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Abstract
We propose a multi-layer cascaded filter architecture consisting of differently sized strictly linear (SL) and widely linear (WL) filters to compensate for the relevant linear impairments in optical fiber communications including in-phase/quadrature (IQ) skew in both transmitter and receiver by using deep unfolding. To control the filter coefficients adaptively, we adopt a gradient calculation with back propagation from machine learning with neural networks to minimize the magnitude of deviation of the filter outputs of the last layer from the desired state in a stochastic gradient descent (SGD) manner. We derive a filter coefficient update algorithm for multi-layer SL and WL multi-input multi-output finite-impulse response filters. The results of a transmission experiment on 32-Gbaud polarization-division multiplexed 64-quadrature amplitude modulation over a 100-km single-mode fiber span showed that the proposed multi-layer SL and WL filters with SGD control could compensate for IQ skew in both transmitter and receiver under the accumulation of chromatic dispersion, polarization rotation, and frequency offset of a local oscillator laser source.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Digital coherent technology in optical fiber communications enables optical fields to be accessed in the digital domain, thus providing a flexible way to compensate for signal distortion [1,2]. Linear impairments incurred during optical fiber propagation, such as chromatic dispersion (CD), polarization mode dispersion (PMD), and polarization rotation, can be compensated by applying the corresponding inverse process in digital signal processing (DSP) at the receiver side [3]. Digital coherent technology also opens up possibilities to use higher-order modulation formats with high spectral efficiency [4–7]. Recently, probabilistic constellation shaping based on higher-order quadrature amplitude modulation (QAM) formats has attracted attention because it can provide both fine rate adaptability and high sensitivity approaching the Shannon limit [8–12].
Higher-order modulation signals are generally susceptible to impairments stemming from imperfections of the transmitter (Tx) or receiver (Rx) devices, including skew between in-phase (I) and quadrature (Q) components, IQ imbalance, and non-uniform IQ frequency characteristics. They thus require rigorous calibration [7] or precise equalization to compensate for the impairments [10]. Since the characteristics of these impairments from device imperfections may vary slightly due to changes of environment, automatically compensating them with adaptive equalization is desirable.
Conventional DSP used in optical fiber communications compensates for the impairments in a block-wise manner based on its physical model [3]. This block-wise compensation provides an efficient filter architecture because impairments have different features. For example, CD is static, independent of polarizations, and yields a large temporal spread, while PMD and polarization rotation are dynamic and cause polarization mixing, and their temporal spread is usually small compared to that of CD. Compensation of the CD and PMD/polarization rotation is achieved by means of complex-valued signals and filters with complex-valued coefficients. Since I and Q components are not treated independently in these complex-valued filters, to compensate for IQ non-uniformity such as skew, gain imbalance, and frequency characteristics, a real-valued multi-input multi-output (MIMO) filter, which has real-valued inputs and outputs of I and Q, has been used instead of a complex-valued MIMO filter for polarization demultiplexing [13–15]. This real-valued MIMO filter is equivalent to an augmented complex-valued MIMO filter in which complex-valued signals and their complex conjugates are inputs. Thus augmented complex-valued MIMO filter is known as a widely linear (WL) filter [16]. In this context, the conventional complex-valued filter is called strictly linear (SL).
However, if there is an accumulation of CD through fiber propagation, and if CD compensation is followed by a WL MIMO filter, IQ non-uniformity can no longer be compensated by using only the WL MIMO filter. In [15], CD compensation was applied to each of the I and Q components of the X and Y polarizations independently and a 4 $\times$ 2 WL MIMO filter with these complex-valued signals as its inputs was used so that IQ mixing did not occur during CD compensation. In [16], a large WL MIMO filter with a large temporal spread was used to compensate for IQ skew and CD simultaneously in one WL filter. Both these approaches require multiple filters to compensate for CD, which usually have a large temporal spread, and thus large circuit resource requirements. These approaches are also affected by the frequency offset (FO) from a laser source used as a local oscillator (LO), as discussed later.
