Multi-impairment monitoring from adaptive finite-impulse-response filters in a digital coherent receiver

Md. Saifuddin Faruk; Yojiro Mori; Chao Zhang; Koji Igarashi; Kazuro Kikuchi

doi:10.1364/OE.18.026929

1. Introduction

A digital coherent optical receiver facilitates effective compensation for various impairments in the optical transmission system by using digital signal processing (DSP) and enables high spectral efficiency with multilevel modulation and polarization multiplexing [1,2]. Hence, it is a promising candidate for the receiver used in next-generation high-capacity optical networks operating at the bit rate over 100 Gbit/s.

In such networks, channel-parameter monitoring is an important issue for diagnosing optical signal quality and providing its information to higher layers for network management such as impairment-aware-routing [3]. The digital coherent receiver allows equalization of all linear impairments, namely, chromatic dispersion (CD), polarization-mode dispersion (PMD) and polarization-dependent loss (PDL) by using four finite-impulse-response (FIR) filters structured in a two-by-two butterfly configuration [4]. After the filters are adapted by a suitable algorithm, we can construct a frequency-dependent two-by-two matrix with four elements, which correspond to transfer functions of the adapted four FIR filters. The inverse of this matrix is essentially the channel transfer matrix and is called the monitoring matrix. Several efforts for monitoring CD, first-order PMD and PDL from this matrix have been demonstrated so far [5–8].

For such monitoring purpose, we need a precise algorithm to separate out CD, PMD and PDL. References [5] and [6] neglect the effect of PDL when determining CD and differential group delay (DGD) between two principal states of polarization (PSPs). Moreover, the proposed DGD estimation process is rather complicated, because it needs some adjustment of the matrix element and also sinusoidal curve fitting. Reference [7] requires matrix decomposition before separating out individual impairment. In [8], an optical signal-to-noise ratio (OSNR) monitor is needed prior to the PDL estimation; however, precise OSNR monitoring is not available in the post-processing unit of the receiver. In addition, none of the above mentioned works has investigated monitoring of the second-order polarization mode dispersion (second-order PMD).

In this paper, we propose a simple and unified algorithm to separate out CD, DGD, PDL, and second-order PMD from the adaptive FIR filter tap coefficients. As to second-order PMD, polarization-dependent chromatic dispersion (PCD) and depolarization (DEP) of principal states of polarization are obtained separately. This algorithm has an advantage that individual impairment can be estimated directly from the monitoring matrix formed from the filter tap coefficients without any matrix decomposition; thus it enables accurate estimation of the impairments, even when the transmitted signal suffers from distortion stemming from various origins. In addition, it should be stressed that no additional hardware is required for our proposed algorithm. Our proposed multi-impairment monitoring algorithm is verified by dual-polarization quadrature phase-shift keying (QPSK) transmission experiments.

In the rest of the paper, vectors and matrices are in boldface letters; superscripts (•)*, (•)^T, (•)⁻¹, and (•)^† denote complex conjugate, transpose, inverse, and Hermitian transpose, respectively; subscript (•)_ω is the numerical differentiation with respect to the angular frequency ω; and functions DFT (•), arg (•), and det (•) denote discrete Fourier transform, argument, and determinant, respectively.

2. Monitoring algorithm

If the input power launched on a transmission link is low enough to operate in the linear or weakly nonlinear region, the transfer function of the link can be modeled as a concatenation of CD, PMD and PDL elements, given as D(ω), U(ω) and K, respectively, and the Jones matrix T expressing the birefringence of the fiber as

H_{f i b e r} (ω) = D (ω) U (ω) K T,

where ω is the angular frequency of the optical carrier.D(ω) is a scalar function and its phase has quadratic dependence on ω as shown by

D (ω) = e^{- j ω^{2} β_{2} z / 2},

where β₂ is the group-velocity dispersion (GVD) parameter and z is the fiber link length. The first-order PMD matrix U(ω) is unitary and is given as

