Statistical moments-based OSNR monitoring for coherent optical systems

Chen Zhu; An V. Tran; Simin Chen; Liang B. Du; Cuong C. Do; Trevor Anderson; Arthur J. Lowery; Efstratios Skafidas

doi:10.1364/OE.20.017711

1. Introduction

Coherent optical communication systems have been intensively investigated in recent years because they enable higher-order modulation formats such as phase shift keying (PSK) and quadrature amplitude modulation (QAM) along with polarization multiplexing (pol-mux) [1]. Coherent reception also allows digital signal processing (DSP) to be used to mitigate fiber impairments such as chromatic dispersion (CD) and polarization-mode dispersion (PMD) [2]. Furthermore, coherent systems have dramatically increased the throughput of a single-mode fiber [3]. Optical performance monitoring (OPM) is a key issue for managing high capacity optical communication systems [4]; and is essential in fault management applications and new channel establishment.

Optical signal-to-noise ratio (OSNR) is an important parameter in OPM and is critical to network management. Previously, OSNR monitoring approaches have been reported based on uncorrelated beat noise [5], polarization-nulling [6], amplitude and phase histograms [7], delay tap sampling [8], and neural networks [9]. Coherent receivers enable OSNR monitoring methods that are based on the empirical moments of the asynchronously sampled signals [10] and the equalized signal [11–14]; this allows a cost-effective and robust in-service estimation of performance. Among these coherent-receiver-based OSNR estimation algorithms, the methods from [11, 12] utilize statistical moments of the equalized signal, which only requires the envelope of the equalized signal. In contrast, other methods [13, 14] require a fully compensated signal, which requires additional phase noise compensation and computational effort. However, existing statistical moments-based methods [11, 12] can only be applied to constant modulus modulation formats such as PSK, so are not suitable for high-order QAM. Moreover, these methods rely on determining two specific moments of the equalized signal, which potentially increases the implementation complexity. This complexity could be reduced if moments calculated for other purposes, such as signal quality evaluation [15] and blind channel equalization [16], could be reused.

In this paper we propose a novel OSNR estimator utilizing the equalized signal from pol-mux coherent single carrier systems. Our contribution is to provide a general class of estimators to monitor OSNR for constant and non-constant modulus constellations with two arbitrary statistical moments of the equalized signal. The proposed method is independent of the equalizer; therefore, it is compatible with frequency domain, time domain, training-aided and blind equalizers. We experimentally demonstrate this OSNR monitor, to show that a frequency domain equalizer with training-aided channel estimation can predict OSNR rapidly. Frequency domain equalizers require much lower computational effort compared with time domain equalizers [17] and training symbols avoid the extra convergence time required form blind equalizers [18]. Our experimental results show that accurate OSNR estimation is possible over a wide range of OSNRs in long distance transmission for all the modulation formats tested. Our simulation results show that the technique is unaffected by CD and PMD. We also show that the technique works well if blind equalization is used.

The paper is organized as follows: Section 2 investigates the theory of the proposed statistical moments-based OSNR estimator. In Sections 3 and 4 the experimental and simulation results respectively are shown and discussed. Finally we summarize the paper in Section 5.

2. Statistical moments-based OSNR estimator

Let us consider a transmitted data symbol, $s_{n}$ , with an amplitude $| s_{n} |$ , that comes from a modulation scheme with m different amplitudes, $Z_{1}, Z_{2}, ..., Z_{m}$ , with associated probabilities p₁, $p_{1}, p_{2}, ..., p_{m}$ . As shown in Fig. 1 , QPSK has only one amplitude, $Z$ , with probability 1, whereas 16-QAM has three different amplitudes with probabilities (0.25 0.5, 0.25) for ( $p_{1}, p_{2}, p_{3}$ ). 64-QAM has nine different amplitudes. Without any loss of generality, we assume that these transmitted symbols have unit average power, i.e. $\sum_{n = 1}^{m} p_{n} \cdot Z_{n}^{2} = 1$ .

Fig. 1 Constellation diagram of (a) QPSK, (b) 16-QAM and (c) 64-QAM.