In this study, we investigate an alternative approach to compensating all the relevant impairments including Tx/Rx IQ skew while considering the order in which they occur. Since IQ skew (compensation) and CD (compensation) are not mutually commutative, the order in which they are compensated is important. We propose and investigate a multi-layer cascaded filter architecture consisting of differently sized SL and WL filters to compensate all the relevant impairments by means of deep unfolding. The coefficients of these filters are controlled adaptively. It is well known that adaptive control of a single SL or WL MIMO filter can be achieved by stochastic gradient descent (SGD) to minimize the magnitude of deviation of the filter outputs from the desired state as a loss function [3,13–16]. However, it cannot be applied to each of multi-layer cascaded filters straightforwardly. When multiple impairments are treated in cascaded compensation blocks, distortions still remain in the outputs of each block until the last block, which makes the individual optimization of the filter coefficients in each block difficult. In addition, the filter outputs are nonlinear in terms of the filter coefficients of the multiple blocks, since the multiplication of two or more filter coefficients appears in the outputs. Therefore, for the SGD of multi-layer filters, we adopt a gradient calculation inspired by machine learning with neural networks. In a supervised learning of neural networks, the gradient of the loss function with respect to each weight is calculated from the last layer by the chain rule, known as back propagation. Gradient calculation with back propagation can be applied not only to neural networks but also to general differentiable input-output relations [17], e.g., the step size optimization of iterative MIMO processing [18]. We apply the concept of the gradient calculation with back propagation to control the filter coefficients of multi-layer SL and WL finite-impulse response (FIR) filters by means of SGD. This multi-layer SL and WL filter architecture, which uses the features and restrictions of impairments occurring in optical fiber communications, enables compensation of IQ skew in both Tx and Rx when a conventional CD compensation block and a polarization demultiplexing block are used. Moreover, the SGD of multi-layer filters bridges conventional block-wise linear compensation DSP and nonlinear compensation based on neural networks [19].
To evaluate the compensation capability of the proposed multi-layer SL and WL filters with adaptive SGD control for IQ skew in both Tx and Rx, we performed a transmission experiment on 32-Gbaud polarization-division multiplexed (PDM) 64QAM over a 100-km single-mode fiber (SMF) span. The results showed that they could compensate for IQ skew in both Tx and Rx in the presence of CD, polarization rotation, and FO.
In this paper, we first introduce the concept of multi-layer SL and WL filters to compensate for all the relevant impairments and derive an update algorithm for adaptive filter coefficients by means of SGD with back propagation from the outputs of the last filter. Then, we run a simulation using a simple model of 32-Gbaud PDM-quadrature phase shift keying (QPSK) after SMF transmission. Finally, we report the experimental results of IQ skew compensation by the proposed multi-layer SL and WL filters with SGD in the 100-km SMF transmission of 32-Gbaud PDM-64QAM.
2. Theory
We first introduce our multi-layer filter architecture, which consists of differently sized SL and WL MIMO filters to compensate for all the relevant impairments including IQ skew in both Tx and Rx. Then, we derive a filter coefficient update algorithm by SGD with back propagation from the last filter outputs.
Let us first consider impairments that occur in optical fiber communication and how to compensate for them. Figure 1(a) shows a schematic diagram of dominant impairments that arise in optical fiber communication with a PDM signal and intradyne coherent reception. At the Tx side, impairments from device imperfections such as IQ skew in a modulator or the frequency characteristics of driver amplifiers occur. CD, PMD, and polarization rotation occur through fiber propagation. Nonlinear distortion due to the Kerr effect may also occur through fiber propagation, though we ignore it here for simplicity. At the Rx side, FO due to the frequency difference between the carrier of the signal and an LO, device impairments such as IQ skew and frequency characteristics of trans-impedance amplifiers in a coherent receiver occur.
Fig. 1. (a) Model of impairments in optical fiber communications with coherent detection, (b) conventional digital processing for demodulation, and (c) ideal inverse processing to compensate all impairments. CD: chromatic dispersion, PMD: polarization mode dispersion.
Except for the Kerr effect, these impairments are all linear phenomena. Some of them, such as the IQ skew, affect IQ components differently. Therefore, they should be treated using either a real-valued MIMO process for the real-valued signal vectors of the I and Q components of the X and Y polarizations [20] or a WL MIMO process [16]. CD, PMD, polarization rotation, and FO are SL (MIMO) processes. Although all of these impairments can be compensated by one large WL MIMO filter with IQ and XY cross-terms and a large temporal spread, as they are all linear, this requires large circuit resources.