U (ω) = R_{1}^{- 1} [\begin{matrix} e^{j ω Δ τ / 2} & 0 \\ 0 & e^{- j ω Δ τ / 2} \end{matrix}] R_{1},

where

Δ τ

is the DGD and R₁ is a unitary matrix converting two PSPs into the x- and y-polarization. Although PDL may randomly distributed over the transmission system, the global PDL can be represented by a single Hermitian matrix K as

K = R_{2}^{- 1} [\begin{matrix} \sqrt{Γ_{\max}} & 0 \\ 0 & \sqrt{Γ_{\min}} \end{matrix}] R_{2},

where Γ_max and Γ_min are maximum and minimum values of the transmission coefficient, respectively, and R ₂ is a unitary matrix converting the eigen modes for PDL into the x- and y-polarization. Finally, T is a two-by-two unitary matrix, whose matrix elements are angular-frequency independent.

In dual-polarization transmission systems, the transmitter sends complex amplitudes of both x- and y-polarized electric fields. While propagating through the fiber, the complex amplitudes suffer from effects of CD, PMD, PDL and birefringence. Then, the complex amplitudes are detected by a phase- and polarization-diverse homodyne receiver that preserves both the amplitude and phase information of the transmitted signal. Next, polarization demultiplexing and equalization can be done in the digital domain by using four complex-valued multi-tap FIR filters arranged in a two-by-two butterfly configuration.

After the equalization algorithm is converged, we take discrete Fourier transforms of filter-tap coefficients of the four FIR filters in the butterfly configuration. Provided that the filter tap length Lis long enough compared to the impulse response of the channel, we can construct the monitoring matrix M(ω) as

M (ω) \approx H_{f i b e r} (ω) = {D F T [\begin{matrix} h_{x x} (n) & h_{x y} (n) \\ h_{y x} (n) & h_{y y} (n) \end{matrix}]}^{- 1} .

The tap coefficient vector is given as $h_{i j} (n) = {[\begin{matrix} h_{i j, 0} (n) & h_{i j, 1} (n) & \cdot \cdot \cdot & h_{i j, L} (n) \end{matrix}]}^{T}$ , where i and j are any one of x and y, and n is the number of iteration.

Now, M(ω) contains the information of all the linear impairments. In the following, we propose a simple algorithm, which can separate out CD, DGD, PDL, and second-order PMD, that is, PCD and DEP, through straightforward algebraic manipulations of this monitoring matrix M(ω). With such an algorithm, multi-impairment monitoring is finally enabled by the adaptive FIR filters.

While deriving equations in this section, we use the general property of the matrix inversion (AB)⁻¹ = B ⁻¹ A ⁻¹, the property of a unitary matrix U ⁻¹ = U ^†, and the property of a Hermitian matrix $K = K^{†}$ without notice.

2.1 CD monitoring

The determinant of the matrix M(ω) can be expressed as

d e t {M (ω)} = D^{2} (ω) d e t {U (ω)} d e t (K) d e t (T) .

Since

d e t (U) = 1

and

d e t (K) = \sqrt{Γ_{\max} Γ_{\min}}

, Eq. (6) becomes

\det {M (ω)} = D^{2} (ω) \sqrt{Γ_{m a x} Γ_{m i n}} .

In view of the fact that

\sqrt{Γ_{\max} Γ_{\min}}

is a real number, Eqs. (2) and (7) show that we can estimate the CD value using a quadratic fitting on the unwrapped phase of det{M(ω)}.

2.2 PMD Monitoring

Conventionally, the fiber PMD is characterized by a frequency-dependent PMD vector $\vec{τ}$ in the three-dimensional Stokes space and can be written as

\vec{τ} = Δ τ \hat{p},

where the unit vector

\hat{p}

points the direction of the slower PSP and the magnitude

Δ τ

represents DGD.

