1. Introduction

High spectral efficiency has been achieved in optical fiber communication systems by using advanced modulation formats after widespread use of digital coherent technologies [1–5]. In particular, probabilistic constellation shaping (PCS) is regarded as a promising technique because it enables high sensitivity and precise rate adaptability for the received signal-to-noise ratio (SNR) from short-reach to long-haul transmission [6–9]. PCS often uses high-order quadrature amplitude modulation (QAM) as a base format and optimizes the probability of occurrence of symbol points for a transmission channel, usually according to a Maxwell-Boltzmann (MB) distribution for an additive white Gaussian noise channel. Remarkable long-haul transmission capability has been achieved with PCS [10–12]. Moreover, high symbol rates such as 100 Gbaud have been actively investigated to increase the data rate per transceiver and reduce the number of optical and electrical components in a transmission system [13–15]. However, high-frequency optical and electrical components for implementing a transmitter (Tx) and receiver (Rx) with a high symbol rate currently have non-negligible imperfections, such as in-phase (I) and quadrature (Q) skew, IQ amplitude imbalance, and IQ phase deviation from $\pi /2$ [3,16]. Accordingly, compensation of such impairments due to Tx and Rx device imperfections is becoming important.

The characteristics of impairments due to Tx and Rx components are usually unknown beforehand, thus, adaptive compensation by digital signal processing (DSP) on the Rx side is often used for IQ impairments that occurred in Rx components [17,18], in Tx components [19], or both [20–24]. Linear IQ impairments such as IQ skew that occur in Tx and Rx components cannot be compensated by complex-valued response linear filters with complex-valued input and output signals in the phasor representation, because the IQ components are not dealt with independently [18]. In this sense, such complex-valued response filters are referred as strictly linear (SL). Instead, compensation of IQ impairments requires a real-valued response multi-input multi-output (MIMO) filter with real-valued input and output signals for the IQ components [19], which is equivalent to a complex-valued response filter with inputs consisting of a complex-valued signal and its complex conjugate [18]. Such complex-valued response filters with inputs consisting of a complex-valued signal and its complex conjugate are referred to as widely linear (WL).

Optical fiber communication systems involve not only Tx and Rx IQ impairments but also other impairments, including chromatic dispersion (CD), polarization mode dispersion (PMD), and phase offset between the optical carrier and the local oscillator (LO), which are SL. As matrix multiplication is not commutative, MIMO processes are not generally commutative. For example, changing the order of polarization rotation and differential group delay affects a signal differently. Tx and Rx IQ impairments are intrinsically MIMO processes of IQ components and are thus not generally commutative with other impairments [23]. Rx IQ impairments are added to a signal after accumulation of CD, but when CD compensation is performed first, the IQ impairments become mixed in a complicated way. On the other hand, it is not straightforward to adaptively compensate for Rx IQ impairments first, because the loss function to be minimized by an adaptive filter is unclear before CD compensation.

For these reasons, several approaches have been investigated to compensate for Rx IQ impairments in optical fiber communication with CD accumulation. Independent CD compensation on I and Q signals can avoid IQ mixing in terms of Rx and a subsequent adaptive 4$\times$2 MIMO filter enables Rx IQ impairment compensation [17,21,22]. Alternatively, a large adaptive WL MIMO filter can compensate for Rx IQ impairments, CD, and polarization effects simultaneously [17]. As for Tx IQ impairments, they can be compensated by adaptive WL filters after most other impairments including CD, polarization effects, and carrier phase/frequency offset are compensated [19,21,22], however, other adaptive filters for polarization demultiplexing or Rx IQ impairment compensation degrade performance when they are positioned before Tx IQ impairment compensation, as they must work under Tx IQ impairments. To simultaneously compensate for Rx and Tx IQ impairments as well as polarization effects after independent CD compensation on I and Q signals, an adaptive 8$\times$2 MIMO filter with phase offset compensation has been proposed [20,24]. In summary, all of these previous works rely on either individual adaptive filter blocks in which residual impairments degrade the performance, or an aggregated large adaptive filter that cannot efficiently compensate for multiple impairments having different characteristics.

Accordingly, for compensation of Tx and Rx IQ impairments as well as other impairments in coherent optical fiber communication systems, we previously proposed adaptive multi-layer (ML) SL&WL filters by considering the order in which impairments occur and their mutual commutativity [23]. In these adaptive ML filters, the filter coefficients in all layers can be controlled by gradient calculation with back propagation from the last layer and stochastic gradient descent. This approach enables precise compensation of all the relevant impairments, including Tx and Rx IQ impairments, to minimize a loss function that is composed of the last layer’s outputs. The filters in each layer can be individually designed according to the impairments of interest in order to minimize the temporal spread and the number of cross terms. Moreover, the compensated impairments can be individually and simultaneously monitored via the obtained filter coefficients in the corresponding layers in the ML filters after the convergence of adaptive control [25], which provides useful diagnostics for transmission systems.

To date, simultaneous compensation of Tx and Rx IQ impairments in optical fiber communication systems has been demonstrated over a relatively short reach from 100 to 1500 km of single-mode fiber (SMF) [20–24]. To extend the adaptive ML filter approach for simultaneous compensation of Tx and Rx IQ impairments to ultra-long-haul optical fiber transmission, over a range such as 10,000 km of SMF in which the accumulated CD becomes large, with an advanced modulation format, we have to consider the computational complexity. In the adaptive ML filters, the gradients of the loss function in terms of the filter coefficients and inputs of each layer are calculated through back propagation successively from the last layer. However, the temporal spread of the gradients progressively increases when going back through the layers because of the temporal spread of the filter response in each layer. Accordingly, filters with a large temporal spread are necessary to compensate for the large CD accumulation in the case of ultra-long-haul transmission, and thus, a CD compensation filter in the ML filters entails a large computational complexity for gradient calculation with back propagation.

In this study, we revisited the mutual non-commutativity of SL and WL filters and we propose an adaptive ML filter architecture to solve the problem described above. In this architecture, CD compensation is performed on the received signal and its complex conjugate, and the two augmented signals are the inputs of 2$\times$1 multi-input single-output (MISO) filters in the first layer, for Rx IQ impairment compensation, of the adaptive ML filters. In this ML architecture, the CD compensation filters, whose coefficients can be static, are swept before the ML filters, while the mutual non-commutativity of Rx IQ impairment compensation and CD compensation is solved. The static CD compensation is performed independently of the adaptive ML filters, whose coefficients are controlled by gradient calculation with back propagation from the last layer and stochastic gradient descent. This architecture enables mitigation of the computational complexity for gradient calculation with back propagation in the case of a large CD accumulation in ultra-long-haul transmission, because the CD compensation is not included in the ML filters, while precise compensation of Tx and Rx IQ impairments can still be achieved. Moreover, this architecture retains the features of adaptive ML filters, e.g., that the filters in each layer can be individually designed, and that impairments can be individually and simultaneously monitored via the adaptive filter coefficients in each layer.

We evaluated the proposed adaptive ML filters with augmented inputs through both numerical simulation and wavelength-division-multiplexed (WDM) transmission experiments of 16 channels of 32-Gbaud polarization-division-multiplexed (PDM)-PCS-64QAM signals with an information rate (IR) of 2.4 b/sbl/pol in a 50-GHz grid over 10,200 km of SMF in which Tx and Rx IQ skews were digitally emulated. The results showed that Tx IQ skew up to $\pm$12 ps and Rx IQ skew over $\pm$ 20 ps could be compensated by the adaptive ML filters with augmented inputs under a large CD accumulation over 10,000 km of SMF. Furthermore, emulated Tx and Rx IQ skews up to $\pm$12 ps could be monitored individually via the adaptive filter coefficients of the corresponding layers with a maximum error of 1.1 ps. This work should pave the way for application of Tx and Rx IQ impairment compensation by the adaptive ML filters to ultra-long-haul transmission systems.

The paper is organized as follows. First, we show the proposed adaptive ML filters and its principle. Then, we assess the performance of the approach through numerical simulations with a simple model with large CD accumulation. Finally, we evaluate the performance in the long-haul WDM transmission experiment with PDM-PCS-64QAM.

