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Ultra-fast RSOP tracking via 3 pilot tones for short-distance coherent SCM systems

Hexun Jiang, Zhiqun Zhai, Xiaobo Zeng, Mengfan Fu, Yunyun Fan, Lilin Yi, Weisheng Hu, and Qunbi Zhuge

Open Access Open Access

Abstract

We propose an algorithm to track the rotation of state of polarization (RSOP) for short-distance coherent subcarrier-multiplexing systems. 3 pilot tones are used to estimate RSOP matrices on a block-by-block basis and recover phase noise as well. An ultra-fast RSOP tracking ability using the proposed algorithm is demonstrated by experiment. Specifically, the bit error rate increases from 2.3×10−3 to 5.6×10−3 when the RSOP speed increases from 0 rad/s to 50 Mrad/s. We also demonstrate the robustness of the algorithm against polarization mode dispersion and polarization dependent loss.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

A point-to-multipoint (PTMP) coherent architecture is an advanced solution for aggregation networks such as 5G-Xhaul and passive optical network [1]. The PTMP architecture divides a high-bandwidth carrier into multiple lower-bandwidth subcarriers utilizing digital subcarrier multiplexing (SCM) [2], and these subcarriers can be independently routed to and from different access nodes. Among access nodes, 5G fronthaul is a special scenario, where some transmission cables close to the radio unit base stations are aerial, making them easily affected by environmental variations. The main influence on the signal is the fast rotation of the state of polarization (RSOP). It is not a problem for the traditional aggregation network based on intensity modulation direct detection systems. However, it is critical for the novel coherent SCM PTMP structure, since digital equalization may fail when ultra-fast RSOP occurs. It has been reported that the RSOP speed can reach hundreds of krad/s under hostile environments [3,4], and even higher than 8 Mrad/s [5] caused by lightning strikes.

In conventional coherent digital signal processing (DSP), RSOP is tracked by a 2×2 multiple-input and multiple-output (MIMO) finite-impulse response (FIR) equalizer, which can tolerate hundreds of krad/s RSOP mainly induced by mechanical vibrations [6], but may fail when facing faster RSOP. To bridge the gap, many ultra-fast RSOP tracking algorithms are proposed, and they can be roughly divided into two categories. The algorithms in the first category employ a Kalman filter to track the fast RSOP. The Kalman filter based algorithms in [7–9] do not consider band-limited influence, and their performances will degrade in the presence of severe filtering impairments. The Kalman filter combined with the 2×2 MIMO FIR equalizer [10–12] can track the fast RSOP and compensate for the band-limited effects simultaneously. However, the complexity is much higher than the conventional 2×2 MIMO equalizer based on a constant-modulus algorithm or least-mean-square (LMS) algorithm. The algorithms in the second category [13–15] modify the 2×2 MIMO structure to a simplified 2-stage equalizer. The RSOP tracking ability in [13] is not sufficient to handle all situations. The methods in [14,15] achieve 18 Mrad/s and 20 Mrad/s SOP tracking speed, respectively, but they require an additional 12.5% overhead for training symbols, reducing transmission rate significantly.

In this paper, we propose a novel ultra-fast polarization tracking algorithm using 3 pilot tones (PTs) for short-distance coherent SCM systems. The PT has been utilized for the compensation of laser noise and fiber nonlinearities [16,17], modulation format identification [18] and polarization recovery [19]. It is worth mentioning that a PT is used to recover a static SOP in [19], but it is not suitable for the fast RSOP tracking. In our algorithm, 3 PTs are used to track the fast RSOP and recover phase noise. At the transmitter (Tx), the 3 PTs with different SOPs are inserted into the spectral gap between subcarriers. With the aid of these PTs, the RSOP matrix and phase noise can be obtained at the receiver (Rx). The performance of the proposed algorithm is evaluated by experiment. Due to the lack of a high-speed polarization scrambler, we add RSOP in the Tx DSP. An ultra-fast RSOP tracking ability is demonstrated. When the RSOP speed increases from 0 rad/s to 50 Mrad/s, the bit error rate (BER) only increases from 2.3×10−3 to 5.6×10−3. Then, we evaluate and show the robustness of the proposed algorithm in the presence of other polarization effects. The performance slightly degrades in the presence of polarization mode dispersion (PMD) equivalent to a fiber length of 100 km, and a polarization dependent loss (PDL) up to 3 dB. Finally, the computational complexity of the proposed algorithm is discussed.

2. Principle

Generally, RSOP can be modeled as a 2${\times} $2 unitary matrix in the Cayley-Klein form [9] as

$${R} = \left[ {\begin{array}{cc} {a + j\textrm{b}}&{c + jd}\\ { - c + jd}&{a - jb} \end{array}} \right]$$
where $a,b,c,d$ are the 4 real values, j is the imaginary unit. The rotation matrix is a unitary matrix with the relationship:
$${a^2} + {b^2} + {c^2} + {d^2} = 1$$

After chromatic dispersion (CD) compensation and frequency offset (FO) compensation in the Rx DSP, the relationship between received samples and transmitted samples can be expressed as

$$\left[ {\begin{array}{c} X\\ Y \end{array}} \right] = {e^{j\varphi }}\left[ {\begin{array}{cc} {a + jb}&{c + jd}\\ { - c + jd}&{a - jb} \end{array}} \right]\left[ {\begin{array}{c} x\\ y \end{array}} \right] + \left[ {\begin{array}{c} {{n_x}}\\ {{n_y}} \end{array}} \right]\textrm{ }$$
where $\varphi$, ${n_x}$ and ${n_\textrm{y}}$ denote phase noise, additive noise in X polarization and Y polarization, respectively.

