Improved polar decoding for optical PAM transmission via non-identical Gaussian distribution based LLR estimation

Honghang Zhou; Yan Li; Yan Li; Tao Dong; Jifang Qiu; Xiaobin Hong; Hongxiang Guo; Yong Zuo; Jie Yin; Yuwei Su; Jian Wu; Jian Wu

doi:10.1364/OE.403117

1. Introduction

In recent years, driven by the increasing demand of virtual reality (VR), artificial intelligence (AI) and the Internet of things (IoT), short-reach optical interconnection has attracted wide-spread interests from industry and academia for metro access, data-center communications, and fifth-generation wireless systems (5G) front haul. Considering the sensitivity to cost and power consumption, the intensity modulation with direct detection (IM/DD) has been adopted as the mainstream solution instead of coherent detection [1]. Among many advanced modulation formats such as carrier-less amplitude and phase modulation (CAP) and discrete multi-tone modulation (DMT), multi-level pulse amplitude modulation (PAM-m) has been widely studied due to low cost and simple configuration [2]. PAM-4 has been ratified by IEEE 802.3bs for 400Gbps Ethernet transmission and PAM-8 is believed as a promising candidate for next-generation 800-Gbps or 1-Tbps Ethernet link [3]. However, there exists two main challenges. First, the low-pass filtering effects induced by the limited bandwidth of the PAM transmitter and receiver will cause severely inter-symbol interference (ISI), which can be eliminated by the linear equalizer [4]. Second, low-cost devices such as radio frequency amplifiers, modulators and photodiodes (PD) produce nonlinearity, also leading to degradation in system performance and reduction of power budgets. Until now, the complexity of nonlinear equalizer is still high to efficiently mitigate nonlinear distortions [5].

Forward error correction (FEC), as a cost-effective method to provide additional code gain, has been widely employed to improve overall performance and guarantee reliable transmission [6]. Recently, the polar code [7] as a novel provable Shannon capacity-achieving code has attracted much attention of academia and industry alike and been chosen as a channel coding scheme in the 5G standardization process of the 3rd Generation Partnership Project (3GPP) [8]. Assisted by the successive cancellation (SC) decoding [7] or successive cancellation list (SCL) decoding [9,10], it can avoid the inherent error floor problem in iterative decoding [11]. Moreover, with low encoding and decoding complexities, the polar code is proved to outperform state-of-the-art LDPC code in short block lengths for latency-constrained systems like short-reach interconnection [12]. Therefore, it could be a potential technique due to its low complexity and high capability for short-reach PAM transmission which requires low cost and low latency FEC scheme. However, current researches of the polar code mainly focus on wireless communications and coherent optical communications, while for short-reach optical interconnection which needs low complexity and performance improvement, polar code is rarely studied. In our previous work [13], for the first time, a polar coded PAM system for short-reach optical link has been conducted, which obtains 30.9 dB and 23.7 dB optical power budgets for PAM-4 and PAM-8 respectively.

With the assumption of additive white Gaussian noise (AWGN) channel, the traditional polar decoder mainly uses the standard symbol points and the consistent noise variance to estimate the input log-likelihood ratio (LLR) of each information bit. However, it has been proved that the code gain of traditional polar decoder in nonlinear domain is much smaller than that in linear domain [14]. For low-cost PAM transmission where the nonlinear distortions exist, the initial assumption of identically distributed signals for AWGN channel mode is not valid anymore, thus degrading the polar decoding performance to a certain extent.

