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A novel compressed sensingbased nonorthogonal multiple access scheme for massive MTC in 5G systems
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 81 (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.
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.
However, nearly all the prior works on CSbased NOMA typically use the pseudo random noise (PN) sequence as the spread signatures and do not consider the general construction method for the spread signatures to ensure the sufficient recovery for users’ signals in CSbased multiuser detection. In our recent work [23], we propose a novel CSbased NOMA scheme for the CDMA and /or the OFDM systems in which CSMUD is deployed to enable joint activity and data detection without knowledge of the activity information of the users in advance. In this paper, we further extend the previous work to the more general case. We present a sufficient condition for the construction of the spreading signatures of users to ensure the successful CSbased multiuser detection. We also consider the case when the channel state information (CSI) is not perfect and present a bound on the performance of the CSbased NOMA scheme. In order to facilitate the understanding of different NOMA schemes in the literature, we compare them for the main different features in Table 1. Note that in the table \(\left  \mathbb {X} \right \) denotes the cardinality of the constellation set \(\mathbb {X}\), and w is the maximum number of nonzero signals superimposed on each chip or subcarrier. In addition, K, S, and N are the number of users, the number of active users, and the spreading factor, respectively.
We summarize the main contribution of this paper as follows:

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.
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 a conventional CDMA system, user separation can be achieved by uniquely assigning orthogonal spreading signature to each user. However, given the gain N, the number of orthogonal spreading signatures cannot exceed N, which means that the total number of users that the system can support at the same time cannot be greater than N. In our proposed CSNOMA scheme shown in Fig. 1, we introduce a new version of spreading signature, called low coherence spreading (LCS) signature, to achieve the system overload. Unlike orthogonal spreading signatures whose mutual coherence is equal to zero, the mutual coherence between the LCS signatures is nonzero, but is very low, such as far less than 1. Specifically, given the gain N, the number of LCS signatures can exceed N, resulting in a system overload, i.e., β>1. Let \({x_{k}} \in \mathbb {X}\) be the transmitted symbol of user \(k\in \mathcal {K}\), where \(\mathbb {X} = \left \{ {\mathcal {A}\cup 0} \right \}\), then let \({\mathbf {x}} = {\left [{x_{1}}, \ldots,{x_{K}}\right ]^{\intercal }} \in {\mathbb {X}^{K \times 1}}\) be the multiuser transmitted signal vector. We assume that at most S users are active at a given time, which means that up to S users transmit symbols while other users transmit nothing at a given time. Thus, based on the CS theory [6], the multiuser transmitted signal vector x is an Ssparse signal vector, i.e., there are at most S nonzero elements in x. Symbol x_{ k } spreads over the LCS signature \({\mathbf {f}_{k}} = \left [{f_{1,k}}, \ldots,{f_{N,k}}\right ]^{\intercal } \in {\mathbb {R}^{N \times 1}}\) for user k. Then, the LCS matrix is defined by \({\mathcal {F}} = \left [{\mathbf {f}_{1}}, \ldots,{\mathbf {f}_{K}}\right ] \in {\mathbb {R}^{N \times K}}\), which contains the LCS signatures of all users.
The proposed CSNOMA scheme is essentially a typical asynchronous CDMA uplink transmission scheme similar to [24]. Furthermore, the spread symbols of each user are distorted by a userspecific frequencyselective channel with impulse response \({\mathbf {h}_{k}} = {\left [{h_{1,k}}, \ldots,{h_{L,k}}\right ]^{\intercal }} \in {\mathbb {C}^{L \times 1}}\) (\(k\in \mathcal {K}\)) of length L which is constant for a whole frame, and let \({\mathcal {H}} = \left [ {\mathbf {h}_{1}}, \ldots,{\mathbf {h}_{K}} \right ] \in \mathbb {C}^{L \times K}\) denote the channel response matrix of all users with the same number of channel paths L. Then, considering the effect of convolution operation of the channels on users’ signals, we can model the channel matrix for user k as
where the values of the entries which are not shown in the matrix H_{ k } in (1) are zeros, and such typical channel model has been widely used in machinetype communication (MTC) [9, 13]. In the BS, the received signal formed by N chips is denoted by \(\mathbf {y}={\left [{y_{1}}, \ldots,{y_{N}}\right ]^{\intercal }} \in {\mathbb {C}^{N \times 1}}\), and the noise vector is defined as \(\mathbf {v}={\left [{v_{1}}, \ldots,{v_{N}}\right ]^{\intercal }} \in {\mathbb {C}^{N \times 1}}\). Thus, the received signal vector is given by
where the effective channel matrix B can be written as
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.