Figure 1(b) shows the conventional DSP for demodulation in optical fiber communications. Each impairment is compensated in a block-wise manner in accordance with its physical model. CD compensation is performed with SL filters for two polarizations individually. Polarization demultiplexing is done with an SL adaptive MIMO filter having a relatively small temporal spread. Carrier recovery, which compensates for the frequency and phase offset between the carrier of the signal and the LO, is performed for each polarization. The order of these blocks can be rearranged because they are all SL, though CD compensation is usually positioned first, since adaptive filter control algorithms for polarization demultiplexing (such as constant modulus algorithm (CMA) and decision directed least mean square (LMS) algorithm) do not work well if a large accumulated CD remains at the outputs of the MIMO filter. Note that IQ skew in Tx and Rx cannot be compensated in these SL MIMO filters.
The WL process and SL process are generally not commutative because the WL process is equivalent to four-dimensional rotation [16,20]. Therefore, to compensate for all the linear impairments including IQ skew in a block-wise manner, each impairment should be compensated in reverse order, as shown in Fig. 1(c). Rx device compensation, polarization demultiplexing, carrier recovery, and Tx device compensation blocks should be adaptively controlled, but distortion still remains at the outputs of each block except for the last block, which makes it difficult to optimize the filter coefficients in each block individually. Therefore, we use a gradient calculation with back propagation from the outputs of the last block to control the filter coefficients so that the magnitude of deviation of the last outputs from the desired state is stochastically minimized.
Figure 2 shows the proposed multi-layer SL and WL MIMO filters corresponding to Fig. 1(c) to compensate for all the linear impairments including IQ skew. We regard all the filters as half-symbol spaced FIR MIMO filters for simplicity. The multi-layer filters shown in Fig. 2 are composed of five layers for Rx device compensation, CD compensation, polarization demultiplexing, carrier recovery, and Tx device compensation, in this order. The first layer consists of two 2 $\times$ 1 WL MIMO filters for two polarizations without polarization cross-term to compensate for IQ non-uniformity in Rx whose coefficients are controlled adaptively. The second layer consists of two 1 $\times$ 1 SL filters for CD compensation whose coefficients are treated as static. The third layer consists of a 2 $\times$ 2 SL MIMO filter for polarization demultiplexing and PMD compensation. These coefficients are adaptively controlled fast enough to track the variation of polarization states in fiber propagation. The fourth layer consists of two 1 $\times$ 1 1-tap SL filters to compensate the phase and frequency offset. The fifth layer consists of two adaptive 2 $\times$ 1 WL MIMO filters for compensation of IQ non-uniformity in Tx. In this multi-layer FIR filter architecture, the input samples of longer duration are related to the last outputs in one symbol time slot, going back over the filter layer.
Fig. 2. Architecture of multi-layer FIR filters for adaptive compensation of impairments corresponding to Fig. 1(c). WL: widely linear, SL: strictly linear.
In the multi-layer filters shown in Fig. 2, the filter coefficients of the first, third, and fifth layers are adaptively controlled. The conventional SGD approach for one MIMO filter is extended here to multi-layer SL and WL MIMO filters. The FIR filter outputs are differentiable in terms of their inputs and filter coefficients. Therefore, gradients of the loss function in terms of the filter coefficients of each layer can be calculated successively with back propagation, the same as in machine learning with neural networks, even if the loss function is composed of the outputs of the last layer. The filter coefficients of the second layer are set by the physical model of the CD as $h_{\mathrm {CD}} = \mathcal {F}^{-1}[H_{\mathrm {CD}}[\omega ]]$ and
where $\mathcal {F}^{-1}$ is the inverse Fourier transform and $c$, $D$, $z$, and $\nu$ are the speed of light, the dispersion coefficient, the transmission distance, and the carrier frequency of the signal, respectively. The filter coefficients of the fourth layer are and the compensated phases $\theta _{i} [k]$ ($i = 1, 2$ for two polarizations, $k$ is the sample time integer) are determined by the decision-directed digital phase-locked loop (PLL) for the output symbols of the last layer outside SGD.In the following, we describe the input-output relation, or forward propagation, of the SL and WL MIMO filters. Then we derive backward propagation for the calculation of gradients.