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The system is assumed to operate at a low-enough launch power to avoid nonlinear effects, which may lead to non-Gaussian characteristic of the equalized signal. We further assume that the channel impairments such as CD and PMD have been fully compensated by a frequency domain multiple input multiple output (FD-MIMO) equalizer, as discussed in Section 3. The equalized signal is:

r_{k} = \sqrt{S} \cdot s_{n} + N

where:

r_{k}

is the k^th equalized symbol,

\sqrt{S}

is a real signal power scaling factor,

N

, is the complex noise sample with zero mean and variance

2 σ^{2}

. The noise is a combination of optical amplified spontaneous emission (ASE) and the background noise, including shot noise, electrical thermal noise and quantization noise. Defining the first parameter we want to estimate as the signal to noise ratio (SNR), which is:

η = SNR = \sum_{n = 1}^{m} p_{n} \cdot \frac{S \cdot Z_{n}^{2}}{2 σ^{2}} = \frac{S}{2 σ^{2}} .

We shall first consider PSK modulation schemes with only one amplitude, $Z$ . The conditional probability density function (PDF) of $r_{k}$ is given by the bivariate Gaussian distribution [12]:

P_{P S K} (r_{k} | Z_{n}, θ, σ) = \frac{1}{2 π σ^{2}} \exp (\frac{- | r_{k} - Z_{n} e^{j θ} |^{2}}{2 σ^{2}})

where

θ

represents the constellation phase, which is normally treated as a uniformly distributed random variable and is unknown to us. We can remove

θ

by integration and only look at the amplitude of the received signal,

| r_{k} |

, where the conditional PDF can be described as [19]:

P_{P S K} (| r_{k} |) = \frac{| r_{k} |}{σ^{2}} \cdot \exp (- η \cdot Z - \frac{| r_{k} |^{2}}{2 σ^{2}}) \cdot I_{0} (\frac{| r_{k} |}{σ} \sqrt{2 η \cdot Z})

where

I_{0} (.)

denotes the modified Bessel function of the first kind, which is a well-known Ricean distribution. Then we can formulize the i^th moment of the Ricean distribution in terms of parameters η and σ as [20]:

ρ_{i, P S K} (σ, η) = E [| r_{k} |^{i}] = {(2 σ^{2})}^{i / 2} \cdot Γ (i / 2 + 1) \cdot \exp (- η \cdot Z^{2}) \cdot_{1} F_{1} (i / 2 + 1; 1; η \cdot Z^{2})

where:

E (.)

denotes the expectation operation,

Γ (.)

indicates the gamma function and

_{1} F_{1} (.)

is the confluent hypergeometric function.

Any high-order modulation format can be decomposed into m different PSK sub-systems; for example, PSK partitioning for 16-QAM and 64-QAM is shown as colored rings in Fig. 1. Therefore, the moments of the signal envelope are m Ricean distributions, one for each ring [21]:

ρ_{i} (σ, η) = \sum_{n = 1}^{m} p_{n} \cdot {(2 σ^{2})}^{i / 2} \cdot Γ (i / 2 + 1) \cdot \exp (- η \cdot Z_{n}^{2}) \cdot_{1} F_{1} (i / 2 + 1; 1; η \cdot Z_{n}^{2}) .

There are only two unknown parameters

η

and

σ

within Eq. (6), where

η

is what we want to estimate. Hence we can define a simple function based on two different statistical moments,

f

and

g

, to eliminate the unknown parameter

σ

:

ξ_{f, g} (η) = \frac{ρ_{f}^{μ / f}}{ρ_{g}^{μ / g}}

where

μ = L C M (f, g)

is the least common multiple (LCM) of

f

and

g

. We collect

L

equalized signals for each OSNR estimation, where

L

needs to be large enough in order to match the random statistics of the theoretical constellations, and an insufficient number of symbols can affect the stability and estimation accuracy of the proposed algorithm.

L

determines the maximum tracking speed of the algorithm, and only the OSNR difference between two consecutive

L

symbol blocks is tractable. We choose

L

equals to 12288 in our experiment and simulation, then we can construct the moments-based estimator by solving Eq. (7), to get:

η_{f, g} = ξ_{f, g}^{- 1} (\frac{ρ_{f}^{μ / f}}{ρ_{g}^{μ / g}})

where

ρ_{i}

is the estimation of the i^th moment of the equalized signal, given by

ρ_{i} = L^{- 1} \cdot \sum_{k = 1}^{L} | r_{k} |^{i}

and

ξ_{f, g}^{- 1} (.)

means the inverse function of

ξ_{f, g} (η)

. The estimation result can be continuously updated with the latest

L

equalized symbols, which is helpful in real-time monitoring. Furthermore, this easy update enables the application of the proposed algorithm to be used in optical networks where the signals are dynamically added and dropped. The estimator will always indicate the SNR (therefore the OSNR) of the current optical path as long as the received signals have been equalized properly.

