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A novel compressed sensingbased nonorthogonal multiple access scheme for massive MTC in 5G systems
 Kesen He^{1}Email authorView ORCID ID profile,
 Yangqing Li^{1},
 Changchuan Yin^{1} and
 Yanbin Zhang^{1}
https://doi.org/10.1186/s1363801810794
© The Author(s) 2018
 Received: 14 November 2017
 Accepted: 5 March 2018
 Published: 19 April 2018
Abstract
The main challenges for massive machine type communication in 5G system are to support random access for massive users and to control signaling overhead and data processing complexity. To address these challenges, we propose a novel compressed sensing (CS)based nonorthogonal multiple access (NOMA) scheme, called CSNOMA, which introduces low coherence spreading (LCS) signatures to enable joint activity and data detection without requiring the activity information of users in advance. We present a sufficient condition for the construction of the LCS signatures to ensure that a CSbased multiuser detection (CSMUD) can be effectively deployed in base station. Furthermore, we study the CSNOMA scheme with imperfect channel state information (CSI) and present a bound for the performance of the CSNOMA scheme. Simulation results show that the proposed scheme achieves a relatively high system overload (up to 4) when the active users are relatively sparse with an activity ratio of 1%, which implies that the CSNOMA scheme can significantly improve the spectral efficiency, avoid the control signaling overhead, and reduce the transmission latency.
Keywords
 Nonorthogonal multiple access
 Compressed sensing
 Sparse multiuser detection
 Massive machine type communication
 Channel state information
1 Introduction
Massive machine type communication (MMTC) characterized by the requirements of low data rates, small packet sizes, and in some cases, tight delay constraints is expected to be one of the major drivers for the 5th generation (5G) wireless communication system. In the 5G system, a single base station will serve 10 to 100 times more machine type devices (MTDs) than the personal mobile phones, which poses great challenges to efficiently support massive users random access [1–3]. According to the statistics of mobile traffics [4], the number of active users is usually much smaller than the number of all possible users even in the busy hours in cellular communications, especially for 5G MMTC applications, where users can sporadically access or leave the system. Thus, the sparsity of user activity naturally exists in massive connectivity. In order to reduce the overhead of the MTD transmission, the sparse multiuser detection in the base station (BS) is expected to be deployed without requiring the activity knowledge of MTD, thereby reducing the control signaling overhead [5].
Recently, compressive sensing (CS) theory [6–8] has been widely used to recover sparse signals and also shows that reliable signal reconstruction far below the Nyquist sampling rate is possible provided that the signal is sparse. In order to address the challenge in future MMTC, the authors in [9–12] propose a novel physical layer signal processing scheme, called CSbased multiuser detection (CSMUD), which takes advantage of the CS technology to detect the received sparse multiuser signals. The CSMUD enables joint activity and data detection, which facilitates a reliable detection of direct random access. In [9, 13], the CSMUD is deployed in a code division multiple access (CDMA) system. Then, a CSMUD algorithm designed for singlecarrier orthogonal frequency division multiplexing (SCOFDM) systems is proposed in [10]. To improve the flexibility and scalability of accessing both time and frequency resources, Monsees et al. [11] and Bockelmann et al. [12] apply the CSMUD to multicarrier transmissions. However, all these multiple access schemes belong to the category of orthogonal multiple access (OMA), which are difficult to meet the requirements of massive connectivity for MMTC in 5G systems.
To address the challenge of spectrum scarcity in 5G communications, recently some novel nonorthogonal multiple access (NOMA) schemes [14–20], e.g., the power domain NOMA [14], the lowdensity spreading (LDS) [15, 16], the sparse code multiple access (SCMA) [17], and the multiuser shared access (MUSA) [18], are proposed. In these wellknown NOMA schemes, MUD is implemented using the successive interference cancellation (SIC) or the message passing algorithm (MPA), which requires the receiver to be exactly notified about the activity information of users in advance, resulting in hightransmission latency and control signaling overhead. In order to address this problem, the authors in [21, 22] proposed grantfree NOMA schemes for the OFDM system without requiring the activity information of users, which significantly reduces the signaling overhead and transmission latency.
Comparison of different NOMA schemes

 1)
We propose a CSbased NOMA scheme, called CSNOMA, and introduce a new version of spreading signature, called low coherence spreading (LCS) signature. We also present a sufficient condition for the construction of the LCS signatures and then theoretically prove the reliability of this condition.
