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Low-complexity and low-latency equalization technique – probabilistic noise cancellation

Nebojša Stojanović, Youxi Lin, Talha Rahman, Stefano Calabrò, Md Sabbir-Bin Hossain, Tom Jonas Wettlin, and Maxim Kuschnerov

Open Access Open Access

Abstract

Both inside data centers (DCs) and in short optical links between data centers (DC campuses), intensity-modulation and direct-detection (IMDD) systems using four-level pulse amplitude modulation (PAM4) will dominate this decade due to low transceiver price and power consumption. The next DC transceiver generation based on 100 Gbaud PAM4 will require advanced digital signal processing (DSP) algorithms and more powerful forward error correction (FEC) codes. Because of bandwidth limitations, the conventional DC DSP based on a few-tap linear feed-forward equalizer (FFE) is likely to be upgraded to more complex but still low-complexity Volterra equalizers followed by a noise whitening filter and either a maximum likelihood sequence estimation (MLSE) or a maximum a posteriori probability (MAP) algorithm. However, stringent power consumption and latency requirements may limit the use of complex algorithms such as decision feedback equalizer (DFE) or MLSE/MAP in DC networks (DCN). In this paper, we introduce a low-complexity, low-latency algorithm based on a feedforward structure, yielding a performance between DFE and MLSE. We call the novel equalization algorithm probabilistic noise cancellation (PNC), since it weights noise patterns based on their probabilities in the presence of bandwidth limitations. The probabilistic weighting is efficiently exploited in correcting correlated errors caused by noise coloring in the FFE.

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

1. Introduction

Permanently increasing data rates in data centers (DC) with transceivers in near future likely carrying 3.2Tbit/s will require improvements in electrical and optical components, including analog-to-digital and digital-to-analog convertors (ADC/DAC), modulators, light detectors, optical multiplexer and demultiplexers, and electrical amplifiers, and the introduction of further components like semiconductor optical amplifiers, dispersion compensation modules, possibly polarization rotators and new fibers [1] for four-wave mixing suppression, etc. Pluggable DC networks (DCN) modules will likely achieve a power efficiency of 10pJ/bit in the 4 × 200 G scenario while it is expected that co-packaged optics (CPO) [2] will improve it by 3 times with the limit of 1pJ/bit rapidly approaching. Another way to relax power requirements and to improve electro-optical density is to avoid DSP in the optical module by deploying advanced DSP in the switch application-specific integrated circuit (ASIC) along with linear (drive) pluggable optics (LPO) [3].

Typically, 50 Gbaud PAM4 DC receiver DSP is based on symbol-rate ADC, simple binary-based decision-aided Mueller and Muller timing phase detector [4], a few-tap linear feed-forward equalizer (FFE), and KP4 forward error correction (FEC) code. Doubling bit rates in next generation of DC optical transceivers (200Gbit/s/lane PAM4) requires dramatic changes such as a new frequency grid in wavelength division multiplexing (WDM) links, advanced equalization algorithms based on either decision feedback equalization (DFE) or maximum likelihood sequence estimation (MLSE) [5], sophisticated clock extraction techniques [6], upgrade of existing standards [7] with the introduction of a new FEC, etc. A new error correction code based on the concatenation of an extended soft-decoded Hamming code and KP4 FEC [8,9] has been already standardized. Including the soft Hamming FEC results in the need for high-quality soft information after the equalization blocks [10].

Improving component bandwidth and nonlinearities, using advanced digital pre-compensation, and the total system optimization improve performance significantly, open doors for higher bit rates, and also relax DSP demands and power consumption. Excellent performance in intensity-modulation and direct-detection (IMDD) bandwidth-limited systems can be achieved by using a nonlinear FFE followed by a matched filter and MLSE. The pipelined MLSE structure requires a very clever design to decrease complexity and latency. On the other hand, the DFE-based signal reconstruction also involves high complexity and latency with performance that can be much worse than that of MLSE. This was the main driver to design a novel algorithm with performance between DFE and MLSE. The novel algorithm, called probabilistic noise cancellation (PNC), is based on a feed-forward structure and introduces negligible latency and complexity. The PNC studies error statistics and uses a clever logic to improve decisions.

This paper is organized as follows: First, an overview of equalization in DCN transceivers is given in Section 2, then we introduce the novel PNC algorithm in Section 3, Sections 4 and 5 present equalization results with simulations and experiments, respectively, Section 6 gives an error analysis, and finally we conclude the paper in Section 7.

