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Frequency Offset Estimation and Carrier Phase Recovery for High-order QAM Constellations Using the Viterbi-Viterbi Monomial Estimator

Christos Spatharakis*, Nikolaos Argyris*, Stefanos Dris*, Hercules Avramopoulos*

*School of Electrical & Computer Engineering, National Technical University of Athens, Greece [email protected], [email protected], [email protected], [email protected]

Abstract—We present two novel, low-complexity feedforward algorithms for frequency offset estimation (FOE) and carrier phase recovery (CPR) in coherent optical M-QAM systems, employing the Viterbi-Viterbi monomial-based estimator. The FOE scheme is an extension of the phase-increment algorithm used for QPSK, while CPR is achieved with a robust, two-stage algorithm that does not rely on a computationally expensive Blind Phase Search (BPS) stage. Simulations show performance close to that of BPS for square constellations up to 256-QAM, while the penalty when both FOE and CPR are operated together is shown to be negligible.

Index Terms—Digital Modulation; Digital Signal Processing; Estimation; Optical Fiber Communications; Quadrature Amplitude Modulation

I. INTRODUCTION Modern coherent optical communication systems rely

on intradyne detection with unlocked transmitter (Tx) and local oscillator (LO) lasers. Even if identical lasers are used, the wavelength mismatch between the two can lead to frequency offsets of a few GHz. Moreover, typical linewidths of commercial distributed feedback (DFB) and external cavity lasers (ECL) range from ~100 kHz to ~10 MHz, potentially leading to significant phase noise in the received QAM constellation. As such, the Frequency Offset Estimation (FOE) and Carrier Phase Recovery (CPR) algorithms are the cornerstones of the digital coherent optical receiver, and substantial research effort has been focused on these.

FOE and CPR are straightforward when dealing with signals where every symbol lies on four equidistant phase axes (e.g., QPSK), since the modulation-stripping property of the ubiquitous Viterbi-Viterbi Fourth Power Estimator (VV4PE) allows isolation of the carrier, and therefore accurate measurement and tracking of the frequency offset and phase noise. In [1], each QPSK symbol is multiplied with the conjugate of the previous symbol, with VV4PE then applied to remove the phase modulation and estimate the frequency. QPSK CPR is achieved in a similar fashion [2].

VV4PE cannot be directly applied to 16-QAM to wipe the modulation; schemes based on QPSK-partitioning have thus been developed [3] [4]. However, this strategy performs poorly with higher-order formats and is not a viable option for 64-QAM and beyond.

Figure 1: Illustrating the contribution of AWGN to the phase error in a

16-QAM constellation (upper-right quadrant shown). AWGN contributes more spread in the phases of symbols with lower amplitudes

( 2> 1).

Phase Entropy-based FOE (PE-FOE) [5] has been shown to provide very good accuracy for arbitrarily-shaped constellations up to 32-QAM, but can only be used as a fine estimator, preceded by a coarse-estimation stage, in order to keep computational complexity at acceptable levels. Accurate CPR can be achieved with the blind phase search (BPS) algorithm [6] that can work with any constellation, even high-order QAM; however, this also comes at the expense of very high computational complexity that requires massive parallelization in a digital implementation.

In this work we explore the use of the generalization of the VV4PE, the Viterbi-Viterbi Monomial estimator (VVMPE) [7], in schemes for both FOE and CPR. Our proposed FOE algorithm (termed VVMFOE in the rest of this paper) is an extension of the phase-increment scheme for QPSK [1]. The crucial differentiation is that in our approach, the exponentiation of the received signal is not only used for modulation stripping (albeit partial, in the case of M-QAM), but also to give an increased weighting to the estimate from symbols with higher amplitudes. In essence, we use VVMPE to apply an even higher weighting to the outer symbols (the relation with the amplitude is exponential).

