Abstract

Tiny mismatches in timing, phase, and/or amplitude between in-phase (I) and quadrature (Q) tributaries in an electro-optic IQ modulator, namely IQ imbalance, can severely affect high baud-rate and/or high modulation-order signals in modern coherent optical communications systems. To maintain such analog impairment within the tight penalty limit over wavelength and temperature during the product lifetime, in-service in-field monitoring and calibration of the IQ imbalance, including its frequency dependence, become increasingly important. In this study, we propose a low-complexity IQ monitoring technique based on direct detection with phase retrieval called a single-pixel optical modulation analyzer (SP-OMA). By reconstructing the optical phase information lost during the detection process computationally via phase retrieval, SP-OMA facilitates the in-service in-field monitoring of the frequency-dependent imbalance profile without sending dedicated pilot tones and regardless of any receiver/monitor-side IQ imbalance. The feasibility of SP-OMA is demonstrated both numerically and experimentally with a 63.25-Gbaud 16QAM signal.

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

1. Introduction

The precise monitoring and careful calibration of analog imperfections inherent in optical transceivers are vital in modern fiber-optic communications systems that employ advanced modulation formats such as > 50-Gbaud > 16-array quadrature amplitude modulation (QAM). Timing, phase, and/or amplitude response mismatch between the in-phase (I) and quadrature (Q) tributaries in an electro-optic IQ modulator, namely IQ imbalance (IQI), is a notable example of such imperfections [15]. A slight mismatch can severely degrade high-speed and high-order QAM signals. Further, it was recently reported that the frequency-dependent variation of IQI, called frequency-dependent IQI (FD-IQI), represents the most stringent limitation in ultra high-order modulation formats [6]. Such IQ impairments must be maintained within the tight penalty limits over wavelength and temperature during the product lifetime. Moreover, modern optical transponders need to be open and optics should be pluggable. To this end, not only factory calibration, but also in-service monitoring and calibration in the field become increasingly important [3,79].

For the factory calibration, IQI is often monitored using a "gold-standard" coherent receiver that is a bulky, costly measurement instrument. This approach can be an in-service in-field solution if it is implemented with the far-end coherent receiver. Such in-service optical modulation analyzers based on coherent receiver (CO-OMAs), however, often suffer from the receiver-side impairments and they require extra efforts for separating the transmitter-side IQI from the receiver-side IQI, carrier frequency offset (CFO), and other transmission impairments [3,7,1012]. In addition, some feedback channel is required for transmitter calibration based on the far-end receiver.

Thus far, some low-cost alternatives based on direct detection have been proposed and demonstrated for the in-field calibration of pluggable optics [8,9,1316]. These direct-detection-based analyzers have low complexity and they are often integrable with transponders. However, a dedicated pilot/reference signal, such as phase-shifted frequency-swept tones [8], is required for determining the phase response through the intensity-only measurement in most cases. The use of a specific pilot signal limits their application in the in-service scenarios. Even a small dither tone may be harmful for the extremely high-order modulation formats. Moreover, most conventional direct-detection-based approaches pertain to the frequency-independent IQI model [9,13,14,16]. To the best of the authors’ knowledge, few works have been reported on a direct-detection-based in-service IQ monitoring solution that can fully characterize the frequency-dependent imbalance profile, including its phase component, without sending a dedicated pilot signal.

In this paper, we propose a low-complexity FD-IQI monitoring technique using a single monitor photodetector with phase retrieval [17], called single-pixel optical modulation analyzer (SP-OMA). The complex-valued impulse/frequency responses of individual IQ tributaries are reconstructed computationally from phase-less measurements by using a novel pilot-aided widely-linear (WL) phase-retrieval technique. SP-OMA can exploit either the information-bearing signal or the standard preamble, such as the overhead bits in 400ZR [5], as the pilot signal; it has a potential to be an in-service in-field monitoring solution for FD-IQI. The validity and feasibility of the proposed single-pixel approach is investigated numerically and experimentally with a 63.25-Gbaud 16QAM signal. In the experiment, despite the lack of the phase information, SP-OMA achieved a comparable, or even better, estimation accuracy than CO-OMA based on a typical linear least-squares channel estimation. The receiver-side FD-IQI and CFO often limit the practical performance of CO-OMA, while SP-OMA is exempt from such impairments. This feature makes SP-OMA not only a poorman’s IQ monitoring solution but also an attractive alternative to CO-OMA for pluggable optics and open transponders.

The remainder of the paper is organized as follows. In Sec. 2, we describe the details of the proposed widely-linear phase retrieval technique for the FD-IQI estimation. To show the validity and feasibility of the proposed approach, some numerical and experimental results are provided in sections 3 and 4, respectively. Finally, Sec. 5 concludes this study.

2. Frequency-dependent IQ imbalance estimation based on computational coherent detection via phase retrieval

2.1 IQ modulator with frequency-dependent IQ imbalance

In this study, we formulate the response of an optical IQ modulator in the presence of FD-IQI via WL modeling [18,19]. Suppose the driving signal is limited to the linear region; then, the output field $E(t)$ of the IQ modulator under FD-IQI is given by

$$E(t)\propto E_0 \left\{ h_+(t)\otimes x(t) + h_-(t)\otimes x^{*}(t)+\rho\right\},$$
where $\otimes$ denotes the linear convolution and $(\cdot )^{*}$ is the complex conjugation. The terms in brackets represent the IQ distorted modulation signal; $E_0\in \mathbb {C}$ is an optical carrier to be modulated. The complex-valued finite impulse response (FIR) filters $h_+(t)$ and $h_-(t)$ represent the IQ-distorted pulse shaping. The driving signal $x(t):=\sum _n s_n\delta (t+nT)$, where $s_n\in \mathbb {C}$ denotes the $n$th information bearing symbol, $\delta (t)$ denotes the Dirac delta function, and $T$ denotes a symbol duration. A complex constant $\rho$ represents the DC bias skew. According to a generic FD-IQI model shown in Fig. 1, the WL filter pair $\{h_{+}(t),h_{-}(t)\}$ is parameterized by
$$h_{\pm}(t) = \frac{h_I(t)\pm (1+\epsilon)h_Q(t)e^{j\theta}}{2},$$
where $h_{I/Q}(t)$ denotes the overall response of the I/Q tributary including such as a digital-to-analog converter (DAC), driver amplifier, pluggable connector, and PCB trace. The real coefficients $\epsilon$ and $\theta$ denote the frequency-independent power and phase mismatches during the optical IQ mixing process, respectively. The timing mismatch between the trajectories is included as part of $h_{I/Q}(t)$ for simplicity. The DC offset $\rho := \rho _I + j\rho _Q$, where $\rho _{I/Q}\in \mathbb {R}$ is the DC-bias skew at the I/Q blanch.

 figure: Fig. 1.

Fig. 1. Schematic of direct-conversion IQ modulator under frequency-dependent IQ imbalance.

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Most previous works investigate frequency non-selective IQ imbalances such as $\epsilon$ and $\theta$ by assuming that the pulse shaping filters are identical, i.e., $h_{I}(t) = h_{Q}(t)$ (or $h_Q(t-\Delta t)$). Here, $h_{I}(t)$ and $h_{Q}(t)$ are slightly detuned from each other and the mismatch makes the overall IQ distortion frequency-selective. Nevertheless, such parameterization may not be necessary in practice; the WL filter pair $\{h_+(t), h_-(t)\}$ and $\rho$ suffice for the implementation of the pre- and post-compensation filters in the complex baseband domain. We directly estimate them (in the digital domain) without resorting to (2).

2.2 FD-IQI monitoring based on coherent detection (CO-OMA)

With coherent detection, one can estimate FD-IQI via conventional pilot-aided WL channel estimation techniques [18]. The baseband-equivalent of the output field (1) after ideal coherent detection and analog-to-digital conversion is given by

$$r_n = \boldsymbol{h}_{+}^\textrm{T}\boldsymbol{s}_n +\boldsymbol{h}_{-}^\textrm{T}\boldsymbol{s}_n^{*}+\rho,$$
where $(\cdot )^\textrm {T}$ denotes the matrix/vector transpose and $\boldsymbol {s}_n := [s_n, s_{n-1}, \ldots , s_{n-L+1}]^\textrm {T}\in \mathbb {C}^{L\times 1}$ ($s_{n}=0$ for $n\leq 0$, for simplicity). The vectors $\boldsymbol {h}_{+}$ and $\boldsymbol {h}_{-}$ are the basedband equivalents of $h_{+}(t)$ and $h_{-}(t)$, respectively, and they are defined as $\boldsymbol {h}_{\pm }:=[h_{\pm }(0),h_{\pm }(T), \ldots , h_{\pm }((L-1)T)]^\textrm {T}\in \mathbb {C}^{L\times 1}$, where we assume $h_{\pm }(t) = 0$ for $t > (L-1)T$. Suppose $N$ transmitted symbols $\{s_1, \ldots , s_{N}\}$ are known to the receiver; then, the WL least-square (LS) estimate of $\boldsymbol {h}_{\pm }$ is given by [18]
$$\begin{bmatrix} \boldsymbol{h}_{\textrm{CO}+}\\ \boldsymbol{h}_{\textrm{CO}-} \end{bmatrix} =\bar{\boldsymbol{S}}^{+}\boldsymbol{r},$$
where $\boldsymbol {h}_{\textrm {CO}\pm }$ denotes the estimate and $\boldsymbol {r} := [r_1, \ldots , r_{N}]^\textrm {T}$. The $N \times 2L$ matrix $\bar {\boldsymbol {S}}$ comprises pilot symbols; the $n$th raw of $\bar {\boldsymbol {S}}$ is given by $[\boldsymbol {s}_n^\textrm {T} \boldsymbol {s}_n^\textrm {H}]$. The pseudo inverse matrix of $\bar {\boldsymbol {S}}$, i.e., $\bar {\boldsymbol {S}}^{+}:= (\bar {\boldsymbol {S}}^\textrm {H}\bar {\boldsymbol {S}})^{-1}\bar {\boldsymbol {S}}^\textrm {H}$, is denoted by $\bar {\boldsymbol {S}}^{+}$. We omit the DC bias term $\rho$. However, the impact of $\rho$ on the LS fitting is almost negligible as $\textrm {E}[s_n]=0$ in most phase-modulation formats. For the given $\boldsymbol {h}_{\textrm {CO}\pm }$, the bias term is then estimated by $\rho _{\textrm {CO}}=\frac {1}{N}\sum _{n=1}^{N}\{r_n-\boldsymbol {h}_{\textrm {CO}+}^\textrm {T}\boldsymbol {s}_n -\boldsymbol {h}_{\textrm {CO}-}^\textrm {T}\boldsymbol {s}_n^{*}\}$. The IQ down-conversion at optical coherent receivers cannot be perfect in practice and CO-OMAs suffer from IQI originating from their own analog front-ends. In addition, CFO, which is inevitable in optical coherent receivers, is known to deteriorate IQI estimation performance [20]. Therefore, CO-OMAs require well-proven optical analog front-ends or additional computation for jointly estimating the transceiver impairments and extracting those originating from the transmitter [3,7,1012].

2.3 FD-IQI monitoring based on direct detection with phase retrieval (SP-OMA)

Meanwhile, with direct detection, we have the intensity of (3) as

$$\psi_n = |\boldsymbol{h}_{+}^\textrm{T}\boldsymbol{s}_n + \boldsymbol{h}_{-}^\textrm{T}\boldsymbol{s}_n^{*}+\rho|^{2}.$$
OMA is required to decouple I and Q channel responses and recover their phases from phase-less measurements $\{\psi _n\}$. To this end, most previous works impose several restrictions on $\{s_n\}$. For instance, $\textrm {Re}\{s_n\}$ and $\textrm {Im}\{s_n\}$ need to be identically periodic "01" patterns in [14] and frequency-swept sinusoidal waves in [8,15]. The use of dedicated pilot signals limits their application in the in-service scenario. In addition, such narrowband reference signals, particularly if they need to be turned on and off during the measurement, can induce some "observer effect" in higher-order modulation formats. Recently, an IQI calibration technique based on a directly detected QAM signal has been proposed in [9] based on a sophisticated genetic algorithm. The machine-learning-aided approach in [16] that relies on computational coherent detection based on the Kramers-Kronig (KK) relationship [21] also enables the IQI estimation from a QAM signal. However, these approaches resort to frequency non-selective IQI models, e.g., fitness functions employed in the genetic algorithm in [9] are derived for the frequency non-selective model. In addition, the KK reception in [16] requires a reference optical carrier at OMA. As far as the authors know, none of the previous direct-detection-based OMAs can characterize frequency-selective IQ distortion from a pilot signal of arbitrary modulation format.