2. Theory

We consider here the deterministic impairments that occur in optical fiber communication systems with PDM and intradyne coherent detection, and the compensation of these impairments by DSP. On the Tx side, both electrical and optical components cause IQ impairments including IQ skew, IQ amplitude imbalance, and IQ phase deviation from $\pi /2$. In optical fiber propagation, CD and PMD, including polarization rotation, accumulate. Nonlinear impairment due to the optical Kerr effect in fiber propagation also occurs, but we ignore it here for simplicity, because CD more dominantly affects the optical signal. On the Rx side, IQ impairments due to optical and electrical components again occur. Tx and Rx IQ skew, imbalance, and phase deviation are WL and intrinsically MIMO processes. SL processes such as CD are also MIMO when they are represented in the IQ basis. Thus, a WL process and an SL process are generally not mutually commutative, accordingly, the order in which these processes occur and are compensated matters when they are compensated in a block-wise manner, like in conventional DSP for optical fiber communications [26]. Given this non-commutativity, certain impairments should be compensated in the reverse order of their occurrence, ideally. For this purpose, we previously proposed the adaptive SL&WL filters shown in Fig. 1 [23]. These filters consist of Rx IQ impairment compensation, CD compensation, polarization demultiplexing, carrier phase offset compensation, and Tx IQ impairment compensation, in this order.

 

Fig. 1. Adaptive SL&WL filters to compensate for Tx/Rx IQ impairments, proposed in a previous work [23].

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We first examine the first and second layers of the adaptive SL&WL filters. The first layer consists of two 2$\times$1 WL filters to compensate for Rx IQ impairments. The input of each of the 2$\times$1 WL filters is the received signal of X or Y polarization. A 2$\times$1 WL filter performs convolution of two filter responses to the input signal and its complex conjugate, and the output is a sum of the two signals after convolution. The term 2$\times$1 indicates the structure of the filter coefficients. The second layer consists of two 1$\times$1 SL filters with two polarizations to compensate for CD. Because both layers are independent in terms of two polarizations, we theoretically analyze one polarization here, without loss of generality. When a 2$\times$1 WL filter and then a 1$\times$1 SL filter for CD compensation operate on an input signal $x(t) \in \mathbb {C}$, where $\mathbb {C}$ is the set of complex numbers, the output of the 2$\times$1 WL filter, $y(t) \in \mathbb {C}$, is given by

Here, $h_{1}(\tau ), h_{\ast 1}(\tau )$ represent the complex-valued impulse response of the 2$\times$1 WL filter, and $h_{\mathrm {CD}}(\tau )$ is the complex-valued impulse response of the 1$\times$1 SL filter for CD compensation, which has a large temporal spread in ultra-long-haul SMF transmission. In addition, the superscript $\ast$ indicates the complex conjugate. By the distributive property, Eq. (2) can be rewritten as

Consequently, the operation of a 2$\times$1 WL filter and a 1$\times$1 SL CD compensation filter on the input of $x$ in this order is equivalent to the operation of a 2$\times$1 SL MISO filter with a response of $h_{1}(\tau ), h_{\ast 1}(\tau )$ after CD compensation with a response of $h_{\mathrm {CD}}(\tau )$ on $x$ and $x^{\ast }$.

This behavior can also be understood from the block diagram shown in Fig. 2. Because of the mutual non-commutativity, simply changing the order of the 2$\times$1 WL filter and the 1$\times$1 SL filter, shown in Fig. 2(a), results in a different operation and does not work for simultaneous Rx IQ impairment and CD compensation. Instead, if the 1$\times$1 SL filter for CD compensation is moved before summation via the distributive property, then the filters with complex-valued impulse responses $h_{1}(\tau ), h_{\ast 1}(\tau )$ and the one with $h_{\mathrm {CD}}(\tau )$ are both 1$\times$1 SL processes, accordingly they are mutually commutative and provide the equivalent configuration shown in Fig. 2(b). In optical fiber transmission systems, CD can be treated as static except in the case of optical path switching [27], and it is unnecessary to adaptively control the coefficients of CD compensation filters. Thus, by using the configuration shown in Fig. 2(b), Rx IQ impairment compensation can be achieved under CD, while CD compensation is swept before the adaptive filters and operated independently.

 

Fig. 2. DSP architecture for Rx IQ impairment and CD compensation (a) by a 2$\times$1 WL filter and a 1$\times$1 SL filter, and (b) by an equivalent configuration.

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We can apply a similar approach to the fourth and fifth layers of the adaptive SL&WL filters. The operation of a 1$\times$1 SL filter to compensate for carrier phase and frequency offset and then a 2$\times$1 WL filter to compensate for Tx IQ impairments, as shown in Fig. 3(a), is equivalent to the configuration shown in Fig. 3(b). This explains the embedded carrier phase offset compensation in an 8$\times$2 WL MIMO filter [20].

 

Fig. 3. DSP architecture for carrier phase offset and Tx IQ impairment compensation (a) by a 1$\times$1 SL filter and a 2$\times$1 WL filter, and (b) by an equivalent configuration.

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Figure 4 shows the proposed adaptive ML filter architecture, based on the above analysis, with augmented inputs consisting of the received signal and its complex conjugate for CD compensation. In this architecture, the CD compensation operates on the signal and its complex conjugate before the ML filters. Then, with the augmented inputs consisting of the signal and its complex conjugate after CD compensation, the adaptive ML filters compensate for Rx IQ impairments, polarization effects, carrier phase and frequency offset, and Tx IQ impairments. The first layer of the ML filters consists of two 2$\times$1 SL MISO filters with two polarizations to compensate for Rx IQ impairments. The subsequent layers are the same as in our previous adaptive ML SL&WL filters: the second layer consists of a 2$\times$2 SL MIMO filter for polarization demultiplexing; the third layer consists of two 1-tap 1$\times$1 SL filters with two polarizations for carrier recovery; and the fourth layer consists of two 2$\times$1 WL filters with two polarizations to compensate for Tx IQ impairments. In this work, these filters are half-symbol-spaced finite impulse response (FIR) filters. The filter coefficients in the first, second, and fourth layers are all adaptively controlled by gradient calculation with back propagation and stochastic gradient descent to minimize the loss function, which is composed of the last layer’s outputs. The coefficients in the third layer for carrier recovery are controlled by a phase-locked loop (PLL) with a second-order loop filter [28] from the last layer’s outputs, although adaptive control for carrier recovery by gradient descent has also been investigated [29].

 

Fig. 4. Architecture of the proposed adaptive ML filters with augmented inputs of the signal and its complex conjugate with CD compensation to compensate for Tx and Rx IQ impairments under a large CD accumulation. CDC: chromatic dispersion compensation; CR: carrier recovery; PLL: phase-locked loop.

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As mentioned above, in the architecture shown in Fig. 4, the CD compensation operates independently before the adaptive ML filters. CD compensation filters require about 5600 half-symbol-spaced taps for transmission over 10,000 km of conventional SMF having a dispersion coefficient of 17 ps/nm/km with a 32 Gbaud signal [26]. Here, we mitigate the computational complexity of gradient calculation by avoiding back propagation through the CD compensation filters, which have a large temporal spread. In the experiments described later, the tap lengths of the filters for Rx/Tx IQ impairment compensation and for polarization demultiplexing were 5 and 21, respectively. With these tap lengths, the required number of complex multiplications for back propagation of the proposed adaptive ML filters with augmented inputs is reduced to 0.2% of that required by our previous adaptive ML SL&WL filters including CD compensation. (See Appendix A for the details of the computational complexity estimation.) As for forward propagation, although this architecture doubles the number of CD compensation filters, they can be implemented in the frequency domain to the reduce the computational complexity [30], which is very effective, especially for filters with a large temporal spread.

It is worth mentioning that applying CD compensation to both the signal and its complex conjugate is not equivalent to applying CD compensation to the signal and then obtaining its complex conjugate. Rather, inputting the augmented signal and its complex conjugate with CD compensation to the adaptive filters is equivalent to inputting independently CD-compensated IQ signals [17]. The augmented signals $y(t), y_{c}(t) \in \mathbb {C}$ after the operation of CD compensation on the received signal $x(t)$ and its complex conjugate are represented as

Meanwhile, consider the case of CD compensation operating independently on the IQ components $x_{\mathrm {I}}(t), x_{\mathrm {Q}}(t) \in \mathbb {R}$ of the received signal $x(t) = x_{\mathrm {I}}(t) + i x_{\mathrm {Q}}(t)$, where $\mathbb {R}$ is the set of real numbers. In this case, the obtained signals $y_{i}(t), y_{q}(t) \in \mathbb {C}$ are represented as

Here, the following relation holds [18]:

Here, $T$ satisfies $T T^{\dagger} = T^{\dagger} T = 2 I$, where $I$ is the identity matrix and $\dagger$ denotes the Hermitian conjugate. As a result, the following relation also holds:

In consequence, the input of $y(t)$ and $y_{c}(t)$ to the adaptive filters can yield the same results as the input of $y_{i}(t)$ and $y_{q}(t)$, with only an additional transformation $T$.