The motivation is twofold: 1) for square QAM, the symbols with the highest amplitude are those at the four corners of the constellation. These lie on four equidistant phase axes, which means that the fourth power operation in the VVMPE will completely remove their modulation;

2014 9th International Symposium on Communication Systems, Networks & Digital Sign (CSNDSP)

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thus, the estimate from the outer ring of the square QAM constellation will be most accurate. 2) Arbitrary White Gaussian Noise (AWGN) corrupts our estimate of the phase of each symbol, by contributing to the phase error (see section III of [2] for a detailed treatment). Even if our Tx and LO lasers had no linewidth (and thus no phase noise contribution), our estimate of the symbols’ phase would be noisy due to AWGN. Referring to Fig. 1 which shows a single quadrant of a 16-QAM constellation, it is obvious that the phase error due solely to AWGN is lower for higher-amplitude symbols ( 1< 2); again, the estimate from outer rings will therefore tend to be more accurate, and giving more weight to these will improve overall estimation. This is corroborated by the simulation results presented in this paper, where performance of the VVMFOE algorithm is assessed for formats up to 256-QAM.

For CPR, we propose a new, two-stage algorithm: VVMPE followed by Maximum Likelihood (ML) estimation. The rationale for using VVMPE (and higher weighting on the outer symbols) is exactly the same as for the FOE case. We show, via simulation, that the linewidth tolerance achieved with this scheme allows the use of commercial DFB/ECL lasers for square constellations up to 256-QAM. Since no BPS stage is required, the implementation complexity is potentially low.

II. VITERBI AND VITERBI MONOMIAL-BASED FREQUENCY OFFSET ESTIMATOR (VVMFOE)

A. Algorithm Description Consider a complex square M-QAM signal whose

symbols lie at points +a jb , where a and b can take on values (2 1)± −k , k being an integer satisfying

1,..., / 2k M= . The received signal, corrupted by AWGN and phase noise due to transmission through fiber and intradyne detection at the coherent Rx, can be represented in polar form as , 0,1,..., 1nj

n nx e n Nϕρ= = − . The proposed estimator is a combination of the phase

increment estimation algorithm and the VVPE algorithm based on the monomial transformations. The goal is to estimate the frequency offset, fΔ , that causes a phase increment, , between two successive symbols:

1

2 sf Tθπ

Δ = Δ ⋅ (1)

where the signal is sampled at the symbol period, Ts. The current symbol, xn, is multiplied by the conjugate of the previous one, yielding the product *

1n nx x −⋅ whose phase is equal to the two symbols’ phase difference ( 1n n nθ ϕ ϕ −= − ), and whose magnitude is 1n n nr ρ ρ −= ⋅ . VVMPE is then performed to obtain the mean phase increment over N symbols:

( )1

4

0

1 arg ,4

nN

l jn

n

r e lθθ−

=Δ = ⋅ ∈ (2)

When l = 4, (2) reduces to the special case of the Fourth Power Viterbi-Viterbi estimator. The crucial difference in our scheme is that we allow l to take on values greater than 4, which therefore enables even higher weighting to be applied to the outer symbols of the constellation, yielding more accurate results. Note that the estimation range of the algorithm is ±1/(8·Ts), due to the fourth power operation. A block diagram for the DSP algorithm is presented in Fig. 2.

Figure 2: DSP implementation of the Viterbi -Viterbi Monomial-based

Frequency Offset Estimator (VVMFOE) The range of the exponent, l, the dependence of its

optimum value on the SNR and linewidth, as well as the optimization of the estimator’s performance, have been extensively examined through simulations described in the next section.

B. Simulation Results

In order to evaluate the performance of the proposed FOE algorithm, transmission of 16-, 64- and 256-QAM signals was simulated in MATLAB. Each transmitted signal consisting of 104 symbols, is corrupted by AWGN and phase noise due to the laser linewidth, modeled as a discrete time random walk, 1n n nϕ ϕ −= + Δ , where n is a zero-mean Gaussian random variable with variance

Figure 3: Contour plots showing RMS Normalized Frequency Error (NFErms), as a function of SNR and linewidth for (a) 16-, (b) 64- and (c) 256

QAM, for different values of the exponent l.