To mitigate the impact of device nonlinearity and improve the decoding performance, for the first time, an improved polar decoder based on LLR estimation of non-identical Gaussian distributions is employed for short-reach optical interconnection in this paper. The transmission of 28-GBaud PAM-4 and PAM-8 signals based on commercial 10-GHz directly modulated laser are both demonstrated to investigate the performance of the novel polar decoder. Experimental results show that, compared to traditional polar decoder, the additional sensitivity gains of 0.7 dB and 1 dB are respectively achieved for PAM-4 and PAM-8 systems after 10-km single-mode fiber (SSMF) transmission. Based on our previous work, the polar coded PAM-8 system obtains a total optical power budget of 24.7 dB after transmission with the improved polar decoder. The rest of this paper is organized as follows. In section 2, we illustrate the principle of the polar coded PAM signal theoretically and introduce the non-identical Gaussian distributions based LLR estimation to resist the performance degradation due to nonlinear distortions. Section 3 describes the experimental setup in detail. The estimated probability density function (PDF) and calculated LLRs are investigated first, and then the bit error rate (BER) performance of the proposed polar decoder is evaluated in section 4. Finally, the conclusion is drawn in section 5.

2. Principle

The schematic diagram of the polar coded PAM system is shown in Fig. 1. At the transmitter, the user’s data is sent to the polar encoder as the information bits. Assuming that the number of information bits for the encoder is K and the polar code is of length N, the number of frozen bits i.e. redundant bits for the encoder is N-K.

Fig. 1. Schematic diagram of polar coded PAM system.

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In the polar encoder, the method of Gaussian approximation [15] is used here for the construction of polar code to obtain the capacity of different bit channels. According to the reliability estimation, the information bits are assigned to bit channels with high reliability while the frozen bits are assigned to bit channels with low reliability to form the bits sequence ${X_N}$. After bits sorting, the bit channels are polarized through a series of modulo-2 addition to get the coded data ${C_N}$, which can be defined as

(1)$${C_N}\textrm{ = }{\textrm{X}_N}{\ast }{\textrm{G}_N}\textrm{ = }{X_N}{\ast }{B_N}{\ast }{\left[ {\begin{array}{cc} 1&0\\ 1&1 \end{array}} \right]^{ {\otimes} n}},$$

where ${G_N}$ is the generator matrix, ${B_N}$ is the bit-reversal permutation and ${\otimes} n$ represents the ${n^{th}}$ Kronecker power of the matrix.

Subsequently, the polar coded bits are Gray mapped to the PAM-4 or PAM-8 constellations every 2-bits or 3-bits as a group. After pulse shaping, the modulated signal is transmitted to the fiber channel. At the receiver, the signal is first extracted by clock recovery and then processed by the equalizer. Finally, the equalized signal is fed into the polar decoder which consists of LLR estimation and SC/SCL decoding. Taking the PAM-8 signal as an example, the LLR in polar decoder can be expressed as

(2)$$LLR({b_i}) = \ln \frac{{P({b_i} = 0|y)}}{{P({b_i} = 1|y)}},i = [1,2,3],$$

where ${b_i}$ donates the ${i^{th}}$ coding bit which mapped to a PAM-8 symbol in the transmitter and y represents the received PAM-8 signals. Note that the probabilities of ${b_i} = 0$ and ${b_i} = 1$ are equal, i.e.

(3)$$P({b_i} = 1) = P({b_i} = 0) = \frac{1}{2}.$$

Thus, according to the characteristic of conditional probability, the posterior probability of $P({b_i} = 0|y)$ can be defined as

(4)$$\begin{array}{l} P({b_i} = 0|y) = \frac{{P(y|{b_i} = 0) \cdot P({b_i} = 0)}}{{P(y)}}\\ = \frac{{P(y|{b_i} = 0) \cdot P({b_i} = 0)}}{{P(y|{b_i} = 0) \cdot P({b_i} = 0) + P(y|{b_i} = 1) \cdot P({b_i} = 1)}}\\ = \frac{{P(y|{b_i} = 0)}}{{P(y|{b_i} = 0) + P(y|{b_i} = 1)}}. \end{array}$$

Also,

(5)$$P({b_i} = 1|y) = \frac{{P(y|{b_i} = 1)}}{{P(y|{b_i} = 0) + P(y|{b_i} = 1)}}.$$

For conventional scheme, it is assumed that the signal is transmitted through the AWGN channel with zero mean and variance ${\sigma ^2}$. The channel transition probability $P(y|{b_i} = k)$ is noted as