As shown in Fig. 2, the coefficients of the 2$\times$1 SL MISO filter in the first layer of the proposed adaptive ML filters straightforwardly correspond to those of the 2$\times$1 WL filter for Rx IQ impairment compensation. Moreover, the 2$\times$1 WL filter for Tx IQ impairment compensation in the fourth layer is the same as in our previous ML SL&WL filters. Thus, the IQ impairment monitoring via the obtained adaptive filter coefficients in the corresponding layers [25] is immediately applicable in the proposed adaptive ML filter architecture.

3. Simulation

We first evaluated the adaptive ML filter architecture described above in simulations using a simple model. We used a 32-Gbaud PDM-PCS-64QAM signal with an IR of 1.6 b/sbl/pol, which corresponds to the IR of quadrature phase shift keying (QPSK) when forward error correction (FEC) is applied with the same code rate. CD corresponding to that of 10,000 km of SMF was added to the signal. We focused on IQ skew as the source of Tx and Rx IQ impairments.

3.1 Simulation model

On the Tx side, a 32-Gbaud PDM-PCS-64QAM signal with an IR of 1.6 b/sbl/pol was generated. Three frames of low-density parity check FEC for DVB-S2 with a frame length of 64,800 and a code rate of 0.8, obtained by loading random bits in the payload, were generated as the transmitted data for the IQ components and two polarizations, which resulted in a total of 12 frames. A distribution matcher with the probabilistic amplitude shaping architecture [6] and constant composition distribution matching (CCDM) [31] mapped the data to the PDM-PCS-64QAM signal. The probability distribution was an MB distribution, and the IR was set to 1.6 b/sbl/pol. To enable use of the pilot-based [32,33] data-aided least-mean-square (LMS) algorithm for coefficient control of the adaptive filters, one pilot symbol was inserted every 15 symbols. The pilot symbols were randomly sampled from the generated PDM-PCS-64QAM signal in the same manner so as to match the statistical characteristics of the pilot symbols and the transmitted signal. To enable the Rx DSP to detect the timing of the pilot symbol sequence’s head, a preamble of 64 symbols was inserted in every FEC frame. The preamble symbols consisted of a random sequence from the inner four symbol points of the 64QAM, which enabled the Rx DSP to detect the preamble on the basis of the signal power. In addition, to match the limitation of an instrument that was used in the experiment described later, dummy data consisting of 448 symbols that were randomly sampled from the PDM-PCS-64QAM signal was added after each preamble.

The resulting signal with the pilot and preamble symbols was oversampled to 32-fold oversampling and shaped with a root-raised cosine filter having a roll-off factor of 0.1. The four IQ components with two polarizations corresponded to the electrical signals that a digital-to-analog converter (DAC) would output. Tx IQ skew was emulated for these signals. Lastly, a laser source with a frequency of 193.3 THz and a linewidth of 100 kHz was modulated by these signals, and a 32 Gbaud PDM-PCS-64QAM signal was thus generated.

Next, CD of 170 ns/nm, which corresponds to 10,000 km of SMF, was added to the generated signal. After adding random polarization rotation, the signal’s optical-signal-to-noise ratio (OSNR) was set to 15 dB/0.1 nm. The optical signal was received through coherent detection with an LO source having a linewidth of 100 kHz, no frequency offset, and low-pass filters whose 3-dB bandwidth was 0.8 times the symbol rate. Rx IQ skew was then emulated for the four IQ components with two polarizations, which corresponded to the electrical signals that would be input to an analog-to-digital converter (ADC). The ADC sampled these signals with two-fold oversampling.

Next, DSP was performed. Specifically, low-pass filtering with a 3-dB bandwidth equal to the symbol rate was performed on the four received signals for resampling, and power normalization was performed individually on the IQ components with two polarizations. The signals were resampled to two-fold oversampling on the basis of the timing error [34], and matched filtering with the root-raised cosine filter was applied. Then, the adaptive ML filters shown in Fig. 4 were applied. The CD compensation on the signal and its complex conjugate operated in the frequency domain.

The adaptive ML filters, with augmented inputs of the received signal and its complex conjugate with CD compensation, consisted of four layers as described in the previous section. The tap length of the 2$\times$1 SL MISO filters in the first layer, for Rx IQ impairment compensation, was set to five. The tap lengths of the 2$\times$2 SL MIMO filter in the second layer, for polarization demultiplexing, and the 2$\times$1 WL filters in the fourth layer, for Tx IQ impairment compensation, were set to 21 and 5, respectively. The coefficients for these filters were initialized as one at the center of the main diagonal, with the remaining values set to zero. The filter coefficients in the first, second, and fourth layers were adaptively updated with back propagation and stochastic gradient descent by the data-aided LMS algorithm at the timings of the preamble, dummy, and pilot symbols, with no update at the data symbol timings. The step sizes were $10^{-2}$ for the first and fourth layers and $5\times 10^{-2}$ for the second layer. The coefficients of the third layer, for carrier phase and frequency offset compensation, were controlled by a PLL from the last layer’s outputs. Here, the phase error was detected from data symbols at the timings of the preamble, dummy, and pilot symbols and from decision-directed symbols at other timings.

For timing alignment with the known pilot sequence, the following procedure was carried out before applying the adaptive ML filters. The preamble timing was roughly detected from the received signal after CD compensation on the basis of a moving average of the sample power. Then, for about $\pm$ 10 symbols around the detected timing, adaptive control of the ML filters were attempted over a duration of about 10,000 symbols. For this adaptive control, the tap length of the 2$\times$2 SL MIMO filter was set to one, and the precise timing of the pilot sequence’s head was detected on the basis of the magnitude of the loss after adaptive control. After the precise timing alignment, the received signal and its complex conjugate with CD compensation were input to the ML filters, and the pilot-based adaptive control was performed again. From the symbol-spaced outputs of the adaptive ML filters after convergence of the filter coefficients, the preamble, dummy, and pilot symbols were removed, and FEC and CCDM decoding were then performed. A sum-product algorithm with maximum 50 iterations was used for FEC decoding. The post-decoding bit error rate (BER) and the normalized generalized mutual information (NGMI) [35] averaged over two polarizations were evaluated.

To provide a reference for conventional DSP, we also evaluated a configuration in which a 2$\times$2 SL MIMO filter embedding carrier recovery [36] for polarization demultiplexing and carrier phase and frequency offset compensation was applied after CD compensation. This is referred to hereafter as the 2$\times$2 SL configuration. The coefficients of the 2$\times$2 SL MIMO filter were controlled by the pilot-based data-aided LMS algorithm, and carrier recovery was controlled by a PLL. The step size was $5\times 10^{-2}$.

3.2 Simulation results

Figure 5 shows the resulting constellations after the adaptive ML filters or the 2$\times$2 SL configuration under several conditions: without both Tx and Rx IQ skew, with Tx IQ skew of 16 ps, with Rx IQ skew of 16 ps, and with both Tx and Rx IQ skews of 16 ps. In this evaluation, the Tx or Rx IQ skew was emulated as the delay of the Q component in the X polarization. The constellations for the X polarization are shown in the figure. When Tx or Rx IQ skew of 16 ps was emulated, a clear PCS-64QAM constellation like in the case without both Tx and Rx IQ skew could not be obtained by the 2$\times$2 SL configuration. In contrast, with Tx or Rx IQ skew, the proposed adaptive ML filters obtained similar constellations to the case without both Tx and Rx IQ skew. Moreover, even in the case with both Tx and Rx skews of 16 ps, the adaptive ML filters still obtained a similar constellation.

 

Fig. 5. Constellations of the X polarization for reception of a 32-Gbaud PDM-64QAM signal with an IR of 1.6 b/sbl/pol by the 2$\times$2 SL configuration or the adaptive ML filters, under several Tx/Rx IQ skew conditions.

