1. Introduction

With the sharp growth in user demand for data-transmission capacity [1], digital coherent transmission with high-order modulation formats and a high symbol rate, such as single-carrier 800G or beyond, has become a research hotspot [2–4]. However, as the modulation order and symbol rate increase as well as the roll-off factor tends to zero, systems become vulnerable to transmitter (Tx) and receiver (Rx) in-phase/quadrature (IQ) imbalances, including amplitude imbalance (Amp. Imb.), phase imbalance (Pha. Imb.), and skew [5,6]. Tx/Rx IQ imbalance compensation (IQIC) has become indispensable to the future development of Nyquist coherent optical systems. Moreover, the Tx/Rx IQ imbalance estimation (IQIE) is another important function that is necessary for diagnosing and locating faults either in the pre-service or in the in-service stages. This paper focuses on Rx IQ imbalances only and applies to the scenarios where Tx IQ imbalances have been compensated at transmitter. In fact, even Rx and Tx IQ imbalances both existing at the receiver, a frequency offset may isolate the Tx and Rx imbalances [7]. Thus, only the Rx IQ imbalances are considered in the paper.

Various approaches have been proposed for Rx IQIC and IQIE [7–18]. The Gram-Schmidt orthogonalization procedure was used to compensate and estimate the Rx IQ Amp. Imb. and Pha. Imb. unbiasedly; however, it was invalid for the Rx IQ skew [7,8]. The modified interpolator in timing recovery was proposed for Rx IQ skew compensation and estimation [9]. However, the compensation range of each Rx IQ imbalance was limited. Previous studies have proposed that adaptive equalizers can be used to perform polarization de-multiplexing and the linear equalization, including Rx IQIC [8,10–18]. For the real-valued time domain 4×4 multi-input and multi-output equalizer (TD 4×4 MIMO), the Rx IQ imbalances are removed by 16 sub-filters, and each of the four-sampling sequences can have a different transfer function through the equalizer [10,11]. Moreover, the Rx IQ imbalances are estimated by analyzing converged coefficients of the equalizer [8]. However, the traditional chromatic dispersion compensation (CDC) mixes the IQ components and degrades the IQIC performance [12,13]. To remove this limitation, the IQ-independent CDC was proposed. Combined with the TD complex-valued 4×2 MIMO equalizer (TD 4×2 MIMO), Rx IQIC was successfully performed, even under a large of accumulated CD [14,15]. Regrettably, the computational complexity of the TD 4×2 MIMO increases with the number of taps, making the TD 4×2 MIMO with a large number of taps difficult to implement [16]. To overcome this challenge, a frequency domain 4×2 MIMO equalizer (FD 4×2 MIMO) and an FD widely linear equalizer based on the constant-modulus algorithm have been proposed [17,18]. The computational complexity was reduced by block-by-block signal processing [19–22]. However, this approach cannot be applied to multi-moduli signal.

In this study, an FD 4×2 MIMO adapted by a radially directed equalizer (RDE) is proposed for Rx IQIC. Moreover, the Rx IQ imbalances are derived from the converged discrete frequency response of the FD 4×2 MIMO. The computational complexity of the FD 4×2 MIMO is lower than that of the TD 4×2 MIMO, particularly when the taps number of the TD 4×2 MIMO, corresponding to the number of discrete frequency responses of the FD 4×2 MIMO, reaches 11 or more; this is typically required for long-haul transmission. Both the proposed blind adaptive equalizer and the derived estimators are numerically confirmed using a 128 GBaud polarization division multiplexing 16-ary quadrature amplitude modulation (PDM-16QAM) Nyquist signal over a 35×80 km optical fiber transmission. Additionally, the equalizer and estimator are experimentally validated with 40 GBaud PDM-16QAM signals over 5×80 km optical fiber transmission. For comparison, the TD 4×2 MIMO and TD 2×2 MIMO are also performed. The results verify that the proposed FD 4×2 MIMO and estimators can fully compensate and precisely estimate Rx IQ imbalances with a wide range, even for long-haul transmission.

