Generalized Multi-user Sparse Superposition Transmission for Massive Machine-type Communications

I. INTRODUCTION

WITH the continuous integration of sensors and wireless communication networks, any object equipped with sensors can become an intelligent device, thus turning into a potential network access user. This has spawned many new applications based on the Internet of things (IoT), such as smart healthcare and home automation [1]. Due to the application of a large number of sensor terminals, data collected in the IoT networks is mainly lightweight to support real-time interaction, short packet communications (SPCs) have gradually become a new business in the wireless communications systems [2]. With limited spectrum resources, how to meet the requirements of massive machine-type communications (mMTC) in the SPCs scenarios will be one of the key issues of the IoT networks.

Combining with channel coding and non-orthogonal multiple access (NOMA) technologies, multi-user superposition transmission can be realized on the time-frequency grids of the existing orthogonal frequency division multiplexing (OFDM), which has been attracted increasing attention in recent years [3]–[7]. In the superimposed transmission systems, the NOMA technologies realize the co-channel access of multiple users, and the channel codes guarantee the reliable communications. At the receiver side, the multi-user detector mostly adopts the successive interference cancellation (SIC)- based decoding algorithm in the power allocation-based superimposed transmission systems [8], [9], whereas joint iteration detection and decoding algorithms are generally used as the multi-user detector in the code-domain superimposed transmission systems [10]–[12]. Therefore, the transmission reliability of these superimposed transmission systems can be improved by using power optimization and modulation constellation design. However, the application of NOMA technologies can not completely eliminate the interference among users. In addition, the polar codes and low-density parity-check (LDPC) codes adopted by the fifth generation (5G) wireless communications have obvious advantages in the encoding of long packets. When the data packet is short, the sparseness of the parity check matrix and channel polarization are difficult to be fully guaranteed, which results in a decreased block error ratio (BLER) performance of NOMA-based superimposed transmission systems.

Recently, sparse superimposition coding (SSC), a novel coding scheme based on position index modulation, has been introduced [13]–[16]. The SSC technology divides the transmission bits into one index bit stream and [TeX:] $$N \geqslant 1$$ modulation bit streams, where the index bit stream determines which N spreading sequences in the codebook are selected for the spread spectrum transmission of the modulation bits. Because the number N of the spreading sequences selected from the codebook is small, SSC can achiever better latency and BLER performance than polar codes, which has been verified in many previous works [17]–[20]. Apart from its implementation as a coding technology, researchers mentioned in [21] divided the orthogonal codebook into multiple small sub-codebooks to serve multiple users simultaneously, thus evolving SSC into a multi-user sparse superimposed transmission (MUSST) scheme. However, this approach restricts each user to select one spreading sequence from their preassigned sub-codebook. Since the number of transmitted bits of each user depends on the number of selected spreading sequences, this undoubtedly limits the frequency efficiency and flexibility of the MUSST. To further improve the coding efficiency of MUSST, this paper proposes a generalized multi-user sparse superposition transmission (GMUSST) scheme based on the SSC. Similar to SSC, GMUSST allows each user to select multiple spreading sequences for the encoding of the index bit stream. When there is only one user, the proposed GMUSST is converted into a SSC scheme. Additionally, the proposed GMUSST can use non-orthogonal codebooks like the one proposed in [22] to provide multi-user superimposed transmission. However, our proposal is a MUSST scheme that combines position index modulation and constellation modulation, whereas the scheme proposed in [22] is only a MUSST scheme with position index modulation. Giving the codebook, our proposal can transmit more bits than [22]. When the number of the transmitted information bits is predetermined, our proposal can use smaller codebooks for multiple users than [22], thereby enhancing transmission reliability.

Furthermore, matching pursuit algorithms, such as orthogonal matching pursuit (OMP) and multi-path matching pursuit (MMP), have been widely adopted as the multi-user detector due to the equivalence of MUSSC to a sparse system [13], [19], [22]. However, matching pursuit algorithms are greedy-based sub-optimal detectors, and the user interference is not completely eliminated during iteration, resulting in lower BLER performance compared to the maximum likeihood (ML) detector, despite the involvement of multiple paths. For this, authors in [23] have been proposed a low-complexity detector based on SIC to replace matching pursuit algorithm. Its BLER performance approaches that of maximum a posteriori probability (MAP) detector when the number of access users is small, but will be fall into error floor due to error accumulation of SIC with the number of access users increasing. To tackle this problem, like MMP, a multi-path searchingbased SIC (MSIC) detector is developed to improve the BLER performance of SIC when the number of access users is large. The decoders mentioned above (including MSIC) are all multi-user joint detection. To the best of our knowledge, there is currently no single-user detection algorithm that can directly detect one user transmitted signal from the superimposed signal of multiple users. Based on this, we attempt to design a single-user SIC detector that incorporates minimum mean square error (MMSE) estimation, which is suitable for downlink GMUSST system with large access users.

The main contributions of this paper can be summarized as follows:

· To reduce the high complexity of ML multi-user detector, a multi-user detector based on SIC (MSIC) with multipath searching is given. It can achieve improved BLER performance than the MMP algorithm, while its computational complexity is comparable to that of MMP.

· Considering that in downlink GMUSST systems a user is primarily concerned with their own transmitted signals, we propose a single-user detector based on MMSE estimation. This detector can efficiently extract an individual user’s transmitted signals from the received superimposed signal of multiple users. We name the proposed single-user detector as MMSE-SIC since its decoding procedures are similar to that of MSIC. The BLER performance of MMSE-SIC does not rely on multi-path searching, yet it can still achieve satisfactory results.

