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Four-dimensional trellis-coded modulation using sphere packing non-cubic constellations
Optik - International Journal for Light and Electron Optics 186 (2019) 84–92
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Original research article
Four-dimensional trellis-coded modulation using sphere packing non-cubic constellations
T
⁎
Ye Zhanga, Guijun Hua, , Xueying Yangb, Yadong Suna a b
College of Communication Engineering, Jilin University, Changchun, 130012, China College of International, Beijing University of Posts and Telecommunications, Beijing, 100876, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Coding gain Four-dimensional modulation Set-partitioning Trellis-coded modulation
In the paper, we introduce the 8-state trellis-coded modulation (TCM) scheme using sphere packing four-dimensional (4D) non-cubic constellations on the basis of a novel set-partitioning (SP) method. Sphere packing four-dimensional formats have a higher asymptotic power efficiency (APE) because the spatial arrangement of constellation points is more dense. The scheme can further alleviate the contradiction between spectral efficiency (SE) and APE in 4D modulation format. The SP process of 4D non-cubic constellations is completed in the condition that the minimum Euclidean distance between the constellation points and the number of nearest neighbors are adequately considered. We compared these 4D TCM formats to their corresponding 4D formats (i.e., dicyclic4_16, b4_32, b4_64, SP-QAM4_128 and 256-D4) in the 800 km standard single-mode fiber (SSMF) transmission system. Through measurements of bit error rate (BER) performance, at the BER = 1 × 10−3, the coding gains are 5.7 dB, 3.8 dB, 3.6 dB, 3.9 dB and 4.1 dB respectively.
1. Introduction In recent years, with the increasing demand for large-capacity long-distance communication, the contradiction between SE and APE in conventional two-dimensional (2D) modulation format becomes increasingly acute. In order to solve this problem effectively, high-dimensional modulation has been investigated and thus gained much attention [1–7]. In particular, the 4D modulation, which using two quadratures in two polarization components of the optical electric field to transmit data, has been attracting attention as it has the advantages of implementation simplicity and compatibility with existing transmission systems [8–15]. Also worth mentioning is the fact that the TCM scheme was first proposed by Ungerboeck. G in 1982 [16]. The TCM scheme can achieve significant coding gain when it is combined with the SP technique. SP technique is the core of TCM scheme. It divides the constellation into several subsets, with increasing the minimum Euclidean distance between the constellation points. On this basis, the convolutional encoding and the subsequent symbol mapping combine to realize the TCM encode scheme. As evident from the above, the performance of the 4D modulation formats could be further improved if it is combined with the TCM scheme. We can optimize the Euclidean distance between 4D constellation points by combining TCM scheme, in order to enable lower BER at the expense of little SE. More attention has been paid to TCM based on 4D-mQAM constellations in recent years, and some results have been achieved. In [17–19], Shota Ishimura and Kazuro Kikuchi applied the 8-state TCM using constellations of 4DmQAM. Significant coding gain can be achieved against the SP-4D-mQAM without sacrificing the SE. In 2015, Emmanuel Le Taillandier de Gabory et al. experimentally evaluated 2, 8, 32-state TCM-QPSK and TCM-16QAM. The results show that TCM offers ⁎
Corresponding author. E-mail address: [email protected] (G. Hu).
https://doi.org/10.1016/j.ijleo.2019.04.012 Received 19 June 2018; Received in revised form 8 March 2019; Accepted 2 April 2019 0030-4026/ © 2019 Published by Elsevier GmbH.
Optik - International Journal for Light and Electron Optics 186 (2019) 84–92
Y. Zhang, et al.
configuration for flexible transponders increasing SE, on 1.1 dB wider ranges [20]. The same year, they experimentally demonstrated the improvement of the transmission distance with 8-state TCM 12QAM, based on a 16QAM constellation, compared to PM-8QAM [21]. In 2017, Saleem Alreesh et al. experimentally investigated 4D TCM based on PDM-16QAM, PDM-32QAM, and PDM-64QAM formats. Flexible SE was realized by using a single fixed encoder and decoder structure [22]. For the foregoing cubic 4D modulation formats, 4D-mQAM, the SP process is all based on 2D-mQAM constellations [16], which have an extremely regular pattern distribution. However, there are lots of non-cubic 4D modulation formats based on sphere packing theory. They have a higher APE than conventional rectangular constellations because the spatial arrangement of constellation points is more dense [9,10], such as dicyclic4_16, b4_32, b4_64, SP-QAM4_128 [23], 256-D4 [24], etc. With the same spectral efficiency, the non-cubic 4D modulation formats has lower BER. Some of their constellations can be given as
Cb4 _ 64 = {ζ ( ± 1 0 0 0), ζ ( ± 1 ± 1 0 0), ζ ( ± 1 ± 1 ± 1 0) }
(1)
ζ ( ± 1 0 0 0), ζ ( ± 1 ± 1 ± 1 0), ζ ( ± 1 ± 2 0 0), ⎫ C256 − D4 = ⎧ ⎨ ⎩ ζ ( ± 1 ± 1 ± 1 ± 2), ζ ( ± 1 ± 2 ± 2 0), ζ ( ± 3 0 0 0) ⎬ ⎭
(2)
where Greek alphabet ζ presents all combinations and permutations of the 4D vectors set. The spatial distribution of these above constellations is far from regular than 4D-mQAM, and they cannot be simply showed in two 2D planes. There is no effective way for SP of non-cubic 4D constellations, with the result that the completion of TCM scheme is limited. In order to realize 4D TCM with sphere packing non-cubic constellations, we propose a novel SP method, which can complete the SP process of the sphere packing non-cubic 4D constellations. Then, the TCM encode can be achieved through convolutional encoding and symbol mapping based on partitioned subsets, so as to achieve significant coding gain and higher performance. 2. The principle and method of the set-partitioning 2.1. The principle of set-partitioning In the AWGN model, the received signal has an isotropic distribution around the transmitted signal in a 4D space [10]. A simple and useful approximation to the upper limit of symbol error rate (SER) can be expressed as
