Digital Four-Dimensional Transceiver IQ Characterization

Signal impairments caused by the relative characteristic differences and crosstalk between electrical in-phase (I) lanes and quadrature (Q) lanes are some of the main obstacles to the development of 100-Gbaud-class high-symbol-rate transceivers. To address such IQ impairments, we propose a method for enabling fully frequency-resolved four-dimensional (4D) characterization of IQ impairments even in the presence of arbitrary crosstalk across four IQ lanes of polarization-multiplexed transceivers. We formulated a 4D signal-propagation model that takes into account inter-polarization IQ crosstalk and found how to separate the 4D IQ characteristics of the transmitter and receiver from the coefficients of a single-layer complex 8×2 multiple-input multiple-output adaptive equalizer on the basis of the model. We also introduce an application of the method for characterizing typical IQ impairments (e.g., IQ skew, IQ amplitude imbalance, phase deviation) from the filter coefficients. We numerically and experimentally tested our method with 96-Gbaud 16QAM signals and demonstrated its feasibility.


I. INTRODUCTION
M ODERN coherent optical communication networks, from datacenter interconnects to long-haul transport systems, require broadband transceivers [1]. As stated in the Shannon-Hartley theorem [2], expanding bandwidth of the transceivers allows for more capacity per channel without raising the signal-to-noise ratio requirement, which is becoming increasingly important as net rates of coherent transmission systems are approaching its channel capacity [3], [4], [5]. Highsymbol-rate transceivers, in combination with constellation shaping techniques, can also provide fine rate-distance flexibility to such transmission systems [5], allowing for more optimal system design for both point-to-point links and reconfigurable optical networks.
One type of limiting factor for increasing the symbol rate of transceivers is in-phase and quadrature (IQ) impairments, i.e., impairments caused by unavoidable relative characteristic differences and crosstalk between IQ lanes in the transceiver, Manuscript  such as IQ skew, IQ amplitude imbalance, IQ phase deviation, and IQ crosstalk [1], [6]. These IQ impairments cannot be compensated for using a conventional complex 2×2 multiple-input multiple-output (MIMO) adaptive equalizer (AEQ) and require additional compensation mechanisms. Various techniques for IQ-impairment mitigation have been investigated and are summarized in a previous study [7]. When considering IQ impairments on only transmitter (Tx)-or receiver (Rx)-side IQ impairments, using a well-known widely linear (4×2) AEQ [8] can be a solution, but impairments of the opposite side remain. For simultaneous compensation, concatenating AEQs before and after the carrier-phase-recovery section of digital signal processing (DSP) has been proposed [9], [10], [11]; however, it has the disadvantage of unstable convergence. Another candidate is the combination of a static filter and IQ-impairment-characteristic measurement techniques. Several techniques have been proposed for characterizing IQ impairments such as using multi-tone signals [12], phase retrieval [13], concatenated AEQs [10], [14], [15], Gram-Schmidt orthogonalization [16], and k-means clustering [17].
