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Abstract
This paper proposes a three-dimensional power sparse code division non-orthogonal multiple access (3D-PSCD-NOMA) scheme with 3D constellation pair mapping. The proposed sparse code is based on a balanced incomplete block design (BIBD). Its correlation matrix performs the overall signal mapping of multi-user information. Power multiplexing is realized by overlaying multi-level power signals with different path losses through pair mapping. Compared with the conventional 2D standard square 32 Quadrature Amplitude Modulation (QAM), the proposed 3D constellation pair mapping can improve the constellation points’ minimum Euclidean distance (MED) by 17.7%, which is beneficial for the performance of the system. Based on obtaining the optimal power distribution ratio (PDR) for different schemes, a 3D-PSCD-NOMA signal with a rate of 15.22 Gb/s over a 25 km single-mode fiber (SMF) is experimentally performed. The experimental results show that 3D-PSCD-NOMA has a clear superiority. At the same rate, 3D-PSCD-NOMA2 can obtain a sensitivity gain of about 1.6 dB and 1.9 dB over the conventional 2D constellation. Moreover, 3D-PSCD-NOMA reduces the system's peak-to-average power ratio (PAPR) by 1.3 dB. The difference in sensitivity of the system before and after sparse code is about 0.15 dB, and no significant degradation occurred. Due to its advantages in transmission performance, 3D-PSCD-NOMA is a potential solution for future optical access systems.
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1. Introduction
As various traffic-demanding services such as metaverse, 8 K video continue to develop, global traffic is growing exponentially. The vast majority of these traffic will eventually converge into optical fibers for transmission. Optical fiber communication networks face new challenges due to the increasing demand for data transfer. Optical access networks link millions of people and are often called the “capillaries” of communication systems. The access network must be upgraded to accommodate the massive bandwidth demand. Large bandwidth, broad coverage, and cheap cost have established passive optical networks (PON) as an essential technology for next-generation flexible optical access network architectures [1,2]. Orthogonal frequency division multiplexing (OFDM) has been widely studied due to its high spectral efficiency. [3–6]. OFDM can be implemented by the inverse fast Fourier transform (IFFT). This IFFT-based multi-carrier multiplexing can improve the spectral efficiency of the system. Still, when the different sub-carriers are all peaked for superposition, it will cause the OFDM signal to peak as well, resulting in a high peak-to-average power ratio (PAPR). High PAPR will lead to non-linear signal distortion and ultimately result in transmission performance degradation [7]. Many studies have been carried out to address the problem of high PAPR in OFDM, such as clipping [8], signal code [9], and constellation expansion [10]. A 3D OFDM scheme with a 3D signal mapper and 2D inverse discrete Fourier transform (IDFT) modulator was proposed [11]. It is a novel paradigm of OFDM, which can reduce PAPR in the generation mechanism. It has been shown that the PAPR of 3D OFDM is 2 dB lower than that of 2D OFDM [12]. In addition, 3D OFDM extends conventional OFDM into three dimensions, which can improve the utilization of the constellation space and increase the minimum Euclidean distance (MED) of the constellation points under the same transmitting power conditions. The higher the modulation order, the more information is transmitted, but the MED of the constellation point decreases, which results in a deterioration of the BER performance. Therefore, for higher-order modulation formats, 3D OFDM shows significantly better performance than 2D constellations [13].