SER ≤
1 M
M
M
∑∑ k=1 j=1 j≠k
dkj ⎞ 1 erfc ⎜⎛ ⎟ 2 2 ⎝ N0 ⎠
(3)
ck − → cj || is the Euclidean distance between the symbols and M denotes the number of constellation points. The where dkj = || → probability of misjudgment from one symbol to a symbol at minimum Euclidean distance d min is higher than making an error to a symbol at d > d min . Besides the distance d min , the total number of nearest neighbors increases the SER proportionally. In this case, the number of nearest neighbors of constellation point χ is denoted by Mdmin χ . Therefore, the average number of nearest neighbors of the constellation can be expressed as ¯ dmin = 1 M M
M
∑ Mdmin i
(4)
i=1
From above analysis, the misjudgment of the 4D symbols mainly occurs between the constellation points with the Euclidean distance d min . In order to reduce the BER caused by the SER, it is necessary to map the bit labelings with the minimum Hamming distance to the constellation points with the minimum Euclidean distance. However, when the number of nearest neighbors of a constellation point is more, especially it is greater than the number of bits of its bit labeling, the average Hamming distance of those bit labelings among those nearest neighbors will be greater, which leads to a higher BER caused by the SER. ¯ dmin is the minimum in these partitioned subsets. And that is, Therefore, we should ensure that d min is the maximum and M ¯ dmin in partitioned subsets”, we complete the SP process. according to the principle of “maximum d min and minimum M For another, any 4D constellation point can be represented by a 4D vector (( a b c d ) ). Similarly, the non-cubic 4D constellations based on sphere packing are usually given by several vector sets (such as Eqs. (1) and (2)). According to the knowledge of permutation and combination, although not as regular as 4D-mQAM, these vector sets also have certain rules. For example, the number of the vectors in each vector set is an integer multiple of eight, and these vectors have universal symmetry properties in terms of the coordinate zero dot and coordinate axes. 2.2. The method of set-partitioning According to the principle of the above SP, taking eight subsets as an example, the specific steps are as follows: 1) All the vector sets contained in the 4D constellation are arranged in layers according to the Euclidean distance with the origin, from near to far. Each layer only stores one vector set. That is, if the distances for different vector sets are the same, place them in the adjacent layers. Define the layer nearer the origin as the lower layer; 2) Divide the lowest layer into eight groups that are consistent with the Euclidean distance space distribution, among which the symmetry vectors about the origin (i.e., ( a1 b1 c1 d1) and (− a1 − b1 − c1 − d1) ) are divided into the same group as a matter of 85
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Y. Zhang, et al.
Fig. 1. The SP process for 256-D4 constellation.
priority; ¯ dmin of divided groups in all possible divided cases and select the case which has maximum d min and 3) Calculate d min and M ¯ dmin in its groups. That is, the SP of this layer is completed; minimum M 4) According to the order from the low to the high layer, the vector set of the next layer is divided into eight groups follow step 2) and then combine with the eight partitioned subsets of the lower layers to get eight divided groups. The partitioned result of this layer is selected follow step 3); 5) Repeat step 4) until partitioning of all layers is completed. In order to visually and concretely explain the method of the SP, the process is illustrated by partitioning 256-D4, as shown in Fig. 1. First of all, the 6 different vector sets contained in the 256-D4 constellation are arranged in layers, which are surrounded by blue dash lines (step 1). After that, these vector sets are partitioned layer by layer, from the lowest to the highest, which are surrounded by red, purple, and green dash lines respectively (step 2, 3, … , 7). It is worth mentioning that it is necessary to integrate with the partitioned result of the lower layers when partitioning the other layers, except the lowest layer. Similarly, Fig. 2 shows the simplified 2D diagram of the SP for non-cubic 4D constellation. Dotted circles represent the different layers and dots on the circle represent the constellation points in this layer. Letter d represents the distance between the layer and the origin, d1 ≤ d2 ≤ d3 ≤ d4 . The dots of the same color are partitioned into the same subset, a total of eight subsets. Take, for example, sphere packing non-cubic 4D modulation formats dicyclic4_16, b4_32, b4_64, SP-QAM4_128 and 256-D4 (corresponding to the SE of 2, 2.5, 3, 3.5 and 4 bits/symb/pol, respectively). The comparison of each modulation format before and after the SP is shown in Fig. 3. Where the hollow marks indicate parameters before the SP and the solid marks indicate parameters after the SP. We can see from ¯ dmin is reduced from 2 to 1, by partitioning dicyclic4_16 the Fig. 3 that the d min is increased from 2 − 2 to 2 + 2 , and the M ¯ dmin is reduced from 3 to 0.5. constellation (red circle marks). For b4_32 (blue star marks), the d min is increased from 1 to 2 , the M
Fig. 2. Simplified 2D diagram of the SP for non-cubic 4D constellation. 86
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¯d Fig. 3. The comparison of d min and M min for each modulation format before and after the SP.