The main idea behind the proposed algorithm is to insert 3 PTs with different SOPs in the spectrum gap between subcarriers and use them to calculate the RSOP matrix. The 3 PTs ${\left[ {\begin{array}{cc} {{X_i}(n )}&{{Y_i}(n )} \end{array}} \right]^T}$ in the digital domain are expressed as

$$\left[ {\begin{array}{c} {{X_i}(n )}\\ {{Y_i}(n )} \end{array}} \right] = \left[ {\begin{array}{c} {{x_i}}\\ {{y_i}} \end{array}} \right]{e^{j2\pi \Delta {f_i}n/fs}}\textrm{, i = 1,2,3}$$
where ${[{{x_i}\textrm{ }{y_i}} ]^T}$ is the Jones vectors of the 3 PTs, $\Delta {f_i}$ is the frequency of the PT, and $f\textrm{s}$ is the sampling rate. In our algorithm, the Jones vectors of 3 PTs are selected as [0 1]T, [1 1]T and [1 0]T, respectively. Mathematically, other choices of Jones vectors might also be applicable, but there exists one constraint in the selection of the three Jones vectors, which will be explained in this section later. As shown in Fig. 1(a), they are inserted between four subcarriers, positioned in the left, middle and right gaps, respectively.

 figure: Fig. 1.

Fig. 1. Power spectrum of the experimental data (a) before adding RSOP in the Tx, (b) before RSOP tracking in the Rx, and (c) after RSOP tracking in the Rx.

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It is straightforward to estimate the FO by comparing the PT position at the Rx to its original position at the Tx. After CD compensation and FO compensation, the 3 PTs are shifted to zero frequency, holding the relationship as

$$\left[ {\begin{array}{c} {X_i^{\prime}(n)}\\ {Y_i^{\prime}(n)} \end{array}} \right] = {e^{j\varphi (n )}}\left[ {\begin{array}{cc} {a + j\textrm{b}}&{c + jd}\\ { - c + jd}&{a - jb} \end{array}} \right]\left[ {\begin{array}{c} {{x_i}}\\ {{y_i}} \end{array}} \right] + \left[ {\begin{array}{c} {{n_x}}\\ {{n_y}} \end{array}} \right]\textrm{, i = 1,2,3}$$

Due to the RSOP in fiber, the Jones vectors of PTs are deviated from the original coordinate. As shown in Fig. 1(b), all the 3 PTs appear in both polarizations. To remove the influence of the phase noise, the PTs of X polarization are divided by the PTs of Y polarization as

$${M_i}(n) = \frac{{X_i^{\prime}(n)}}{{Y_i^{\prime}(n)}}\textrm{ = }\frac{{({a + jb} ){x_i} + ({c + jd} ){y_i} + {n_x}}}{{({ - c + jd} ){x_i} + ({a - j\textrm{b}} ){y_i} + {n_y}}}\textrm{, i = 1,2,3}$$

We divide the signal and PTs into many blocks with the same length. The sampling rate is much larger than the RSOP speed. Thus, the SOP matrix is regarded as constant in a short block. We average Eq. (6) within a block as

$$\overline {{M_i}} = \sum\limits_{n = 1}^N {{M_i}(n)/N} \textrm{, i = 1,2,3}$$
where N denotes the block length. We substitute the three Jones vectors [0 1]T, [1 1]T and [1 0]T into Eq. (7). Then the 3 equations can be obtained as follows:
$$\overline {{M_1}} \approx \frac{{c + jd}}{{a - jb}}\textrm{ }$$
$$\overline {{M_2}} \approx \frac{{a + jb + c + jd}}{{ - c + jd + a - jb}}\textrm{ }$$
$$\overline {{M_3}} \approx \frac{{a + jb}}{{ - c + jd}}\textrm{ }$$

Note that the average operation decreases the influence of the additive noise, which is ignored in Eqs. (8–10). The constraint of selecting three Jones vectors is that Eqs. (8–10) should be linearly independent, which is valid for [0 1]T, [1 1]T and [1 0]T. Due to the linearly independent relationship, we can represent 4 unknown parameters $\textrm{a,b,c,d}$ with one parameter. Then the RSOP matrix $\textrm{R}$ can be obtained as follows:

$$R = \left[ {\begin{array}{cc} {a + j\textrm{b}}&{c + jd}\\ { - c + jd}&{a - jb} \end{array}} \right] = \left[ {\begin{array}{cc} {aa}&{bb}\\ {cc}&{dd} \end{array}} \right]$$
$$\begin{array}{l} aa = d{d^\ast } = a + jb = \\ \pm \frac{{j\overline {{M_3}} \times ({\overline {{M_2}} - \overline {{M_1}} } )}}{{{{[{({\overline {{M_1}} - \overline {{M_2}} } )\times ({\overline {{M_1}} - \overline {{M_3}} } )\times ({\overline {{M_2}} - \overline {{M_3}} } )} ]}^{0.5}}}} \end{array}$$
$$\begin{array}{l} bb ={-} c{c^\ast } = c + jd = \\ \pm \frac{{j\overline {{M_1}} \times ({\overline {{M_3}} - \overline {{M_2}} } )}}{{{{[{({\overline {{M_1}} - \overline {{M_2}} } )\times ({\overline {{M_1}} - \overline {{M_3}} } )\times ({\overline {{M_2}} - \overline {{M_3}} } )} ]}^{0.5}}}} \end{array}$$
where the specific calculation process is shown in the Appendix. Two matrices with opposite elements can be obtained and the signs of Eq. (12) and Eq. (13) must be selected either both positive or both negative. No matter which matrix $\textrm{R}$ we select, the correct result can be achieved. This is because the 180 degree phase difference will be removed in the process of phase recovery. Then the RSOP inverse matrix ${R^{ - 1}}$ is calculated and multiplied with the signal to rotate the signal to the standard X/Y coordinate. As shown in Fig. 1(c), the SOPs of these 3 PTs are the same as those in Fig. 1(a). The RSOP tracking process is processed on a block-by-block basis, which means that the RSOP in one block is regarded as a constant matrix but the RSOP of every sample within one block is not identical in reality. Thus, a faster tracking ability will be obtained by a shorter block length. Nonetheless, the shorter block length means the more block number and the higher complexity induced by the calculation of ${R^{ - 1}}$. In addition, there are less samples in the calculation of Eq. (7), which might increase the penalty of the additive noise. It is worth mentioning that the values of $\overline {{M_1}} ,\overline {{M_2}} ,\overline {{M_3}}$ might be very different in the calculation of Eq. (8)–(10). For example, when the RSOP matrix is close to $\left[ {\begin{array}{cc} 1&0\\ 0&1 \end{array}} \right]$, $\overline {{M_1}}$ is very small but $\overline {{M_3}}$ is very large. In this case, an arithmetic overflow might occur when it is implemented in an application-specific integrated circuit with limited fixed-point precision. Thus, it must be designed carefully to decrease the arithmetic overflow effect in practice.

After the RSOP tracking, we perform the phase recovery with one of the 3 PTs. The PT based phase recovery is proposed in [16]. In our algorithm, the only difference from [16] is that the RSOP effect needs to be removed first. It can be performed by multiplying the inverse RSOP matrix ${R^{ - 1}}$ to the PT on a block-by-block basis. Then, the phase of the PT can be extracted directly to recover the phase noise.

3. Experimental setup, results and discussions

3.1 Experimental setup and DSP flow

Figure 2 depicts the experimental setup and DSP flow. In the Tx DSP, four subcarriers are generated and shaped by a root raised cosine (RRC) filter. They are all modulated with 4 Gbaud 16 quadrature amplitude modulation (16-QAM) signals and the frequency spacing is 5 GHz. 3 PTs whose Jones vectors are [1 0]T, [1 1]T and [0 1]T are inserted at −5 GHz, 100 MHz and +5 GHz, respectively. To avoid direct current blocking, the PT is not placed at zero frequency. Differential coding is not required attributed to the use of the PT for recovering phase noise. Owing to the lack of a high-speed polarization scrambler, we add the polarization effects in the Tx DSP, which will be explained in detail later. Afterwards, the pre-compensation including pre-emphasis and Tx in-phase and quadrature (IQ) skew and XY skew compensation are performed. The skew calibration process follows the scheme in [20]. The signal is then generated via an 80 GSa/s arbitrary waveform generator (AWG) with the length of 262144. The dual polarization IQ modulator (DP-IQM) is driven by the signal using an external cavity laser (ECL) with a central wavelength of 1550 nm and a linewidth less than 100 kHz. After being boosted by an Erbium-doped fiber amplifier (EDFA) to 0 dBm, the signal enters a 10 km single mode fiber (SMF). Our proposed algorithm is designed for coherent access systems, in which the performance is evaluated by the power budget and received optical power (ROP). Thus, a variable optical attenuator (VOA) is used to adjust the ROP. After the coherent detection using another ECL with a similar linewidth and 9 dBm power, a four-channel real-time digital oscilloscope (DSO) with a sampling rate of 100 GSa/s is employed to digitize the waveform. Finally, the captured waveforms are processed offline in MATLAB. The calculation based on MATLAB has an ideal accuracy, so the possible arithmetic overflow does not occur.

 figure: Fig. 2.

Fig. 2. Experimental setup and DSP flow. PC: polarization controller, PBS/PBC: polarization beam splitter/combiner.

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The polarization model added in the Tx DSP consists of 3 parts:

$$J(\omega )= \left[ {\begin{array}{cc} {\sqrt {1 + \gamma } }&0\\ 0&{\sqrt {1 - \gamma } } \end{array}} \right]\left[ {\begin{array}{cc} {\cos \alpha {e^{j\xi }}}&{ - \sin \alpha {e^{j\eta }}}\\ {\sin \alpha {e^{ - j\eta }}}&{\cos \alpha {e^{ - j\xi }}} \end{array}} \right]\left[ {\begin{array}{cc} {{e^{j\omega \tau /\textrm{2}}}}&0\\ 0&{{e^{ - j\omega \tau /2}}} \end{array}} \right]$$
where the first, second, and third matrix denotes the PDL matrix, RSOP matrix and PMD matrix, respectively. In the PDL matrix, $\gamma$ is the magnitude of the PDL, which is generally defined in dB as
$$PDL(dB) = 10{\log _{10}}(\frac{{1 + \gamma }}{{1 - \gamma }})$$