3D OFDM effectively improves the transmission performance of the system, but it is still an orthogonal multiple access method that satisfies strict orthogonality between different subcarriers. It isn't easy to meet future development needs for the comprehensive coverage and large-scale access required by 5 G. Non-orthogonal multiple access (NOMA) technology has become a hot research topic. As one of the critical technologies for 5 G, NOMA breaks the independence between different users in traditional access methods and achieves higher spectral efficiency and more users by overlaying signals of multiple users on the same resource block [14–16]. There are two main types of NOMA. The first is non-orthogonal access through the superposition of different sparse codebooks. Sparse code multiple access (SCMA) is the classical code-domain NOMA [14]. Another approach is based on the power domain, where non-orthogonal access is achieved by assigning users with different powers and superimposing them on the same spectrum of resources [17]. A power domain NOMA scheme for next-generation PON was reported, which can get a good balance between throughput and fairness and a higher system capacity for a more significant number of users [18]. It effectively demonstrates the feasibility of introducing NOMA into PON. A new multi-level NOMA architecture has been proposed, as shown in Fig. 1. It groups users according to their distance to the optical line terminal (OLT) and assigns different signal powers to different groups to ensure system fairness [19]. Further, a power-code division NOMA scheme has been proposed, in which the power is allocated according to the distance from the OLT, and users with the same power are assigned with SCMA. This method requires a complex message passing algorithm (MPA) for demodulation at the receiver side with high computational complexity [20]. In our previous work, we proposed a multi-power multiplexing scheme based on sparse code, similar to the Ref. [20], of which power allocation is based on distance and sparse code for the same power level [21]. However, in our scheme, the sparse code treats the signals of different users as a whole for coding and constellation mapping. The receiver only needs to demodulate according to the constellation, and it does not need the complex MPA, which significantly reduces the computational complexity. However, this work is still based on a 2D-NOMA implementation. In fact, NOMA can be enhanced to high dimensional by multi-power superposition of 3D OFDM. We have implemented a 3D-NOMA transmission with a 2D-IFFT and a 3D constellation design. Its experimental results show that this 3D NOMA scheme improves the MED of the constellation by 15.48% and achieves a sensitivity gain of 1 dB [22]. The current research on 3D-NOMA is mainly confined to the structure of 16 constellation points, and research on higher-order modulation formats has not yet been carried out.
In this paper, we propose a 3D-PSCD-NOMA scheme with 3D constellation pair mapping. This is a joint power and sparse code access method. The power is allocated according to the distance between the user and the OLT. As shown in Fig. 1, users closer to the OLT are allocated lower power, while users further away from the OLT are allocated higher power. The same power levels are overloaded in a sparse code manner, which treats the constellations with the information of different users as a whole, avoiding the computational complexity introduced by the irreversibility of the traditional SCMA codebook overlay. Moreover, two square-based 3D constellation pair mapping methods are proposed, which can form a 3D spatial structure of 32 constellation points. Compared to the 2D square constellation, the 3D square extension can improve the MED of the constellation points by 17.7% under the same transmit power. 3D-PSCD-NOMA also has a significantly lower PAPR than 2D-PSCD-NOMA. To verify the system performance of the proposed 3D-PSCD-NOMA, a 15.22 Gb/s 3D-PSCD-NOMA transmission over 25 km SMF is successfully carried out in the intensity modulation direct detector (IMDD) system. Experimental results show that the proposed 3D-PSCD-NOMA scheme has very good performance and a wide range of applications in future optical access systems.
2. Principle
Figure 2 shows the flow chart of 3D-PSCD-NOMA, which mainly includes sparse code, 3D constellation mapping, 2D-IFFT, and power allocation. In terms of sparse code, unlike the traditional SCMA, our proposed sparse coding approach uses a balanced incomplete block design (BIBD) [23], where the signals of different users are coded as a whole. Specifically, BIBD is usually expressed as D (ν, b, r, k, λ). ν is the number of the subcarriers. b represents the number of users. r is the number of the users per subcarrier carries and k is the subcarriers occupied by each user and λ represents the overlapping number of users on different subcarriers. It is worth noting that the user mentioned in this paper is the ONU. The most critical part of BIBD is the design of the correlation matrix, which is a binary ν × b matrix. It can be expressed as
Assume that the information for each of the b users is
Then the transmission matrix after sparse coding can be represented as
Equation (6) is the correlation matrix of our method. This matrix has three non-zero elements in each row and two non-zero elements in each column, respectively. It indicates that each resource block carries three user messages, and each user occupies two resource blocks, which allows for an overload of 6/4 = 150%. By labeling the non-zero elements, the Y matrix is obtained, given as:
Each row of the Y matrix has exactly three elements corresponding to the bit information of three different users. These three bits are used to map eight constellation points in three dimensions. By superimposing two different power signals, a 32 QAM constellation is obtained. In our previous work, two NOMA schemes with 32 constellation points were designed as shown in Fig. 4. Figure 4(a) is a normalized 32 QAM signal composed of QPSK and a geometric shaping signal, which we named 2D-PSCD-NOMA1. Figure 4(b) shows an irregular shaping signal with the origin as the base element and a positive hexagonal extension at the periphery named 2D-PSCD-NOMA2. 3D constellation design can fully use the constellation decision space and effectively alleviate the problem of more petite MED in higher-order modulations. Specifically, the three-dimensional constellation points can be expressed as
${S_{x,{N_s}}}$, ${S_{y,{N_s}}}$, ${S_{z,{N_s}}}$ represents the three-dimensional coordinates of constellation points, ${N_s}$ is the index of the subcarriers, and N is the total number of subcarriers. The symbol of 3D OFDM consists of the signals of N subcarriers, which can be expressed as
In addition to 3D constellation mapping, 2D-IFFT is employed to convert 3D-OFDM signals in the frequency domain into the time domain. 2D-IFFT can be realized by Eq. (10).
