Gradient-descent noise whitening techniques for short-reach IM-DD optical interconnects with severe bandwidth limitation

Qi Wu; Qi Wu; Zhaopeng Xu; Zhaopeng Xu; Yixiao Zhu; Yixiao Zhu; Honglin Ji; Yu Yang; Gang Qiao; Lulu Liu; Shangcheng Wang; Junpeng Liang; Jinlong Wei; Zhixue He; Qunbi Zhuge; Weisheng Hu; Weisheng Hu

doi:10.1364/OE.509262

1. Introduction

The rapid expansion of broadband applications across various domains including the Metaverse, video streaming services, and the Internet of Things, has led to a significant increase in data traffic within and between data centers. How to cope with the surge of traffic has become a major challenge for data center interconnects (DCIs), necessitating innovative solutions to meet the growing demand. In the realm of short-reach DCIs, optical interconnects employing intensity modulation and direct detection (IM-DD) have emerged as the primary solution over the past decade [1–11]. These IM-DD systems are recognized for their cost-effectiveness, low power consumption, and seamless integration. For 400 Gigabit Ethernet (GbE), established configurations encompass either 8 lanes of 50 Gbit/s or 4 lanes of 100 Gbit/s 4-level pulse-amplitude modulation (PAM-4) signals, all operating within the O-band [12]. As the next generation of Ethernet links aims for 800-GbE, or potentially even 1.6-TbE [13], there is a heightened emphasis on elevating single-wavelength data rates. However, achieving a single-lane data rate of 200 Gbit/s and beyond presents challenges for IM-DD optical interconnects [14–16], particularly when utilizing commercial components with limited bandwidth. Furthermore, a critical concern for IM-DD systems is the phenomenon of chromatic dispersion-induced power fading [17]. This phenomenon leads to frequency-selective dips and severe inter-symbol interference (ISI), which can significantly impact the overall system performance.

Although the ISI resulting from bandwidth restrictions and fiber dispersion can be alleviated through receiver-side channel equalization techniques such as linear feedforward equalization (FFE), these compensation approaches inevitably amplify noise while rectifying high-frequency attenuation. This leads to the emergence of equalization-enhanced colored noise (EECN) and subsequent degradation of system performance [18–21]. To circumvent this predicament, strategies like pre-equalization or pre-emphasis can be employed on the transmitter side, proving effective in mitigating the noise amplification associated with post-equalization techniques [22–25]. By distributing substantial power to high-frequency components, pre-emphasis counteracts the issue of excessive power loss. However, pre-emphasis can lead to an increase in the peak-to-average power ratio of the signal, which consequently results in a compromise on the effective resolution of the digital-to-analog converter. Moreover, their effectiveness is limited when dealing with severe bandwidth constraints, such as a brick-wall frequency-amplitude response, which entirely eliminates frequency components beyond the device's bandwidth. To address the challenges posed by severe bandwidth limitations and power fading, the integration of FFE and decision feedback equalizer (FFE-DFE) [26–28] is shown to avoid the EECN. In [29], a modification of a DFE with zero-forcing equalization is proposed to serve as a noise whitening filter (NWF) to track the fast variations of the channels. However, the schemes originating from the DFE are susceptible to error propagation, leading to relatively unstable performance. As a solution, Tomlinson-Harashima precoding has been implemented in IM-DD optical interconnects at the transmitter side to prevent error propagation, albeit at the cost of precoding loss [30–32]. However, compared with the pre-processing, receiver-only digital signal processing (DSP) techniques are more suitable for co-packaged optics because they allow for the direct driving of the optical transmitter using the output of switch Serializer/Deserializer.

From the perspective of DSP techniques at the receiver, the concept of direct detection faster than Nyquist (DD-FTN) has been proposed and explored within bandwidth-limited IM-DD systems [33,34]. This technique involves a digital one-tap post-filter and maximum likelihood sequence estimation (MLSE) [35–37] to demodulate the signal. The post-filter, positioned after the equalizer to suppress in-band EECN as an NWF, is employed alongside MLSE to decode the symbol sequence, accounting for artificially introduced ISI from the post-filter. However, the effectiveness of the one-tap post-filter is limited, and improving its impact on noise whitening (NW) by increasing linear taps to better approximate the channel response is beneficial. The coefficients of these finite impulse response (FIR) taps can be obtained through Yule-Walker equations [38], a method rooted in auto-regressive modeling. Similarly, these tap coefficients are also utilized in the MLSE or maximum a posteriori probability (MAP) decoder. The introduction of a multi-tap noise whitening significantly enhances the resistance to bandwidth limitations, enabling ultra-high baud-rate faster-than-Nyquist (FTN) [39–43] signaling. In [40], a transmission experiment involving 570-GBd on-off-keying (OOK) and 300-GBd PAM-4 signals under a 100-GHz brick-wall bandwidth is conducted using multi-tap NW and MAP decoder to approach the Shannon limit.

