIMO systems. While for uplink multiuser MIMO systems, the multiuser detection plays an important role for mitigating the multiantenna interference.
A review of various detection techniques for uplink multiuser MIMO was provided in [6]. The optimal nonlinear detectors, for example, the maximum a posterior (MAP) or maximumlikelihood (ML) detector, performs an exhaustive search in the whole solution space, and thus is practically prohibitive because its large computational complexity which usually exponentially increases with respect to the number of antennas in MIMO systems. On the contrary, the linear detectors, such as the minimummeansquareerror (MMSE) and zeroforcing (ZF) detectors have very low complexity but their performances are generally far from the performance bound, in particular for largescale MIMO systems. Hence, it is desired to develop new multiuser detection methods for achieving a better tradeoff between the performance and complexity for largescale MIMO systems.
Recently, numerous sparseaware (SA) detectors in term of sparse signal processing, have been presented for uplink multiuser MIMO systems [7,8,9,10]. The SA detectors utilized the hidden sparsity of a residual error vector to refine the results achieved by a lowcomplexity linear detector (e.g., ZF or MMSE). More specifically, at the receiver with the channel matrix \({{\mathbf{H}}}\) being known, the transmitted signal vector \({{\mathbf{x}}}\) is first detected by using ZF or MMSE associated with a slicing function. By letting the original received signals (\({{\mathbf{y}}}={{\mathbf{H}}}{{\mathbf{x}}}+{{\mathbf{n}}}\)) minus the product of the detected symbol vector \(\hat{{{\mathbf{x}}}}\) and the channel matrix (i.e., \(\hat{{{\mathbf{y}}}}={{\mathbf{y}}}{{\mathbf{H}}}\hat{{{\mathbf{x}}}}={{\mathbf{H}}}{{\mathbf{e}}}+{{\mathbf{n}}}\) where \({{\mathbf{e}}}={{\mathbf{x}}}\hat{{{\mathbf{x}}}}\) and \({{\mathbf{n}}}\) represents a noise vector), we can achieve a sparse MIMO system with the residual error vector \({{\mathbf{e}}}\) being the inputs. Note that the residual error vector \({{\mathbf{e}}}\) is usually sparse because the symbol error rates (SERs) achieved by the underlying linear detector is generally small for practical SNR regimes, and then a SA detector is employed to acquire \(\hat{{{\mathbf{e}}}}\) which is used in refining \(\hat{{{\mathbf{x}}}}\). The entire process is illustrated in Fig. 1. Intuitively, we may consider to apply compressive sensing (CS) based techniques for detecting the sparse residual error vector \({{\mathbf{e}}}\). For example, in [7], a multipath matching pursuit (MMP) method [14] was considered to detect the residual error vector. [8] first identified the support of the sparse error vector using a CS based method [11]. Then, a very overdetermined MIMO system is created by removing the detected supports. In the resultant system, the number of transmit dimensions is much less than the number of receive dimensions, such that a lowcomplexity linear detector (e.g., ZF or MMSE) is able to detect the remaining nonzero error symbols. Simulation results demonstrated that those CSbased SA detectors were efficient for improving the performance of the underlying linear detectors. However, most of the CS based methods were originally proposed for underdetermined systems where the number of receive dimensions (or antennas) is much less than that of transmit dimensions (or antennas) and an iterative detecting process is usually required [12, 13], but MIMO systems, in particular uplink MIMO systems, are generally overdetermined, which refers that the number of transmit antennas is smaller than that of receive antennas. Therefore, the CSbased SA detectors are not appropriate for the overdetermined MIMO systems in sense of power consumption, execution time etc., especially in ultrareliable lowlatency communication (URLLC) service in 5G communication [15]. Taking the issue into account, a linear sparseaware detector, named SAMMSE, was presented in [9, 10], and has achieved attractive performance with a comparable complexity. Leveraging the hidden equality between the pseudo norm (\(\Vert .\Vert _0\)) and the second norm (\(\Vert .\Vert _2\)) when a lowerorder modulation BPSK was considered, the conventional nonconvex optimization problem on detecting the residual error vector \({{\mathbf{e}}}\) was converted into a convex one after relaxing the finitealphabet constraints. An extension of the SAMMSE for higherorder modulations was presented in [10]. After relaxing the finitealphabet constraints of the residual error vector, a closedform solution on the estimation of the residual error vector was achieved. Then, a welldesigned slicing function was employed to map the estimations into the finite alphabet for achieving final decisions. Like the conventional ZF and MMSE detectors, the SAMMSE detector follows a linear execution manner resulting in low computational complexity. However, this also renders the performance of SAMMSE is suboptimal because it relaxed the finitealphabet constrains as the ZF/MMSE does, and its performance heavily depends on how precise the threshold of the slicing function is. In practical, it is extremely difficult to acquire the exactly accurate threshold due to a lot of unexpected dynamics of wireless channels. As SNR increases, the SAMMSE detector probably suffers an errorfloor. If the SAMMSE detector takes the finitealphabet constraints into account, its detection complexity will exponentially increase as the size of the finite alphabet again.
Therefore, in this paper, we aim at designing a novel detection scheme which not only adheres to the finite alphabet constraints for achieving better performance but also costs a comparable computational complexity. In the development of the proposed method, we can consider to utilize the QR decomposition to convert a multipledimensional search into a singledimensional search as MIMO systems are generally overdetermined. Meanwhile, the unfavourable matrix inversion, required in the SAMMSE detector proposed in [9], can also be avoided. Therefore, we can guarantee low computational complexity of the proposed method. Since we only need to do search in a single dimension, it is not necessary to relax the finitealphabet constraints. Thus, no information will be lost due to the relaxation. Accordingly, we can expect that the performance of the proposed method should be better than the SAMMSE detector. We verify through simulations that the proposed method achieves relatively better tradeoff between the complexity and performance compared to other existing detectors. The key contributions of this paper are summarized as follows.