2. Equalization in IMDD optical systems

DCN optical transceivers use simple equalization schemes based on a few linear and nonlinear FFE taps (reduced complexity Volterra equalizers) and one sample per symbol (1 sps). DFE and MLSE may bring less than 0.5 and 1 dB, respectively, in 50 Gbaud transceivers as the transceiver bandwidth is satisfactory and the FFE BER floor is well below the KP4 FEC threshold. However, for future transceivers based on 100 Gbaud PAM4 both chromatic dispersion and bandwidth limitations will require stronger DSP structure either with DFE or MLSE.

A Pth order discrete 1 sps Volterra filter with input x, output y, and memory length M can be described as

$$y(k )= {w_{dc}} + \mathop \sum \nolimits_{r = 1}^p \mathop \sum \nolimits_{{k_1} = 0}^{M - 1} \cdots \mathop \sum \nolimits_{{k_r} = {k_r} - 1}^{M - 1} {w_r}({{k_1},{k_2}, \ldots ,{k_r}} )\times x({{k_1} - {k_1}} )\cdots x({{k_1} - {k_r}} ),$$
where wr are rth order Volterra kernels [11]. A Volterra filter orthogonalized by the Wiener model for PAM4 is described in [12]. The coefficient wdc is responsible for the dc component. We omit the dc kernel as it can be realized in different ways and represent a linear 1 sps FFE with 2N + 1 coefficients by
$$y(n )= \mathop \sum \nolimits_{i ={-} N}^N a(i )x({n - i} ),$$
where a(i) represents the ith FFE coefficient. Adaptive equalizers are used in many applications such as noise and echo cancellation, sinusoidal rejection, equalization etc. [13,14]. The mean square error (MSE) criterion, also known as least mean square (LMS) algorithm [15–18]. The coefficient adaptation for the MSE algorithm is defined by
$${a_j}({k + 1} )= {a_j}(k )+ \mu {e_k}x(k ),\; {e_k} = {d_k} - {y_k},$$
where µ denotes a small constant that controls the magnitude of the weight adjustment and the error ${e_k}$ is the difference between the decision dk and equalizer output ${y_k}$ (see Fig. 1). In systems suffering from severe bandwidth limitations, nonlinearities, and eye skew [19], the equalizer coefficient acquisition success depends on the initializing sampling phase and the starting taps. In PAM4 DC receivers, timing recovery exploits the phase detector (PDET) output signal to establish the stable sampling clock. Two PDETs that can reliably extract the clock in PAM4 signals are described in [4] and [6]. A binary-based Mueller and Müller PDET [4] is commonly used in ASIC supported timing recovery schemes while the abs PDET [6] can support any modulation format and has better jitter performance in bandwidth-limited systems. To improve the channel acquisition, another PAM4 blind algorithm, the so-called Sato algorithm [20], can be helpful with the error calculation defined by
$${e_k} = {y_k} - \gamma \; sgn({y_k}),\; \; \gamma = \frac{{E[{{s^2}} ]}}{{E[s ]}},$$
where E denotes the expectation and s represents the PAM symbol.

 figure: Fig. 1.

Fig. 1. Basic equalization techniques. Q – quantizer.

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As the FFE changes noise statistics, the FFE output signal suffers from colored noise that can be partially flattened by a feedback structure shown in Fig. 2. In this case, the output signal becomes

$$y(n )= \mathop \sum \nolimits_{i ={-} \textrm{N}}^0 a(i )x({n - i} )+ \mathop \sum \nolimits_{i = 1}^K b(i )d({n - i} ),$$
where the FFE compensates for the precursor (N + 1 taps) and the DFE part is responsible for the postcursor tail (K taps). In some (ideal) channels, in which the received pulse shape has only the postcursor, the FFE is not necessary and the DFE is responsible for the signal restoration. However, in real systems this is much more complicated. Some of taps of 41-tap FFE are shown in Fig. 3 (106 Gbaud experiment shown in Fig. 17) together with the channel impulse response. The channel impulse response clearly shows strong pre and postcursor that cannot be compensated only by the DFE. Therefore, the hybrid FFE-DFE structure should be used to improve the performance. FFE taps with and without 1-tap DFE are shown in Fig. 4 (FFE is extended by the postcursor to achieve better performance). The FFE-DFE architecture slightly improves the FFE BER performance from 7.6e-4 to 6.2e-4. Adding more DFE taps degrades performance as the DFE taps are small and the decisions are not “reliable enough”.

 figure: Fig. 3.

Fig. 3. FFE taps and impulse response in experiments.

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 figure: Fig. 4.

Fig. 4. 41 FFE taps with and without 1-tap DFE in experiments.