· The BLER performance of MSIC is verified compared with the ML and MMP multi-user detectors. Simulation results show that the BLER performance of the MSIC detector approaches that of the ML detector with signalto- noise ratio (Eb/N0) increasing. Moreover, the MSIC detector exhibits superior BLER performance compared to the MMP detector. In the downlink GMUSST systems, the BLER performance of the MMSE-SIC detector gradually approaches and even exceeds the BLER performance of the MSIC detector, as the number of users increases.

· Finally, employing the selected coding parameters, we compare the BLER performance of GMUSST with the existing polar codes sparse code multiple access (PCSCMA) systems. Simulation results show that GMUSST can obtain better BLER performance than PC-SCMA. With hybrid automatic repeat request (HARQ) mechanism [24], GMUSST will have more obvious advantages in terms of BLER performance and retransmission times.

The remainder of this paper is organized as follows. Section II introduces the system model of GMUSST systems, including the transmitter and receiver explanation in the uplink and downlink GMUSST systems. Afterwards, we further discuss the conditions to support signal recovery with a nonorthogonal codebook, and the optimal coding parameter selection problems for a large packet in Section III. The simulation results are shown in Section IV, followed by conclusions in Section V.

II. THE SYSTEM MODEL OF GENERALIZED MULTI-USER SPARSE SUPERPOSITION TRANSMISSION

Consider that there is a base station (BS) serving J users, and these J users share K time-frequency resources (or subcarriers) with a [TeX:] $$K \times L$$ codebook [TeX:] $$\mathbf{C}=\left[\mathbf{C}_1, \mathrm{C}_2, \cdots, \mathbf{C}_J\right]$$, where [TeX:] $$\mathbf{C}_j, j \in[1, J]$$ is the sub-codebook for the jth user. Each column in the codebook [TeX:] $$\mathbf{C}$$ denotes a spreading sequence, and each sub-codebook will have [TeX:] $$L_e=L / J$$ spreading sequences by using an equal distribution method. The transmitted bit stream [TeX:] $$b^j$$ of jth user is split into one index bit stream [TeX:] $$b_0^j$$ and N modulation bit streams [TeX:] $$b_n^j, n \in[1, N].$$ For convenience, we assume that the number of information bits for arbitrary user is the same, i.e., [TeX:] $$b=b^1=\cdots=b^J, b_0=b_0^1=\cdots=b_0^J$$ and [TeX:] $$b_n=b_n^1=\cdots=b_n^J .$$ Meanwhile, the label of each bit stream also denotes the number of information bits of itself in this paper. For example [TeX:] $$b^j$$ is not only the label but also the number of transmitted bits of the jth user.

Moreover, the index bit stream [TeX:] $$b_0^j$$ determines which N spreading sequences are selected from the sub-codebook [TeX:] $$\mathbf{C}_j$$. According to the one-to-one mapping criterion, the relationship between [TeX:] $$L_e \text { and } b_0^j$$. can be thus given as

where [TeX:] $$\lfloor\cdot\rfloor$$ is the round-down operation.

The remaining N modulation bit streams [TeX:] $$b_n^j, n \in[1, N]$$ have the same number of bits bn, and they will respectively generate N modulated signals [TeX:] $$s_n^j, n \in[1, N]$$ via the M-order quadrature amplitude modulation (QAM) constellations, where [TeX:] $$M=2^{b_n}.$$ Let [TeX:] $$\Omega_n, n \in[1, N]$$ denote the column index of the nth selected spreading sequence in the codebook [TeX:] $$\mathbf{C}_j$$. With the selected spreading sequences [TeX:] $$\mathbf{c}_{j, \Omega_n}, n \in[1, N]$$, the transmitted signal vector is

where [TeX:] $$\mathbf{s}_j$$ is message vector of the jth user, and it only has N non-zero elements [TeX:] $$\left\{s_n^j\right\}_{n=1}^N.$$ We generally have [TeX:] $$N \ll L_e$$ in the message vector [TeX:] $$\mathbf{s}_j$$, and (2) is thus called SSC in many previous studies [19], [25], [26]. Extending (2) to a multi-user scenario, the system model for GMUSST of N = 2 can be given as Fig. 1, in which [TeX:] $$\mathbf{s}=\left[\mathbf{s}_1, \mathbf{s}_2, \cdots, \mathbf{s}_J\right]^T$$ is the message vector of J users.

A. Uplink GMUSST Systems

1) Transmitter: Let [TeX:] $$\mathbf{h}_j=\left[h_{1, j}, h_{2, j}, \cdots, h_{K, j}\right]$$ denote the channel gain of jth user. Assuming that all users are synchronized, the received vector of the BS will be

where [TeX:] $$\operatorname{diag}\left(\mathbf{h}_j\right)$$ is the diagonal matrix of [TeX:] $$\mathbf{h}_j,$$ and [TeX:] $$\mathbf{z}=\left[z_1, z_2, \cdots, z_K\right]^T$$ is noise vector, which obeys a complex Gaussian distribution with zero mean and variance [TeX:] $$\sigma^2.$$ Let [TeX:] $$\mathbf{H}_j=\operatorname{diag}\left(\mathbf{h}_j\right) \mathbf{C}_j$$ and [TeX:] $$\phi=\left[\mathbf{H}_1, \mathbf{H}_2, \cdots, \mathbf{H}_J\right],$$ the matrix representation of (3) is

