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An efficient codebook optimization algorithm is proposed to maximize mutual information in sparse code multiple access (SCMA). At first, SCMA signal model is given according to superposition modulation structure, in which the channel matrix is column-extended. The superposition model can well describe the relationship between the codebook matrix and received signal. Based on the above model, an iterative codebook optimization algorithm is proposed to maximize mutual information between discrete input and continuous output. This algorithm can efficiently adapt to multiuser channels with arbitrary channel coefficients. The simulation results show that the proposed algorithm has good performance in both AWGN and non-AWGN channels. In addition, message passing algorithm (MPA) works well with the codebook optimized according to the proposed algorithm.

Nonorthogonal multiple access (NOMA) [

Code-domain NOMA is considered as an important candidate technique [

Recently, sparse code multiple access (SCMA) [

In this paper, SCMA signal model is given according to superposition modulation structure, where the channel matrix is column-extended. The proposed model can well describe the relationship between the codebook of each user and received signal. Therefore, the codebook optimization algorithm is derived according to maximizing mutual information between the discrete input and continuous output. Our analysis shows that the proposed algorithm can efficiently adapt to multiuser channels with random channel coefficients.

This paper is organized as follows. In Section

In the following parts, lower and upper boldface letters denote the vector and matrix, respectively. For the matrix

In this section, SCMA signal model is given according to superposition modulation structure. The analysis shows that the channel matrix is column-extended in the proposed structure. In addition, the relationship between the codebook and received signal is given. The analysis in this section lays foundation for the codebook optimization. For clearness, the typical SCMA signal model is detailed in the first subsection.

A typical SCMA factor graph is given in Figure

Factor graph with

In the factor graph shown in Figure

The matrix

Based on the channel matrix in (

In the next subsection, the superposition modulation SCMA signal model is carefully analyzed.

It can be seen from Figure

In this paper, the superposition modulation structure is introduced. In [

By extending the above model to

In order to adapt to the transmit signal expression in (

For example, in the first column of

Based on the above analysis, the column-extended channel matrix

Based on

According to block diagonal property of

Based on the above analysis, the received signal

In this section, the codebook optimization to maximize mutual information is carefully analyzed. We assume that the number of users and the channel responses are known by the transmitter. In practical wireless communication systems, this assumption is possible for the downlink transmission with channel state information feedback but not possible for the uplink.

At first, the concrete expression of mutual information between the bit vector

Similar to that in [

When the bit vector

Under the equal probability input assumption, the mutual information in (

From the above analysis, it can be seen that mutual information is the function of codebook matrices. In the following subsection, the gradient of mutual information with respect to codebook matrix of each user is analyzed and the KKT conditions are introduced to maximize mutual information.

To optimize mutual information, the gradient of mutual information with respect to

According to the results in [

In SCMA, we assume that the codebook matrix of each user satisfies individual power constraint. This requires the gradient with respect to each user’s codebook matrix

In order to maximize mutual information between

With the power constraint of each user, the KKT conditions corresponding to problem (

According to the result in [

Depending on the KKT conditions, the line search method shown in [

In addition, it can be seen that when calculating the gradient with respect to

In Section

In the first subsection, the line search applied in the iterative codebook optimization algorithm is described. Afterwards, the steps of the proposed algorithm are elaborated. Because the mutual information and mean squared error do not have closed-form expressions, the optimization is implemented based on their Monte Carlo simulation results.

Based on the line search method in [

There are two nested loops in the proposed algorithm. The outer-loop index denotes the iteration number and the inner loop index denotes the user index from 1 to

In the

In addition,

Based on the gradient expression in (

In addition, the codebook matrix of each user should satisfy the power constraint. Assuming that the maximum transmit power of user

Afterwards, the

If the above constraint is not satisfied, the calculations in (

In the next subsection, the detailed steps of the proposed iterative codebook optimization algorithm are given.

From the analysis in Section

According to the above analysis, concrete steps of the proposed algorithm are given in Algorithm

Input: Randomly select codebook matrix

(

(

(

(a) Perform monte-carlo simulations to calculate

(b) Calculate the gradient

Do

(c) Update

(d) Calculate

(e) Perform monte-carlo simulations to calculate

(f) Update step size

While

(g) Generate the updated codebook matrix

(

(h) Generate

(

It should be noted that the performance of backtracking line search method depends on the initial values of the codebook matrices. Therefore, in simulations, the iterative optimization shown in Algorithm

In order to evaluate upper bound of the proposed algorithm, Gaussian channel capacity with the same channel coefficient matrix should be calculated. According to [

In this section, the simulation results are given. With the factor graph in Figure

In Figure

Mutual information performance in non-AWGN channel. The SCMA structure is given in Figure

According to the analysis in Section

In Figure

Furthermore, in Figure

Convergence performance of the proposed iterative optimization algorithm in non-AWGN channel. The SNR is set equal to 0 dB, 2 dB, and 4 dB, respectively.

In the following, the optimized codebook with mutual information equal to 6 bits is applied. The concrete codebook expressions are given in Appendix

Uncoded BER performance of maximum likelihood detection algorithm (ML) and message passing algorithm (MPA) in non-AWGN channel.

In Figure

Coded BER performance with the optimized codebook matrix in non-AWGN channel. Two channel coding schemes are involved.

In addition, the performance of outer-loop iteration between channel decoder and message passing algorithm (MPA) with scheme 1 is given in Figure

Coded BER performance with outer iteration between channel decoding and message passing algorithm (MPA) in non-AWGN channel. The SCMA structure is given in Figure

In order to improve the credibility, we further give the simulation results averaging over 1000 Rayleigh fading channels in Figure

Mutual information performance averaging over 1000 Rayleigh fading channels. The SCMA structure is given in Figure

In this subsection, simulation results in AWGN channel are given. Figure

Mutual information performance in AWGN channel. The SCMA structure is given in Figure

Furthermore, the uncoded bit error rate (uncoded BER) of the optimized codebook in AWGN channel is given in Figure

Uncoded BER performance of message passing algorithm (MPA) in AWGN channel. The SCMA structure is given in Figure

The above simulations’ results are all based on the factor graph in Figure

Factor graph with

The proposed column-extended channel model can well describe the codebook optimization problem with

Mutual information performance with

In this paper, an efficient SCMA codebook optimization algorithm is proposed according to maximizing mutual information between the discrete input and continuous output. Firstly, SCMA signal model is given based on the superposition modulation structure, which can well represent the relationship between the codebook matrix and received signal. Based on the superposition model, the iterative codebook optimization algorithm is proposed, where the line search method is applied to find locally optimal codebooks. It is shown that the superposition model can be applied in multiuser channel with random channel coefficients. In AWGN channel, the proposed optimization codebook can approach Gaussian capacity upper bound in low and medium SNR regime. In non-AWGN channel, the performance loss compared with upper bound is not very large. In addition, with the optimized codebook, message passing algorithm (MPA) at the receiver exhibits good performance.

Based on the result in [

Furthermore, expression (

For the first part, we have

For the second part, we have

It can be seen that the third part and the fourth part have the same result as (

With equal probability input assumption, the expression of

The above analysis shows that it is difficult to derive the closed-form expression of

The channel responses applied in non-AWGN scenario are given by

The optimized codebook matrices from

In AWGN channel, the optimized codebook matrices for factor graph in Figure

In addition, Huawei codebook proposed in [

Based on the factor graph in Figure

With

In addition, the multiuser access model can be further denoted by

According to the above expression, the proposed iterative codebook optimization algorithm can be implemented.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (61601047, 61671080, and 61771066).