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Experimental demonstration of low-complexity trigonometric-memory-polynomial decision-feedback equalizer for nonlinear compensation in IM/DD optical systems

Xing Liu, Yun Liu, Chengju Hu, and Jian Zhao

Open Access Open Access

Abstract

We propose a simple but high-performance trigonometric-memory-polynomial decision-feedback equalizer (TMP-DFE) to cope with the nonlinear distortions in intensity-modulation direct-detection (IM/DD) systems. The proposed method employs sine and cosine operations of received samples, which can be implemented by the efficient CORDIC algorithm using only additions and shifts, to fit odd- and even-order nonlinearities with the effect of different nonlinear orders adjusted by the nonlinear factor. We further propose TMP improved-weighted DFE (TMP-IWDFE) to reduce the error propagation probability of decision feedback. We experimentally evaluate the performance of the proposed schemes in a C-band Erbium-doped-fiber-amplifier-free 56-80Gbit/s four-level pulse-amplitude-modulation (PAM-4) IM/DD system over 30-50 km standard single-mode fiber (SSMF) transmission. The results show that TMP-DFE exhibits better bit error rate performance than Volterra decision-feedback equalizer (V-DFE), diagonally-pruned V-DFE (DP-V-DFE), and diagonally-pruned absolute-term V-DFE (DPAT-V-DFE) while only requiring real multiplications 20.04%, 43.25%, and 74.12% of these conventional schemes. TMP-IWDFE further improves the performance and is better than V-IWDFE, DP-V-IWDFE, and DPAT-V-IWDFE in terms of both performance and complexity. Therefore, the proposed schemes have great potential for high-performance and low-cost IM/DD optical transmission systems.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Intensity modulation and direct detection (IM/DD) systems have been widely used to support the continuous growth of data traffic in optical data-center interconnects and access networks, while meeting the requirements of low cost, low power consumption, and simple structure [1,2]. For IM/DD transmission systems, signal distortions are mainly due to nonlinearity associated with the interplay of dispersion and square-law detection, which causes signal-to-signal beating interference and frequency-selective power fading [3–5]. To improve the achievable transmission distance and capacity for IM/DD systems, many digital signal processing (DSP) based equalization schemes have been proposed [6–21].

Feed-forward equalizer and decision-feedback equalizer (FFE-DFE) [6] are simple and widely used in IM/DD systems but they commonly have limited compensation performance. Volterra nonlinear equalizers (VNLEs) [7–11] improve the nonlinear compensation capability but at the expense of complexity. In addition, feedforward-based VNLEs cannot compensate for dispersion-induced spectral nulls, which can be solved by the combination of VNLEs and DFE (V-DFE). A straightforward strategy to reduce the complexity of VNLEs is to prune unimportant cross-beating terms. Diagonally-pruned VNLEs (DP-VNLEs) as well as joint DP-VNLEs and DFE (DP-V-DFE), where the cross-beating nonlinear terms with a large time delay between beating elements are pruned, were proposed for PAM-4 IM/DD systems [12,13]. The simplest form of DP-VNLE is the polynomial nonlinear equalizer [14], which contains only self-beating terms without any cross-beating terms. To further reduce the complexity, diagonally-pruned absolute-term based VNLE (DPAT-VNLE) was proposed to replace the nonlinear beating terms in DP-VNLE with absolute terms [15]. The absolute operations can be implemented using additions instead of multiplications, thus reducing the complexity. However, this method exhibits a performance penalty. Orthogonal matching pursuit greedy algorithm and weight-sharing strategy based on k-means clustering algorithm were proposed to obtain the significant kernels of nonlinear terms [16–18]. However, they require additional sparse processing units and are therefore not friendly to hardware implementation. On the other hand, although DFE can compensate for chromatic dispersion (CD) induced spectral nulls, it inevitably suffers from error propagation from wrong decisions. In [19], an improved-weighted DFE was proposed and investigated in FFE-DFE and V-DFE to mitigate this effect.

In this paper, we propose a trigonometric-memory-polynomial decision-feedback equalizer (TMP-DFE) to further reduce the complexity and improve the performance of IM/DD systems. The proposed equalizer employs trigonometric operations of received samples together with a nonlinear adjustment factor to improve the fitting capability for nonlinearities of different orders. The trigonometric operations can be implemented by the efficient coordinate rotation digital computer (CORDIC) algorithm using only additions and shifts, thus reducing the complexity. We further propose TMP improved-weighted DFE (TMP-IWDFE) to alleviate the error-propagation effect. The proposed methods are evaluated and compared with conventional nonlinear equalizers in a C-band Erbium-doped-fiber-amplifier (EDFA) free 56-80Gbit/s PAM-4 IM/DD system over 30-50 km SSMF transmission. Experimental results show that the proposed TMP-DFE outperforms V-DFE, DP-V-DFE and DPAT-V-DFE, with the required real multiplications only 20.04%, 43.25%, and 74.12% of these methods. TMP-IWDFE further improves the performance and also exhibits lower BER and complexity than V-IWDFE, DP-V-IWDFE and DPAT-V-IWDFE.