2) Receiver: At the receiver side, the ML multi-user detector can be used to obtain the transmitted message vector s, i.e.,

where [TeX:] $$\|\cdot\|$$ is 2-norm, [TeX:] $$\mathcal{S}$$ is the set of candidate message vector, and [TeX:] $$\mathbf{s}^*$$ is the optimal transmitted vector obtained from [TeX:] $$\mathcal{S}$$ according to (5). Obviously, [TeX:] $$\left(M^N \times 2^{b_0}\right)^J$$ candidate message vectors in [TeX:] $$\mathcal{S}$$ should be searched to obtain [TeX:] $$\mathbf{s}^*$$. Although N and M are usually small, its complexity is very high due to large L, which can reach hundreds even thousands. Therefore, designing a low-complexity detector is inevitable.

According to the encoding procedures of GMUSST, the spreading sequences selected from [TeX:] $$\mathbf{C}$$ can be identified, and then the modulation signals of s can be recovered. Thus, we can obtain all uses’ bit information with the help of SIC. Let U = JN denote the total number of modulation symbols in the message vector [TeX:] $$\mathbf{s} \text {, and } \mathbf{r}_{u-1}$$ is the residual vector of the uth (u ∈ [1,U]) iterations of SIC. Each iteration of SIC-based multi-user can be divided into three steps. The first step is to calculate the correlation between the residual vector [TeX:] $$\mathbf{r}_{u-1}$$ and the codebook [TeX:] $$\mathbf{C}$$ to find out which one spreading is selected, i.e.,

where [TeX:] $$\left\langle\phi, \mathbf{r}_{u-1}\right\rangle$$ denotes the inner product, and [TeX:] $$\tilde{\pi}_u$$ is spreading with the largest correlation coefficient. With the column index [TeX:] $$\tilde{\pi}_u$$, the second step is to obtain the estimate of uth modulation signal by

and

where [TeX:] $$\phi^{\dagger}=\left(\phi^T \phi\right)^{-1} \phi^T$$ is the pseudo-inverse of [TeX:] $$\phi \text {, and }\mathbf{G}$$ is the modulation constellation.

Finally, the residual vector for the next (u + 1)th iteration will be

By the decoding procedures from (6) to (9), we can obtain U column index [TeX:] $$\left\{\tilde{\pi}_u\right\}_{u=1}^U$$ of codebook [TeX:] $$\mathbf{G}$$ and U modulation signal [TeX:] $$\left\{s_u^*\right\}_{u=1}^U$$ of the message vector [TeX:] $$\mathbf{s}$$. However, the reliability of SIC may be low due to the incompleteness of interference cancellation. To address this issue, a multi-path searching-based MSIC detector is introduced, drawing on the principle of MMP.

Assume that [TeX:] $$D \geqslant 1$$ candidate column indices of codebook [TeX:] $$\mathbf{C}$$ can be found out for each spreading sequences’ estimation. Given a searching path [TeX:] $$\ell=1+\sum_{u=1}^U\left(d_u-1\right) D^{u-1},$$ where [TeX:] $$d_u \in[1, D] \text { for } \forall u \in[1, U] \text {, }$$ (6) will be rewritten as

where [TeX:] $$\mathbf{r}_{d_{u-1}}^{\ell}$$ is the residual vector of the [TeX:] $$d_{u-1} \text {th }$$ candidate column index of codebook [TeX:] $$\mathbf{C}.$$

Afterwards, the [TeX:] $$d_u \text {th }$$ candidate column index [TeX:] $$\tilde{\pi}_{d_u}^{\ell}$$ is selected from [TeX:] $$\left\{\tilde{\pi}_{d_u}^{\ell}\right\}_{d_u=1}^D$$ to estimate the modulated symbol, which can be given as

With [TeX:] $$\tilde{\pi}_{d_u}^{\ell} \text { and } s_{d_u}^{l, *} \text {, }$$ the residual vector [TeX:] $$\mathbf{r}_{d_u}^{\ell}$$ with SIC is expressed as

After Uth iteration, the desired column indices [TeX:] $$\left\{\tilde{\pi}_{d_u}^{\ell}\right\}_{u=1}^U$$ and modulation symbols [TeX:] $$\left\{s_{d_u}^{\ell, *}\right\}_{u=1}^U$$ can be obtained if [TeX:] $$\left\|\mathbf{r}_{d_U}^{\ell}\right\| \leqslant \varepsilon$$ or [TeX:] $$\ell \leqslant \ell_{\max },$$ where [TeX:] $$\varepsilon$$ is the termination threshold, and [TeX:] $$\ell_{\max }$$ is the maximum number we set. Otherwise, the next [TeX:] $$(\ell+1) \text {th }$$ searching path should be constructed. The detection procedure is summarized as Algorithm 1.

B. Downlink GMUSST Systems

1) Transmitter: For downlink systems, the transmitted vectors of all users are first superimposed and then transmitted at the BS side. With his channel gain [TeX:] $$\mathbf{h}_j,$$ the jth [TeX:] $$(\forall j \in[1, J])$$ users’ receiving signal can be expressed as

Similar to (3), the matrix representation of (13) is

where [TeX:] $$\phi=\operatorname{diag}\left(\mathbf{h}_j\right) \mathbf{C}.$$

2) Receiver: While MSIC can serve as a multi-user detector in both uplink and downlink GMUSST systems, its suitability for downlink systems is questionable. That’s because all users’ transmitted signals should be detected by MSIC for arbitrary devices, which obviously results in an increased computational complexity. In the downlink GMUSST systems, where the transmitted signals of the jth user is only relevant to that specific user, a detector that get the jth user’s transmitted signal directly from the superimposed signal of J users emerges as the most suitable candidate. To address this, a single-user detector named MMSE-SIC, based on MMSE estimation, is proposed for the downlink GMUSST systems.