Figure 1(a) shows two PMD vectors $\vec{τ} (ω)$ and $\vec{τ} (ω + Δ ω)$ in the three-dimensional Stokes space, where $Δ ω$ is a small change in the angular frequency, usually given as the DFT angular-frequency resolution. Then, the second-order PMD vector ${\vec{τ}}_{ω}$ can be defined as the derivative of $\vec{τ}$ with respect to ω as shown in Fig. 1 (b). The second-order PMD vector can be resolved into two components as

{\vec{τ}}_{ω} = \frac{d \vec{τ}}{d ω} = Δ τ_{ω} \hat{p} + Δ τ {\hat{p}}_{ω} :

One is the DEP component

{\vec{τ}}_{ω ⊥} = Δ τ {\hat{p}}_{ω}

, indicating that the pointing direction of the PMD vector varies with ω. The other is the PCD component

{\vec{τ}}_{ω ∥} = Δ τ_{ω} \hat{p}

, which represents the change in DGD with. As shown in Fig. 1(b), the DEP component is perpendicular to

\vec{τ} (ω)

, while the PCD component is parallel to

\vec{τ} (ω)

.

Fig. 1 Definition of second-order PMD vector. (a): Two PMD vectors at angular frequencies ω and $ω + Δ ω$ are shown in the Stokes space. (b): Corresponding second-order PMD vector with its perpendicular and parallel components. This is given as the derivative of the first-order PMD vector with respect to the angular frequency ω.

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Now, we consider the matrix $M (ω + Δ ω) M^{- 1} (ω)$ . Equation (1) yield

M (ω + Δ ω) M^{- 1} (ω) = D (ω + Δ ω) U (ω + Δ ω) K T {D (ω) U (ω) K T}^{- 1}

= D (ω + Δ ω) D^{*} (ω) U (ω + Δ ω) K T {(K T)}^{- 1} U^{- 1} (ω)

= c U (ω + Δ ω) U^{†} (ω),

where the scalar term

D (ω + Δ ω) D {(ω)}^{*}

is expressed as c. Substituting the explicit form of U(ω) given by Eq. (3) into Eq. (10), we have

M (ω + Δ ω) M^{- 1} (ω) = c R_{1}^{- 1} [\begin{matrix} e^{j Δ ω Δ τ / 2} & 0 \\ 0 & e^{- j Δ ω Δ τ / 2} \end{matrix}] R_{1} .

Equation (11) shows that eigen values of the matrix

M (ω + Δ ω) M^{- 1} (ω)

,

ρ_{1, 2}

, are associated with DGD while corresponding eigen vectors,

| t_{1, 2} 〉

, locate the PSPs. The estimation of eigen values and eigen vectors from this matrix enables DGD and second-order PMD monitoring in the following manners.

2.2.1 DGD Monitoring

From Eq. (11), DGD at ω can be estimated as

Δ τ = | \frac{\arg (ρ_{1} / ρ_{2})}{Δ ω} | .

The condition

Δ τ Δ ω < π

is necessary for avoiding ambiguities that arise from the multi-valued argument function.

2.2.2 Second-order PMD Monitoring

Two DGD values at frequencies $ω + Δ ω$ and ω enable the calculation of PCD by using the following equation:

| {\vec{τ}}_{ω ∥} | = \frac{Δ τ (ω + Δ ω) - Δ τ (ω)}{Δ ω} .

Let the eigen vector $| t_{1} 〉 = [s_{x}, s_{y}]$ relates to the slower PSP. Then, the corresponding Stokes vector S can be calculated as

S = [s_{x} s_{x}^{*} - s_{y} s_{y}^{*}, s_{x} s_{y}^{*} + s_{x}^{*} s_{y}, j (s_{x} s_{y}^{*} - s_{x}^{*} s_{y})],

and the unit vector

\hat{p}

can be found as

\hat{p} = S / | S |

. Thus, unit vectors

\hat{p} (ω + Δ ω)

and

\hat{p} (ω)

enable the estimation of

| {\hat{p}}_{ω} |

as

| {\hat{p}}_{ω} | = \frac{1}{2} \cdot \frac{\cos^{- 1} {\hat{p} (ω + Δ ω) \cdot \hat{p} (ω)}}{Δ ω} .