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Another but more complicated way to improve performance is shown in Fig. 1 as the third option using the FFE followed by a post filter (PF) to decorrelate the FFE output noise. The PF block generates the controlled ISI that can be solved by the complex MLSE [21]. The M-tap PF output is defined by

$$z(n )= \mathop \sum \nolimits_{i = 0}^M \alpha (i )y({n - i} )= y(n )+ \mathop \sum \nolimits_{i = 1}^M \alpha (i )y({n - i} ),$$
where the first tap is normally equal to 1 (α(0) = 1). Other PF taps can be found by using linear prediction techniques to predict noise samples ${e_k} = {d_k} - {y_k}$ [22]. To simplify PAM4 MLSE and to achieve quite good performance, the 2-tap post filter can be used with $\alpha (0 )= 1$ and $\alpha (1 )= \frac{{E[{{e_k}{e_{k - 1}}} ]}}{{E[{{e_k}{e_k}} ]}}\textrm{}$. After the PF, a 4-state PAM4 MLSE brings BER below 1e-5 for the case shown in Fig. 3. In this experiment, $\alpha (1 )\textrm{}$ was larger than 0.4 and the MLSE is known to bring a significant improvement in strong ISI scenarios as will be demonstrated later (larger $\alpha (1 )$ leads to larger MLSE gain). In the next section, we introduce techniques that can bring performance close to those of MLSE for medium $\alpha (1 )$ values.

3. Probabilistic noise cancellation

The advantage of DFE is that it uses decisions to improve the performance and it does not introduce noise enhancement as the FFE does. However, the DFE can produce long error bursts, which, however, can be partially prevented by differential encoding. In fast optical systems, the ASIC works at much lower speed than the symbol rate so that hundreds of symbols are processed in parallel. To solve the feedback loop in parallel DFE equalizers [23], sophisticated algorithms as lookahead techniques are required. A 3-tap DFE design at very high speeds can be found in [24], whereas a 28 Gbit/s 4-tap FFE/15-tap DFE serial link transceiver was presented in [25].

To avoid the DFE feedback, some solutions are presented in [26] and [27]. The noise cancellation (NC) algorithm uses weighted noise samples e from neighboring symbols to whiten noise at the considered symbols and to get the samples w with better noise statistics by (see Fig. 5)

$$w(n )= y(n )+ \mathop \sum \nolimits_{i = 1}^M [\beta (i )e({n - i} )+ \beta ({ - i} )e({n + i} )],$$
where coefficients $\beta $ should be optimized (precursor and postcursor noise samples are involved). The NC using only precursor noise samples is not so effective, but it improves the BER, the NC using postcursor behaves similarly as the DFE while the third one exploiting both precursor and postcursor noise samples behaves similarly as the DFE but with some differences. For the sake of simplicity, we call the third option NC in the following. The best NC and DFE performance can be achieved when the neighboring decisions are correct (lower bound). However, in blind mode, some decisions are erroneous and these two algorithms show significant performance degradation at higher BER values.

 figure: Fig. 5.

Fig. 5. Simple noise cancellation.

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The NC decisions can be improved by analyzing error patterns in a multi-stage architecture that improves decisions after each stage, which brings us to the PNC. The coefficients $\beta $ in (7) can be perfectly known (optimized) but errors cannot. The idea in multi-stage PNC is to use mild coefficients in the early stages to correct the most suspicious error locations. At later stages, when the decisions are improved we can use the right/optimum $\beta $ coefficients. The N-stage PNC is shown in Fig. 6 where each PNC stage improves BER and BERn + 1 ≤ BERn holds.

 figure: Fig. 6.

Fig. 6. Multi-stage PNC.

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Most communication systems have a low-pass characteristic that can be approximated by a 1 + aD function, where D is a delay of one symbol interval and a is a value between 0 and 1. In 106 Gbaud transmission systems using state-of-the-art components, the value of a is typically in a range from 0.4 to 0.6 (see Fig. 7). Therefore, the noise after a full-response FFE has a shape ∼ 1/(1 + aD) with additive Gaussian noise at the FFE input. Let us denote noise samples derived by using the transmitted symbols s by es = s-y where y is the FFE output. By ed = d-y, we denote noise samples obtained by using the decisions. The signs of these two types of errors are ses = sign(es) and sed = sign(ed) are binary encoded (-1=>0 and +1=>1). Let us consider only three consecutive samples at error decision positions for a = 0.5. The numbers of these events (8 different sign patterns) are presented in Fig. 8, where the error pattern corresponds to the binary representation of the pattern number. Because of noise amplification at higher frequencies, errors ses(-+-,+-+) (error patterns of type 2 and 5) are converted to sed(—,+++) and therefore they have higher probabilities. To avoid error propagation, in the multi-stage PNC we should start repairing these locations. The K-stage PNC uses the following rule to get improved the nth sample w(n,m) after the stage m:

$$\begin{array}{l} w({n,m} )= y(n )+ \\ \mathop \sum \nolimits_{ep = 0}^{{2^{M + 1}} - 1} \mathop \sum \nolimits_{i = 1}^M [\beta ({i,m,ep} )e({n - i,m - 1} )+ \beta ({ - i,m,ep} )e({n + i,m - 1} )], \end{array}$$
where m = 1,…,K denotes the PNC stage and ep denotes an error pattern index from 0 to

 figure: Fig. 7.