Assuming that the candidate spreading and modulation symbol before nth non-zero elements of message vector [TeX:] $$\mathbf{s}_j$$ has been successfully find out, and [TeX:] $$\mathbf{r}_{j, n-1}$$ denotes the residual vector after SIC. Let [TeX:] $$\mathbf{x}_n^j=\mathbf{c}_{j, \Omega_n} s_n^j \text { and } \hat{\mathbf{x}}_n^j$$ be the estimate of [TeX:] $$\mathbf{x}_n^j.$$ As we know, the purpose of MMSE estimation is to eliminate the influence of noise, , thereby ensuring that the estimated [TeX:] $$\hat{\mathbf{x}}_n^j$$ closely approximates the true value [TeX:] $$\mathbf{x}_n^j.$$ Ideally, the mean square error is equal to 0. Therefore, our purpose is to let

where [TeX:] $$\mathbb{E}(\cdot)$$ is the mean function, [TeX:] $$(\cdot)^H$$ denotes the conjugate transpose, and the gain [TeX:] $$G_{m m s e}$$ of MMSE can be thus calculated as

Moreover, let the signals [TeX:] $$\mathbf{x}_n^j \text { for } \forall j \in[1, J], \forall n \in[1, N]$$ be independent of each other, [TeX:] $$\mathbb{E}\left\{\mathbf{x}_n^j \mathbf{r}_{j, n-1}^H\right\}$$ and [TeX:] $$\mathbb{E}\left\{\mathbf{r}_{j, n-1} \mathbf{r}_{j, n-1}^H\right\}$$ can be further respectively written as

and

Furthermore, assuming that the average power of [TeX:] $$\mathbf{x}_n^j$$ is 1, i.e., [TeX:] $$\mathbb{E}\left(\mathbf{x}_n^j\left(\mathbf{x}_n^j\right)^H\right)=1,$$ we have

and the estimate [TeX:] $$\hat{\mathbf{x}}_n^j$$ will be

where [TeX:] $$\mathbf{I}$$ is identity matrix.

Subsequently, we can obtain the candidate column indices in [TeX:] $$\mathbf{C}$$ by calculating the correlation between [TeX:] $$\mathbf{C}_j$$ and [TeX:] $$\hat{\mathbf{x}}_n^j,$$ i.e.,

With the candidate column index [TeX:] $$\tilde{\pi}_{j, n},$$ the related modulation symbol [TeX:] $$s_n^{j, *}$$ will be estimated as

where [TeX:] $$\mathbf{C}_j^{\dagger}$$ is the pseudo inverse of [TeX:] $$\mathbf{C}_j .$$

The above procedures can help us successfully find out its related spreading and symbol of nth non-zero elements. Given [TeX:] $$N \geqslant 2$$, the estimate [TeX:] $$s_n^{j, *}$$ will be treated as interference, and should be removed to detect the next non-zero element in [TeX:] $$\mathbf{s}_j.$$ Therefore, after SIC, the residual vector [TeX:] $$\mathbf{r}_{j,n}$$ will be

Similar to (10), the multi-path parameter D can also be used in (21). However, due to the estimation error of MMSE, the BLER performance of MMSE-SIC is inferior to that of MSIC, and increasing the multi-path parameter D to improve the BLER performance of MMSE-SIC is not significantly. Therefore, we default D = 1 in the MMSE-SIC detector. When D = 1, the detection procedures of the proposed MMSE-SIC are shown in Algorithm 2.

C. Complexity Analysis

According to Algorithm 1, the complexity of MSIC detector mainly focuses on the [TeX:] $$\mathcal{O}(L)$$ of (10) and [TeX:] $$\mathcal{O}(M)$$ of (11) at one path. Therefore, the maximum searching complexity of MSIC detector is [TeX:] $$\mathcal{O}\left(\ell_{\max } U(L+M)\right).$$ Compared to the complexity [TeX:] $$\mathcal{O}\left(2^{J b_0} M^U\right)$$ of ML detector, the proposed MSIC has low complexity as J and N increase. Additionally, the sparse recovery algorithm has been widely used as the decoder for SSC, such as the MMP algorithm. For GMUSST system, the MMP algorithm can also be used. The differences between the proposed MSIC and MMP are the estimation of modulation symbols and the calculation of the residual vector. In the MMP detector, given the searching path ℓ, the estimation of the uth modulation symbol is

and the residual vector [TeX:] $$\mathbf{r}_{d_u}^{\ell}$$ is thus updated as

The estimates [TeX:] $$\left\{\hat{s}_{d_u}^{\ell}\right\}_{u=1}^U$$ will be used to obtain the desired [TeX:] $$\left\{s_{d_u}^*\right\}_{u=1}^U$$ by [TeX:] $$s_{d_u}^*=\underset{s_u \in \mathbf{G}}{\arg \min }\left|\hat{s}_{d_u}^{\ell}-s_u\right|$$ until the condition [TeX:] $$\left\|\mathbf{r}_{d_U}^{\ell}\right\| \leqslant \varepsilon \text { or } \ell=\ell_{\max }$$ is satisfied. Therefore, the maximum complexity of MMP is [TeX:] $$\mathcal{O}\left(\ell_{\max } U L+U M\right).$$ Generally, L is much larger than U and M. The complexity of MMP and MSIC can be approximately [TeX:] $$\mathcal{O}\left(\ell_{\max } U L\right)$$ for large L.