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Figure 6 shows the NGMI after the 2$\times$2 SL configuration or the adaptive ML filters when the Tx or Rx IQ skew in the X polarization was swept from -31 to +31 ps, and Fig. 7 shows the post-decoding BER under the same conditions. The error-free BER results are plotted at a level of $10^{-5}$ for readability. In the case of the 2$\times$2 SL configuration, which could not compensate for Tx or Tx IQ impairments, both the Tx and Rx IQ skews degraded the NGMI. Moreover, the post-decoding BER was not error-free when the Tx IQ skew was beyond $\pm$14 ps, and when the Rx IQ skew was beyond $\pm$ 20 ps. In contrast, with the adaptive ML filters, NGMI results similar to those in the case without IQ skew and error-free post-decoding BER results were achieved with up to $\pm$23 ps of Tx IQ skew, and with over $\pm$31 ps of Rx IQ skew. Both figures also show the results obtained by the adaptive ML filters when the tap lengths of the filters for Rx and Tx IQ impairment compensation in the first and fourth layers were nine. These results show that the adaptive ML filters’ tolerance to Tx IQ skew did not increase when the tap length of the filters for Tx IQ impairment compensation increased to nine. After several evaluations, the PLL that determined the filter coefficients of the third layer became unstable when the Tx IQ impairment was large. This behavior will be a future issue for us to resolve, although Tx IQ skew compensation over $\pm$10 ps will still be useful in practice for current optical fiber communication systems.

 

Fig. 6. NGMI for reception of a 32-Gbaud PDM-64QAM signal with an IR of 1.6 b/sbl/pol by the 2$\times$2 SL configuration or the adaptive ML filters, with a tap length of five or nine taps for Tx and Rx IQ impairment compensation. The results are shown for variation of (a) the Tx IQ skew in the X polarization and (b) the Rx IQ skew in the X polarization.

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Fig. 7. Post-decoding BER for reception of a 32-Gbaud PDM-64QAM signal with an IR of 1.6 b/sbl/pol by the 2$\times$2 SL configuration or the adaptive ML filters, with a tap length of five or nine taps for Tx and Rx IQ impairment compensation, when (a) the Tx X-IQ skew and (b) the Rx X-IQ skew was varied.

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We also evaluated the capability for simultaneous compensation of Tx and Rx IQ skews. Random Tx and Rx skews in two polarizations from a uniform distribution within $\pm$16 ps were emulated simultaneously, with a total of 1000 random realizations for evaluation. A histogram of the NGMI in the cases with the 2$\times$2 SL configuration and the adaptive ML filters is shown in Fig. 8. In the 2$\times$2 SL case, the NGMI distribution was spread to lower values by the Tx and Rx IQ skews. In contrast, the adaptive ML filters achieved a stable NGMI of around one. These results confirm that the proposed adaptive ML filters with augmented inputs could sufficiently and simultaneously compensate for Tx and Rx IQ skews under a large CD accumulation corresponding to 10,000 km of SMF.

 

Fig. 8. Histogram of the NGMI for reception of a 32-Gbaud PDM-64QAM signal with an IR of 1.6 b/sbl/pol by the 2$\times$2 SL configuration or the adaptive ML filters under 1000 random realizations of simultaneous Tx and Rx IQ skew in two polarizations.

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Figure 9 shows examples of the time evolution of the loss with the adaptive ML filters under 10 random realizations of Tx and Rx IQ skew. The loss at the timings of the preamble, dummy, and pilot symbols was taken with a moving average over 10 symbols. The results show that the adaptive control converged within a few thousand symbols.

 

Fig. 9. Convergence of the loss at the pilot symbol timings with the adaptive ML filters. Examples of the time evolution are plotted for 10 random realizations of simultaneous Tx and Rx IQ skew.

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We also monitored the Tx and Rx IQ skews from the filter coefficients of the fourth and first layers in the ML filters after convergence. Fig. 10 shows the monitored values from the corresponding layers [25] for 1000 random realizations in which the Tx and Rx IQ skews were simultaneously emulated in two polarizations. For all of the Tx and Rx IQ skews in two polarizations, the monitored values agreed well with the emulated skews, as the error was within $\pm$1 ps. Thus, we confirmed that impairment monitoring via the adaptive filter coefficients works for the proposed adaptive ML filters with augmented inputs under a large CD accumulation.

 

Fig. 10. Impairment monitoring via the coefficients of the adaptive ML filters for (a) Tx IQ skew in the X polarization, (b) Tx IQ skew in the Y polarization, (c) Rx IQ skew in the X polarization, and (d) Rx IQ skew in the Y polarization, under 1000 random realizations of simultaneous Tx and Rx IQ skew.

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4. Ultra-long-haul transmission experiment

We also evaluated the adaptive ML filter architecture with augmented inputs of the received signal and its complex conjugate with CD compensation in a transmission experiment over 10,000 km of SMF. In this experiment, we transmitted 16-channel WDM signals of 32-Gbaud PDM-PCS-64QAM with an IR of 2.4 b/sbl/pol, which corresponds to the IR of 8QAM when FEC is applied with the same code rate. The signals were transmitted over a recirculating loop configuration consisting of five spans of 60-km pure-silica-core (PSC) SMF. As in the simulation described above, we focused on IQ skew as the source of Tx and Rx impairments.

4.1 Experimental setup

Figure 11 shows a schematic diagram of the experimental setup. On the Tx side, 32-Gbaud PDM-PCS-64QAM signals were generated, with frequencies ranging from 192.90 to 193.65 THz in a 50-GHz grid. The signal at a frequency of 193.3 THz, which was received and evaluated, was generated by modulating a laser source having a linewidth of about 100 kHz with the outputs of a four-channel DAC at a sampling rate of 92 GS/s and a vertical resolution of eight bits. The transmitted data were generated by the same procedure that was described above for the simulation. The IR of the PDM-PCS-64QAM signal was set to 2.4 b/sbl/pol. The signal, with pilot and preamble symbols included, was oversampled to two-fold oversampling and shaped with a root-raised cosine filter having a roll-off factor of 0.1. Tx IQ skew was digitally emulated in the X polarization as a delay of the Q component. A DAC outputted the four IQ components with two polarizations after resampling to 92 GS/s. The 15 remaining channels were generated by using 15 channels of laser sources combined with a polarization-maintaining arrayed waveguide grating (AWG) and a four-channel DAC at a sampling rate of 64 GS/s without IQ skew. The signal under evaluation and the other 15 channels were then combined after power equalization by a wavelength selective switch (WSS), and low-speed polarization scrambling (10$\times 2\pi$ rad/s) was performed.

 

Fig. 11. Experimental setup for WDM transmission of 32-Gbaud PDM-PCS-64QAM signals with an IR of 2.4 b/sbl/pol over a loop configuration consisting of five spans of 60-km SMF. LD: laser diode; AWG: arrayed waveguide grating; DAC: digital-to-analog converter; MOD: modulator; WSS: wavelength selective switch; PS: polarization scrambler; AOM: acousto-optic modulator; EDFA: erbium-doped fiber amplifier; SMF: single-mode fiber; OBPF: optical band-pass filter; CRx: coherent receiver; ADC: analog-to-digital converter.

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The generated WDM signals were transmitted to the transmission line through an acousto-optic loop switch. As mentioned above, the transmission line had a loop configuration composed of five spans of 60-km PSC SMF, with erbium-doped fiber amplifiers (EDFAs). The dispersion coefficient and effective core area of the SMF were about 21 ps/nm/km and 153 $\mu$m$^{2}$, respectively. The span input power was -1 dBm/ch, which was an optimized value in a preliminary experiment. After five spans, the nonuniform loss and gain in the loop for the WDM channels were equalized by the WSS. After 34 loops, or 10,200 km, the OSNR was 18.7 dB/0.1 nm.

After transmission over the loop configuration, the evaluated signal was demultiplexed by an optical band-pass filter with a 3-dB bandwidth of 50 GHz, and then received by a polarization-diversity coherent receiver. A laser source with a linewidth of about 100 kHz was used as an LO, which was free-running with the Tx laser source and had an average frequency offset of about 70 MHz. The four outputs of the coherent receiver were sampled by a four-channel oscilloscope, which was used as an ADC, at a sampling rate of 80 GS/s and a vertical resolution of eight bits. A received signal consisting of 2 MS of successive data was obtained three times under each condition. Rx IQ skew was digitally emulated in the X polarization after sampling as a delay of the Q component. The initial Tx and Rx IQ skews in the experimental setup when IQ skew was not emulated were calibrated as closely to zero as possible. After emulating the Rx IQ skew, DSP was performed offline. We evaluated the same configurations that were described above for the simulation, namely, the adaptive ML filters with the augmented inputs (ML) and the reference (2$\times$2 SL).