Fig. 7. 1 + aD channel amplitude transfer function for different a value.

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 figure: Fig. 8.

Fig. 8. Error pattern distribution.

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2M + 1-1. The PNC is based on the following rules and steps:

  • 1. The total number of considered different noise patterns (sign binary notation) is 2M + 1.
  • 2. The parameter β depends on the noise pattern ep, the stage m, and the noise sample position + i (past error patterns) or -i (future error sample) relative to the current position n. The noise patterns can be divided into subsets in each stage to simplify the PNC realization.
  • 3. The PNC works on a long sample block (e.g., ASIC block) and a few first and last errors of ASIC block will not be updated, which introduces negligible BER degradation. Alternatively, ASIC blocks can be overlapped by a few symbols so that this degradation can be avoided.
  • 4. The parameters $\beta ({i,m,ep} )$ and $\beta ({ - i,m,ep} )$ can be different and should be carefully optimized. They depend on the channel transfer function and the noise spectrum after the FFE.

The main problem is to find the parameters M, K, and β. Although, this seems complicated, we provide a very simple PNC variant that can be used in most practical cases. The parameter α(i) in (6) can be derived by the procedure described in [22] (linear prediction theory). Normally, α(1) is the strongest coefficient while other coefficients can be neglected. In 1 + aD channels, the parameter a is equal to α(1) when M = 1 in (6). A very simple PNC variant can be derived by using the following steps:

  • • Consider 8 error patterns (M = 1) and 3-stage PNC (K = 3).
  • • Use the following rule to update the FFE outputs
    $$w({n,m} )= y(n )+ {\beta _B}({m,ep} )e({n + 1,m - 1} )+ {\beta _A}({m,ep} )e({n - 1,m - 1} )$$
  • • Stage 1 (m = 1) – Use (9) to update samples with error patterns $sign({e({n - 1,0} )} )= sign({e({n + 1,0} )} )$, where the errors are obtained after the FFE, and ${\beta _B} = {\beta _A} = \alpha (1 )/2$.
  • • Stage 2 (m = 2) – Use the Stage 1 decisions and errors to correct error patterns sign$({e({n - 1,0} )} )= sign({e({n + 1,0} )} )$ with ${\beta _B} = {\beta _A} = \alpha (1 )$. Correct the other error patterns $sign({e({n - 1,0} )} )\sim{=} sign(e({n + 1,0} )\; $ by
    $$if\; abs({e({n + 1,1} )} )\ge abs({e({n - 1,1} )} )$$
    $${\beta _B} = 0,\; \; {\beta _A} = \frac{{\alpha (1 )}}{2}$$
    else
    $${\beta _A} = 0,\; \; {\beta _B} = \frac{{\alpha (1 )}}{2}$$
    end i.e., add only the smaller error sample e weighted by $\alpha (1 )/2$ in (9).
  • • Stage 3 (m = 3) – Use the Stage 2 decisions and errors and repeat the Stage 2.

In the following text and figures, we simply use α to denote α(1). Due to its high efficiency and low complexity, we call the implementation above as “Golden” PNC (G-PNC). We applied the G-PNC to the case shown in Fig. 8. The FFE symbol error rate (SER) was 1e-2 but after the G-PNC Stage 1 it was lowered to 4.8e-3, i.e., SER was improved more than twice. The next G- PNC stage achieves SER of 3.7e-3 while the last stage provides SER = 3.2e-3. It is also visualized in Fig. 9 where one can see the significant improvement after the G-PNC Stage 1. The G-PNC Stage 2 further flattens SER per error patterns. The last stage generates almost equally probable error patterns. The effect of SER improvement is also visible in Fig. 10. The FFE histograms around the symbol levels (±1,±3) are significantly improved and one can observe fewer samples close to the thresholds (±2,0). The G-PNC effect in spectral domain is visualized in Fig. 11. The noise PSD after the FFE shows a clear high-frequency noise enhancement that is significantly improved after the G-PNC Stage 1. The G-PNC stages 2 and 3 improve SER but they also compress noise more at higher frequencies. Noise after the G-PNC is not white and there are some samples excursions that are a bit worse than before, however, the G-PNC improves SER by more than 3 times. Odd and even samples are used to generate a 2-dimensional (2D) constellation shown in Fig. 12 to highlight G-PNC effects. As one can see from Fig. 12, the FFE scattered samples are now much more compressed after the G-PNC. The G-PNC 2D symbols are not concentrated within cycles/balls but rather in ellipses that also says that the PNC is not perfect because we established very simple PNC rules (ASIC friendly). We observe that, if only Stage 1 and Stage 2 are implemented (simplified variant with also good performance) only two multiplications and quantizers are necessary. The best G-PNC configuration can be detected, e.g., by counting corrected FEC errors. It should be mentioned that we did not apply any optimization of β parameters in (8). To construct the PNC approaching the NC lower bound it is necessary to study the channel characteristic and consider noise added by the transmission system. Although the PNC concept and algorithms are discussed for noncoherent systems it is straightforward to apply this idea in coherent receivers. For example, in dual-polarization coherent systems the (multi-stage) equalizer compensates for many impairments such as chromatic dispersion, ISI, polarization mode dispersion etc. [28–31]. The equalizer also boosts noise at higher frequencies as the transmission channel is bandwidth-limited. With complex constellations, the PNC will use complex samples and β coefficients. The PNC rules can be established based on the concrete transmission scenario. In general, the PNC concept can be applied in any system to whiten colored noise.