 2)
We present a generating algorithm for the LCS matrix that consists of the LCS signatures of all users.
 3)
We present a bound on the performance of signal reconstruction under the case when the channel state information is not perfect.
The rest of the paper is structured as follows. In Section 2, we introduce the system model. In Section 3, we provide sufficient condition for the design of LCS signatures to ensure that CSMUD can be effectively deployed in the BS, and present the generating algorithm for the LCS matrix. In Section 4, we discuss the CSNOMA scheme under imperfect CSI. In Section 5, we present the simulation results. Finally, we draw the conclusions in Section 6.
Notation: Throughout this paper, vectors and matrices will be represented by boldfaced lowercase and uppercase letters (e.g., x and X), respectively. All vectors are defined as column vectors. Variables and constants are denoted in lowercase and uppercase letters (e.g., x and X), respectively. Superscript \(\intercal \) and † represent the transpose and the MoorePenrose pseudoinverse of a matrix, respectively. The notation \(\mathcal {N}\left ({0,1} \right)\) is denoted as the Gaussian distribution with zero mean and unit variance, and \(\mathbb {E}\left \{{x}\right \}\) represents the mean of x.
2 System model
We consider an MMTC scenario where a set of users (i.e., MTDs), \(\mathcal {K}=\{1,...,K\}\), sporadically accesses to a single BS over multipath wireless channels in the presence of additive white Gaussian noise (AWGN). All users share N (N≪K) CDMA chips at the same time, which means that the gain of the spreading signature is N. Furthermore, we consider that at most S (S≪N) users are active in a given time, and let ρ_{a}=S/K (ρ_{a}≪1) and β=K/N (β>1) denote the activity ratio and the system overload, respectively. We assume that the active users transmit symbols from a symbol alphabet \(\mathcal {A}\), and the inactive users transmit nothing, i.e., the transmitted symbols are equal to zero. In the following, we present the proposed CSNOMA scheme for the CDMA system.
In order to enable joint activity and data detection without requiring the activity knowledge of users, the CSMUD is deployed in the BS to recover the sparse signal x in (3). In Fig. 1, \({\hat {\mathbf {x}}} = {\left [{\hat x_{1}}, \ldots,{\hat x_{K}}\right ]^{\intercal }} \in {\mathbb {X}^{K \times 1}}\) is the estimation of x; and it can be achieved by the orthogonal matching pursuit (OMP) algorithm [25, 26], which is a typical greed algorithm for CSMUD. Based on the CS theory [6, 7], the effective channel matrix B in (3) is expected to be an effective sensing matrix which guarantees the recovery of the sparse signal x. In (4), B is a synthetic matrix generated by the channel matrices (i.e., H_{1},…,H_{ K }) and the LCS signatures (i.e., f_{1},…,f_{ K }) of all users. Specifically, due to the physical propagation property of the channel, the channel matrices naturally exist in some form and can not be artificially controlled. Thus, we expect that the effective channel matrix B is an effective sensing matrix. Therefore, in the next section, we will discuss how to design the LCS signatures (i.e., the design of \(\mathcal {F}\)) to ensure that the effective channel matrix B guarantees the recovery of the sparse signal x.
3 Design of the LCS signatures
In this section, we discuss how to design the LCS signatures to ensure that the effective channel matrix B in (4) to be an effective sensing matrix which can guarantee the recovery of the sparse signal x. According to the CS theory [27, 28], coherence of the sensing matrix (i.e., B) can provide sufficient condition for guaranteeing recovery of the sparse signal. Therefore, by analyzing the coherence of matrix B, we present sufficient conditions at which the LCS signatures should satisfy. Then, based on these conditions for the design of the LCS signatures, we present a generating algorithm to construct the LCS matrix \(\mathcal {F}\). Note that in this section we only consider the design of LCS signatures and the effective channel matrix in real number value, which can be extended directly to the complex number value case.
3.1 Coherence of the effective channel matrix
It can be shown that \(\mu (\mathbf {A}) \in \left [ {\sqrt {\frac {{K  N}}{{N(K  1)}}},1} \right ]\) [32, 33]. Note that when K≫N, the lower bound, known as the Welch bound [32], is approximately \({{\mu (\mathbf {A}) \ge 1} \left / {\sqrt N }\right.}\).
where the notation A(:,t) represents the tth column of A.