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.
Coherence of the effective channel matrix
Now, we first give the definition of the mutual coherence between two nonzero vectors \(\mathbf {a}_{1},\mathbf {a}_{2} \in \mathbb {R}^{N \times 1}\), i.e.,
where \(\left \langle {\mathbf {a}_{1},\mathbf {a}_{2}} \right \rangle \) denotes the inner product between a_{1} and a_{2}, i.e., \(\left \langle {\mathbf {a}_{1},\mathbf {a}_{2}} \right \rangle = {\mathbf {a}_{1}^{\intercal }}\mathbf {a}_{2}\). Then, the coherence of \(\mathbf {A} = \left [{\mathbf {a}_{1}}, \ldots,{\mathbf {a}_{K}}\right ] \in {\mathbb {R}^{N \times K}}\), denoted by \(\mu \left (\mathbf {A} \right)\), is defined as [28–31]
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 \({{\mu (\mathbf {A}) \ge 1} \left / {\sqrt N }\right.}\).
In order to facilitate analysis, we rewrite (4) as
where \({\mathbf {F}_{k}} \in \mathbb {R}^{N \times L}\) (1≤k≤K) is generated by f_{ k } in the following form, i.e.,
Then, we denote the inner product between the mth column of F_{ k } and the nth column of F_{ j } (1≤m,n≤L,1≤k,j≤K) as
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
Given two matrices \({\mathcal {H}} = \left [ {{\mathbf {h}_{1}}, \ldots,{\mathbf {h}_{K}}} \right ] \in {\mathbb {R}^{L \times K}}\) and \({\mathcal {F}} = \left [ {{\mathbf {f}_{1}}, \ldots,{\mathbf {f}_{K}}} \right ] \in {\mathbb {R}^{N \times K}}\), 2<L<N. \(\mathcal {F}\) has unitnorm columns, i.e., \( \left \ {{\mathbf {f}_{k}}} \right \_{2}^{2} = 1\), and h_{ k }≠0, 1≤k≤K. \({\mathbf {F}_{k}}\in {\mathbb {R}^{N \times L}}\) is full rank and generated by f_{ k } in (8). If there exist constants \(\mu _{1}, {\mu _{2}}, \theta \in \left [0,1\right)\) satisfying the following properties for 1≤m,n≤L and 1≤k,j≤K:
then, for arbitrary θ+μ_{1}<1, the coherence of \(\mathbf {B} = \left [{\mathbf {F}_{1}}{\mathbf {h}_{1}}, \ldots,{\mathbf {F}_{K}}{\mathbf {h}_{K}}\right ] \in \mathbb {R}^{N \times K}\), i.e., \(\mu \left (\mathbf {B} \right)\), satisfies the following inequality
The proof of Theorem 1 is presented in Appendix 1.
From Theorem 1, we have an upper bound for the coherence of B, which is denoted as
Moreover, from the proof of Theorem 1 in Appendix 1, we have that the upper bound μ_{up} is only determined by the LCS signatures f_{1},…,f_{ K } (i.e., matrix \(\mathcal {F}\)) and has nothing to do with the channel response vectors h_{1},…,h_{ K } (i.e., matrix \(\mathcal {H}\)). In [27, 28], the authors have proved that the coherence of a sensing matrix B (i.e., μ(B)) can guarantee the recovery of Ssparse signal when S satisfies
Based on Theorem 1, the LCS signatures satisfying the properties (10)(12) ensure that \(\mu \left (\mathbf {B} \right) \le \mu _{\text {up}}= \frac {{ \mu _{2} }}{{1  \theta  \mu _{1} }}\), so we have the minimum sparsity determined by B as follows
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.
In order to further interpret the proposed CSNOMA scheme with the LCS signatures satisfying Theorem 1, we introduce the restricted isometry property (RIP) [7] which is another necessary and sufficient condition for guaranteeing the recovery of the sparse signal x in the presence of noise [7, 27]. First, we give the definition of RIP, i.e., a matrix A satisfies the RIP of order S if there exists a δ_{ S }∈(0,1) such that
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
In Theorem 1, when \({\mathcal {H}}\)has unitnorm columns, i.e., \(\left \ {{\mathbf {h}_{k}}} \right \_{2}^{2}= 1\), 1≤k≤K, then B has the RIP of order S with constant δ_{ S }=θ+μ_{1}+(S−1)μ_{2} for arbitrary S<1+1/μ_{up}, i.e.,
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.