¯ dmin of b4_64, SP-QAM4_128 and 256-D4 are also improved. In the case of dicyclic4_16 and b4_32, the Similarly, the d min and the M specific partitioned results are shown in Tables 1 and 2, respectively. 3. The realization of 4D TCM The block diagram of encoder structure for 8-state TCM is shown in Fig. 4. In the case of the generation of TCM-256-D4, the 4D TCM encoder receives seven information bits (b1, b2, ..., b7). Two bits (b1, b2), which are surrounded by dash lines in Fig. 4, out of the information bits are fed into the convolutional encoder with the constraint length v = 3 and the code rate R = 2/3. Then, three output bits (b0, b1, b2) including one parity bit are used to select one of 23 = 8 subsets (S0 - S7) based on the Table 3. The remaining five bits (b3, b4, b5, b6, b7) are mapped on the selected subset constellation. Up to this point, the convolution encoding and symbol mapping are completed. The mapping rule of all of the modulation formats is based on the erroneous bits minimization (EBM) mapping [25]. The receiving end of the 4D TCM system is mainly composed of two parts. The first part is the subset decoding. Firstly, calculate the Euclidean distance from the received signal to all the reference signal points in every subset, respectively. And then select the closest reference signal point in every subset and record the Euclidean distance value. The second part is the Viterbi decoding. Those selected reference signal points and their Euclidean distance values are used in Viterbi decoding process. Thus the convolutional input bits, output bits and the constellation points of the transmitted signals are determined. 4. Performance analysis of 4D TCM 4.1. Basic simulation system The simulation system of 4D TCM is shown in Fig. 5. At the transmitter, the TCM encoder module is included as presented in Fig. 4. Here, the module mainly generates 8-state 4D TCM with non-cubic constellations. The 4D symbols are split into four parts and modulated to the real and imaginary parts of each of the two polarizations with a dual-polarization I/Q modulator. In the simulation, a continuous wave laser (CWL) is used as the laser source. The system is working in the third communication window and laser power is 10 mW. Optical carrier from the laser is split into x- and y-polarization with a polarization beam splitter (PBS). The pseudo random binary sequence (PRBS) with a length of 216 is used as the data bit sequence. The fiber link consists of a number of identical sections with 80 km spans SSMF, followed by an erbium-doped fiber amplifier (EDFA). We used the following parameter values in the transmission simulations: attenuation is 0.2 dB/km, dispersion coefficient is 16 ps/nm/km, polarization mode dispersion is 0.2 ps/ km1/2 and the EDFA noise figure is set to 5.0 dB. At the receiver, polarization diversity is obtained by splitting the signal with a PBS and mixing the light in the x- and the y-polarization with the output from a local oscillator (LO) laser in two optical 90° hybrids having integrated balanced detectors. After photo-detection, the received signals are sampled by ADCs and sent to the offline DSP Table 1 4D vectors in each subset partitioned from dicyclic4_16. partitioned subset
4D vectors in subset
S0
(1 0 0 0)
S1
(− 1 0 0 0)
( 2 2
S2
(0 1 0 0)
( 2 2 −
2 2 0 0)
S3
(0 − 1 0 0)
(−
2 2 0 0)
(−
2 2 −
2 2
S4
(0 0 0 1)
(0 0 −
S5
(0 0 0 − 1)
(0 0
S6
(0 0 1 0)
(0 0 −
S7
(0 0 − 1 0)
87
(0 0
2 2 0 0)
2 2 0 0)
2 2 −
2 2 2 2 2 2 −
2 2)
2 2) 2 2) 2 2)
Optik - International Journal for Light and Electron Optics 186 (2019) 84–92
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Table 2 4D vectors in each subset partitioned from b4_32. partitioned subset
4D vectors in subset
S0 S1 S2 S3 S4 S5 S6 S7
(0 0 0 − 1) (0 0 1 0) (0 0 0 1) (0 0 − 1 0) (− 1 0 0 0) (1 0 0 0) (0 − 1 0 0) (0 1 0 0)
(− 1 0 − 1 0) (0 − 1 0 − 1) (0 − 1 1 0) (− 1 0 0 1) (0 − 1 0 1) (0 − 1 − 1 0) (− 1 0 0 − 1) (1 0 − 1 0)
(1 0 1 0) (0 1 0 1) (0 1 − 1 0) (1 0 0 − 1) (0 1 0 − 1) (0 1 1 0) (1 0 0 1) (− 1 0 1 0)
(0 0 − 1 1) (0 0 − 1 − 1) (0 0 1 − 1) (0 0 1 1) (1 − 1 0 0) (− 1 1 0 0) (1 1 0 0) (− 1 − 1 0 0)
Fig. 4. Encoder structure for 8-state 4D TCM. Table 3 Assignment of three output bits to the partitioned subset. Output bits (b2 b1 b0)
(000)
(001)
(010)
(011)
(100)
(101)
(110)
(111)
Selected subset
S0
S1
S2
S3
S4
S5
S6
S7
Fig. 5. 4D TCM simulation system.
circuit. In this paper, we follow the same approach as [26] for impairment compensation. 4.2. Simulation results and analyses Finally, we present the measured BER as a function of optical signal noise ratio (OSNR) for the formats under the same baud rate, in a back-to-back (B2B) configuration. The comparisons of the BER in terms of OSNR between 4D TCM modulation and their corresponding 4D modulation formats, respectively, is shown in Fig. 6. At a BER of 10−3, the required OSNRs for dicyclic4_16, b4_32, b4_64, SP-QAM4_128 and 256-D4 are 12.7 dB, 12.6 dB, 14 dB, 15.2 dB and 16.9 dB respectively. At the same of BER = 1 × 10−3, the required OSNRs for TCM-dicyclic4_16, TCM-b4_32, TCM-b4_64, TCM-SP-QAM4_128 and TCM-256-D4 are 7.3 dB, 9 dB, 10.8 dB, 11.7 dB and 13.7 dB respectively. In other words, the required OSNRs for these modulation formats improved 5.4 dB, 3.6 dB, 3.2 dB, 3.5 dB and 3.2 dB respectively, sacrificing 0.5 bits/symb/pol spectral efficiency. As shown in Fig. 7, under the same SE, we compare the BER of different modulation formats. We use the same color line to represent the same SE of the different modulation formats. The lines of the TCM scheme are marked with diamonds. For example, the SE of both dicyclic4_16 and TCM-b4_32 is 2 bits/symb/pol. At a BER of 10−3, the required OSNR of TCM-b4_32 is 3.7 dB less than dicyclic4_16. In order to compare the data, at a BER of 10−3, the required OSNRs for different modulation formats in B2B simulation is shown in Table 4. Fig. 8 shows simulation results of BERs for all modulation formats after transmission over the 800 km SSMF link. We compared Fig. 8(a) and Fig. 6(a), at a BER of 10−3. The transmission penalties are 0.5 dB for dicyclic4_16 and 0.2 dB for TCM-dicyclic4_16, 88
Optik - International Journal for Light and Electron Optics 186 (2019) 84–92
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Fig. 6. BER vs OSNR in B2B simulation at 7 Gbaud.