However, these techniques cannot handle electrical interpolarization IQ crosstalk. That is, it is assumed that each of the polarization tributaries is independent within the transceiver, and that there is no crosstalk between IQ signals of different polarization tributaries, for instance, the Q lane of X-polarization (XQ) and the I lane of Y-polarization (YI). Since such crosstalk cannot be expressed as a combination of intra-polarization IQ impairments and changes in the state of polarization, the presence of such crosstalk directly results in signal-quality degradation. Due to the high losses in high-frequency electrical transmission lines and the ever-increasing demand for smaller transceivers [18], the density of electrical wiring is expected to continue to increase, and the assumption of no inter-polarization IQ crosstalk may be overly optimistic in the future. Integrated transceiver components, such as coherent driver modules (CDM) [19], intradyne coherent receiver (ICR), and co-packaged optics, will be sources of performance-limiting in-band inter-polarization IQ crosstalk [20].
The 8×2 MIMO AEQ was proposed to handle interpolarization IQ crosstalk [21]. It can compensate for IQ impairments including inter-polarization crosstalk with a single-layer MIMO structure by adding degrees of freedom (DOFs) of complex conjugates of the signals and local oscillator (LO) frequency offsets to a MIMO AEQ. Although the 8×2 MIMO AEQ has been shown to enable high-information-rate transmission over 1 Tb/s [21], [22], it is still important to separately characterize transceiver IQ impairments for designing an efficient DSP architecture. This has not been accomplished due to the lack of both proper modeling of inter-polarization IQ crosstalk and understanding of its relationship with the 8×2 MIMO AEQ.
Recently, we proposed a transceiver characterization method for supporting inter-polarization IQ crosstalk [23]. As shown in Fig. 1. IQ impairments including inter-polarization IQ crosstalk can be expressed by frequency-dependent four-dimensional (4D) matrices H R (ω) and H T (ω), each of which represents the IQ characteristics of the Rx and Tx, acting on x/y-polarized signals s x/y and their complex conjugates. From this formulation, we first constructed a 4D signal-propagation model accounting for arbitrary IQ crosstalk. On the basis of the model and its connection to the 8×2 MIMO AEQ, we found that inversed characteristics H −1 R (ω) and H −1 T (ω) can be calculated from filter coefficients of the 8×2 MIMO AEQ through simple calculation. Our method can simultaneously and separately characterize IQ impairments occurring within the Tx and Rx in a fully frequencyresolved manner to compensate for IQ impairments. We also presented an application of the method of characterizing the typical IQ impairments (e.g., IQ skew, IQ amplitude imbalance, IQ phase deviation) from the 8×2 MIMO filter coefficients. We conducted numerical simulations and an experiment to validate the proposed method, and found it to be valid for 96-Gbaud 16QAM (quadrature amplitude modulation) signals. As an extension of our previous work [23], this paper describes the detailed description of the theory and experiment. Also, we show the results of proof-of-principle numerical experiments.