1.
We consider the same sparse MIMO system as that defined in [9] after applying a underlying detector (ZF or MMSE) and propose the SDSBSA detector. Compared with several linear detectors, including the conventional ZF, MMSE and the SAMMSE proposed in [9], our proposed SDSBSA detector has superiority in terms of performance while costing a comparable computational complexity.

2.
The QR decomposition is utilized to in the development of the SDSBSA detector, which circumvents the matrix inversion operation required in the SAMMSE detector and converts joint multidimensional search into a singledimensional search in order to keep low computational complexity. Meanwhile, the singledimensional search can preserve the finitealphabet constraints for achieving better performance.

3.
In order to avoid the possible error propagation for higherorder modulations, the layered SDSBSA detector is developed by leveraging the hierarchical structure of the residual error vector.
This rest of paper is organized as follows. Section 2 describes the system model for an uplink MIMO transmission system and briefly overviews the main idea of SAMMSE. The proposed singledimensional searchbased SA detector is presented in section 3. Simulation results and discussion are illustrated to verify the efficiency of the proposed detection methods in Sect. 4. Conclusion is given in Sect. 5.
Notation
The uppercase and lowercase boldface letters represent matrices and vectors, respectively. \({\text{Im}}(x)\) and \({\text{Re}}\,(x)\) state the imaginary and real parts of a complexvalued vector x, respectively. \({{\mathbf{X}}}^T\) represents the complex conjugate transpose of the matrix \({{\mathbf{X}}}\). \({\mathbb{C}}\) and \({\mathbb{R}}\) represent the entire complex and real domains, respectively.
System model
In this section, we first introduce an uplink multiuser MIMO system, and then briefly overview the main idea of SA detection.
Uplink multiuser MIMO
An uplink multiuser MIMO system is considered, as illustrated in Fig. 2, in which there is one base station (BS) with M antennas and the BS can service K singleantenna users (UEs) simultaneously. We define \({{{\mathbf{x}}}} \in {\mathbb{C}}^K\) as the symbol vector transmitted by the K UEs with all entries being selected from the same finite constellation set \({{\mathcal{A}}}\) (chosen according to a modulation scheme used in the system). At the BS, the received signal vector, denoted by \(\bar{{\mathbf{y}}}\in {\mathbb{C}}^M\), is expressed as
where \(\bar{{\mathbf{H}}}\in {\mathbb{C}}^{M \times K}\) is the channel matrix whose elements are assumed to be independent and identically distributed with zero mean and unit variance over a richscattering communication environment as in [16], and \(\bar{{\mathbf{n}}} \in {\mathbb{C}}^M\) is the additive noise whose entries are drawn from the complexvalued circularlysymmetric Gaussian distribution \({\mathbb{CN}}(0,\sigma ^2)\). In this paper, we assume the channel matrix \(\bar{{{\mathbf{H}}}}\) is perfectly known at the BS.
For the ease of representation and accommodating to the requirement of the proposed algorithm, we rewrite the complex inputoutput relationship in (1) into an equivalent real representation as,
where \({\mathbf{y }}=\left[ {\text{Re}}(\bar{{\mathbf{y}}})\ {\text{Im}}(\bar{{\mathbf{y}}})\right] ^T\), \({\mathbf{x }}=\left[ {\text{Re}}(\bar{{\mathbf{x}}})\ {\text{Im}}(\bar{{\mathbf{x}}})\right] ^T\), \({\mathbf{n }}=\left[ {\text{Re}}(\bar{{\mathbf{n}}}) \ {\text{Im}}(\bar{{\mathbf{n}}})\right] ^T\) and
This real system representation will be used in the sequel.
The sparseaware detecting
In this section, we briefly overview the methodology of the sparseaware detectors. Given the received vector \(\mathbf{y }\), the lowcomplexity underlying detector (e.g., ZF or MMSE) is first employed to detect the outputs as
where \({\text{Q}}(.)\) refers to a slicing function and \(\mathbf{W }\) is the weighted matrix of the underlying detector. The sparse MIMO system is achieved by taking the difference from the received signal to the product of the channel matrix and the output vector of the underlying detector, i.e.,
where \({{\mathbf{e}}}={{\mathbf{x}}}\hat{{{\mathbf{x}}}}\), named a residual error vector (or symbol error vector). Note that the conventional ZF (or MMSE) detector associated with the symbol quantization can successively detect symbols with a certain possibility, namely, the detected symbol vector may be closed to but not always the same as the original transmit vector. Consequently, the residual error vector is sparse since the most of entries in \(\mathbf{e }\) should be zeros with an overwhelming possibility in practical communication systems. Therefore, the sparse error vector \(\mathbf{e }\) can be detected via the following optimization problem