2. Principle of the proposed equalizer

2.1 Compensation principle

The proposed FD 4×2 MIMO is adapted using RDE for PDM-16QAM, as shown in Fig. 1. The four sampling sequences, denoted by x_I, x_Q, y_I, and y_Q, are output by an analog-to-digital converter (ADC) with twofold oversampling. After the IQ-independent CDC, the I and Q tributaries remain independent because four separate filters are utilized. Each of the four data-streams is then divided into even and odd tributaries, thereby obtaining eight symbol-spaced sampling sequences. Using serial-to-parallel (S/P) converters, the even and odd tributaries are further divided into several L point data blocks. The k_th block is represented as $\left(\vec{u}_{q}^{e, o}\right)_{k}=\left[\vec{u}_{q}^{e, o}(1), \vec{u}_{q}^{e, o}(2) \ldots \vec{u}_{q}^{e, o}(n) \ldots \vec{u}_{q}^{e, o}(L)\right]$, where L is the length of the data blocks equaling to the number of discrete frequency responses of the FD 4×2 MIMO denoted by N_FD, n is the sampling index, $q\in\{x_I,x_Q,y_I,y_Q\} $ and (${\bullet} $)^e, (${\bullet} $)° represent the even and odd tributaries, respectively. After a fast Fourier transform (FFT) using the overlap-save method with 50% overlap, the data blocks are fed into the finite-impulse-response (FIR) filters to achieve FD adaptive equalization. As shown in Fig. 1, the even sub-filters are denoted by $H_{pq}^e$ and the odd sub-filters are denoted by $H_{pq}^\textrm{o}$. They comprise the discrete frequency response of the FD 4×2 MIMO, denoted by H_FD-4×2, where $p\in\{x,y\}$.

 

Fig. 1. Schematic of the proposed FD 4×2 MIMO for PDM-16QAM.

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To update the H_FD-4×2, the RDE is implemented for higher order QAM, which does not require training symbols and can be updated using a blind algorithm. A decision-directed algorithm can be considered to discriminate of multiple levels for very high-order modulation formats. Taking the PDM-16QAM signal as an example, the three reference-moduli are denoted by R₁, R₂, and R₃. The key problem of error calculation is to find the reference modulus that is nearest to the output of the equalizer denoted by v_p(n). As shown in Fig. 1, the entire constellation diagram of the 16QAM can be divided into three regions A, B, and C, which are separated by two circles with radius RR₁= (R₁+ R₂)/2 and RR₂= (R₂+ R₃)/2. The reference modulus is found according to the region where v_p(n) is located in. If |v_p(n)|< RR₁, v_p(n) is located in region A, and the reference modulus is R₁. If |v_p(n)|> RR₁, v_p(n) is located in region C and the reference modulus is R₃. In other cases, v_p(n) is located in region B, and the reference modulus is R₂. Then, the error value of RDE (e_p)_k is calculated in TD as follows:

Then, (e_p)_k =[e_p(1), e_p(2)…. e_p(n)…e_p(L)] are converted into FD which is denoted by (E_p)_k to calculate the gradient vectors as follows:

2.2 Estimation principle

By analyzing the relationships between the TD 4×4 MIMO and FD 4×2 MIMO, the estimation function for the IQIE can be derived from H_FD-4×2 in the FD. Under 50% overlapping, the corresponding TD tap coefficients vector of $H_{pq}^{e,o}$ can be derived as follows:

The last N_FD elements of IFFT ($H_{pq}^\textrm{o} + H_{pq}^e$) are discarded for the using of the overlap method. After processing the 16 sub-filters of the FD 4×2 MIMO, the corresponding TD tap coefficients of the FD 4×2 MIMO are denoted by the lowercase letter h_FD-4×2.

By separating the I path and the Q path, the TD 4×4 MIMO and FD 4×2 MIMO can be considered as equivalent equalizers. The TD tap coefficients of the TD 4×4 MIMO are denoted by m_TD-4×4, and the corresponding FD discrete frequency response is M_TD-4×4. The m_TD-4×4 can be equivalent to h_FD-4×2 as follows:

According to the FFT, the zero-frequency point value of each sub-filter is the sum of its corresponding TD tap coefficients. Combined with Eq. (5), the relationship between the FD 4×2 MIMO and TD 4×4 MIMO at the zero-frequency point can be obtained. M_11|w = 0 is taken as an example:

By bringing Eq. (6) into Eq. (7), the Rx IQ Amp. Imb. and Pha. Imb. estimation functions based on the FD 4×2 MIMO can be represented by H_FD-4×2 as follows:

The Rx IQ skew can be derived from the group delay difference between sub-filters [24]. For each sub-filter of the FD 4×2 MIMO,

By an equivalent transformation,

Then, according to the properties of the Fourier transform,

Combining with Eq. (12), Eq. (11) is equal as follows:

Combining with Eq. (13), the group delay of a sub-filter denoted by GD_pq(k) can be obtained as follows:

2.3 Computational complexity

Generally, the computational complexity of a multiplier is approximately three times that of a comparator and ten times that of an adder in implementation [25,26]. For ease of comparison, the computational complexity is evaluated in terms of the required number of real multiplications per symbol. The TD 4 × 2 MIMO and FD 4 × 2 MIMO both worked on the twofold oversampling input sequences.