2.1 Forward propagation
We consider the filter of the $l$-th layer. In the multi-layer FIR filter architecture, the samples of longer duration are related to the outputs of the last layer in one time slot. If we define $M^{[l]}_{\mathrm {out}}$ as the length of the outputs of the $l$-th layer required to obtain the outputs of the last layer in one symbol time slot, the outputs of the filter of the $l$-th layer at the sample time integer $k$ with the length of $M^{[l]}_{\mathrm {out}}$ are vectorized as
Strictly linear MIMO FIR filter
When the filter of the $l$-th layer is an SL MIMO filter, its output samples are described as
Widely linear MIMO FIR filter
When the filter of the $l$-th layer is a WL MIMO filter, its output samples are described as
2.2 Back propagation
At the last $L (=5)$-th layer, $M^{[L]}_{\mathrm {out}} = 1$ and is described as $y_{i}[k] = \boldsymbol{u}^{[L]}_{i} [k]$. Here, we consider the symbol timing integer $k = 2 l$. The loss function $\phi$ to be minimized is composed of $y_{i}[k]$. The coefficient $\xi$ in any filter is controlled with SGD. Since we are dealing with complex-valued signals and filter coefficients, we utilize Wirtinger derivatives. Gradient descent becomes
where $\alpha$ is the step size.In the case of CMA with the amplitude criterion of $r$, the loss function is
The gradients in terms of the last outputs $y_{i}[k]$ are calculated as where $e_{i} = r^{2} - |y_{i}[k]|^{2}$.In the case of decision-directed LMS, the loss function is
and the gradients in terms of the last outputs are where $e_{i} = d(y_{i}[k]) - y_{i}[k]$ and $d$ is the decision operation. These are the gradients in terms of the outputs of the last $L$-th layer.Given the gradients in terms of the outputs of the filter of the $l$-th layer, we can use back propagation to calculate the gradients in terms of its inputs, or the outputs of the ($l-1$)-th layer, and its filter coefficients can be calculated with back propagation. We derive back propagation for the SL and WL MIMO FIR filters as follows.
Strictly linear MIMO FIR filter
After calculating differentials and arranging them, the gradients of the loss function $\phi$ in terms of the inputs and filter coefficients of an SL MIMO filter are
Widely linear MIMO FIR filter
Similarly, the gradients of the loss function $\phi$ in terms of the inputs and filter coefficients of a WL MIMO filter are
Now, we have an adaptive filter coefficient update algorithm for all the filters. The gradients of the loss function in terms of the filter coefficients back to the first layer can be calculated from the last $l$-the layer with these back propagation equations successively.
2.3 Simulation
First, we evaluated the proposed multi-layer SL and WL filters with SGD to compensate for all the relevant impairments by numerical simulation using a simple model. The reception of a 32-Gbaud PDM-QPSK signal with coherent detection was simulated. In Tx and Rx, IQ skew was imposed in the X polarization where the Q component was delayed. We assumed no polarization rotation, PMD, laser phase noise, or FO. After coherent reception, DSP was applied to the two-fold oversampled signals.
We evaluated three types of DSP architecture. The first was a conventional DSP for demodulation in optical fiber communications, as shown in Fig. 1(b), which is hereinafter referred to as 2 $\times$ 2 SL after the MIMO filter for polarization demultiplexing. The second was a reference of the conventional method with IQ skew compensation capability, which helps clarify the results. With this one, a 4 $\times$ 2 WL MIMO filter was used for the polarization demultiplexing block in Fig. 1(b) instead of a 2 $\times$ 2 MIMO filter (corresponding to the real 4 $\times$ 4 filter in [15]), which is referred to as 4 $\times$ 2 WL. The third was the proposed multi-layer SL and WL filters with SGD, which we call Multi-layer SL&WL. CD compensation was performed by a 61-tap FIR filter enabling compensation of accumulated CD over 100-km SMF in all cases. The polarization demultiplexing block had the tap length of 21. In the case of Multi-layer SL&WL, Rx/Tx device compensation was performed by two 2 $\times$ 1 WL filters with five taps for two polarizations. The loss function was decision-directed LMS.
The received constellations of the X and Y polarizations after DSP for demodulation under the back-to-back condition are shown in Fig. 3. The received optical signal-to-noise ratio (OSNR) was set to 30 dB/0.1 nm. Figure 3(a) shows the simulation results under the condition with the X-IQ skew of 5 ps in Tx and the DSP of 2 $\times$ 2 SL. Figure 3(b) shows that with the X-IQ skew of 5 ps in Rx and the DSP of 2 $\times$ 2 SL. Since the SL MIMO filter does not have the capability of IQ skew compensation, signal distortion occurred when there was IQ skew in either Tx or Rx. Figures 3(c) and (d) show the results with 4 $\times$ 2 WL with the X-IQ skew of 5 ps in Tx and Rx, respectively. In contrast to the case of 2 $\times$ 2 SL, IQ skew was compensated and good constellations were obtained. Figures 3(e) and (f) show the results with Multi-layer SL&WL with the X-IQ of 5 ps skew in Tx and Rx. IQ skew was fully compensated here as well.