2. Principle

2.1 Conventional DP-V-DFE and DPAT-V-DFE

In DP-V-DFE, the cross-beating terms with a larger time delay between beating elements in the feedforward path are pruned to reduce the complexity while the DFE is used to avoid noise amplification in the compensation of CD-induced spectral nulls. When the received signal is two samples per symbol, the output of the 2nd-order DP-V-DFE can be expressed as:

$$\begin{aligned} y(n) &= \sum\limits_{k ={-} \lfloor{({N_1} - 1)/2} \rfloor }^{\lfloor{({N_1} - 1)/2} \rfloor } {{w_1}(k)x(2n - k)} + \sum\limits_{k = 1}^{{N_2}} {{w_2}(k)d(n - k)} \\ &+ \sum\limits_{q = 0}^{Q - 1} {\sum\limits_{k ={-} \lfloor{({N_3} - 1 - q)/2} \rfloor }^{\lfloor{({N_3} - 1 - q)/2} \rfloor } {{w_3}(k,q)x(2n - k)x(2n - k - q)} } \end{aligned}$$
where ⌊·⌋ denotes the rounding-down operation, x(2n-k) is the received samples with T/2 per sample, d(n) is the decision output, wk and Nk are the tap coefficients and the memory lengths of the linear, decision-feedback, and nonlinear parts of the DP-V-DFE, respectively, Q represents the pruning factor. When Q is equal to N3, DP-V-DFE degenerates to V-DFE.

By replacing the cross-beating terms in DP-V-DFE with the absolute value of the sum of the two input samples, the output of the 2nd-order DPAT-V-DFE can be written as [15]:

$$\begin{aligned} y(n) &= \sum\limits_{k ={-} \lfloor{({N_1} - 1)/2} \rfloor }^{\lfloor{({N_1} - 1)/2} \rfloor } {{w_1}(k)x(2n - k)} + \sum\limits_{k = 1}^{{N_2}} {{w_2}(k)d(n - k)} \\ &+ \sum\limits_{q = 0}^{Q - 1} {\sum\limits_{k ={-} \lfloor{({N_3} - 1 - q)/2} \rfloor }^{\lfloor{({N_3} - 1 - q)/2} \rfloor } {{w_3}(k,q)|{x(2n - k) + x(2n - k - q)} |} } \end{aligned}$$

From Eq. (2), it is seen that DPAT-V-DFE can employ additions to implement the absolute terms instead of multiplications, thus reducing the complexity. However, Eqs. (1,2) only consider 2nd-order nonlinearity. When higher-order nonlinearities are included [21], the complexity would increase significantly.

2.2 Proposed TMP-DFE and TMP-IWDFE

We propose TMP-DFE to include higher-order nonlinear effects to improve the performance without increasing the complexity. It consists of linear, decision-feedback, and nonlinear parts:

$$\begin{aligned} y(n) &= \sum\limits_{k ={-} \lfloor{({N_1} - 1)/2} \rfloor }^{\lfloor{({N_1} - 1)/2} \rfloor } {{w_1}(k)x(2n - k)} + \sum\limits_{k = 1}^{{N_2}} {{w_2}(k)d(n - k)} \\ &+ \sum\limits_{k ={-} \lfloor{({N_3} - 1)/2} \rfloor }^{\lfloor{({N_3} - 1)/2} \rfloor } {{w_3}(k)\sin (\alpha \cdot x(2n - k))} + \sum\limits_{k ={-} \lfloor{({N_3} - 1)/2} \rfloor }^{\lfloor{({N_3} - 1)/2} \rfloor } {{w_4}(k)\cos (\alpha \cdot x(2n - k))} \\ &= {{\textbf W}_1}{\textbf x}(2n) + {{\textbf W}_2}{\textbf d}(n) + {{\textbf W}_3}{{\bf e}_1}(2n) + {{\textbf W}_4}{{\bf e}_2}(2n) \end{aligned}$$
where W1 and W2 represent the tap coefficient vectors for the linear and decision-feedback parts, respectively. W3 and W4 denote the tap coefficient vectors for the nonlinear parts, and the parameter α is the nonlinear adjustment factor to improve the nonlinear fitting ability. In practice, these coefficients can be obtained using the least-mean square (LMS) or recursive least square (RLS) algorithm. x(2n) = [x(2n+⌊(N1-1)/2⌋),…, x(2n),…, x(2n-⌊(N1-1)/2⌋)]T is the linear input vector, where [·]T represents the transpose. d(n) = [d(n-1), d(n-2),…, d(n-N2)]T is the input vector of the decision-feedback part. The nonlinear inputs can be obtained by using the nonlinear expansion block as shown in Fig. 1, where each received sample x(2n-k) is expanded into sin(α·x(2n-k)) and cos(α·x(2n-k)). The sin(·) and cos(·) have been used in the functional link artificial neural network [20], which exhibits better performance than multilayer perceptron and polynomial perceptron networks.