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Figure 4: 95% ±Normalized Frequency Error range for (a) 16-, (b) 64- and (c) 256-QAM. See text for explanation.

(normalized to the symbol rate) 2 2 sv Tσ πΔ = ⋅ Δ ⋅ . For each run a frequency offset, sf TΔ ⋅ (also normalized to the symbol rate), was added, ranging between ±0.1 to ensure that the algorithm will not exceed the limit of ±1/(8·Ts) mentioned above. For example, the range of the frequency offset values for a 10 GBaud signal is between -1 and +1 GHz. 103 runs (Nsims) were carried out for each SNR and linewidth value simulated. Fig. 3 shows contour plots for the Root Mean Squared value of the Normalized Frequency Error (NFErms) between the actual ( fΔ ) and

estimated ( fΔ ) normalized frequency offsets, as a function of the laser linewidth and received SNR. Different contour lines correspond to different values of the exponent l, for 16-, 64-, and 256-QAM. We define NFErms as:

( ){ }2

NFE sims

sN

rmssims

f f T

N

Δ − Δ (3)

Only specific values of NFErms were chosen for plotting, so as to ease illustration of the general dependence on the SNR and linewidth. The value of l was found to play a crucial role in the estimation accuracy. Its value was swept, and the one that gave the best results was found for each modulation format. For clarity, only values close to this ‘optimum’ are depicted in Fig. 3. For the case where l = 4 (i.e. the classic VV4PE-based phase increment algorithm), performance was extremely poor and the NFErms was found to be significantly higher compared to VVMPE with optimal l. The contour plots for l = 4 are not within the intervals of SNR and linewidth set for the simulations and thus they are not included in Fig. 3.

Fig. 4 shows the range of the Normalized Frequency Error for 95% of the simulation runs (Nsims = 103 runs) as a function of the linewidth, for selected (best) values of l, at SNRs close to the expected operating point of a real system (the SNRs were chosen to be close to those that ideally would yield BERs near the 10-3 FEC limit). The values of chosen were l = 9, 12, and 10, for 16-, 64-, and 256-QAM respectively (for the latter, any value above 10 gave similar results). Excellent accuracy is achieved for all three modulation formats. For 16-QAM at linewidths up to ~5·10-4, 95% of the time the estimate lies between ±0.002 of the actual

normalized offset (for a 28 Gbaud system, this translates to just ±0.056 GHz). For 64-QAM the 95% estimation range is less than ±0.003, while for 256-QAM it is less than ±0.004.

The frequency offset estimation range is illustrated in Fig. 5 for the case of 16-QAM. The NFErms is plotted as a function of the normalized frequency offset added to the signal, with SNR and normalized linewidth values of 12dB and 2·10-4 respectively.

Figure 5: Estimation accuracy as a function of frequency offset

added to a 16-QAM signal.

Figure 6: Contour plots showing RMS Normalized Frequency Error

(NFErms), for 32-QAM with positive and negative values of the exponent l.

The results discussed so far concern only square QAM constellations. In order to explore the algorithm’s behavior for cross-shaped constellations, simulations for 32-QAM were also carried out. It is important to note that the basic principle of using higher exponents to achieve better modulation stripping does not apply to non-square


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constellations such as 32- or 128-Qoutermost symbols do not lie on the fouaxes. As shown in Fig. 6 we atteperformance by allowing l to take on neis equivalent to inverting the cequivalently, giving more weight to t(i.e. the ones with lower amplitude). Ththe innermost symbols of the 32-QAM a QPSK partition, and the fourth poweVVMPE completely wipes the moinnermost ring. While this proved to dperformance compared to using the(l = 4), the estimation is still an order accurate than in the case of thconstellations. The issue here is one introduction: Since the inner symboamplitude, the phase estimation error corrupted by AWGN. For completenessalgorithm exhibited the best performa32-QAM.