From Algorithm 2, it can be known that the complexity of MMSE-SIC relies on the (21) and (23), which are respectively [TeX:] $$\mathcal{O}\left(L_e\right) \text { and } \mathcal{O}(M)$$ for each modulation symbol of jth user. Therefore, the complexity of MMSE-SIC is [TeX:] $$\mathcal{O}\left(N\left(L_e+M\right)\right)$$ when D = 1.

III. IMPLEMENTATION ISSUE

In this section, we discuss the implementation issues including the conditions to support signal recovery with a nonorthogonal codebook in the GMUSST systems, and the selection of the optimal coding parameters for a large transmitted bit stream.

A. The Codebook with Non-orthogonality

When considering the proposed GMUSST in the context of compressed sensing (CS), as illustrated in Fig. 1 and described by (2), it becomes evident that the GMUSST can be regarded as a CS model when the transmitted sparse vector s exhibits sparsity. The principle of CS can be used to guard us on how to better build a GMUSST system, and the codebook [TeX:] $$\mathbf{C}$$ does not require strict orthogonality. In other words, the principles of CS provide direction for the enhanced construction of the GMUSST, where the codebook [TeX:] $$\mathbf{C}$$ is not bound by the requirement of strict orthogonality. Based on the principle of CS, the receiver is possible to correctly recovery the desired sparse vector as long as K, L and U = NJ satisfy [TeX:] $$K=\mathcal{O}\left(U \log _2 L\right)$$ [22]. Furthermore, with the following formula

where (a) follows

Equation (1) can be rewritten as

From (29), [TeX:] $$U \log _2 L$$ satisfies

By plugging (30) into [TeX:] $$K=c\left(U \log _2 L\right),$$ the lower bound of K that guarantees a high probability of sparse recovery will be

where c is a positive integer. Generally, c = 2 is set. Since [TeX:] $$U \ll L,$$ we can derive from (31) that the value of K generally is not greater than L. This indicates that we can use a non-orthogonal codebook in GMUSST to transmit more information with high reliability. Fig. 2 shows the BLER performance of uplink GMUSST systems with non-orthogonal codebooks, in which μ is the maximum correlation coefficient among spreading sequences in the codebook [TeX:] $$\mathbf{C}$$ and other parameters satisfy (31). The larger μ, the greater the nonorthogonality of the codebook. Therefore, we can see from Fig. 2 that when other parameters are the same, a large μ leads to a decreased BLER performance, but its BLER performance is still appreciable, as evidenced by cases such as K = 84, μ = 0.38 and K = 84, μ = 0.43.

B. The Selection of Optimal Coding Parameters

From (1) and (31), it can be easily obtained that with [TeX:] $$b_0$$ increasing, K and L will be increased. When the increase of [TeX:] $$b_0$$ reaches a certain degree, the K and L will be too large for the system to withstand. For instance, when N = 2 and J = 2, we have [TeX:] $$b_0=10,$$ L = 92, K = 52 and [TeX:] $$b_0=14,$$ L = 364, K = 68. As we know, excessive K and L will lead to a large codebook, which complicates the task of designing a large codebook that adheres to the restricted isometry property (RIP) of CS. As a result, the BLER performance of GMUSST will deteriorate. Additionally, the decoder need to search more spreading sequences from a large codebook to find the desired results. That is to say, the larger the codebook, the higher the decoding complexity. Therefore, we should select appropriate coding parameters to ensure high transmission reliability, while the decoding complexity is also within an acceptable range. Especially when a large number of transmitted bits is given, multiple GMUSST encoding blocks should be adopted for transmission. This will involve the selection of coding parameters to obtain the optimal BLER performance.

To tackle this challenge, we introduce a method aimed at minimizing BLER, tailored for the selection of coding parameters. For the sake of simplicity, we consider the downlink GMUSST system with MSIC detector. Assume that when the MSIC to find the uth spreading sequence index [TeX:] $$\Omega_u,$$ all spreading sequences’ indices [TeX:] $$\left\{\Omega_i\right\}_{i=1}^{u-1}$$ before [TeX:] $$\Omega_u$$ have been found out without error. Therefore, according to (10), the success probability of [TeX:] $$\Omega_u$$ will be

where [TeX:] $$\mathbf{h}=\mathbf{h}_1=\mathbf{h}_2=\cdots=\mathbf{h}_J$$ in the downlink GMUSST systems.

Moreover, we further assume that all modulation symbols [TeX:] $$\left\{s_i\right\}_{i=1}^{u-1}$$ before [TeX:] $$s_u$$ are correctly detected. That is to say

In this way, the lower bound of MSIC can be expressed as Theorem 1 following the approach of [26].