Generating algorithm of the LCS matrix \(\mathcal {F}\)
Based on the results in Theorem 1, the LCS matrix \(\mathbf {\mathcal {F}} \in {\mathbb {R}}^{N \times K}\) can be obtained by system searching over the set {ξ} by using Algorithm 1, where the entries of \({\xi } \in \mathbb {R}^{N \times 1}\) are independent realizations of random variables with following distribution
where ς_{ n } is the nth entry of ξ. In this paper, we consider N=M, which implies that the entries of ξ are independent realizations of ± Bernoulli random variables with
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 \(\mathbf {\mathcal {F}}\).
Effect of the imperfect channel state information (CSI)
Since the CSMUD is applied to perform joint activity and data detection, the BS needs to know the sensing matrix \(\mathbf {B} = \left [{\mathbf {F}_{1}}{\mathbf {h}_{1}}, \ldots,{\mathbf {F}_{K}}{\mathbf {h}_{K}}\right ]\) based on the CS theory [7]. Because the LCS matrix \(\mathcal {F}\) is designed in advance, i.e., F_{ k } (1≤k≤K) is known to the BS. The channel response vector h_{ k } (1≤k≤K) is generally obtained by using a channel estimation algorithm in BS. However, the channel state information (CSI) is generally not perfect, which means that there exists some error in the estimation of h_{ k } in BS. Let \({\hat {\mathbf {h}}_{k}}\) and e_{ k } denote the estimated channel response vector and the error vector of user k, respectively. Then, we have
Correspondingly, let \(\hat {\mathbf {B}} = \big [ {{\mathbf {F}_{1}}{\hat {\mathbf {h}}_{1}}, \ldots,{\mathbf {F}_{K}}{\hat {\mathbf {h}}_{K}}} \big ]\) denote the estimated matrix of B in the BS. From Theorem 1, we know that the upper bound of μ(B), i.e., μ_{up}, is only determined by the LCS signatures f_{1},…,f_{ K } (i.e., the LCS matrix \(\mathcal {F}\)) and has nothing to do with the channel response vectors h_{1},…,h_{ K } (i.e., the channel response matrix \(\mathcal {H}\)). Specifically, the differences between B and \(\hat {\mathbf {B}}\) only exist in the channel response vectors, i.e., B and \(\hat {\mathbf {B}}\) have the same upper bound of coherence μ_{up}. Then, the coherence of \(\hat {\mathbf {B}}\), i.e., \(\mu (\hat {\mathbf {B}})\), satisfies the inequality
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 3.1 in [36]). Suppose that the coherence of \(\mathbf {A} \in \mathbb {R}^{N \times K}\) (N≪K) is μ_{ A } and \(\mathbf {x} \in \mathbb {R}^{K \times 1}\) is an Ssparse signal, where \(S < \left (1 / \mu _{\mathbf {A}} + 1\right)/4\). Furthermore, suppose that the measurement \(\mathbf {y}\in \mathbb {R}^{N \times 1}\) is obtained by y=Ax+n, where \(\mathbf {n}\in \mathbb {R}^{N \times 1}\) is a noise vector. Then, the estimation of x is given by
where \(\mathfrak {B} \left (\mathbf {y} \right) = \left \{ {\mathbf {z}:{{\left \ {{\mathbf {Az}}  \mathbf {y}} \right \}_{2}} \le \epsilon } \right \}\). Then, we have that
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.