Fig. 7. BER vs OSNR in B2B simulation with the same SE.
hence coding gain for TCM-dicyclic4_16 against dicyclic4_16 is 5.7 dB at 7 Gbaud. Similarly, through comparison and analysis for Fig. 8 (b) - (e) and Fig.6 (b) - (e), the transmission penalties of b4_32, b4_64, SP-QAM4_128, 256-D4 and TCM-b4_32, TCM-b4_64, TCM-SP-QAM4_128, TCM-256-D4 are 0.5 dB, 0.8 dB, 1 dB, 1.5 dB and 0.3 dB, 0.4 dB, 0.5 dB, 0.6 dB respectively. The coding gains are 3.8 dB, 3.6 dB, 4 dB and 4.1 dB for TCM-b4_32, TCM-b4_64, TCM-SP-QAM4_128 and TCM-256-D4, respectively. 89
Optik - International Journal for Light and Electron Optics 186 (2019) 84–92
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Table 4 At a BER of 10−3, the required OSNRs in B2B simulation. SE [bits/symb/pol]
modulation format
the required OSNRs [dB]
coding gain [dB]
1.5 2
TCM-dicyclic4_16 dicyclic4_16 TCM-b4_32 b4_32 TCM-b4_64 b4_64 TCM-SP-QAM4_128 SP-QAM4_128 TCM-256-D4 256-D4
7.3 12.7 9 12.6 10.8 14 11.7 15.2 13.7 16.9
5.4
2.5 3 3.5 4
Fig. 8. BER vs OSNR after a 800 km transmission distance at 7 Gbaud.
90
3.6 3.2 3.5 3.2
Optik - International Journal for Light and Electron Optics 186 (2019) 84–92
Y. Zhang, et al.
Fig. 9. BER vs OSNR with the same SE after a 800 km transmission distance.
Table 5 At a BER of 10−3, the required OSNRs after a 800 km transmission distance. SE [bits/symb/pol]
modulation format
the required OSNRs [dB]
coding gain [dB]
1.5 2
TCM-dicyclic4_16 dicyclic4_16 TCM-b4_32 b4_32 TCM-b4_64 b4_64 TCM-SP-QAM4_128 SP-QAM4_128 TCM-256-D4 256-D4
7.5 13.2 9.3 13.1 11.2 14.8 12.2 16.2 14.3 18.4
5.7
2.5 3 3.5 4
3.8 3.6 4 4.1
As described above, Fig. 9 shows the BER of different modulation formats with the same SE after a 800 km transmission distance. At a BER of 10−3, the required OSNRs for different modulation formats is shown in Table 5. 5. Conclusions In this paper, we propose the TCM scheme based on non-cubic 4D modulation formats. Above all, we partition these 4D con¯ dmin in partitioned subsets”. This method can effectively increase the stellations under the principle “maximum d min and minimum M Euclidean distance between the constellation points and reduce the average number of nearest neighbors of the constellation. The minimum Euclidean distance between the constellation points can be increased by a factor of 3 + 2 2 , and the average nearest ¯ dmin neighbors are reduced from 2 to 1, by partitioning dicyclic4_16 constellation. For b4_32, the d min is enlarged from 1 to 2 , the M ¯ dmin of b4_64, SP-QAM4_128 and 256-D4 are also improved. To our is drastically reduced from 3 to 0.5. Similarly, the d min and M knowledge for the first time, we propose a SP method for non-cubic 4D constellations. The method is simple and convenient, and only by the calculation of Euclidean distance and numerical comparison. On the basis of SP, these non-cubic 4D constellations can be combined with the TCM scheme. To verify the performance of noncubic 4D TCM, we conduct computer simulations. The results show that in the B2B system, at the BER = 1 × 10−3, the required OSNRs for TCM-dicyclic4_16, TCM-b4_32, TCM-b4_64, TCM-SP-QAM4_128 and TCM-256-D4 were reduced by 5.4 dB, 3.6 dB, 3.2 dB, 3.5 dB and 3.2 dB compared with the corresponding dicyclic4_16, b4_32, b4_64, SP-QAM4_128 and 256-D4, respectively. Likewise, in the 800 km SSMF transmission system, the coding gains are 5.7 dB, 3.8 dB, 3.6 dB, 4 dB and 4.1 dB respectively at the BER = 1 × 10−3. Moreover, under the same SE, we compare the BER of different modulation formats. At a BER of 10−3, the required OSNRs of 4D TCM modulation formats are less than 4D modulation formats. Thus, the TCM scheme based on non-cubic 4D modulation formats can achieve significant coding gain with retaining the advantage of non-cubic 4D modulation. The communication system enable higher sensitivity at the expense of more complex signal generation. Acknowledgements This work was supported in part by the grant from the National Natural Science Foundation of China (NSFC) (No. 61575078), and in part by Jilin Provincial Science and Technology Department (No. 20190302014GX). References [1] D. Millar, T. Koike-Akino, S. Arik, K. Kojima, K. Parsons, T. Yoshida, T. Sugihara, High-dimensional modulation for coherent optical communications systems, Opt. Express 22 (7) (2014) 8798–8812. [2] D.S. Millar, T. Koike-Akino, K. Kojima, K. Parsons, Multidimensional Modulation For Next-Generation Transmission Systems, SPIE, 2017.