In the sequel, we first present a theorem to clarify the conditions at which the LCS signatures should satisfy to ensure the sparse signal x can be recovered, then based on this theorem, we propose a construction algorithm for the LCS matrix F in Section 3.2.
Theorem 1
The proof of Theorem 1 is presented in Appendix 1.
From (15), we have that when S<S_{min}, the effective channel matrix B absolutely guarantees the recovery of the Ssparse signal x.
In summary, Theorem 1 presents a sufficient condition for the construction of the LCS signatures. Specifically, when the LCS signatures or the matrix \(\mathcal {F}\) satisfy the properties in (10)(12), based on the above discussion, we can ensure that the effective channel matrix B in (3) is an effective sensing matrix which can guarantee the recovery of S_{min}sparse signal x.
holds for all Ssparse signal x. In the following, we present a corollary of Theorem 1, which analyzes the RIP of B consisting of \({\mathcal {H}}\) with unitnorm columns.
Corollary 1
where \(\mathbf {x} \in {\mathbb {R}^{K \times 1}}\) is an Ssparse signal.
The proof of Corollary 1 is presented in Appendix 2.
In this paper, we assume that each user has the same transmit power and then normalize the channel response vectors of all users h_{1},…,h_{ K }, then the channel response matrix \({\mathcal {H}}\) will have unitcolumns. In addition, the authors of [34, 35] have proved that a matrix A satisfying the RIP of order 2S can guarantee the recovery of the Ssparse signal. Thus, in our CSNOMA scheme, from Corollary 1, the LCS signatures satisfying Theorem 1 ensure that B has the RIP of order 2S<2S_{min}=1+1/μ_{up} which guarantees the recovery of Ssparse signal in the presence of noise.
Based on the above analysis, the LCS signatures satisfying Theorem 1 can guarantee the recovery of a sparse signal either in terms of the coherence property or RIP. In the following, we present a generating algorithm for the LCS matrix \(\mathcal {F}\) based on Theorem 1.
3.2 Generating algorithm of the LCS matrix \(\mathcal {F}\)
Thus, from Theorem 1, we have θ=L/M. In the following, we give some interpretation on Algorithm 1.
In Algorithm 1, procedure 1 and procedure 2 ensure that the designed LCS signatures satisfy the properties in (10) and (11), respectively. In procedure 1, we randomly generate ξ and let f_{ j }=ξ, repeating this process until finding a solution of ξ that ensures \(\mu _{\max }^{jj} = {\max _{m}}\sum \limits _{n = 1,n \ne m}^{L} { {\mu _{m,n}^{jj}} } \leq \mu _{1}\), so the property in (10) is satisfied. In procedure 2, the LCS matrix \(\mathbf {\mathcal {F}}\) is initialized as f_{ j } obtained in procedure 1. K_{ iter } denotes the number of the columns of \(\mathbf {\mathcal {F}}\) and is initialized as 1. N_{ exce } is defined as the number of the columns of \(\mathbf {\mathcal {F}}\) that makes \(\mu _{\max }^{kj} = {\max _{n}}\sum \limits _{m = 1}^{L} { {\mu _{m,n}^{kj}} } > \mu _{2}\) with f_{ j } and is initialized as 0 in each repeat of the while at line 11. Then, f_{ j } and F_{ j } are updated by implementing procedure 1. We compute the \(\mu _{\max }^{kj} \) between f_{ k } (1≤k≤K_{ iter }) and f_{ j }. If \(\mu _{\max }^{kj} > \mu _{2}\), i.e., N_{ exce }=1, we stop the while at line 14 and then repeat the while at line 11. If \(\mu _{\max }^{kj} \leq \mu _{2}\) for arbitrary k, i.e., N_{ exce }=0, we update \(\mathbf {\mathcal {F}}=\mathbf {\mathcal {F}}\cup \mathbf {f}_{j}\), K_{ iter }=K_{ iter }+1, and then repeat the while at line 11. We repeat the above operations until K=K_{ iter }, and finally output the LCS matrix \(\mathbf {\mathcal {F}}\).
4 Effect of the imperfect channel state information (CSI)
which implies that \(\hat {\mathbf {B}}\) is also an effective sensing matrix.
In the sequel, we analyze the performance of signal reconstruction under the imperfect CSI case for the CSNOMA scheme. We first introduce Theorem 2 which plays an important role in analysis of the performance.