 figure: Fig. 1.

Fig. 1. Schematic diagrams of the proposed TMP-DFE structure. W1 and W2 denote the tap coefficients of the linear and decision-feedback parts, W3 and W4 represent the tap coefficients of the nonlinear part.

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Based on the Taylor expansion, sin(α·x(2n-k)) and cos(α·x(2n-k)) represent the odd- and even-order nonlinear terms, respectively:

$$\sin ({\alpha \cdot x({2n - k} )} )\sim \alpha \cdot x(2n - k) - {\alpha ^3} \cdot {x^3}(2n - k)/6 \ldots $$
$$\cos ({\alpha \cdot x({2n - k} )} )\sim 1 - {\alpha ^2} \cdot {x^2}(2n - k)/2 \ldots$$

From Eqs. (4,5), it is seen that the parameter α can be used to control the effect of different nonlinear orders. For example, increasing α would enhance the effect of high-order terms. In the implementation, α can be updated adaptively:

$${\alpha _{new}} = {\alpha _{old}} + \mu \cdot \varepsilon (n)[{{\textbf x}_1^T(2n), - {\textbf x}_1^T(2n)} ]\odot [{{\textbf e}_2^T(2n),{\textbf e}_1^T(2n)} ]{[{{{\textbf W}_3},{{\textbf W}_4}} ]^T}$$
where μ is the step size, ɛ(n)=d(n)-y(n) is the error, x1(2n) = [x(2n+⌊(N3-1)/2⌋),…, x(2n),…, x(2n-⌊(N3-1)/2⌋)]T is the input vector of the nonlinear expansion block, ${\odot} $ denotes the Hadamard product.

In addition to the control of the effect of different nonlinear orders via α, Eq. (3) shows that the proposed TMP-DFE also have different tap coefficients, W3 and W4, for sin(α·x(2n-k)) and cos(α·x(2n-k)), thus having the freedom to control the odd- and even-order nonlinearities. Therefore, TMP-DFE is expected to achieve better nonlinear fitting ability than the conventional memory-polynomial nonlinear equalizer. Also note that although Eq. (3) neglects the cross-beating terms to reduce the complexity, as will be shown, it can still achieve better performance than the second-order VNLE even without any pruning.

On the other hand, additional investigation shows that the DFE part in Eq. (3) is essential to compensate CD-induced spectral nulls and cannot be neglected. However, conventional DFE may suffer from error propagation due to wrong decisions. To alleviate this effect, we further propose TMP-IWDFE, which replaces the hard decision in DFE with the soft decision:

$$\begin{aligned} y(n) &= \sum\limits_{k ={-} \lfloor{({N_1} - 1)/2} \rfloor }^{\lfloor{({N_1} - 1)/2} \rfloor } {{w_1}(k)x(2n - k)} + \sum\limits_{k = 1}^{{N_2}} {{w_2}(k)\widehat d(n - k)} \\ &+ \sum\limits_{k ={-} \lfloor{({N_3} - 1)/2} \rfloor }^{\lfloor{({N_3} - 1)/2} \rfloor } {{w_3}(k)\sin (\alpha \cdot x(2n - k))} + \sum\limits_{k ={-} \lfloor{({N_3} - 1)/2} \rfloor }^{\lfloor{({N_3} - 1)/2} \rfloor } {{w_4}(k)\cos (\alpha \cdot x(2n - k))} \end{aligned}$$
where $\hat{d}(n - k)$ is the soft-decision output and is obtained from the equalizer output y(n) and the hard-decision output d(n) as [19]:
$$\hat{d}(n) = y(n) + f({r_n})[{d(n) - y(n)} ]$$
where rn is:
$${r_n} = \left\{ {\begin{array}{{l}} {1 - |{y(n) - d(n)} |,if|{y(n)} |< M - 1}\\ {1,else} \end{array}} \right.$$
where M is the PAM-M level.