Fig. 3. Simulation results for compensation of transmitter and receiver IQ skew under the back-to-back condition. Received constellations of PDM-QPSK by ((a) and (b)) 2 $\times$ 2 SL, by ((c) and (d)) 4 $\times$ 2 WL, and by ((e) and (f)) Multi-layer SL&WL. IQ skew of the X polarization signal was introduced at the transmitter side in (a), (c), and (e), and at the receiver side in (b), (d), and (f). The left side shows X polarization and the right side shows Y polarization.
The received constellations of the X and Y polarizations after demodulation DSP in 100-km SMF transmission are shown in Fig. 4. The upper images (Figs. 4(a), (c), and (e)) are the results with the X-IQ skew of 5 ps in Tx, and the lower ones (Figs. 4(b), (d), and (f)) are those with the X-IQ skew of 5 ps in Rx. In the case of 2 $\times$ 2 SL (Figs. 4(a) and (b)), the IQ skew was not compensated at all, the same as under the back-to-back condition. In the case of 4 $\times$ 2 WL (Figs. 4(c) and (d)), the IQ skew in Rx was not compensated, in contrast to the results under the back-to-back condition shown in Fig. 3(d). This is because the 4 $\times$ 2 WL MIMO filter is not commutative to the CD compensation block (as can be seen in Fig. 1(c)), which causes IQ mixing to occur in the CD compensation. (If the CD compensation was applied to both I and Q independently to avoid IQ mixing through the CD compensation, the IQ skew in Rx can be compensated [15], while it requires additional large circuit resources.) In contrast, in the case of Multi-layer SL&WL (shown in Figs. 4(e) and (f)), good constellations were obtained even when the IQ skew in both Tx and Rx was imposed. This result demonstrates that the proposed multi-layer SL and WL filters with SGD can effectively compensate for IQ skew in both Tx and Rx under the accumulation of CD.
Fig. 4. Simulation results for compensation of transmitter and receiver IQ skew after 100-km SMF transmission and CD compensation: Received constellations of PDM-QPSK by ((a) and (b)) 2 $\times$ 2 SL, by ((c) and (d)) 4 $\times$ 2 WL, and by ((e) and (f)) Multi-layer SL&WL. IQ skew of the X polarization signal was introduced at the transmitter side in (a), (c), and (e), and at the receiver side in (b), (d), and (f). The left side shows X polarization and the right side shows Y polarization.
Whether the loss function of decision-directed LMS or data-aided LMS in terms of the adaptive filter coefficients is down-convex is not well known (or hard to guarantee to be down-convex), though polarization demultiplexing with an adaptive filter has been succeeded in optical fiber communications. Thus, to show that the proposed multi-layer SL and WL filters with SGD works not only in a specific condition, we evaluated the performance in the case where IQ skew exists in all Tx/Rx and X/Y by numerical simulation. The simulation model is the same as the previous one of 100-km transmission except that the received OSNR was set to 15 dB/0.1 nm. Each of Tx/Rx X/Y IQ skew was a random value from a zero-mean Gaussian distribution with a standard deviation equal to 5 ps. The averaged error vector magnitude (EVM) of the received signals over two polarizations was evaluated 100 times. Figure 5 shows the results of the histogram of the averaged EVM. Since 2 $\times$ 2 SL cannot compensate for IQ skew in both Tx and Rx and 4 $\times$ 2 WL cannot compensate for IQ skew in Rx in this condition, the averaged EVM with these two was spread and 4 $\times$ 2 WL provided slightly better results. In contrast, Multi-layer SL&WL provided good and almost constant averaged EVM, since it can compensate for IQ skew in both Tx and Rx. This result demonstrates that the proposed Multi-layer SL&WL works even in the complicated condition where IQ skew exists in all Tx/Rx and X/Y.
3. Experimental results
Next, we experimentally evaluated the effectiveness of the proposed technique with 32-Gbaud PDM-64QAM. We focused on the capability of the IQ skew compensation in Tx and Rx, the same as in the simulation.