In the perfect CSI case, from Theorem 2, the CSNOMA scheme has a coherencebased bound on the performance for the bounded noise v, which is given by
where μ_{ B } is the coherence of B, and \(S < \left (1/\mu _{\text {up}}+1\right)/4\). In this paper, we denote the perfect LCS signatures as ones that are designed with the parameters \(\mu _{1}\rightarrow 0\), \(\mu _{2}\rightarrow 0\), and θ≠1 (i.e., \(\mu _{\mathbf {B}}\rightarrow 0\)). From (25), the CSNOMA scheme using the perfect LCS signatures has a bound, i.e.,
In fact, the channel response estimation error can eventually be considered as a noise error. Thus, based on Theorem 2, we analyze the bound on the performance of the CSNOMA scheme under imperfect CSI for bounded e_{ k }, 1≤k≤K. For the corresponding \(\hat {\mathbf {B}} = \big [ {{\mathbf {F}_{1}}{{\hat {\mathbf {h}}}_{1}}, \ldots,{\mathbf {F}_{K}}{{\hat {\mathbf {h}}}_{K}}} \big ]\), we rewrite it as \(\hat {\mathbf {B}}=\big [{{\mathbf {F}_{1}}{{\mathbf {h}}_{1}}+{\mathbf {F}_{1}}{{\mathbf {e}}_{1}}, \ldots,{\mathbf {F}_{K}}{{\mathbf {h}}_{K}}+{\mathbf {F}_{K}}{{\mathbf {e}}_{K}}} \big ]\). Thus, the received signal in (3) is rewritten as
where \({\mathbf {B}_{\mathbf {e}}}=\left [{{\mathbf {F}_{1}}{{\mathbf {e}}_{1}}, \ldots,{\mathbf {F}_{K}}{{\mathbf {e}}_{K}}} \right ]\). Let n=B_{ e }x+v denotes the total noise. From Theorem 2, when μ(B) and S are fixed, the upper bound of \({\left \ {\mathbf {x}  \hat {\mathbf {x}}} \right \_{2}}\) is determined by the upper bound of \({\left \ \mathbf {n} \right \_{2}}\), i.e., \({\left \ {{\mathbf {B}_{\mathbf {e}}}\mathbf {x} + \mathbf {v}} \right \_{2}}\). Thus, based on the triangle inequality [38], we have
In order to obtain an upper bound for \({\left \ \mathbf {n} \right \_{2}}\), we need to analyze the upper bound of \(\left \ {{\mathbf {B}_{\mathbf {e}}}\mathbf {x}} \right \_{2}\). Similar to (46) in Appendix 1, the bound of \(\left \ {{\mathbf {B}_{\mathbf {e}}}\mathbf {x}} \right \_{2}^{2}\) is given by
where \({\lambda _{\min }^{\mathbf {D}}}\) and \({\lambda _{\max }^{\mathbf {D}}}\) are the minimum and maximum eigenvalues of \(\mathbf {D}={\mathbf {B}_{\mathbf {e}}}^{\intercal } {\mathbf {B}_{\mathbf {e}}} \in \mathbb {R}^{K \times K}\), respectively. D is represented by
Since \({\left ({{\mathbf {F}_{k}}{\mathbf {e}_{k}}} \right)^{T}}{\mathbf {F}_{k}}{\mathbf {e}_{k}} = \left \ {{\mathbf {F}_{k}}{\mathbf {e}_{k}}} \right \_{2}^{2}, 1\leq k \leq K\), from (48) in Appendix 1, we have the bound for \(\left \ {{\mathbf {F}_{k}}{\mathbf {e}_{k}}} \right \_{2}^{2}\), i.e.,
Furthermore, because \({ {{\left ({{\mathbf {F}_{k}}{\mathbf {e}_{k}}} \right)}^{T}}{\mathbf {F}_{j}}{\mathbf {e}_{j}} }={\left  {\left \langle {{\mathbf {F}_{k}}{\mathbf {e}_{k}},{\mathbf {F}_{j}}{\mathbf {e}_{j}}} \right \rangle } \right }\), 1≤k,j≤K,k≠j, from (54) in Appendix 1, we have
where \({\left \ \mathbf {e} \right \_{2}}{\text {= }}{\max _{k}}{\left \ {{\mathbf {e}_{k}}} \right \_{2}}\). Let d_{k,j} denote the (k,j)th entry of D, then d_{k,j} satisfies the following properties:

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

\(\sum \limits _{j = 1,j \ne k }^{K} {\left  {d_{k,j}} \right } \le (K1) \mu _{2} {\left \ \mathbf {e} \right \_{2}^{2}}, ~\forall k.\)
From Theorem 3 in Appendix 1, we have that
where φ=1+μ_{1}+μ_{2}(K−1). This implies that \({\lambda _{\max }^{\mathbf {D}}}\leq \varphi \left \ {{\mathbf {e}}} \right \_{2}^{2}\). Therefore, we have
Insert (34) into (28), we have
Thus, from Theorem 2, we have the coherencebased bound on the performance of the CSNOMA scheme in the case that CSI is not perfect, i.e.,
where \( S < \left (1/\mu _{\text {up}}+1\right)/4 \). Furthermore, when \(\mu _{1},\mu _{2} \rightarrow 0\), from (36), we have
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.