II. PRINCIPLE
In this section, we describe the principle of our proposed method. We focus on electrical IQ impairments and simply ignore IQ impairments in the optical domain, such as optical crosstalk within transceivers, for simplicity. We also ignore the effect of nonlinearity within the entire transmission line. An example of the signal processing flow of the method is shown in Fig. 2. With our method, we apply an auxiliary 8×2 MIMO AEQ to the signal to obtain its filter coefficients. We then obtain the 4D IQ characteristics H −1 R (ω) and H −1 T (ω)from the coefficients by simple linear characteristics. The obtained characteristics can be fed back to static filters in the Rx-side DSP to compensate for the impairments before and after the conventional 2×2 MIMO AEQ; otherwise, the Tx characteristics can be fed back to the Tx-side DSP.
Subsection A describes the 4D IQ-signal-propagation model used in this study and its relationship with the 8×2 MIMO AEQ. Subsection B then presents how the 4D IQ impairments are estimated, and Subsection C presents how each of typical IQ impairments are characterized by using the proposed method.

A. Four-dimensional IQ Signal Modeling and Physical Meaning of 8×2 MIMO AEQ
In this subsection, we present a model of 4D optical fiber transmission that can describe IQ impairments including interpolarization IQ crosstalk, which is necessary to derive the proposed method. The purpose of this subsection is to understand the correspondence between the IQ impairments defined in the model and the filter coefficients of the 8×2 MIMO AEQ.
Considering that general IQ signals can be represented by a linear superposition of a complex signal and its complex conjugates, we define the 4D input and output signal vectors s in (ω) , s out (ω) in the frequency domain as where s x/y, in/out refers to the x/y-polarized input/output signals of the entire system. The even-numbered rows represent complex-conjugated signals in the time domain. The effects on the signals along the entire transmission process, including transceiver IQ impairments, fiber transmission characteristics, signal up/down-conversion to optical/RF frequency, can be modeled using 4D matrices acting on the input vectors in the time and frequency domains. We express these effects as where FT refers to Fourier transformation. Matrix describes LO of the Rx, ω R is a frequency of the LO, describes the chromatic dispersion of the transmission line, describes the state of polarization (SOP) fluctuation characteristics within the transmission line including polarization rotation, polarization mode dispersion (PMD), and polarization dependent loss (PDL), describes the carrier frequency of the Tx, ω T x , ω T y is a frequency of the X-/Y-polarization tributaries, and is a straightforward 4D extension of the Jones matrix, and describes combined effects of PMD, PDL, and frequency-dependent SOP rotations. Assuming there are no IQ impairments in the fiber transmission line, combining the Jones matrix with its complex conjugate gives the formulation of H −1 SOP (ω). To make the equation more concise, we define Δω x , Δω y as Δω and Then, s in (ω) can be rewritten using A(ω) and B(ω) as Additionally, we define a matrix T by and obtain One may realize this simplified equation directly corresponds to the structure of the 8×2 MIMO AEQ illustrated in Fig. 3. This equation explains the working principle of the signalimpairment equalization using the 8×2 MIMO AEQ: It adaptively emulates the concatenated functions A(ω)T and B(ω)T . The filter coefficients of the AEQ, therefore, should become equal to components of the first and third rows of matrices A(ω)T and B(ω)T under ideal conditions without noise and nonlinearity.
We define the frequency-domain tap coefficients of the 8×2 MIMO AEQ h 1 (ω), . . . , h 16 (ω) as illustrated in Fig. 3. It can be shown that Since A(ω) contains information of H −1 R (ω) and H −1 T (ω) by its definition, the IQ characteristics are supposed to be retrieved from A(ω), as discussed in the next subsection.

B. Characterization of Four-Dimensional IQ Impairments
In this subsection, we describe how 4D IQ impairment characteristics can be separated from A(ω). Thus far, we have assumed full 16 physically a 2 (ω) = a * 1 (−ω) and a 4 (ω) = a * 3 (−ω) must always be satisfied. Therefore, elements in the even columns of H T,Δω (ω) can be expressed using elements in the odd columns of H T,Δω (ω): In addition, multiplying matrix R(ω), defined as (h a , …, h d are arbitrary complex functions) by H −1 T,Δω from the right side, will not affect IQ-impairment compensation since it can be absorbed by the 2×2 MIMO AEQ. In other words, we modify A(ω) by and redefine H −1 T,Δω (ω)R(ω) as a new H −1 T,Δω (ω) and We can then reduce the net DOFs of H T (ω) per frequency bin to 4 by with each component redefined. Similarly, the net DOFs of H R (ω) can be reduced to 4 by expressing it as Therefore, we obtain which contains all the DOFs of IQ impairments we need. The last step is to obtain H T (ω) from H T, Δω (ω). This can be accomplished using (11) and Δω x , Δω y estimated in the carrier-phase-recovery section of the AEQ.
In summary, the procedure to obtain 4D IQ characteristics is as follows.

. Obtain H T from H T, Δω
We discussed the computational complexity of static filtering with the proposed 4D IQ characterization method in comparison to MIMO AEQs in Appendix A.
Note that the obtained value will diverge to infinity if ) is not regular in any of frequency components. Such a situation occurs, for instance, when the Tx/Rx has large IQ skews. This is because the frequency components of these matrices which satisfy ωτ π (τ : IQ skew) will become nearly equal to zero matrices under a condition of low inter-polarization IQ crosstalk. To prevent this, we recommend to roughly compensate for intra-polarization IQ impairments beforehand.