Theorem 2
Theorem 2 provides a bound for the worstcase performance given a bounded noise n, i.e., \({\left \ \mathbf {n} \right \}_{2} \leq C\), where C is an absolute constant [36, 37]. Note that this theorem holds for the case when ε=0 as well as ∥n∥_{2}=0. Thus, it also applies to the noisefree setting. Furthermore, there is no restriction on ∥n∥_{2}≤ε. In fact, this theorem is valid even when ε=0 but ∥n∥_{2}≠0 [37]. However, as noted in [36], Theorem 2 is the result of a worstcase analysis and will typically overestimate the actual error.

\(\left (1  \theta  \mu _{1}\right){\left \ \mathbf {e} \right \_{2}^{2}} \le d_{k,k} \le \left (1 + \mu _{1}\right){\left \ \mathbf {e} \right \_{2}^{2}},~\forall k\);

\(\sum \limits _{j = 1,j \ne k }^{K} {\left  {d_{k,j}} \right } \le (K1) \mu _{2} {\left \ \mathbf {e} \right \_{2}^{2}}, ~\forall k.\)
which is supported by the perfect LCS signatures. Note that (36)(37) are the results of the worstcase analysis and typically overestimate the actual error.
5 Simulation results
where \(\hat {\mathbf {x}}\) is an estimate of x.
Based on the analysis in Section 4, we present the MSE performance for the cases that the CSI is perfect and imperfect.
Note that (39)(42) are the worstcase results and typically overestimate the actual MSEs. In addition, we consider ε=0 for these MSEs in (39)(42).
5.1 Validation for the coherences with different LCS signatures
where I and \(\mathbf {\mathcal {F}}\) are the identity matrix and the LCS matrix generated by Algorithm 1, respectively. For the identity matrix I, we have that θ=1 and μ_{2}=1, which implies that the LCS signatures in \(\mathbf {\mathcal {\bar {F}}}\) do not satisfy the properties (11) and (12) in Theorem 1.
5.2 Performance in perfect CSI case
From the simulation results in Figs. 3, 4, 5, and 6, we can see that the designed LCS signatures based on Theorem 1 can provide guarantee for the recovery of at least S_{min}sparse signal x. In addition, both MSE_{ μ } and MSE_{1} provide bounds for the MSE for the proposed CSNOMA scheme. It also admits the validity of the proposed CSNOMA scheme. Furthermore, the proposed scheme achieves a relatively high system overload when the active users are relatively sparse, which implies that the CSNOMA scheme can significantly improve the spectral efficiency.
5.3 Performance in imperfect CSI case
The results in Figs. 7 and 8 imply that the LCS matrix \(\mathcal {F}\) designed based on Theorem 1 ensures effective recovery of sparse signal even in the imperfect CSI case. In addition, the proposed CSNOMA scheme enables joint activity users and data detection through CSMUD, even if the CSI is not perfect.
5.4 Convergence and computational complexity of the proposed algorithm
6 Conclusions
In this paper, a novel CSNOMA scheme for MMTC in 5G was proposed to enable joint detection of active users and their data. Due to the introduction of the LCS signature, the proposed CSNOMA scheme can achieve a relatively high system overload. Then, we presented a theorem for guiding the design of the LCS signatures which provided the theoretical guarantee for ensuring that the CSMUD could be effectively deployed in the base station. Furthermore, we presented a construction algorithm for the LCS matrix. We also discussed the imperfect CSI case and presented a corresponding bound for the performance of signal recovery for the CSNOMA scheme. The simulation results shown that the CSNOMA scheme not only supported massive access, but also had high spectral efficiency and low transmission latency.
7 Appendix 1
In order to prove Theorem 1, we first provide Theorem 3 that plays an important role in the proof of Theorem 1.
Theorem 3
(Gershgorin Circle Theorem, Theorem 2 in [39]). The eigenvalues of an N×N matrix M with entries m_{ ij }, 1≤i,j≤N, lie in the union of N discs \({d_{i}} = {d_{i}}\left ({c_{i}},{r_{i}}\right)\), 1≤i≤N, centered at c_{ i }=m_{ ii } and with radius \({r_{i}} = {\sum \nolimits }_{i \ne j} {\left  {{m_{i}}_{j}} \right }\).
Now, we prove Theorem 1 in the following.