For the TD 4 × 2 MIMO adapted by RDE with N_TD taps, the delay spacing of the FIR filter is T_s/2; however, the filter-tap coefficients are typically adapted every two samples. To obtain N_TD output symbols from the X and Y output ports of the equalizer, 24N_TD+4 complex multiplications are needed, including 16 N_TD complex multiplications for output calculations, 8 N_TD complex multiplications for tap coefficients updating, and extra 4 complex multiplications for multi-modulus error calculations. Moreover, 16N_TD +8 real multiplications are needed, including 16N_TD real multiplications for tap coefficients updating and 8 real multiplications for multi-modulus error calculations. In total, the computational complexity per symbol of the TD 4 × 2 MIMO C_TD-4×2 is calculated as follows:

For the proposed FD 4 × 2 MIMO adapted by RDE with N_FD discrete frequency responses, the delay spacing of the FIR filters is T_s. To obtain N_FD output symbols from the X and Y ports, 66N_FD complex multiplications are needed including 32N_FD complex multiplications for output calculations, 32N_FD complex multiplications for tap updating, and 2N_FD complex multiplications for multi-modulus error calculation. It is worth noting that the use of a 50% overlap results in an FFT/IFFT that is 2N_FD-point in length. Each 2N_FD-point FFT operation requires N_FDlog₂(2N_FD) complex multiplications under a radix-2 FFT operation [20]. Therefore, 44N_FDlog₂(2N_FD) complex multiplications for 44 FFT/IFFTs processes are needed, which includes 8 FFTs for inputs, 2 IFFTs for outputs, 2 FFTs for multi-modulus error calculation, and 32 FFT/IFFTs for employing the gradient constraint. Moreover, 68N_FD real multiplications are required, including 64N_FD real multiplications for tap coefficients updating and 4N_FD real multiplications for multi-modulus error calculation. Finally, the complexity per symbol of the FD 4 × 2 MIMO C_FD-4×2 is calculated as follows:

To compare the computational complexity, it is assumed that the number of discrete frequency response for the FD 4×2 MIMO is equal to the number of taps for the TD 4×2 MIMO under the same performance. As shown in Fig. 2, the computational complexity of both equalizers increases with N_TD or N_FD; however, the proposed FD 4×2 MIMO provides significantly lower computational complexity than the TD 4×2 MIMO when N_TD and N_FD reaches as far as 11 and beyond. Moreover, this benefit enhances sharply as N_TD or N_FD increase Therefore, an FD 4×2 MIMO with a similar Rx IQIC performance to that of a TD 4×2 MIMO, but with a lower computational complexity, is more advantageous for long-haul transmission.

 

Fig. 2. Computational complexity of TD/FD 4 × 2 MIMO Vs N_TD and N_FD.

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3. Simulation results and discussions

A 128 GBaud PDM-16QAM Nyquist transmission system simulation platform is constructed to verify the effectiveness of the proposed FD 4×2 MIMO adapted by an RDE. The roll-off factor of a root-raised-cosine (RRC) pulse shaping, frequency offset, transceiver linewidth, transmission distance, and optical signal-to-noise ratio (OSNR) are 0.1, 1 GHz, 100 kHz, 35×80 km, 23.3 dB, respectively. The Rx IQ Pha. Imb. is set within the 90° Hybrid and the Rx IQ skew and Rx IQ Amp. Imb. are controlled before the ADCs.

The performance of the proposed FD 4×2 MIMO is evaluated under the existence of only one Rx IQ imbalances, as shown in Fig. 3(a-c). For comparison, the performances of the TD 4×2 MIMO and TD 2×2 MIMO are also evaluated. The number of taps is 31 for the TD 2×2/4×2 MIMO, and the number of directed frequency response is 21 for the FD 4×2 MIMO. It can be observed that the FD 4×2 MIMO exhibits a similar performance to the TD 4×2 MIMO. The estimation ranges of the skew, Amp. Imb., and Pha. Imb. are [-70%, 70%] T_s, [-7.5, 7.5] dB, and [-40, 40] deg within a 0.1 dB Q-factor penalty. However, the performance of the TD 2×2 MIMO is degraded under any Rx IQ imbalances. Without compensation, a 20% T_s skew, 3.5 dB Amp. Imb, or 20 deg Pha. Imb. induce an approximately 2 dB Q-factor penalty. Comprehensively, the optimal performance is maintained by the TD/FD 4×2 MIMO, which are more suitable for Rx IQIC than the TD 2×2 MIMO.