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Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Four-dimensional trellis-coded modulation using sphere packing non-cubic constellations
T
⁎
Ye Zhanga, Guijun Hua, , Xueying Yangb, Yadong Suna a b
College of Communication Engineering, Jilin University, Changchun, 130012, China College of International, Beijing University of Posts and Telecommunications, Beijing, 100876, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Coding gain Four-dimensional modulation Set-partitioning Trellis-coded modulation
In the paper, we introduce the 8-state trellis-coded modulation (TCM) scheme using sphere packing four-dimensional (4D) non-cubic constellations on the basis of a novel set-partitioning (SP) method. Sphere packing four-dimensional formats have a higher asymptotic power efficiency (APE) because the spatial arrangement of constellation points is more dense. The scheme can further alleviate the contradiction between spectral efficiency (SE) and APE in 4D modulation format. The SP process of 4D non-cubic constellations is completed in the condition that the minimum Euclidean distance between the constellation points and the number of nearest neighbors are adequately considered. We compared these 4D TCM formats to their corresponding 4D formats (i.e., dicyclic4_16, b4_32, b4_64, SP-QAM4_128 and 256-D4) in the 800 km standard single-mode fiber (SSMF) transmission system. Through measurements of bit error rate (BER) performance, at the BER = 1 × 10−3, the coding gains are 5.7 dB, 3.8 dB, 3.6 dB, 3.9 dB and 4.1 dB respectively.
1. Introduction In recent years, with the increasing demand for large-capacity long-distance communication, the contradiction between SE and APE in conventional two-dimensional (2D) modulation format becomes increasingly acute. In order to solve this problem effectively, high-dimensional modulation has been investigated and thus gained much attention [1–7]. In particular, the 4D modulation, which using two quadratures in two polarization components of the optical electric field to transmit data, has been attracting attention as it has the advantages of implementation simplicity and compatibility with existing transmission systems [8–15]. Also worth mentioning is the fact that the TCM scheme was first proposed by Ungerboeck. G in 1982 [16]. The TCM scheme can achieve significant coding gain when it is combined with the SP technique. SP technique is the core of TCM scheme. It divides the constellation into several subsets, with increasing the minimum Euclidean distance between the constellation points. On this basis, the convolutional encoding and the subsequent symbol mapping combine to realize the TCM encode scheme. As evident from the above, the performance of the 4D modulation formats could be further improved if it is combined with the TCM scheme. We can optimize the Euclidean distance between 4D constellation points by combining TCM scheme, in order to enable lower BER at the expense of little SE. More attention has been paid to TCM based on 4D-mQAM constellations in recent years, and some results have been achieved. In [17–19], Shota Ishimura and Kazuro Kikuchi applied the 8-state TCM using constellations of 4DmQAM. Significant coding gain can be achieved against the SP-4D-mQAM without sacrificing the SE. In 2015, Emmanuel Le Taillandier de Gabory et al. experimentally evaluated 2, 8, 32-state TCM-QPSK and TCM-16QAM. The results show that TCM offers ⁎
Corresponding author. E-mail address: [email protected] (G. Hu).
https://doi.org/10.1016/j.ijleo.2019.04.012 Received 19 June 2018; Received in revised form 8 March 2019; Accepted 2 April 2019 0030-4026/ © 2019 Published by Elsevier GmbH.
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configuration for flexible transponders increasing SE, on 1.1 dB wider ranges [20]. The same year, they experimentally demonstrated the improvement of the transmission distance with 8-state TCM 12QAM, based on a 16QAM constellation, compared to PM-8QAM [21]. In 2017, Saleem Alreesh et al. experimentally investigated 4D TCM based on PDM-16QAM, PDM-32QAM, and PDM-64QAM formats. Flexible SE was realized by using a single fixed encoder and decoder structure [22]. For the foregoing cubic 4D modulation formats, 4D-mQAM, the SP process is all based on 2D-mQAM constellations [16], which have an extremely regular pattern distribution. However, there are lots of non-cubic 4D modulation formats based on sphere packing theory. They have a higher APE than conventional rectangular constellations because the spatial arrangement of constellation points is more dense [9,10], such as dicyclic4_16, b4_32, b4_64, SP-QAM4_128 [23], 256-D4 [24], etc. With the same spectral efficiency, the non-cubic 4D modulation formats has lower BER. Some of their constellations can be given as
Cb4 _ 64 = {ζ ( ± 1 0 0 0), ζ ( ± 1 ± 1 0 0), ζ ( ± 1 ± 1 ± 1 0) }
(1)
ζ ( ± 1 0 0 0), ζ ( ± 1 ± 1 ± 1 0), ζ ( ± 1 ± 2 0 0), ⎫ C256 − D4 = ⎧ ⎨ ⎩ ζ ( ± 1 ± 1 ± 1 ± 2), ζ ( ± 1 ± 2 ± 2 0), ζ ( ± 3 0 0 0) ⎬ ⎭
(2)
where Greek alphabet ζ presents all combinations and permutations of the 4D vectors set. The spatial distribution of these above constellations is far from regular than 4D-mQAM, and they cannot be simply showed in two 2D planes. There is no effective way for SP of non-cubic 4D constellations, with the result that the completion of TCM scheme is limited. In order to realize 4D TCM with sphere packing non-cubic constellations, we propose a novel SP method, which can complete the SP process of the sphere packing non-cubic 4D constellations. Then, the TCM encode can be achieved through convolutional encoding and symbol mapping based on partitioned subsets, so as to achieve significant coding gain and higher performance. 2. The principle and method of the set-partitioning 2.1. The principle of set-partitioning In the AWGN model, the received signal has an isotropic distribution around the transmitted signal in a 4D space [10]. A simple and useful approximation to the upper limit of symbol error rate (SER) can be expressed as