(6)$$P(y|{b_i} = k) = \sum\limits_{n = 1}^4 {P(y|s_n^{i,k})} = \sum\limits_{n = 1}^4 {\frac{1}{{\sqrt {2\pi } \sigma }}\exp } \left( { - \frac{{{{(y - s_n^{i,k})}^2}}}{{2{\sigma^2}}}} \right),$$

where $k = 0,1$ and $s_n^{i,k}$ represents the standard symbols whose mapped ${i^{th}}$ bit equals to k. For instance, when ${b_i}\textrm{ = }0$, there are four possible cases of the other two bits and the $P(y|{b_i} = 0)$ is the sum of these four probabilities. According to Eq. (2) and Eq. (6), we can calculate LLR as follow

(7)$$LLR({b_i}) = \ln \frac{{\sum\limits_{n = 1}^4 {\exp ( - \frac{{{{(y - s_n^{i,0})}^2}}}{{2{\sigma ^2}}})} }}{{\sum\limits_{n = 1}^4 {\exp ( - \frac{{{{(y - s_n^{i,1})}^2}}}{{2{\sigma ^2}}})} }}.$$

The noise variance can be evaluated by adding training sequences using the method mentioned in [16]. At the beginning of the transmitted data, we send a training sequence T with a length of ${L_t}$ and the noise variance can be estimated as

(8)$${\sigma ^2} = \frac{{\sum\limits_{t = 1}^{{L_t}} {{{({y_t} - {T_t})}^2}} }}{{{L_t}}}.$$

Thus, the traditional estimation is based on identical Gaussian distributions where all amplitude levels of PAM signal are influenced by the same noise variance. After LLR estimation, the SC decoding utilizes the estimated LLR as initialized input on the right side of the butterfly decoding structure for recursive operation [7]. The upper nodes and lower nodes of the butterfly structure are then respectively calculated to obtain the LLR value of each bit and the decision is made according to the output LLR at last. The SCL decoding [9,10] with a list size of L is also employed in this paper to avoid the error propagation problem of SC decoding.

However, for the implementation of PAM transmission, the nonlinear modulation curve of DML results in amplitude nonlinearity and relative intensity noise also causes amplitude-dependent noise in different amplitudes. Thus, the received PAM signals affected by the nonlinearity-induced impairments are not in accordance with the conventional estimated distribution. As shown in the schematic diagram of PAM-8 system in Fig. 2, the PDF with identical Gaussian distributions performed as the black broken line depicts is mismatched to the real occurrence of signals represented by blue bars. Thus, the LLR estimated by (7) deviates from the real LLR and the conventional polar decoder employing identical Gaussian distributions based LLR estimation is not desirable for low-cost PAM transmission.

Fig. 2. Schematic diagram of conventional LLR estimation with identical Gaussian distributions and proposed LLR estimation with non-identical Gaussian distributions in polar decoder for PAM-8 system.

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To achieve accurate LLR estimation where nonlinearity exists, the non-identical Gaussian distributions based LLR estimation is proposed to be employed in the polar decoder. As illustrated by the red line in Fig. 2, all amplitude levels are supposed as non-identical Gaussian distributions with different means and standard deviations to replace the identical distributions in conventional estimation. The PDF of improved estimation is more consistent with the distribution of real occurrence and the correspond LLR estimation of the first bit in PAM-8 symbol can be performed as

(9)$$LLR({b_1}) = \ln \frac{{\sum\limits_{j = 0}^3 {\exp ( - \frac{{{{(y - {u_j})}^2}}}{{2{\sigma _j}^2}})} }}{{\sum\limits_{j = 4}^7 {\exp ( - \frac{{{{(y - {u_j})}^2}}}{{2{\sigma _j}^2}})} }},$$