f(rn) is the compressed sigmoid nonlinear function:

$$f({r_n}) = 0.5 \times \{{{{\{{1 - \textrm{exp} [{ - a({r_n}/b - 1)} ]} \}} / {\{{1 + \textrm{exp} [{ - a({r_n}/b - 1)} ]} \}}} + 1} \}$$
where a is a positive integer and b is a compression factor, 0 < b ≤ 1. Figure 2 shows f(rn) for different a and b. It is seen that f(rn) is monotonically increasing. a determines the minimal f(rn) while b affects the steepness of f(rn) given the parameter a. In practice, Eq. (10) can be implemented using a look-up table associated with parameters a and b, which maps the input rn to the output f(rn) without multiplication operations. The schematic diagram of the TMP-IWDFE is illustrated in Fig. 3. Compared with TMP-DFE, TMP-IWDFE adds a reliability block to calculate rn using Eq. (9) and a use block to obtain $\hat{d}(n)$ using Eq. (8) and Eq. (10).

 figure: Fig. 2.

Fig. 2. f(rn) versus rn for different a and b.

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 figure: Fig. 3.

Fig. 3. Schematic diagrams of TMP-IWDFE, y(n) is the equalizer output, d(n) is the hard-decision output, and $\hat{d}(n)$ is the soft-decision output.

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2.3 Complexity analysis

We will evaluate the complexity of different nonlinear equalizers using the required number of real multiplications per output symbol. The complexity at the convergence stage is not included. Table 1 compares the required number of real multiplications for different equalizers. For all equalizers, the numbers of multiplications for the linear and decision-feedback parts are N1 and N2, respectively. For the nonlinear part, the number of nonlinear terms for the proposed TMP-DFE in Eq. (3) is 2N3, and each nonlinear term requires two real multiplications, i.e. the multiplication with α and with w3(k) or w4(k). Therefore, TMP-DFE requires N1 + N2 + 4N3 real multiplications. Note that sine and cosine functions can be realized using the efficient CORDIC algorithm that only requires shifts and additions [22]. For V-DFE and DP-V-DFE, from Eq. (1), the number of nonlinear terms is N3(N3 + 1)/2 and Q(2N3-Q + 1)/2 respectively and each nonlinear term requires two multiplications. Therefore, the total real multiplications for these two methods are N1 + N2 + N3(N3 + 1) and N1 + N2 + Q(2N3-Q + 1), respectively. On the other hand, from Eq. (2), the number of nonlinear terms in DPAT-V-DFE is Q(2N3-Q + 1)/2 and each term needs one multiplication, so the total required number of multiplications is N1 + N2 + Q(2N3-Q + 1)/2. Finally, compared to DFE, IWDFE needs only one more multiplication in Eq. (8). Note that Eq. (10) can be implemented using a look-up table. Therefore, if the DFE is replaced by IWDFE, the required number of real multiplications is increased by 1 for all equalizers. Figure 4 shows the required number of multiplications for different equalizers versus the nonlinear memory length N3. Other parameters N1, N2, and Q are set to be 61, 12, and 7, respectively. These values are optimal for the experiments in the next section. It is seen that the proposed TMP-DFE exhibits the least complexity. For example, when N3 is 29 as will be used in the experiments, the number of real multiplications in TMP-DFE is only 20.04%, 43.25%, and 74.12% of those in V-DFE, DP-V-DFE, and DPAT-V-DFE, respectively.

 figure: Fig. 4.

Fig. 4. The number of real multiplications versus the nonlinear memory length N3 for different nonlinear equalizers. In the figure, N1, N2, and Q are set to be 61, 12, and 7, respectively.

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Tables Icon

Table 1. The complexity of different nonlinear equalizers

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3. Experimental setup

The proposed TMP-DFE and TMP-IWDFE outperform conventional nonlinear equalizers besides the reduced complexity. We verified their performance advantages in a C-band EDFA-free PAM-4 IM/DD system. Figure 5 depicts the experimental setup. At the transmitter, the PAM-4 signal was generated in the offline processing and re-shaped to a square-root-raised-cosine (SRRC) spectral profile with a roll-off factor of 0.4. The PAM-4 signal was loaded into an arbitrary waveform generator (AWG, Keysight M9502A) with a 3-dB bandwidth of 25 GHz operating at a sample rate of 56 GSa/s. Different baud rates were investigated, and unless specified, the default baud rate was 28 Gbaud. The generated PAM-4 electrical signal was amplified by an electrical amplifier (EA) with a bandwidth of 40 GHz and a gain of 16 dB and drove a 23-GHz Mach-Zehnder modulator (MZM) for electrical-optical conversion. The optical carrier was generated from a laser with a center wavelength of 1550 nm. The transmission distance was 40 km unless otherwise stated and the power into the SSMF was around 5 dBm. After transmission, the received optical power (ROP) at the photodetector (PD) was adjusted using a variable optical attenuator (VOA). The signal was then detected and amplified by an EA with 30-dB gain and sampled by a digital storage oscilloscope (DSO, LabMaster 10-36Zi-A) operating at 80 GSa/s. The bandwidth of the DSO was set as 13 GHz using the internal setting of the DSO so that the system was bandwidth limited. The offline DSP was implemented using MATLAB and included resampling to 2 samples per symbol, symbol synchronization, matched SRRC filtering, equalization, and BER calculation. The coefficients of the equalizer, W1, W2, W3, and W4, were obtained using the RLS algorithm with 10,000 training symbols.