 figure: Fig. 9.

Fig. 9. Errors after three G-PNC stages.

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 figure: Fig. 10.

Fig. 10. PAM4 histograms after FFE and G-PNC.

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 figure: Fig. 11.

Fig. 11. Noise spectrum after FFE and G-PNC.

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 figure: Fig. 12.

Fig. 12. 2D constellation.

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The G-PNC flow chart is shown in Fig. 13. In each stage, N symbols/samples are processed in parallel to improve N-2 decisions (“for loops” are used just for illustration purposes). High-speed DCN transceivers utilize high-level of parallelization (e.g., N = 100 symbols are processed simultaneously). Under the assumption that a single multiplication and quantization can be performed in one ASIC clock, the G-PNC processes requires 3 ASIC clocks and 2 multiplications per symbol. The MLSE processes independent blocks of N + M symbols (M symbols are overhead/decoding depth) and requires N + M ASIC clocks if the processing is symbol-by-symbol based [32]. The processing time can be shortened if the N symbols are divided into shorter blocks, but that adds more complexity as an overhead of M symbols must be processed before each short block. The 4-state PAM4 MLSE requires 16 multiplications per symbol and ∼(N + M)2/2 buffers. The 1-tap DFE can be realized, similar to the MLSE, by using independent blocks of N + M symbols (latency of N + M ASIC clocks), one multiplication, and ∼(N + M)2/2 buffers. The PAM4 DFE unfolding technique can also be used to decrease the latency, however, it requires enormous number of multiplexers (and large fanout) [33].

 figure: Fig. 13.

Fig. 13. G-PNC flow chart.

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4. PNC performance in simulations

In this section, we present simulation results in 1 + aD channels with a = 0.3, 0.4, 0.5, and 0.6. A unipolar PAM4 signal is first distorted by ISI and later by additive Gaussian noise. The FFE block uses 5 linear taps while the DFE equalizer is realized via a single feedback tap. One can complain about the complexity, i.e., the number of taps required for FFE and DFE, however, the real channel impulse response includes pre and postcursor. More FFE taps are also required to compensate for near (electrical) MPI as well as short optical link reflections. In general, the channel group delay is frequency dependent and the precursor compensation is necessary. The simulation results are shown in Fig. 14 for FFE, DFE (FFE-DFE), NC, G-PNC, and MLSE. The NC-lower bound (LB) is also presented in the same figure. The MLSE results are also added to indicate the best performance that can be achieved but with the maximum complexity (4-state Euclidian distance MLSE is used). We compared different schemes at KP4 FEC threshold equal to 2.4e-4. It is interesting that the NC LB is always below the MLSE BER that is also emphasized in Fig. 15, which shows the G-PNC gain vs other schemes at the KP4 FEC threshold. It also indicates the great potential of PNC as a multi-stage NC, i.e., there is high probability that an appropriate PNC exists for the specific transmission scenario. We presented the results of the G-PNC and NC with the optimum α(1) parameter value. The PNC results are quite good but it loses a race with the MLSE for larger values of a (visible in Fig. 15). For medium a values of 0.3 and 0.4, the simple NC always outperforms the FFE-DFE while for a = 0.5, their BERs are very similar. The FFE-DFE outperforms the NC only for a = 0.6 at higher signal-to-noise ratio (SNR) values, e.g., at KP4 FEC threshold. On the other side, the G-PNC beats the FFE-DFE for any a and SNR value. For a = 0.3, the G-PNC is better by 0.3 dB than the DFE while for larger a values this gap goes above 0.6 dB. For a = 0.4, the MLSE outperforms the G-PNC only by 0.25 dB and it grows to 0.4 dB for a = 0.5 while 1 dB difference is observed for a = 0.6. According to the current electrical and optical components bandwidth (ADC)/DAC, modulator, demodulator/photo detector), it is expected that in 100 Gbaud PAM4 optical systems the value of a will be between 0.4 and 0.5 where the PNC enables a gain of more than 1 dB after the FFE block that is not too far from the MLSE performance but better than the DFE and the gain is achieved with negligible complexity and latency. We have presented PAM4 simulation results, however, the same trend can be observed for any PAM modulation format except that for other modulation formats such as PAM2 or PAM8 the SNR values are different.

 figure: Fig. 14.