In this paper, we propose a gradient-descent NW (GD-NW) algorithm to address the severe bandwidth limitations in IM-DD systems with receiver-only processing. By formulating the cost function and calculating the gradient for updating the tap coefficients of the FIR filter, the system performance can be enhanced. Furthermore, we extend this approach by introducing nonlinear kernels to further optimize performance, addressing both EECN and nonlinear impairments simultaneously. This extended approach is referred to as gradient-descent nonlinear NW (GD-NNW). As a proof-of-concept, we validate the proposed GD-NW and GD-NNW in bandwidth-limited IM-DD experiments, enabling 360-GBd OOK and 180-GBd PAM-4 back-to-back (BTB) signaling under a 62-GHz brick-wall bandwidth limitation. 280-GBd OOK and 150-GBd PAM-4 signals are also transmitted over 1-km standard single-mode fiber (SSMF) with a bit error rate (BER) below 7% hard-decision forward error correction (HD-FEC) to validate the effectiveness of the proposed algorithms in the presence of bandwidth limitation and power fading. Compared with GD-NW, the receiver optical power (ROP) sensitivity has been enhanced by 0.5 dB and 1 dB for OOK and PAM-4 signaling, respectively, with the help of GD-NNW. The rest of the paper is organized as follows. In Section 2, the principle and concept of GD-NW and GD-NNW approaches are introduced in detail. In Section 3, high-speed experiments are conducted, and the performance of the proposed methods is analyzed and compared with traditional algorithms. Finally, Section 4 provides a summary of the paper.

2. Principle of gradient-descent linear and nonlinear noise whitening

In this section, we present the theoretical foundation of the proposed GD-NW and GD-NNW algorithms. We delve into the process of constructing the cost function, calculating the gradient, and updating the NWF.

Fig. 1. Schematic of proposed gradient-descent noise whitening algorithm. r(n): The received signals; $\hat{y}$ (n): The equalized symbols free from ISI; x(n): The transmitted symbols; O(n): The output of NW filter; TS(n): The training symbols undergoing NW filter.

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Figure 1 illustrates the underlying principle of proposed GD-NW and GD-NNW techniques. Typically, FFE is employed to counteract the severe bandwidth limitation in the IM-DD system. However, the implementation of FFE can result in an increased noise level within the high-frequency domain, particularly when applied to FTN transmission subject to a brick-wall electrical bandwidth constraint. For the purpose of our discussion, we consider a scenario where the number of taps in the FFE is ample to manage ISI. Thus, the equalized signal can be represented as follows:

(1)$$\hat{y}(n) = x(n) + \hat{n}(n), $$

where $\hat{y}$ (n) represents the equalized symbols free from ISI, x(n) denotes the transmitted symbols, and $\hat{n}$ (n) signifies the EECN that requires whitening. To alleviate the impact of EECN, a joint of post-filter and MLSE is proposed in [33], known as DD-FTN. The post-filter functions as an NWF, positioned after the FFE to introduce controlled ISI and smooth the noise spectrum. Its Z transformation is expressed as:

(2)$${H_{DD - FTN}}(Z) = 1 + \alpha {Z^{ - 1}}. $$

However, the potential of this single-tap post-filter to effectively whiten noise becomes limited as the bandwidth constraints become more severe. Therefore, to achieve more robust noise whitening, the implementation of a multi-tap filter becomes necessary. In practical IM-DD systems (or coherent detection), in addition to linear effects, nonlinear impairments are often present due to factors like nonlinearity originating from devices such as modulators and photodetectors (PDs) [44–46]. Therefore, we incorporate nonlinear kernels, representing controlled nonlinear ISI, into the NW filter to address the challenge posed by these nonlinear impairments. The equalized symbols are passed through the nonlinear NWF, which can be mathematically expressed as:

(3)$$O(n) = \hat{y}(n) + \sum\limits_{{k_L} = 1}^M {{\alpha _{{k_L}}}\hat{y}(n - {k_L})} + \sum\limits_{{k_{NL}} = 0}^L {{\beta _{{k_{NL}}}}{{\hat{y}}^2}(n - {k_{NL}})}, $$

${\alpha _{{k_L}}}\textrm{ }({k_L} = 1,2, ... ,M)$ and ${\beta _{{k_{NL}}}}\textrm{ }({k_{NL}} = 0,1, ... ,L)$are the tap coefficients of linear and nonlinear kernels, respectively. M and L denote the number of linear and nonlinear taps, respectively. If ${\beta _{{k_{NL}}}}\textrm{ = 0 (}{k_{NL}} = 0,1, ... ,L)$, the NWF only takes the linear ISI into consideration, denoted as GD-NW. If the weights of nonlinear kernels are not equal to zeros, the nonlinear ISI can also be addressed, which is called GD-NNW in this paper. Note that the nonlinear terms reflected in Eq. (3) could be extended to cubic or absolute terms to provide a more accurate estimation of the nonlinear ISI. Subsequently, the transmitted symbols, often referred to as training sequences, also undergo the NW process, which can be represented as:

(4)$$TS(n) = x(n) + \sum\limits_{{k_L} = 1}^M {{\alpha _{{k_L}}}x(n - {k_L})} + \sum\limits_{{k_{NL}} = 0}^L {{\beta _{{k_{NL}}}}{x^2}(n - {k_{NL}})}. $$

Next, we define the error as the difference between the output at time n and the corresponding value in the training sequence, given by

(5)$$\begin{array}{l} e(n) = O(n) - TS(n) = \left[ {\hat{y}(n) + \sum\limits_{{k_L} = 1}^M {{\alpha_{{k_L}}}\hat{y}(n - {k_L})} + \sum\limits_{{k_{NL}} = 0}^L {{\beta_{{k_{NL}}}}{{\hat{y}}^2}(n - {k_{NL}})} } \right] - \\ \quad \quad \quad \quad \quad \quad \quad \quad \;\left[ {x(n) + \sum\limits_{{k_L} = 1}^M {{\alpha_{{k_L}}}x(n - {k_L})} + \sum\limits_{{k_{NL}} = 0}^L {{\beta_{{k_{NL}}}}{x^2}(n - {k_{NL}})} } \right]\\ \quad \quad \quad \quad \quad \quad \quad \;\;\, = \hat{n}(n) + \sum\limits_{{k_L} = 1}^M {{\alpha _{{k_L}}}\hat{n}(n - {k_L})} + \sum\limits_{{k_{NL}} = 0}^L {{\beta _{{k_{NL}}}}{{\hat{n}}^2}(n - {k_{NL}})} \end{array}$$

The cost function is defined as the mean squared error (MSE), calculated as:

(6)$$MSE = {\mathbb E}\{{e{{(n)}^2}} \}$$

where ${\mathbb E}\left\{ \cdot \right\}$ denotes expectation. To minimize the MSE, in this study, we instantiate the algorithm using gradient descent to estimate the tap coefficients of the NWF. The gradients of error at time n with respect to $\boldsymbol{\mathrm{\alpha}} = {[{\alpha _1},{\alpha _2}, ... ,{\alpha _M}]^T}$and $\boldsymbol{\mathrm{\beta}} = {[{\beta _0},{\beta _1}, ... ,{\beta _L}]^T}$ are calculated as:

(7)$$\begin{array}{l} \frac{{\partial e{{(n)}^2}}}{{\partial \boldsymbol{\mathrm{\alpha}}}} = {[\frac{{\partial e{{(n)}^2}}}{{\partial {\alpha _1}}},\frac{{\partial e{{(n)}^2}}}{{\partial {\alpha _2}}}, ... ,\frac{{\partial e{{(n)}^2}}}{{\partial {\alpha _M}}}]^T}\\ \quad \quad \quad {\kern 1pt} {\kern 1pt} \;{\kern 1pt} = 2({O(n) - TS(n)} )\times {[\hat{n}(n - 1),\hat{n}(n - 2), ... ,\hat{n}(n - M)]^T}\\ \frac{{\partial e{{(n)}^2}}}{{\partial \boldsymbol{\mathrm{\beta}}}} = {[\frac{{\partial e{{(n)}^2}}}{{\partial {\beta _0}}},\frac{{\partial e{{(n)}^2}}}{{\partial {\beta _1}}}, ... ,\frac{{\partial e{{(n)}^2}}}{{\partial {\beta _L}}}]^T}\\ \quad \quad \quad {\kern 1pt} {\kern 1pt} \;{\kern 1pt} = 2({O(n) - TS(n)} )\times {[{{\hat{n}}^2}(n),{{\hat{n}}^2}(n - 1), ... ,{{\hat{n}}^2}(n - L)]^T} \end{array}. $$

Finally, the tap coefficients in the (n + 1)-th iteration are updated using

(8)$$\begin{array}{l} {\boldsymbol{\mathrm{\alpha}}^{(n + 1)}} = {\boldsymbol{\mathrm{\alpha}}^{(n)}} - \mu \frac{{\partial e{{(n)}^2}}}{{\partial {\boldsymbol{\mathrm{\alpha}}^{(n)}}}} = {\boldsymbol{\mathrm{\alpha}}^{(n)}} - \mu \times 2({O(n) - TS(n)} )\times {[\hat{n}(n - 1),\hat{n}(n - 2), ... ,\hat{n}(n - M)]^T}\\ {\boldsymbol{\mathrm{\beta}}^{(n + 1)}} = {\boldsymbol{\mathrm{\beta}}^{(n)}} - \mu \frac{{\partial e{{(n)}^2}}}{{\partial {\boldsymbol{\mathrm{\beta}}^{(n)}}}}\\ \quad \quad {\kern 1pt} = {\boldsymbol{\mathrm{\beta}}^{(n)}} - \mu \times 2({O(n) - TS(n)} )\times {[{{\hat{n}}^2}(n),{{\hat{n}}^2}(n - 1), ... ,{{\hat{n}}^2}(n - L)]^T} \end{array}, $$

where μ represents the step size of the gradient descent method. The cost function of MSE is minimized when the noise samples become colorless. Once the tap coefficients are obtained using this approach, they will all be inputted into the conventional decoder for sequence detection, including the MLSE and MAP decoder. To decode the symbol sequence while taking into account both the introduced linear and nonlinear ISI, modifications are required for the conventional MLSE decoder and MAP decoder to accommodate the nonlinear branch metric [22]. The adaptation and flexibility of GD-NNW enable the processing of EECN and nonlinear impairments in a simultaneous manner. It is noteworthy that, for practical optical transceivers, the linear and nonlinear tap coefficients can often be calculated in advance due to the relatively stable characteristics of device bandwidth and channel response. As such, the computational complexity of acquiring the tap coefficients can be omitted for a fixed IM-DD transmission scenario. The dominant computational complexity comes from the MLSE or MAP decoding. The complexity of the MLSE and NW filters is briefly outlined as follows: Each N-level PAM symbol involves 2(L + 1)+M multiplications and M + L + 1 additions in the NW process, which can be implied from Eq. (3). Furthermore, based on the analysis in [47], the conventional MLSE necessitates (M + L + 1)×N^M^+ L^+ 1 multiplications, (M + L + 3)×N^M^+ L^+ 2 additions, and N^M^+ L^+ 1 comparisons. It is noteworthy that some methods can be utilized to reduce the state of MLSE [47] and MAP [18,19]. The complexity grows linearly rather than exponentially with M and L [18,19]. In addition, they can also be performed using the look-up-tables [36,44], which are beneficial for practical implementation.