Simulation results
In our simulations, we consider an MMTC scenario in which K users sporadically access to a single BS over the multipath wireless channels in the presence of AWGN. All users share N CDMA chips, i.e., the gain of LCS signature is N. We adopt BPSK modulation and assume that all users’ wireless channels have L paths. Here, we consider that the entries of the channel response vector h_{ k } (1≤k≤K) follow i.i.d. \(\mathcal {N}\left ({0,1} \right)\) for user k. Furthermore, we also discuss the case that the CSI is not perfect and assume that the entries of the channel response estimated error vector e_{ k } (1≤k≤K) are i.i.d. Gaussian variables with \(\mathcal {N}\left ({0,\sigma ^{2}} \right)\) (σ^{2}≪1). In addition, the LCS matrix \(\mathcal {F} \in \mathbb {R}^{N \times K}\) is generated by Algorithm 1 with the predetermined parameters M,μ_{1},μ_{2}. The OMP [25, 26] are employed to demonstrate the effectiveness of the proposed scheme. We evaluate the meansquareerror (MSE) performance, which is defined as
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.
In the case that CSI is perfect, from (25) we have
where S<(1/μ_{up}+1)/4. For the CSNOMA scheme using the perfect LCS signatures (i.e., \({\mu _{\mathbf {B}}}\rightarrow 0\)), we have
In the case that CSI is not perfect, from (36) we have
When \(\mu _{1},\mu _{2}\rightarrow 0\), from (37) we have
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).
Validation for the coherences with different LCS signatures
For ease of reference, we denote μ(B)_{max} as the maximum coherence of different B that consist of different \(\mathbf {\mathcal {H}}\) and a fixed \(\mathbf {\mathcal {F}}\), and \(\mu (\hat {\mathbf {B}})_{\text {max}}\) is defined as the maximum coherence of different \(\hat {\mathbf {B}}\) which consist of different \(\mathbf {{\mathcal {\hat {H}}}}=\left [\hat {\mathbf {h}}_{1}, \ldots, \hat {\mathbf {h}}_{K}\right ]\) and a fixed \(\mathbf {\mathcal {F}}\). Furthermore, for comparison, we give a counterexample of the LCS matrix \(\mathbf {\mathcal {F}}\), which is given by
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.
Figure 2 shows μ(B) versus μ_{1}, μ_{2} with N=M=800, β=1.5, L=4. Note that \(\mathbf {B}_{\mathbf {\mathcal {\overline {F}}}}\) and \(\mathbf {B}_{\mathbf {{\mathcal {F}}}}\) respectively represent the cases that consist of \(\mathbf {\mathcal {\overline {F}}}\) and \({\mathbf {{\mathcal {F}}}}\), and \(\hat {\mathbf {B}}_{\mathbf {{\mathcal {F}}}}\) denotes the matrix \(\hat {\mathbf { B}}\) that consists of \({\mathbf {{\mathcal {F}}}}\) in the imperfect CSI case with 1/σ^{2}=10 dB. From the figure, we see that, for different μ_{1},μ_{2}, the plane of μ_{up} is higher than the planes of \(\mu \left (\mathbf {B}_{\mathbf {{\mathcal {F}}}}\right)_{\text {max}}\) and \(\mu \left (\hat {\mathbf {B}}_{\mathbf {{\mathcal {F}}}}\right)_{\text {max}}\), but lower than the plane of \(\mu \left (\mathbf {B}_{\mathbf {{\mathcal {\overline {F}}}}}\right)_{\text {max}}\). Thus, when μ_{1} and μ_{2} are given (i.e., given \(\mathcal {F}\)), \(\mu \left (\mathbf {B}_{\mathbf {{\mathcal {F}}}}\right)_{\text {max}}\) and \(\mu \left (\hat {\mathbf {B}}_{\mathbf {{\mathcal {F}}}}\right)_{\text {max}}\) are always less than μ_{up}, i.e., μ_{up} is the upper bound of \(\mu \left (\mathbf {B} \right)\) and \(\mu \left (\hat {\mathbf {B}} \right)\). These results verify that the designed LCS matrix \(\mathbf {\mathcal {F}}\) can ensure that the coherence of the effective channel matrix, i.e., μ(B), has an upper bound μ_{up} that has nothing to do with the channel response matrix \(\mathcal {H}\). Since the differences between \(\hat {\mathbf {B}} \) and B only exist in the channel response matrix, \(\mu (\hat {\mathbf {B}})\) and μ(B) have the same upper bound μ_{up}. In addition, since the LCS signatures in \({\mathbf {{\mathcal {\overline {F}}}}}\) do not satisfy Theorem 1, \(\mu \left (\mathbf {B}_{\mathbf {{\mathcal {\overline {F}}}}}\right)_{\text {max}}\) is always equal to 1 for different μ_{1} and μ_{2}. In summary, these simulation results verify the validity of the designed LCS signatures based on Theorem 1.
