Low-complexity nonlinear equalizer based on absolute operation for C-band IM/DD systems

Yukui Yu; Tianwai Bo; Yi Che; Deaho Kim; Hoon Kim

doi:10.1364/OE.394961

1. Introduction

Intensity-modulation (IM)/direct-detection (DD) system, due to its simplicity and cost-effectiveness, has long been the optical transmission system of choice for short- and intermediate-haul applications [1–4]. Even though the coherent systems are now expanding their territory from long-haul to metro applications, cost-sensitive applications are still dominated by the simplest of all optical transmission systems, IM/DD system. For the implementation of small-footprint low-cost IM/DD systems, directly modulated lasers (DMLs) or electro-absorption modulated lasers (EMLs) are typically employed at the transmitter [5–8].

The transmission distance of high-speed IM/DD systems operating in the C-band is limited by fiber chromatic dispersion. This is because the square-law detection gives rise to nonlinear waveform distortions when the optical signal is directly detected after transmission over dispersive optical fiber. The distortions become even worse when the optical transmitter has non-zero frequency chirp [9–11]. For example, the DML-based optical transmission systems suffer from serious performance degradation caused by large transient and adiabatic frequency chirp of DML [11]. Although these distortions can be compensated by using a dispersion compensation module [12,13], it is not desirable to utilize the optical component on the link due to its relatively high cost and insertion loss.

One of popular approaches to coping with those waveform distortions is to utilize the electrical equalization technique at the receiver [14–21]. Since the waveform distortions in IM/DD systems are inherently nonlinear, the Volterra nonlinear equalizers (VNLEs) can be employed to compensate for the distortions [14–16]. The 2^nd-order VNLE is composed of beating terms formulated by a product of two inputs having a relative delay w (i.e., x_k·x_k-w, where x_k is the discrete-time sample of the equalizer’s input at a time index k). Thus, it can be used to compensate for the 2^nd-order nonlinear distortion components at the receiver. However, the number of beating terms to be included in the equalizer grows rapidly with the channel memory length. Thus, the high implementation complexity of VNLE makes it an impractical option for cost-sensitive IM/DD systems. A straightforward strategy to reduce the equalizer’s complexity is to prune some beating terms which are considered to be less important. For example, the diagonally-pruned (DP) VNLE discards the beating terms having a large relative delay [16,17]. The simplest form of DP-VNLE is the polynomial nonlinear equalizer (PNLE). It includes only self-beating terms (i.e., x_k·x_k), without any cross-beating terms (i.e., x_k·x_k-w, where w ≠ 0). Thus, it could be used for O-band DML-based IM/DD systems where the waveform distortions are mainly caused by the nonlinear modulation dynamics of DML. To lower the complexity of PNLE, we have recently proposed and demonstrated the nonlinear equalizer based on absolute operation [18]. In this work, we replaced the square operation in the PNLE with the absolute one. Since the absolute operation can be realized simply by flipping the sign bit of a negative number, we can reduce the number of multiplications, and thus implementation complexity. The proposed equalizer performs similar to the conventional PNLE, but reduces the number of multiplications by ∼30% in O-band DML-based IM/DD systems [18].

Unlike the O-band IM/DD systems where fiber chromatic dispersion is close to zero and thus the channel memory length is short, the performance of C-band IM/DD systems is limited by a large accumulated fiber chromatic dispersion. A considerable number of cross-beating terms should be included in DP-VNLE to compensate for the waveform distortions arising from long memory length of fiber channel [10,21]. Thus, DP-VNLE would be still too complicated and power-hungry to be used for cost-sensitive C-band IM/DD systems. Recently, the absolute operation-based nonlinear equalizer was proposed for C-band IM/DD system [22]. In this equalizer, the self-beating terms of x_k·|x_k| and |x_k|·x_k² are added to the conventional VNLE. Thus, it exhibits better signal-to-noise ratios than the VNLE, but at the expense of implementation complexity.