Theorem 1: For a large Eb/N0, i.e., [TeX:] $$P / \sigma^2,$$ the lower bound of BLER of MSIC will be

where [TeX:] $$d_{\min }^s=\left[d_{\min }+\frac{U}{2}(1-\mu)\right]^2$$ with the minimum Euclidean distance [TeX:] $$d_{\min }$$ of the modulation constellation and the maximum mutual coefficient [TeX:] $$\max _{q_1 \neq q_2}\left|\left\langle\frac{\phi_{q_1}}{\left\|\phi_{q_1}\right\|_2}, \frac{\phi_{q_2}}{\left\|\phi_{q_2}\right\|_2}\right\rangle\right|, q_1, q_2 \in[1, U]$$ of codebook.

Proof: The derivation of Theorem 1 is given as follows.

From Section II, we know that the spreading sequences selected from [TeX:] $$\mathbf{C}$$ are identified with (10), and then the modulation symbols of s are recovered with (11). Therefore, the derivation of Theorem 1 can be divided into two aspects, including the error performance of spreading sequences’ column indices estimation and the error performance of modulation symbols estimation.

1) Error performance of spreading sequences’ column indices estimation: According to (33), [TeX:] $$\left|\left\langle\phi_{\Omega_u}, \mathbf{r}_u\right\rangle\right| \text { and }\left|\left\langle\phi_q, \mathbf{r}_u\right\rangle\right|$$ in (32) will be

and

where [TeX:] $$\beta_{q_1, q_2}=\frac{\boldsymbol{\phi}_{q_1}^T \boldsymbol{\phi}_{q_2}}{\left\|\boldsymbol{\phi}_{q_1}\right\|_2\left\|\boldsymbol{\phi}_{q_2}\right\|_2}, q_1, q_2 \in[1, L], q_1 \neq q_2$$ denote the correlation coefficient (i.e, inner product) of arbitrary two vectors in [TeX:] $$\phi, z_u=\frac{\phi_{\Omega_u}^T}{\left\|\phi_{\Omega_u}\right\|_2} \mathbf{z} \text { and } z_q=\frac{\phi_q^T}{\left\|\phi_q\right\|_2} \mathbf{z}.$$ Both of them obey the same Gaussian distribution [TeX:] $$\mathcal{C N}\left(0, \sigma^2\right)$$ [26].With (35) and (36) in hand, the probability [TeX:] $$\operatorname{Pr}\left(\left|\left\langle\phi_{\Omega_u}, \mathbf{r}_u\right\rangle\right|\gt \left|\left\langle\phi_q, \mathbf{r}_u\right\rangle\right|\right)$$ can be given as

by using the two inequalities

and

Furthermore, assume that the power of each modulation symbol is equality, i.e., [TeX:] $$\left|s_u\right|=\sqrt{P / U}, \forall u \in[1, U],$$ where P denotes the total power, we then have the following inequality

where [TeX:] $$d_{\min }=\min _{i \neq u}\left|s_i-s_u\right|$$ denotes the minimum distance between any two modulation symbols. By utilizing this inequality, the first term in (37) can be further rewritten as

where [TeX:] $$Q(\cdot)$$ is the Q-function, [TeX:] $$z_{i q}=z_i+z_q \sim \mathcal{C N}\left(0,2 \sigma^2\right).$$ Similarly, the second term in (37) will be

For a Rayleigh fading channel model, each element of channel gain [TeX:] $$\mathbf{h}$$ can be regarded as a complex Gaussian distribution [27], and the sum of square of K random variables with i.i.d. standard Gaussian form a Chi-squared distribution, the PDF of [TeX:] $$\|\mathbf{h}\|_2^2$$ is [TeX:] $$f_{\|\mathbf{h}\|_2^2}(t)=t^{K-1} \exp (-t) /(K-1)!$$ [18]. Given a exponential function [TeX:] $$f(t)=\exp (-a t)$$ with constant a, the conditional expectation is [TeX:] $$\mathrm{E}[f(t) \mid \mathbf{h}]=1 /(a+1)^K.$$ With [TeX:] $$Q(t) \leqslant \exp \left(-t^2 / 2\right),$$ by plugging (41) and (42) into (32), the success probability of [TeX:] $$\Omega_u$$ can be finally given as

where [TeX:] $$\mathrm{E}\left[\operatorname{Pr}\left(\left|\phi_{\Omega_i}^T \mathbf{r}_i\right| \geqslant\left|\phi_q^T \mathbf{r}_i\right|\right)\right]$$ satisfies

According to (43), the average success probability [TeX:] $$\operatorname{Pr}\left(\Omega_U\right)$$ of all non-zero positions can be given as

where [TeX:] $$d_{\min }^s=d_{\min }+\frac{N}{2}(1-\mu)$$ by using the half of U (i.e., U/2) to replace U − i in (45), since U − i decreases from U − 1 to zero with i increasing.

2) Error performance of modulation symbols estimation: Assuming that all the spreading sequences’ column indices can be successful find out with 100% probability, for a large Eb/N0, the error probability of one modulation symbol can be expressed as [28]

where [TeX:] $$E_s$$ is the average power of symbol [TeX:] $$s_u,$$ and [TeX:] $$N_sI$$ is the noise power suffered by [TeX:] $$s_u,$$ which has the distribution with [TeX:] $$\mathcal{C N}\left(0, \sigma^2 / K-1\right)$$ [26]. The lower bound of success probability for determining the desired [TeX:] $$s_u$$ will be

With (45) and (47) in hand, the final BLER can be given by [TeX:] $$\operatorname{Pr}_B=1-\operatorname{Pr}\left(\Omega_U\right) \operatorname{Pr}\left(s_u \mid \hat{s}_u\right)^U .$$ This completes the proof of Theorem 1.