Proof
Therefore, we first analyze the bound of \({\left \ {{\mathbf {F}_{k}}{\mathbf {h}_{k}}} \right \}_{2} {\left \ {{\mathbf {F}_{j}}{\mathbf {h}_{j}}} \right \}_{2}\) in Step 1. Then the upper bound of \({\left  {\left \langle {{\mathbf {F}_{k}}{\mathbf {h}_{k}},{\mathbf {F}_{j}}{\mathbf {h}_{j}}} \right \rangle } \right }\) will be discussed in Step 2. □

\(1  \theta \le g_{m,m}^{k} \le 1,~1 \le m \le L,~\forall k\);

\(\sum \limits _{n = 1,n \ne m }^{L} {\left  {g_{m,n}^{k}} \right  }=\sum \limits _{n = 1,n \ne m }^{L} {\left  {\mu _{m,n}^{kk}} \right } \le \mu _{1},~ \forall m,k.\)
In addition, since all columns of F_{ k } are linearly independent (i.e., F_{ k } is full rank), G^{ k } is a Gram matrix which means that G^{ k } is positive definite. Thus, from Theorem 3, the inequality θ+μ_{1}<1 is necessary to be held.
which implies that \(\mu \left (\mathbf {B} \right) \le \frac {{ \mu _{2} }}{{1  \theta  \mu _{1} }}\). Thus, we complete the proof of Theorem 1.
8 Appendix 2
Now we prove Corollary 1 using the similar tricks in the proof of Theorem 1 as follows.
Proof
Let Λ be an arbitrary subset of \(\left \{ {1, \ldots,K} \right \}\) and \(\left  \Lambda \right {{= }}S\), then \(\mathbf {B}_{\Lambda } \in {\mathbb {R}^{N \times S}}\) is obtained by only keeping the columns of B corresponding to Λ. Similar to this, \({\mathcal {H}}_{\Lambda }\) and \({\mathcal {F}}_{\Lambda }\) are obtained by only keeping the columns of \({\mathcal {H}}\) and \({\mathcal {F}}\) corresponding to Λ, respectively. Thus, B_{ Λ } is generated by \({\mathcal {H}}_{\Lambda }\) and \({\mathcal {F}}_{\Lambda }\). Let h_{ s } and f_{ s } be the scolumn (1≤s≤S) of \({\mathcal {H}}_{\Lambda }\) and \({\mathcal {F}}_{\Lambda }\), respectively. Accordingly, F_{ s } is generated by f_{ s } in the form of (8). Hence, we have that \(\mathbf {B}_{\Lambda } = \left [{\mathbf {F}_{1}}{\mathbf {h}_{1}}, \ldots,{\mathbf {F}_{S}}{\mathbf {h}_{S}}\right ] \). Now, we analyze the bound of the eigenvalues of \(\mathbf {C}=\mathbf {B}^{\intercal }_{\Lambda } \mathbf {B}_{\Lambda } \in {\mathbb {R}^{S \times S}}\). □

1−θ−μ_{1}≤c_{i,i}≤1+μ_{1},1≤i≤S;

\(\left  {{c_{i,s}}} \right  \le {\mu _{2}},i \ne s,1 \le i,s \le S.\)
where δ_{ S }=θ+μ_{1}+ (S−1)μ_{2} and \(\mathbf {x}\in {\mathbb {R}^{S \times 1}}\). Note that Theorem 3 needs the inequality 1−δ_{ S }>0 to be held, which implies \(S<1+ \left (1\theta \mu _{1}\right)/\mu _{2}\), i.e., S<1+1/μ_{up}. In addition, because (57) holds for arbitrary \(\Lambda \in \{1,\dots,K \}\) with Λ=S, so \((1  {\delta _{S}})\left \ \mathbf {x} \right \_{2}^{2} \le \left \ {{\mathbf {B} }\mathbf {x}} \right \_{2}^{2} < (1 + {\delta _{S}})\left \ \mathbf {x} \right \_{2}^{2}\), where \(\mathbf {x}\in {\mathbb {R}^{K \times 1}}\) is an Ssparse signal. Thus, we complete the proof.
Declarations
Acknowledgements
This work is supported in part by the National Natural Science Foundation of China under Grants 61671086 and 61629101, the 111 Project (NO.B17007), and the Director Funds of Beijing Key Laboratory of Network System Architecture and Convergence (NO.2017BKLNSACZJ04).