Among them, ${n_1} \in [{0,2} ],{n_2} \in [{0,{N_{IFFT}} - 1} ]$ represent the columns and rows of the time domain signal after 2D IFFT, respectively. In fact, 2D-IFFT can be implemented with two 1D IFFT. One 3D-OFDM signal can be obtained by adding cyclic prefix (CP).
The 3D-PSCD-NOMA signal can be obtained by the superposition of two sparse code 3D-OFDM channels with different power. It can be expressed as:
In Eq. (11), ${\textrm{s}_1}(\textrm{t} ),{\textrm{s}_2}(\textrm{t} )$ are two different channels, indicating different power distributions. P1 is the user with a high-power level far away from the OLT in Fig. 1, and P2 is the user with a low power level near the OLT. Figure 5 shows the two new 3D-PSCD-NOMA schemes. The mapping rules and 3D constellation coordinates of the two schemes for different powers are shown in Table 1. In the 3D constellation, cube extension is the primary way. In Fig. 5(a), the high-power signal is the QPSK constellation distribution in the XOZ plane of 3D space, while the low-power signal is the cuboid structure. The constellation after the superposition of the two signals is shown in Fig. 5(a). It can be found that the whole constellation map presents a structure of two 16 QAM. In order to make full use of low power point resources, 3D-PSCD-NOMA2 is designed, as shown in Fig. 5(b). The high-power signal still adopts QPSK constellation distribution in the XOZ plane of three-dimensional space, and the low-power signal adopts pair mapping. The quadrantal distribution of high-power signals determines the signal mapping rule and constellation distribution. Table 1 shows the mapping rules of low-power signals in case of case 1. The other three cases of P2 can be obtained according to the constellation distribution of the case1, which can be expressed as:
Table 1. Parameters of 3D-PSCD-NOMA
As the 3D-PSCD-NOMA adopts multi-power multiplexing mode at the transmitter, SIC (successive interference cancellation) algorithm is needed to demodulate multistage signals. The flowchart of SIC is shown in Fig. 6. Firstly, channel estimation is performed to obtain the channel response h1, and then the signal is equalized. At this time, R(t) actually contains the information of both the high and low-power signals, but the low-power signals are regarded as noise in the process of the first demodulation. Taking 2D-PSCD-NOMA1 as an example, the constellation of R(t) at the receiver is the distribution of 32 constellation points. In the demodulation process of high-power signal, QPSK corresponding to P1 is adopted to demodulate. After the demodulation of the P1 signal is completed, the signal is re-modulated to obtain the reconstruction of signal R1(t) of P1. The low-power user signal R2(t) can be obtained by subtracting the high-power signal R1(t) from the receiver signal R(t), which removes the interference from the data stream with the high power. It can be found from Fig. 6 that due to the power allocated by R2(t) being smaller than that of R1(t), the amplitude of R2(t) is significantly smaller than that of R1(t). This is also the reason why low-power signals can be regarded as noise in the demodulation process of a high-power signal. After performing channel estimation and equalization on the obtained low-power signal, the R2(t) information can be demodulated according to the matching mapping rules corresponding to the originating end.