In this paper, we propose and demonstrate a low-complexity diagonally-pruned nonlinear equalizer based on the absolute operation for C-band IM/DD transmission systems. The key idea is to replace the cross-beating terms (i.e., x_k·x_k-w) in the 2^nd-order DP-VNLE with the absolute operation of a sum of two inputs, i.e., |x_k + x_k-w|. Since the multiplication operation has a much higher implementation complexity and consumes much more power than the absolute and addition operations in digital signal processing (DSP), the proposed equalizer lowers the complexity and power consumption considerably in comparison with the DP-VNLE. We evaluate the performance of the proposed equalizer using 56-Gb/s 4-ary pulse amplitude modulation (PAM-4) signals generated from a DML or EML, both operating in the C-band. These two optical transmitters are widely used for short- and intermediate-haul applications. However, their waveform behavior with respect to transmission distance is markedly different since they have quite distinct chirp characteristics. We show that, for both optical transmitters, the proposed equalizer exhibits similar performance to the conventional 2^nd-order DP-VNLE, but reduces the implementation complexity by >20%. We also show that our proposed equalizer based on absolute operation outperforms the DP-VNLE when the implementation complexities of the two nonlinear equalizers are similar.

2. Principle

2.1 Conventional 2^nd-order VNLE and DP-VNLE

The Voterra series theory offers a general way to design a nonlinear equalizer. The 2^nd-order VNLE can be expressed as

(1)$${y_n} = \sum\limits_{k = 0}^{{L_1} - 1} {h_k^{(1 )}{x_{n - k}}} + \sum\limits_{w = 0}^{{L_2} - 1} {\sum\limits_{k = 0}^{{L_2} - 1 - w} {h_{k,w}^{(2 )}{x_{n - k}}{x_{n - k - w}}} }$$

where x_n and y_n are the n^th received and recovered samples, respectively. Also, L_p and h^(p) are the memory length and equalizer coefficient of p^th (p=1, 2) order terms, respectively. The first term on the right-hand side of Eq. (1) represents a linear feedforward equalizer (FFE). The second term denotes a quadratic equalizer, which is composed of self- and cross-beating terms. It is well recognized that these cross-beating terms are resposible for a major implemention complexity of VNLE. The DP-VNLE offers a popular strategy to eliminate some of them. Typically, the cross-beating terms produced by input samples separated far away in time (i.e., a large w) are excluded in DP-VNLE. The rationale behind this pruning is that the coefficient values for cross-beating terms decrease as w increases in most practial systems. If a 2^nd-order DP-VNLE prunes the cross-beating terms with w larger than W-1, it can be written as [16,21],

(2)$${y_n} = \sum\limits_{k = 0}^{{L_1} - 1} {h_k^{(1 )}{x_{n - k}}} + \sum\limits_{w = 0}^{W - 1} {\sum\limits_{k = 0}^{{L_2} - 1 - w} {h_{k,w}^{(2 )}{x_{n - k}}{x_{n - k - w}}} }$$

Since the second term of Eq. (2) can be regarded as a linear combination of quadratic products between input samples, DP-VNLE can be easily implemented by using parallel finite impulse response (FIR) filters, as shown in Fig. 1(a). Thus, the coefficient values can be obtained by using the traditional algorithms, such as the least mean square (LMS) method with training symbols.

Fig. 1. FIR implementation of (a) 2^nd-order DP-VNLE and (b) the proposed equalizer.

