*4.2. Generalized Mutual Information and BER Results*

The Generalized mutual information (GMI) can be used to express the achievable information rate for bit-metric decoding. In this section, the generalized mutual information and bit error rate performance of the probability shaping signal using parallel distributed matcher are analyzed using the PAS scheme. The advantages of the proposed structure are demonstrated when compared with the PS-QAM signal using CCDM.

The GMI for bit-interleaved coded modulation is represented by (8) (Equation (13) of [3] and Equation (7) of [28]). In Monte Carlo simulations of N samples, GMI can be expressed as (9) (Equation (8) of [28]). Considering the rate loss of DM, the achievable information rate is shown in (10) (Equation (15) of [24]).

$$GMI(X;Y) = H(X) - \sum\_{j=1}^{m} H(\mathcal{B}\_j|Y) \tag{8}$$

$$\begin{split} \text{GMI} &\approx \frac{1}{N} \sum\_{k=1}^{N} \left[ -\log\_{2} P\_{X}(\mathbf{x}\_{k}) \right] - \frac{1}{N} \sum\_{k=1}^{N} \sum\_{i=1}^{m} \left[ \log\_{2} (1 + e^{(-1)^{\frac{b\_{i}}{b\_{k}} \Lambda\_{k,i}})} \right. \\ \Lambda\_{k,i} &= \log \frac{\sum\_{\mathbf{x} \in \mathcal{X}\_{1}^{i}} p\_{Y|X}(y\_{k}|\mathbf{x}) P\_{X}(\mathbf{x})}{\sum\_{\mathbf{x} \in \mathcal{X}\_{0}^{j}} p\_{Y|X}(y\_{k}|\mathbf{x}) P\_{X}(\mathbf{x})} = \log \frac{p\_{Y|B\_{i}}(y\_{k}|\mathbf{1})}{p\_{Y|B\_{i}}(y\_{k}|\mathbf{0})} + \log \frac{P\_{B\_{i}}(\mathbf{1})}{P\_{B\_{i}}(\mathbf{0})} \end{split} \tag{9}$$

$$\rm{GMI}\_{\rm{DM}} = \rm{GMI} - \rm{R}\_{\rm{loss}} \tag{10}$$

We consider a dual-polarization PS-64QAM modulation system with P(1, 3, 5, 7) = (0.4, 0.3, 0.2, 0.1), the baud rate per polarization is 28 GBaud. The simulation block diagram and parameters in VPI are shown in Figure 4. In Figure 5, GMI in bits per symbol is shown over OSNR in dB for PS-64QAM. Under the same OSNR, the greater n (n = 30, 50, 100) is, the smaller the rate loss is, and the larger the generalized mutual information is for the PS-64QAM signal utilizing the parallel distribution matcher. The OSNR required by the parallel distribution matcher is less than that using CCDM when n is 100 and the GMI is the same. When GMI is 4 bit/symbol, compared with the uniformly distributed 64QAM signal, the OSNR of the PS-64QAM signal using CCDM is reduced by 0.35 dB, and the OSNR of the PS-64QAM signal using the parallel distribution matcher is reduced by 0.47 dB, which is a 0.12 dB improvement compared with CCDM. When the block length of CCDM increases from n = 100 to n = 50,000, the OSNR only reduces by 0.33 dB; so, the parallel structure has a relatively large improvement of 0.12 dB.

**Figure 4.** The GMI simulation block diagram and parameters in VPI.

**Figure 5.** GMI in bit/symbol over OSNR in dB for bit-metric decoding and PS-64QAM. The inset zooms into the region around GMI = 4 bit/symbol, where the parallel distribution matcher of length n = 100 is 0.47 dB more power-efficient than uniform 64QAM, and is 0.12 dB more power-efficient than PS-64QAM using CCDM.

Figure 6 demonstrates the maximum and minimum GMI values for PS-64QAM signals using this parallel distribution matcher at different block lengths when the OSNR is 15, 16, and 17 dB. When OSNR = 15 dB, GMI = 3.655 bit/symbol; OSNR = 16 dB, GMI = 3.948 bit/symbol; and OSNR = 17 dB, GMI = 4.23 bit/symbol. The needed block length n of the parallel distribution matcher is only 60, although the required block length of the CCDM is 100; so, the block length can be saved by 40%. The matcher can attain the same performance as CCDM while using fewer blocks.

**Figure 6.** GMI in bit/symbol over block length n when OSNR = 15, 16, and 17 dB. When the GMI is the same, the parallel distribution matcher required block length n is only 60, while the required block length of CCDM is 100; so, the block length can be saved by 40%.

In terms of system bit error rate, the probability distribution of the final PS-QAM signal is the same whether utilizing the parallel distribution matcher or CCDM; the only difference is the number of PS-64QAM symbols generated. If n = 50 and the input bit sequence length is 58,128, the parallel structure can generate 33,600 PS-64QAM symbols, while the CCDM can generate 34,600 PS-64QAM symbols. When n = 100 and the input bit sequence length is 62,300, the parallel structure can generate 35,000 PS-64QAM symbols and the CCDM can generate 35,600 PS-64QAM symbols. Consequently, the bit error rate performance of the two is the same in the back-to-back situation.