Performance in perfect CSI case
Figure 3 shows the MSE versus μ_{1},μ_{2} for different sparsities with N=M=800, L=4, β=1.5, and SNR = 25 dB. In the figure, the notation \({\text {MSE}}\_\mathbf {\mathcal {\overline {F}}}\) represents the MSE performance of the CSNOMA scheme using the LCS matrix \(\mathbf {\mathcal {\overline {F}}}\) in (43) with ρ_{a}=1%. Note that the CSNOMA scheme always adopts the LCS matrix \(\mathbf {\mathcal {{F}}}\) satisfying Theorem 1 unless otherwise specified in this paper. It can be seen from the figure that for different μ_{1} and μ_{2} (within a certain range, e.g., 0.03≤μ_{1}≤0.09 and 0.26≤μ_{2}≤0.32), the MSE performances are almost the same when the sparsity of the signal x is given. In addition, the MSE performance of the CSNOMA scheme using the LCS matrix \(\mathbf {\mathcal {{F}}}\) is the best when the sparsity of x is S_{min}, and it decreases with the increase of ρ_{a}; when ρ_{a}=6%, it is worse than the MSE_{1} performance which is achieved by using perfect LCS signatures. As for comparison, the \({\text {MSE}}\_\mathbf {\mathcal {\overline {F}}}\) performance is the worst, which implies that the CSNOMA scheme using \(\mathbf {\mathcal {\overline {F}}}\) can not work.
Furthermore, we also simulate the MSE versus SNR for different sparsities with N=M=800, L=4, β=1.5, μ_{1}=0.03, and μ_{2}=0.26, as shown in Fig. 4. It can be seen that the MSE performance of the CSNOMA scheme with \(\mathbf {\mathcal {{{F}}}}\) improves with the increase of SNR and decreases with the increase of ρ_{a}; when ρ_{a}≥5%, it is close to the MSE_{1} performance at high SNR, but always better than the MSE_{ μ } performance. However, as for comparison, the \({\text {MSE}}\_\mathbf {\mathcal {\overline {F}}}\) performance is almost unchanged and quite poor for different SNR, even worse than both MSE_{1} and MSE_{ μ } performances at high SNR.
In Fig. 5, we show the MSE versus SNR for different β with N=M=800, L=4, ρ_{a}=1%, μ_{1}=0.03, and μ_{2}=0.28. From the figure, it can be seen that the MSE performance of the CSNOMA scheme improves with the reduction of β at the same conditions. In addition, the MSE performance of the CSNOMA scheme is always better than the MSE_{1} performance when β≤4. But, when β=5, it is worse than the MSE_{1} performance at high SNR, e.g., SNR ≥20 dB. Thus, the proposed scheme can achieve a system overload of 4 when the active users are relatively sparse with an activity ratio of 1%.
In addition, we also provide the BER performance of the CSNOMA scheme in Fig. 6 with N=M=800, L=4, β=1.5, μ_{1}=0.03, and μ_{2}=0.26. In the figure, we compare the BER performances of the CSNOMA scheme using the LCS signature with that using the PN signature which is the typical spreading sequence in CSbased MTC [9, 13, 21, 22]. Here, after the column vector normalization of the PN matrix which contains the PN signatures of all users, we obtain θ=L/M=1/200, μ_{1}=0.31, and μ_{2}=0.47 for the PN matrix with L=4, which indicates that the PN matrix satisfies Theorem 1. Therefore, these PN signatures used in the simulation are also LCS signatures, as such the BER performance of the CSNOMA schemes using the LCS signature are almost the same with that using the PN signature at the same sparsity. These results further validate the availability of our proposed scheme.
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.
Performance in imperfect CSI case
Figure 7 shows the MSE versus μ_{1},μ_{2} in the imperfect CSI case with N=M=800, β=1.5, L=4, ρ_{a}=1%, and SNR = 25 dB. In the figure, the notations 15 dB represents 1/σ^{2}=15 dB. We can see that for different μ_{1},μ_{2} (within a certain range, e.g., 0.03≤μ_{1}≤0.09 and 0.26≤μ_{2}≤0.32), the MSE performances are almost the same when 1/σ^{2} is given. Additionally, the MSE performance in the imperfect CSI case improves with the increase of 1/σ^{2} and is better than the MSE_{2} performances, but it is worse than the MSE performance in the perfect CSI case with the same conditions. These results imply that even in the imperfect SCI case, the LCS matrix \(\mathcal {F}\) ensures that B is an effective sensing matrix.