Funding
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Authors’ contributions
The authors have contributed jointly to all parts of the preparation of this manuscript, and all authors read and approved the final manuscript.
Competing interests
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References
 3GPP, Service requirements for machinetype communications (MTC). Rel. 12:, 2–27 (2013).Google Scholar
 JG Andrews, S Buzzi, W Choi, SV Hanly, A Lozano, AC Soong, JC Zhang, What will 5G be?. IEEE J Sel. Areas Commun.32(6), 1065–1082 (2014).View ArticleGoogle Scholar
 N Alliance, 5G white paper. Next Gener. Mob. Netw. White Pap (2015). https://www.ngmn.org/5gwhitepaper/5gwhitepaper.html.
 JP Hong, W Choi, BD Rao, Sparsity controlled random multiple access with compressed sensing. IEEE Trans. Wirel. Commun.14(2), 998–1010 (2015).View ArticleGoogle Scholar
 M Hasan, E Hossain, D Niyato, Random access for machinetomachine communication in LTEadvanced networks: Issues and approaches. IEEE Commun. Mag.51(6), 86–93 (2013).View ArticleGoogle Scholar
 DL Donoho, Compressed sensing. IEEE Trans. Inf. Theory. 52(4), 1289–1306 (2006).MathSciNetView ArticleMATHGoogle Scholar
 EJ Candès, T Tao, Nearoptimal signal recovery from random projections: Universal encoding strategies?. IEEE Trans. Inf. Theory. 52(12), 5406–5425 (2006).MathSciNetView ArticleMATHGoogle Scholar
 W Chen, MR Rodrigues, IJ Wassell, Projection design for statistical compressive sensing: A tight frame based approach. IEEE Trans. Signal Process. 61(8), 2016–2029 (2013).View ArticleGoogle Scholar
 C Bockelmann, HF Schepker, A Dekorsy, Compressive sensing based multiuser detection for machinetomachine communication. Trans. Emerg. Telecommun. Technol. 24(4), 389–400 (2013).View ArticleGoogle Scholar
 Y Beyene, C Boyd, K Ruttik, C Bockelmann, O Tirkkonen, R Jäntti, in AFRICON, 2015. Compressive sensing for MTC in new LTE uplink multiuser random access channel (IEEEAddis Ababa, 2015), pp. 1–5.Google Scholar
 F Monsees, M Woltering, C Bockelmann, A Dekorsy, in 2015 IEEE 81st Vehicular Technology Conference (VTC Spring). Compressive sensing multiuser detection for multicarrier systems in sporadic machine type communication (IEEEGlasgow, 2015), pp. 1–5.Google Scholar
 C Bockelmann, F Monsees, M Woltering, A Dekorsy, Hardwareintheloop measurements of the multicarrier compressed sensing multiuser detection (mcsm) system. arxiv preprint arxiv:1511.09278 (2015).Google Scholar
 HF Schepker, A Dekorsy, in 2011 8th International Symposium on Wireless Communication Systems (ISWCS). Sparse multiuser detection for cdma transmission using greedy algorithms (IEEEAachen, 2011), pp. 291–295.View ArticleGoogle Scholar
 Y Saito, Y Kishiyama, A Benjebbour, T Nakamura, A Li, K Higuchi, in 2013 IEEE 77th Vehicular Technology Conference (VTC Spring). Nonorthogonal multiple access (NOMA) for cellular future radio access (IEEEDresden, 2013), pp. 1–5.Google Scholar
 R Hoshyar, FP Wathan, R Tafazolli, Novel lowdensity signature for synchronous cdma systems over awgn channel. IEEE Trans. Signal Proc. 56(4), 1616–1626 (2008).MathSciNetView ArticleGoogle Scholar
 M AlImari, P Xiao, MA Imran, R Tafazolli, in 2014 11th International Symposium on Wireless Communications Systems (ISWCS). Uplink nonorthogonal multiple access for 5g wireless networks (IEEEBarcelona, 2014), pp. 781–785.View ArticleGoogle Scholar
 H Nikopour, H Baligh, in 2013 IEEE 24th International Symposium on Personal Indoor and Mobile Radio Communications (PIMRC). Sparse code multiple access (IEEELondon, 2013), pp. 332–336.View ArticleGoogle Scholar
 Z Yuan, G Yu, W Li, Multiuser shared access for 5g. Telecommun. Netw. Technol.5(5), 28–30 (2015).Google Scholar