When the transmission distance is long in the optical fiber transmission link, the amplifier spontaneous emission accumulates continuously and the number of symbols can have a slight impact on the bit error rate. Simulation block diagram and parameters of standard single-mode optical fiber transmission are shown in Figure 7. Figure 8 shows the BER performance under different transmission distances; the fiber is a standard singlemode fiber (SSMF) with *<sup>α</sup>* = 0.2 dB/km, *<sup>γ</sup>* <sup>=</sup> 1.3 (W · km)−<sup>1</sup> , and D = 17 ps/nm/km; each span of length 100 km is followed by an Erbium-doped fiber amplifier with a noise figure of 3.8 dB. Laser phase noise and polarization mode dispersion are not included in the simulation as perfect compensation is assumed. The transmission distance of PS-64QAM signals using the paired optimized parallel distribution matcher is marginally greater than that of PS-64QAM signals using CCDM at the forward error correction threshold of −2.42 for n = 50 and n = 100.

**Figure 7.** Simulation block diagram and parameters of standard single-mode optical fiber transmission.

**Figure 8.** The BER performance over transmission distance in the standard single-mode fiber transmission link. The transmission distance of PS-64QAM signals using the parallel distribution matcher is slightly larger than that of PS-64QAM signals using CCDM.

### **5. Conclusions and Discussion**

As a key part of probability shaping technology, the distribution matcher has significant importance and application value for research. At present, the widely used constant composition distribution matcher is a progressive optimization scheme. It is only when the length of the output symbols is infinity long that the rate loss of CCDM is zero. In addition, it adopts arithmetic coding, which is a highly serial coding method. It represents the input and output sequences, respectively, by dividing intervals between intervals. When the number of input and output sequences is large, the intervals to be divided become greater and the boundary between intervals becomes blurred, which is difficult to distinguish and easy to make mistakes when mapping. Therefore, it is necessary to propose a distribution matcher structure with good performance under short block length.

In this paper, we have proposed a novel parallel architecture distribution matcher, which is an improvement over conventional CCDM at short output block lengths. The output sequence has various compositions compared with the constant composition of the CCDM because of the parallel structure and pairwise optimization of the two CCDMs. As a result, the proposed structure has a lower rate loss when the output block lengths (n is less than 100) are the same and the output block lengths can be lowered by up to 30% with the same rate loss. In the simulation of an optical communication system of PS-64QAM signal, compared with a CCDM signal with the same generalized mutual information, the PS-64QAM signal employing this structure requires a lower OSNR, the block length can be reduced by 40%, and the transmission distance is increased. In detail, When GMI is 4 bit/symbol, compared with the uniformly distributed 64QAM signal, the OSNR of the PS-64QAM signal using CCDM is reduced by 0.35 dB, and the OSNR of the PS-64QAM signal using the parallel distribution matcher is reduced by 0.47 dB, which is a 0.12 dB improvement. The improvement in transmission distance is small, and it is expected to further improve the transmission distance in combination with digital signal processing technology, which will be our future research direction.

Other parallel distribution matcher schemes, such as parallel-amplitude distribution matcher [24], realize the function of symbol-level DM through multiple-bit-level DM. Therefore, M-1 bit-level DM is required to map sequences with M output amplitude values. For high-order QAM format, the number of bit-level DM required increases with the increase in modulation order, and the overall structure of the system is complex. In addition, the multiset-partition distribution [20], which also adopts the idea of pairwise optimization, needs to build a Huffman tree structure to index different complementary pairs and uses serial processing for the input data. The coding process is complex and

error-prone. In contrast, the scheme proposed in this paper is a two-stage parallel structure. No matter how high the order of the QAM signal is, only two CCDMs are required. At the same time, the input data are processed in the two CCDMs after serial–parallel conversion. The receiver performs a corresponding parallel–serial conversion. There is no complex coding process; so, the structure is simple.

**Author Contributions:** Conceptualization, Y.Z. (Yao Zhang); methodology, Y.Z. (Yao Zhang) and H.W.; software and validation, Y.Z. (Yao Zhang); writing—original draft preparation, Y.Z. (Yao Zhang); supervision and project administration, H.W., Y.J. and Y.Z. (Yu Zhang); funding acquisition, H.W., Y.J. and Y.Z. (Yu Zhang). All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Key Research and Development Program of China (Grant No. 2019YFB1803601) and the National Natural Science Foundation of China (Grant No. 62021005).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank the editors and anonymous reviewers for giving valuable suggestions that have helped to improve the quality of the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


**Shulin Tang 1,2,3, Chang Chang <sup>1</sup> , Peiji Zhou <sup>1</sup> and Yi Zou 1,4,\***


**Abstract:** Plasmonic sensors have attracted intensive attention due to their high sensitivity. However, due to intrinsic metallic loss, plasmonic sensors usually have a large full width at half maximum (FWHM) that limits the wavelength resolution. In this paper, we numerically investigate and propose a dielectric grating-assisted plasmonic device, leveraging the bound states in the continuum (BIC) effect to suppress the FWHM of the resonance. We initiate quasi-SP-BIC modes at 1559 nm and 1905 nm wavelengths by slightly tilting the incident angle at 2◦ to break the symmetry, featuring a narrow linewidth of 1.8 nm and 0.18 nm at these two wavelengths, respectively. Refractive index sensing has also been investigated, showing high sensitivity of 938 nm/RIU and figure of merit (FOM) of 521/RIU at 1559 nm and even higher sensitivity of 1264 nm/RIU and FOM of 7022/RIU at 1905 nm.

**Keywords:** bound states in the continuum; plasmonic; sensor