Fig. 14. Simulations results of 1 + aD channel for a value 0.3, 0.4, 0.5, and 0.6.

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 figure: Fig. 15.

Fig. 15. G-PNC gain at KP4 FEC threshold in 1 + aD channel for a value of 0.3, 0.4, 0.5, and 0.6.

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To get more realistic simulation results, we modeled our lab components and ran simulations for 106 Gbaud PAM4 modulation format. Our simulation setup is similar to the experimental setup shown in Fig. 17 that will be explained in more details later. We tuned some parameters such as components bandwidth to get similar a value in simulations and experiments. Some experimental imperfections such as MPI and nonlinear phase characteristics of components were not modeled. The performance of different equalization schemes, i.e., their BER values vs input power (Pin) are shown in Fig. 16(a) for the optimum α value, i.e., α(1) G-PNC value. Also, we indicated the KP4 FEC threshold at BER = 2.4e-4. The FFE-DFE enables 1 dB gain at the FEC threshold comparing to the FFE while the NC beats the FFE-DFE by 0.45 dB. The G-PNC and MLSE outperform the DFE by 0.95 dB. Compared to the G-PNC, the MLSE shows slightly better performance above the KP4 FEC threshold and a bit worse performance below this threshold. The NC-LB indicates that there is a room for the PNC performance improvement. The estimated value of α (unknown in practice) using three estimation methods is shown in Fig. 16(b). By using symbols to calculate errors similar values of α are derived by $\alpha = \frac{{E[{{e_k}{e_{k - 1}}} ]}}{{E[{{e_k}{e_k}} ]}}\; $ and by using 1-tap DFE (FFE-DFE). At high input power values, these estimations are similar, however, more noise makes them slightly different. On the other side, using the erroneous decisions underestimates α. Does it make problems to the G-PNC and NC? Figure 16(c) presents the G-PNC, NC, and NC-LB performance vs α at three Pin values. The NC-LB and G-PNC optimum α values are identical while the NC may suffer a bit when the optimum α value is not found (see, e.g., Pin = -4 dBm results). Blind α estimation using decisions does not degrade the G-PNC performance as the G-PNC BER curves are not very steep around the optimum values. For example, the blind α value at -7 dBm is equal to 0.327, the best value is 0.4, and BERs at for these two α values are 2.228e-3 and 2.221e-3.

 figure: Fig. 16.

Fig. 16. Simulation results. (a) BER vs Pin. (b) α estimation at different Pin values. (c) BER vs α.

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5. PNC performance in experiments

To verify the PNC in experiments, we carried out the 106 Gbaud PAM4 back-to-back (B2B) experiments with the experimental setup shown in Fig. 17 that also includes information about the bandwidth of the components. The main bandwidth limitations come from the DAC and a transmitter optical subassembly (TOSA) that integrates an electrical amplifier (driver) and an electro-absorption modulated laser (EML). The EML bias variations were compensated by a dc tap included in the FFE along with the linear taps. The O-band EML output signal is attenuated by a variable optical attenuator (VOA) and then amplified by a praseodymium-doped fiber amplifier (PDFA), which is necessary because the photo diode (PD) does not include an electrical amplifier (not commercially available transimpedance amplifiers at this speed). After resampling, timing recovery [6] is used to remove the clock offset and find the best sampling phase and then the signal is down-sampled to 1 sps. The FFE consists of 41 linear taps and a dc tap to control dc variations. The performance of different equalization schemes i.e., their BER values vs Pin are shown in Fig. 18(a) for the optimum α value (three sets of data with 1.7e6 symbols per set at different time are captured). The FFE-DFE achieves 2 dB gain at the FEC threshold comparing to the FFE while the NC beats the FFE-DFE by 0.4 dB. The G-PNC outperforms the FFE-DFE by 1 dB while the MLSE outperforms the G-PNC by 0.5 dB. At high Pin values, α is around 0.44 (see Fig. 18(b)) while in simulations it was 0.4 (see Fig. 16(b)). Likely, it was one of reasons why MLSE and G-PNC differ in experiments at the FEC threshold. Also, in simulations we modelled all components by Bessel filter of 4th order (smooth and friendly transfer function) while in experiments transfer functions of some components (especially DAC transfer function) show a significant roll-off close to Nyquist frequency. Compared to the G-PNC, the MLSE shows slightly better performance above the KP4 FEC threshold and a bit worse performance below this threshold. The NC-LB is close to the MLSE curve and indicates that there is a room for the PNC performance improvement. The value of α using three estimation methods are shown in Fig. 18(b) that is very similar to Fig. 16(b) but with some exceptions. For example, the α values derived by symbols are always larger than those equal to the DFE tap. Figure 18(c) presents the G-PNC, NC, and NC-LB performance vs α at three Pin values (not the same as in Fig. 16(c)). The NC-LB and G-PNC optimum α values are identical while again as seen in simulations the NC may suffer a bit if the optimum α value is not found (see, e.g., Pin = -2 dBm results). Blindly derived α parameter can be used in the G-PNC (optimization not necessary) without a noticeable performance degradation. Note that both in simulations and experiments, less noise (low BER) increases the optimum α value.