3. Experimental setup and results
Figure 7 is the 3D-PSCD-NOMA transmission system based on IMDD. At the OLT side, the signal of 3D-PSCD-NOMA is firstly encoded by offline DSP. Table 2 shows the parameter settings of 2D-PSCD-NOMA and 3D-PSCD-NOMA. Both 2D-PSCD-NOMA and 3D-PSCD-NOMA adopted the mapping method of 32 constellation points. The overload rate of sparse code is 150%. The signal after 3D constellation mapping is assigned to 216 subcarriers. The 512-point IFFT converts the frequency domain signal into the time domain. 240 PSCD-NOMA signals are used, 10 of which are used for channel equalization. The length of CP is set to 1/16 of the PSCD-NOMA symbol. After offline DSP, the 3D-PSCD-NOMA signal is injected into the arbitrary waveform generator (AWG) for digital-to-analog conversion. The 3D signal uses three elements to represent its three-dimensional coordinates, whose transmission rate is reduced to one-third of the 2D signals. According to Table 2, the data rate is entropy × AWG Sampling rate/3 × Number of subcarriers/IFFT points × Number of information symbols/Number of symbol/(1 + CP) = 5 × 24/3 × 216/512 × 230/240 × 16/17 = 15.22Gb/s. In order to ensure the same transmission rate, Deng et al. adopted the method of 2D signal repeat transmission [24]. Due to the strong correlation of signals, this method also had a particular impact on the experimental results. We adjust the sampling rate of 3D-PSCD-NOMA to three times that of 2D-PSCD-NOMA to ensure consistent transmission rates, which is 15.22 Gb/s. The generating electrical signal passes through a radio frequency amplifier (RFA) and is modulated by a Mach-Zehnder modulator (MZM). The laser used in electro-optic modulation is an optical carrier with a wavelength of 1550 nm supplied by a tunable laser. After 25 km of single-mode fiber (SMF) transmission, the signals are distributed to the different ONUs by a power splitter (PS). A variable optical attenuator (VOA) is set in the front of each ONU to adjust signal attenuation. The received optical signals will be collected by a photodetector (PD) and converted into electrical signals. A mixed signal oscilloscope (MSO, TekMSO73304DX) connected to PD will capture the electric signal. Finally, the demodulation of 3D-PSCD-NOMA is performed in offline DSP.
Fig. 7. Experimental setup (AWG: arbitrary waveform generator; RFA: radio frequency amplifier; MZM: Mach-Zehnder modulator; SMF: Single mode fiber; PS: power splitter; VOA: variable optical attenuator; PD: photodetector; MSO: mixed signal oscilloscope).
Table 2. Parameters of 2D-PSCD-NOMA and 3D-PSCD-NOMA
NOMA is multiplexed through signal superposition of different power, so the power allocation ratio is the key parameter that affects the performance of NOMA. Finding optimal PDR is an integral part of NOMA. In this scheme, we adopt the way of traversal to minimize the total BER as the goal to find the optimal PDR. Figure 8 shows the BER curves of 2D-PSCD-NOMA and 3D-PSCD-NOMA under different PDR. As seen from Fig. 8, although the four schemes are different, they all have similar BER trends. The BER of the high-power signal shows a decreasing trend. With the increase of PDR, more power is allocated to high-power signals. Low-power signals have less influence on high-power signals, so the performance of high-power signals is improved. The BER of the low-power signal is low in the middle and high on both sides. When the PDR is small, the difference between the two signals is slight. There is an overlap between the 32 constellation points after the superposition of the two signals, resulting in more erroneous bits in the decision of high-power signals. In the high power remodulation, erroneous bits are carried out according to the result of the high power decision. In other words, the R1(t) signal in Fig. 6 is not the original correct signal. In the demodulation of low-power signals, the receiver signal subtracts the incorrect R1(t), thus increasing the BER of low-power signals. This is the error propagation effect of SIC [25]. When the PDR is large, the power allocated to the high-power signal gradually increases, the error propagation effect is weakened, and the impact on the low-power signal is reduced, so the BER of the low-power signal begins to decrease. However, when the PDR is too large, the power of the low-power signal is meager and gradually becomes noisy, so the BER rises again. The optimal PDR is the critical point of the decrease and increase of the BER of the low-power signal. It can be seen from Fig. 8 that the optimal PDR of 2D-PSCD-NOMA is nine, and that of 3D-PSCD-NOMA is four.