C. Characterization of Typical IQ Impairments (IQ skew, Amplitude imbalance, and Phase deviation)
Although frequency-resolved IQ characteristics in the form of matrix functions can be obtained by the procedure described above, it is convenient to obtain an explicit value of a typical IQ impairment, such as IQ skew, IQ amplitude imbalance, IQ phase deviation. We describe how such typical IQ impairments can be estimated from the filter coefficients of the 8×2 MIMO AEQ. We assume negligible inter-polarization crosstalk. We parametrize all these impairments by P T /R x/y , τ T /R x/y , θ T /R x/y such that in the Tx/Rx (Re(s x/y ) Im(s x/y )) becomes ( Re(s x/y ) P T /R x/y exp(iωτ T /R x/y + θ T /R x/y )Im(s x/y ) ) within the Tx/Rx, where P T /R x/y represents IQ amplitude imbalance, τ T /R x/y represents IQ skew, and θ T /R x/y represents IQ phase deviation. The corresponding components of H −1 T,Δω (ω) and H −1 R (ω) will then become (29) shown at the bottom of the next page, and all the other components will become zeros. We then define k T /R x/y (ω) by These quantities approximately equal P T /R x/y exp(iωτ T /R x/y + θ T /R x/y ) when there are no noise and inter-polarization crosstalk. For example, we obtain from which we can calculate P Rx , τ Rx , and θ Rx straightforwardly from k Rx (ω). One can also show which approximates P T x e i(ωτ T x +θ T x ) when the impairments are sufficiently small: Suppose θ T x 1, we obtain Suppose tan(ωτ T x ), θ T x 1 and P T x 1, we obtain Since the definition of k T /R x/y above uses the filtercoefficients of X-polarized input, estimation quality will degrade if signal power is concentrated on the Y-polarized input. One can where we assumed negligible system noise and interpolarization crosstalk for the approximation. We can use either expression to calculate k T /R x/y , depending on the SOP of the signal.

III. NUMERICAL SIMULATION
To confirm the validity of the procedure described above, we conducted numerical simulations of high baud-rate transmission assuming various situations. Throughout the simulations, we assumed 80-km single mode fiber transmission at 96 Gbaud. We first assumed one typical impairment at a time: IQ skew such that signal (Re(s x ), Im(s x ), Re(s y ), Im(s y )) T develops into (Re(s x ), Im(s x ) cos θ XQY I − Re(s y ) sin θ XQY I , Im(s x ) sin θ XQY I + Re(s y ) cos θ XQY I , Im(s y )) T . We defined the amount of XQ-YI crosstalk XT XQY I [dB] by XT XQY I = 20log 10 tan(θ XQY I ). The characteristic of such crosstalk is which can be obtained by a change of the matrix basis from the basis where a signal vector is defined as (Re(s x ), Im(s x ), Re(s y ), Im(s y )) T to the basis where a signal is defined as ( s x s * x s y s * y ) T and then separating polarization-related DOFs using (23,24). Once H −1 T (ω) and H −1 R (ω) are estimated, θ can be retrieved by We determined θ XQY I for each XT XQY I by θ XQY I = tan −1 (10 XT XQY I /20 ) .
At the Tx, we prepared 16QAM signals Nyquist filtered with roll-off factor of 0.1. We simulated the chromatic dispersion of 18 ps/nm/km, PMD of 0.1 ps/km 1/2 , and optical SNR of 30 dB with the concatenated waveplate model [24]. At the Rx, we set the frequency offset and linewidth at 100 MHz and 20 kHz, assuming a Wiener process [25], and estimated the imposed impairments by using a 64-tap time-domain 8×2 MIMO AEQ. To update the filter coefficients of the equalizer, we employed a decision-directed least mean square algorithm with periodically inserted pilot symbols. The pilot overhead is 1.8%. For carrier-phase-recovery, we used a phase-locked loop algorithm whose details are shown in [25]. Note that we used the same algorithms in the experiments described in Section IV. We repeated the simulation ten times for each impairment value. The results are shown in Fig. 4(a). The dots indicate estimation results and solid lines indicate the true values for reference. The

IV. EXPERIMENT
In this section, we describe the experimental demonstration of the proposed 4D IQ-impairment-characterization method by 80-km 96-GBaud transmission, same as with the numerical simulations discussed above.