The RSOP matrix including 3 parameters $\alpha ,\textrm{ }\xi ,\textrm{ }\eta$ can fully describe the rotations in a fiber channel [9]. To model the ultra-fast rotation effects, we add an increment $\Delta (k )$ to $\alpha ,\textrm{ }\xi ,\textrm{ }\eta$

$$\Delta (k )= \Omega /fs \times k$$
$$\alpha (k )= {\alpha _0} + \Delta (k )\textrm{ }\xi (k )= {\xi _0} + \Delta (k )\textrm{ }\eta (k )= {\eta _0} + \Delta (k )$$
where $\Omega $ is the RSOP speed, $fs$ is the sampling rate, k is the sample index, and ${\alpha _0}$, ${\xi _0}$, ${\eta _0}$ are the corresponding initial values. It is worth mentioning that the increment is applied to the 3 parameters meanwhile, the same as [7–9].

In the PMD matrix, $\tau$ is the differential group delay (DGD) and $\omega$ is the angular frequency. Since our algorithm is designed for a short-distance transmission system, it is reasonable to only consider the first order PMD as shown in Eq. (14). The algorithm will be ineffective in the presence of large PMD, which causes a frequency-dependent RSOP change. Thus, the DGD range needs to be discussed

For the Rx offline DSP, CD compensation is not included since the transmission distance is only 10 km. First, the Gram-Schmidt orthogonalization procedure (GSOP) algorithm [21] is used to compensate for IQ imbalance. Then we extract the 3 PTs with low-pass filters, which are ideal 50 MHz brick-wall filters. In practice, the filter shape and bandwidth should be designed to balance performance and complexity, and it is left for future study. Based on the 3 PTs, we perform the FO estimation, RSOP tracking, phase recovery sequentially as explained in Section 2. Note that we average every samples within a block in the calculation of Eq. (7). Then the subcarriers are demultiplexed and processed separately. After the RRC based matched filtering, LMS based single-input and single-output (SISO) equalization is implemented to separately compensate for other linear effects for each polarization. Then, maximum likelihood (ML) phase recovery is performed to compensate for residual phase noise. The RSOP is added in the Tx DSP before the Tx IQ imbalance, and thus there still exists some IQ crosstalk between the two polarizations after RSOP tracking. An 8×8 real-value MIMO equalizer with only one tap is employed to compensate for the IQ crosstalk in the Tx [22]. Since in real systems the RSOP in a fiber link is after the Tx IQ imbalance, the IQ crosstalk will not be mixed in the two polarizations after RSOP tacking. Therefore, two independent 4×4 real-value MIMO equalizers can be used to separately compensate for the IQ crosstalk within two polarizations if needed. Finally, signal de-mapping and decoding are applied.

3.2 Experimental results and discussions

Generally, for PT-aided systems, the pilot-to-signal ratio (PSR) needs to be optimized. This is because with a low PSR, the ability of RSOP tracking and phase recovery is limited while with a high PSR the power of signal is reduced. Therefore, we first experimentally investigate the impact of the PSR on system performance. Figure 3 shows the relationship between BER and PSR when the ROP is −25 dBm, indicating that the optimal PSR is −18 dB, both for 0 rad/s and 10 Mrad/s RSOP speeds. Therefore, a PSR of −18 dB is chosen in the following experiments.

 figure: Fig. 3.

Fig. 3. BER vs. PSR.

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In the following discussions, we select the algorithm in [16] for comparison, which uses one PT to recover the phase noise and a 2×2 MIMO equalizer to track the RSOP. The PSR of the PT is −20 dB, which is also an optimized value. For convenience, our proposed algorithm is named as PT-RT&PR and the algorithm in [16] is named as PT-PR, in which RT denotes RSOP tracking and PR denotes phase recovery. The adaptive step size of the equalizer is an important parameter related to the RSOP tracking ability. After optimization, the step size is 10−4 in the MIMO equalizer of the PT-PR algorithm and 10−5 in the SISO equalizer of our proposed PT-RT&PR algorithm. The tap coefficients are updated on a sample-by-sample basis. Based on the forward error correction (FEC) of 400G ZR systems [23], the BER threshold is set to 0.0125.

First, we investigate the RSOP tracking ability. PMD and PDL are not added, and the ROP is fixed at −25 dBm. As shown in Fig. 4(a), the PT-PR algorithm exceeds the BER threshold when the RSOP speed is larger than 500 krad/s and completely collapses when the RSOP speed is larger than 1 Mrad/s. We test our proposed PT-RT&PR algorithm with various block sizes from 27 to 210, all of which can recover a 10 Mrad/s RSOP speed with slight penalties. Note that the reported fastest RSOP speed is 8.1 Mrad/s in [5], which is much slower than the tracking speed of our algorithm. The induced penalty as the RSOP speed increases may come from the incomplete separation of two polarizations. On the other hand, the ability of RSOP tracking increases with the decrease of the block size. When the RSOP speed increases from 0 rad/s to 50 Mrad/s, the BER increases from 2.3×10−3 to 5.6×10−3 and 7.6×10−3 for the block size of 27 and 28, respectively. When SOP speed is lower than 100 krad/s, the performance of the PT-PR algorithm is better than the PT-RT&PR algorithm. This result may originate from two reasons. First, since 3 PTs with a −18 dB PSR are used in the PT-RT&PR algorithm while only one PT with a −20 dB PSR is used in the PT-PR algorithm, the power of signal in the PT-RT&PR algorithm is smaller.

 figure: Fig. 4.