Here, we propose reconstructing the phase information lost in the direct-detection process computationally by using phase retrieval . In fact, it is straightforward to cast the pilot-aided channel estimation problem in (5) into a common form of the phase-retrieval problem. By introducing augmented vectors $\bar {\boldsymbol {h}} := [\boldsymbol {h}^\textrm {T}_+, \boldsymbol {h}^\textrm {T}_-, \rho ]^\textrm {T}$ and $\bar {\boldsymbol {s}}_{n} := [\boldsymbol {s}_{n}^\textrm {T}, \boldsymbol {s}_{n}^\textrm {H}, 1]^\textrm {T}$ (Here, $(\cdot )^\textrm {H}$ denotes the Hermitian transpose), the channel estimation problem can be written by

$$\mathrm{find}\quad \bar{\boldsymbol{h}} \quad\mathrm{s.t.}\quad \psi_{n} = |\bar{\boldsymbol{s}}_{n}^\textrm{T}\bar{\boldsymbol{h}}|^{2}\quad (n = 1,\ldots, N).$$
Eq. (6) is in a standard form of the generalized phase-retrieval problem [17]: reconstructing complex-valued unknown $\bar {\boldsymbol {h}}$ from multiple intensity-only samples $\{\psi _n\}$ with general non-Fourier (typically random) measurement vectors $\{\bar {\boldsymbol {s}}_n\}$. Phase retrieval, which is an ill-posed nonlinear inverse problem, arises in many areas such as crystallography, astronomy, and imaging, and it has been intensively studied over several decades [17]. In particular, in the last decade, notable progress was made on the theory and algorithm of generalized phase retrieval because of advances in algebraic geometry, matrix completion, and compressed sensing. It has been conjectured that $4m-4$ generic measurements such as i.i.d. Gaussian measurements are both necessary and sufficient to reconstruct $m$ unknown phases uniquely [22,23]. Further, contemporary phase-retrieval algorithms such as those based on direct non-convex optimization [2427] enable robust and computationally efficient phase-recovery even if the number of measurements is close to the aforementioned theoretical sample-complexity limit [17,28]. These advances allows us to estimate FD-IQI by directly solving (6) with realistic computational complexity. As the measurement vectors $\{\bar {\boldsymbol {s}}_n\}$ comprise pilot symbols, the $4m-4$ conjecture indicates that $\bar {\boldsymbol {h}}\in \mathbb {C}^{(2L+1)\times 1}$ can be uniquely determined by sending a random sequence of $N>8L$ symbols long (in the absence of noise); the information-bearing symbols and standard preamble can be exploited as the pilot $\{s_n\}$. Therefore, the proposed phase-retrieval-based approach has the potential to be a low-cost in-service monitoring solution for FD-IQI. Hereafter, we call this approach the single-pixel optical modulation analyzer (SP-OMA) after single-pixel imaging.

The resulting schematic architecture of SP-OMA is shown in Fig. 2. SP-OMA comprises a broadband monitor photodetector (PD) and an analog-to-digital converter (ADC) at the symbol rate or higher. Further, there is a secondary path for sharing the transmitted signal information; the path may not be necessary if a predetermined sequence such as the overhead bits in 400ZR [5] is used. The FD-IQI is estimated from the single PD output by solving the pilot-aided WL channel estimation problem in (6) with a contemporary phase retrieval algorithm. SP-OMA can be viewed as a WL extension of the pilot-aided channel estimation technique based on phase retrieval that was recently proposed for the characterization of multimode fiber (MMF) channels [2931].

 figure: Fig. 2.

Fig. 2. Schematic architecture of proposed single-pixel optical modulation analyzer.

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2.3.1 Robust algorithm for widely-linear convolutional phase retrieval

Although SP-OMA relies on generalized phase retrieval, measurement vectors $\{\bar {\boldsymbol {s}}_n\}$ in (6) cannot be purely random; $\{\bar {\boldsymbol {s}}_n\}$ are structured even if $\{s_n\}$ is a random sequence. The system model in (1) is convolutional and the resulting measurement vectors $\{\bar {\boldsymbol {s}}_n\}$ are strongly correlated, i.e., $\boldsymbol {s}_{n}$ is a (time) shifted version of $\boldsymbol {s}_{n-1}$. Furthermore, each vector $\bar {\boldsymbol {s}}_{n}$ has an internal structure: $\bar {\boldsymbol {s}}_{n} = [\boldsymbol {s}_{n}^\textrm {T}, \boldsymbol {s}_{n}^\textrm {H}, 1]^\textrm {T}$. Thus, $\{\bar {\boldsymbol {s}}_n\}$ are parameterized by only $N$ independent random variables and a constant, whereas the purely random measurement model involves $(2L+1)N$ random variables.

Such structured random measurements require more intensity samples for stable phase recovery compared with purely random measurements. A careful selection and parameter tuning of the algorithms is required for stable phase recovery. The study of structured random measurements for phase retrieval remains limited, but there has been a certain attention to the (standard linear) convolutional measurement model motivated by applications such as those for wireless channel estimation [32] and optical space-division multiplexed (SDM) transmission [29,31]. In [33], the global convergence property of a gradient-descent type algorithm, a prominent example of the recent non-convex optimization-based phase retrieval algorithms, in the convolutional measurement model was revealed. In our previous works [34,35], the performance of some non-convex approaches were investigated numerically and experimentally for the convolutional phase retrieval problem in the carrier-less coherent detection of the optical SDM signal. In the experiment, the phase retrieval algorithm based on the alternating direction method of multipliers (ADMM) [27] outperformed other algorithms such as amplitude-based gradient descent [25,31] and approximate message passing (AMP) [26] in the low sample complexity regime, and it achieved the sample complexity of $7.5m$ for the detection of $m$ QPSK symbols. In this study, we employ the ADMM-based approach called PhareADMM [27] for solving the phase retrieval problem in the biased WL convolutional measurement model in (6).

Here, we briefly introduce the PhareADMM algorithm for FD-IQI estimation. To solve (6), PhareADMM minimizes the amplitude-based loss function

$$l(\bar{\boldsymbol{h}})=\sum_{n=1}^{N} \left(\sqrt{\psi_{n}}-|\bar{\boldsymbol{s}}_{n}^\textrm{T}\bar{\boldsymbol{h}}|\right)^{2}$$
by the augmented Lagrangian method that uses sequential partial updates. Note that, the least-squares loss on amplitude values often offers lower sample- and computational-complexities than the intensity values, i.e., $\psi _{n}-|\bar {\boldsymbol {s}}_{n}^\textrm {T}\bar {\boldsymbol {h}}|^{2}$, in phase retrieval [36]. By introducing auxiliary amplitude variables $u_n\in \mathbb {R}_+$ and phase variables $\phi _n\in (0, 2\pi ]$ ($n = 1, \ldots , N$), the minimization problem can be re-written as
$$\begin{array}{c} \min\limits_{\bar{\boldsymbol{h}}, \{u_n\}, \{\phi_n\}}\sum\limits_{n=1}^{N} \left(\sqrt{\psi_{n}}-u_{n}\right)^{2} \\ \mathrm{s.t.}\quad \bar{\boldsymbol{s}}_{n}^\textrm{T}\bar{\boldsymbol{h}}=u_n e^{j\phi_n},\ u_{n}\geq~0,\ (n = 1,\ldots, N). \end{array}$$
The augmented Lagrangian function of (8) is
$$\begin{aligned}\mathcal{L}&\left(\bar{\boldsymbol{h}}, \left\{u_n, \phi_n, \lambda^\textrm{(r)}_n, \lambda^\textrm{(i)}_n\right\}\right) = \sum_{n=1}^{N} \left(\sqrt{\psi_{n}}-u_{n}\right)^{2}\\ &+\sum_{n=1}^{N}\left\{\lambda^\textrm{(r)}_n \left\{\bar{\boldsymbol{s}}_{n}^\textrm{T}\bar{\boldsymbol{h}}-u_n e^{j\phi_n}\right\}+\frac{\eta}{2}\left(\operatorname{Re}\left\{\bar{\boldsymbol{s}}_{n}^\textrm{T}\bar{\boldsymbol{h}}-u_n e^{j\phi_n}\right\}\right)^{2}\right\}\\ &+\sum_{n=1}^{N}\left\{\lambda^\textrm{(i)}_n\operatorname{Im}\left\{\bar{\boldsymbol{s}}_{n}^\textrm{T}\bar{\boldsymbol{h}}-u_n e^{j\phi_n}\right\}+\frac{\eta}{2}\left(\operatorname{Im}\left\{\bar{\boldsymbol{s}}_{n}^\textrm{T}\bar{\boldsymbol{h}}-u_n e^{j\phi_n}\right\}\right)^{2}\right\}, \end{aligned}$$
where $\lambda ^\textrm {(r)}_n\in \mathbb {R}$ and $\lambda ^\textrm {(i)}_n\in \mathbb {R}$ are Lagrange multipliers for real and imaginary components, respectively, and $\eta >0$ is the step size. In ADMM, the Lagrangian function is minimized sequentially via alternating minimization; in the $t$th iteration,
$$\{u_n(t+1), \phi_n(t+1)\}=\mathop{\mathrm{argmin}}_{\{u_n, \phi_n\}}\mathcal{L}\left(\bar{\boldsymbol{h}}(t), \left\{u_n, \phi_n, \lambda^\textrm{(r)}_n(t), \lambda^\textrm{(i)}_n(t)\right\}\right),$$
$$\bar{\boldsymbol{h}}(t+1)=\mathop{\mathrm{argmin}}_{\bar{\boldsymbol{h}}}\mathcal{L}\left(\bar{\boldsymbol{h}}, \left\{u_n(t+1), \phi_n(t+1), \lambda^\textrm{(r)}_n(t), \lambda^\textrm{(i)}_n(t)\right\}\right),$$
with the multiplier updates
$$\lambda^\textrm{(r)}_n(t+1)=\lambda^\textrm{(r)}_n(t) + \eta\operatorname{Re}\left\{\bar{\boldsymbol{s}}^\textrm{T}_{n}\bar{\boldsymbol{h}}(t+1)-u_n(t+1) e^{j\phi_n(t+1)}\right\},$$
$$\begin{aligned} \lambda^\textrm{(i)}_n(t+1)&=\lambda^\textrm{(i)}_n(t) + \eta\operatorname{Im}\left\{\bar{\boldsymbol{s}}^\textrm{T}_{n}\bar{\boldsymbol{h}}(t+1)-u_n(t+1) e^{j\phi_n(t+1)}\right\} .\\ &\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad(n = 1, \ldots ,N) \end{aligned}$$
As in [27], Eqs. (10) and 11 have closed-form solutions
$$\phi_n(t+1) = \angle\{{g_n(t)}\},$$
$$u_n(t+1)=\frac{\eta\left|g_n(t)\right|+2\sqrt{\psi_n}}{2+\eta},$$
$$\bar{\boldsymbol{h}}(t+1)=\tilde{\boldsymbol{S}}^{+}\left(\boldsymbol{v}(t+1)-\frac{1}{\eta}\boldsymbol{\lambda}(t)\right),$$
where $g_n(t):=\bar {\boldsymbol {s}}_{n}^\textrm {T}\bar {\boldsymbol {h}}(t) + \frac {\lambda _n(t)}{\eta }$ and $\angle (\cdot )$ denotes the phase angle operator. The composite version of the Lagrange multipliers is defined as $\lambda _n(t) := \lambda ^\textrm {(r)}_n(t) + j\lambda ^\textrm {(i)}_n(t)$ and $\boldsymbol {\lambda }(t):=[\lambda _1(t), \ldots , \lambda _N(t)]$. The $n$th raw of the measurement matrix $\tilde {\boldsymbol {S}}\in \mathbb {C}^{N\times (2L+1)}$ is given by $[\boldsymbol {s}_{n}^\textrm {T}, \boldsymbol {s}_{n}^\textrm {H}, 1]$ and $\boldsymbol {v}(t)\in \mathbb {C}^{N\times 1}$ represents the reconstructed received signal, i.e., $[\boldsymbol {v}(t)]_n := u_n(t)e^{j\phi _n(t)}$. Using $\lambda _n(t)$, Eqs. (12) and 13 are summarized as
$$\lambda_n(t+1)=\lambda_n(t) + \eta\{\bar{\boldsymbol{s}}^\textrm{T}_{n}\bar{\boldsymbol{h}}(t+1)-u_n(t+1) e^{j\phi_n(t+1)}\}.$$