4.2 Transmission results

Under the condition of transmission over 34 loops, or 10,200 km, the Tx or Rx IQ skew was varied from -20 to +20 ps. Figure 12 shows the NGMI after the 2$\times$2 SL configuration or the adaptive ML filters, and Fig. 13 shows the post-decoding BER. The results were averaged over the three received waveforms that were obtained. With the 2$\times$2 SL configuration, the post-decoding BER was error-free only within $\pm$4 ps of Tx IQ skew and within $\pm$8 ps of Rx IQ skew, whereas post-decoding errors occurred outside those ranges. In contrast, with the adaptive ML filters, error-free post-decoding BER results and NGMI results similar to those in the case without IQ skew were achieved with up to $\pm$12 ps of Tx IQ skew, and with over $\pm$20 ps of Rx IQ skew. As was observed in the simulation, when the Tx IQ skew was around $\pm$20 ps, the NGMI and post-decoding BER deteriorated with the adaptive ML filters. Except for that case, however, these results experimentally confirm that the proposed adaptive ML filters could compensate for both Tx and Rx IQ skews under a large CD in ultra-long-haul transmission.

 

Fig. 12. NGMI for reception of a 32-Gbaud PDM-64QAM signal with an IR of 2.4 b/sbl/pol by the 2$\times$2 SL configuration or the adaptive ML filters after 10,200 km of SMF transmission. The results were obtained while emulating (a) Tx IQ skew in the X polarization or (b) Rx IQ skew in the X polarization.

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Fig. 13. Post-decoding BER for reception of a 32-Gbaud PDM-64QAM signal with an IR of 2.4 b/sbl/pol by the 2$\times$2 SL configuration or the adaptive ML filters after 10,200 km of SMF transmission, with emulation of (a) Tx IQ skew in the X polarization or (b) Rx IQ skew in the X polarization.

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Next, we evaluated simultaneous compensation of the Tx and Rx IQ skews for transmission over 10,200 km. Figure 14 shows the NGMI after the 2$\times$2 SL configuration and the adaptive ML filters when Tx and Rx IQ skews in the X polarization were simultaneously emulated. With the 2$\times$2 SL configuration, as shown in Fig. 14(a), both the Tx and Rx IQ skews degraded the NGMI, although the degradation was not smooth when the Tx and Rx skews became large. The reason was that the adaptive 2$\times$2 SL MIMO filter could not converge stably in this region. The NGMI variation was relatively smaller when the Rx IQ skew was varied than when the Tx IQ skew was varied in this region, because the Rx IQ skew was emulated for the same three received waveforms that were obtained for a certain emulated Tx IQ skew. In contrast, except in the region around a Tx IQ skew of $\pm$20 ps, a similar NGMI was achieved with the adaptive ML filters even when the Tx and Rx IQ skews were simultaneously emulated.

 

Fig. 14. NGMI for reception of a 32-Gbaud PDM-64QAM signal with an IR of 2.4 b/sbl/pol by (a) the 2$\times$2 SL configuration or (b) the adaptive ML filters, after 10,200 km of SMF transmission when Tx and Rx IQ skew in the X polarization were simultaneously emulated.

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We also evaluated impairment monitoring via the adaptive ML filter coefficients in the corresponding layers. For 13 conditions each of Tx and Rx IQ skews of 0, $\pm$1, …, $\pm$4, $\pm$8, $\pm$12 ps, giving a total of 169 conditions, we obtained three results under each condition. Figure 15 shows the monitoring results for Tx and Rx IQ skews in the X polarization, which show that the monitored values agreed well with the emulated skews. The monitored values for the Rx IQ skew showed a relatively large error, though the maximum error was 1.1 ps. Consequently, these results experimentally confirm that IQ impairment in both Tx and Rx could be monitored in ultra-long-haul transmission via the coefficients of the proposed adaptive ML filters with augmented inputs.

 

Fig. 15. Impairment monitoring via the coefficients of the adaptive ML filters after 10,200 km of SMF transmission for (a) Tx and (b) Rx IQ skew in the X polarization.

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Finally, we evaluated the optimization of the PCS-QAM signal’s IR with fixed FEC, for transmission over 10,200 km, by the adaptive ML filters. The IR was varied from 1.6 to 3.2 b/sbl/pol in steps of 0.1 b/sbl/pol. Figure 16 shows the NGMI and the post-decoding BER after 10,200 km of transmission. Post-decoding error-free transmission was achieved up to 2.8 b/sbl/pol. In addition, Fig. 17 shows constellations of the signal after the adaptive ML filters for several IRs up to 2.8 b/sbl/pol.

 

Fig. 16. (a) NGMI and (b) post-decoding BER for reception of a 32-Gbaud PDM-64QAM signal by the adaptive ML filters after 10,200 km of SMF transmission while varying the IR with fixed FEC.

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Fig. 17. Constellations of a 32-Gbaud PDM-64QAM signal with IRs of (a) 1.6, (b) 2.0, (c) 2.4, and (d) 2.8 b/sbl/pol, as received by the adaptive ML filters after 10,200 km of SMF transmission.

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5. Conclusion

We have proposed an adaptive ML filter architecture for Tx and Rx IQ impairment compensation in ultra-long-haul transmission. In this architecture, CD compensation is performed on a received signal and its complex conjugate. The two signals are then given as augmented inputs to 2$\times$1 SL MISO filters in the first layer of the proposed ML filters, which is for Rx IQ impairment compensation. The coefficients of the ML filters are controlled by gradient calculation with back propagation from the last layer and stochastic gradient descent. The CD compensation filters are swept before the ML filters, while the mutual non-commutativity of Rx IQ impairments and CD compensation is solved. This considerably mitigates the computational complexity of the back propagation in the case with the large CD accumulation. The proposed architecture leverages the features of adaptive ML filters: namely, the filters in each layer can be individually designed, and impairments can be individually and simultaneously monitored from the converged coefficients. We evaluated the proposed adaptive ML filters with augmented inputs through both numerical simulation and in WDM transmission experiments. The experimental evaluation used 16 channels of 32-Gbaud PDM-PCS-64QAM signal with an IR of 2.4 b/sbl/pol in a 50-GHz grid over 10,200 km of SMF transmission, in which Tx and Rx IQ skews were digitally emulated. The results showed that the adaptive ML filters could compensate Tx IQ skew up to $\pm$12 ps and Rx IQ skew over $\pm$ 20 ps under a large CD accumulation over the 10,000 km of SMF. Furthermore, emulated Tx and Rx IQ skews up to $\pm$12 ps could be monitored individually from the adaptive filter coefficients of the corresponding layers with a maximum error of 1.1 ps.

Appendix A: Forward propagation and back propagation of adaptive ML filters

In this study. the forward and back propagation of the layers in the adaptive ML filters shown in Fig. 4 are basically the same as in our previous study [23]; hence, we review them here. The filter coefficients are adaptively controlled through gradient calculation with back propagation. We explicitly handle updating of the filter coefficients, and first, we describe the case when all the intermediate outputs of the layers in the ML filters are recalculated for every coefficient update. Next, we give an approximated calculation. Because the adaptive ML filters deal with time-series signals and a coefficient update is usually small in one update of stochastic gradient descent, a portion of the intermediate outputs of layers other than the last layer can be approximately reused after a coefficient update. This approximation reduces the computational complexity for forward propagation of the layers to that of a conventional FIR filter. Accordingly, we used the approximated calculation in this study.

Full calculation

For the $l$-th layer, the related output and input signals to obtain the last layer’s output samples at timing integer $k$ are given by

(10)$$\boldsymbol{u}_{i}^{[l]}[k] = (u_{i}^{[l]}[k; k], u_{i}^{[l]}[k-1; k], \ldots, u_{i}^{[l]}[k-M_{l}+1; k])^{\mathrm{T}},$$

and

(11)$$\boldsymbol{u}_{i}^{[l-1]}[k] = (u_{i}^{[l-1]}[k; k], u_{i}^{[l-1]}[k-1; k], \ldots, u_{i}^{[l-1]}[k-M_{l-1}+1; k])^{\mathrm{T}}.$$

Here, T denotes the transpose, $u_{i}^{[l]}[k-m; k]$ denotes the $m$-th component obtained by the filter coefficients at timing integer $k$, and $i = 1, 2$ is an index for the two polarizations. In the ML filters, the input signals of the $l$-th layer correspond to the output signals of the $(l-1)$-th layer. The length of the $l$-th layer’s output signals $M_{l}$, the length of its input signals $M_{l-1}$, and the filter tap length $M^{[l]}$ satisfy the following relation:

(12)$$M^{[l]} = M_{l-1} - M_{l} +1,$$

which is based on convolution. Because of this relation, the length of the input signals progressively increases toward the beginning layers.