where \(\hat{{{\mathcal{A}}}}\) represents the finite alphabet for symbol errors \(e_i\) , (e.g., \(\hat{{{\mathcal{A}}}}=\{\pm 2,0\}\) for BPSK), \(\lambda\) is a regularization factor related to the sparsity of the symbol error vector which is determined in term of the bit or symbol error rate achieved by the underlying linear detector. The theoretical analysis of ZF or MMSE on error rates is available in [17, 18] and can be used to calculate the value of \(\lambda\). Finally, \(\Vert {{\mathbf{e}}}\Vert _0\) states the pseudonorm of \(\mathbf{e }\) returning the number of nonzero entries of \({{\mathbf{e}}}\).
It is extremely difficult to tackle (5) since it is a nonconvex function. Therefore, [7] and [8] had relaxed the \(l_0\) norm to the \(l_1\) norm and removed the finitealphabet constraints. Then, a greedy iterative algorithm was employed to estimate the symbol error vector with the attendant high power and time consumption, which in turn preclude their practical implementation. In [9], when BPSK is considered, the mathmatical equality that \(\Vert {{\mathbf{e}}}\Vert _0=\frac{1}{4}\Vert {{\mathbf{e}}}\Vert _2^2\) holds. Therefore, the above optimization problem can be equivalently rewritten as
By relaxing the finitealphabet constraints \(\mathbf{e }\in \hat{{{\mathcal{A}}}}^{2K}\), the estimation of \({{\mathbf{e}}}\), named \(\tilde{{{\mathbf{e}}}}\) can be achieved via
which is named SAMMSE as it has a similar form as MMSE does except that \(\lambda\) represents the sparsity level of \({{\mathbf{e}}}\) rather than the noise variance. Finally, the detection of \({{\mathbf{e}}}\) is acquired via \(\hat{{{\mathbf{e}}}}={{\mathcal{Q}}}_{\theta }(\tilde{{{\mathbf{e}}}})\) where \({{\mathcal{Q}}}_{\theta }(.)\) is a slicing function with \(\theta\) being a threshold. Similar to the ZF and MMSE detectors, the SAMMSE detector follows a linear manner guaranteeing low computational complexity. However, it is also suboptimal in sense of performance as the finitealphabet constraints are relaxed. Therefore, in this paper, we develop a novel sparseaware detector which not only adheres to the finitealphabet constraints for better performance but also has low computational complexity.
Method of singledimensional searchbased sparseaware (SDSBSA) detector
In this section, we propose the singledimensional searchbased sparseaware detector for largescale uplink MIMO systems. First, we consider the case of lowerorder modulation schemes. Then, we propose a layered SDSBSA detector for higherorder modulation schemes in order to mitigate the possible effect of error propagation.
SDSBSA for lowerorder modulation schemes
Recall the sparse MIMO system model given in (4)
Note that the channel matrix \({{\mathbf{H}}}\) is a tall matrix since the number of receive dimensions is larger than that of transmit dimensions for an uplink multiuser MIMO system. Therefore, the QR decomposition is implementable for the channel matrix \({{\mathbf{H}}}\) resulting in \({{\mathbf{H}}}={{\mathbf{Q}}}{{\mathbf{R}}}\) where \({{\mathbf{R}}}\) is a \(2K \times 2K\) upper triangular matrix and \({{\mathbf{Q}}}\) is an \(2M \times 2K\) unitary matrix [19].
Let (8) be multiplied by \({{\mathbf{Q}}}^T\) at both sides, and we can achieve the following expression
Since the unitary matrix \({{\mathbf{Q}}}\) does not change the Gaussian noise distribution, the \(\tilde{{{\mathbf{n}}}}\) still satisfies Gaussian distribution with zero mean and covariance matrix \(\sigma ^2{{\mathbf{I}}}_M\). Consequently, the independence between the symbol error vector \({{\mathbf{e}}}\) and the noise vector \(\tilde{{{\mathbf{n}}}}\) is preserved. By following the similar processes given in [9], the MAP detector can be equivalently rewritten as
Let \({\tilde{y}}_k\) and \(r_{k,i}\) represent the kth element of \(\tilde{{{\mathbf{y}}}}\) and the element at the kth row and ith column of \({{\mathbf{R}}}\), respectively. Then, by taking use of the triangular structure of \({{\mathbf{R}}}\), (10) can be equivalently represented as
From the above equation, we can observe that, the upper triangular form of \({{\mathbf{R}}}\) enables to successively detect \(e_k\) following a reversed order. Therefore, the original detection problem given in (10), which requires a multidimensional search in the finite alphabet \(\hat{{{\mathcal{A}}}}\), can be equivalently decoupled into a set of singledimensional searchbased subproblems. Specifically, given the detected symbols \(\{{\hat{e}}_i\}_{i=k+1}^{2K}\), the \(k\)th symbol can be detected via
This minimization problem entails only one scalar variable taking one of (\(2\times 2^n1\)) possible elements in \(\hat{{{\mathcal{A}}}}\). However, since \({{\mathbf{R}}}\) usually is not a diagonal matrix, the proposed SDSB algorithm in (12) possibly suffers from error propagation, in particular, when a higherorder modulation scheme is employed. As a compromise, a layered singledimensional searchbased method is developed next, to further improve performance by adhering to the finitealphabet constraints, at the price of slightly increased complexity compared to (12).