The resulting algorithm is summarized in Algorithm 1 in a vector and matrix form. In the pseudo code, $\boldsymbol {\xi }:=[\sqrt {\psi _{1}}, \ldots ,\sqrt {\psi _{N}}]^\textrm {T}$ and $\circ$ denotes the Hadamard product; further, $\frac {\boldsymbol {g}(t)}{|\boldsymbol {g}(t)|}\in \mathbb {C}^{N\times 1}$, which corresponds to $\{e^{j\phi _n(t+1)}\}$, is defined by

$$\left[\frac{\boldsymbol{g}(t)}{|\boldsymbol{g}(t)|}\right]_n=\begin{cases} 0 & \textrm{for}\ g_n(t)=0,\\ \frac{g_n(t)}{|g_n(t)|} & \textrm{otherwise}. \end{cases}$$
We employ the stopping criterion $\Delta \leq \Delta _0$ for a prescribed accuracy $\Delta _0$ (e.g., $\Delta _0\approx 10^{-6}$) in Algorithm 1.

For the computational complexity, PhareADMM requires $\mathcal {O}(LN)$ multipliers per iteration, whereas the matrix inversion of $\tilde {\boldsymbol {S}}\in \mathbb {C}^{N\times (2L+1)}$ in the initialization step needs $\mathcal {O}(L^{2}N)$ ($N>L$ in general). Thus the overall complexity of PhareADMM is $\mathcal {O}(L^{2}N)$. It is on the same order as the complexity of CO-OMA based on WL-LS, which involves the matrix inversion of $\bar {\boldsymbol {S}}\in \mathbb {C}^{N\times (2L)}$. $\tilde {\boldsymbol {S}}^{+}$ can be processed in advance if a known preamble sequence is used as $\{s_n\}$.

In our previous work on a phase-retrieval coherent receiver [31] and the preliminary report on SP-OMA [37], we employed an alternative approach wherein some of the received samples are discarded purposely and the following phase retrieval problem is solved

$$\mathrm{find}\quad \bar{\boldsymbol{h}} \quad\mathrm{s.t.}\quad \psi_{kL} = |\bar{\boldsymbol{h}}^\textrm{T}\bar{\boldsymbol{s}}_{kL}|^{2}\quad (k = 1,\ldots, K),$$
where $\psi _{kL}$ denotes the $L$-fold decimated version of $\psi _{n}$. After the decimation, $\boldsymbol {s}_{kL} (k=1,\ldots ,K)$ do not overlap each other and the measurement becomes less structured. This approach eases the task for solving the structured phase retrieval problem and offers broader choices of algorithms at the expense of an increased pilot overhead.

2.4 Initialization and resolution of ambiguities

Most iterative phase-retrieval algorithms with the non-convex loss functions require careful initialization such as spectral initialization [28]. Fortunately, the unknown $\bar {\boldsymbol {h}}$ is a detuned version of the ideal IQ pulse-shaping filter, i.e., $\bar {\boldsymbol {h}}_{\textrm {ideal}}=[\boldsymbol {h}_{\textrm {ideal}}^\textrm {T},\boldsymbol {0}^{1\times (L+1)}]^\textrm {T}$, where $\boldsymbol {h}_{\textrm {ideal}}\in \mathbb {C}^{L\times 1}$ denotes an ideal pulse-shaping filter response such as a raised-cosine function. Therefore, “warm start” is possible by choosing the initial value $\bar {\boldsymbol {h}}_0=\bar {\boldsymbol {h}}_{\textrm {ideal}}$. As we show later in the numerical and experimental results, informative initialization with the robust algorithm facilitates a stable phase recovery.

Phase-retrieval algorithms suffer from global phase ambiguity, i.e., $l(e^{j\alpha }\bar {\boldsymbol {h}})=l(\bar {\boldsymbol {h}})$ for any $\alpha \in (0, 2\pi ]$. The WL phase retrieval also suffers from the ambiguity of complex conjugation; $\boldsymbol {h}_{\mp }^{*}$ can also be a solution if $\boldsymbol {h}_{\pm }$ is a global minimizer because $|\boldsymbol {h}_+^\textrm {T}\boldsymbol {s}_n+\boldsymbol {h}_-^\textrm {T}\boldsymbol {s}^{*}_n|=|\boldsymbol {h}_-^\textrm {H}\boldsymbol {s}_n+\boldsymbol {h}_+^\textrm {H}\boldsymbol {s}^{*}_n|$. However, such ambiguities will not be a problem for the modulation analyser as it is interested in only the mismatch between $\boldsymbol {h}_{+}$ and $\boldsymbol {h}_{-}$. For instance, the relative amplitude and phase responses are sufficient for implementing pre-compensation filters at the transmitter DSP. Therefore, we can resolve the ambiguities without the loss of practicality such as by adjusting the peak of $\boldsymbol {h}_{+}$ to be 1, i.e., $\boldsymbol {h}_{+}\leftarrow \frac {\beta _0^{*}}{|\beta _0|^{2}}\boldsymbol {h}_{+}$, $\boldsymbol {h}_{-}\leftarrow \frac {\beta _0^{*}}{|\beta _0|^{2}}\boldsymbol {h}_{-}$, and $\rho \leftarrow \frac {\beta _0^{*}}{|\beta _0|^{2}}\rho$, where $\beta _0$ denotes the peak value of $\boldsymbol {h}_{+}$. Meanwhile, since $\|\boldsymbol {h}_{+}\|_2^{2} \gg \|\boldsymbol {h}_{-}\|_2^{2}$ in practice, the complex conjugation can be detected and mitigated by replacing $\{\boldsymbol {h}_{+}, \boldsymbol {h}_{-}\}\leftarrow \{\boldsymbol {h}^{*}_{-}, \boldsymbol {h}^{*}_{+}\}$ if the estimate $\|\boldsymbol {h}_{+}\|_2^{2} < \|\boldsymbol {h}_{-}\|_2^{2}$.

3. Numerical results

The performance of the proposed single-pixel approach is evaluated via numerical simulation. The modulation format is 16QAM, and FD-IQI is induced based on (2) in the baseband domain. The IQ distorted signal is transmitted over a complex additive white Gaussian noise (AWGN) channel and then received via an ideal square-law detector and an ideal coherent detector for SP-OMA and CO-OMA, respectively. We test the SP-OMA with PhareADMM and the CO-OMA based on the WL-LS criterion in (4). Hereafter, CO-OMA and SP-OMA denote these configurations unless otherwise noted. Other optical channel impairments are omitted because SP-OMA is intended to be implemented close to (or integrated with) the transmitter.

In FD-IQI generation, each distortion parameter in (2) is selected as follows. Pulse shaping filters $\boldsymbol {h}_{I}$ and $\boldsymbol {h}_{Q}$ are randomly detuned raised-cosine filters: $\boldsymbol {h}_{I,Q}\sim$ $\mathcal {N}_c(\boldsymbol {h}_\textrm {ideal},$ $0.005\boldsymbol {I}_{13\times 13})$, where $\mathcal {N}_c$ denotes the circularly symmetric complex Gaussian distribution and the mean vector $\boldsymbol {h}_\textrm {ideal}$ comprises of the truncated ($L=13$) impulse response of a raised-cosine filter with a roll-off factor of 0.2. The frequency independent amplitude imbalance $\epsilon \sim \mathcal {N}(0,0.0016)$, where $\mathcal {N}$ denotes the (real-valued) Gaussian distribution, and the phase imbalance $\theta \sim \mathcal {N}(0,0.063)$ in radians. The WL filter representation $\boldsymbol {h}_\textrm {dist}\pm := \frac {\beta _0^{*}}{|\beta _0|^{2}}\{\boldsymbol {h}_I\pm (1+\epsilon )\boldsymbol {h}_Qe^{j\theta }\}$, where $\beta _0$ is introduced to adjust the peak of $\boldsymbol {h}_ {{\textrm{dist}}_{+}}$ to 1 for convenience. The DC bias $\rho _\textrm {dist}(=\rho _I+j\rho _Q)\sim \mathcal {N}_c(0,0.02)$ whereas the driving signal power $\textrm {E}[|s_n|^{2}]$ is normalized to 1.

We evaluate the normalized mean-square error (NMSE) performance of SP-OMA and CO-OMA over 5,000 independent trials. Here, the MSEs for $\boldsymbol {h}_{\pm }$ and $\rho$ are normalized based on the IQ-distorted pulse shaping filter gain, i.e.,

$$NMSE(\boldsymbol{h}_{{\pm}}) = \textrm{E}\left[\frac{\|\boldsymbol{h}_+{-}\boldsymbol{h}_\textrm{dist+}\|_2^{2}+\|\boldsymbol{h}_-{-}\boldsymbol{h}_\textrm{dist-}\|_2^{2}}{\|\boldsymbol{h}_\textrm{dist+}\|_2^{2}+\|\boldsymbol{h}_\textrm{dist-}\|_2^{2}}\right],$$
$$NMSE(\rho) = \textrm{E}\left[\frac{2L|\rho-\rho_\textrm{dist}|^{2}}{\|\boldsymbol{h}_\textrm{dist+}\|_2^{2}+\|\boldsymbol{h}_\textrm{dist-}\|_2^{2}}\right].$$
As for the PhareADMM parameters, we choose $\eta = 0.6$, $\Delta _0 =10^{-6}$, and $\bar {\boldsymbol {h}}_{0}=[\boldsymbol {h}_{\textrm {ideal}}^\textrm {T},\boldsymbol {0}^{1\times (L+1)}]^\textrm {T}$.

The convergence property of PhareADMM in SP-OMA is investigated. Figure 3(a) shows the MSE performance of SP-OMA against the number of ADMM iterations $T$. Here, the signal-to-noise power ratio (SNR) is fixed to 20 dB. The pilot length $N=27,040$. As shown in Fig. 3(a), the channel responses $\boldsymbol {h}_{\textrm {SP}\pm }$ and DC bias $\rho _{\textrm {SP}}$ are stably and quickly converged to the ground truth. The NMSE of less than $0.3\times 10^{-4}$ is achieved for $T>20$. No reconstruction failure is observed over the trials.

 figure: Fig. 3.

Fig. 3. NMSE performance of SP-OMA and CO-OMA versus a) number of PhareADMM iterations, b) SNR, c) pilot length, and d) IRR.