 figure: Fig. 5.

Fig. 5. Experimental setup of the C-band EDFA-free 56-80Gbit/s PAM-4 IM/DD system.

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4. Results and discussions

We first optimize the spectral roll-off of the PAM-4 signal under the utilized experimental setup. Figure 6 shows (a) peak-to-average power ratio (PAPR) and (b) bit error rate (BER) of FFE versus the spectral roll-off factor at back-to-back and 56 Gbit/s. The ROP is -11.6 dBm and the peak-to-peak voltage of the AWG output is 400 mV. It can be observed that the signal PAPR tends to decrease as the roll-off factor increases and becomes stable when the roll-off factor is larger than 0.4. Because the signal power, i.e. the AC power, increases as the PAPR decreases, the BER first reduces with the roll-off factor. However, a larger roll-off also results in a wider signal bandwidth and the signal would be distorted by the limited device bandwidth. Therefore, the BER then increases for larger roll-off factors and there is an optimal roll-off value of 0.4.

 figure: Fig. 6.

Fig. 6. (a) PAPR and (b) BER versus the spectral roll-off factor of the PAM-4 signal at 0 km.

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Then we optimize the peak-to-peak output voltage of the AWG. Figure 7 shows BER versus the output voltage (a) at back-to-back and -11.6 dBm ROP and (b) at 40 km and -5.6 dBm ROP. In both figures, N1, N2, N3, and Q are 61, 12, 29 and 7 for all equalizers. From Fig. 7(a), the BER performance first improves with the voltage for all methods due to the increasing signal power. When the voltage is higher than 400 mV, the distortion due to modulator nonlinearity becomes prominent and the BER starts to increase. On the other hand, in contrast to Fig. 7(a), the performances of different equalizers are much more distinct after 40-km transmission while the proposed TMP-DFE exhibits the best performance. This implies that the proposed TMP-DFE is indeed more effective in compensating transmission impairments. It is also seen that the optimal voltages for FFE and FFE-DFE are 150 mV, while V-DFE, DP-V-DFE, DPAT-V-DFE, and the proposed TMP-DFE can increase the optimal voltage to 200 mV due to better compensation capability. Compared to the back-to-back case, the optimal voltages of all equalizers after transmission are reduced. This is because the PAPR of the signal after transmission increases due to chromatic dispersion, as shown in Fig. 8. Under a fixed average power, the peak amplitude of the signal becomes higher and results in more prominent nonlinearity under the square-law detection. Therefore, for the case of 40 km, the optimal voltage should be reduced to re-balance the signal power and the nonlinearity. In the following, we set the voltage of the AWG as 200 mV.

 figure: Fig. 7.

Fig. 7. BER versus the AWG output voltage (a) after 0-km transmission, ROP is -11.6 dBm, (b) after 40-km transmission, ROP is -5.6 dBm. The data rate is 56 Gbit/s.

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 figure: Fig. 8.

Fig. 8. The complementary cumulative distribution function (CCDF) of the PAPR for the signal before and after 40-km SSMF transmission. In the calculation of each PAPR, the number of used symbols is 300 and the upsampling factor is 20.

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Next, we investigate the effect of parameters N1, N2, N3, and Q at 40 km and the ROP of -5.6 dBm. Figure 9 shows BER versus (a) N1 for FFE, (b) N2 for FFE-DFE, (c) N3 for V-DFE and TMP-DFE, and (d) Q for DP-V-DFE and DPAT-V-DFE. In (b)-(d), N1 = 61; In (c)-(d), N2 = 12; In (d), N3 = 29. From (a), it is seen that the performance of FFE improves as N1 increases and becomes saturated when N1 is larger than 60. In (b), the BER reduces rapidly when N2 increases from 0 to 2 and then decreases smoothly afterward. In order to balance the performance and the complexity, N2 = 12 is chosen. In (c), the BER reduces significantly with N3 initially and then becomes stable when N3 is larger than 29. Therefore, the nonlinear memory length N3 is selected as 29. Finally, in Fig. 9(d), the performances of DP-V-DFE and DPAT-V-DFE improve slightly for Q < 7 and then become unchanged afterward. Therefore, N1, N2, N3, and Q are set to be 61, 12, 29, and 7 respectively, unless otherwise stated.

 figure: Fig. 9.