3.1 Experimental setup
A schematic diagram of the experimental setup is shown in Fig. 6. On the Tx side, a laser source at the frequency of 193.3 THz having a linewidth of about 100 kHz was modulated to 32-Gbaud PDM-64QAM by waveforms generated with a four-channel digital-to-analog converter (DAC) at the sampling rate of 64 GS/s with a vertical resolution of eight bits. Forward error correction (FEC) of low-density parity-check code (LDPC) for DVBS-2 with a frame length of 64,800 and a code rate of 4/5 was used. Eight FEC frames were generated for each polarization by loading random bits to their payload and were then mapped to PDM-64QAM with gray mapping. In this experiment, a pilot sequence was inserted for each polarization to perform a pilot-based DSP for demodulation in Rx [21,22]. One pilot symbol of QPSK was inserted every 25 symbols. In addition, due to the restriction of the DAC used in the experiment, an overhead of QPSK symbols was also inserted to ensure the periodicity of the waveforms generated by DAC. QPSK symbols in the pilot and the overhead, which were about 7% in total, were set to the outer symbol points of 64QAM for simplicity of the decision processing in DSP, which yields 0.7 dB of the power penalty. The data generated in this way were upsampled to two-fold oversampling and the root raised cosine filter with a roll-off factor of 0.1 was performed. Frequency characteristics in the Tx devices were pre-compensated.
Fig. 6. Experimental setup for compensation of transmitter and receiver IQ skew. LD: laser diode, DAC: digital-to-analog converter, MOD: modulator, PS: polarization scrambler, SMF: single-mode fiber, ASE: amplified spontaneous emission, EDFA: erbium-doped fiber amplifier, OBPF: optical bandpass filter, ADC: analog-to-digital converter.
The modulated optical signal was transmitted to a 100-km SMF span after slow polarization scrambling at a rate of 10 $\times$ 2$\pi$ rad/s. The span input optical power was set to 0 dBm. After transmission, amplified spontaneous emission was added to the signal to set an OSNR value. The optical signal was amplified by an erbium-doped fiber amplifier and filtered by an optical bandpass filter having a 3-dB bandwidth of 50 GHz, and then received by a polarization diversity coherent receiver with a laser source used as an LO having a linewidth of about 100 kHz. The laser sources of the signal and the LO were free-running and had frequency differences fluctuated within about $\pm$100 MHz. The four outputs of the coherent receiver were sampled with a digital oscilloscope at a sampling rate of 80 GS/s with a vertical resolution of eight bits. IQ skew in Tx and Rx was emulated digitally as a delay of the Q components of the X polarization.
DSP was performed offline. The received signals were normalized and resampled to two-fold oversampling. After matched root raised cosine filtering, detection of the pilot position and alignment was performed before the main DSP. This procedure was done in the following manner. First, CD compensation was applied to the received signals and the pilot position was detected according to the difference of the averaged powers between the 64QAM signal and the QPSK pilot. Then, polarization demultipexing was performed by CMA for the pilot. After FO compensation based on the fourth power of the signal [23], timing alignment was done by using correlation to the known pilot sequence. The main DSP for demodulation, for which we evaluated the same three methods as in the previous simulation (2 $\times$ 2 SL, 4 $\times$ 2 WL, and Multi-layer SL&WL) was applied to the received signal after the matched filter with timing alignment before performing CD compensation again. In the experiment with PDM-64QAM, the filter coefficient update was carried out using the symbols of the QPSK pilot with data-aided LMS, with no updates by the 64QAM signal. The tap lengths of the filters were the same as in the simulation: 61 taps for CD compensation, 21 taps for polarization demultiplexing, and five taps for compensation of Tx/Rx device impairments. Carrier recovery was performed using both the signal and the pilot symbols with decision-directed PLL. After the main DSP for demodulation and removal of the pilot and the overhead, the normalized generalized mutual information (NGMI) as a performance indicator was calculated [24], and the post FEC bit error rate (BER) was calculated with FEC decoding. The received waveforms were acquired three times under each condition and about 0.8 Mbits were evaluated for the post-FEC BER in each acquisition. The averaged NGMI was also evaluated. As the post-FEC BER largely fluctuated around the FEC cliff, we based our evaluation on its median.