Fig. 4. (a) BER vs. the speed of RSOP from 10 krad/s to 50 Mrad/s. (b) Required ROP at the BER of 0.0125 vs. the speed of RSOP from 10 krad/s to 10 Mrad/s. RT: RSOP tracking, PR: phase recovery, BS: block size.

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Next, we evaluate the required ROP at the BER threshold when the block size is 29. The result is shown in Fig. 4(b). When the RSOP speed is lower than 10 krad/s, our PT-RT&PR algorithm has approximately 1 dB penalty compared to the PT-PR algorithm. However, the PT-RT&PR algorithm has a 3.6 dB gain over the PT-PR algorithm at the RSOP speed of 500 krad/s. The PT-RT&PR algorithm still work well with a 1.6 dB penalty at the RSOP speed of 10 Mrad/s, but the PT-PR algorithm totally collapses when the RSOP speed is higher than 500 krad/s.

Finally, we study the influence of PMD and PDL when the ROP is −25 dBm and the RSOP speed is 10 Mrad/s. PMD and PDL are added in the Tx DSP according to the description in Section 3.1. For PMD, DGD is a time-variant value in a certain fiber link, whose probability density function (PDF)$p(\tau )$ follows a Maxwellian distribution as [24]

$$p(\tau ) = \frac{{32}}{{{\pi ^2}}}\frac{{{\tau ^2}}}{{{{\bar{\tau }}^3}}}exp ( - \frac{{4{\tau ^2}}}{{{{\bar{\tau }}^2}}})$$
$$\bar{\tau } = pm{d_{coff}} \times \sqrt L$$
where $\bar{\tau }$, $pm{d_{coff}}$, and L denote the mean DGD, PMD coefficients and fiber length, respectively. We add the DGD in the Tx DSP, assuming the optical signal propagating through a virtual fiber of various lengths. This allows us to cover low probability DGD values in our study. The real fiber length is always 10 km in the setup. $pm{d_{coff}}$ is selected as 0.04 ps/km1/2 based on the typical value of commercial G.652 fibers, and the equivalent fiber length ranges from 2 km to 100 km. According to Eq. (18) and Eq. (19), we can obtain the PDF of the instant DGD. However, it is not practical to test all the instant DGD in the PDF, so we set a sufficiently large DGD to verify the worst cases. International standard guidelines recommend an outage probability of 6.5×10−8 [25], which is equivalent to 2 seconds of outage per year. We adopt this probability to set the DGD values for various virtual fiber lengths, which means the probability of an instant DGD value in real systems exceeding the added DGD value is 6.5×10−8. The BER performance and the set DGD versus the virtual fiber length are shown in Fig. 5(a). In the short distance scenario, the virtual fiber accumulates a small DGD, and the BER keeps almost unchanged when the virtual fiber length increases to 100 km. For PDL, we change the PDL from 0 to 3 dB in the Tx DSP as well and the result is shown in Fig. 4(b). PDL will change the signal power of X and Y polarizations, so the performance gets worse as the PDL value increases. All the BERs are below 9×10−3, indicating that PDL will not influence the correct demultiplexing of two polarizations, and thereby our proposed algorithm can still track the fast RSOP in the presence of PDL.

 figure: Fig. 5.

Fig. 5. (a) BER and set DGD vs. virtual fiber length. (b) BER vs. PDL.

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3.3 Complexity analysis

In this section, we discuss the computational complexity of the proposed algorithm, which consists of two parts: 1) insertion and extraction of the 3 PTs; 2) RSOP tracking and phase recovery using the 3 PTs.

In a SCM system, it is straightforward to insert the 3 PTs in the gap between subcarriers at the Tx, and thus the complexity is ignorable. The extraction of the 3 PTs at the Rx requires low-pass filters, which can be implemented using the overlap-and-add or overlap-and-save approaches in the frequency domain. In this way, fast Fourier transform (FFT), multiplication with the filter coefficients and inverse FFT (IFFT) are performed sequentially. In the SCM system, the process of subcarrier demultiplexing has already included a pair of FFT and IFFT. The FFT can transform the subcarriers and 3 PTs into the frequency domain together, so there is no additional complexity of FFT for extracting the PTs. The 3 PTs are then shifted to zero frequency and multiplied by low-pass filter coefficients. Afterwards, 3 additional IFFTs are needed to convert them back to the time domain. Leveraging the characteristics of the low-pass filtering operation, a smaller size IFFT can be used, which is equivalent to downsampling in the time domain. Since the RSOP tracking is implemented on a block-by-block basis, the down-sampled PTs can also achieve the same task, but might deliver the worse performance because the number of samples in Eq. (7) is decreased. In addition, 1 PT requires interpolation to have the same length as the signal, in order to recover phase noise on a sample-by-sample basis.