where ${u_j}$ represents the different amplitude levels and $\sigma _j^2$ represents different noise variances for eight symbols. Compared to traditional distribution, the non-identical Gaussian distributions based estimation can also be accomplished by the training method at the cost of multiplied training length. To avoid the large amount of data redundancy, K-means clustering [17] is employed to blindly estimate the parameters for dynamic resistance to nonlinear impairments in short-reach interconnection here, although the computational complexity increases to $O(knt)$ relative to the training method with $O(n)$. Note that n is the symbol length for estimation, k represents the number of clustering centers and t is the number of iterations. First, the standard symbols are selected as initial centroids and the received PAM signal is classified by the K-means algorithm. After convergence, the clustering centers are regarded as estimated amplitude levels and the within-cluster sums of point-to-center distances for every cluster are calculated as estimated amplitude-dependent noise variances. Note that the LLR values of the second bit and the third bit can be obtained respectively using the same calculation method. Once the LLR is estimated, the SC and SCL decoding methods will be adopted to recover the original bit sequence. Since the proposed method for initialized LLR input takes the nonlinearity into consideration, it is expected to improve decoding precision compared with conventional polar decoder.

3. Experimental results

To investigate the performance of the improved polar decoder, the experiment is carried out as illustrated in Fig. 3. The 2¹³ pseudo random bit sequence (PRBS) is firstly generated as the original data and then sent to the polar encoder. The code length N is set to 1024 and the code rate is set to 1/2 to reduce the complexity and date latency. After encoding, the bits are mapped to 28-Gbaud PAM signals and up-sampled for loading into the arbitrary wave generator (AWG, Keysight 8195) with 65-GSa/s sampling rate to circularly transmit the coded data. Subsequently, the output electrical signals after radio frequency amplifier is modulated by a commercial 10-GHz DML and launched into the SSMF. The central wavelength is 1549.39-nm and the transmitted optical signal power is fixed at 7-dBm. In the receiver, a variable optical attenuator (VOA) is utilized to vary the received optical power (ROP). Since no inline trans-impedance amplifier (TIA) is cascaded behind the 50-GHz photodetector (PD, Finisar XPDV2120R), the combination of erbium-doped fiber amplifier (EDFA) and an optical passband filter (OPBF) is inserted before PD to amplify the signal and remove out-of-band noise. Finally, the detected signals are sampled by a real-time digital sampling oscilloscope (DSO, LeCory LabMaster) with 80GSa/s sampling rate and restored for digital signal processing (DSP) offline in MATLAB.

Fig. 3. Experimental setup. (AWG: arbitrary waveform generator; DML: directly modulated generator; SSMF: standard single-mode fiber; VOA: variable optical attenuator; EDFA: erbium-doped fiber amplifier; OPBF: optical passband filter; PD: photodetector; TIA: trans-impedance amplifier; DSO: digital sampling oscilloscope).

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In the offline DSP of receiver, the signals are firstly re-sampled to 56-GSa/s to achieve two samples per symbol. Then the timing offset and jitter are eliminated by the clock recovery (CR) with a Gardener phase detector [18]. The feed-forward equalizer (FFE) based on the least-mean square (LMS) algorithm is employed to mitigate the inter-symbol interference (ISI), which is mainly induced by limited bandwidth. After that, the equalized signals are fed into the polar decoder for LLR estimation and SC/SCL decoding. Both the conventional polar decoder (CPD) and the improved polar decoder (IPD) are utilized for comparison. Finally, the recovered bit sequence with a length of about 500k is circularly compared with origin bits for BER calculation.

As shown in Fig. 4, the estimated PDF for CPD and IPD are first investigated after back-to-back transmission. It can be observed from Fig. 4(a) that the input signal of polar decoder has different mean values and noise variances for four amplitude levels of PAM-4. However, the amplitude levels of CPD are equally spaced and the estimated noise variance mismatches the input signal, which is plotted as dashed black line. When using the IPD instead, the estimated PDF is highly consistent with the distribution of real occurrence due to the consideration of nonlinearity. The same phenomenon is found in the case of PAM-8 from Fig. 4(b), which means that the IPD with non-identical Gaussian distributions can make more precision estimation for real occurrence.

Fig. 4. The probability density functions (PDF) of conventional polar decoder (CPD) and improved polar decoder (IPD) after back-to-back transmission for (a) PAM-4 at ROP= -23 dBm and (b) PAM-8 at ROP=-10 dBm.