Now, Eqs. (12) and (15) yield DEP as

| {\vec{τ}}_{ω ⊥} | = Δ τ | {\hat{p}}_{ω} |

Finally, the magnitude of the total second-order PMD is determined from

| {\vec{τ}}_{ω} | = \sqrt{{| τ_{ω | |} |}^{2} + {| τ_{ω ⊥} |}^{2}} .

2.3 PDL Monitoring

The matrix M ^†(ω)M(ω) can be expressed from Eq. (1) as

M^{†} (ω) M (ω) = {D (ω) U (ω) K T}^{†} {D (ω) U (ω) K T}

= D^{*} (ω) D (ω) {K T}^{†} U {(ω)}^{†} U (ω) {K T}

= T^{†} K^{†} K T

Hence, Eq. (18) can be rewritten by using Eq. (4) as

M^{†} (ω) M (ω) = {(T R_{2})}^{- 1} [\begin{matrix} Γ_{\max} & 0 \\ 0 & Γ_{\min} \end{matrix}] (T R_{2}) .

Equation (19) shows that eigen values of the matrix M ^†(ω)M(ω),

α_{1, 2}

, can give PDL in dB as

P D L_{d B} = 10 \log_{10} (\frac{α_{1}}{α_{2}}) .

3. Experiments

3.1 Experimental setup

In order to validate our proposed algorithm, we conducted experiments employing a coherent optical receiver as shown in Fig. 2 . The transmitter laser was a distributed-feedback laser diode (DFB-LD) having a center wavelength of 1552 nm and a 3-dB linewidth of 150 kHz. The laser for LO had the same characteristics. A NRZ-QPSK signal was generated using a LiNbO₃ optical IQ modulator (IQM) from two streams of precoded data from an arbitrary waveform generator (AWG) with 2⁹-1 pseudo-random binary sequences (PRBS). The dual-polarization signal was then produced in the split-delay-combine manner by using a polarization beam splitter (PBS) and a polarization beam combiner (PBC). The delay length was long enough so that the signals in two polarization tributaries were uncorrelated. We generated PDL by attenuating one polarization tributary using a variable optical attenuator (VOA). Then, the signal passed through a PMD emulator (PMDE). We used a commercially available PMDE. It consisted of three programmable DGD sections separated by polarization controllers and was capable of generating all-order PMD with tunable statistics. A standard single-mode fiber (SMF) accumulated CD. After the transmitted signal was pre-amplified by an erbium-doped fiber amplifier (EDFA), it was incident on a phase and polarization diverse coherent optical receiver. The received power was controlled by VOA in such a manner that the polarization tributary suffered more from PDL had a BER about 3×10⁻⁴. Outputs of the receiver were sampled and digitized at twice the symbol rate with analog-to-digital converters (ADCs), and stored for offline digital signal processing.

Fig. 2 Schematics of the dual-polarization QPSK transmission system for verifications of the proposed impairment-monitoring algorithm.

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In the DSP circuit, the signals were polarization-demultiplexed and equalized by using four 33-tap butterfly-structured FIR filters with tap spacing of a half-symbol duration and adapted by the constant-modulus algorithm (CMA). The singularity problem of CMA was avoided by using the training mode prior to the blind CMA mode [9]. After filter tap coefficients were converged, we obtained the monitoring matrix from them and determined the impairments based on the proposed algorithm.

3.2 Results and discussions

The spectrum of the transfer function of a T/2-spaced filter covers the range from -B to + B, where B is the symbol rate. However, the parameter estimation is concentrated to several center taps only where low-pass electrical filters in the transmitter and the receiver have the linear phase response. First, to prove the effectiveness of the algorithm monitoring CD, DGD and PDL, we conducted 40 Gbit/s dual-polarization unrepeated QPSK transmission experiments with the setup shown in Fig. 2. We used only one DGD section of our PMDE to generate a fixed DGD value and also the amount of PDL was set to a constant value.