SER ≤
1 M
M
M
∑∑ k=1 j=1 j≠k
dkj ⎞ 1 erfc ⎜⎛ ⎟ 2 2 ⎝ N0 ⎠
(3)
ck − → cj || is the Euclidean distance between the symbols and M denotes the number of constellation points. The where dkj = || → probability of misjudgment from one symbol to a symbol at minimum Euclidean distance d min is higher than making an error to a symbol at d > d min . Besides the distance d min , the total number of nearest neighbors increases the SER proportionally. In this case, the number of nearest neighbors of constellation point χ is denoted by Mdmin χ . Therefore, the average number of nearest neighbors of the constellation can be expressed as ¯ dmin = 1 M M
M
∑ Mdmin i
(4)
i=1
From above analysis, the misjudgment of the 4D symbols mainly occurs between the constellation points with the Euclidean distance d min . In order to reduce the BER caused by the SER, it is necessary to map the bit labelings with the minimum Hamming distance to the constellation points with the minimum Euclidean distance. However, when the number of nearest neighbors of a constellation point is more, especially it is greater than the number of bits of its bit labeling, the average Hamming distance of those bit labelings among those nearest neighbors will be greater, which leads to a higher BER caused by the SER. ¯ dmin is the minimum in these partitioned subsets. And that is, Therefore, we should ensure that d min is the maximum and M ¯ dmin in partitioned subsets”, we complete the SP process. according to the principle of “maximum d min and minimum M For another, any 4D constellation point can be represented by a 4D vector (( a b c d ) ). Similarly, the non-cubic 4D constellations based on sphere packing are usually given by several vector sets (such as Eqs. (1) and (2)). According to the knowledge of permutation and combination, although not as regular as 4D-mQAM, these vector sets also have certain rules. For example, the number of the vectors in each vector set is an integer multiple of eight, and these vectors have universal symmetry properties in terms of the coordinate zero dot and coordinate axes. 2.2. The method of set-partitioning According to the principle of the above SP, taking eight subsets as an example, the specific steps are as follows: 1) All the vector sets contained in the 4D constellation are arranged in layers according to the Euclidean distance with the origin, from near to far. Each layer only stores one vector set. That is, if the distances for different vector sets are the same, place them in the adjacent layers. Define the layer nearer the origin as the lower layer; 2) Divide the lowest layer into eight groups that are consistent with the Euclidean distance space distribution, among which the symmetry vectors about the origin (i.e., ( a1 b1 c1 d1) and (− a1 − b1 − c1 − d1) ) are divided into the same group as a matter of 85
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Fig. 1. The SP process for 256-D4 constellation.
priority; ¯ dmin of divided groups in all possible divided cases and select the case which has maximum d min and 3) Calculate d min and M ¯ dmin in its groups. That is, the SP of this layer is completed; minimum M 4) According to the order from the low to the high layer, the vector set of the next layer is divided into eight groups follow step 2) and then combine with the eight partitioned subsets of the lower layers to get eight divided groups. The partitioned result of this layer is selected follow step 3); 5) Repeat step 4) until partitioning of all layers is completed. In order to visually and concretely explain the method of the SP, the process is illustrated by partitioning 256-D4, as shown in Fig. 1. First of all, the 6 different vector sets contained in the 256-D4 constellation are arranged in layers, which are surrounded by blue dash lines (step 1). After that, these vector sets are partitioned layer by layer, from the lowest to the highest, which are surrounded by red, purple, and green dash lines respectively (step 2, 3, … , 7). It is worth mentioning that it is necessary to integrate with the partitioned result of the lower layers when partitioning the other layers, except the lowest layer. Similarly, Fig. 2 shows the simplified 2D diagram of the SP for non-cubic 4D constellation. Dotted circles represent the different layers and dots on the circle represent the constellation points in this layer. Letter d represents the distance between the layer and the origin, d1 ≤ d2 ≤ d3 ≤ d4 . The dots of the same color are partitioned into the same subset, a total of eight subsets. Take, for example, sphere packing non-cubic 4D modulation formats dicyclic4_16, b4_32, b4_64, SP-QAM4_128 and 256-D4 (corresponding to the SE of 2, 2.5, 3, 3.5 and 4 bits/symb/pol, respectively). The comparison of each modulation format before and after the SP is shown in Fig. 3. Where the hollow marks indicate parameters before the SP and the solid marks indicate parameters after the SP. We can see from ¯ dmin is reduced from 2 to 1, by partitioning dicyclic4_16 the Fig. 3 that the d min is increased from 2 − 2 to 2 + 2 , and the M ¯ dmin is reduced from 3 to 0.5. constellation (red circle marks). For b4_32 (blue star marks), the d min is increased from 1 to 2 , the M
Fig. 2. Simplified 2D diagram of the SP for non-cubic 4D constellation. 86
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¯d Fig. 3. The comparison of d min and M min for each modulation format before and after the SP.