III. VITERBI AND VITERBI MONOMIAPHASE RECOVERY

A. Stage 1: Viterbi and Viterbi MonomEstimator (VVMPE)

Consider again a sequence of QAMrepresentation, , 0,1,...,nj

n nx e n Nϕρ= =amplitudes normalized as in SViterbi-Viterbi phase estimator based transformations is given by [7]:

( )1 1

4

0

1ˆ arg ,4

nN

l jvvmpe n

n

e ϕθ ρ−

=

= ⋅

Equation (4) is very similar to (2), estimates the phase of each symbol, ratshift between two consecutive symboloperation in (4) can be performed symbols with the same phase correctsymbols in the block, or using a slidinThe latter amounts to filtering with Response (FIR) structure whose coefficA more optimal phase estimate can bemploy approximate Wiener filteringcoefficients of the L-tap FIR approxima

Figure 7: DSP implementation of the VVMPE-

QAM, where the ur equidistant (90°) empt to improve egative values; this constellation, or, the inner symbols he idea here is that constellation form

er operation in the odulation in the

drastically improve e classic VV4PE of magnitude less

he square QAM mentioned in the

ols are lowest in is more severely

s, we note that the ance at l = −2 for

AL-BASED CARRIER

ial-based Phase

M symbols in polar

1 1N − , and with ection II. The on the monomial

l ∈ (4)

except here v̂vmpeθ ther than the phase s. The summation on blocks of N1 ion applied to all

ng-block approach. a Finite Impulse

cients are all equal. be obtained if we . If w(k) are the

ation to the Wiener

filter [2], the phase estimate( )1 / 2LΔ = − ) becomes:

1

1ˆ ( ) arg4

L

vvmpe kk

n wθ ρ=

− Δ =

B. Stage 2: Maximum LikelihoThe QAM symbols, x(n), a

corresponding block- or Wien

v̂vmpeθ , and fed into the secondML phase estimate on 2 symb

2 1

*

0

ˆ argN

ML n nn

c xθ−

=

=

where *ˆnc is the conjugate of thnth symbol corrected by the phamore simply, Stage 2 dconstellation point is nearesprocessed, and computes the etwo. Smoothing of the estimaveraging. After the second stthe phase error are subtractedoutput of Stage 1, yielding constellation at the output of Stimplementation in DSP.

C. Simulation Results Numerical simulations of 5

symbols were carried out to asas a function of the combined to the symbol rate ( sv TΔ ⋅ ), andnoise was again modeled as a d

1n n nϕ ϕ −= + Δ , where n israndom variable with varianfrequency offset was added, inof the phase noise and evaluateThe performance of VVMPagainst that of the single-stagephases for 16-QAM and 64 teQAM [6]. Approximate WienStage 1 (5), and block averagstage).

-ML phase recovery algorithm. Example 64-QAM constellations shstage are included (SNR=22.5, v·Ts=2.7·10-5).

e at symbol n − Δ (where

4 ,n kjln k e lϕρ −

− ⋅ ∈ (5)

ood Estimation are derotated by using the ner-filtered blind estimate,

d stage which computes the bols:

v̂vmpejn e θ⋅ (6)

he receiver's decision at the ase estimate in Stage 1. Put determines which ideal st to the symbol being error in phase between the ate is achieved via block tage, the ML estimates of

d from the symbols at the the final phase-recovered tage 2. Fig. 7 illustrates our

·105 differentially-encoded ssess the BER performance laser linewidth normalized d received SNR. The phase discrete time random walk,s a zero-mean Gaussian nce 2 2 sv Tσ πΔ = ⋅ Δ ⋅ . No

n order to isolate the effect e the CPR algorithm alone.