For the error performance derivation of MMSE-SIC, we first derive the estimation error of MMSE to estimate [TeX:] $$\hat{\mathbf{x}}_n^j$$ in (20), which can be found in [29]. Afterwards, using (21) and (22), we derive the error performance of spreading sequences and modulation symbols’ estimation, which can be referred to the derivation of MSIC provided in this paper. From this, we can know that the derivation procedure of MMSE-SIC is similar to that of MSIC. Therefore, the complete derivation of the MMSE-SIC is omitted for brevity.

From (34), some insights can be observed.

1) An increase in the parameter K can improve the decoding probability of both spreading sequences’ column indices and modulation symbols, thereby enhancing the success probability of the MSIC detector.

2) Increasing any one of the parameters U = NJ, [TeX:] $$L=J L_e \text { and } M$$ will let the success probability of decoder down, where U and M affect both the probability of column indices detection and modulation symbols detection.

3) The larger the minimum Euclidean distance [TeX:] $$d_{\min}$$ of the modulation constellation, the greater the success probability of decoder.

4) By appropriately increasing the value of parameter L to reduce the value of parameter U, the success probability of the detection algorithm can be improved.

Based on the above insights, we propose a BLER minimized-based coding parameters selection method. Giving the total number [TeX:] $$K_t$$ of sub-carriers, the total number [TeX:] $$b_t$$ of information bits, the minimum number [TeX:] $$K_{\min}$$ of sub-carriers in each GMUSST coding block, modulation mode [TeX:] $$b_n=\log _2(M)$$ and the user number J, the proposed coding parameter selection method can be summarized as Algorithm 3.

From Algorithm 3, we can obtain some approaches to provide the implementation of GMUSST when [TeX:] $$K_t, b_t \text { and } J$$ are given. Afterwards, these approaches will be further plugged into (34) to select which one has the optimal BLER performance. For examples, (1) giving [TeX:] $$K_t=128, b_t=32, J=4,$$ we can obtain [TeX:] $$K=32, N=2, L_e=7, K=16, N=1, L_e=4$$ by setting [TeX:] $$b_n=2 \text { and } K=64, N=3, L_e=17,$$ [TeX:] $$K=32, N=2, L_e=4, K=16, N=1, L_e=2$$ by setting [TeX:] $$b_n=3.$$ In this case, the optimal coding approach is [TeX:] $$K=32, N=2, b_n=2 \text { and } L_e=7 \text { with } V=4$$ GMUSST coding blocks. (2) Similarly, giving [TeX:] $$J=6, K_t=256$$ [TeX:] $$K=128, N=3, L_e=20, K=64, N=2, L_e=7,$$ [TeX:] $$K=32, N=1, L_e=4 \text { and } K=128, N=3, L_e=11,$$ [TeX:] $$K=64, N=2, L_e=4$$ can be respectively obtained by setting [TeX:] $$b_n=2 \text { and } b_n=3.$$ Finally, [TeX:] $$K=128, N=3, b_n=2 \text{ and } L_e=20$$ are selected as the optimal coding parameters when [TeX:] $$J=6, \text{ and } K_t=256.$$ Comparison results are shown in Fig. 6 among different coding approaches selected by Algorithm 3.

IV. SIMULATION RESULTS

In this section, we first set [TeX:] $$K=32, \text{ and } L_e=8$$ to compare the BLER performance of the proposed MSIC, ML and MMP detectors in the uplink GMUSST systems, as well as the BLER performance of MSIC and MMSE-SIC detectors in the downlink GMUSST systems. Afterwards, the designed coding parameter selection algorithm (i.e., Algorithm 3) based on BLER minimization is verified by setting [TeX:] $$b_t=32$$ for each user. Employing the coding parameters selected by Algorithm 3, the BLER performance of GMUSST with HARQ mechanism [24] is studied compared with the current PC-SCMA systems when the number of transmitted bits and the number of time-frequency resources are the same. In the PC-SCMA system, the joint iterative detection and decoding receiver proposed in [10] is utilized. Moreover, we further investigated the performance of MSIC and MMSE-SIC in the downlink HARQ-MUSST systems. In these simulations, orthogonal codebooks generated by Walsh codes are adopted. Because this article does not address the issues of pilot insertion and channel estimation, we assume that the channel parameter [TeX:] $$\mathbf{h}_j=32$$ is known to the receiver, and the Rayleigh channel model is adopted. Interested readers can refer to the studies [30], [31] to explore the channel estimation of GMUSST systems. Additionally, HARQ is a retransmission mechanism that combines cyclic redundancy check (CRC). When CRC of one user’s decoding data fails, the transmitter is notified to retransmit and the receiver then decodes the user’s data again. Therefore, the probability of this user can be improved due to the reduction of inter-user interference during retransmission.

A. BLER Performance without HARQ

In Fig. 3, the BLER performance for ML, MMP and MSIC detectors is compared when J = 2. In order to obtain the desired BLER performance, we set D = 6 for MMP and MSIC detectors. Fig. 3 reveals that ML detector has the best BLER performance, whereas MMP has the worst BLER performance. The BLER performance of MSIC surpasses that of MMP. When the Eb/N0 is high, the BLER performance of MSIC approaches that of the ML detector. Consequently, the proposed MSIC algorithm can achieve improved BLER performance with reduced complexity.