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The eye diagrams of LLRs for CPD and IPD are presented in Fig. 5. After estimation, the LLR values of the first bit and the second bit are respectively distributed as two and four means due to the gray mapping of PAM-4. The parallel bits of PAM signal are converted to one bit stream after SC/SCL decoding and the LLRs of information bits are also depicted. Note that the zero position of LLR is depicted as a broken line and the magnitude value represents the reliability of a given bit being one or being zero [19]. Compared to the case of CPD, the magnitude values in IPD are larger both after LLR estimation and after SC/SCL decoding, which means that the IPD can offer more certainty to determine the decoded bits. This phenomenon also corresponds to the matched distribution of IPD in Fig. 4 where more accurate LLR values can be provided.

Fig. 5. The eye diagrams of LLRs for CPD and IPD.

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Finally, the BER performance comparison between the CPD and IPD is illustrated in Fig. 6. Note that an outer Bose–Chaudhuri–Hocquenghem (BCH) code having a threshold at 1 × 10⁻³ to realize the final BER of 1 × 10⁻¹⁵ is assumed here [20]. From the figure we can observe that the system performance of SCL decoding becomes much better than the SC decoding with the increasing decoding list size L. Considering the tradeoff between computational complexity and system performance, the best choice of list size is 4 due to the similar performance between L=4 and L=8. For PAM-4 system in Figs. 6(a) and 6(b), compared to CPD, the IPD achieves additional sensitivity gains of 0.3 dB and 0.5 dB @BER=1 × 10⁻³ in SC decoding after back-to-back (BTB) and 10-km transmission, respectively. When optimal SCL decoding (L=4) is employed, with the help of the improved LLR estimation, the increased optical power budget of about 0.7 dB can be obtained for both BTB and 10-km transmission. For PAM-8 system, it can be observed from Fig. 7(c) that about 0.9 dB additional sensitivity gain is achieved by using the IPD in back-to-back case for both SC and optimal SCL decoding. In Fig. 7(d), after 10-km transmission, the performance decreases significantly and even the BER after SC decoding cannot reach the threshold of 1 × 10⁻³. Thanks to optimal SCL decoding, BER is below threshold when ROP is larger than -16.8 dBm and additional optical power budget of 1 dB is improved by employing IPD. Note that the performance improvement of PAM8 for IPD is large than that of PAM4 signal because PAM8 is more susceptible to device nonlinearity than PAM4. Based on our previous results in [13], by replacing CPD with IPD, the total optical power budget of 24.7 dB is achieved after 10-km SSMF transmission for polar coded PAM-8 system.

Fig. 6. The BER performance versus received optical power (ROP) (a) after back-to-back and (b) 10-km transmission for PAM-4 system; (c) after back-to-back (BTB) and (d) 10-km transmission for PAM-8 system where SC decoding and SCL decoding with different decoding list sizes L is employed.

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4. Conclusion

To eliminate nonlinearity and improve FEC performance, an improved polar decoding method via non-identical Gaussian distributions based LLR estimation is proposed and experimentally demonstrated for PAM-4 and PAM-8 system in this paper. It is validated that, aided by the improved polar decoder, the additional sensitivity gains of 0.7 dB and 1 dB are respectively obtained for the 28-Gbaud polar coded PAM-4 and PAM-8 system based on 10-GHz DML. We believe that the improved polar decoding method with satisfactory BER performance provides an efficient and practical way for low-cost PAM transmission, which could be a potential FEC technique in future short-reach optical links.

Funding

National Key Research and Development Program of China (2019YFB1803601); National Natural Science Foundation of China (61675034, 61875019); Fundamental Research Funds for the Central Universities; Open Research Fund of State Key Laboratory of Space-Ground Integrated Information Technology (2018_SGIIT_KFJJ_TX_04).

Disclosures

The authors declare no conflicts of interest.

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Improved polar decoding for optical PAM transmission via non-identical Gaussian distribution based LLR estimation

Abstract

1. Introduction

2. Principle

3. Experimental results

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (6)

Equations (9)

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