 

Fig. 3. Q-factors with Rx IQ (a) skew, (b) Amp. Imb., (c) Pha. Imb., (d) skew and Amp. Imb. under 40 deg Pha. Imb., (e) skew and Pha. Imb. under 7.5 dB Amp. Imb., and (f) Pha. Imb. and Amp. Imb. under 70%T_s skew after 35×80 km transmission.

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Considering the practical case wherein all three IQ imbalances coexist, the proposed FD 4×2 MIMO is performed for Rx IQIC by maintaining one of the three types Rx IQ imbalances as a constant and varying the last two. For simplicity, only the positive values of the Rx IQ imbalances are analyzed. As shown in Fig. 3(d-f), the maximum Q-factor penalty is less than 0.2 dB, even if the three Rx IQ imbalances coexist.

Next, the performance of each estimator is assessed. Figure 4 shows the estimated value and corresponding estimated error which is defined as the difference between the estimated value and exact value under one of the Rx IQ imbalances. It can be observed that each Rx IQ imbalance can be correctly estimated across a wide range with only a very small error. For each estimator, a skew of up to 70% Ts may be estimated with a 0.4 ps error, an Amp. Imb. of up to 7.5 dB may be estimated with a 0.3 dB error, and a Pha. Imb. estimator is accurate to within 0.8 deg from -40 to 40 deg. The corresponding estimated errors vary with the exact values randomly for each Rx IQ imbalance. This is because the iteration step of the adaptive algorithm cannot always provide the optimal value which contributes to the minimum cost function. In addition, random noise may also induce an error fluctuation.

 

Fig. 4. Estimated values and corresponding estimated errors of Rx IQ (a) skew, (b) Amp. Imb., and (c) Pha. Imb. after a 35×80 km transmission.

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Under 5×80, 10×80, and 35×80 km fiber transmission, the N_TD values of the TD 4×2 MIMO are 25, 27, and 31, and the N_FD values of the FD 4×2 MIMO are 13, 15, and 21. The corresponding computational complexity is shown in Fig. 5. After 35×80 km transmission, the computational complexity of the proposed FD 4×2 MIMO is reduced by approximately 63.3% compared with the TD 4×2 MIMO, without sacrificing performance. The transmission distance is positively correlated with N_TD or N_FD; therefore, the FD 4×2 MIMO with a lower computational complexity is more suitable for Rx IQIC for long-haul fiber transmission.

 

Fig. 5. Computational complexity comparisons between TD/FD 4×2 MIMO.

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4. Experimental results and discussions

Limited by the experimental conditions in our lab, the 40 GBaud PDM-16QAM signal Nyquist coherent transmission experiments are conducted to further investigate the validity and performance of the proposed algorithm for IQIC and IQIE. As shown in Fig. 6, in the offline transmitter (Tx) digital-signal-processing (DSP) blocks, 16QAM symbol mapping, threefold oversampling, and Nyquist pulse shaping with a 0.1 roll-off factor are carried out sequentially. Then, the oversampled signals are fed into an arbitrary waveform generator (AWG, 8194A, Agilent, USA) operating at 120 GSa/s to generate 40 GBaud signals. The carrier laser is generated by an external cavity laser with a 50 kHz measured linewidth inserted in the dual polarization IQ modulators, which are driven by the AWG to generate the I and Q data. The fiber loop consists of an 80 km standard single-mode fiber (SSMF) span, two erbium-doped fiber amplifiers (EDFA), two optical switches, an optical coupler, and an optical bandpass filter (OBPF) with a 0.3 nm bandwidth. At the Rx, the linewidth of the local oscillator (LO) laser inserted in the coherent receiver (IQS70, Teledyne LeCroy, USA) is less than 100 kHz. The optical signals after the fiber transmission and LO are coherently detected by the coherent receiver at 70 GHz and then converted into four electrical baseband signals that are digitized by an 80 GSa/s real-time sampling oscilloscope (LabMaster 10-59 Zi-A, Teledyne LeCroy, USA). Finally, the samples are processed by an offline Rx-DSP that includes including the proposed FD 4×2 MIMO for polarization de-multiplexing, Rx IQIC, and Rx IQIE.

 

Fig. 6. Experimental setup of 40 GBaud PDM-16QAM Nyquist transmission system.