In terms of the number of multiplications, the computational complexity of the proposed algorithm is $(f + g - 2) \cdot L$ . As the lowest-order moments, (1, 2) require the least computational effort, they should be the natural choice for ( $f, g$ ). Although, in this case, the analytical function of $ξ_{1, 2}^{- 1} (.)$ cannot be derived, we can find a solution by searching for the nearest result with a lookup list that covers all possible pairs for ( $η, ξ_{f, g}$ ) within the range 0 dB ≤ η ≤ 30 dB with, say, a step size 0.01 dB. However, there exists a simpler approach when (2, 4) is chosen for ( $f, g$ ). From Eq. (6) we can write:

ρ_{2} = (2 σ^{2}) \cdot (η + 1) a n d ρ_{4} = {(2 σ^{2})}^{2} \cdot (α η^{2} + 4 η + 2)

where

α = \sum_{n = 1}^{m} p_{n} Z_{n}^{4}

. In this case

μ = 4

and the inverse function for

ρ_{2}^{2} / ρ_{4} = {(η + 1)}^{2} / (α η^{2} + 4 η + 2)

is tractable. Therefore, the estimator can be written as a formula:

η_{2, 4} = \frac{ρ_{4} - 2 ρ_{2}^{2} - \sqrt{(2 - α) (2 ρ_{2}^{4} - ρ_{2}^{2} ρ_{4})}}{α ρ_{2}^{2} - ρ_{4}} .

For any constant-amplitude modulation scheme (

α = 1

), Eq. (10) is equivalent to the estimator proposed in [11, 12] for QPSK systems. In Section 4, we will investigate the performance of the estimator based on the

η_{1, 2}

and

η_{2, 4}

methods.

As mentioned before, $η$ is the estimation of SNR. For a balanced coherent receiver the electrical noise contains local oscillator × ASE beat noise, thermal noise and shot noise. In order to separate the noise due to ASE noise from the total noise, we set a calibration point SNR_RF which is the SNR value measured by the back-to-back transmission without the ASE noise, after taking the noise reference bandwidth into account [12]. The OSNR estimator can then be described as:

O S N R_{d B} = 10 \log_{10} (\frac{B_{r e f}}{R_{s}}) - 10 \log_{10} (\frac{1}{η} - \frac{1}{S N R_{R F}})

where

R_{s}

is the symbol rate and

B_{r e f}

is the measurement bandwidth of the ASE noise. In optical networks, although the optical paths keep changing, we can use the same reference value for calibration as they are sharing the same coherent receiver, and the estimation performance is expected to be stable for different paths as long as the received signals get proper equalization.

3. Experimental demonstration

Figure 2 shows the experimental setup for QPSK and 16-QAM. Two independent single-carrier signals were generated by driving a pair of I/Q modulators with two 10-G symbols/s arbitrary waveform generators (AWG1 and AWG2). The output of a 100-kHz linewidth external cavity laser (ECL) was split into two paths with a 50:50 coupler. These paths were both modulated to generate two optical QPSK/16-QAM signals. These two signals were combined using a polarization beam combiner (PBC) to generate a dual-polarization signal. After transmission through five spools of standard single mode fiber, an ASE source consisting of two EDFAs and one optical band-pass filter (OBPF) was used to control the OSNR level during the transmission. After ASE noise injection, the signal was split into two paths using a 90:10 coupler. The 10% path was connected to an optical spectrum analyzer (OSA) for OSNR monitoring and the 90% path was first filtered by an OBPF before being detected by a coherent receiver with a local oscillator (LO) and balanced receivers. The signals were digitized for offline processing using a 40-G samples/s real-time oscilloscope.

Fig. 2 Experimental setup, LPF: low-pass filter, AMP: amplifier, PBS: polarization beam splitter.

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Figure 3(a) illustrates the DSP procedure at the transmitter [22]. The binary bits were first mapped into QPSK/16-QAM signals. The overlap-FDE (O-FDE) approach [23] was used to eliminate the inter block interference, rather than inserting a cyclic prefix (CP) to achieve circular convolution. A preamble was inserted before the information symbol. The preamble contained training symbols TS1 and TS2 for the X polarization and TS3 and TS4 for the Y polarization. These symbols were chosen to be orthogonal to each other, as shown at the bottom of Fig. 3(a). Their spectra were scheduled according to the Alamouti coding matrix [24], where (.)* denotes the complex conjugate operation.

Fig. 3 DSP procedures at the (a) transmitter and (b) receiver.