A. Experimental Setup
A schematic representation of the experimental setup is illustrated in Fig. 5. During the experiment detailed below, we emulated IQ impairments digitally. In Tx-side DSP, we prepared 16QAM signal Nyquist filtered with a roll-off factor of 0.1 then imposed the IQ impairments on the signal.
Our transmitter consisted of a 193-THz external cavity laser (ECL), two 22 GHz bandwidth Mach-Zehnder-type IQ modulators (IQM), 120 GSample/s digital-to-analog converters (DACs) within a 45 GHz bandwidth arbitrary waveform generator  (AWG), and drivers. We modulated orthogonally polarized continuous wave light from the ECL independently and combined it with a polarization beam combiner (PBC).
Our 80-km transmission line is composed of two 40-km-long single-mode fibers (SMFs) between which a 40-m-long polarization maintaining fiber (PMF) was inserted. A polarization scrambler (PS) was also placed before the transmission line. The reason of insertion of the PMF and PS was to check if our proposed method worked correctly in the presence of PMD and with switching of state of polarizations. The optical loss of the transmission line was compensated for using two erbium-doped fiber amplifier (EDFA) before and after the transmission line, followed by optical band-pass filters (OBPFs) that attenuate amplified spontaneous emission (ASE) noise outside the modulation bandwidth. The input power to the transmission line was 18.5 dBm, which made the electrical SNR of the entire system 14.7 (estimated from the demodulated signals), limited by the electrical noise of the transceiver.
Our Rx coherently detected the signals with an LO, 70 GHz bandwidth balanced photodetector, and 70 GHz bandwidth digital oscilloscope. In the Rx-side DSP, Rx IQ impairments were emulated, then concatenated signal impairments of the transceiver and transmission line were equalized using the 128tap time-domain 8×2 MIMO AEQ, the filter coefficients of which were fed into our proposed method. The intrinsic IQ impairments of the transceiver, such as variations in the radio-frequency cable, were compensated for before the measurement using the proposed method. As a demonstration, we demodulated the signal by using a conventional 2×2 MIMO AEQ before and after this calibration. For comparison, the signal was demodulated also by using 2×2, 4×2, and 8×2 MIMO AEQ without the calibration. The constellations of the demodulated signals are shown in Fig. 6, which shows that the proposed method has competing IQ impairment compensation ability (normalized generalized mutual information (NGMI) 0.961) with that of the 8×2 MIMO AEQ (NGMI 0.975). Also, it performed better than the 4×2 MIMO AEQ (NGMI 0.934) and the 2×2 MIMO AEQ (NGMI 0.729).

B. Results
We first applied one impairment at a time to the transceiver and estimated it. During the measurement, one of the IQ impairment (−5-5 ps IQ skews, 0.6-1.4 IQ amplitude imbalance −10-10°p hase deviation, and −28 to −20 dB unitary XQ-YI crosstalk) was chosen and digitally emulated. For XQ-YI crosstalk estimation, the average of 100 filter coefficients obtained at 1000 symbol intervals was used. Fig. 7(a) shows the estimation results, indicated with red dots, and true values indicated with blue lines as references. For all impairments, the estimated and true values were in good agreement. Next, we imposed multiple IQ impairments at once on both Tx/Rx: IQ skew of 5 ps, IQ imbalance of 1.2, and phase deviation of 5°on the X-polarized signal, along with the XQ-YI crosstalk of -20 dB. We then compared the outcomes with the reference simulation, which involved no ASE noise. For experiment, the average of 100 filter coefficients obtained at 1000 symbol intervals was used. The other assumptions of the experiment were kept the same as the simulations' described in Section III. Fig. 7(b) shows the results, where those of the experiment are represented with solid lines and those of the simulations are represented with dotted lines. They also are in good agreement for all the matrix components. To quantify the estimation accuracy of each impairment, we repeated the measurements 50 times and made histograms of the estimation values, which are shown in Fig. 7(c). We succeeded in estimating all the measurements with standard deviationσ. For example, sub-ps-level σ was achieved for IQ skew estimation, which is shorter than the pulse duration of 100 GBaud-class high-symbol-rate signals, thus sufficient for the skew compensation. Note that discrepancy of the mean estimation and true values can be observed for some histograms. This can be attributed to the use of (29) and (38), assuming the non-coexistence of intra-polarization IQ impairments and inter-polarization crosstalk. However, we argue that this will not be the final quality of IQ impairment compensation since 4D characteristics obtained with (28), which will be used for the compensation, do not assume such approximation.