Here, we consider the case in which the $l$-th layer consists of a WL MIMO filter. Other type of filters correspond to a limited case of the WL MIMO filter. This approach cannot be applied for a carrier recovery layer, but we separately examine that case later. The filter coefficients of the WL MIMO filter at timing integer $k$ are as follows:

(13)$$\boldsymbol{h}_{ij}^{[l]}[k] = (h_{ij}^{[l]}[0; k], h_{ij}^{[l]}[1; k], \ldots, h_{ij}^{[l]}[M^{[l]}-1; k])^{\mathrm{T}},$$

(14)$$\boldsymbol{h}_{{\ast} ij}^{[l]}[k] = (h_{{\ast} ij}^{[l]}[0; k], h_{{\ast} ij}^{[l]}[1; k], \ldots, h_{{\ast} ij}^{[l]}[M^{[l]}-1; k])^{\mathrm{T}}.$$

The WL MIMO filter’s forward propagation is given by

(15)$$\boldsymbol{u}_{i}^{[l]}[k] = \sum_{j=1}^{2} H_{ij}^{[l]}[k] \boldsymbol{u}_{j}^{[l-1]}[k] + \sum_{j=1}^{2} H_{{\ast} ij}^{[l]}[k] \boldsymbol{u}_{j}^{[l-1] \ast}[k],$$

where

(16)$$H_{ij}^{[l]}[k] = \begin{pmatrix} h_{ij}^{[l]}[0; k] & h_{ij}^{[l]}[1; k] & \cdots & h_{ij}^{[l]}[M^{[l]}-1; k] & 0 & \cdots & 0\\ 0 & \ddots & \ddots & & \ddots & \ddots & \vdots \\ \vdots & & & & & & 0 \\ 0 & \cdots & 0 & h_{ij}^{[l]}[0; k] & h_{ij}^{[l]}[1; k] & \cdots & h_{ij}^{[l]}[M^{[l]}-1; k] \\ \end{pmatrix}$$

and

(17)$$H_{{\ast} ij}^{[l]}[k] = \begin{pmatrix} h_{{\ast} ij}^{[l]}[0; k] & h_{{\ast} ij}^{[l]}[1; k] & \cdots & h_{{\ast} ij}^{[l]}[M^{[l]}-1; k] & 0 & \cdots & 0\\ 0 & \ddots & \ddots & & \ddots & \ddots & \vdots \\ \vdots & & & & & & 0 \\ 0 & \cdots & 0 & h_{{\ast} ij}^{[l]}[0; k] & h_{{\ast} ij}^{[l]}[1; k] & \cdots & h_{{\ast} ij}^{[l]}[M^{[l]}-1; k] \\ \end{pmatrix}$$

are matrices of size $M_{l} \times M_{l-1}$. Note that we adopt a notation without the complex conjugate for the filter coefficients here. Regarding the right side of Eq. (15), if the second term is omitted, then it represents the SL case. If the summation over $j$ is omitted, then it represents the case without MIMO. Lastly if the index $j$ is extended to denote the augmented signals, then the right side incorporates the case of SL MISO filters. Equation (15) can be rewritten as

(18)$$\boldsymbol{u}_{i}^{[l]}[k] = \sum_{j=1}^{2} U_{j}^{[l-1]}[k] \boldsymbol{h}_{ij}^{[l]}[k] + \sum_{j=1}^{2} U_{j}^{[l-1] \ast}[k] \boldsymbol{h}_{{\ast} ij}^{[l]}[k],$$

where

(19)$$U_{j}^{[l-1]}[k] = \begin{pmatrix} u_{j}^{[l-1]}[k; k] & u_{j}^{[l-1]}[k-1; k] & \cdots & u_{j}^{[l-1]}[k-M^{[l]}+1; k] \\ u_{j}^{[l-1]}[k-1; k] & u_{j}^{[l-1]}[k-2; k] & \cdots & u_{j}^{[l-1]}[k-M^{[l]}; k] \\ \vdots & & & \vdots \\ u_{j}^{[l-1]}[k-M_{l}+1; k] & u_{j}^{[l-1]}[k-M_{l}; k] & \cdots & u_{j}^{[l-1]}[k-M_{l-1}+1; k] \\ \end{pmatrix},$$

is a matrix of size $M_{l} \times M^{[l]}$.

In the case when the $l$-th layer consists of filters for carrier recovery, the carrier phase and frequency offset are described via multiplication in the time domain, rather than convolution in the time domain. Consequently, the carrier phase and frequency offset or their compensation are not mutually commutative with other SL processes such as CD, strictly speaking, when the phase offset varies over time, which results in equalization-enhanced phase noise [37,38] . On the other hand, suppose that the phase offset can be assumed to be static and common to the two polarizations, which roughly holds for the carrier recovery filters in the adaptive ML filters because they do not deal with a large temporal spread. In this case, the phase offset or its compensation is commutative with CD, but not commutative with WL processes such as Tx and Rx impairments. The carrier recovery filters in the adaptive ML filters are 1-tap SL filters. When the compensation values at timing integers ranging from $k-M_{l}+1$ to $k$ are

(20)$$\boldsymbol{h}_{i}^{[l]}[k: k-M_{l}+1] = (h_{i}^{[l]}[0; k], h_{i}^{[l]}[0; k-1], \ldots, h_{i}^{[l]}[0; k-M_{l}+1])^{\mathrm{T}},$$

the output signals of the $l$-th layer are given by

(21)$$\boldsymbol{u}_{i}^{[l]}[k] = \boldsymbol{h}_{i}^{[l]}[k: k-M_{l}+1] \circ \boldsymbol{u}_{i}^{[l-1]}[k],$$

where $\circ$ denotes the Hadamard product. By applying the above forward propagation from the first layer to the last ($L$-th) layer, we obtain the outputs $u_{i}^{[L]}[k]$ of the adaptive filters at timing integer $k$, where the length of the last layer’s outputs is $M_{L} = 1$.

For adaptive control of the ML filters, the loss function to be minimized is composed of the last $L$-th layer’s outputs. When the data-aided LMS algorithm is used, the instantaneous loss to be minimized by stochastic gradient descent is

(22)$$\phi [k] = \sum_{i=1}^{2} |d_{i}[k] - u_{i}^{[L]}[k]|^{2},$$

for symbol integer $k$, where the $d_{i}[k]$ are the known training symbols. A filter coefficient $\xi \in \mathbb {C}$ is updated by gradient descent. By using Wirtinger derivatives, the coefficient update is given by

(23)$$\xi \rightarrow \xi - 2 \alpha \frac{\partial \phi}{\partial \xi^{{\ast}}},$$

where $\alpha$ is a step size.

For the last ($L$-th) layer, the gradients of the loss in terms of the outputs are

(24)$$\frac{\partial \phi}{\partial u_{i}^{[L]}[k]} ={-}e_{i}^{{\ast}},$$

where $e_{i} = d_{i}[k] - u_{i}^{[L]}[k]$. When these gradients are given in terms of the $l$-th layer’s outputs, they can be obtained in terms of the filter coefficients and inputs of the $l$-th layer by back propagation via the chain rule for derivatives. When the $l$-th layer consists of a WL MIMO filter, the gradients of the loss in terms of the filter coefficients and inputs are

(25)$$\frac{\partial \phi}{\partial \boldsymbol{h}_{ij}^{[l] \ast}[k]} = U_{j}^{[l-1] \dagger}[k] \frac{\partial \phi}{\partial \boldsymbol{u}_{i}^{[l] \ast} [k]},$$

(26)$$\frac{\partial \phi}{\partial \boldsymbol{h}_{{\ast} ij}^{[l] \ast}[k]} = U_{j}^{[l-1] \mathrm{T}}[k] \frac{\partial \phi}{\partial \boldsymbol{u}_{i}^{[l] \ast} [k]},$$

(27)$$\frac{\partial \phi}{\partial \boldsymbol{u}_{j}^{[l-1]}[k]} = \sum_{i=1}^{2} \left( H_{ij}^{[l] \mathrm{T}}[k] \frac{\partial \phi}{\partial \boldsymbol{u}_{i}^{[l]} [k]} + H_{{\ast} ij}^{[l] \dagger}[k] \frac{\partial \phi}{\partial \boldsymbol{u}_{i}^{[l] \ast} [k]} \right),$$

where

(28)$$\frac{\partial \phi}{\partial \boldsymbol{u}_{i}^{[l] \ast}[k]} = \left( \frac{\partial \phi}{\partial \boldsymbol{u}_{i}^{[l]}[k]} \right)^{{\ast}},$$

because the loss function $\phi \in \mathbb {R}$. The back propagation for the case when the $l$-th layer consists of SL filters or is without MIMO can be obtained from these representations. As for the case when the $l$-th layer consists of filters for carrier recovery, the gradients in terms of the inputs are

(29)$$\frac{\partial \phi}{\partial \boldsymbol{u}_{i}^{[l-1]}[k]} = \boldsymbol{h}_{i}^{[l]}[k: k-M_{l}+1] \circ \frac{\partial \phi}{\partial \boldsymbol{u}_{i}^{[l]} [k]}.$$

Here, note that the filter coefficients for carrier recovery are not controlled by gradient descent in the adaptive ML filters in this study. Finally by applying the above back propagation from the last layer to the first layer, the gradients are obtained in terms of all the coefficients in all the layers, and the coefficients are updated according to Eq. (23).