In the second part, the algorithm relies on the calculation of the inverse RSOP matrix with $\overline {{M_i}}$. According to Eq. (12) and Eq. (13), it requires 6 multiplications, 8 additions and 2 square roots to calculate the RSOP matrix R, and additional complexity in the calculation of the inverse matrix ${R^{ - 1}}$. However, since the RSOP tracking is performed on a block-by-block basis, the complexity per sample should be divided by the BS. When the BS is as large as several hundred samples, the number of multiplications, additions and square roots is less than 0.1 per sample, which can be ignored compared to the other DSP algorithm. Then, we multiply the inverse matrix ${R^{ - 1}}$ to the signal and one of the 3 PTs. Last, we use the phase of the processed PT to recover the phase noise of signal directly.

It is worth mentioning that our proposed algorithm can save complexity in the equalization. Specifically, we use two SISO equalizers to replace one traditional MIMO equalizer, saving 50% complexity in the process of convolution. In addition, the complexity in the process of updating the tap coefficients can also be significantly reduced. The SISO mainly compensates the static impairments such as band-limited effects, so it does not require a fast updating. As shown in Fig. 6, the absolute value of the middle tap coefficient of MIMO changes with the iterations after initial convergence. In our algorithm, the middle tap coefficient of the SISO equalization is almost unchanged. Therefore, even though our proposed algorithm induces additional computational complexity, but more than 50% complexity can be saved in the equalization.

 figure: Fig. 6.

Fig. 6. (a) The absolute value of middle taps coefficients vs. iterations of SISO equalization and MIMO equalization. (b) The zoomed-in version of the SISO curve.

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4. Conclusion

Coherent SCM PTMP access architecture shows advantages over traditional point-to-point optical transmission solutions in terms of flexibility and cost. However, coherent systems require tracking fast RSOP, which often occurs in some access scenarios such as fronthaul. To overcome this challenge, we propose a novel RSOP tacking algorithm for SCM short-distance systems using 3 PTs. Through experimental verification, it is shown that when the RSOP speed increases from 0 rad/s to 50 Mrad/s, the BER increases from 2.3×10−3 to 5.6×10−3, indicating an ultra-fast RSOP tracking ability. The robustness of the algorithm against PMD and PDL is demonstrated, and the complexity is analyzed.

Appendix

The derivation process of the RSOP matrix R is shown in this section. First, based on Eqs. (8–10), we can represent $b,c,d$ using a as follows:

$$b = \frac{{ja \times ({\overline {{M_2}} - \overline {{M_3}} - \overline {{M_1}} \times \overline {{M_3}} + \overline {{M_2}} \times \overline {{M_3}} } )}}{{\overline {{M_2}} - \overline {{M_3}} + \overline {{M_1}} \times \overline {{M_3}} - \overline {{M_2}} \times \overline {{M_3}} }}$$
$$c = \frac{{ - a \times ({\overline {{M_1}} - \overline {{M_2}} - \overline {{M_1}} \times \overline {{M_2}} + \overline {{M_1}} \times \overline {{M_3}} } )}}{{\overline {{M_2}} - \overline {{M_3}} + \overline {{M_1}} \times \overline {{M_3}} - \overline {{M_2}} \times \overline {{M_3}} }}$$
$$d = \frac{{ - ja \times ({\overline {{M_1}} - \overline {{M_2}} + \overline {{M_1}} \times \overline {{M_2}} - \overline {{M_1}} \times \overline {{M_3}} } )}}{{\overline {{M_2}} - \overline {{M_3}} + \overline {{M_1}} \times \overline {{M_3}} - \overline {{M_2}} \times \overline {{M_3}} }}$$

Then we combine Eqs. (20–22) and Eq. (2):

$$\begin{array}{l} {a^2} + {b^2} + {c^2} + {d^2}\\ = {a^2}\{ 1 + {\left[ {\frac{{j({\overline {{M_2}} - \overline {{M_3}} - \overline {{M_1}} \times \overline {{M_3}} + \overline {{M_2}} \times \overline {{M_3}} } )}}{{\overline {{M_2}} - \overline {{M_3}} + \overline {{M_1}} \times \overline {{M_3}} - \overline {{M_2}} \times \overline {{M_3}} }}} \right]^2} + \\ {\left[ {\frac{{({\overline {{M_1}} - \overline {{M_2}} - \overline {{M_1}} \times \overline {{M_2}} + \overline {{M_1}} \times \overline {{M_3}} } )}}{{\overline {{M_2}} - \overline {{M_3}} + \overline {{M_1}} \times \overline {{M_3}} - \overline {{M_2}} \times \overline {{M_3}} }}} \right]^2} + {\left[ {\frac{{j \times ({\overline {{M_1}} - \overline {{M_2}} + \overline {{M_1}} \times \overline {{M_2}} - \overline {{M_1}} \times \overline {{M_3}} } )}}{{\overline {{M_2}} - \overline {{M_3}} + \overline {{M_1}} \times \overline {{M_3}} - \overline {{M_2}} \times \overline {{M_3}} }}} \right]^2}\} \\ = 1 \end{array}$$

Solving Eq. (23), we can obtain $a$

$$a ={\pm} \frac{{({\overline {{M_2}} - \overline {{M_3}} + \overline {{M_1}} \times \overline {{M_3}} - \overline {{M_2}} \times \overline {{M_3}} } )}}{{2j \times {{[{({\overline {{M_1}} - \overline {{M_2}} } )\times ({\overline {{M_1}} - \overline {{M_3}} } )\times ({\overline {{M_2}} - \overline {{M_3}} } )} ]}^{0.5}}}}$$