Layered SDSBSA for higherorder modulation schemes
We consider a \(2^{2n}\)order QAM constellation (e.g., 16QAM, 64QAM etc) which results in that the residual error vector takes value from the finite alphabet \({{\mathcal{A}}}=\{0, \pm 2, \pm 4,\dots \pm 2i,\dots \pm 2(2^n1)\). The proposed layered SDSBSA detector is inspired by the hierarchical structure of the residual error vector, according to which, the residual error vector can be decomposed into a set of orthogonal suberror vectors, i.e.,
where \({\mathbf{v}}_i \in {{\mathcal{A}}}_i^{2K}\) with \({{\mathcal{A}}}_i=\{\pm 2i,0\}^{2K}\) for \(i\in \{1,...,2^n1\}\). Note that \(\hat{{{\mathcal{A}}}}={{\mathcal{A}}}_1\cup {{\mathcal{A}}}_2\dots \cup {{\mathcal{A}}}_i \dots \cup {{\mathcal{A}}}_{2^{n}1}\), and the subsymbol error vectors \({\mathbf{v}}_i\) are orthogonal to each other, i.e., \({\mathbf{v}}_i{\mathbf{v}}_j^T=0\) when \(i\ne j\).
Therefore,
\(^{(a)}\) comes from that \({\mathbf{v}}_i{\mathbf{v}}_j^T=0\) for \(i\ne j\) and \(\Vert \tilde{{{\mathbf{y}}}}\Vert _2^2\) is known and regarded as a constant, adding this term to (15) will not affect the final result of (15). Consequently, the detection problem given in (10) is equivalent to
With the relaxation on the orthogonality constraint, the objective function associated with \({\mathbf{v}}_i\) is given as
Similarly, the kth entry of the suberror vector \({\mathbf{v}}_i\) can be detected as
Define
Then, the optimization problem in (16) can be simplified as
Remark 1
The main difference between (12) and (19) is that the former one entails one scalar variable taking one of possible \((2 \times 2^n1)\) values in a full finite alphabet set \(\hat{{{\mathcal{A}}}}\), whereas the later one only considers one of
three values in a subset \({{\mathcal{A}}}_i.\)
A twolayer reserved successive execution fashion is required to realize the entire detection processes. The innerlayer successive execution is to detect the subsymbol errors \({\mathbf{v }}_i\) via (19) following a reversed order. Once the estimation \(\hat{\mathbf{v }}_i\) is achieved, it is used to update the sparse system (4) via
Then, the resulting sparse system (20) is used in detecting the subsymbol error vector \({\mathbf{v}}_{i1}\) in the outer layer through (19) again. The outer layer reversed successive fashion can guarantee that the detected subsymbol error vector \(\hat{\mathbf{v }}_i\) and \(\hat{\mathbf{v }}_{i1}\) can be orthogonal to each other as all previous subsymbol error vectors do not effect on the detection of \({\mathbf{v}}_{i1}\). Moreover, it may reduce the effect of error propagation by letting the subsymbol error vector with a larger power level be detected first. That is because according to the results on probability of error for Mary PAM or QAM in [20], the subsymbol error vector with a larger power level has a smaller error probability than the subsymbol error vector with a smaller power level. Thus, detecting the subsymbol error vector with a larger power level first can reduce the effect of error propagation for the following subsymbol error vector detection.
The proposed layered SDSBSA detector for MIMO systems is officially summarized in Algorithm I given in Table 1.
Numerical results and discussion
In this section, the performances of the proposed SDSBSA detector is evaluated for uplink multiuser MIMO systems. The conventional MMSE and ZF detectors are used as the underlying detectors providing the initialization for the SAMMSE and the proposed SDSBSA detectors, and also provides one of baselines for performance comparison. Another baseline is provided by the SAMMSE detector.
In Fig. 3, we consider an overdetermined uplink MIMO system where \(M=32\), \(K=20\) and 4QAM is applied.We compared the bit error rates (BERs) of the conventional ZF/MMSE detectors, the SAMMSE with ZF/MMSE initialization and the proposed SDSBSA detector with ZF/MMSE initialization. Since the modulation order is small (i.e., \(n=1\)), we only consider to employ the proposed SDSBSA detector given in (12). Simulation results demonstrate that the proposed SDSBSA detector greatly improves the performance compared with the ZF and MMSE detectors. In particular, at BER \(=10^{3}\), the SDSBSA detector using either ZF or MMSE initialization achieves a 2 dB gain over the conventional ZF and MMSE detectors. Furthermore, the proposed SDSBSA detector with the MMSE initialization performs as the same as the SAMMSE with the same MMSE initialization in low SNR regimes, but becomes superior to the SAMMSE detector in high SNR regimes. That is because the proposed SDSBSA detector counts for the finitealphabet constraints of error symbols. The similar results are also observed in the comparison with the SAMMSE with the ZF initialization that the SDSBSA detector with the ZF initialization performs the same as the SAMMSE in low