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Next, we compare the performance of SP-OMA and CO-OMA. Figures 3(b), (c), and (d) show the NMSE performance of the modulation analyzers against the optical noise level, pilot length, and IQ distortion level, respectively. In these figures, red solid/dashed lines denote the NMSEs with SP-OMA and blue solid/dashed lines are those with CO-OMA. Hereafter, the number of ADMM iterations is $T=50$, SNR is 20 dB, and pilot length $N=27,040$, unless otherwise noted. Figure 3(b) shows the NMSEs of SP-OMA and CO-OMA against SNR. The IQ distortion parameters are obtained successfully via SP-OMA across a practical SNR range. There exists a certain SNR penalty compared with CO-OMA because of the lack of the phase information. However, the penalty is only $\approx 3$ dB for SNR $> 15$ dB, although SP-OMA reconstructs the lost information via non-convex optimization. More importantly, the NMSEs decrease at the same rate in SP-OMA and CO-OMA in the high SNR region. Figure 3(c) shows the impact of pilot length $N$ (or equivalently the sample complexity) on the NMSE performance at SNR = 20 dB. The NMSEs of SP-OMA quickly decreases down to $10^{-4}$ for $N > 5,000$. The estimation variances decrease as the sample size increases at the same rate in SP-OMA and CO-OMA. Finally, $NMSE(\boldsymbol {h}_{\textrm {SP}\pm })$ and $NMSE(\boldsymbol {h}_{\textrm {CO}\pm })$ versus the image rejection ratio (IRR) in each trial are plotted in Fig. 3(d). IRR represents the IQ distortion level and is defined by $IRR(\boldsymbol {h}_{\pm }) = \|\boldsymbol {h}_{+}\|^{2}_2/ \|\boldsymbol {h}_{-}\|^{2}_2$. The IRR varies from around 5 to 20 dB during the trials. SP-OMA can converge to the ground truth almost independent of the IQI level, as with CO-OMA, and it achieves $NMSE(\boldsymbol {h}_{\textrm {SP}\pm }) < 0.5\times 10^{-4}$ for all trials. The bias skew is also reconstructed stably, although it is omitted in the figure for simplicity.

These numerical results show the validity of the proposed phase-retrieval based approach. Moreover, besides the 3-dB SNR penalty, SP-OMA exhibits very similar characteristics as CO-OMA, a typical least-squares estimator based on coherent detection, against the background noise, sample size, as well as IQI level. This feature makes SP-OMA a promising low-cost alternative to CO-OMA.

4. Experimental results

The numerical results presented in the previous section do not consider the fundamental sensitivity penalty of direct detection compared with coherent detection; it is not easy for SP-OMA to achieve the same SNR level as CO-OMA in practice. On the other hand, analog impairments inherent in coherent receivers such as CFO and FD-IQI are omitted in CO-OMA. To further demonstrate the feasibility of the proposed single-pixel approach, we evaluate the performance of SP-OMA and CO-OMA experimentally with a 63.25-Gbaud 16QAM signal.

Figure 4 shows the experimental setup. The optical transmitter comprises a 1550.92-nm laser with a 100-kHz linewidth, 4-channel 92-Gs/s arbitrary waveform generator (AWG), and a dual-polarization (DP) LiNO3 Mach-Zehnder IQ modulator. The modulation format was 63.25-Gbaud DP-16QAM. To simulate FD-IQI, the 16QAM signal on each polarization blanch was pre-distorted in the digital domain by appending a ($13\times 2$)-tap WL filter $\bar {\boldsymbol {h}}_{\textrm {dist}} = [\boldsymbol {h}_{\textrm {dist}+}^\textrm {T},\boldsymbol {h}_{\textrm {dist}-}^\textrm {T} ]^\textrm {T}$. The DC bias skew was omitted in this experiment because of the difficulty for precisely tuning the optical bias component inducing its phase in the current setup. The modulator output was amplified via an erbium-doped fiber amplifier (EDFA) and then input to SP-OMA or CO-OMA. SP-OMA is implemented with a 70-GHz PD and a 160-GSa/s digital storage oscilloscope (DSO). A polarization beam splitter (PBS) was installed in front of SP-OMA to extract a polarization channel. CO-OMA was based on standard analog coherent optics with polarization diversity configuration and a 4-channel 80-GSa/s DSO. The linewidth of the local oscillator laser was 100 kHz. The input powers were + 10 dBm and −10 dBm for SP-OMA and CO-OMA, respectively. The polarization states of the modulator and the OMAs were manually adjusted via polarization controllers (PCs). In both OMAs, the received signals were immediately re-sampled down to the symbol rate, i.e., 63.25 GSa/s. The pilot sequence for channel estimation comprised 84,000 random 16QAM symbols. SP-OMA required the input power of +10 dBm for directly digitizing the PD output without a transimpedance amplifier (TIA). Although the received power was 20 dB lower than SP-OMA, the coherent receiver was operated at its standard operation range, and therefore, the received SNR of CO-OMA was not that low; the error vector magnitude (EVM) performance observed at the coherent receiver in the absence of FD-IQI was around 10%.

 figure: Fig. 4.

Fig. 4. Experimental setup for SP-OMA and CO-OMA with a 63.25-Gbaud 16QAM signal.

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SP-OMA estimated FD-IQI as a $(13\times 2)$-tap WL filter $\bar {\boldsymbol {h}}_{\textrm {SP}}:=[\boldsymbol {h}_{\textrm {SP}+}^\textrm {T},\boldsymbol {h}_{\textrm {SP}-}^\textrm {T} ]^\textrm {T}$ using PhareADMM with the step size of 0.6. The number of ADMM iterations was set to 50 with the stopping criterion $\Delta _0 = 10^{-6}$. Meanwhile, CO-OMA derived the WL filter $\bar {\boldsymbol {h}}_{\textrm {CO}}:=[\boldsymbol {h}_{\textrm {CO}+}^\textrm {T},\boldsymbol {h}_{\textrm {CO}-}^\textrm {T} ]^\textrm {T}$ with (4). In each OMA, the estimated WL filter was calibrated using a WL equalizer (EQ) to mitigate the inherent IQ distortion and frequency response of the equipment. The equalizer weights were obtained from $\bar {\boldsymbol {h}}_{\textrm {SP}}$ and $\bar {\boldsymbol {h}}_{\textrm {CO}}$ for SP-OMA and CO-OMA, respectively, in the absence of the digital pre-distortion filter at the transmitter. In CO-OMA, the WL equalizer was intended to mitigate FD-IQIs both in the transmitter and receiver equipment. In addition, the CFO and the phase noise were mitigated in CO-OMA before IQI estimation based on periodic pilot symbols; the overhead for the symbols was not considered in the following results.

First, we tested the OMAs for three "eye-friendly" IQ distortion patterns to show the feasibility of SP-OMA intuitively. The original, received, and reconstructed constellation maps for the three cases are depicted in Fig. 5. The original maps in the first column are constellation diagrams of the IQ pre-distorted driving signals. The maps in the second column represent the received signal at the coherent receiver after CFO compensation, phase noise cancellation, and WL equalization; the three IQ distortion patterns were generated properly. The reconstructed maps in the third/fourth column represent the simulated constellation diagrams by appending the estimated WL filters $\bar {\boldsymbol {h}}_{\textrm {CO}}$/$\bar {\boldsymbol {h}}_{\textrm {SP}}$ to a random (IQI-free) 16QAM sequence. The impulse and frequency responses of the pre-distortion and estimated WL filters are plotted in Fig. 6. The WL filter pair $\boldsymbol {h}_{\textrm {SP}\pm }$ properly converged to the ground truth $\boldsymbol {h}_{\textrm {dist}\pm }$, as with CO-OMA, and the IQ distorted constellation maps were drawn exactly without coherent detection as shown in Fig. 5.

 figure: Fig. 5.

Fig. 5. Received and simulated constellation maps for the three FD-IQI patterns.

Download Full Size | PPT Slide | PDF

 figure: Fig. 6.

Fig. 6. Estimated impulse responses and frequency responses with CO-OMA and SP-OMA for the three FD-IQI patterns.

Download Full Size | PPT Slide | PDF

Next, the statistical performance was evaluated by randomly changing the pre-distortion filter response. The distributions of $\boldsymbol {h}_{\textrm {dist}+}$ and $\boldsymbol {h}_{\textrm {dist}-}$ were the same as in Sec. 3. We measured the NMSE performances over 20 different FD-IQI patterns. Figure 7(a) shows the NMSE of CO-OMA and SP-OMA for each distortion pattern; each bar is the average of 3 independent trials with $T=50$ and $N=84,000$. SP-OMA achieved a comparable or even better NMSE for some patterns. The averaged NMSEs over 20 patterns were $7.71\times 10^{-4}$ and $9.09\times 10^{-4}$ for SP-OMA and CO-OMA, respectively. An example of the estimated impulse/frequency response is shown in Fig. 8; the IRR was 12.8 dB in the example.

 figure: Fig. 7.

Fig. 7. a) NMSE performance of CO-OMA and SP-OMA for the 20 random FD-IQI patterns: a) NMSE per pattern and b) averaged NMSE against pilot length.

Download Full Size | PPT Slide | PDF

 figure: Fig. 8.

Fig. 8. An example of estimated impulse responses and frequency responses with CO-OMA and SP-OMA for the random FD-IQI patterns.

Download Full Size | PPT Slide | PDF

Figure 7(b) shows the averaged NMSEs against the pilot length $N$. The NMSEs converged quickly but suffered from error floors. Unlike in Fig. 3(a), the longer pilot sequence did not offer the better NMSE for $N>10000$. This indicates that, even if the received SNRs were not well-coordinated between CO-OMA and SP-OMA, the optical noise did not dominate the performance of the analyzers in the experiment. In fact, a notable performance-limiting factor was the residual transceiver analog impairment. CO-OMA suffered from residual IQI, CFO, and phase noise at the coherent receiver, while SP-OMA was free from these receiver-side IQ impairments. This enabled SP-OMA to achieve an even better performance despite the fundamental 3-dB SNR penalty.

These experimental results demonstrate the practicality of SP-OMA. Moreover, IQ monitoring independent of the receiver-side IQ impairments, which severely limited the performance of CO-OMA, makes SP-OMA not only a low-cost alternative to CO-OMA but also an attractive solution for the optical transceiver monitoring and calibration in open and disaggregated optical transmission systems.

5. Conclusions

A low-complexity optical modulation analyser using a single monitor photodetector (SP-OMA), was proposed for the in-service and in-field monitoring of optical IQ modulators. Based on a novel pilot-aided widely-linear phase-retrieval technique, SP-OMA can characterize the frequency-dependent amplitude, phase, and timing mismatches between the I and Q tributaries (FD-IQI) in an optical IQ modulator using the intensity-only measurements. The performance of SP-OMA was evaluated numerically and experimentally in comparison with the conventional modulation analyzer based on coherent detection (CO-OMA). The SP-OMA achieved performance comparable to CO-OMA, even though the lack of the optical phase information. The simple optical analog front-end of SP-OMA allows the FD-IQI estimation to be free from the receiver-side IQ impairments, which is a notable performance limiting factor in CO-OMA. This feature makes SP-OMA not only a low-cost alternative to CO-OMA but also an attractive solution for the on-line calibration of pluggable optics and open transponders. An extension of the proposed single-pixel approach to dual-polarization IQ modulators will be reported in our forthcoming paper (See [38] for the preliminary results).

Funding

Japan Society for the Promotion of Science (JP19K04384); National Institute of Information and Communications Technology (205).

Acknowledgments

The authors thank Dr. Takahito Tanimura, Hitachi Ltd., for the insightful comments and suggestions.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. I. Fatadin, S. J. Savory, and D. Ives, “Compensation of quadrature imbalance in an optical QPSK coherent receiver,” IEEE Photonics Technol. Lett. 20(20), 1733–1735 (2008). [CrossRef]  

2. Y. Yue, Q. Wang, and J. Anderson, “Transmitter skew tolerance and spectral efficiency tradeoff in high baud-rate QAM optical communication systems,” Opt. Express 26(12), 15045–15058 (2018). [CrossRef]  

3. J. Liang, Y. Fan, Z. Tao, X. Su, and H. Nakashima, “Transceiver imbalances compensation and monitoring by receiver DSP,” IEEE/OSA J. Lightwave Technol. (Early access) p. 1 (2019).

4. T. Kobayashi, M. Nakamura, F. Hamaoka, M. Nagatani, H. Yamazaki, H. Nosaka, and Y. Miyamoto, “Long-haul WDM transmission with over-1-Tb/s channels using electrically synthesized high-symbol-rate signals,” in 2020 Optical Fiber Communications Conference and Exhibition, (2020), paper M4K.1.