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Figure 3(b) shows the DSP procedure at the receiver [22]. After frequency offset compensation, the timing detection was realized using sample autocorrelation based on the same training sequences used for channel estimation, the received training sequences were extracted to estimate the 2 × 2 channel matrix using a zero forcing algorithm. At the same time, the received information data were passed through the FD-MIMO equalizer with block-wise processing and fast Fourier transform (FFT). The output of the equalizer was then used for OSNR estimation based on both of the $η_{1, 2}$ and $η_{2, 4}$ methods. Figures 4 and 5 show the constellation diagrams of the equalized signals without phase correction for both QPSK and 16-QAM transmission after 400-km of fiber. In each figure, the constellations for high, moderate, and low OSNR are indicated by (a), (b), and (c), respectively.

Fig. 4 Constellation diagrams of equalized QPSK signals after 400-km with different OSNR: (a) 20.4 dB (b) 14.1dB and (c) 10.5 dB.

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Fig. 5 Constellation diagrams of equalized 16-QAM signals after 400-km with different OSNR: (a) 21.8 dB (b) 17.3 dB and (c) 13 dB.

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It is difficult to estimate very high OSNRs due to thermal, shot and other sources of ASE-independent noise. On the other hand, a very low OSNR would severely affect the estimation due to the large degradation in equalization performance. Thus, the OSNR value to be estimated is set to be between 7 - 22 dB for QPSK and 10 - 22 dB for 16-QAM. Figures 6(a) and 6(b) plot the estimation error versus the OSNR value measured using the OSA, for QPSK and 16-QAM modulation formats, respectively, for back to back (b2b) and after 400-km transmission for OSNR estimators based on $η_{1, 2}$ and $η_{2, 4}$ methods. These results show that the two estimators perform almost equally for QPSK and the errors are all below 0.5 dB. For 16-QAM, the differences between these two estimators are slightly larger. The reason is that we are assuming the constellation is uniformly distributed for all modulation formats, i.e. 1/4 for each constellation in QPSK and 1/16 in 16-QAM, but in practice the constellation will not be perfectly uniformly distributed. This does not affect the envelope distribution of QPSK because there is only one amplitude. However, this will lead to a slightly different envelope distribution for 16-QAM, which causes the inconsistency between different estimators. Most of the errors are still less than 0.5 dB showing the algorithm is also robust for non-constant modulus signals. Although the $η_{1, 2}$ estimator requires less complexity, it essentially needs extra time to find the best matched result. Therefore, $η_{2, 4}$ is more efficient in real-time systems.

Fig. 6 Estimation error versus true OSNR for (a) QPSK and (b) 16-QAM.

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4. Simulation results

We conducted simulations using VPItransmissionMaker 8.6 to further investigate the performance of the proposed OSNR estimator against different system impairments. η_2,4 was used for all simulations unless specified. The system was set to have 20-Gbaud, which supports 160-Gb/s for a pol-mux 16-QAM system, and the DSP algorithm was the same as that in the experiments except that the length of the equalizer was adjusted according to the bit rate and different fiber impairments. Moreover, we investigated a much wider OSNR range of 1 - 23 dB. The standard deviations of the estimation errors after several trials were used to evaluate the performance.

In our first simulation, the impact of PMD and CD on the OSNR estimation performance was investigated for a 16-QAM system. Figure 7(a) shows the results for the system operated with mean differential group delay (DGD) values between 0 and 50-ps, for all-order PMD after 1120-km. The estimator performs very accurately with the error always less than 0.3 dB. Then we fixed the mean DGD value to 30-ps and run the system over different transmission distances up to 1920-km. Figure 7(b) shows that most of the errors are still below 0.3 dB; only a few estimations of very high (> 20dB) or very low (< 5dB) OSNR values after long transmission distances (1520-km and 1920-km) have larger errors (still less than 0.5 dB) due to degraded equalization performance.

Fig. 7 Estimation performance for 16-QAM transmission with: (a) different mean DGD values with 1120-km transmission, (b) different transmission distances with 30-ps mean DGD.

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The tolerance of the OSNR estimation when the system impairments were undercompensated was studied by changing the number of equalizer taps. With 2 samples/symbol equalization, 1120-km transmission, 30-ps DGD and at 20-Gbaud, the theoretical fewest number of taps is 230 [2]. Table 1 shows the simulated received bit error rate (BER) for 25-dB OSNR when using different numbers of equalizer taps. The equalization performance is severely degraded for fewer than 512 taps. However, as shown in Fig. 8 , the OSNR estimator still has good performance with 256 taps if a proper calibration, accounting for the under-compensated part, is used. If we further reduce the number of taps to 128, the estimation errors become very large at low OSNRs. This is because the equalization performance becomes much poorer with lower OSNR values, which cannot be corrected by the calibration.

