1. Introduction
Compared with the 4G system, the spectrum efficiency of the 5G system in the future is increased by 5∼15 times [
1]. Driven by the urgent demand, non-orthogonal multiple access technologies have become a research hotspot in recent years. Sparse code multiple access (SCMA) proposed in [
2] is a promising candidate technology which can achieve massive capacity and connectivity [
3].
In general, a constellation with a larger minimum Euclidean distance (MED) can achieve better performance when no collisions occur among users or layers over a tone in [
4]. As a result, the codebook design is the key to the SCMA system, which can improve the system capacity and bit error rate (BER) performance. In [
4], the unitary rotation was applied on the mother constellation (MC) to control dimension dependency and power variation. In addition, this scheme separated the MC from the operator. However, the MED between the superposed symbols (SS-MED) in the superposed constellation (SC) is small. A novel SCMA codebook design scheme to maximize the sum-rate was proposed in [
5] by rotating the symbols on a 1-dimension complex constellation. A 1-dimension searching algorithm to minimize the upper bound of pairwise error probability (PEP) under amplitude and phase rotation setting was proposed in [
6]. Ref. [
7] presented a method to design codebooks for downlink SCMA systems based on constellation rotation. Constellation rotation and coordinate interleaving were also used to a design multi-dimension SCMA (MD-SCMA) codebook in [
8]. Based on [
8], Mheich et al. optimized the SCMA codebook design process by optimizing the initial phase of the golden angle modulation (GAM) constellation in [
9]. In order to satisfy the criteria of PEP, Yu et al. designed SCMA codebooks to achieve large MED by rotating the star-QAM and optimizing the ratio between the phase shift keying (PSK) rings in [
10]. By rotating the quadrature amplitude modulation (QAM) or pulse amplitude modulation (PAM) constellation employed by the colliding user on each resource, Ref. [
11] optimized the SC to minimize the metric obtained from the upper bounded error probability in the additive white Gaussian noise (AWGN) channel.
A 1-dimension searching algorithm to minimize the upper bound of pairwise error probability (PEP) under amplitude and phase rotation setting was proposed in [
6]. Ref. [
7] presented a method to design codebooks for downlink SCMA systems based on constellation rotation. Constellation rotation and coordinate interleaving were also used to a design multi-dimension SCMA (MD-SCMA) codebook in [
8]. Based on [
8], Mheich et al. optimized the SCMA codebook design process by optimizing the initial phase of the golden angle modulation (GAM) constellation in [
9]. In order to satisfy the criteria of PEP, Yu et al. designed SCMA codebooks to achieve large MED by rotating the star-QAM and optimizing the ratio between the phase shift keying (PSK) rings in [
10]. By rotating the quadrature amplitude modulation (QAM) or pulse amplitude modulation (PAM) constellation employed by the colliding user on each resource, [
11] optimized the SC to minimize the metric obtained from the upper bounded error probability in the additive white Gaussian noise (AWGN) channel. Wang et al. maximized of MED between symbols of conflicting users on each resource node and proposed an efficient sub-optimal SCMA codebook design method for large scale codebooks in [
12]. Huang et al. in [
13] proposed an optimization problem to maximize MED of superimposed codewords under power constraints. Zhang et al. in [
14] pointed out that several small constellations can be superimposed to form a large constellation, namely, the SC. If each point in the SC can be distinguished uniquely, the transmitted codewords can be inferred. In [
15], a novel multiplexing and multiple access scheme named as SCMA based on OFDM-IM (SCMA-IM) was proposed. The SCMA-IM can be implemented in uplink system in an efficient way with similar computational complexity as SCMA.
Since SCMA is a self interference system, these codebook design methods do not aim to reduce the interference between users. However, those schemes adopting the phase rotation are inefficient and the distribution of symbols in the SC is non-uniform which makes it difficult to distinguish some adjacent symbols. In this study, we propose a novel SCMA codebooks design method named decomposition of the superposed constellation (DCSC). First, we decompose the superposed QAM constellation with a large MED and small power into several sets of user symbols (USs) which form QAM or PAM constellations. Second, a few of the sets are assigned to different dimensions to balance the power of the MC. Third, the USs on different dimensions are paired to increase the MED between the codewords (CW-MED).
The rest of this paper is arranged as follows.
Section 2 reviews preliminary backgrounds of the SCMA.
Section 3 describes the SCMA codebook design method.
Section 4 implements several simulations to verify the effectiveness of the DCSC.
Section 5 concludes this letter.
In order to make it clear for readers to understand this letter, we stipulate the following definitions in advance. A codebook is a complex matrix and is formed by several codewords. A codeword is a complex column vector and is formed by several symbols or constellation points. Unless otherwise specified, user is equivalent to user node (UN) and resource is equivalent to resource node (RN).