 Z Ding, X Lei, GK Karagiannidis, R Schober, J Yuan, VK Bhargava, A survey on nonorthogonal multiple access for 5g networks: Research challenges and future trends. IEEE J. Sel. Areas Commun.35(10), 2181–2195 (2017).View ArticleGoogle Scholar
 L Dai, B Wang, Y Yuan, S Han, I ChihLin, Z Wang, Nonorthogonal multiple access for 5G: Solutions, challenges, opportunities, and future research trends. IEEE Commun. Mag.53(9), 74–81 (2015).View ArticleGoogle Scholar
 B Wang, L Dai, T Mir, Z Wang, Joint user activity and data detection based on structured compressive sensing for noma. IEEE Commun. Lett.20(7), 1473–1476 (2016).Google Scholar
 B Wang, L Dai, Y Zhang, T Mir, J Li, Dynamic compressive sensingbased multiuser detection for uplink grantfree noma. IEEE Commun. Lett.20(11), 2320–2323 (2016).View ArticleGoogle Scholar
 K He, Y Li, C Yin, in 2016 IEEE 27th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC). Sparse multiuser detection for nonorthogonal multiple access in 5G systems (IEEEValencia, 2016), pp. 1–5.Google Scholar
 L Applebaum, WU Bajwa, MF Duarte, R Calderbank, Asynchronous codedivision random access using convex optimization. Phys. Commun.5(2), 129–147 (2012).View ArticleGoogle Scholar
 YC Pati, R Rezaiifar, PS Krishnaprasad, in 1993 Conference Record of The TwentySeventh Asilomar Conference on Signals, Systems and Computers. Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition (IEEEPacific Grove, 1993), pp. 40–44.Google Scholar
 JA Tropp, AC Gilbert, Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory. 53(12), 4655–4666 (2007).MathSciNetView ArticleMATHGoogle Scholar
 MF Duarte, YC Eldar, Structured compressed sensing: From theory to applications. IEEE Trans. Signal Process. 59(9), 4053–4085 (2011).MathSciNetView ArticleGoogle Scholar
 DL Donoho, M Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ _{1} minimization. Proc. Natl. Acad. Sci.100(5), 2197–2202 (2003).MathSciNetView ArticleMATHGoogle Scholar
 DL Donoho, X Huo, Uncertainty principles and ideal atomic decomposition. EEE Trans. Inf. Theory. 47(7), 2845–2862 (2001).MathSciNetView ArticleMATHGoogle Scholar
 JA Tropp, Greed is good: Algorithmic results for sparse approximation. EEE Trans. Inf. Theory. 50(10), 2231–2242 (2004).MathSciNetView ArticleMATHGoogle Scholar
 R Gribonval, M Nielsen, Sparse representations in unions of bases. EEE Trans. Inf. Theory. 49(12), 3320–3325 (2003).MathSciNetView ArticleMATHGoogle Scholar
 L Welch, Lower bounds on the maximum cross correlation of signals. EEE Trans. Inf. Theory. 20(3), 397–399 (1974).View ArticleMATHGoogle Scholar
 T Strohmer, RW Heath, Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal.14(3), 257–275 (2003).MathSciNetView ArticleMATHGoogle Scholar
 EJ Candès, T Tao, Decoding by linear programming. IEEE Trans. Inf. Theory. 51(12), 4203–4215 (2005).MathSciNetView ArticleMATHGoogle Scholar
 MA Davenport, Random observations on random observations: Sparse signal acquisition and processing. PhD thesis, Rice University (2010).Google Scholar
 DL Donoho, M Elad, VN Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inf. Theory. 52(1), 6–18 (2006).MathSciNetView ArticleMATHGoogle Scholar
 MA Davenport, MF Duarte, YC Eldar, G Kutyniok, Introduction to compressed sensing. Preprint. 93(1), 2 (2011).Google Scholar
 N Bourbaki, Topological Vector Spaces: Chapters 1–5 (Springer Science & Business Media, France, 2013).Google Scholar
 SA Geršhgorin, Über die abgrenzung der eigenwerte einer matrix. Akad. Nauk SSSR Ser. Fiz.Mat. (6), 749–754 (1931).Google Scholar
 PS Bullen, DS Mitrinovic, M Vasic, Means and Their Inequalities, vol. 31. (Springer Science & Business Media, Dordrecht, 2013).MATHGoogle Scholar