PE-ML was benchmarked e BPS scheme with 32 test st phases for 64- and 256-ner filtering was used for ging for Stage 2 (the ML

howing the evolution after each


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Figure 8: Contour plots obtained via simulation showing the BER's dependence on SNR and linewidth for (a) 16-, (b) 64-, and (c) 256-QAM. Solid

lines indicate BER contours of the VVMPE-ML scheme, dotted lines correspond to the BPS algorithm

The filter coefficients and appropriate block lengths per SNR-linewidth pair were determined by brute-force, and the ones that gave the best results were selected. For example, in the case of 16-QAM and SNR and linewidth values 18 dB and 10-5 respectively, the block length for Stage 2 was 33.

The value of the exponent l in the first stage of VVMPE-ML was varied to obtain the best results for each constellation simulated. In general, using higher powers resulted in better performance, and this tended to level off, with no obvious benefit after a certain value. Thus, for the simulation, l was set to 10, 11 and 13, for 16-, 64- and 256-QAM respectively.

Fig. 8 shows contour plots of the BER as a function of SNR and linewidth, for all simulated formats. The solid lines correspond to the VVMPE-ML, and the dotted ones to the BPS algorithm. Table I gives the linewidth tolerances for a 1 dB penalty at two different FEC limits: 1·10-3 (11% overhead) and 1.9·10-2 (28% overhead). VVMPE-ML outperforms BPS for 16-QAM, and manages to roughly equal the performance for 64-QAM, while being slightly worse for 256-QAM. The contour plots show that the performance of BPS deteriorates faster with increasing BER, implying that the linewidth tolerance of VVMPE-ML can benefit more from operation at lower SNRs, with a stronger FEC.

These results show that, when operating at the higher FEC limit, there is a clearer advantage to using VVMPE-ML for 16-QAM and 64-QAM, while the performance gap from BPS at 256-QAM is significantly smaller. Taking the worst case (256-QAM) and assuming 32 GBaud (including 28% overhead), the maximum allowed laser linewidth is 262 kHz with VVMPE-ML

(285 kHz with BPS), well within the reach of commercially available ECLs.

In terms of hardware complexity, and leaving the filtering operation aside for the moment, VVMPE-ML is similar to the 2SC scheme (the latter being one of the lowest-complexity schemes available according to Gao et al. [8]). The difference is in the first stage: An additional power operation on the magnitude of the received signal is employed, but the computational effort required for this is significantly less than that needed in the QPSK-partition and normalization operations of 2SC. We can conclude that when VVMPE-ML is used with block-averaging, the resulting algorithm is one of the simplest to date. Complexity of VVMPE-ML is higher when using FIR-approximated Wiener filtering, but is still much lower than block-averaged BPS. Although more work is needed to compare with other non-BPS schemes, one should also account for the fact that VVMPE-ML performs well for higher order formats, not just 16-QAM.

IV. COMBINED SIMULATION AND RESULTS In this section we evaluate the joint performance of the

two algorithms presented in this paper. The aim is to examine whether the two DSP subsystems can be efficiently put together and achieve performance equivalent to the standalone BER result presented in Section III. The motivation is that both schemes share the basic concept of the monomial transformations of the VVMPE. It follows that in a digital implementation they could potentially share common hardware blocks, thus significantly decreasing the cost, area and power consumption of the ASIC developed.

TABLE I. LINEWIDTH TOLERANCES OF THE VVMPE-ML AND BPS ALGORITHMS

Linewidth Tolerance

1dB penalty @ BER = 1·10-3 Normalized (1/Ts=28GBaud)

1dB penalty @ BER = 1.9·10-2 Normalized (1/Ts=32GBaud)

Constellation BPS VVMPE-ML BPS VVMPE-ML

16-QAM 1.3·10-4 (3.6 MHz) 1.7·10-4 (4.8 MHz) 2.0·10-4 (6.4 MHz) 2.9·10-4 (9.3 MHz)

64-QAM 2.8·10-5 (784 kHz) 2.4·10-5 (672 kHz) 4.5·10-5 (1.4 MHz) 4.9·10-5 (1.6 MHz)

256-QAM 7.8·10-6 (218 kHz) 3.6·10-6 (101 kHz) 8.9·10-6 (285 kHz) 8.2·10-6 (262 kHz)