2. Preliminary Backgrounds
Considering the uplink SCMA system for the AWGN channel, as shown in
Figure 1, it consists of
J user nodes (UNs) and
K resource nodes (RNs), e.g.,
K OFDMA tones,
K time slots, ⋯. Each UN is able to access
N RNs (
N <
K), while each RN is shared by
UNs (
<
J). SCMA encoder maps the incoming
bits binary data stream directly to
K-dimension complex codewords, which can be defined as [
2,
3]:
where
M is the modulation order and
is the codebook of
j-th user.
represents the incoming data streams.
is the transmitted codeword.
is the codebook employed by the
j-th user. Each complex codeword
is a sparse vector with
non-zero symbols. The sparsity of the codewords limits the number of UN colliding over the same resource, which, in turn, reduces the complexity of multi-user detection. By removing all of the zero elements from
, we get an
N-dimension constellation
. The mapping from
binary bits to
is defined as [
2,
3]:
where
represents a column vector. Thus, the encoder in (
1) can be rewritten as
, where
represents the binary low density spreading matrix for the
j-th user or layer and maps the
N-dimension constellation points to a
K-dimension SCMA codeword.
The whole structure of the SCMA code can be represented by a factor graph matrix
where
, and
k-RN is used by
j-UN if, and only if,
. Then, the factor graph matrix is equivalent to the factor graph as shown in
Figure 2. For the case where
,
, and
, one of the possible factor graph matrices can be expressed as follows [
2,
3]:
4. Numerical Results and Analysis
In this section, we give the simulations considering the cases where
,
, and
. The factor graph in
Figure 2 and the corresponding factor graph matrix given in (
4) are adopted to generate the codebooks. The number of iterations of MPA is set to 6. The binary switch algorithm (BSA) proposed in [
11] is run 20 times to optimize the label of the codeword. The BER performances of DCSC and S-DCSC are estimated by comparing previous designs, such as MD-SCMA proposed in [
8], GAM-SCMA proposed in [
9], and MUO-SCMA proposed in [
11].
Table 1 shows the comparison of SS-MED and CW-MED of the codebook with power normalization. The CW-MED of DCSC and S-DCSC is smaller than that of MUO and GAM, because MD and GAM codebook design schemes focus on designing an MC for excellent CW-MED. Then, the phase rotation is applied to the mother codebook to get the user’s codebook, which does not change the CW-MED. Based on the optimized mother codebook, Ref. [
11] further pairs the symbols on different dimensions in codebooks, hence, the CW-MED of MUO is the largest. On the contrary, whether the modulation order is 4 or 8, the SS-MED of DCSC and S-DCSC is much larger than that of MUO, GAM, and MD. This is because the DCSC decomposes the QAM constellation in which symbols are distributed on the I-Q plane uniformly. Compared with the SC proposed by other schemes, the symbols in the SC proposed in this study are easier to distinguish due to the larger SS-MED. In the demodulation process, the larger SS-MED will certainly determine the USs with a higher probability.
Since the power of
is much lower than
and
, we deleted
from (
5). Therefore,
and
can be allocated more energy. For
, the SS-MED and CW-MED of S-DCSC are slightly larger than those of DCSC. Compared with SC of DCSC which is a rectangle-QAM for
, the SC of S-DCSC is a square-QAM that can save energy, so the SS-MED and CW-MED of S-DCSC are obviously larger than those of DCSC. Combined with the BER performance shown in
Figure 4, it can be concluded that SS-MED can make more contributions to the promotion of BER performance. In other words, we need to maximize CW-MED on the basis of maximized SS-MED. The sets of user symbols are demonstrated in
Table 2.
Figure 4 shows the comparison of BER performance of different codebook design schemes. The DCSC and S-DCSC SCMA codebooks perform better than previous schemes. This profoundly proves that larger SS-MED can improve BER performance more effectively. In the high SNR region, the user interference is the major factor that deteriorates BER performance; however, larger SS-MED can suppress the interference between users more effectively. Since the SS-MED and CW-MED of DCSC and S-DCSC are close for
in
Table 1, BER performance of S-DCSC in
Figure 4a is similar to that of DCSC. Meanwhile, because the SS-MED and CW-MED of S-DCSC are larger than that of DCSC for
, BER performance of S-DCSC in
Figure 4b is better than that of DCSC.
For a special modulation order in
Figure 5, the SS-MED of the two SCs are the same. Because CW-MED of the optimized codebook is larger than that of the other, its BER performance is better in the high SNR region. Especially in the case where
, the optimized CW-MED codebook achieves a gain of 3.3dB at a BER value of
than the other. This is because the USs in the codeword of the latter are paired randomly so that the CW-MED is small. All these show that the CW-MED can be used as an evaluation metric of BER performance and the metric provided in (
9) is useful to increase the CW-MED.