3. Experiment setup and results

3.1 Experimental setup and DSP procedures

In this section, we experimentally validate the principle of the proposed GD-NW algorithm for the IM-DD system operating under severe bandwidth limitations. Figure 2(a) depicts the experimental setup, while the DSP procedures executed at the transmitter and receiver sides are outlined in Fig. 2(b). 2¹⁷ random bits are generated in MATLAB and mapped to OOK/PAM-4 symbols. After passing through a pulse shaping filter with a roll-off factor of 0.01, the signals are resampled to align with the sampling rate of an arbitrary waveform generator (AWG, Keysight 8199B), operating at 224 GSa/s. The AWG's electrical output drives a Mach-Zehnder modulator (MZM) with a 50-GHz 3-dB bandwidth, modulating the light from an external cavity laser with a linewidth of ∼100 kHz. The relationship between the optical power of the MZM's output and the direct component bias is illustrated in Fig. 2(c) and the MZM is biased at a quadrature point of around 4.1 V. Thus, the power of the optical signal is about 4 dBm and its spectrum is shown in Fig. 2(d). After transmission through 1-km SSMF, the optical signal is detected by a PD (Keysight N4377A, 60-GHz), and the resulting electrical signal is digitalized by a real-time oscilloscope (RTO, Keysight UXR0594AP) with a sampling rate of 256 GSa/s.

Fig. 2. (a) Experimental setup. RRC: Root-raised cosine; AWG: Arbitrary waveform generator; MZM.: Mach-Zehnder modulator; SSMF: Standard single-mode fiber; VOA: Variable optical attenuator; PD: Photodetector; RTO: Real-time oscilloscope. (b) DSP procedures implemented at the transmitter and receiver-side. (c) Output power versus DC bias of MZM. (d) Optical spectrum of the signal.

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In the receiver DSP, the digitalized signals are resampled to 2 samples-per-symbol and then synchronized using the frame head. After the matched filtering, a linear FFE with 401 taps is employed to ensure the eradication of ISI, and the equalized samples are down-sampled to one sample-per-symbol. The tap coefficients of the equalizer are updated based on 1024 training symbols utilizing the recursive least square algorithm. Fewer taps result in decreased computational complexity with an acceptable penalty on BER performance. When implementing noise whitening, we can strike a balance between computational complexity and system performance by tuning the number of FFE taps. Subsequently, the proposed GD-NW and GD-NNW algorithm are used to deal with the EECN, and the introduced ISI by the NWF is handled by the MLSE decoder. Finally, the BER is calculated by averaging ∼7.8 × 10⁵ bits.

3.2 PAM-4 in the BTB case

We first validate the principle of the proposed GD-NW using PAM-4 modulation in the BTB configurations, where the bandwidth limitation is less severe. The received electrical spectrum of the 150-GBd PAM-4 signal is shown in Fig. 3(a), and the channel response of our experimental setup is depicted in Fig. 3(b), illustrating that the 3-dB bandwidth is about 50 GHz and the brick-wall bandwidth limitation for this transceiver is approximately 62 GHz. The 62-GHz electrical bandwidth is challenging for transmitting a 150-GBd signal, leading to the emergence of EECN due to the operation of the FFE, particularly within the frequency range of 62 GHz to 75 GHz. The key parameters of GD-NW encompass the step size μ, the required number of training symbols (i.e., the number of iterations), and the number of NWF taps M on the NW performance. The measured BER results as a function of the number of training symbols are displayed in Fig. 3(c), which show the gradual enhancement of BER performance with the increasing of training symbols since the tap coefficients are approaching the optimum values for NW. The number of the filter taps M is set as 7 here. Then, we focus on the step size. Figure 3(c) indicates a step size < 5E-3 results in the slow convergence of this algorithm, necessitating an increased number of training symbols and introducing frame redundancy. Conversely, a step size beyond 1E-1 causes challenges for the gradient descent algorithm to locate the global optimum, leading to less stable system performance. Consequently, in subsequent investigations, we set the step size to 1E-2 and employ 1200 training symbols to estimate the tap coefficients.

Fig. 3. (a) Received electrical spectrum of 150-GBd PAM-4 signal in the BTB case. (b) Channel response of the experimental setup in the BTB case. (c) Measured BERs as a function of number of training symbols.