3.2 Back-to-back condition
We first evaluated the capability of IQ skew compensation under the back-to-back condition. The received OSNR was set to 30 dB/0.1 nm. The results of the post-FEC BER while changing X-IQ skew from $-$10 ps to +10 ps in Tx and Rx are shown in Fig. 7(a) and (b), respectively. The error-free results are plotted at 10$^{-5}$ for visibility. In the case of the IQ skew in Tx shown in Fig. 7(a), the proposed Multi-layer SL&WL could compensate for it, while 2 $\times$ 2 SL and 4 $\times$ 2 WL could not. In the case of the IQ skew in Rx shown in Fig. 7(b), 4 $\times$ 2 WL and Multi-layer SL&WL could compensate for it. In contrast to the simulation results (Fig. 3(c) and (d)), the IQ skew in Tx was not compensated by 4 $\times$ 2 WL, while IQ skew in Rx was. This is because FO existed in the experiment, rather than being set to zero as in the simulation, which renders the carrier recovery block and the Rx device compensation non-commutative, as shown in Fig. 1(c). Figures 8(a) and (b) show the corresponding evaluation results of the NGMI. The proposed Multi-layer SL&WL could compensate for IQ skew in both Tx and Rx. Figure 9 shows the received constellations after demodulation with Multi-layer SL&WL for several IQ skew values in Rx.
Fig. 7. Experimental results for post-FEC BER under back-to-back condition with (a) Tx X-IQ skew and (b) Rx X-IQ skew.
Fig. 8. Experimental results for NGMI under back-to-back condition with (a) Tx X-IQ skew and (b) Rx X-IQ skew.
Fig. 9. Received constellations after demodulation DSP with Multi-layer SL&WL under back-to-back condition with Rx X-IQ skew of (a) $-$5 ps, (b) 0 ps, and (c) +5 ps. The left side shows X polarization and the right side shows Y polarization.
Figure 10 shows the dependence of the NGMI on IQ skew in Tx and Rx when they were added simultaneously. The 2 $\times$ 2 SL in Fig. 10(a) had no tolerance to IQ skew in either Tx or Rx. The 4 $\times$ 2 WL in Fig. 10(b) had tolerance to IQ skew in Tx but not in Rx. In contrast, Multi-layer SL&WL had tolerance to IQ skew in both Tx and Rx, and was able to deliver a stable performance.
Fig. 10. Dependence of NGMI on Tx X-IQ skew and Rx X-IQ skew in the cases of (a) 2 $\times$ 2 SL, (b) 4 $\times$ 2 WL, and (c) Multi-layer SL&WL under back-to-back condition.
3.3 100-km SMF transmission condition
Finally, we evaluated the capability of IQ skew compensation after transmission over a 100-km SMF span. The received OSNR was set to 30 dB/0.1 nm. The results of the post-FEC BER while changing X-IQ skew from $-$10 ps to +10 ps in Tx and Rx are shown in Figs. 11(a) and (b), respectively. The corresponding results of the NGMI are shown in Fig. 12. Both 2 $\times$ 2 SL and 4 $\times$ 2 WL were unable to compensate for IQ skew in both Tx and Rx, in contrast to the results for the back-to-back condition, due to the existence of accumulated CD and FO. Multi-layer SL&WL was able to compensate for them both, the same as in the back-to-back condition. Figure 13 shows the dependence of the NGMI on IQ skew in Tx and Rx after 100-km SMF transmission. These results show that the proposed Multi-layer SL&WL could compensate for IQ skew in both Tx and Rx even in the presence of CD accumulation, polarization rotation, and FO.
Fig. 11. Experimental results for post-FEC BER after 100-km SMF transmission with (a) Tx X-IQ skew and (b) Rx X-IQ skew.
Fig. 12. Experimental results for NGMI after 100-km SMF transmission with (a) Tx X-IQ skew and (b) Rx X-IQ skew.
Fig. 13. Dependence of NGMI on Tx X-IQ skew and Rx X-IQ skew in the cases of (a) 2 $\times$ 2 SL, (b) 4 $\times$ 2 WL, and (c) Multi-layer SL&WL after 100-km SMF transmission.
In this study, we examined multi-layer SL and WL FIR filters. However, conventional DSP uses frequency domain filters as well, especially for CD compensation since the required circuit resources can be reduced [25,26]. Thus, our future work will examine the treatment of multi-layer SL and WL filters in mixed time and frequency domains.
4. Conclusion
We have developed a multi-layer filter architecture consisting of SL and WL MIMO filters to compensate for relevant impairments including Tx/Rx IQ skew and adaptive control of their filter coefficients. Taking inspiration from the idea of gradient calculation with back propagation in machine learning with neural networks, we derived an adaptive filter coefficient algorithm for multi-layer SL and WL MIMO FIR filters with SGD based on a loss function composed of the outputs of the last layer. We evaluated the compensation capability of the proposed multi-layer SL and WL filters with adaptive SGD control for IQ skew in both Tx and Rx through simulations and an experiment on 32-Gbaud PDM-64QAM transmission over a 100-km SMF span. The results showed that they could compensate for IQ skew in both Tx and Rx in the presence of CD, polarization rotation, and FO.