Figure 3(a) shows the estimation result of CD for 50-km-, 100-km- and 150-km-long SMFs with 20-ps DGD and 3-dB PDL. Estimated values are in good agreement with values measured by a standard CD measuring instrument that uses the modulation phase shift method. As shown in Fig. 3(b), DGD values are estimated with good accuracy even in the presence of 1600-ps/nm CD and 3-dB PDL. Figure 3(c) shows PDL measurement results with 1600-ps/nm CD and 20-ps DGD, which are also in good agreement with set values of PDL.

Fig. 3 Monitoring results. (a): CD monitoring result with 20-ps DGD and 3-dB PDL. (b): DGD monitoring result with 1600-ps/nm CD and 3-dB PDL. (c): PDL monitoring result with 1600-ps/nm CD and 20-ps DGD.

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Generation of a fixed and known value of second-order PMD is not available in our PMDE; hence, to verify the second-order PMD monitoring algorithm, we estimate the statistical behavior of second-order PMD and compare it with the theoretical expectation. In this experiment, we transmitted a 100-Gbit/s dual-polarization QPSK signal through PMDE, which was set to generate a Maxwellian-distributed DGD with the mean value of 35 ps and a corresponding second-order PMD with a refresh rate of 10 ms, while CD and PDL were set to zero. We estimate $Δ τ$ , $| {\vec{τ}}_{ω} |$ , $| {\vec{τ}}_{ω ∥} |$ , and $| {\vec{τ}}_{ω ⊥} |$ from 700 different experimental data repeatedly taken in three seconds using our proposed algorithm.

Estimated probability densities of all the four parameters are shown by bars in Figs. 4(a) -4(d). On the other hand, solid curves in these figures represent theoretical probability densities calculated from the measured mean DGD value based on the theoretical model given in [10–12], which assumes a constant DGD value and randomly fluctuating birefringence along the fiber.

Fig. 4 Probability densities of the first- and second-order PMD. (a): DGD $Δ τ$ , (b): PCD $| {\vec{τ}}_{ω ∥} |$ , (c): DEP $| {\vec{τ}}_{ω ⊥} |$ , and (d): the magnitude of second-order PMD $| {\vec{τ}}_{ω} |$ . Bars show those estimated from monitored values and solid curves are theoretical ones.

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For all the cases, the measured probability density well matches with the theoretical one. The estimated mean DGD $Δ τ$ of 36 ps is very close to the set value of 35 ps, whereas the measured mean second-order PMD value $| {\vec{τ}}_{ω} |$ is 606 ps², which is about 20% less than the theoretical value determined from the estimated mean DGD of 36 ps.

The discrepancy between the estimated second-order PMD and the theoretical prediction may stem from the inadequate number of statistical samples. In addition, three DGD sections in our PMDE may be insufficient to generate the second-order PMD similar to that of the real fiber, which is usually emulated by more than several hundreds of the birefringence section [13].

4. Conclusion

We have proposed a novel algorithm that enables monitoring of linear impairments of an optical transmission system, such as CD, PDL, DGD and second-order PMD from the adaptive FIR-filter tap coefficients in a digital coherent receiver. We have verified our proposed algorithm with dual-polarization QPSK transmission experiments. Multi-impairment monitoring demonstrated here has the advantage that it can be realized in an efficient way with just a small additional complexity in the DSP circuit of the digital coherent receiver.

Acknowledgments

This work was supported in part by Strategic Information and Communications R&D Promotion Programme (SCOPE) (081503001), the Ministry of Internal Affairs and Communications, Japan, and Grant-in-Aid for Scientific Research (A) (22246046), the Ministry of Education, Science, Sports and Culture, Japan.

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Multi-impairment monitoring from adaptive finite-impulse-response filters in a digital coherent receiver

Abstract

1. Introduction

2. Monitoring algorithm

2.1 CD monitoring

2.2 PMD Monitoring

2.2.1 DGD Monitoring

2.2.2 Second-order PMD Monitoring

2.3 PDL Monitoring

3. Experiments

3.1 Experimental setup

3.2 Results and discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (24)

Optics Express