¯ dmin of b4_64, SP-QAM4_128 and 256-D4 are also improved. In the case of dicyclic4_16 and b4_32, the Similarly, the d min and the M specific partitioned results are shown in Tables 1 and 2, respectively. 3. The realization of 4D TCM The block diagram of encoder structure for 8-state TCM is shown in Fig. 4. In the case of the generation of TCM-256-D4, the 4D TCM encoder receives seven information bits (b1, b2, ..., b7). Two bits (b1, b2), which are surrounded by dash lines in Fig. 4, out of the information bits are fed into the convolutional encoder with the constraint length v = 3 and the code rate R = 2/3. Then, three output bits (b0, b1, b2) including one parity bit are used to select one of 23 = 8 subsets (S0 - S7) based on the Table 3. The remaining five bits (b3, b4, b5, b6, b7) are mapped on the selected subset constellation. Up to this point, the convolution encoding and symbol mapping are completed. The mapping rule of all of the modulation formats is based on the erroneous bits minimization (EBM) mapping [25]. The receiving end of the 4D TCM system is mainly composed of two parts. The first part is the subset decoding. Firstly, calculate the Euclidean distance from the received signal to all the reference signal points in every subset, respectively. And then select the closest reference signal point in every subset and record the Euclidean distance value. The second part is the Viterbi decoding. Those selected reference signal points and their Euclidean distance values are used in Viterbi decoding process. Thus the convolutional input bits, output bits and the constellation points of the transmitted signals are determined. 4. Performance analysis of 4D TCM 4.1. Basic simulation system The simulation system of 4D TCM is shown in Fig. 5. At the transmitter, the TCM encoder module is included as presented in Fig. 4. Here, the module mainly generates 8-state 4D TCM with non-cubic constellations. The 4D symbols are split into four parts and modulated to the real and imaginary parts of each of the two polarizations with a dual-polarization I/Q modulator. In the simulation, a continuous wave laser (CWL) is used as the laser source. The system is working in the third communication window and laser power is 10 mW. Optical carrier from the laser is split into x- and y-polarization with a polarization beam splitter (PBS). The pseudo random binary sequence (PRBS) with a length of 216 is used as the data bit sequence. The fiber link consists of a number of identical sections with 80 km spans SSMF, followed by an erbium-doped fiber amplifier (EDFA). We used the following parameter values in the transmission simulations: attenuation is 0.2 dB/km, dispersion coefficient is 16 ps/nm/km, polarization mode dispersion is 0.2 ps/ km1/2 and the EDFA noise figure is set to 5.0 dB. At the receiver, polarization diversity is obtained by splitting the signal with a PBS and mixing the light in the x- and the y-polarization with the output from a local oscillator (LO) laser in two optical 90° hybrids having integrated balanced detectors. After photo-detection, the received signals are sampled by ADCs and sent to the offline DSP Table 1 4D vectors in each subset partitioned from dicyclic4_16. partitioned subset
4D vectors in subset
S0
(1 0 0 0)
S1
(− 1 0 0 0)
( 2 2
S2
(0 1 0 0)
( 2 2 −
2 2 0 0)
S3
(0 − 1 0 0)
(−
2 2 0 0)
(−
2 2 −
2 2
S4
(0 0 0 1)
(0 0 −
S5
(0 0 0 − 1)
(0 0
S6
(0 0 1 0)
(0 0 −
S7
(0 0 − 1 0)
87
(0 0
2 2 0 0)
2 2 0 0)
2 2 −
2 2 2 2 2 2 −
2 2)
2 2) 2 2) 2 2)
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Table 2 4D vectors in each subset partitioned from b4_32. partitioned subset
4D vectors in subset
S0 S1 S2 S3 S4 S5 S6 S7
(0 0 0 − 1) (0 0 1 0) (0 0 0 1) (0 0 − 1 0) (− 1 0 0 0) (1 0 0 0) (0 − 1 0 0) (0 1 0 0)
(− 1 0 − 1 0) (0 − 1 0 − 1) (0 − 1 1 0) (− 1 0 0 1) (0 − 1 0 1) (0 − 1 − 1 0) (− 1 0 0 − 1) (1 0 − 1 0)
(1 0 1 0) (0 1 0 1) (0 1 − 1 0) (1 0 0 − 1) (0 1 0 − 1) (0 1 1 0) (1 0 0 1) (− 1 0 1 0)
(0 0 − 1 1) (0 0 − 1 − 1) (0 0 1 − 1) (0 0 1 1) (1 − 1 0 0) (− 1 1 0 0) (1 1 0 0) (− 1 − 1 0 0)
Fig. 4. Encoder structure for 8-state 4D TCM. Table 3 Assignment of three output bits to the partitioned subset. Output bits (b2 b1 b0)
(000)
(001)
(010)
(011)
(100)
(101)
(110)
(111)
Selected subset
S0
S1
S2
S3
S4
S5
S6
S7
Fig. 5. 4D TCM simulation system.
circuit. In this paper, we follow the same approach as [26] for impairment compensation. 4.2. Simulation results and analyses Finally, we present the measured BER as a function of optical signal noise ratio (OSNR) for the formats under the same baud rate, in a back-to-back (B2B) configuration. The comparisons of the BER in terms of OSNR between 4D TCM modulation and their corresponding 4D modulation formats, respectively, is shown in Fig. 6. At a BER of 10−3, the required OSNRs for dicyclic4_16, b4_32, b4_64, SP-QAM4_128 and 256-D4 are 12.7 dB, 12.6 dB, 14 dB, 15.2 dB and 16.9 dB respectively. At the same of BER = 1 × 10−3, the required OSNRs for TCM-dicyclic4_16, TCM-b4_32, TCM-b4_64, TCM-SP-QAM4_128 and TCM-256-D4 are 7.3 dB, 9 dB, 10.8 dB, 11.7 dB and 13.7 dB respectively. In other words, the required OSNRs for these modulation formats improved 5.4 dB, 3.6 dB, 3.2 dB, 3.5 dB and 3.2 dB respectively, sacrificing 0.5 bits/symb/pol spectral efficiency. As shown in Fig. 7, under the same SE, we compare the BER of different modulation formats. We use the same color line to represent the same SE of the different modulation formats. The lines of the TCM scheme are marked with diamonds. For example, the SE of both dicyclic4_16 and TCM-b4_32 is 2 bits/symb/pol. At a BER of 10−3, the required OSNR of TCM-b4_32 is 3.7 dB less than dicyclic4_16. In order to compare the data, at a BER of 10−3, the required OSNRs for different modulation formats in B2B simulation is shown in Table 4. Fig. 8 shows simulation results of BERs for all modulation formats after transmission over the 800 km SSMF link. We compared Fig. 8(a) and Fig. 6(a), at a BER of 10−3. The transmission penalties are 0.5 dB for dicyclic4_16 and 0.2 dB for TCM-dicyclic4_16, 88
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Fig. 6. BER vs OSNR in B2B simulation at 7 Gbaud.