Table 1. BER for Different Equalizer Length for 16-QAM System

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Fig. 8 Estimation performance for 16-QAM with different length of equalizer.

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We further investigate the estimation performance against fiber non-linearity with 512 equalizer taps. The nonlinear coefficient of the fiber was set to 2.6 × 10⁻²⁰ m²/W. As can be seen in Fig. 9 , without extra non-linear compensation, the estimation performance decreases as the launched power becomes larger, this is due to the increasing non-Gaussian characteristic of the equalized signal caused by the fiber non-linearity. The estimation error is more than 1 dB when the launched power is ≥ 4 dBm.

Fig. 9 Estimation performance with different launched powers for 16-QAM transmission.

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We changed the modulation format to 64-QAM while keeping all other DSP parameters the same as before. In this case the BERs versus different equalizer length with 25-dB OSNR are included in Table 2 . With equalizer taps of 512, Fig. 10(a) shows the constellation diagrams of the fully equalized X-polarization signal after 1120-km transmission and 30-ps mean DGD with 25-dB OSNR. This transmission condition gives a BER of about 0.03, which is similar to the previous 256-tap equalizer simulation for 16-QAM system. Then we performed the OSNR estimation with both the $η_{1, 2}$ and the $η_{2, 4}$ methods. Figure 10(b) shows that the simulated results are all below 0.3 dB, which is consistent with the previous simulation. This again illustrates the effectiveness and high accuracy of the proposed estimator.

Table 2. BER for Different Equalizer Length for 64-QAM System

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Fig. 10 (a) Fully equalized X-polarization constellation diagrams for 64-QAM signal after 1120-km transmission and 30-ps DGD; (b) Estimation performance.

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We compared the performance of the OSNR estimator when training-aided and blind equalization techniques were used. The blind adaptation equalizer used the frequency domain constant modulus algorithm (CMA) [18]. Due to the short-reach limit of the blind algorithm, we only show the standard deviation of estimation errors for back-to-back and 320-km transmission cases in Fig. 11 . It is clear that the estimation is accurate and similar for both equalization schemes, which demonstrates that the OSNR estimator is independent of the equalization technique.

Fig. 11 Estimation performance using training-aided and blind equalization techniques.

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5. Conclusions

We have proposed and demonstrated a new and effective statistical moments-based OSNR monitoring algorithm for coherent optical transmission systems. The method provides a general approach to estimate SNR by using two arbitrary statistical moments of the equalizer output before phase compensation. After a proper reference value is set for calibration, the ASE noise can be separated from the background RF electronic noise to give an accurate estimation of OSNR. Furthermore, this approach can be applied to any modulation format, including constant and non-constant modulus formats, and is suitable for either data-aided or blind equalization systems. We have experimentally demonstrated that the proposed technique, combining with training-aided frequency domain equalization, achieves fast and accurate estimation of OSNR for both QPSK and 16-QAM systems after 400-km. Furthermore, simulation results verify that the technique can still perform very well in the presence of large amounts of CD and PMD, and is also robust with a certain amount of impairments not being fully equalized.

Acknowledgments

NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications, the Digital Economy, and the Australian Research Council through the ICT Centre of Excellence program. The research is also partially supported by the Australian Research Council Centre of Excellence for Ultrahigh Bandwidth Devices for Optical Systems.

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Equalizer length	BER
1024	3.57 × 10⁻⁵
512	1.24 × 10⁻⁴
256	2.80 × 10⁻²
128	1.54 × 10⁻¹

Equalizer length	BER
1024	1.03 × 10⁻²
512	2.79 × 10⁻²
256	7.65 × 10⁻²
128	5.04 × 10⁻¹

Equalizer length	BER
1024	3.57 × 10⁻⁵
512	1.24 × 10⁻⁴
256	2.80 × 10⁻²
128	1.54 × 10⁻¹

Equalizer length	BER
1024	1.03 × 10⁻²
512	2.79 × 10⁻²
256	7.65 × 10⁻²
128	5.04 × 10⁻¹

Statistical moments-based OSNR monitoring for coherent optical systems

Abstract

1. Introduction

2. Statistical moments-based OSNR estimator

3. Experimental demonstration

4. Simulation results

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (11)

Tables (2)

Equations (11)

Optics Express