 figure: Fig. 17.

Fig. 17. 106 Gbaud PAM4 B2B experimental setup. DPD – digital predistortion, DAC – digital-to-analog converter, TOSA – transmitter optical subassembly, VOA – variable optical amplifier, PDFA – Praseodymium-doped fiber amplifier, PD – photo diode, BW – bandwidth.

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 figure: Fig. 18.

Fig. 18. Experimental results. (a) BER vs Pin. (b) α estimation at different Pin values. (c) BER vs α.

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6. Error analyses in experiments

In this Section, we studied errors behavior in experiments. Error analyses were done on one set of captured data (∼1.7e6 symbols) at Pin = -7 dBm. We use 2D constellations to show signals after the FFE, NC, G-PNC, and FFE-DFE that are presented in Fig. 19. The 2D FFE constellation shown in Fig. 19(a) shows that after the FFE the samples clouds are not confined in circles around the 2D symbols as it should be when the FFE noise is Gaussian. The FFE error spectrum, which is presented in Fig. 19(a), is quite irregular, i.e., not smooth as in Fig. 11 obtained with smooth 1 + aD transfer function. However, the FFE noise amplification at higher frequencies is well emphasized and can be partially repaired by the FFE-DFE confining 2D symbols within balls, as shown in Fig. 19(d). The FFE-DFE makes the output noise “flatter” (see Fig. 20(a)) although due to the quite irregular transfer function it is impossible to make it perfectly flat. The FFE-DFE balls produce less errors than the FFE ellipses but there are still a lot of errors close to the 2D thresholds. Also, there are some errors concentrated in irregular manner, e.g., between the (1,3) and (3,3) balls. The NC improves BER after the FFE and produces fewer errors than the FFE-DFE. The NC 2D ellipses, which are shown in Fig. 19(b) still suffer from strange noise regions like the FFE-DFE. But, the NC noise is over-suppressed at higher frequencies. The G-PNC error correction is the best and corresponds to very regular 2D ellipses in Fig. 19(c). The samples between ellipses are much more improved comparing to other techniques. The G-PNC noise spectrum, shown in Fig. 20(a), is also over-suppressed at higher frequencies as in the NC case, however, it produces the least errors. The 2D FFE symbols are elliptical due to noise enhancement at high frequencies (Fig. 19(a)), the 2D FFE-DFE symbols are less elliptical (noise is “whiter”), and the 2D NC/G-PNC symbols are elliptical due to strong noise suppression at high frequencies (opposite ellipticity to the FFE one). A cumulative sum (CS) of histogram bins of errors around thresholds (only samples producing symbol errors are considered; Fig. 20(a) shows the spectrum of errors derived as a difference between symbols and samples) are shown in Fig. 20(b). In this figure, the threshold is 0 and an error is calculated as a difference between the closest threshold and samples when a symbol error is generated. The last CS value is the number of erroneous symbols that also indicates the best algorithm (G-PNC). When the absolute error value increases above 0.4 (Fig. 20(b) only shows the case when the error value is less than -0.4; the optimum thresholds are placed around -2, 0, and 2 with the optimum levels close to -3, -1, 1, and 3) the FFE-DFE and G-PNC generate more errors than the FFE. For example, at bin value of -0.625 the FFE-DFE generates 76 errors, the G-PNC delivers 57 errors, and the FFE has only 20 errors. Large FFE-DFE outliers are caused by erroneous FFE-DFE decisions used in the feedback loop while large G-PNC outliers are generated by incorrect G-PNC input noise samples used in the noise decorrelation algorithm. It may cause some problems to soft FEC codes (large outliers and error burst statistics) that will be a subject of future investigations.

 figure: Fig. 19.

Fig. 19. 2D constellations in experiment at -7 dBm. (a) FFE. (b) NC. (c) G-PNC. (d) FFE-DFE.