5. Y. Miyamoto, “OIF-400ZR-01.0: Implementation agreement 400ZR,” Optical Internetworking Forum (2020).

6. X. Chen, J. Cho, A. Adamiecki, and P. Winzer, “16384-QAM transmission at 10 GBd over 25-km SSMF using polarization-multiplexed probabilistic constellation shaping,” in 45th European Conference on Optical Communication, (2019), paper PD3.3.

7. C. Ju, N. Liu, and C. Li, “In-service blind transceiver IQ imbalance and skew monitoring in Long-Haul Non-Dispersion managed coherent optical systems,” IEEE Access 7, 150051–150059 (2019). [CrossRef]  

8. C. R. S. Fludger, T. Duthel, P. Hermann, and T. Kupfer, “Low cost transmitter self-calibration of time delay and frequency response for high baud-rate QAM transceivers,” in 2017 Optical Fiber Communications Conference and Exhibition, (2017), paper Th1D.3.

9. J. C. M. Diniz, F. Da Ros, E. P. da Silva, R. T. Jones, and D. Zibar, “Optimization of DP-M-QAM transmitter using cooperative coevolutionary genetic algorithm,” J. Lightwave Technol. 36(12), 2450–2462 (2018). [CrossRef]  

10. Q. Zhang, Y. Yang, C. Guo, X. Zhou, Y. Yao, A. P. T. Lau, and C. Lu, “Algorithms for blind separation and estimation of transmitter and receiver IQ imbalances,” J. Lightwave Technol. 37(10), 2201–2208 (2019). [CrossRef]  

11. M. Arikawa and K. Hayashi, “Transmitter and receiver impairment monitoring using adaptive multi-layer linear and widely linear filter coefficients controlled by stochastic gradient descent,” Opt. Express 29(8), 11548–11561 (2021). [CrossRef]  

12. Q. Zhang, Y. Yang, C. Gu, Y. Yao, A. P. T. Lau, and C. Lu, “Multi-Dimensional, Wide-Range, and Modulation-Format-Transparent transceiver imbalance monitoring,” J. Lightwave Technol. 39(7), 2033–2045 (2021). [CrossRef]  

13. Q. Wang, Y. Yue, and J. Anderson, “Detection and compensation of power imbalance, modulation strength, and bias drift in coherent IQ transmitter through digital filter,” Opt. Express 26(18), 23069–23083 (2018). [CrossRef]  

14. Y. Yue, B. Zhang, Q. Wang, R. Lofland, J. O’Neil, and J. Anderson, “Detection and alignment of dual-polarization optical quadrature amplitude transmitter IQ and XY skews using reconfigurable interference,” Opt. Express 24(6), 6719–6734 (2016). [CrossRef]  

15. Y. Fan, Z. Tao, L. Dou, Y. Zhao, H. Chen, S. Taku, K. Kousuke, T. Hoshida, and J. C. Rasmussen, “Overall frequency response measurement of DSP-based optical transmitter using built-in monitor photodiode,” in 42nd European Conference on Optical Communication, (2016), paper Th.2.P2.SC4.

16. X. Dai, X. Li, M. Luo, and S. Yu, “Numerical simulation and experimental demonstration of accurate machine learning aided IQ time-skew and power-imbalance identification for coherent transmitters,” Opt. Express 27(26), 38367–38381 (2019). [CrossRef]  

17. Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015). [CrossRef]  

18. L. Anttila, M. Valkama, and M. Renfors, “Frequency-Selective I/Q mismatch calibration of wideband Direct-Conversion transmitters,” IEEE Trans. Circuits Syst. II 55(4), 359–363 (2008). [CrossRef]  

19. E. P. da Silva and D. Zibar, “Widely linear equalization for IQ imbalance and skew compensation in optical coherent receivers,” J. Lightwave Technol. 34(15), 3577–3586 (2016). [CrossRef]  

20. P. F. Horlin and P. A. Bourdoux, Digital Compensation for Analog Front-Ends: A New Approach to Wireless Transceiver Design (John Wiley & Sons, 2008).

21. A. Mecozzi, C. Antonelli, and M. Shtaif, “Kramers–Kronig receivers,” Adv. Opt. Photonics 11(3), 480–517 (2019). [CrossRef]  

22. A. S. Bandeira, J. Cahill, D. G. Mixon, and A. A. Nelson, “Saving phase: Injectivity and stability for phase retrieval,” Appl. Comput. Harmon. Anal. 37(1), 106–125 (2014). [CrossRef]  

23. A. Conca, D. Edidin, M. Hering, and C. Vinzant, “An algebraic characterization of injectivity in phase retrieval,” Appl. Comput. Harmon. Anal. 38(2), 346–356 (2015). [CrossRef]  

24. E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Trans. Inf. Theory 61(4), 1985–2007 (2015). [CrossRef]  

25. G. Wang, G. B. Giannakis, Y. Saad, and J. Chen, “Phase retrieval via reweighted amplitude flow,” IEEE Trans. Signal Process. 66(11), 1 (2018). [CrossRef]  

26. J. Ma, J. Xu, and A. Maleki, “Optimization-based AMP for phase retrieval: The impact of initialization and ℓ2 regularization,” IEEE Trans. Inf. Theory 65(6), 3600–3629 (2019). [CrossRef]  

27. J. Liang, P. Stoica, Y. Jing, and J. Li, “Phase retrieval via the alternating direction method of multipliers,” IEEE Signal Process. Lett. 25(1), 5–9 (2018). [CrossRef]  

28. Y. Chen and E. J. Candès, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Comm. Pure Appl. Math. 70(5), 822–883 (2017). [CrossRef]  

29. S. Ö. Arik and J. M. Kahn, “Direct-detection mode-division multiplexing in modal basis using phase retrieval,” Opt. Lett. 41(18), 4265–4268 (2016). [CrossRef]  

30. K. Choutagunta, S. Ö. Arik, K. Ho, and J. M. Kahn, “Characterizing mode-dependent loss and gain in multimode components,” J. Lightwave Technol. 36(18), 3815–3823 (2018). [CrossRef]  

31. Y. Yoshida, T. Umezawa, A. Kanno, and N. Yamamoto, “A phase-retrieving coherent receiver based on two-dimensional photodetector array,” J. Lightwave Technol. 38(1), 90–100 (2020). [CrossRef]  

32. P. Walk, H. Becker, and P. Jung, “OFDM channel estimation via phase retrieval,” in 2015 49th Asilomar Conference on Signals, Systems and Computers, (2015), pp. 1161–1168.

33. Q. Qu, Y. Zhang, Y. C. Eldar, and J. Wright, “Convolutional phase retrieval via gradient descent,” IEEE Trans. Inf. Theory 66(3), 1785–1821 (2020). [CrossRef]  

34. Y. Yoshida, T. Umezawa, A. Kanno, K. Inagaki, N. Yamamoto, and T. Kawanishi, “On the sample complexity of phase-retrieval receiver based on 2-D arrayed photodetectors,” in 2020 Optical Fiber Communication Conference and Exhibition, (2020), paper W4A.2.

35. Y. Yoshida, T. Umezawa, A. Kanno, and N. Yamamoto, “Phase-retrieving coherent reception and its sample complexity,” in 46th European Conference on Optical Communications, (2020), paper Th2D-1.

36. M. Soltanolkotabi, “Structured signal recovery from quadratic measurements: Breaking sample complexity barriers via nonconvex optimization,” IEEE Trans. Inf. Theory 65(4), 2374–2400 (2019). [CrossRef]  

37. Y. Yoshida, T. Tanimura, S. Yoshida, T. Hoshida, and N. Yamamoto, “Single-pixel IQ monitor via computational coherent reception with widely linear phase retrieval,” in 46th European Conference on Optical Communications, (2020), paper Th2D-4.

38. Y. Yoshida, S. Yoshida, T. Oda, T. Hoshida, and N. Yamamoto, “Simultaneous monitoring of frequency-dependent IQ imbalances in a dual-polarization IQ modulator by using a single photodetector: A phase retrieval approach,” in 2021 Optical Fiber Communication Conference and Exhibition, (2021), paper Th5D.2.

References

  • View by:

  1. I. Fatadin, S. J. Savory, and D. Ives, “Compensation of quadrature imbalance in an optical QPSK coherent receiver,” IEEE Photonics Technol. Lett. 20(20), 1733–1735 (2008).
    [Crossref]
  2. Y. Yue, Q. Wang, and J. Anderson, “Transmitter skew tolerance and spectral efficiency tradeoff in high baud-rate QAM optical communication systems,” Opt. Express 26(12), 15045–15058 (2018).
    [Crossref]
  3. J. Liang, Y. Fan, Z. Tao, X. Su, and H. Nakashima, “Transceiver imbalances compensation and monitoring by receiver DSP,” IEEE/OSA J. Lightwave Technol. (Early access) p. 1 (2019).
  4. T. Kobayashi, M. Nakamura, F. Hamaoka, M. Nagatani, H. Yamazaki, H. Nosaka, and Y. Miyamoto, “Long-haul WDM transmission with over-1-Tb/s channels using electrically synthesized high-symbol-rate signals,” in 2020 Optical Fiber Communications Conference and Exhibition, (2020), paper M4K.1.
  5. Y. Miyamoto, “OIF-400ZR-01.0: Implementation agreement 400ZR,” Optical Internetworking Forum (2020).
  6. X. Chen, J. Cho, A. Adamiecki, and P. Winzer, “16384-QAM transmission at 10 GBd over 25-km SSMF using polarization-multiplexed probabilistic constellation shaping,” in 45th European Conference on Optical Communication, (2019), paper PD3.3.
  7. C. Ju, N. Liu, and C. Li, “In-service blind transceiver IQ imbalance and skew monitoring in Long-Haul Non-Dispersion managed coherent optical systems,” IEEE Access 7, 150051–150059 (2019).
    [Crossref]
  8. C. R. S. Fludger, T. Duthel, P. Hermann, and T. Kupfer, “Low cost transmitter self-calibration of time delay and frequency response for high baud-rate QAM transceivers,” in 2017 Optical Fiber Communications Conference and Exhibition, (2017), paper Th1D.3.
  9. J. C. M. Diniz, F. Da Ros, E. P. da Silva, R. T. Jones, and D. Zibar, “Optimization of DP-M-QAM transmitter using cooperative coevolutionary genetic algorithm,” J. Lightwave Technol. 36(12), 2450–2462 (2018).
    [Crossref]
  10. Q. Zhang, Y. Yang, C. Guo, X. Zhou, Y. Yao, A. P. T. Lau, and C. Lu, “Algorithms for blind separation and estimation of transmitter and receiver IQ imbalances,” J. Lightwave Technol. 37(10), 2201–2208 (2019).
    [Crossref]
  11. M. Arikawa and K. Hayashi, “Transmitter and receiver impairment monitoring using adaptive multi-layer linear and widely linear filter coefficients controlled by stochastic gradient descent,” Opt. Express 29(8), 11548–11561 (2021).
    [Crossref]
  12. Q. Zhang, Y. Yang, C. Gu, Y. Yao, A. P. T. Lau, and C. Lu, “Multi-Dimensional, Wide-Range, and Modulation-Format-Transparent transceiver imbalance monitoring,” J. Lightwave Technol. 39(7), 2033–2045 (2021).
    [Crossref]
  13. Q. Wang, Y. Yue, and J. Anderson, “Detection and compensation of power imbalance, modulation strength, and bias drift in coherent IQ transmitter through digital filter,” Opt. Express 26(18), 23069–23083 (2018).
    [Crossref]
  14. Y. Yue, B. Zhang, Q. Wang, R. Lofland, J. O’Neil, and J. Anderson, “Detection and alignment of dual-polarization optical quadrature amplitude transmitter IQ and XY skews using reconfigurable interference,” Opt. Express 24(6), 6719–6734 (2016).
    [Crossref]
  15. Y. Fan, Z. Tao, L. Dou, Y. Zhao, H. Chen, S. Taku, K. Kousuke, T. Hoshida, and J. C. Rasmussen, “Overall frequency response measurement of DSP-based optical transmitter using built-in monitor photodiode,” in 42nd European Conference on Optical Communication, (2016), paper Th.2.P2.SC4.
  16. X. Dai, X. Li, M. Luo, and S. Yu, “Numerical simulation and experimental demonstration of accurate machine learning aided IQ time-skew and power-imbalance identification for coherent transmitters,” Opt. Express 27(26), 38367–38381 (2019).
    [Crossref]
  17. Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
    [Crossref]
  18. L. Anttila, M. Valkama, and M. Renfors, “Frequency-Selective I/Q mismatch calibration of wideband Direct-Conversion transmitters,” IEEE Trans. Circuits Syst. II 55(4), 359–363 (2008).
    [Crossref]
  19. E. P. da Silva and D. Zibar, “Widely linear equalization for IQ imbalance and skew compensation in optical coherent receivers,” J. Lightwave Technol. 34(15), 3577–3586 (2016).
    [Crossref]
  20. P. F. Horlin and P. A. Bourdoux, Digital Compensation for Analog Front-Ends: A New Approach to Wireless Transceiver Design (John Wiley & Sons, 2008).
  21. A. Mecozzi, C. Antonelli, and M. Shtaif, “Kramers–Kronig receivers,” Adv. Opt. Photonics 11(3), 480–517 (2019).
    [Crossref]
  22. A. S. Bandeira, J. Cahill, D. G. Mixon, and A. A. Nelson, “Saving phase: Injectivity and stability for phase retrieval,” Appl. Comput. Harmon. Anal. 37(1), 106–125 (2014).
    [Crossref]
  23. A. Conca, D. Edidin, M. Hering, and C. Vinzant, “An algebraic characterization of injectivity in phase retrieval,” Appl. Comput. Harmon. Anal. 38(2), 346–356 (2015).
    [Crossref]
  24. E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Trans. Inf. Theory 61(4), 1985–2007 (2015).
    [Crossref]
  25. G. Wang, G. B. Giannakis, Y. Saad, and J. Chen, “Phase retrieval via reweighted amplitude flow,” IEEE Trans. Signal Process. 66(11), 1 (2018).
    [Crossref]
  26. J. Ma, J. Xu, and A. Maleki, “Optimization-based AMP for phase retrieval: The impact of initialization and ℓ2 regularization,” IEEE Trans. Inf. Theory 65(6), 3600–3629 (2019).
    [Crossref]
  27. J. Liang, P. Stoica, Y. Jing, and J. Li, “Phase retrieval via the alternating direction method of multipliers,” IEEE Signal Process. Lett. 25(1), 5–9 (2018).
    [Crossref]
  28. Y. Chen and E. J. Candès, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Comm. Pure Appl. Math. 70(5), 822–883 (2017).
    [Crossref]
  29. S. Ö. Arik and J. M. Kahn, “Direct-detection mode-division multiplexing in modal basis using phase retrieval,” Opt. Lett. 41(18), 4265–4268 (2016).
    [Crossref]
  30. K. Choutagunta, S. Ö. Arik, K. Ho, and J. M. Kahn, “Characterizing mode-dependent loss and gain in multimode components,” J. Lightwave Technol. 36(18), 3815–3823 (2018).
    [Crossref]
  31. Y. Yoshida, T. Umezawa, A. Kanno, and N. Yamamoto, “A phase-retrieving coherent receiver based on two-dimensional photodetector array,” J. Lightwave Technol. 38(1), 90–100 (2020).
    [Crossref]
  32. P. Walk, H. Becker, and P. Jung, “OFDM channel estimation via phase retrieval,” in 2015 49th Asilomar Conference on Signals, Systems and Computers, (2015), pp. 1161–1168.
  33. Q. Qu, Y. Zhang, Y. C. Eldar, and J. Wright, “Convolutional phase retrieval via gradient descent,” IEEE Trans. Inf. Theory 66(3), 1785–1821 (2020).
    [Crossref]
  34. Y. Yoshida, T. Umezawa, A. Kanno, K. Inagaki, N. Yamamoto, and T. Kawanishi, “On the sample complexity of phase-retrieval receiver based on 2-D arrayed photodetectors,” in 2020 Optical Fiber Communication Conference and Exhibition, (2020), paper W4A.2.
  35. Y. Yoshida, T. Umezawa, A. Kanno, and N. Yamamoto, “Phase-retrieving coherent reception and its sample complexity,” in 46th European Conference on Optical Communications, (2020), paper Th2D-1.
  36. M. Soltanolkotabi, “Structured signal recovery from quadratic measurements: Breaking sample complexity barriers via nonconvex optimization,” IEEE Trans. Inf. Theory 65(4), 2374–2400 (2019).
    [Crossref]
  37. Y. Yoshida, T. Tanimura, S. Yoshida, T. Hoshida, and N. Yamamoto, “Single-pixel IQ monitor via computational coherent reception with widely linear phase retrieval,” in 46th European Conference on Optical Communications, (2020), paper Th2D-4.
  38. Y. Yoshida, S. Yoshida, T. Oda, T. Hoshida, and N. Yamamoto, “Simultaneous monitoring of frequency-dependent IQ imbalances in a dual-polarization IQ modulator by using a single photodetector: A phase retrieval approach,” in 2021 Optical Fiber Communication Conference and Exhibition, (2021), paper Th5D.2.

2021 (2)

2020 (2)

Y. Yoshida, T. Umezawa, A. Kanno, and N. Yamamoto, “A phase-retrieving coherent receiver based on two-dimensional photodetector array,” J. Lightwave Technol. 38(1), 90–100 (2020).
[Crossref]

Q. Qu, Y. Zhang, Y. C. Eldar, and J. Wright, “Convolutional phase retrieval via gradient descent,” IEEE Trans. Inf. Theory 66(3), 1785–1821 (2020).
[Crossref]

2019 (6)

M. Soltanolkotabi, “Structured signal recovery from quadratic measurements: Breaking sample complexity barriers via nonconvex optimization,” IEEE Trans. Inf. Theory 65(4), 2374–2400 (2019).
[Crossref]

J. Ma, J. Xu, and A. Maleki, “Optimization-based AMP for phase retrieval: The impact of initialization and ℓ2 regularization,” IEEE Trans. Inf. Theory 65(6), 3600–3629 (2019).
[Crossref]

C. Ju, N. Liu, and C. Li, “In-service blind transceiver IQ imbalance and skew monitoring in Long-Haul Non-Dispersion managed coherent optical systems,” IEEE Access 7, 150051–150059 (2019).
[Crossref]

X. Dai, X. Li, M. Luo, and S. Yu, “Numerical simulation and experimental demonstration of accurate machine learning aided IQ time-skew and power-imbalance identification for coherent transmitters,” Opt. Express 27(26), 38367–38381 (2019).
[Crossref]

Q. Zhang, Y. Yang, C. Guo, X. Zhou, Y. Yao, A. P. T. Lau, and C. Lu, “Algorithms for blind separation and estimation of transmitter and receiver IQ imbalances,” J. Lightwave Technol. 37(10), 2201–2208 (2019).
[Crossref]

A. Mecozzi, C. Antonelli, and M. Shtaif, “Kramers–Kronig receivers,” Adv. Opt. Photonics 11(3), 480–517 (2019).
[Crossref]

2018 (6)

2017 (1)

Y. Chen and E. J. Candès, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Comm. Pure Appl. Math. 70(5), 822–883 (2017).
[Crossref]

2016 (3)

2015 (3)

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

A. Conca, D. Edidin, M. Hering, and C. Vinzant, “An algebraic characterization of injectivity in phase retrieval,” Appl. Comput. Harmon. Anal. 38(2), 346–356 (2015).
[Crossref]

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Trans. Inf. Theory 61(4), 1985–2007 (2015).
[Crossref]

2014 (1)

A. S. Bandeira, J. Cahill, D. G. Mixon, and A. A. Nelson, “Saving phase: Injectivity and stability for phase retrieval,” Appl. Comput. Harmon. Anal. 37(1), 106–125 (2014).
[Crossref]

2008 (2)

L. Anttila, M. Valkama, and M. Renfors, “Frequency-Selective I/Q mismatch calibration of wideband Direct-Conversion transmitters,” IEEE Trans. Circuits Syst. II 55(4), 359–363 (2008).
[Crossref]

I. Fatadin, S. J. Savory, and D. Ives, “Compensation of quadrature imbalance in an optical QPSK coherent receiver,” IEEE Photonics Technol. Lett. 20(20), 1733–1735 (2008).
[Crossref]

Adamiecki, A.

X. Chen, J. Cho, A. Adamiecki, and P. Winzer, “16384-QAM transmission at 10 GBd over 25-km SSMF using polarization-multiplexed probabilistic constellation shaping,” in 45th European Conference on Optical Communication, (2019), paper PD3.3.

Anderson, J.

Antonelli, C.

A. Mecozzi, C. Antonelli, and M. Shtaif, “Kramers–Kronig receivers,” Adv. Opt. Photonics 11(3), 480–517 (2019).
[Crossref]

Anttila, L.

L. Anttila, M. Valkama, and M. Renfors, “Frequency-Selective I/Q mismatch calibration of wideband Direct-Conversion transmitters,” IEEE Trans. Circuits Syst. II 55(4), 359–363 (2008).
[Crossref]

Arik, S. Ö.

Arikawa, M.

Bandeira, A. S.

A. S. Bandeira, J. Cahill, D. G. Mixon, and A. A. Nelson, “Saving phase: Injectivity and stability for phase retrieval,” Appl. Comput. Harmon. Anal. 37(1), 106–125 (2014).
[Crossref]

Becker, H.

P. Walk, H. Becker, and P. Jung, “OFDM channel estimation via phase retrieval,” in 2015 49th Asilomar Conference on Signals, Systems and Computers, (2015), pp. 1161–1168.

Bourdoux, P. A.

P. F. Horlin and P. A. Bourdoux, Digital Compensation for Analog Front-Ends: A New Approach to Wireless Transceiver Design (John Wiley & Sons, 2008).

Cahill, J.

A. S. Bandeira, J. Cahill, D. G. Mixon, and A. A. Nelson, “Saving phase: Injectivity and stability for phase retrieval,” Appl. Comput. Harmon. Anal. 37(1), 106–125 (2014).
[Crossref]

Candès, E. J.

Y. Chen and E. J. Candès, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Comm. Pure Appl. Math. 70(5), 822–883 (2017).
[Crossref]

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Trans. Inf. Theory 61(4), 1985–2007 (2015).
[Crossref]

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Chen, H.

Y. Fan, Z. Tao, L. Dou, Y. Zhao, H. Chen, S. Taku, K. Kousuke, T. Hoshida, and J. C. Rasmussen, “Overall frequency response measurement of DSP-based optical transmitter using built-in monitor photodiode,” in 42nd European Conference on Optical Communication, (2016), paper Th.2.P2.SC4.

Chen, J.

G. Wang, G. B. Giannakis, Y. Saad, and J. Chen, “Phase retrieval via reweighted amplitude flow,” IEEE Trans. Signal Process. 66(11), 1 (2018).
[Crossref]

Chen, X.

X. Chen, J. Cho, A. Adamiecki, and P. Winzer, “16384-QAM transmission at 10 GBd over 25-km SSMF using polarization-multiplexed probabilistic constellation shaping,” in 45th European Conference on Optical Communication, (2019), paper PD3.3.

Chen, Y.

Y. Chen and E. J. Candès, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Comm. Pure Appl. Math. 70(5), 822–883 (2017).
[Crossref]

Cho, J.

X. Chen, J. Cho, A. Adamiecki, and P. Winzer, “16384-QAM transmission at 10 GBd over 25-km SSMF using polarization-multiplexed probabilistic constellation shaping,” in 45th European Conference on Optical Communication, (2019), paper PD3.3.

Choutagunta, K.

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Conca, A.

A. Conca, D. Edidin, M. Hering, and C. Vinzant, “An algebraic characterization of injectivity in phase retrieval,” Appl. Comput. Harmon. Anal. 38(2), 346–356 (2015).
[Crossref]

Da Ros, F.

da Silva, E. P.

Dai, X.

Diniz, J. C. M.

Dou, L.