Fig. 7. BER vs OSNR in B2B simulation with the same SE.
hence coding gain for TCM-dicyclic4_16 against dicyclic4_16 is 5.7 dB at 7 Gbaud. Similarly, through comparison and analysis for Fig. 8 (b) - (e) and Fig.6 (b) - (e), the transmission penalties of b4_32, b4_64, SP-QAM4_128, 256-D4 and TCM-b4_32, TCM-b4_64, TCM-SP-QAM4_128, TCM-256-D4 are 0.5 dB, 0.8 dB, 1 dB, 1.5 dB and 0.3 dB, 0.4 dB, 0.5 dB, 0.6 dB respectively. The coding gains are 3.8 dB, 3.6 dB, 4 dB and 4.1 dB for TCM-b4_32, TCM-b4_64, TCM-SP-QAM4_128 and TCM-256-D4, respectively. 89
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Table 4 At a BER of 10−3, the required OSNRs in B2B simulation. SE [bits/symb/pol]
modulation format
the required OSNRs [dB]
coding gain [dB]
1.5 2
TCM-dicyclic4_16 dicyclic4_16 TCM-b4_32 b4_32 TCM-b4_64 b4_64 TCM-SP-QAM4_128 SP-QAM4_128 TCM-256-D4 256-D4
7.3 12.7 9 12.6 10.8 14 11.7 15.2 13.7 16.9
5.4
2.5 3 3.5 4
Fig. 8. BER vs OSNR after a 800 km transmission distance at 7 Gbaud.
90
3.6 3.2 3.5 3.2
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Fig. 9. BER vs OSNR with the same SE after a 800 km transmission distance.
Table 5 At a BER of 10−3, the required OSNRs after a 800 km transmission distance. SE [bits/symb/pol]
modulation format
the required OSNRs [dB]
coding gain [dB]
1.5 2
TCM-dicyclic4_16 dicyclic4_16 TCM-b4_32 b4_32 TCM-b4_64 b4_64 TCM-SP-QAM4_128 SP-QAM4_128 TCM-256-D4 256-D4
7.5 13.2 9.3 13.1 11.2 14.8 12.2 16.2 14.3 18.4
5.7
2.5 3 3.5 4
3.8 3.6 4 4.1
As described above, Fig. 9 shows the BER of different modulation formats with the same SE after a 800 km transmission distance. At a BER of 10−3, the required OSNRs for different modulation formats is shown in Table 5. 5. Conclusions In this paper, we propose the TCM scheme based on non-cubic 4D modulation formats. Above all, we partition these 4D con¯ dmin in partitioned subsets”. This method can effectively increase the stellations under the principle “maximum d min and minimum M Euclidean distance between the constellation points and reduce the average number of nearest neighbors of the constellation. The minimum Euclidean distance between the constellation points can be increased by a factor of 3 + 2 2 , and the average nearest ¯ dmin neighbors are reduced from 2 to 1, by partitioning dicyclic4_16 constellation. For b4_32, the d min is enlarged from 1 to 2 , the M ¯ dmin of b4_64, SP-QAM4_128 and 256-D4 are also improved. To our is drastically reduced from 3 to 0.5. Similarly, the d min and M knowledge for the first time, we propose a SP method for non-cubic 4D constellations. The method is simple and convenient, and only by the calculation of Euclidean distance and numerical comparison. On the basis of SP, these non-cubic 4D constellations can be combined with the TCM scheme. To verify the performance of noncubic 4D TCM, we conduct computer simulations. The results show that in the B2B system, at the BER = 1 × 10−3, the required OSNRs for TCM-dicyclic4_16, TCM-b4_32, TCM-b4_64, TCM-SP-QAM4_128 and TCM-256-D4 were reduced by 5.4 dB, 3.6 dB, 3.2 dB, 3.5 dB and 3.2 dB compared with the corresponding dicyclic4_16, b4_32, b4_64, SP-QAM4_128 and 256-D4, respectively. Likewise, in the 800 km SSMF transmission system, the coding gains are 5.7 dB, 3.8 dB, 3.6 dB, 4 dB and 4.1 dB respectively at the BER = 1 × 10−3. Moreover, under the same SE, we compare the BER of different modulation formats. At a BER of 10−3, the required OSNRs of 4D TCM modulation formats are less than 4D modulation formats. Thus, the TCM scheme based on non-cubic 4D modulation formats can achieve significant coding gain with retaining the advantage of non-cubic 4D modulation. The communication system enable higher sensitivity at the expense of more complex signal generation. Acknowledgements This work was supported in part by the grant from the National Natural Science Foundation of China (NSFC) (No. 61575078), and in part by Jilin Provincial Science and Technology Department (No. 20190302014GX). References [1] D. Millar, T. Koike-Akino, S. Arik, K. Kojima, K. Parsons, T. Yoshida, T. Sugihara, High-dimensional modulation for coherent optical communications systems, Opt. Express 22 (7) (2014) 8798–8812. [2] D.S. Millar, T. Koike-Akino, K. Kojima, K. Parsons, Multidimensional Modulation For Next-Generation Transmission Systems, SPIE, 2017.
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