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Moving forward, we proceed to evaluate the impact of the number of taps on the NW performance. Figure 4(a) shows the measured BERs as a function of M, which represents the number of taps utilized in the GD-NW algorithm. The results for various baud rates are also plotted in Fig. 4(a) to explore the effect of bandwidth limitation severity on the optimal number of taps. It is evident that a higher number of taps results in improved NW performance and consequently, a lower BER is achieved. It's worth mentioning that M = 1 corresponds to the performance of traditional one-tap post-filter, namely the DD-FTN algorithm. We also observe that higher baud rates necessitate a greater number of taps to effectively whiten the EECN. This requirement arises due to the more severe bandwidth limitation associated with higher baud rates. In other words, as the baud rate increases, the constraints posed by bandwidth limitations become more pronounced, prompting the need for a larger number of tap coefficients to mitigate the effects of EECN and achieve effective noise whitening. We perform a comprehensive comparison between GD-NW and traditional algorithms designed to counteract bandwidth limitations, specifically FFE-DFE and DD-FTN, across various baud rates shown in Fig. 4(b). Under a 7% HD-FEC of 3.8E-3, the maximum achievable baud rates for DD-FTN and FFE-DFE are constrained to 150 GBd and 153 GBd respectively. In contrast, through the application of GD-NW with different numbers of taps (M = 3/5/7), significant improvements are achieved, resulting in maximum attainable baud rates of 168/177/180 GBd respectively.

Fig. 4. (a) Measured BERs as a function of number of taps M for different baud rates with PAM-4 modulation. (b) Measured BERs as a function of baud rate using different algorithms with PAM-4 modulation. (c) Noise spectra of 150-GBd PAM-4 signal using GD-NW with M = 0/1/3/5/7. (d) Tap coefficients of GD-NW (M = 1/3/5/7) for 150-GBd PAM-4 signal. (e) Noise spectra of 180-GBd PAM-4 signal using GD-NW with M = 0/1/3/5/7. (f) Tap coefficients of GD-NW (M = 1/3/5/7) for 180-GBd PAM-4 signal.

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To intuitively demonstrate the NW capability of the proposed GD-NW, we plot the noise spectra with and without GD-NW (M = 1/3/5/7) in Fig. 4(c). The black line represents the scenario where no NW is applied, and it is evident that the noise beyond 62 GHz is notably amplified by the FFE. The DD-FTN scheme (M = 1) showcases limited NW capacity and fails to fully flatten the noise spectrum. With the increasing number of taps in GD-NW, the noise spectrum becomes progressively more level. The corresponding tap coefficients of GD-NW are shown in Fig. 4(d). The noise spectra of a 180-GBd PAM-4 signal, obtained through different NW schemes, are visually presented in Fig. 4(e). It shows the NWF with M = 7 performs the best to flatten the noise spectrum. Furthermore, the corresponding tap coefficients are depicted in Fig. 4(f), providing insights into the behavior and adjustment of the GD-NW algorithm in the presence of bandwidth limitations. The results imply that the tap coefficients get smaller with the tap index increasing since the remote symbols have less impact on the current symbol than the adjacent symbols.

3.3 OOK in the BTB case

In this subsection, we transition from PAM-4 to OOK modulation to further enhance the baud rate. Consequently, the bandwidth limitation becomes more severe, resulting in significant FTN signaling. Similar to the operations for PAM-4 signals, we also investigate the required number of taps under different baud rates for OOK modulation. The measured BERs as a function of the number of taps M are shown in Fig. 5(a). With the increase of taps, the BER performance gets better. When the baud rate gradually increases from 300 to 400 GBd, more taps are required to flatten the noise spectrum. Utilizing more taps in GD-NW implies that the decoder needs to accommodate more severe ISI, resulting in increased computational complexity and hardware resource demands. For transmitting a 300-GBd OOK signal, a 12-tap GD-NW yields the optimal BER performance with additional taps not contributing to improvement. Compared with PAM4, more taps are required here for OOK due to the more pronounced bandwidth limitation. Figure 5(b) provides the measured BERs as a function of the baud rate. Additionally, the performance of the traditional FFE, FFE-DFE, and DD-FTN schemes is included for a simple comparison. Notably, it can be observed that FFE, FFE-DFE, and DD-FTN schemes fail to achieve the desired BER threshold of 7% HD-FEC. In contrast, the proposed GD-NW with different numbers of taps (M = 5/10/15) demonstrates enhanced performance. Specifically, the maximum baud rates achieved are 320, 340, and 360 GBd respectively for GD-NW with M = 5/10/15. Under a baud rate of 300 GBd, the noise spectra using GD-NW with M = 1/5/10/15 are depicted in Fig. 5(c). However, it's challenging to flatten the noise around the frequency of 62 GHz due to the difficulty in approximating the cut-off edge of the brick-wall channel response, which is consistent with findings from Ref. [18]. The corresponding tap coefficients are also displayed in Fig. 5(d). Under a baud rate of 360 GBd, Fig. 5(e) displays the noise spectra using GD-NW with varying tap configurations (M = 1/5/10/15). It shows a 15-tap linear FIR filter enables the NW. Additionally, the tap coefficients are displayed in Fig. 5(f). Figure 5(f) reveals that as the index increases, the tap weights approach zero, indicating that further taps do not contribute to significant performance improvement.