In order to analyze the MSE performance in the imperfect CSI case for different SNRs, Fig. 8 shows the MSE versus SNR with N=M=800, β=1.5, L=4, ρ_{a}=1%, μ_{1}=0.03, and μ_{2}=0.26. From the figure, we can see that the MSE performance improves with the increase of SNR, and it also improves with the increase of 1/σ^{2} in the imperfect CSI case. Furthermore, the MSE_{2} results are better than other MSE results at the same SNR.
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.
Convergence and computational complexity of the proposed algorithm
It is hard to theoretically analyze the convergence performance and the complexity of Algorithm 1; therefore, we use simulation approach to numerically quantify these performances. Figure 9 shows the simulation time of Algorithm 1 versus μ_{2} for different μ_{1}. From the figure, we can see that when μ_{1} are fixed, the simulation time slowly increases with decrease of μ_{2}. But it will tend to be infinite when μ_{2} is below a certain value, e.g., μ_{2}<0.26 for μ_{1}>0.03. This indicates that Algorithm 1 may not be convergent when μ_{2} is below a certain value. Furthermore, when μ_{2} is fixed, the simulation time decreases with increase of μ_{1}, but is almost saturated when μ_{1} is greater than a certain value, e.g., μ_{1}>0.03. And, the convergence time of Algorithm 1 also increases with decrease of μ_{1}. The results imply that the convergence and computational complexity of Algorithm 1 are significantly affected by μ_{1} and μ_{2}, especially when μ_{1} and μ_{2} take low values.
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.
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
Based on the definition of coherence for a matrix, \(\mu \left (\mathbf {B} \right)\) can be written as
From (44), in order to obtain the upper bound of \(\mu \left (\mathbf {B} \right)\), we need to analyze the upper bound of \(\mu \left ({{\mathbf {F}_{k}}{\mathbf {h}_{k}},{\mathbf {F}_{j}}{\mathbf {h}_{j}}} \right)\) which is given by
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. □
Step 1. For arbitrary \(k \in \left \{ {1, \ldots,K} \right \}\), the bound of \({\left \ {{\mathbf {F}_{k}}{\mathbf {h}_{k}}} \right \_{2}^{2}}\) satisfies
where \({\lambda _{\min }^{k}}\) and \({\lambda _{\max }^{k}}\) are the minimum and maximum eigenvalues of \(\mathbf {G}^{k}=\mathbf {F}_{k}^{\intercal } {\mathbf {F}_{k}} \in \mathbb {R}^{L \times L}\), respectively. Let \(g_{m,n}^{k}\) denotes the (m,n)th entry of G^{k}, 1≤m,n≤L. Since we have \(\sum \limits _{n = 1, n \neq m}^{L} {\left  {\mu _{m,n}^{kk}} \right } \leq \mu _{1}, \forall m\), and \(\sum \limits _{l = 1}^{L{\text { }}1} {f_{N  l + 1,k}^{2}} \le \theta \), ∀k, then \(g_{m,n}^{k}\) satisfies the following properties:

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

\(\sum \limits _{n = 1,n \ne m }^{L} {\left  {g_{m,n}^{k}} \right  }=\sum \limits _{n = 1,n \ne m }^{L} {\left  {\mu _{m,n}^{kk}} \right } \le \mu _{1},~ \forall m,k.\)
From Theorem 3, all the eigenvalues of G^{k} satisfy the following inequality
where \(\lambda \left ({\mathbf {A}} \right)\) represents the eigenvalues of A. This implies that \(1\theta  \mu _{1} \leq {\lambda _{\min }^{k}}<{\lambda _{\max }^{k}} \leq 1 + \mu _{1}\). Thus, for arbitrary k, from (46), the bound of \(\left \ {{\mathbf {F}_{k}}{\mathbf {h}_{k}}} \right \_{2}^{2}\) is given by
Therefore, we have the bound of \({{{\left \ {{\mathbf {F}_{k}}{\mathbf {h}_{k}}} \right \}_{2}}} {{{\left \ {{\mathbf {F}_{j}}{\mathbf {h}_{j}}} \right \}_{2}}}\), i.e.,
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.