Approximate calculation

Next, we consider forward propagation of the ML filters at timing integer $k+1$ after a coefficient update at timing integer $k$. From the full calculation above, the output signals of the $l$-th layer at timing integer $k+1$ are given by

(30)$$\boldsymbol{u}_{i}^{[l]}[k+1] = (u_{i}^{[l]}[k+1; k+1], u_{i}^{[l]}[k; k+1], \ldots, u_{i}^{[l]}[k-M_{l}+2; k+1])^{\mathrm{T}},$$

except when the $l$-th layer is for carrier recovery. Here, $\boldsymbol{u}_{i}^{[l]}[k+1]$ does not share any elements with $\boldsymbol{u}_{i}^{[l]}[k]$. However, because the input signals are time series and the step sizes for coefficient updates are usually small for stochastic gradient descent, we can assume that

(31)$$u_{i}^{[l]}[k-n; k+1] \sim u_{i}^{[l]}[k-n; k-n],$$

for an integer $n$. By this approximation, the output signals of the $l$-th layer become

(32)$$\boldsymbol{u}_{i}^{[l]}[k+1] = (u_{i}^{[l]}[k+1; k+1], u_{i}^{[l]}[k; k], \ldots, u_{i}^{[l]}[k-M_{l}+2; k-M_{l}+2])^{\mathrm{T}},$$

in which case they shares elements other than the first element with $\boldsymbol{u}_{i}^{[l]}[k]$. As a result, we only need to calculate $u_{i}^{[l]}[k+1; k+1]$ in forward propagation of the $l$-th layer at timing integer $k+1$.

Under this approximation, the forward propagation in the case when the $l$-th layer consists of a WL MIMO filter is given by

(33)$$\boldsymbol{u}_{i}^{[l]}[k] = \sum_{j=1}^{2} H_{ij}^{[l]}[k:k-M_{l}+1] \boldsymbol{u}_{j}^{[l-1]}[k] + \sum_{j=1}^{2} H_{{\ast} ij}^{[l]}[k:k-M_{l}+1] \boldsymbol{u}_{j}^{[l-1] \ast}[k],$$

where

(34)$$\begin{aligned} &H_{ij}^{[l]}[k:k-M_{l}+1] =\\ &\begin{pmatrix} h_{ij}^{[l]}[0; k] & \cdots & h_{ij}^{[l]}[M^{[l]}-1; k] & 0 & \cdots & 0\\ 0 & \ddots & & \ddots & \ddots & \vdots \\ \vdots & \ddots & & & & 0 \\ 0 & \cdots & 0 & h_{ij}^{[l]}[0; k-M_{l}+1] & \cdots & h_{ij}^{[l]}[M^{[l]}-1; k-M_{l}+1] \\ \end{pmatrix}, \end{aligned}$$

(35)$$\begin{aligned} &H_{{\ast} ij}^{[l]}[k:k-M_{l}+1] =\\ &\begin{pmatrix} h_{{\ast} ij}^{[l]}[0; k] & \cdots & h_{{\ast} ij}^{[l]}[M^{[l]}-1; k] & 0 & \cdots & 0\\ 0 & \ddots & & \ddots & \ddots & \vdots \\ \vdots & \ddots & & & & 0 \\ 0 & \cdots & 0 & h_{{\ast} ij}^{[l]}[0; k-M_{l}+1] & \cdots & h_{{\ast} ij}^{[l]}[M^{[l]}-1; k-M_{l}+1] \\ \end{pmatrix}. \end{aligned}$$

As described above, most elements can be reused from the results at timing integer $k-1$, which greatly mitigates the computational complexity for forward propagation of the $l$-th layer. The same discussion also holds when the $l$-th layer consists of SL filters or is without MIMO. Equation (33) also handles the case when the $l$-th layer consists of filters for carrier recovery, because $H_{ij}^{[l]}[k:k-M_{l}+1] = \mathrm {diag}(\boldsymbol{h}_{i}^{[l]}[k: k-M_{l}+1])$ in this case.

Next the back propagation in the case when the $l$-th layer consists of a WL MIMO filter is calculated as follows. The gradients of the loss in terms of the inputs are given by

(36)$$\frac{\partial \phi}{\partial \boldsymbol{u}_{j}^{[l-1]}[k]} = \sum_{i=1}^{2} \left( H_{ij}^{[l] \mathrm{T}}[k:k-M_{l}+1] \frac{\partial \phi}{\partial \boldsymbol{u}_{i}^{[l]} [k]} + H_{{\ast} ij}^{[l] \dagger}[k:k-M_{l}+1] \frac{\partial \phi}{\partial \boldsymbol{u}_{i}^{[l] \ast} [k]} \right).$$

For the gradients in terms of the filter coefficients, although we can calculate $\partial \phi /\partial \boldsymbol{h}_{ij}^{[l] \ast }[k]$ and $\partial \phi /\partial \boldsymbol{h}_{\ast ij}^{[l] \ast }[k]$ for the coefficients at timing integer $k$, these expressions do not work for gradient descent. They include $\partial \phi /\partial u_{i}^{[l] \ast }[k]$, but this term can be near zero because the magnitude of the first coefficient $h_{ij}^{[l]}[0; k]$ or $h_{\ast ij}^{[l]}[0; k]$ for a conventional causal FIR filter can be very small in Eqs. (34) and (35). Accordingly, instead of obtaining the gradients in terms of the coefficients at timing integer $k$, we modify the coefficient update by using the gradients in terms of the filter coefficients that are related in calculating all the output signals of the $l$-th layer, as follows:

(37)$$\boldsymbol{h}_{ij}^{[l]}[k] \rightarrow \boldsymbol{h}_{ij}^{[l]}[k] - 2 \alpha \sum_{n=0}^{M_{l}-1} \frac{\partial \phi}{\partial \boldsymbol{h}_{ij}^{[l] \ast}[k-n]},$$

(38)$$\boldsymbol{h}_{{\ast} ij}^{[l]}[k] \rightarrow \boldsymbol{h}_{{\ast} ij}^{[l]}[k] - 2 \alpha \sum_{n=0}^{M_{l}-1} \frac{\partial \phi}{\partial \boldsymbol{h}_{{\ast} ij}^{[l] \ast}[k-n]}.$$

In this case, the summations of the gradients are written as

(39)$$\sum_{n=0}^{M_{l}-1} \frac{\partial \phi}{\partial \boldsymbol{h}_{ij}^{[l] \ast}[k-n]} = U_{j}^{[l-1] \dagger}[k:k-M_{l-1}+1] \frac{\partial \phi}{\partial \boldsymbol{u}_{i}^{[l] \ast} [k]},$$

(40)$$\sum_{n=0}^{M_{l}-1} \frac{\partial \phi}{\partial \boldsymbol{h}_{{\ast} ij}^{[l] \ast}[k-n]} = U_{j}^{[l-1] \mathrm{T}}[k:k-M_{l-1}+1] \frac{\partial \phi}{\partial \boldsymbol{u}_{i}^{[l] \ast} [k]},$$

where

(41)$$\begin{aligned}&U_{j}^{[l-1]}[k:k-M_{l-1}+1] =\\ &\begin{pmatrix} u_{j}^{[l-1]}[k; k] & \cdots & u_{j}^{[l-1]}[k-M^{[l]}+1; k-M^{[l]}+1] \\ u_{j}^{[l-1]}[k-1; k-1] & \cdots & u_{j}^{[l-1]}[k-M^{[l]}; k-M^{[l]}] \\ \vdots & & \vdots \\ u_{j}^{[l-1]}[k-M_{l}+1; k-M_{l}+1] & \cdots & u_{j}^{[l-1]}[k-M_{l-1}+1; k-M_{l-1}+1] \\ \end{pmatrix}. \end{aligned}$$

In this way, we can obtain the back propagation for gradient descent to control the filter coefficients of the ML filters in the case of the approximation.