Fig. 5. (a) Measured BERs as a function of number of taps M for different baud rates with OOK modulation. (b) Measured BERs as a function of baud rate using different algorithms with OOK modulation. (c) Noise spectra of 300-GBd OOK signal using GD-NW with M = 0/1/5/10/15. (d) Tap coefficients of GD-NW (M = 1/5/10/15) for 300-GBd OOK signal. (e) Noise spectra of 360-GBd OOK signal using GD-NW with M = 0/1/5/10/15. (f) Tap coefficients of GD-NW (M = 1/5/10/15) for 360-GBd OOK signal.

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3.4 1-km SSMF transmission and the performance of GD-NNW

In the previous BTB experiment, we primarily address transceiver bandwidth limitations. However, it's important to note that chromatic dispersion-induced power fading also contributes significantly to overall system impairments, resulting in a reduction in achievable baud rates. In this subsection, we evaluate the resilience of GD-NW and GD-NNW under a bandwidth constraint caused by both the transceiver and the fiber channel. For this investigation, we maintain OOK and PAM-4 modulation baud rates at 280 and 150 GBd, respectively. The received electrical spectra are displayed in Fig. 6(a), revealing a frequency notch at approximately 55 GHz due to the power fading effect. In addition to the frequency dip, this phenomenon leads to a gradually attenuated frequency response from the baseband to the high-frequency region. Consequently, equalization amplifies noise in the attenuated frequency region, resulting in more pronounced EECN. Similarly, we investigate the required number of training symbols and the optimal step size for the proposed GD-NNW. For 150-GBd PAM-4 modulation, we set M as 7, while L is set to 2 to reduce the computational complexity in the subsequent decoding process. Figure 6(b) displays measured BERs as a function of the number of training symbols under different step sizes. Comparing these results with those in Fig. 3(c) in the BTB case, we find that the optimal step size remains around 1E-2. However, due to the introduction of nonlinear kernels and the more severe nonlinear impairments, GD-NNW requires a larger number of training symbols to calculate the tap coefficients. We also examine the measured BERs versus the number of training symbols with different step sizes for 280-GBd OOK modulation where M = 15 and L = 2. The results are shown in Fig. 6(c). It is evident that OOK modulation demands fewer training symbols to ensure convergence since the operation for OOK is simpler than PAM-4 modulation, resulting in a faster convergence speed. Therefore, to guarantee the performance of GD-NNW in the 1-km transmission case, we employ 2000 training symbols and a step size of 1E-2. These adjustments in the number of training symbols and step size are essential to ensure the effectiveness of GD-NNW in the 1-km transmission scenario.

Fig. 6. (a) Received electrical spectrum of 300-GBd OOK signal in the 1-km SSMF case. (b) Measured BERs as a function of number of training symbols for 150-GBd PAM-4 signal. (c) Measured BERs as a function of number of training symbols for 280-GBd OOK signal.

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Subsequently, we investigate the improvements achieved by GD-NNW in comparison to traditional DSP techniques and the proposed GD-NW. Figure 7(a) presents measured BERs as a function of the ROP for 280-GBd OOK signaling. In this configuration, we set M as 15, similar to GD-NW. The number of nonlinear kernels, L, is set to 2 to strike a balance between NW performance and computational complexity. The results indicate that only GD-NW (M = 7) and GD-NNW (M = 7, L = 2) can reach the BER threshold of 7% HD-FEC. By incorporating nonlinear kernels, GD-NNW accounts for the presence of nonlinear impairments during noise whitening. Notably, the introduction of nonlinear kernels results in an improvement of over 0.5 dB in terms of ROP sensitivity, surpassing the benefits of GD-NW alone. However, as the ROP decreases from 3 dBm to -5 dBm, the improvements provided by both GD-NW and GD-NNW become less significant. This is because, in the low-ROP region, the dominant factor affecting system performance is the electrical noise from the PD and RTO. Figure 7(b) provides insight into the tap coefficients of GD-NNW for 280-GBd OOK modulation. The blue curve represents the weights of the linear kernels (α), while the red curve represents the weights of the nonlinear kernels (β). The results show that the weights of the nonlinear kernels are considerably less than those of the linear kernels. This is primarily because linear ISI has a more significant impact than nonlinear ISI on the system performance. Nevertheless, introducing several nonlinear kernels is indeed beneficial in improving the entire system performance, as shown in Fig. 7(a).