We tested the transmission performance of different schemes with different power signals under the condition of optimal PDR. Figure 9 shows the BER curve of the different schemes when the PDR of 2D-PSCD-NOMA is 9, and the PDR of 3D-PSCD-NOMA is 4. The black line in Fig. 9 represents the BER curve of the ONU far from the OLT in Fig. 1, and the red line represents the BER curve of the ONU close to the OLT. With the increased optical power, the transmission performance of the four schemes is constantly improved. At the same time, due to the non-uniform distribution of power, the transmission performance of high-power signals in the four schemes is significantly better than that of low-power signals, which is a reasonable result. For the high-power signal with a far path in Fig. 1, the user who is closer to the OLT is assigned a lower power to ensure fairness among users with different path losses. At the hard-decision forward error correction (HD-FEC) BER limit of 3.8 × 10−3, 2D-PSCD-NOMA1-P1, 2D-PSCD-NOMA2-P1, 3D-PSCD-NOMA1-P1, and 3D-PSCD-NOMA2-P1 have sensitivity gains of 1.8 dB, 1.4 dB, 1.6 dB, and 0.9 dB, when compared with their corresponding low-power signals. In addition, the error of the 2D-PSCD-NOMA scheme began to appear at −10dBm, while the error bit of 3D-PSCD-NOMA1 and 3D-PSCD-NOMA2 low-power signal began to appear at −11dBm and −12dBm. It can be predicted that 3D-PSCD-NOMA is superior to 2D-PSCD-NOMA.
Fig. 9. BER curves of different power levels under optimized PDR((a) 2D-PSCD-NOMA1, (b) 2D-PSCD-NOMA2, (c)3D-PSCD-NOMA1, (d)3D-PSCD-NOMA2).
Further, the transmission performance of 2D-PSCD-NOMA1, 2D-PSCD-NOMA2, 3D-PSCD-NOMA1, 3D-PSCD-NOMA2 and corresponding OFDM schemes are compared. The BER curves of the schemes under different optical power are shown in Fig. 10. It is worth noting that the BER curve here is the total BER formed by the superposition of two signals.The results show that there is no significant difference in receiver sensitivity between the NOMA scheme and the traditional OFDM transmission scheme. Due to the BER of high power is an order of magnitude lower than that of low power under the same power, the BER of high power signal has little influence on the BER of superimposed signal. The dominant effect is the BER of low power signal. Therefore, the BER curve in Fig. 10 is very close to the curve in Fig. 9. As can be seen from the figure, the 3D-PSCD-NOMA scheme is superior to the 2D-PSCD-NOMA scheme. Error bit occurs at −12dBm for 3D-PSCD-NOMA2, and its performance is the best among the four schemes. Compared to 3D-PSCD-NOMA1, 3D-PSCD-NOMA2 has a sensitivity gain of about 0.75 dB at the BER limit of 3.8 × 10−3, which can be attributed to the fact that 3D-PSCD-NOMA2 places the constellation point closer to the origin and has a lower average transmit power. Compared with 2D-PSCD-NOMA1 and 2D-PSCD-NOMA2, 3D-PSCD-NOMA2 can obtain a sensitivity gain of 1.6 dB and 1.9 dB, which can be attributed to the bigger MED of 3D-PSCD-NOMA. Figure 10 also shows the constellation point distribution when the received optical power is −12 dBm. The constellation point distribution is not particularly clear, but it still has a low BER due to the superiority of three-dimensional space in constellation point decision.