The BLER performance between MSIC and MMP detector for various D is shown in Fig. 4 when J = 2. It can be seen that the BLER performance of both MMP and MSIC will be improved with the increase of D. However, the increase of D does not improve the BLER performance infinitely. For example, MSIC exhibits identical BLER performance at D = 4 and D = 6 when N = 1. Compared to MMP, the proposed MSIC has the better BLER performance when D and N are the same. Specifically, the proposed MSIC can achieve 1 dB gain than MMP detector when [TeX:] $$\text { BLER }=10^{-5}.$$ In addition, we can see that both MMP and MSIC with D = 1 cannot achiever the desired BLER performance. Therefore, D is a key performance factor for MMP and MSIC in the GMUSST systems.

In the J = 3 downlink GMUSST system, the BLER performance of MMSE-SIC detector is simulated compared to MSIC detector, which is shown in Fig. 5. In this figure, we use MMSE-SIC as a multi-user detector. We can see that the BLER performance of MMSE-SIC won’t change much with D increasing. Although MSIC with large D has the better BLER performance than MMSE-SIC, its BLER performance may be fall into error floor given a large N and a large J. In this case, MMSE-SIC will have superior BLER performance than MSIC.

Fig. 6 shows the BLER comparison of different coding parameter. Giving [TeX:] $$J=4, K_t=128 \text { and } J=6, K_t=256,$$ we can obtain some selection results by setting [TeX:] $$b_n=2$$ and [TeX:] $$b_n=3.$$ For [TeX:] $$J=4, \text { and } K_t=128,$$ the optimal coding parameters are [TeX:] $$K=32, N=2, b_n=2$$ and [TeX:] $$L_e=7 \text { with } V=4$$ GMUSST coding blocks. And [TeX:] $$K=128, N=3, b_n=2 \text { and } L_e=20$$ are selected as the optimal coding parameters when [TeX:] $$J=6 \text { and } K_t=256.$$ From Fig. 6, it can be seen that the BLER performance is decreased when N or [TeX:] $$b_n$$ increases. Given the total number of sub-carriers [TeX:] $$K_t$$ and information bits [TeX:] $$b_t,$$ the GMUSST coding block with large K may be not brought an increased BLER performance. When K and N are constants, reducing the value of [TeX:] $$L_e$$ by increasing [TeX:] $$b_n$$ will result in a decreased BLER performance, and to increase [TeX:] $$b_n$$ can let the BLER performance flatten out with Eb/N0 increasing, such as [TeX:] $$K=32, N=2, b_n=3$$ when J = 4 and [TeX:] $$K=64, N=2, b_n=3$$ when J = 6. However, increasing K can mitigate the negative impact on BLER performance caused by increasing N and M. For example, [TeX:] $$K=64, N=2, \text{ and } b_n=3$$ has lower BLER than [TeX:] $$K=128, N=3, \text{ and } b_n=3$$ when J = 6. On the whole, [TeX:] $$K=32, N=2, \text{ and } b_n=3$$ and [TeX:] $$K=128, N=3, \text{ and } b_n=2$$ has the best BLER performance for J = 4 and J = 6, respectively. This is consistent with the final results of Algorithm 3.

B. BLER Performance with HARQ

Fig. 7 depicts the BLER comparison between GMUSST and PC-SCMA systems equipped with a HARQ mechanism. In this simulation, Q denotes the maximum retransmission times. Each user transmits 26 information bits and 6 CRC bits. The corresponding coding rates of PC-SCMA with J = 4 and J = 6 are respectively 0.5 and 0.25. From this figure, we can see that when J = 4, the total 32 bits should be transmitted by employing K = 64 time-frequency blocks. The BLER performance of GMUSST will be lower than that of PC-SCMA without HARQ assistance (Q = 0) in high Eb/N0 regime. But its BLER performance will be obviously improved with the help of HARQ (Q = 1 and Q = 2). When J = 4 and Q = 1, a 4 dB gain can be achieved with the GMUSST system compared to the PC-SCMA system. This advantage of GMUSST will be further enhanced with the increase of Q. It is worth noting that the BLER performance of GMUSST with J = 6 and Q = 1 is better than that of PC-SCMA with J = 6 and Q = 2. As K increases, GMUSST can carry more user with low retransmission times.

In Fig. 8, the effect of the HARQ mechanism on the MMSE-SIC-based multi-user detector is studied compared with MSIC multi-user detector. From this figure, we can see that when J = 4 and Q = 2, MMSE-SIC can achieve BLER performance close to MSIC. When J is large, the BLER performance of MSIC will be worse than that of MMSE-SIC, such as J = 6 and Q = 2. Combining Fig. 5, we can thus conclude that low-complexity MMSE-SIC detector is more suitable for downlink GMUSST systems with large J and N.

V. CONCLUSION

In this paper, a generalized multi-user sparse superimposed transmission system, named GUMMT, was proposed based on SSC. The low-complexity MSIC multi-user detector and the MMSE-SIC single-user detector were respectively provided for uplink and downlink GMUSST systems. The proposed MSIC can obtain near-ML detector’s BLER performance, and superior BLER performance than MMP detector. Compared to the MSIC multi-user detector, the BLER performance of MMSE-SIC-based multi-user detector is inferior when the number J of access users is small. But it will be superior than MSIC detector with a large J. With these detectors, GMUSST systems can achieve higher transmission reliability than the existing PC-SCMA systems given the transmitted bits and time-frequency blocks. The integration of the HARQ mechanism further accentuates the BLER advantage of the GMUSST system. Meanwhile, the BLER performance of MMSE-SIC is close to that of MSIC when J is small.