Step 2. Now, we discuss the upper bound of \({\left  {\left \langle {{\mathbf {F}_{k}}{\mathbf {h}_{k}},{\mathbf {F}_{j}}{\mathbf {h}_{j}}} \right \rangle } \right }\) which can be rewritten as \(\left  {\left \langle {{{\mathbf {h}}_{k}},{{\mathbf {F}}_{k}^{\intercal } } {{\mathbf {F}}_{j}}{{\mathbf {h}}_{j}}} \right \rangle } \right \), then we have
where \({\lambda _{\max }^{kj}}\) represents the maximum eigenvalue of \(\mathbf {P}^{kj}= {\left ({{\mathbf {F}}_{k}^{\intercal }{{\mathbf {F}}_{j}}} \right)^{\intercal }} {{\mathbf {F}}_{k}^{\intercal }} {{\mathbf {F}}_{j}} \in \mathbb {R}^{L \times L}\). Let \(p_{t,r}^{kj}\) denotes the (t,r)th entry of P^{kj}, 1≤t,r≤L, then \(p_{t,t}^{kj} = \sum \limits _{l = 1}^{L} {(\mu {{_{l,t}^{kj}})^{2}}} \ge 0\) and \(p_{t,r}^{kj} = \sum \limits _{l = 1}^{L} {\mu _{l,t}^{kj}\mu _{l,r}^{kj}} \le \sum \limits _{l = 1}^{L} {\left  {\mu _{l,t}^{kj}} \right \left  {\mu _{l,r}^{kj}} \right },~r \ne t\). Let \({\sum \nolimits }_{t}\) denotes the sum of the absolute values of elements in trow of P^{kj}, i.e.,
From Theorem 3, we know that \({\lambda _{\max }^{kj}} \le {\text {mi}}{{\text {n}}_{t}}{\sum \nolimits }_{t} \). Furthermore, from the rootmean squarearithmetic meangeometric mean (RMSAMGM) inequality [40], we have
when a_{1}=a_{2}=⋯=a_{ L }, where \(a_{l} =\left ({ {\mu _{l,t}^{kj}}  +  {\mu _{l,r}^{kj}} }\right)/2\). Since \(\sum \limits _{l = 1}^{L} {\left  {\mu _{l,t}^{kj}} \right } \le {\mu _{2}}\) for arbitrary t, (52) is rewritten as \(\sum \limits _{l = 1}^{L} {\left  {\mu _{l,t}^{kj}} \right \left  {\mu _{l,r}^{kj}} \right } \leq \mu ^{2}_{2}/L, ~\forall t,r\). Then, we have
which implies \({\lambda _{\max }^{kj}} \le {\mu ^{2}_{2}}\). From (50), the upper bound of \(\left  {\left \langle {{\mathbf {F}_{k}}{\mathbf {h}_{k}},{\mathbf {F}_{j}}{\mathbf {h}_{j}}} \right \rangle } \right  \) is given by
Therefore, combining (49) and (54), we have the upper bound of \(\mu \left ({{\mathbf {F}_{k}}{\mathbf {h}_{k}},{\mathbf {F}_{j}}{\mathbf {h}_{j}}} \right)\) for arbitrary k,j, k≠j, i.e.,
which implies that \(\mu \left (\mathbf {B} \right) \le \frac {{ \mu _{2} }}{{1  \theta  \mu _{1} }}\). Thus, we complete the proof of Theorem 1.
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 \(\mathbf {C}=\mathbf {B}^{\intercal }_{\Lambda } \mathbf {B}_{\Lambda } \in {\mathbb {R}^{S \times S}}\). □
Let c_{i,s} be the (i,s)th entry of C, 1≤i, s≤S, then \(c_{i,i}=\ {\mathbf {F}_{i}}{\mathbf {h}_{i}}\^{2}_{2}\) and c_{i,s}=〈F_{ i }h_{ i },F_{ s }h_{ s }〉, i≠s. From (48) and (54), and recall \(\left \ {{\mathbf {h}_{k}}} \right \_{2}^{2}= 1,~ \forall k\), the entries of C satisfy the following properties:

1−θ−μ_{1}≤c_{i,i}≤1+μ_{1},1≤i≤S;

\(\left  {{c_{i,s}}} \right  \le {\mu _{2}},i \ne s,1 \le i,s \le S.\)
Then, \(\sum \limits _{s = 1,s \ne i}^{S} {\left  {{c_{i,s}}} \right  \le } (S  1)\mu _{2},~1 \le i \le S\), from Theorem 3, we have
which implies that
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.
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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).
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He, K., Li, Y., Yin, C. et al. A novel compressed sensingbased nonorthogonal multiple access scheme for massive MTC in 5G systems. J Wireless Com Network 2018, 81 (2018). https://doi.org/10.1186/s1363801810794
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Keywords
 Nonorthogonal multiple access
 Compressed sensing
 Sparse multiuser detection
 Massive machine type communication
 Channel state information