PAPR is an essential index of a multi-carrier system. Figure 11 is the complementary cumulative distribution function (CCDF) of the PAPR of the four transmission schemes. It can be seen that the schemes of 3D-PSCD-NOMA are significantly lower than those of 2D-PSCD-NOMA. The maximum PAPR of the two 3D-PSCD-NOMA schemes is 11.4 dB and 12.2 dB, respectively, while the maximum PAPR of 2D-PSCD-NOMA1 and 2D-PSCD-NOMA2 is 12.9 dB and 13.9 dB respectively. When the CCDF is 10−2, the PAPR of the 3D-PSCD-NOMA1 scheme is at least 1.3 dB lower than that of the 2D-PSCD-NOMA scheme. This indicates that the proposed 3D-PSCD-NOMA scheme can effectively reduce the PAPR of the system.
In order to verify the influence of the proposed sparse code, we compared the system performance graphs before and after the sparse code. 3D-NOMA2 adopts the same constellation design as 3D-PSCD-NOMA2 to carry out multi-power superposition. The only difference is that 3D-NOMA2 does not adopt sparse code. 3D-NOMA2 does not have sparse code. It only needs to perform simple OFDM multi-power superposition. In other words, the encoding process of 3D-NOMA2 mainly includes S/P, 3D constellation mapping, 2D-IFFT, add CP, P/S, power allocation and superposition. Figure 12 shows the system transmission performance before and after sparse code. It can be observed that the two BER curves show an interweaving trend. When the BER is 3.8 × 10−3, the received optical power of 3D-NOMA2 and 3D-PSCD-NOMA2 is −14.7 dBm and −14.55 dBm, respectively. Since no additional noise is introduced by sparse code, their transmission performance is consistent between sparse and non-sparse coded signal. However, due to measurement accuracy in the experimental test, there is 0.15 dB difference was measured.
The same power level adopts sparse code for multi-user access, and different users occupy the same resource block. To verify the fairness of different users, we test the error performance of different users. The BER curves of different users after sparse code are displayed in Fig. 13. It can be observed from Fig. 13 that the BER curves of different users almost overlap. At a BER of 3.8 × 10−3, the difference in sensitivity between the best and worst users is about 0.4 dB, which is an acceptable range. This proves that the proposed sparse code can ensure fairness between different users. This fairness comes from the sparse code process, which regards the information of different users as a whole, and carries out resource block allocation and constellation mapping. No additional noise is introduced between different users, so the influence between each other can be ignored. In addition, the constellations adopted also present a symmetrical distribution structure. The transmitting power among the peripheral constellations is also consistent, so the difference between users can be effectively ensured to be small.
4. Conclusion
This paper presents a higher-order modulation scheme for 3D-PSCD-NOMA based on 3D constellation pair mapping. It uses sparse code and multi-power signal superposition to realize multiple access. The users are divided into groups according to the path loss and allocates different power. Same power level adopts sparse code to achieve 150% overload. The proposed sparse code method takes the signals of multiple users as a whole to map constellation points according to the encoded correlation matrix without introducing additional signal damage. This three-dimensional constellation mapping method can make full use of constellation space, and the MED of the constellation can be improved by 17.7%. A 15.22 Gb/s 3D-PSCD-NOMA transmission over 25 km SMF was demonstrated. Experimental results show that no matter which PSCD-NOMA scheme, the performance of a high-power signal is significantly better than that of a low-power signal, so it is reasonable to allocate power according to the path loss. The transmission performance of 3D-PSCD-NOMA is better than that of 2D-PSCD-NOMA under the optimized PDR condition. Compared with the two 2D PSCD-NOMA schemes, the sensitivity gain of 3D-PSCD-NOMA2 is about 1.6 dB and 1.9 dB. The PAPR of 3D-PSCD-NOMA2 is at least 1.3 dB lower than that of 2D-PSCD-NOMA. The performance difference between users in sparse code is less than 0.4 dB, which can ensure fairness between different users. Based on the promising experimental results, we believe that 3D-PSCD-NOMA has an extensive application prospect in the next generation of optical access systems.
Funding
National Key Research and Development Program of China (2022YFB2903104); National Natural Science Foundation of China (61835005, 61935005, 62205151, 62225503, 62275127, U22B2009, U22B2010); Jiangsu Provincial Key Research and Development Program (BE2022055-2, BE2022079); The Natural Science Foundation of the Jiangsu Higher Education Institutions of China (22KJB510031); The Startup Foundation for Introducing Talent of NUIST.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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