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Considering the possible Rx IQ imbalances of the actual devices, the Rx IQ skew and Amp. Imb. are swept from -6 to 6 ps and -5.5 to 5.5 dB, respectively, by configuring the Optical Modulation Analyzer within an oscilloscope. Because the 90° Hybrid and coherent balanced receiver are integrated into one module, the Pha. Imb. cannot easily be set within optical 90°hybrid; instead, it is set in the offline Rx-DSP before the CDC. The Pha. Imb. of the x polarization is modeled as follows [27]:

Table 1. Three Rx IQ imbalances coexisting

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As observed in Fig. 7, the FD 4×2 MIMO exhibits an almost identical performance to that of the TD 4×2 MIMO and a significantly better performance than that of the TD 2×2 MIMO for Rx IQIC, which is similar to the simulation results. Under 5×80 km transmission and 25.8 dB OSNR, the maximum Q-factor penalties are 0.36, 0.32, and 0.1 dB within the [-6, 6] ps skew, [-5.5, 5.5] dB Amp. Imb., and [-40, 40] deg Pha. Imb. for the TD/FD 4×2 MIMO. Moreover, the maximum Q-factor penalty is 0.38 dB for the cases of Rx1-Rx7, where these three imbalances coexist. Unlike the simulation results, the Q-factor penalty for the Pha. Imb. compensation is smaller than the other penalties. This is because the Pha. Imb. are added in the offline Rx-DSP before the IQ-independent CDC, which avoids the degradation of non-ideal devices. Moreover, the fluctuations of the Q-factors within the compensation range are greater than those in the simulations. This is because non-ideal experimental devices and non-optimal sampling time result in more severely deteriorated signals than those in the simulations. In contrast, for the TD 2×2 MIMO, the ±4 ps skew, ±3.5 dB Amp. Imb., or ±30 deg Pha. Imb. induce a Q-factor penalty larger than 2 dB.

 

Fig. 7. Q-factors Vs Rx IQ (a) skew, (b) Amp. Imb., (c) Pha. Imb., and (d) three imbalances coexisting after 5×80 km transmission.

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As shown in Fig. 8, the estimated values of each Rx IQ imbalance are in good agreement with the exact values, which is similar to the simulation results. For each estimator, an IQ skew of up to 6 ps is estimated with a 0.4 ps error, the Amp. Imb. of up to 5.5 dB is with a 0.3 dB error, and the Pha. Imb. of up to 40 deg with a 0.7 deg error. Similar to simulation results, all the estimated error curves fluctuate with the exact values for each Rx IQ imbalance because the random noise during each test is different.

 

Fig. 8. Estimated values and corresponding estimated errors of Rx IQ (a) skew, (b) Amp. Imb., and (c) Pha. Imb.

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5. Conclusion

In this paper, a low computational complexity and blind FD 4×2 MIMO based on RDE is proposed to compensate and estimate the Rx IQ imbalances of high-order M-QAM signals. This algorithm can fully eliminate the Rx IQ imbalances, and significantly reduce the computational complexity in comparison to the conventional TD 4×2 MIMO, particularly for long-haul transmission. Moreover, the Rx IQ imbalances are derived from the converged discrete frequency response of the proposed equalizer. The proposed algorithm and estimator are assessed via simulation and experiment. In the simulation, 128 GBaud PDM-16QAM signals after 35×80 km optical fiber transmission, 0.7 Ts skew, 7.5 dB Amp. Imb., or 40 deg Pha. Imb. resulted in an approximately 0.1 dB Q penalty by the proposed equalizer. For each estimator, an IQ skew of up to 70% Ts may be estimated with a 0.4 ps error, the Amp. Imb. of up to 7.5 dB with a 0.3 dB error, and the Pha. Imb. of up to 40 deg with a 0.8 deg error. Moreover, the FD 4×2 MIMO can reduce the computational complexity by approximately 63.3% compared with the TD 4×2 MIMO without sacrificing performance. In the experiment, 40 GBaud PDM-16QAM signals after 5×80 km optical fiber transmission, 6 ps skew, 5.5 dB Amp. Imb. and 40 deg Pha. Imb. give rise to approximately 0.36, 0.32, and 0.1 Q penalties by the proposed equalizer. Moreover, the maximum Q-factor penalty is 0.38 dB for the cases of Rx1-Rx7, in which these three imbalances coexist. For each estimator, an IQ skew of up to 6 ps may be estimated with a 0.4 ps error, an Amp. Imb. of up to 5.5 dB with a 0.3 dB error, and a Pha. Imb. of up to 40 deg with a 0.7 deg error. The results demonstrate that the Rx IQ imbalances of high-order M-QAM signals that vary over a wide range can be fully compensated by the proposed equalizer and precisely estimated by the estimators even for long-haul transmission.