Replacing a in Eqs. (20–22, we can get $b,c,d$

$$b ={\pm} \frac{{({\overline {{M_2}} - \overline {{M_3}} - \overline {{M_1}} \times \overline {{M_3}} + \overline {{M_2}} \times \overline {{M_3}} } )}}{{2 \times {{[{({\overline {{M_1}} - \overline {{M_2}} } )\times ({\overline {{M_1}} - \overline {{M_3}} } )\times ({\overline {{M_2}} - \overline {{M_3}} } )} ]}^{0.5}}}}$$
$$c ={\pm} \frac{{j \times ({\overline {{M_1}} - \overline {{M_2}} - \overline {{M_1}} \times \overline {{M_2}} + \overline {{M_1}} \times \overline {{M_3}} } )}}{{2 \times {{[{({\overline {{M_1}} - \overline {{M_2}} } )\times ({\overline {{M_1}} - \overline {{M_3}} } )\times ({\overline {{M_2}} - \overline {{M_3}} } )} ]}^{0.5}}}}$$
$$d ={\mp} \frac{{({\overline {{M_1}} - \overline {{M_2}} + \overline {{M_1}} \times \overline {{M_2}} - \overline {{M_1}} \times \overline {{M_3}} } )}}{{2 \times {{[{({\overline {{M_1}} - \overline {{M_2}} } )\times ({\overline {{M_1}} - \overline {{M_3}} } )\times ({\overline {{M_2}} - \overline {{M_3}} } )} ]}^{0.5}}}}$$

Then we substitute $a,b,c,d$ into Eq. (1), and the RSOP matrix R can be expressed by the following equations:

$$R = \left[ {\begin{array}{cc} {a + j\textrm{b}}&{c + jd}\\ { - c + jd}&{a - jb} \end{array}} \right] = \left[ {\begin{array}{cc} {aa}&{bb}\\ {cc}&{dd} \end{array}} \right]$$
$$\begin{array}{l} aa = d{d^\ast } = a + jb = \\ \pm \frac{{j\overline {{M_3}} \times ({\overline {{M_2}} - \overline {{M_1}} } )}}{{{{[{({\overline {{M_1}} - \overline {{M_2}} } )\times ({\overline {{M_1}} - \overline {{M_3}} } )\times ({\overline {{M_2}} - \overline {{M_3}} } )} ]}^{0.5}}}} \end{array}$$
$$\begin{array}{l} bb ={-} c{c^\ast } = c + jd = \\ \pm \frac{{j\overline {{M_1}} \times ({\overline {{M_3}} - \overline {{M_2}} } )}}{{{{[{({\overline {{M_1}} - \overline {{M_2}} } )\times ({\overline {{M_1}} - \overline {{M_3}} } )\times ({\overline {{M_2}} - \overline {{M_3}} } )} ]}^{0.5}}}} \end{array}$$

Funding

National Key Research and Development Program of China (2018YFB1801200); Shanghai Rising-Star Program (19QA1404600); National Natural Science Foundation of China (61801291).

Disclosures

The authors declare no conflicts of interest.

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Figures (6)
Fig. 1.
Fig. 1. Power spectrum of the experimental data (a) before adding RSOP in the Tx, (b) before RSOP tracking in the Rx, and (c) after RSOP tracking in the Rx.

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Fig. 2.
Fig. 2. Experimental setup and DSP flow. PC: polarization controller, PBS/PBC: polarization beam splitter/combiner.

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Fig. 3.
Fig. 3. BER vs. PSR.

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Fig. 4.
Fig. 4. (a) BER vs. the speed of RSOP from 10 krad/s to 50 Mrad/s. (b) Required ROP at the BER of 0.0125 vs. the speed of RSOP from 10 krad/s to 10 Mrad/s. RT: RSOP tracking, PR: phase recovery, BS: block size.

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Fig. 5.
Fig. 5. (a) BER and set DGD vs. virtual fiber length. (b) BER vs. PDL.

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Fig. 6.
Fig. 6. (a) The absolute value of middle taps coefficients vs. iterations of SISO equalization and MIMO equalization. (b) The zoomed-in version of the SISO curve.

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Equations (30)