Y. Fan, Z. Tao, L. Dou, Y. Zhao, H. Chen, S. Taku, K. Kousuke, T. Hoshida, and J. C. Rasmussen, “Overall frequency response measurement of DSP-based optical transmitter using built-in monitor photodiode,” in 42nd European Conference on Optical Communication, (2016), paper Th.2.P2.SC4.

Duthel, T.

C. R. S. Fludger, T. Duthel, P. Hermann, and T. Kupfer, “Low cost transmitter self-calibration of time delay and frequency response for high baud-rate QAM transceivers,” in 2017 Optical Fiber Communications Conference and Exhibition, (2017), paper Th1D.3.

Edidin, D.

A. Conca, D. Edidin, M. Hering, and C. Vinzant, “An algebraic characterization of injectivity in phase retrieval,” Appl. Comput. Harmon. Anal. 38(2), 346–356 (2015).
[Crossref]

Eldar, Y. C.

Q. Qu, Y. Zhang, Y. C. Eldar, and J. Wright, “Convolutional phase retrieval via gradient descent,” IEEE Trans. Inf. Theory 66(3), 1785–1821 (2020).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Fan, Y.

J. Liang, Y. Fan, Z. Tao, X. Su, and H. Nakashima, “Transceiver imbalances compensation and monitoring by receiver DSP,” IEEE/OSA J. Lightwave Technol. (Early access) p. 1 (2019).

Y. Fan, Z. Tao, L. Dou, Y. Zhao, H. Chen, S. Taku, K. Kousuke, T. Hoshida, and J. C. Rasmussen, “Overall frequency response measurement of DSP-based optical transmitter using built-in monitor photodiode,” in 42nd European Conference on Optical Communication, (2016), paper Th.2.P2.SC4.

Fatadin, I.

I. Fatadin, S. J. Savory, and D. Ives, “Compensation of quadrature imbalance in an optical QPSK coherent receiver,” IEEE Photonics Technol. Lett. 20(20), 1733–1735 (2008).
[Crossref]

Fludger, C. R. S.

C. R. S. Fludger, T. Duthel, P. Hermann, and T. Kupfer, “Low cost transmitter self-calibration of time delay and frequency response for high baud-rate QAM transceivers,” in 2017 Optical Fiber Communications Conference and Exhibition, (2017), paper Th1D.3.

Giannakis, G. B.

G. Wang, G. B. Giannakis, Y. Saad, and J. Chen, “Phase retrieval via reweighted amplitude flow,” IEEE Trans. Signal Process. 66(11), 1 (2018).
[Crossref]

Gu, C.

Guo, C.

Hamaoka, F.

T. Kobayashi, M. Nakamura, F. Hamaoka, M. Nagatani, H. Yamazaki, H. Nosaka, and Y. Miyamoto, “Long-haul WDM transmission with over-1-Tb/s channels using electrically synthesized high-symbol-rate signals,” in 2020 Optical Fiber Communications Conference and Exhibition, (2020), paper M4K.1.

Hayashi, K.

Hering, M.

A. Conca, D. Edidin, M. Hering, and C. Vinzant, “An algebraic characterization of injectivity in phase retrieval,” Appl. Comput. Harmon. Anal. 38(2), 346–356 (2015).
[Crossref]

Hermann, P.

C. R. S. Fludger, T. Duthel, P. Hermann, and T. Kupfer, “Low cost transmitter self-calibration of time delay and frequency response for high baud-rate QAM transceivers,” in 2017 Optical Fiber Communications Conference and Exhibition, (2017), paper Th1D.3.

Ho, K.

Horlin, P. F.

P. F. Horlin and P. A. Bourdoux, Digital Compensation for Analog Front-Ends: A New Approach to Wireless Transceiver Design (John Wiley & Sons, 2008).

Hoshida, T.

Y. Fan, Z. Tao, L. Dou, Y. Zhao, H. Chen, S. Taku, K. Kousuke, T. Hoshida, and J. C. Rasmussen, “Overall frequency response measurement of DSP-based optical transmitter using built-in monitor photodiode,” in 42nd European Conference on Optical Communication, (2016), paper Th.2.P2.SC4.

Y. Yoshida, T. Tanimura, S. Yoshida, T. Hoshida, and N. Yamamoto, “Single-pixel IQ monitor via computational coherent reception with widely linear phase retrieval,” in 46th European Conference on Optical Communications, (2020), paper Th2D-4.

Y. Yoshida, S. Yoshida, T. Oda, T. Hoshida, and N. Yamamoto, “Simultaneous monitoring of frequency-dependent IQ imbalances in a dual-polarization IQ modulator by using a single photodetector: A phase retrieval approach,” in 2021 Optical Fiber Communication Conference and Exhibition, (2021), paper Th5D.2.

Inagaki, K.

Y. Yoshida, T. Umezawa, A. Kanno, K. Inagaki, N. Yamamoto, and T. Kawanishi, “On the sample complexity of phase-retrieval receiver based on 2-D arrayed photodetectors,” in 2020 Optical Fiber Communication Conference and Exhibition, (2020), paper W4A.2.

Ives, D.

I. Fatadin, S. J. Savory, and D. Ives, “Compensation of quadrature imbalance in an optical QPSK coherent receiver,” IEEE Photonics Technol. Lett. 20(20), 1733–1735 (2008).
[Crossref]

Jing, Y.

J. Liang, P. Stoica, Y. Jing, and J. Li, “Phase retrieval via the alternating direction method of multipliers,” IEEE Signal Process. Lett. 25(1), 5–9 (2018).
[Crossref]

Jones, R. T.

Ju, C.

C. Ju, N. Liu, and C. Li, “In-service blind transceiver IQ imbalance and skew monitoring in Long-Haul Non-Dispersion managed coherent optical systems,” IEEE Access 7, 150051–150059 (2019).
[Crossref]

Jung, P.

P. Walk, H. Becker, and P. Jung, “OFDM channel estimation via phase retrieval,” in 2015 49th Asilomar Conference on Signals, Systems and Computers, (2015), pp. 1161–1168.

Kahn, J. M.

Kanno, A.

Y. Yoshida, T. Umezawa, A. Kanno, and N. Yamamoto, “A phase-retrieving coherent receiver based on two-dimensional photodetector array,” J. Lightwave Technol. 38(1), 90–100 (2020).
[Crossref]

Y. Yoshida, T. Umezawa, A. Kanno, K. Inagaki, N. Yamamoto, and T. Kawanishi, “On the sample complexity of phase-retrieval receiver based on 2-D arrayed photodetectors,” in 2020 Optical Fiber Communication Conference and Exhibition, (2020), paper W4A.2.

Y. Yoshida, T. Umezawa, A. Kanno, and N. Yamamoto, “Phase-retrieving coherent reception and its sample complexity,” in 46th European Conference on Optical Communications, (2020), paper Th2D-1.

Kawanishi, T.

Y. Yoshida, T. Umezawa, A. Kanno, K. Inagaki, N. Yamamoto, and T. Kawanishi, “On the sample complexity of phase-retrieval receiver based on 2-D arrayed photodetectors,” in 2020 Optical Fiber Communication Conference and Exhibition, (2020), paper W4A.2.

Kobayashi, T.

T. Kobayashi, M. Nakamura, F. Hamaoka, M. Nagatani, H. Yamazaki, H. Nosaka, and Y. Miyamoto, “Long-haul WDM transmission with over-1-Tb/s channels using electrically synthesized high-symbol-rate signals,” in 2020 Optical Fiber Communications Conference and Exhibition, (2020), paper M4K.1.

Kousuke, K.

Y. Fan, Z. Tao, L. Dou, Y. Zhao, H. Chen, S. Taku, K. Kousuke, T. Hoshida, and J. C. Rasmussen, “Overall frequency response measurement of DSP-based optical transmitter using built-in monitor photodiode,” in 42nd European Conference on Optical Communication, (2016), paper Th.2.P2.SC4.

Kupfer, T.

C. R. S. Fludger, T. Duthel, P. Hermann, and T. Kupfer, “Low cost transmitter self-calibration of time delay and frequency response for high baud-rate QAM transceivers,” in 2017 Optical Fiber Communications Conference and Exhibition, (2017), paper Th1D.3.

Lau, A. P. T.

Li, C.

C. Ju, N. Liu, and C. Li, “In-service blind transceiver IQ imbalance and skew monitoring in Long-Haul Non-Dispersion managed coherent optical systems,” IEEE Access 7, 150051–150059 (2019).
[Crossref]

Li, J.

J. Liang, P. Stoica, Y. Jing, and J. Li, “Phase retrieval via the alternating direction method of multipliers,” IEEE Signal Process. Lett. 25(1), 5–9 (2018).
[Crossref]

Li, X.

Liang, J.

J. Liang, P. Stoica, Y. Jing, and J. Li, “Phase retrieval via the alternating direction method of multipliers,” IEEE Signal Process. Lett. 25(1), 5–9 (2018).
[Crossref]

J. Liang, Y. Fan, Z. Tao, X. Su, and H. Nakashima, “Transceiver imbalances compensation and monitoring by receiver DSP,” IEEE/OSA J. Lightwave Technol. (Early access) p. 1 (2019).

Liu, N.

C. Ju, N. Liu, and C. Li, “In-service blind transceiver IQ imbalance and skew monitoring in Long-Haul Non-Dispersion managed coherent optical systems,” IEEE Access 7, 150051–150059 (2019).
[Crossref]

Lofland, R.

Lu, C.

Luo, M.

Ma, J.

J. Ma, J. Xu, and A. Maleki, “Optimization-based AMP for phase retrieval: The impact of initialization and ℓ2 regularization,” IEEE Trans. Inf. Theory 65(6), 3600–3629 (2019).
[Crossref]

Maleki, A.

J. Ma, J. Xu, and A. Maleki, “Optimization-based AMP for phase retrieval: The impact of initialization and ℓ2 regularization,” IEEE Trans. Inf. Theory 65(6), 3600–3629 (2019).
[Crossref]

Mecozzi, A.

A. Mecozzi, C. Antonelli, and M. Shtaif, “Kramers–Kronig receivers,” Adv. Opt. Photonics 11(3), 480–517 (2019).
[Crossref]

Miao, J.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Mixon, D. G.

A. S. Bandeira, J. Cahill, D. G. Mixon, and A. A. Nelson, “Saving phase: Injectivity and stability for phase retrieval,” Appl. Comput. Harmon. Anal. 37(1), 106–125 (2014).
[Crossref]

Miyamoto, Y.

T. Kobayashi, M. Nakamura, F. Hamaoka, M. Nagatani, H. Yamazaki, H. Nosaka, and Y. Miyamoto, “Long-haul WDM transmission with over-1-Tb/s channels using electrically synthesized high-symbol-rate signals,” in 2020 Optical Fiber Communications Conference and Exhibition, (2020), paper M4K.1.

Y. Miyamoto, “OIF-400ZR-01.0: Implementation agreement 400ZR,” Optical Internetworking Forum (2020).

Nagatani, M.

T. Kobayashi, M. Nakamura, F. Hamaoka, M. Nagatani, H. Yamazaki, H. Nosaka, and Y. Miyamoto, “Long-haul WDM transmission with over-1-Tb/s channels using electrically synthesized high-symbol-rate signals,” in 2020 Optical Fiber Communications Conference and Exhibition, (2020), paper M4K.1.

Nakamura, M.

T. Kobayashi, M. Nakamura, F. Hamaoka, M. Nagatani, H. Yamazaki, H. Nosaka, and Y. Miyamoto, “Long-haul WDM transmission with over-1-Tb/s channels using electrically synthesized high-symbol-rate signals,” in 2020 Optical Fiber Communications Conference and Exhibition, (2020), paper M4K.1.

Nakashima, H.

J. Liang, Y. Fan, Z. Tao, X. Su, and H. Nakashima, “Transceiver imbalances compensation and monitoring by receiver DSP,” IEEE/OSA J. Lightwave Technol. (Early access) p. 1 (2019).

Nelson, A. A.

A. S. Bandeira, J. Cahill, D. G. Mixon, and A. A. Nelson, “Saving phase: Injectivity and stability for phase retrieval,” Appl. Comput. Harmon. Anal. 37(1), 106–125 (2014).
[Crossref]

Nosaka, H.