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2.2 Proposed equalizer

Here we propose to replace the beating polynomial terms in the 2^nd-order DP-VNLE with the terms formed of the absolute value of a sum of two input samples. The proposed 2^rd-order equalizer can be expressed as

(3)$${y_n} = \sum\limits_{k = 0}^{{L_1} - 1} {h_k^{(1 )}{x_{n - k}}} + \sum\limits_{w = 0}^{W - 1} {\sum\limits_{k = 0}^{{L_2} - 1 - w} {h_{k,w}^{(2 )}|{{x_{n - k}} + {x_{n - k - w}}} |} }$$

where |·| donates the absolute operator. The quadratic term in Eq. (2) is now expressed as a first-order term with the absolute operation. Thus, our proposed equalizer reduces the order of DP-VNLE by one. Using the Taylor series expension, we can write |x|≈|a|+(x²-a)/2-(x²-a)²/8 around x = a, where we drop the terms higher than 4^th order. Thus, |x_k+x_k-w| shown in Eq. (3) contains not only self- and cross-beating terms (i.e., x_k·x_k-w), but also higher-order beating terms such as (x_k·x_k-w)².

Since the sampled signals at the receiver in DD system are real-valued numbers, the absolute operation can be realized by an addition opeartion [23]. Therefore, for the calculation of each nonlinear term, our proposed equalizer requires one multiplication and two addition operations, whereas two multipication operations are needed for DP-VNLE. Please note that multiplication operation is much costlier and consumes more power than addition operation [24]. Thus, the implementation complexity of the proposed equalizer is considerably lower than that of DP-VNLE. We can implement the proposed equalizer using the FIR structure, as shown in Fig. 1(b).

The implementation complexity of equalizer could be quantified by the number of real-valued multiplications. The proposed equalizer needs L₁+W(2L₂−W+1)/2 multiplications, whereas L₁+W(2L₂-W+1) and L₁+ L₂(L₂+1) multiplications are required for the 2^nd-order DP-VNLE and VNLE, respectively. We summarize the implementation complexity of the equalizers in Fig. 2 when L₁=L₂=L. Apparently, the 2^nd-order VNLE has the highest complexity, and the pruned scheme of DP-VNLE can reduce the complexity considerably. Our proposed equalizer further lowers the complexity. For instance, the proposed equalizer needs ∼40% less real-valued multipliers than DP-VNLE.

Fig. 2. Implementation complexity measured in terms of the number of real-valued multipliers. The memory lengths L₁ and L₂ are both set to be L.

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3. Experimental setup

We evaluate the performance of the proposed nonlinear equalizer on a 56-Gb/s PAM-4 link implemented by using DML or EML. Figure 3 shows the experimental setup. We first generate a 56-Gb/s PAM-4 signal using a digital-to-analog converter (DAC) which combines two 28-Gb/s uncorrelated pseudo-random binary sequences (length = 2¹⁵-1) produced from a two-channel pulse pattern generator. The signal is then fed directly to the DML or EML, operating at 1549 and 1538 nm, respectively. The DML is biased at six times its threshold current and emits the output power of 8.1 dBm. Such a high bias current benefits the modulation bandwidth of DML [6]. Figure 4(a) shows the measured E/O response of the DML used in our experiment. The 3-dB bandwidth is measured to be 25.9 GHz. Also, the linewidth enhancement factor and adiabatic chirp parameter are measured to be 2.9 and 13 GHz/mW, respectively. The E/O response of the EML is plotted in Fig. 4(b). The bias voltage of the electro-absorption modulator (EAM) inside the EML is set to be −1.5 V, where the transmission curve of EAM is the most linear. At this bias voltage, the 3-dB bandwidth is measured to 24 GHz, but the EML has a gentle roll-off characteristics in comparison with the DML. The linewidth enhancement factor of the EML is measured to be 0.6. After transmission over standard single-mode fiber (SSMF), the PAM-4 signal is detected by using a PIN-TIA detector (bandwidth = 33 GHz) and then digitized at 80 Gsample/s using a real-time oscilloscope. The captured waveforms are processed offline. The post-detection DSP includes resampling, synchronization, and half-symbol-spaced electrical equalization. The bit-error ratio (BER) measurement is performed by direct error counting for Gray-coded symbols. All the equalizers have an asymmetric structure and we optimize the number of the pre- and post-cursors for a given number of equalizer taps. The LMS based on training sequences is used to determine the tap coefficients. The coefficients remain unchanged after the algorithm converges.