Fig. 9. BER versus (a) N1 for FFE, (b) N2 for FFE-DFE with N1 = 61, (c) N3 for V-DFE and TMP-DFE with N1 = 61 and N2 = 12, (d) Q when N1, N2, and N3 are 61, 12 and 29, respectively.

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We then investigate the convergence speed of different equalizers. Figure 10 shows the BER versus the length of the training sequence for different nonlinear equalizers when the tap coefficients are obtained using (a) the LMS algorithm and (b) the RLS algorithm. In Fig. 10(b), the star and asterisk represent the ones where the nonlinear adjustment factor α is obtained adaptively using Eq. (6) or using manual optimization. Comparison between Fig. 10(a) and (b) shows that the RLS algorithm achieves much faster convergence and better performance than the LMS algorithm. 10,000 training symbols are sufficient to obtain the optimal BER performance for all methods and the proposed TMP-DFE exhibits the best performance. Figure 10(b) also verifies that the adaptive algorithm of Eq. (6) can obtain near-optimal value for the parameter α. Therefore, the RLS algorithm is chosen and α is obtained using Eq. (6).

 figure: Fig. 10.

Fig. 10. BER versus the number of training symbols when the tap coefficients are obtained using (a) the LMS algorithm and (b) the RLS algorithm.

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After parameter optimization, we compare the performance of different nonlinear equalizers. Figure 11 shows BER versus (a) ROP at 56 Gbit/s and (b) data rate at an ROP of -5.6 dBm. In (b), N1, N2, N3, and Q are optimized for each bit rate using the same strategy as Fig. 9. It is seen that the performance of FFE-DFE is the worst. DP-V-DFE with the optimized pruning factor Q can achieve similar performance as V-DFE without any pruning while DPAT-V-DFE exhibits a penalty by using absolute terms to replace the nonlinear beating terms. On the other hand, the proposed TMP-DFE performs the best. It is worth noting that TMP-DFE also has the lowest complexity among the investigated nonlinear equalizers, as shown in Fig. 4. Figure 11 is based on the DSO with a bandwidth of 13 GHz. Additional results of 100Gbit/s PAM-4 at 10 km using a 36-GHz DSO show that the same conclusion as Fig. 11 can be drawn.

 figure: Fig. 11.

Fig. 11. BER versus (a) ROP at 56 Gbit/s (b) bit rate at an ROP of -5.6 dBm. In (a) and (b), the fiber length is 40 km.

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We further compare the performance of different equalizers under different transmission distances and complexity, as shown in Fig. 12. The ROP is -5.6 dBm. In (a), N1, N2, N3, and Q are optimized for each fiber length using the same strategy as Fig. 9. In (b), the fiber length is 40 km and the complexity is varied by adjusting N3 while fixing N1, N2 and Q as 61, 12, and 7, respectively. Figure 12(a) shows that the performance advantage of the proposed TMP-DFE maintains for all investigated fiber lengths, while Fig. 12(b) confirms that TMP-DFE can achieve the best performance with the least complexity.

 figure: Fig. 12.

Fig. 12. BER versus (a) the transmission distance and (b) the number of multiplications for different nonlinear equalizers.

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The above results are based on the TMP-DFE, where error propagation may affect the performance. Figure 13 shows BER versus ROP without error propagation, by replacing the decided symbols in DFE with the transmitted symbols. Comparison of Fig. 13 and Fig. 11(a) shows that the BER is reduced by nearly one order of magnitude, indicating that error propagation is indeed an important issue. Therefore, we will investigate TMP-IWDFE and also compare it with FFE-IWDFE, V-IWDFE, DP-V-IWDFE, and DPAT-V-IWDFE. The optimal values of parameters a and b in Eq. (10) are first identified. Figure 14(a) shows the BER as a function of the compression factor b for TMP-IWDFE under different a. Note that the case of b = 0 corresponds to TMP-DFE. It is seen that TMP-IWDFE under the optimized a and b indeed improves the performance due to the reduction of the error propagation probability, and the optimal a and b are 2 and 0.15, respectively. Figure 14(b) shows the BER versus b for V-IWDFE. It is seen that the optimized V-IWDFE also outperforms V-DFE and the optimal a and b are 3 and 0.2, respectively. Similarly, we can optimize a and b for other methods.

 figure: Fig. 13.

Fig. 13. BER versus ROP at 56 Gbit/s and 40 km without error propagation in the DFE.