SNR regime and becomes better in high SNR regime. When the MIMO system becomes further overdetermined, i.e., \(M \gg K\), as illustrated in Fig. 4, where \(M=32\) and \(K=8\), the conventional SAMMSE detector performs even worse than the conventional MMSE/ZF when SNR is over 10 dB. That is because, for a very overdetermined MIMO system, the underlying MMSE or ZF detector can provide acceptable performances resulting in that the residual error vector is extremely sparse. Therefore, the SAMMSE detector become more sensitive to the threshold used in the slicing function, and thus might be severally degraded by the suboptimality caused by the relaxation of finitealphabet constraints. Whereas, the proposed SDSBSA detector provides much better performance than either the MMSE detector or the SAMMSE detector since it adheres to the finitealphabet constraints. In particular, the proposed SDSBSA detector with either MMSE or ZF initialization, at BER \(=10^{3}\), achieves around 1.2 dB gain over the SAMMSE detector, and around 0.8 dB gain over the conventional ZF/MMSE detectors.
Consider a higherorder modulation scheme, e.g.,16QAM, for the uplink MIMO system with \(M=32\) and \(K=20\). In this case, we only used the MMSE detector for the underlying detector providing the initialization for the SDSBSA as the MMSE and ZF performs almost identically in an overdetermined MIMO system. We used SERs as the performance measurement for the comparison of the conventional MMSE detector, the SAMMSE and the proposed SDSBSA detector. From simulation results illustrated in Fig. 4, we still can observe that the proposed SDSBSA detector given in (12) achieves better performance than the conventional MMSE detector, but becomes inferior to the SAMMSE detector as SNR increases. That is because it may suffer from severe error propagation. Whereas, the proposed layered SDSBSA detector given in (19) performs the best among all the existing detectors since it reduces the effects of error propagation by decomposing the residual error vector into a set of orthogonal suberror vectors. More specifically, the proposed layered SDSBSA detector achieves around 1.2 dB gain over the conventional MMSE detector, 0.8 dB over the SAMMSE detector and 1 dB over the SDSBSA detector given in (17). Moreover, the total \({{\mathcal{O}}}(64 \times 3 \times 3)\) number of complexities are required for the proposed layered SDSBSA detector in (19) which is slightly increased compared with \({{\mathcal{O}}}(64 \times 7)\) required for the nonlayered SDSBSA detector in (12). The increment of computation cost is negligible. For reference, the complexities required by either the conventional MMSE/ZF or SAMMSE are \({{\mathcal{O}}}(64 \times 1600)\).
The similar results can be also observed for a \(32 \times 8\) MIMO system in Fig. 5. The proposed SDSBSA detector performs better than the conventional MMSE detector but worse than the SAMMSE detector due to the possible error propagation. The proposed layered SDSBSA detector outperforms both the conventional MMSE and SAMMSE detectors as it utilizes the residual error vector decomposition to mitigate the possible error propagation. Furthermore, its computational complexities (\({{\mathcal{O}}}(64 \times 3 \times 3)\)) are much less than the complexities (\({{\mathcal{O}}}(64 \times 256)\)) cost by either the MMSE or SAMMSE detector (Fig. 6).
Finally, the complexity comparison of the conventional ZF/MMSE, the SAMMSE and the proposed SDSBSA detectors are officially summarized in Table 2.
Conclusion
In this paper, we exploit the sparsity of the residual error vector to develop a novel SDSBSA detector for Largescale MIMO systems. By utilizing the QR decomposition, the unfavourable matrix inversion which is required by the conventional MMSE and SAMMSE detector is avoided. Meanwhile, the multipledimensional searchbased detection problem is converted into the singledimensional searchbased one, so the computational complexity of the proposed SDSBSA is modest compared to that of the MMSE and SAMMSE detectors. Simulation results show that when lowerorder modulations are considered, the proposed SDSBSA outperforms the MMSA and SAMMSE detectors because it takes the finitealphabet constraints into account. When higherorder modulations are considered, the proposed layered SDSBSA detector achieves better performance than the MMSE and SAMMSE as it not only adheres to the finitealphabet constraints and also utilizes the hierarchical structure of the residual error vector to mitigate the possible error propagation.
As an extension of this work, we will consider the more practical factors that channel is imperfect or spatially correlated, and the application of the SDSBSA detector for downlink multiuser MIMO systems.
Availability of data and materials
The data used to support the findings of this study are available from the corresponding author upon request.
Abbreviations
 SA:

Sparseaware
 MIMO:

Multipleinput multipleoutput
 IoT:

Internet of things
 ZF:

Zeroforcing
 SDSB:

Singledimensional searchbased
 URLLC:

Ultrareliable lowlatency communication
 ML:

Maximumlikelihood
 MAP:

Maximum a posteriori
 MMSE:

Minimummeansquareerror
 BER:

Bit error rates
 SER:

Symbol error rate
 CS:

Compressive sensing
 MMP:

Multipath matching pursuit
 AWGN:

Additive white Gaussian noise
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Funding
This research was supported by the Ajou Research Fund, the frame of International Cooperation Program managed by the National Research Foundation of Korea, Grant No. 2017K2A9A2A06016102, and the Basic Research Foundation Program through the National Research Foundation of Korea funded by the Ministry of Education, Grant No. 2017R1D1A1B03035557. This research was partly supported by Sunkyunkwan University Fund, and the frame of Intenatinal Cooperation Program by the National Reseach Fundation of Korea, Grant number 2017K2A9A2A06016102.
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RR as the principal investigator takes the primary responsibility for this research. HO analyzed the results. All authors read and approved the final manuscript.
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Ran, R., Oh, H. Lowcomplexity sparseaware multiuser detection for largescale MIMO systems. J Wireless Com Network 2021, 58 (2021). https://doi.org/10.1186/s13638021019048
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Keywords
 Multiuser detector
 Sparseaware detector
 MIMO systems