Fig. 3. Experimental setup

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Fig. 4. E/O responses of (a) DML and (b) EML.

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We first optimize the performance of DML-based IM/DD transmission system (hereafter referred to as DML/DD system). Figure 5(a) shows the measured BER as a function of the extinction ratio (ER) of the signal. The received signal power is −4 dBm. As exptected, the results show that the BER peformance improves with the ER of PAM-4 signal in the back-to-back transmission. Due to the absence of dispersion-induced nonlinear distortions, the FFE performs simliar to the 2^nd-order VNLE. However, after 40-km transimssion over SSMF, the BER performance of FFE is degraded when the ER is larger than 0.9 dB. This is because large frequency chirp is produced from the DML, and as a result the system performance is limited by severe dispersion-induced waveform distortions [9,21], which the linear FFE cannot compensate effectively. The 2^nd-order VNLE is effective in compensating such distortions. By using the VNLE, the BER performance is improved by more than an order of magnitude. We observe that, in the case of using VNLE, the BER performance saturates when the ER is larger than ∼1.2 dB. Thus, we set the ER to be 1.2 dB in our experiment using DML.

Fig. 5. Parameter optimization of DML/DD system. (a) BER performance versus ER of the signal, (b) BER performance as a function of the memory length of linear equalizer, L₁. (c) BER versus the memory length of 2^nd-order VNLE, L₂. L₁ is set to be 43. (d) BER performance as a function of pruning parameter W. L₁ and L₂ are set to be 43 and 33, respectively.

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Next, we optimize the memory lengths of linear and quadratic equalizers in the DML/DD system after 40-km long transmission. The length of linear memory, L₁, is first determined by running only the linear part of VNLE [i.e., the first term in Eq. (1)]. The results are depicted in Fig. 5(b). It shows that the BER performance is improved as we increase L₁, but levels off beyond ∼40. Then, we measure the BER performance for 2^nd-order VNLE as a function of memory length L₂ with L₁ fixed to 43, as shown in Fig. 5(c). As expected, the nonlinear equalizer improves the BER performance more than an order of magnitude. It is observed that the BER performance saturates when L₂ is larger than 41. We adopt L₂=33 to strike a balance between complexity and performance. Finally, we optimize the pruning parameter, W, in DP-VNLE. Figure 5(d) shows the BER performance for DP-VNLE as a function of W when L₁ and L₂ are set to be 43 and 33, respectively. The system performance is improved as W increases, but saturates beyond W=9. Thus, we set W to be 9 in our experiment.

The EML-based IM/DD transmission system (hereafter referred to as EML/DD system) also suffers from nonlinear waveform distortions induced by fiber chromatic dispersion. However, the channel memory length of EML/DD system should be shorter than that of DML/DD system due to the relatively small frequency chirp of EAM. Thus, we employ the same memory lengths and pruning parameter as the ones selected for the DML/DD system (i.e., L₁=43, L₂=33, and W=9). The ER of the PAM-4 signal in EML/DD system is measured to be 5.5 dB.