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 figure: Fig. 14.

Fig. 14. Measured BER as a function of the compression factor b for (a) TMP-IWDFE and (b) V-IWDFE. The bit rate is 56 Gbit/s, the fiber length is 40 km and the ROP is -5.6 dBm.

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Figure 15 shows BER versus (a) ROP and (b) bit rate after 40-km SSMF transmission. Comparison between Fig. 15 and Fig. 11 shows that the use of IWDFE instead of DFE improves the performance of all methods, due to the reduced probability of error propagation. On the other hand, it is confirmed that the proposed TMP-IWDFE still exhibits the best performance, even better than V-DFE without any pruning. It is worth noting that compared to TMP-DFE, TMP-IWDFE requires only one more real multiplication to obtain the feedback symbols, which is negligible in complexity. Therefore, TMP-IWDFE is a promising solution for low-cost and high-performance IM/DD optical transmission systems.

 figure: Fig. 15.

Fig. 15. BER versus (a) ROP at 56 Gbit/s and (b) bit rate when the ROP is -5.6 dBm. In both (a) and (b), the fiber length is 40 km.

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5. Conclusion

We have proposed and experimentally demonstrated low-complexity TMP-DFE and TMP-IWDFE for nonlinear distortion compensation in IM/DD optical transmission systems. The proposed methods employ TMP with more freedom to improve the fitting capability for not only odd- and even-order but also low- and high-order nonlinearities. The trigonometric operations can be implemented by the efficient CORDIC algorithm using only additions and shifts, thus reducing the required number of multiplications. We have experimentally evaluated the performance of the proposed methods in a C-band EDFA-free 56-80Gbit/s PAM-4 IM/DD system over 30-50 km SSMF transmission. It is shown that the proposed TMP-DFE outperforms V-DFE, DP-V-DFE, and DPAT-V-DFE while the required number of multiplications is only 20.04%, 43.25% and 74.12% of these methods. TMP-IWDFE further improves the performance by reducing the probability of error propagation and exhibits better performance and lower complexity than V-IWDFE, DP-V-IWDFE and DPAT-V-IWDFE. Therefore, the proposed methods can be promising solutions for nonlinear impairment compensation in low-cost and high-performance IM/DD optical transmission systems.

Funding

National Key Research and Development Program of China (2022YFB2903002); National Natural Science Foundation of China (U23A20282); National Natural Science Foundation of China (61971199).

Disclosures

The author declares no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (15)
Fig. 1.
Fig. 1. Schematic diagrams of the proposed TMP-DFE structure. W1 and W2 denote the tap coefficients of the linear and decision-feedback parts, W3 and W4 represent the tap coefficients of the nonlinear part.

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Fig. 2.
Fig. 2. f(rn) versus rn for different a and b.

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Fig. 3.
Fig. 3. Schematic diagrams of TMP-IWDFE, y(n) is the equalizer output, d(n) is the hard-decision output, and $\hat{d}(n)$ is the soft-decision output.

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Fig. 4.
Fig. 4. The number of real multiplications versus the nonlinear memory length N3 for different nonlinear equalizers. In the figure, N1, N2, and Q are set to be 61, 12, and 7, respectively.

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Fig. 5.
Fig. 5. Experimental setup of the C-band EDFA-free 56-80Gbit/s PAM-4 IM/DD system.

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Fig. 6.
Fig. 6. (a) PAPR and (b) BER versus the spectral roll-off factor of the PAM-4 signal at 0 km.

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Fig. 7.
Fig. 7. BER versus the AWG output voltage (a) after 0-km transmission, ROP is -11.6 dBm, (b) after 40-km transmission, ROP is -5.6 dBm. The data rate is 56 Gbit/s.

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Fig. 8.
Fig. 8. The complementary cumulative distribution function (CCDF) of the PAPR for the signal before and after 40-km SSMF transmission. In the calculation of each PAPR, the number of used symbols is 300 and the upsampling factor is 20.

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Fig. 9.
Fig. 9. BER versus (a) N1 for FFE, (b) N2 for FFE-DFE with N1 = 61, (c) N3 for V-DFE and TMP-DFE with N1 = 61 and N2 = 12, (d) Q when N1, N2, and N3 are 61, 12 and 29, respectively.

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Fig. 10.
Fig. 10. BER versus the number of training symbols when the tap coefficients are obtained using (a) the LMS algorithm and (b) the RLS algorithm.

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Fig. 11.
Fig. 11. BER versus (a) ROP at 56 Gbit/s (b) bit rate at an ROP of -5.6 dBm. In (a) and (b), the fiber length is 40 km.

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Fig. 12.
Fig. 12. BER versus (a) the transmission distance and (b) the number of multiplications for different nonlinear equalizers.