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 figure: Fig. 20.

Fig. 20. Experiment at -7dBm. (a) Error power spectrum density (PSD). (b) Cumulative sum of histogram bins of errors around thresholds.

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7. Conclusion

A novel equalization algorithm called G-PNC is developed to whiten noise after the FFE equalizer. Its performance is approaching that of MLSE in medium ISI channels where low latency, and low complexity make the G-PNC a serious candidate for DCN receivers.

Disclosures

The authors declare that there are 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.

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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 (20)
Fig. 1.
Fig. 1. Basic equalization techniques. Q – quantizer.

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Fig. 2.
Fig. 2. FFE-DFE.

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Fig. 3.
Fig. 3. FFE taps and impulse response in experiments.

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Fig. 4.
Fig. 4. 41 FFE taps with and without 1-tap DFE in experiments.

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Fig. 5.
Fig. 5. Simple noise cancellation.

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Fig. 6.
Fig. 6. Multi-stage PNC.

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Fig. 7.
Fig. 7. 1 + aD channel amplitude transfer function for different a value.

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Fig. 8.
Fig. 8. Error pattern distribution.

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Fig. 9.
Fig. 9. Errors after three G-PNC stages.

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Fig. 10.
Fig. 10. PAM4 histograms after FFE and G-PNC.

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Fig. 11.
Fig. 11. Noise spectrum after FFE and G-PNC.

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Fig. 12.
Fig. 12. 2D constellation.

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Fig. 13.
Fig. 13. G-PNC flow chart.

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Fig. 14.
Fig. 14. Simulations results of 1 + aD channel for a value 0.3, 0.4, 0.5, and 0.6.

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Fig. 15.
Fig. 15. G-PNC gain at KP4 FEC threshold in 1 + aD channel for a value of 0.3, 0.4, 0.5, and 0.6.

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Fig. 16.
Fig. 16. Simulation results. (a) BER vs Pin. (b) α estimation at different Pin values. (c) BER vs α.

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Fig. 17.
Fig. 17. 106 Gbaud PAM4 B2B experimental setup. DPD – digital predistortion, DAC – digital-to-analog converter, TOSA – transmitter optical subassembly, VOA – variable optical amplifier, PDFA – Praseodymium-doped fiber amplifier, PD – photo diode, BW – bandwidth.

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Fig. 18.
Fig. 18. Experimental results. (a) BER vs Pin. (b) α estimation at different Pin values. (c) BER vs α.

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Fig. 19.
Fig. 19. 2D constellations in experiment at -7 dBm. (a) FFE. (b) NC. (c) G-PNC. (d) FFE-DFE.

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Fig. 20.
Fig. 20. Experiment at -7dBm. (a) Error power spectrum density (PSD). (b) Cumulative sum of histogram bins of errors around thresholds.

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Equations (12)

Equations on this page are rendered with MathJax. Learn more.

y ( k ) = w d c + ∑ r = 1 p ⁡ ∑ k 1 = 0 M − 1 ⁡ ⋯ ∑ k r = k r − 1 M − 1 ⁡ w r ( k 1 , k 2 , … , k r ) × x ( k 1 − k 1 ) ⋯ x ( k 1 − k r ) ,
y ( n ) = ∑ i = − N N ⁡ a ( i ) x ( n − i ) ,
a j ( k + 1 ) = a j ( k ) + μ e k x ( k ) , e k = d k − y k ,
e k = y k − γ s g n ( y k ) , γ = E [ s 2 ] E [ s ] ,
y ( n ) = ∑ i = − N 0 ⁡ a ( i ) x ( n − i ) + ∑ i = 1 K ⁡ b ( i ) d ( n − i ) ,
z ( n ) = ∑ i = 0 M ⁡ α ( i ) y ( n − i ) = y ( n ) + ∑ i = 1 M ⁡ α ( i ) y ( n − i ) ,
w ( n ) = y ( n ) + ∑ i = 1 M ⁡ [ β ( i ) e ( n − i ) + β ( − i ) e ( n + i ) ] ,
w ( n , m ) = y ( n ) + ∑ e p = 0 2 M + 1 − 1 ⁡ ∑ i = 1 M ⁡ [ β ( i , m , e p ) e ( n − i , m − 1 ) + β ( − i , m , e p ) e ( n + i , m − 1 ) ] ,
w ( n , m ) = y ( n ) + β B ( m , e p ) e ( n + 1 , m − 1 ) + β A ( m , e p ) e ( n − 1 , m − 1 )
i f a b s ( e ( n + 1 , 1 ) ) ≥ a b s ( e ( n − 1 , 1 ) )
β B = 0 , β A = α ( 1 ) 2
β A = 0 , β B = α ( 1 ) 2
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