Appendix A: Convergence speed of the Multi-layer SL&WL
The convergence speed is one of the important characteristics of an adaptive filter. We compared the convergence speed in the cases of 2 $\times$ 2 SL, 4 $\times$ 2 WL, and Multi-layer SL&WL by numerical simulation. The simulation model is the same as the one of 100-km SMF transmission of 32-Gbaud PDM-QPSK described in Section 2.3 while the received OSNR was set to 15 dB/0.1 nm. The filter coefficients at the center of the diagonal position of the MIMO FIR filters were initialized to one and the rest were zero. Figure 14 shows the results of the time development of the loss function under the condition with the X-IQ skew of 5 ps in Rx. The convergence speed of Multi-layer SL&WL was similar to that of 2 $\times$ 2 SL and 4 $\times$ 2 WL in this case.
Appendix B: Computational complexity
The computational complexity is another important characteristic of an adaptive filter to implement an embedded digital signal processor. The number of complex multiplications for outputs in one symbol time slot is a rough indicator of the computational complexity. We consider the example of the five-layer architecture of Fig. 2 and the length of each FIR filter is the same as used in Section 2.3.
In the case of the architecture of Fig. 2, the filter coefficients of the second layer (CD compensation) are assumed to be fixed. Those of the first layer (Rx device compensation) and the fifth layer (Tx device compensation) can be adaptively controlled, while changes of Tx/Rx device impairment are usually slow. Thus, these filter coefficients can be controlled by an off-chip manner, that is, an implementation outside of an embedded digital signal processor, and they can be assumed as constant in terms of the embedded one. The third layer (polarization demultiplexing) is only time-varying and to be adaptively controlled in the embedded processor.
According to Section 2.1, for forward propagation of the proposed multi-layer SL and WL filters, the $l$-th layer requires $4 M^{[l]} M^{[l]}_{\mathrm {out}}$ complex multiplications if it is 2 $\times$ 2 SL, and $8 M^{[l]} M^{[l]}_{\mathrm {out}}$ if it is 4 $\times$ 2 WL. Before the time-varying layer, they reduce to $4 M^{[l]}$ and $8 M^{[l]}$, respectively. Using the parameters of $M^{[1]}=5, M^{[2]}=61, M^{[3]}=21, M^{[4]}=1, M^{[5]}=5$, the proposed multi-layer SL and WL filter architecture requires 592 complex multiplications in total for forward propagation. In contrast, if one large 4 $\times$ 2 WL FIR filter is used to compensate for all the relevant impairments, it requires more complex multiplications of 712 in total for forward propagation since the corresponding FIR length is 89.
For back propagation, since the loss function is real-valued, the $l$-th layer requires $4 M^{[l]} M^{[l]}_{\mathrm {out}}$ complex multiplications for the gradients in terms of the inputs, and $4 M^{[l]} M^{[l]}_{\mathrm {out}}$ for the gradients in terms of the filter coefficients if it is 2 $\times$ 2 SL. If the $l$-th layer is 4 $\times$ 2 WL, it requires $8 M^{[l]} M^{[l]}_{\mathrm {out}}$ complex multiplications for the gradients in terms of the inputs, and $8 M^{[l]} M^{[l]}_{\mathrm {out}}$ for the gradients in terms of the filter coefficients. Since the fourth and fifth layer do not require the gradients in terms of the filter coefficients in the embedded processor, the proposed multi-layer SL and WL filter architecture requires 450 complex multiplications in total for back propagation. In the case of one large 4 $\times$ 2 WL FIR filter, it requires 712 complex multiplications in total for back propagation.
These results show that the proposed multi-layer SL and WL filter architecture is more effective to compensate for all the relevant impairments than one large WL MIMO filter. We acknowledge that further work is required in algorithm simplification and circuit design in order to enable practical implementation on an embedded processor, even though advances in process technology has been facilitating the adoption of new and always more effective compensation schemes in coherent optical communication for more than a decade.
Acknowledgments
We thank Hidemi Noguchi, Masaki Sato, Kohei Hosokawa, Norifumi Kamiya, and Emmanuel Le Taillandier de Gabory for insightful discussions.
Disclosures
The authors declare no conflicts of interest.
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