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Fig. 13.
Fig. 13. BER versus ROP at 56 Gbit/s and 40 km without error propagation in the DFE.

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Fig. 14.
Fig. 14. Measured BER as a function of the compression factor b for (a) TMP-IWDFE and (b) V-IWDFE. The bit rate is 56 Gbit/s, the fiber length is 40 km and the ROP is -5.6 dBm.

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Fig. 15.
Fig. 15. BER versus (a) ROP at 56 Gbit/s and (b) bit rate when the ROP is -5.6 dBm. In both (a) and (b), the fiber length is 40 km.

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Tables (1)

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Table 1. The complexity of different nonlinear equalizers

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Equations (10)

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y ( n ) = ∑ k = − ⌊ ( N 1 − 1 ) / 2 ⌋ ⌊ ( N 1 − 1 ) / 2 ⌋ w 1 ( k ) x ( 2 n − k ) + ∑ k = 1 N 2 w 2 ( k ) d ( n − k ) + ∑ q = 0 Q − 1 ∑ k = − ⌊ ( N 3 − 1 − q ) / 2 ⌋ ⌊ ( N 3 − 1 − q ) / 2 ⌋ w 3 ( k , q ) x ( 2 n − k ) x ( 2 n − k − q )
y ( n ) = ∑ k = − ⌊ ( N 1 − 1 ) / 2 ⌋ ⌊ ( N 1 − 1 ) / 2 ⌋ w 1 ( k ) x ( 2 n − k ) + ∑ k = 1 N 2 w 2 ( k ) d ( n − k ) + ∑ q = 0 Q − 1 ∑ k = − ⌊ ( N 3 − 1 − q ) / 2 ⌋ ⌊ ( N 3 − 1 − q ) / 2 ⌋ w 3 ( k , q ) | x ( 2 n − k ) + x ( 2 n − k − q ) |
y ( n ) = ∑ k = − ⌊ ( N 1 − 1 ) / 2 ⌋ ⌊ ( N 1 − 1 ) / 2 ⌋ w 1 ( k ) x ( 2 n − k ) + ∑ k = 1 N 2 w 2 ( k ) d ( n − k ) + ∑ k = − ⌊ ( N 3 − 1 ) / 2 ⌋ ⌊ ( N 3 − 1 ) / 2 ⌋ w 3 ( k ) sin ⁡ ( α ⋅ x ( 2 n − k ) ) + ∑ k = − ⌊ ( N 3 − 1 ) / 2 ⌋ ⌊ ( N 3 − 1 ) / 2 ⌋ w 4 ( k ) cos ⁡ ( α ⋅ x ( 2 n − k ) ) = W 1 x ( 2 n ) + W 2 d ( n ) + W 3 e 1 ( 2 n ) + W 4 e 2 ( 2 n )
sin ⁡ ( α ⋅ x ( 2 n − k ) ) ∼ α ⋅ x ( 2 n − k ) − α 3 ⋅ x 3 ( 2 n − k ) / 6 …
cos ⁡ ( α ⋅ x ( 2 n − k ) ) ∼ 1 − α 2 ⋅ x 2 ( 2 n − k ) / 2 …
α n e w = α o l d + μ ⋅ ε ( n ) [ x 1 T ( 2 n ) , − x 1 T ( 2 n ) ] ⊙ [ e 2 T ( 2 n ) , e 1 T ( 2 n ) ] [ W 3 , W 4 ] T
y ( n ) = ∑ k = − ⌊ ( N 1 − 1 ) / 2 ⌋ ⌊ ( N 1 − 1 ) / 2 ⌋ w 1 ( k ) x ( 2 n − k ) + ∑ k = 1 N 2 w 2 ( k ) d ^ ( n − k ) + ∑ k = − ⌊ ( N 3 − 1 ) / 2 ⌋ ⌊ ( N 3 − 1 ) / 2 ⌋ w 3 ( k ) sin ⁡ ( α ⋅ x ( 2 n − k ) ) + ∑ k = − ⌊ ( N 3 − 1 ) / 2 ⌋ ⌊ ( N 3 − 1 ) / 2 ⌋ w 4 ( k ) cos ⁡ ( α ⋅ x ( 2 n − k ) )
d ^ ( n ) = y ( n ) + f ( r n ) [ d ( n ) − y ( n ) ]
r n = { 1 − | y ( n ) − d ( n ) | , i f | y ( n ) | < M − 1 1 , e l s e
f ( r n ) = 0.5 × { { 1 − exp [ − a ( r n / b − 1 ) ] } / { 1 + exp [ − a ( r n / b − 1 ) ] } + 1 }
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