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

R = [ a + j b c + j d − c + j d a − j b ]
a 2 + b 2 + c 2 + d 2 = 1
[ X Y ] = e j φ [ a + j b c + j d − c + j d a − j b ] [ x y ] + [ n x n y ]  
[ X i ( n ) Y i ( n ) ] = [ x i y i ] e j 2 π Δ f i n / f s , i = 1,2,3
[ X i ′ ( n ) Y i ′ ( n ) ] = e j φ ( n ) [ a + j b c + j d − c + j d a − j b ] [ x i y i ] + [ n x n y ] , i = 1,2,3
M i ( n ) = X i ′ ( n ) Y i ′ ( n )  =  ( a + j b ) x i + ( c + j d ) y i + n x ( − c + j d ) x i + ( a − j b ) y i + n y , i = 1,2,3
M i ¯ = ∑ n = 1 N M i ( n ) / N , i = 1,2,3
M 1 ¯ ≈ c + j d a − j b  
M 2 ¯ ≈ a + j b + c + j d − c + j d + a − j b  
M 3 ¯ ≈ a + j b − c + j d  
R = [ a + j b c + j d − c + j d a − j b ] = [ a a b b c c d d ]
a a = d d ∗ = a + j b = ± j M 3 ¯ × ( M 2 ¯ − M 1 ¯ ) [ ( M 1 ¯ − M 2 ¯ ) × ( M 1 ¯ − M 3 ¯ ) × ( M 2 ¯ − M 3 ¯ ) ] 0.5
b b = − c c ∗ = c + j d = ± j M 1 ¯ × ( M 3 ¯ − M 2 ¯ ) [ ( M 1 ¯ − M 2 ¯ ) × ( M 1 ¯ − M 3 ¯ ) × ( M 2 ¯ − M 3 ¯ ) ] 0.5
J ( ω ) = [ 1 + γ 0 0 1 − γ ] [ cos ⁡ α e j ξ − sin ⁡ α e j η sin ⁡ α e − j η cos ⁡ α e − j ξ ] [ e j ω τ / 2 0 0 e − j ω τ / 2 ]
P D L ( d B ) = 10 log 10 ( 1 + γ 1 − γ )
Δ ( k ) = Ω / f s × k
α ( k ) = α 0 + Δ ( k )   ξ ( k ) = ξ 0 + Δ ( k )   η ( k ) = η 0 + Δ ( k )
p ( τ ) = 32 π 2 τ 2 τ ¯ 3 e x p ( − 4 τ 2 τ ¯ 2 )
τ ¯ = p m d c o f f × L
b = j a × ( M 2 ¯ − M 3 ¯ − M 1 ¯ × M 3 ¯ + M 2 ¯ × M 3 ¯ ) M 2 ¯ − M 3 ¯ + M 1 ¯ × M 3 ¯ − M 2 ¯ × M 3 ¯
c = − a × ( M 1 ¯ − M 2 ¯ − M 1 ¯ × M 2 ¯ + M 1 ¯ × M 3 ¯ ) M 2 ¯ − M 3 ¯ + M 1 ¯ × M 3 ¯ − M 2 ¯ × M 3 ¯
d = − j a × ( M 1 ¯ − M 2 ¯ + M 1 ¯ × M 2 ¯ − M 1 ¯ × M 3 ¯ ) M 2 ¯ − M 3 ¯ + M 1 ¯ × M 3 ¯ − M 2 ¯ × M 3 ¯
a 2 + b 2 + c 2 + d 2 = a 2 { 1 + [ j ( M 2 ¯ − M 3 ¯ − M 1 ¯ × M 3 ¯ + M 2 ¯ × M 3 ¯ ) M 2 ¯ − M 3 ¯ + M 1 ¯ × M 3 ¯ − M 2 ¯ × M 3 ¯ ] 2 + [ ( M 1 ¯ − M 2 ¯ − M 1 ¯ × M 2 ¯ + M 1 ¯ × M 3 ¯ ) M 2 ¯ − M 3 ¯ + M 1 ¯ × M 3 ¯ − M 2 ¯ × M 3 ¯ ] 2 + [ j × ( M 1 ¯ − M 2 ¯ + M 1 ¯ × M 2 ¯ − M 1 ¯ × M 3 ¯ ) M 2 ¯ − M 3 ¯ + M 1 ¯ × M 3 ¯ − M 2 ¯ × M 3 ¯ ] 2 } = 1
a = ± ( M 2 ¯ − M 3 ¯ + M 1 ¯ × M 3 ¯ − M 2 ¯ × M 3 ¯ ) 2 j × [ ( M 1 ¯ − M 2 ¯ ) × ( M 1 ¯ − M 3 ¯ ) × ( M 2 ¯ − M 3 ¯ ) ] 0.5
b = ± ( M 2 ¯ − M 3 ¯ − M 1 ¯ × M 3 ¯ + M 2 ¯ × M 3 ¯ ) 2 × [ ( M 1 ¯ − M 2 ¯ ) × ( M 1 ¯ − M 3 ¯ ) × ( M 2 ¯ − M 3 ¯ ) ] 0.5
c = ± j × ( M 1 ¯ − M 2 ¯ − M 1 ¯ × M 2 ¯ + M 1 ¯ × M 3 ¯ ) 2 × [ ( M 1 ¯ − M 2 ¯ ) × ( M 1 ¯ − M 3 ¯ ) × ( M 2 ¯ − M 3 ¯ ) ] 0.5
d = ∓ ( M 1 ¯ − M 2 ¯ + M 1 ¯ × M 2 ¯ − M 1 ¯ × M 3 ¯ ) 2 × [ ( M 1 ¯ − M 2 ¯ ) × ( M 1 ¯ − M 3 ¯ ) × ( M 2 ¯ − M 3 ¯ ) ] 0.5
R = [ a + j b c + j d − c + j d a − j b ] = [ a a b b c c d d ]
a a = d d ∗ = a + j b = ± j M 3 ¯ × ( M 2 ¯ − M 1 ¯ ) [ ( M 1 ¯ − M 2 ¯ ) × ( M 1 ¯ − M 3 ¯ ) × ( M 2 ¯ − M 3 ¯ ) ] 0.5
b b = − c c ∗ = c + j d = ± j M 1 ¯ × ( M 3 ¯ − M 2 ¯ ) [ ( M 1 ¯ − M 2 ¯ ) × ( M 1 ¯ − M 3 ¯ ) × ( M 2 ¯ − M 3 ¯ ) ] 0.5
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