To validate this approach, we compared the computational complexities of forward propagation and back propagation for the full and approximated calculations in the case when the $l$-th layer consists of a WL MIMO filter. The number of complex multiplications required in the $l$-th layer for the last layer’s outputs at one timing integer was evaluated as an indicator of the computational complexity.

The results are listed in Table 1. The full calculation requires $8 M^{[l]} M_{l}$ complex multiplications for forward propagation, whereas the approximated calculation requires $8 M^{[l]}$ complex multiplications, which corresponds to the convolution for the $M^{[l]}$-tap WL MIMO filter itself. As for back propagation, the full and approximated calculations require the same number of complex multiplications: $8 M^{[l]} M_{l}$ for the inputs and $8 M^{[l]} M_{l}$ for the coefficients, giving a total of $16 M^{[l]} M_{l}$. Forward propagation of the ML filters is necessary for every symbol integer, whereas back propagation can be performed intermittently if the temporal variation of the impairments in question is slow. Therefore, the approximation’s mitigation of the computational complexity for forward propagation of the ML filters is significant.

Table 1. Number of the complex multiplications required in $l$-th layer for the last layer’s outputs at one timing integer when the $l$-th layer consists of a WL MIMO filter.

View Table

Appendix B: Dependence of IQ skew impact on IR of PCS-QAM

In this study, we investigated ultra-long-haul transmission with high-order QAM-based PCS. In ultra-long-haul transmission, the received OSNR tends to be low, and thus, PCS-QAM with a higher IR is not applicable. Hence, we evaluated the dependence of the impact of IQ skew on the IR of PCS-64QAM through simulation by using the reference 2$\times$2 SL configuration. The simulation model was the same as in the main text, except that the accumulated CD was 17 ns/nm, which corresponds to 1000 km of SMF transmission, and the received OSNR was different. Tx or Rx IQ skew was again emulated in the X polarization.

Figure 18 shows the NGMI results for reception of a 32-Gbaud PDM-64QAM signal with several different IRs by the 2$\times$2 SL configuration while the Tx or Rx IQ skew was varied from -31 ps to +31 ps. The received OSNR was 30 dB/0.1 nm. The optimum step size for stochastic gradient descent depended on the PCS-QAM IR, and it was roughly optimized. In the case of PCS-64QAM with an IR of 4.8 b/sbl/pol, which corresponds to the IR of 64QAM when FEC is applied with the same code rate, an NGMI above 0.85 was achieved within $\pm$4 ps of Tx IQ skew, whereas the range expanded to about $\pm$22 ps with an IR of 1.6 b/sbl/pol. As for Rx IQ skew, an NGMI above 0.85 was achieved within $\pm$4 ps with an IR of 4.8 b/sbl/pol, and within about $\pm$26 ps with an IR of 1.6 b/sbl/pol. These results show that the tolerance to IQ skew increases as the PCS-QAM IR decreases, even in the same base QAM format.

 

Fig. 18. Simulated NGMI results for reception of a 32-Gbaud PDM-64QAM signal with several IRs by the 2$\times$2 SL configuration when (a) the Tx X-IQ skew or (b) the Rx X-IQ skew was varied.

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Next, Fig. 19 shows the NGMI results for reception of a 32-Gbaud PDM-64QAM signal with an IR of 1.6 b/sbl/pol and several differenct received OSNRs by the 2$\times$2 SL configuration while the Tx or Rx IQ skew was varied. For several of the received OSNRs, the NGMI degradation due to IQ skew showed a similar behavior. However, regardless of the IQ skew, the NGMI deteriorated as the received OSNR decreased. This caused the region in which the NGMI was above a certain threshold to become narrow when the received OSNR was low after all, which indicates that precise Tx and Rx IQ impairment compensation is still important for ultra-long-haul transmission.

 

Fig. 19. Simulated NGMI results for reception of a 32-Gbaud PDM-64QAM signal with an IR of 1.6 b/sbl/pol and several received OSNRs by the 2$\times$2 SL configuration when (a) the Tx X-IQ skew or (b) the Rx X-IQ skew was varied.

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Appendix C: Influence of residual CD

In the adaptive ML filter architecture shown in Fig. 4, CD compensation is applied before the adaptive ML filters. In practical optical fiber transmission systems, the accumulated CD in the received signal is estimated on the Rx side [27]. When such CD estimation is not fully accurate, the residual CD after CD compensation can also be compensated in a subsequent adaptive 2$\times$2 SL MIMO filter for polarization demultiplexing in a conventional DSP configuration. Here, we evaluated the influence of such residual CD on the proposed adaptive ML filters with augmented inputs consisting of the received signal and its complex conjugate with CD compensation. The simulation model was the same as in the main text, except that the compensated CD value for the signal and its complex conjugate before the adaptive ML filters did not precisely match the true value.

Figure 20 shows the NGMI when the Tx or Rx IQ skew in the X polarization was swept. The compensated CD value for the signal and its complex conjugate before the adaptive ML filters was set to $\pm$500 ps/nm and zero, which can be compensated by the 2$\times$2 SL MIMO filter in the second layer, from the true value of 170 ns/nm. When the residual CD was $\pm$500 ps/nm, an NGMI similar to that in the case without residual CD was obtained, regardless of the existence of Tx and Rx IQ skew. Thus, the Tx and Rx IQ impairment compensation by the adaptive ML filters with augmented inputs can work if the residual CD after CD compensation and before the ML filters is within the effective compensation range of the second layer in the ML filters.

 

Fig. 20. Simulated NGMI results for reception of a 32-Gbaud PDM-64QAM signal with an IR of 1.6 b/sbl/pol by the adaptive ML filters when the compensated CD value before the ML filters did not precisely match the true value, while varying (a) the Tx X-IQ skew or (b) the Rx X-IQ skew.

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Appendix D: Compensation of IQ imbalance and phase deviation in Tx and Rx

We focused on IQ skew as the source of Tx and Rx IQ impairments in the previous evaluations. A WL filter can compensate for other linear impairments that occur in Tx and Rx, such as IQ imbalance and phase deviation [25]. Here, we evaluated compensation of IQ imbalance and phase deviation by the proposed adaptive ML filters with augmented inputs consisting of the received signal and its complex conjugate with CD compensation. The simulation model was the same as in the main text, except that IQ imbalance and IQ phase deviation were emulated in Tx and Rx. In DSP, power normalization was performed on the complex-valued received signal, rather than performed individually on the IQ components, to evaluate compensation of IQ imbalance.

Figure 21 shows the results of constellations by the 2$\times$2 SL configuration or the adaptive ML filters under Tx/Rx IQ imbalance and phase deviation. IQ imbalance of 30% and phase deviation of 10$^{\circ }$ were emulated in X polarization in Tx/Rx. The constellations for the X polarization are shown in the figure. In the case of the 2$\times$2 SL configuration, all Tx/Rx IQ imbalance and phase deviation distorted the constellation, though a little effect of Rx phase deviation of 10$^{\circ }$ in X polarization was observed. In contrast, the adaptive ML filters obtained similar constellations to the case without IQ impairments shown in Fig. 5. Since Tx IQ imbalance distorted added Gaussian noise, the observed constellation looked slightly different. However, the adaptive ML filters could compensate for Tx IQ imbalance, compared to the case of the 2$\times$2 SL configuration. Therefore, we confirmed that the adaptive ML filters can also compensate for IQ imbalance and phase deviation in Tx/Rx.

 

Fig. 21. Simulated results of constellations of the X polarization for reception of a 32-Gbaud PDM-64QAM signal with an IR of 1.6 b/sbl/pol by the 2$\times$2 SL configuration or the adaptive ML filters under Tx/Rx IQ imbalance and phase deviation.

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Funding

Ministry of Education, Culture, Sports, Science and Technology (19H02138).

Acknowledgments

We thank Emmanuel Le Taillandier de Gabory for his insightful discussions with us.

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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