4. Experimental results and discussions

4.1 DML/DD system

We first investigate the performance of proposed equalizer in DML/DD systems. Figure 6 shows the BER performance of DML/DD system for various transmission distances. Also shown in this figure are the BER performances obtained by using linear FFE, PNLE and DP-VNLE. The eye diagrams measured after applying equalizer in 40-km transmission are also shown. For fair comparison, the memory lengths and pruning parameter are set to be identical among the equalizers. In the back-to-back transmission depicted in Fig. 6(a), all the equalizers perform similarly and no BER floor is observed. It clearly shows that the system performance is not limited by nonlinear distortions. After transmission over SSMF, however, the FFE always exhibits the worst performance, and we cannot reach the forward error correction (FEC) threshold of 3.8 × 10⁻³ after 40-km transmission. We observe a severe eye skew in this case. Evidently, such a skew is mainly caused by the nonlinear waveform distortions induced by the interplay between the DML’s chirp and fiber chromatic dispersion. Nonlinear equalizers can mitigate the eye skew effectively, and thus improve the BER performance. For example, the DP-VNLE lowers the BER from 2.3×10⁻² to 1.7×10⁻³ after 40-km long transmission, and shows wide eye opening without eye skew. It is worth mentioning that the PNLE brings a relatively little performance improvement. This confirms that the cross-beating terms mainly govern the performance of the C-band DML/DD system [20]. The results also show that our proposed equalizer (L₂=33) exhibits a slight worse performance than DP-VNLE after transmission. For example, we observe a sensitivity penalty of 0.7 dB for the proposed equalizer with respect to the DP-VNLE after 40-km transmission. However, it should be noted that our proposed equalizer lowers the number of real-valued multipliers required for its implementation. To compare the performance between our proposed equalizer and DP-VNLE under similar implementation complexity, we increase the memory length L₂ from 33 to 62 in our proposed equalizer. Then, both the proposed equalizer and DP-VNLE requires 565 real-valued multiplication operations. Figure 6 shows that our proposed equalizer (L₂=62) outperforms DP-VNLE. For example, the proposed equalizer achieves a receiver sensitivity improvement of ∼0.3 dB with respect to the DP-VNLE after 40-km transmission. Compared to DP-VNLE, our proposed equalizer (L₂=62) exhibits similar eye opening without eye skew. This confirms that the proposed equalizer is highly effective in compensating for the nonlinear distortions in DML/DD.

Fig. 6. Measured BER curves of 56-Gb/s PAM-4 signal in DML/DD system after (a) 0-, (b) 20-, and (c) 40-km transmssions. The eye-diagrams are measured after 40-km transmission. The pruning parameter W is 9.

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As shown in the previous results, the nonlinear memory length, L₂, greatly affects not only the performance of nonlinear equalizer, but also the implementation complexity. Thus, we compare the performance between the proposed equalizer and DP-VNLE when their implementation complexities are similar. Figures 7(a), 7(b), and 7(c) show the BER performance versus the number of multiplications required in the equalizers when W is set to be 3, 7 and 9, respectively. The transmission distance and the received power are 40 km and −4 dBm, respectively. The results show that the proposed equalizer outperforms the DP-VNLE, regardless of W. For example, we achieve the FEC threshold of 3.8 × 10⁻³ by using the proposed equalizer having 210 multiplication operations (W=3), but the DP-VNLE having the same number of multiplication operations gives us a BER of 6 × 10⁻³. The result show that the proposed equalizer reduces the implementation complexity considerably when compared to DP-VNLE. We also notice that our proposed equalizer underperforms the DP-VNLE very slightly when the number of multiplication is sufficiently large. For example, when the multiplication number is 750 and the pruning parameter W is 9, the DP-VNLE achieves a BER of 1.1 × 10⁻³, which is slightly better than the BER achieved by the proposed equalizer, 1.3 × 10⁻³. We summarize the complexity reduction, defined by the ratio between the numbers of multiplications required for the two equalizers to reach the FEC threshold, in Fig. 7(d). The results show that the ratio decreases with W, but levels off at around 11. Nevertheless, the proposed equalizer has 24∼40% lower complexity than DP-VNLE.

Fig. 7. Measured BER curves of DML/DD system versus the number of multiplications required for the equalizers when (a) W=3, (b) 7, and (c) 9. (d) The complexity reduction versus the pruning parameter W. The received power is −4 dBm and transmission distance is 40 km.