V. CONCLUSION
To attain 100 Gbaud-class high symbol rate transmission, we a method for characterizing transceiver IQ impairments in a separated and frequency-resolved manner, on the basis of a four-dimensional signal propagation model that accounts for inter-polarization IQ crosstalk. Using an auxiliary 8×2 MIMO AEQ, the method can obtain the characteristics through a simple calculation. We also presented a method for separately estimating values of typical IQ impairments (e.g., IQ skew, IQ amplitude impairments, and IQ phase deviation) from the filter coefficients. We conducted numerical simulations and an experiment, the results of which support the feasibility of the method. This work will contribute to both understanding and developing over-100-Gbaud-class high-baud-rate systems in the future.

APPENDIX A COMPARISON OF THE COMPUTATIONAL COMPLEXITY
To discuss the computational complexity of the filters, we compared the required number of complex multiplication for signal impairment equalization with a tap length of L, and summarized the results in Table I. Here, the computational complexity of complex addition, CD compensation, and carrier-phaserecovery were not included. For the coefficient update, the least mean square method was assumed. Also, we assumed that L was sufficiently large so that multiplication regarding the step size multiplication for the coefficient update was negligible. Note that the required multiplication number for the 4D filtering is 1/4 of a full 4D matrix-vector multiplication because coefficients of the filter is defined as (25), (26) where half of them are 0 or 1, and multiplication results of even-numbered rows were always equal to complex conjugates of the results of odd-numbered rows and thus actual multiplication operations for the even-numbered rows were not needed. The comparison showed that the use of static filter can half the multiplication compared to the 8×2 MIMO AEQ and make complexity comparable to the 4×2 MIMO AEQ. Yutaka Miyamoto (Member, IEEE) received the B.E. and M.E. degrees in electrical engineering from Waseda University, Tokyo, Japan, in 1986 and 1988, respectively, and the Dr. Eng. degree in electrical engineering from the University of Tokyo, Tokyo, in 2016. In 1988, he joined the NTT Transmission Systems Laboratories, Yokosuka, Japan, where he was engaged in the research and development of high-speed optical communications systems, including the 10-Gbit/s first terrestrial optical transmission system (FA-10G) using erbium-doped fiber amplifiers (EDFA) inline repeaters. From 1995 to 1997, he was with the NTT Electronics Technology Corporation, Yokohama, Japan, where he was engaged in the planning and product development of high-speed optical module at the data rate of 10 Gb/s and beyond. Since 1997, he has been with the NTT Network Innovation Laboratories, Yokosuka, where he has contributed to the research and development of optical transport technologies based on 40/100/400-Gbit/s channel and beyond. He is currently an NTT Fellow and the Director of the Innovative Photonic Network Research Center, NTT Network Innovation Laboratories, where he has been investigating and promoting the future scalable optical transport network with the Pbit/s-class capacity based on innovative transport technologies, such as digital signal processing, space division multiplexing, and cutting-edge integrated devices for photonic preprocessing. He is a fellow of the Institute of Electronics, Information and Communication Engineers (IEICE).