T. Kobayashi, M. Nakamura, F. Hamaoka, M. Nagatani, H. Yamazaki, H. Nosaka, and Y. Miyamoto, “Long-haul WDM transmission with over-1-Tb/s channels using electrically synthesized high-symbol-rate signals,” in 2020 Optical Fiber Communications Conference and Exhibition, (2020), paper M4K.1.

O’Neil, J.

Oda, T.

Y. Yoshida, S. Yoshida, T. Oda, T. Hoshida, and N. Yamamoto, “Simultaneous monitoring of frequency-dependent IQ imbalances in a dual-polarization IQ modulator by using a single photodetector: A phase retrieval approach,” in 2021 Optical Fiber Communication Conference and Exhibition, (2021), paper Th5D.2.

Qu, Q.

Q. Qu, Y. Zhang, Y. C. Eldar, and J. Wright, “Convolutional phase retrieval via gradient descent,” IEEE Trans. Inf. Theory 66(3), 1785–1821 (2020).
[Crossref]

Rasmussen, J. C.

Y. Fan, Z. Tao, L. Dou, Y. Zhao, H. Chen, S. Taku, K. Kousuke, T. Hoshida, and J. C. Rasmussen, “Overall frequency response measurement of DSP-based optical transmitter using built-in monitor photodiode,” in 42nd European Conference on Optical Communication, (2016), paper Th.2.P2.SC4.

Renfors, M.

L. Anttila, M. Valkama, and M. Renfors, “Frequency-Selective I/Q mismatch calibration of wideband Direct-Conversion transmitters,” IEEE Trans. Circuits Syst. II 55(4), 359–363 (2008).
[Crossref]

Saad, Y.

G. Wang, G. B. Giannakis, Y. Saad, and J. Chen, “Phase retrieval via reweighted amplitude flow,” IEEE Trans. Signal Process. 66(11), 1 (2018).
[Crossref]

Savory, S. J.

I. Fatadin, S. J. Savory, and D. Ives, “Compensation of quadrature imbalance in an optical QPSK coherent receiver,” IEEE Photonics Technol. Lett. 20(20), 1733–1735 (2008).
[Crossref]

Segev, M.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Shechtman, Y.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Shtaif, M.

A. Mecozzi, C. Antonelli, and M. Shtaif, “Kramers–Kronig receivers,” Adv. Opt. Photonics 11(3), 480–517 (2019).
[Crossref]

Soltanolkotabi, M.

M. Soltanolkotabi, “Structured signal recovery from quadratic measurements: Breaking sample complexity barriers via nonconvex optimization,” IEEE Trans. Inf. Theory 65(4), 2374–2400 (2019).
[Crossref]

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Trans. Inf. Theory 61(4), 1985–2007 (2015).
[Crossref]

Stoica, P.

J. Liang, P. Stoica, Y. Jing, and J. Li, “Phase retrieval via the alternating direction method of multipliers,” IEEE Signal Process. Lett. 25(1), 5–9 (2018).
[Crossref]

Su, X.

J. Liang, Y. Fan, Z. Tao, X. Su, and H. Nakashima, “Transceiver imbalances compensation and monitoring by receiver DSP,” IEEE/OSA J. Lightwave Technol. (Early access) p. 1 (2019).

Taku, S.

Y. Fan, Z. Tao, L. Dou, Y. Zhao, H. Chen, S. Taku, K. Kousuke, T. Hoshida, and J. C. Rasmussen, “Overall frequency response measurement of DSP-based optical transmitter using built-in monitor photodiode,” in 42nd European Conference on Optical Communication, (2016), paper Th.2.P2.SC4.

Tanimura, T.

Y. Yoshida, T. Tanimura, S. Yoshida, T. Hoshida, and N. Yamamoto, “Single-pixel IQ monitor via computational coherent reception with widely linear phase retrieval,” in 46th European Conference on Optical Communications, (2020), paper Th2D-4.

Tao, Z.

Y. Fan, Z. Tao, L. Dou, Y. Zhao, H. Chen, S. Taku, K. Kousuke, T. Hoshida, and J. C. Rasmussen, “Overall frequency response measurement of DSP-based optical transmitter using built-in monitor photodiode,” in 42nd European Conference on Optical Communication, (2016), paper Th.2.P2.SC4.

J. Liang, Y. Fan, Z. Tao, X. Su, and H. Nakashima, “Transceiver imbalances compensation and monitoring by receiver DSP,” IEEE/OSA J. Lightwave Technol. (Early access) p. 1 (2019).

Umezawa, T.

Y. Yoshida, T. Umezawa, A. Kanno, and N. Yamamoto, “A phase-retrieving coherent receiver based on two-dimensional photodetector array,” J. Lightwave Technol. 38(1), 90–100 (2020).
[Crossref]

Y. Yoshida, T. Umezawa, A. Kanno, K. Inagaki, N. Yamamoto, and T. Kawanishi, “On the sample complexity of phase-retrieval receiver based on 2-D arrayed photodetectors,” in 2020 Optical Fiber Communication Conference and Exhibition, (2020), paper W4A.2.

Y. Yoshida, T. Umezawa, A. Kanno, and N. Yamamoto, “Phase-retrieving coherent reception and its sample complexity,” in 46th European Conference on Optical Communications, (2020), paper Th2D-1.

Valkama, M.

L. Anttila, M. Valkama, and M. Renfors, “Frequency-Selective I/Q mismatch calibration of wideband Direct-Conversion transmitters,” IEEE Trans. Circuits Syst. II 55(4), 359–363 (2008).
[Crossref]

Vinzant, C.

A. Conca, D. Edidin, M. Hering, and C. Vinzant, “An algebraic characterization of injectivity in phase retrieval,” Appl. Comput. Harmon. Anal. 38(2), 346–356 (2015).
[Crossref]

Walk, P.

P. Walk, H. Becker, and P. Jung, “OFDM channel estimation via phase retrieval,” in 2015 49th Asilomar Conference on Signals, Systems and Computers, (2015), pp. 1161–1168.

Wang, G.

G. Wang, G. B. Giannakis, Y. Saad, and J. Chen, “Phase retrieval via reweighted amplitude flow,” IEEE Trans. Signal Process. 66(11), 1 (2018).
[Crossref]

Wang, Q.

Winzer, P.

X. Chen, J. Cho, A. Adamiecki, and P. Winzer, “16384-QAM transmission at 10 GBd over 25-km SSMF using polarization-multiplexed probabilistic constellation shaping,” in 45th European Conference on Optical Communication, (2019), paper PD3.3.

Wright, J.

Q. Qu, Y. Zhang, Y. C. Eldar, and J. Wright, “Convolutional phase retrieval via gradient descent,” IEEE Trans. Inf. Theory 66(3), 1785–1821 (2020).
[Crossref]

Xu, J.

J. Ma, J. Xu, and A. Maleki, “Optimization-based AMP for phase retrieval: The impact of initialization and ℓ2 regularization,” IEEE Trans. Inf. Theory 65(6), 3600–3629 (2019).
[Crossref]

Yamamoto, N.

Y. Yoshida, T. Umezawa, A. Kanno, and N. Yamamoto, “A phase-retrieving coherent receiver based on two-dimensional photodetector array,” J. Lightwave Technol. 38(1), 90–100 (2020).
[Crossref]

Y. Yoshida, T. Umezawa, A. Kanno, K. Inagaki, N. Yamamoto, and T. Kawanishi, “On the sample complexity of phase-retrieval receiver based on 2-D arrayed photodetectors,” in 2020 Optical Fiber Communication Conference and Exhibition, (2020), paper W4A.2.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)

Fig. 1.
Fig. 1. Schematic of direct-conversion IQ modulator under frequency-dependent IQ imbalance.
Fig. 2.
Fig. 2. Schematic architecture of proposed single-pixel optical modulation analyzer.
Fig. 3.
Fig. 3. NMSE performance of SP-OMA and CO-OMA versus a) number of PhareADMM iterations, b) SNR, c) pilot length, and d) IRR.
Fig. 4.
Fig. 4. Experimental setup for SP-OMA and CO-OMA with a 63.25-Gbaud 16QAM signal.
Fig. 5.
Fig. 5. Received and simulated constellation maps for the three FD-IQI patterns.
Fig. 6.
Fig. 6. Estimated impulse responses and frequency responses with CO-OMA and SP-OMA for the three FD-IQI patterns.
Fig. 7.
Fig. 7. a) NMSE performance of CO-OMA and SP-OMA for the 20 random FD-IQI patterns: a) NMSE per pattern and b) averaged NMSE against pilot length.
Fig. 8.
Fig. 8. An example of estimated impulse responses and frequency responses with CO-OMA and SP-OMA for the random FD-IQI patterns.

Equations (21)

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E ( t ) E 0 { h + ( t ) x ( t ) + h ( t ) x ( t ) + ρ } ,
h ± ( t ) = h I ( t ) ± ( 1 + ϵ ) h Q ( t ) e j θ 2 ,
r n = h + T s n + h T s n + ρ ,
[ h CO + h CO ] = S ¯ + r ,
ψ n = | h + T s n + h T s n + ρ | 2 .
f i n d h ¯ s . t . ψ n = | s ¯ n T h ¯ | 2 ( n = 1 , , N ) .
l ( h ¯ ) = n = 1 N ( ψ n | s ¯ n T h ¯ | ) 2
min h ¯ , { u n } , { ϕ n } n = 1 N ( ψ n u n ) 2 s . t . s ¯ n T h ¯ = u n e j ϕ n ,   u n   0 ,   ( n = 1 , , N ) .
L ( h ¯ , { u n , ϕ n , λ n (r) , λ n (i) } ) = n = 1 N ( ψ n u n ) 2 + n = 1 N { λ n (r) { s ¯ n T h ¯ u n e j ϕ n } + η 2 ( Re { s ¯ n T h ¯ u n e j ϕ n } ) 2 } + n = 1 N { λ n (i) Im { s ¯ n T h ¯ u n e j ϕ n } + η 2 ( Im { s ¯ n T h ¯ u n e j ϕ n } ) 2 } ,
{ u n ( t + 1 ) , ϕ n ( t + 1 ) } = a r g m i n { u n , ϕ n } L ( h ¯ ( t ) , { u n , ϕ n , λ n (r) ( t ) , λ n (i) ( t ) } ) ,
h ¯ ( t + 1 ) = a r g m i n h ¯ L ( h ¯ , { u n ( t + 1 ) , ϕ n ( t + 1 ) , λ n (r) ( t ) , λ n (i) ( t ) } ) ,
λ n (r) ( t + 1 ) = λ n (r) ( t ) + η Re { s ¯ n T h ¯ ( t + 1 ) u n ( t + 1 ) e j ϕ n ( t + 1 ) } ,
λ n (i) ( t + 1 ) = λ n (i) ( t ) + η Im { s ¯ n T h ¯ ( t + 1 ) u n ( t + 1 ) e j ϕ n ( t + 1 ) } . ( n = 1 , , N )
ϕ n ( t + 1 ) = { g n ( t ) } ,
u n ( t + 1 ) = η | g n ( t ) | + 2 ψ n 2 + η ,
h ¯ ( t + 1 ) = S ~ + ( v ( t + 1 ) 1 η λ ( t ) ) ,
λ n ( t + 1 ) = λ n ( t ) + η { s ¯ n T h ¯ ( t + 1 ) u n ( t + 1 ) e j ϕ n ( t + 1 ) } .
[ g ( t ) | g ( t ) | ] n = { 0 for   g n ( t ) = 0 , g n ( t ) | g n ( t ) | otherwise .
f i n d h ¯ s . t . ψ k L = | h ¯ T s ¯ k L | 2 ( k = 1 , , K ) ,
N M S E ( h ± ) = E [ h + h dist+ 2 2 + h h dist- 2 2 h dist+ 2 2 + h dist- 2 2 ] ,
N M S E ( ρ ) = E [ 2 L | ρ ρ dist | 2 h dist+ 2 2 + h dist- 2 2 ] .