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4.2 EML/DD system

Next, we investigate the performance of proposed equalizers in EML/DD system. In this case, we set the FEC threshold to be 2×10⁻⁴ since this system suffers less from the nonlinear waveform distortions than the DML/DD system due to a relatively small frequency chirp of EML [5]. Figure 8 shows the BER performance for various transmission distances, and the eye-diagrams measured after 20-km transmission. In the back-to-back transmission, all the equalizers perform similarly, except that the proposed equalizer having L₂=62 slightly outperforms the others. We believe that this is because residual nonlinear distortions of EML are compensated by the equalizer having a long memory length. The figure shows the BER curves measured after 10- and 20-km transmissions. The linear FFE always shows the worst performance when compared to the other equalizers, which is similar to the case of DML/DD system. Unlike DML/DD system, however, we have no eye skew for EML/DD system in the eye diagram. Instead, we observe uneven eye opening between symbol levels. This should be attributed to the interplay between fiber dispersion and the driving signal-dependent linewidth enhancement factor of EML [7]. Thus, nonlinear nature of these waveform distortions makes it attractive to apply nonlinear equalizers. For example, the PNLE improves the BER performance, but we cannot achieve a BER better than 2×10⁻⁴ after 20-km long transmission. Substantial performance improvement is achieved when we utilize the equalizers containing the cross-beating terms. Even though the EML has low frequency chirp, the fiber chromatic dispersion spreads the data symbol over adjacent symbols, and consequently creates numerous cross-beating terms when directly detected at the receiver. The results show that the proposed equalizer (L₂=33) and DP-VNLE exhibit a similar performance, achieving a BER of 3.7×10⁻⁵ and 2.5×10⁻⁵, respectively, after 20-km transmission at the received optical power of −4 dBm. However, we should note that our proposed equalizer requires less multiplications than the DP-VNLE. When we set both the equalizers to have similar complexity, the proposed equalizer (L₂=62) outperforms the DP-VNLE, and exhibits an equally spaced eye diagram. The results show that the proposed equalizer (L₂=62) achieves the receiver sensitivity 0.7 dB less than that of DP-VNLE.

Fig. 8. Measured BER curves of EML/DD system after (a) 0-, (b) 10-, and (c) 20-km transmssions. The eye-diagrams are measured after 20-km transmission. The pruning parameter W is 9.

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Figures 9(a), 9(b), and 9(c) show the measured BER curves of EML/DD system versus the number of multiplications required for the equalizers when we set W to be 3, 7, and 9, respectively. The transmission distance and the received power are 20 km and −4 dBm, respectively. Similar to the cases of DML/DD system, the proposed equalizer outperforms the DP-VNLE when their complexities are similar. In other words, when the proposed equalizer is employed for EML/DD system, we can achieve performance similar to the DP-VNLE with reduced implementation complexity. In Fig. 9(d), we plot the complexity reduction ratio in percentage as a function of W. The results show that we can save the number of multiplications by >20% by using the proposed equalizer when compared to the DP-VNLE.

Fig. 9. BER curves of EML/DD system versus the number of multiplications required for the equalizers when (a) W=3, (b) 7, and (c) 9. (d) The complexity reduction versus the pruning parameter W. Transmission distance is 20 km.

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5. Summary

We have proposed and experimentally demonstrated a low-complexity nonlinear equalizer based on the absolute operation for IM/DD transmission systems. By replacing the cross-beating terms in the conventional Volterra nonlinear equalizer with the absolute operation of a sum of two input samples, we reduce the number of multiplication operations required in the nonlinear equalizer. We carry out the experimental demonstration in 56-Gb/s PAM-4 systems implemented by using DML or EML, both operating at 1.5 µm. The results show that the proposed equalizer performs similar to the Volterra nonlinear equalizer for both DML- and EML-based IM/DD systems, but reduces the complexity by >20%. The results also show that the proposed equalizer outperforms the Volterra equalizer when the complexity of the two equalizers is similar. These results imply that the proposed equalizer could be used to compensate for the waveform distortions induced by fiber chromatic dispersion at the direct-detection receiver in a cost-effective manner, instead of complicated Volterra nonlinear equalizer.

Funding

Institute for Information and Communications Technology Promotion (2016-0-00083).

Disclosures

The authors declare no conflicts of interest.

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Low-complexity nonlinear equalizer based on absolute operation for C-band IM/DD systems

Abstract

1. Introduction