Fig. 7. (a) Measured BERs as a function of ROP with different NW schemes for 280-GBd OOK signal. (b) Tap coefficients of GD-NNW (M = 15, L = 2) for 280-GBd OOK signal.

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Finally, we validate the effectiveness of the proposed algorithm using 150-GBd PAM-4 modulation in the 1-km transmission scenario. Figure 8(a) presents measured BERs as a function of ROP with different DSP schemes. For GD-NNW, we set M as 7 and L as 2 to maintain moderate computational complexity. Under the 7% HD-FEC threshold, the required ROPs are approximately 1 dBm and 2 dBm for GD-NW and GD-NNW, respectively. With PAM-4 modulation, GD-NNW exhibits an enhancement in ROP sensitivity of about 1 dB. This gain is especially pronounced for higher-order modulation formats due to their heightened sensitivity to nonlinear impairments. Figure 8(b) displays the tap coefficients of linear (blue line) and nonlinear (red line) kernels for 150-GBd PAM-4 modulation. Similar to the OOK modulation case, the weights of the nonlinear kernels are also less than those of the linear kernels, and the coefficients approach zero as the tap index increases. This consistency with the OOK modulation further underscores the effectiveness of the proposed GD-NNW algorithm in mitigating nonlinear impairments and enhancing system performance. To summarize, the proposed gradient-descent methods offer flexibility and adaptability in acquiring tap coefficients and demonstrate the capability to smoothly incorporate nonlinear kernels, addressing nonlinear impairments during noise whitening with improved BER performance.

Fig. 8. (a) Measured BERs as a function of ROP with different NW schemes for 150-GBd PAM-4 signal. (b) Tap coefficients of GD-NNW (M = 7, L = 2) for 150-GBd PAM-4 signal.

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4. Conclusion

In this work, we propose two flexible noise whitening algorithms called GD-NW and GD-NNW, which employ a cost function framework instantiated through the gradient descent method. We conduct experimental validation of proposed algorithms in IM-DD systems subjected to remarkably severe bandwidth limitations. Our focus lies on characterizing the crucial parameters of the proposed algorithm, encompassing step size, the required number of training symbols, and the number of taps. The experimental outcomes underscore the efficacy of the proposed algorithms in efficiently whitening EECN. Furthermore, the integration of nonlinear kernels enables noise flattening while simultaneously mitigating nonlinear impairments. The experimental results show a substantial 1 dB improvement in ROP sensitivity, particularly notable for PAM-4 IM-DD transmission. Under the BER threshold of 7% HD-FEC, we successfully demonstrate 180-GBd PAM-4 and 360-GBd OOK signaling in the BTB configuration. Additionally, we achieve transmission rates of 150-GBd PAM-4 and 280-GBd OOK over 1 km of SSMF. Beyond the scope of IM-DD systems, the applicability of these algorithms can be extended to other optical communication systems, such as coherent detection setups where bandwidth limitations originate from wavelength-selective switches and transceivers. We believe that the proposed GD-NW and GD-NNW algorithms offer a robust DSP solution to combat the challenges posed by severe bandwidth constraints in high-speed optical communications.

Funding

National Key Research and Development Program of China (2020YFB1806400); China Scholarship Council (202306230183), National Natural Science Foundation of China (62001287, 62271305).

Disclosures

The authors declare 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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Gradient-descent noise whitening techniques for short-reach IM-DD optical interconnects with severe bandwidth limitation

Abstract

1. Introduction

2. Principle of gradient-descent linear and nonlinear noise whitening

3. Experiment setup and results

3.1 Experimental setup and DSP procedures

3.2 PAM-4 in the BTB case

3.3 OOK in the BTB case

3.4 1-km SSMF transmission and the performance of GD-NNW

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (8)

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