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The simulation carried out was similar to that used for the standalone evaluation of the FOE and CPR algorithms. For a total number of 105 differentially-encoded 16-QAM symbols, the BER performance was assessed as a function of the combined laser linewidth normalized to the symbol rate ( sv TΔ ⋅ ), and the received SNR. The normalized frequency offset values added to the symbols ranged between ±0.1, as described in Section II. Fig. 9 shows the resulting BER contour plot. The solid contour lines correspond to the case where both algorithms are used in sequence (FOE first, followed by CPR). In order to benchmark this result, the corresponding contour plot of the standalone VVMPE-ML CPR simulation with no added frequency offset (Fig. 8 (a)), is included in the same plot.

Figure 9: Contour plot showing the BER's dependence on SNR and

linewidth for standalone VVMPE-ML (dashed line) without frequency offset, and combined VVMFOE + VVMPE-ML run (solid line) with

frequency offset, for 16QAM.

TABLE II.

LINEWIDTH TOLERANCE OF THE VVMFOE AND VVMPE-ML ALGORITHMS WHEN RUN TOGETHER FOR 16-QAM (AND COMPARISON

TO STANDALONE VVMPE-ML WITHOUT FREQUENCY OFFSET).

Linewidth Tolerance

Algorithm

1dB penalty @ BER = 1·10-3

Normalized (1/Ts=28GBaud)

1dB penalty @ BER = 1.9·10-2

Normalized (1/Ts=32GBaud)

VVMPE-ML 1.7·10-4 (4.8 MHz) 2.9·10-4 (9.3 MHz) VVMFOE

+ VVMPE-ML 1.5·10-4 (4.2 MHz) 2.7·10-4 (8.6 MHz)

V. CONCLUSION Two feed-forward algorithms for Frequency Offset

Estimation and Carrier Phase Recovery were presented. Both algorithms take advantage of the monomial-based Viterbi-Viterbi estimator, and can be implemented with relatively low computational complexity. Numerical simulations with square QAM constellations showed robust performance when each algorithm was run

independently, as well as when the two were run in sequence. In the case of the FOE algorithm, simulation of 104 symbols yielded NFErms of ~10-3 for 16-, 64-, and 256-QAM, for SNRs and linewidths which would be expected in a real coherent optical system. CPR with the proposed VVMPE-ML scheme achieves excellent linewidth tolerance for square QAM up to 256-QAM, making it a viable alternative to the computationally intensive BPS algorithm.

ACKNOWLEDGEMENTS This work was supported by the European

Commission through the FP7 projects ICT-SPIRIT and ICT-PANTHER.

REFERENCES

[1] A. Leven, N. Kaneda, U.-V. Koc and Y.-K. Chen, "Frequency Estimation in Intradyne Reception," IEEE Photonics Technology Letters, vol. 19, no. 6, pp. 366-368, March 2007.

[2] E. Ip and J. M. Kahn, "Feedforward Carrier Recovery for Coherent Optical Communications," Journal of Lightwave Technology, vol. 25, no. 9, pp. 2675-2692, September 2007.

[3] I. Fatadin and S. Savory, "Compensation of Frequency Offset for 16-QAM Optical Coherent Systems Using QPSK Partitioning," IEEE Photonics Technology Letters, vol. 23, no. 17, pp. 1246-1248, September 2011.

[4] I. Fatadin, D. Ives and S. Savory, "Laser Linewidth Tolerance for 16-QAM Coherent Optical Systems Using QPSK Partitioning," IEEE Photonics Technology Letters, vol. 22, no. 9, pp. 631-633, May 2010.

[5] S. Dris, I. Lazarou, P. Bakopoulos and H.Avramopoulos, "Frequency Offset Estimation in M-QAM Coherent," in Conference on Lasers and Electro-Optics 2012, OSA Technical Digest (online) (Optical Society of America, 2012), paper CF1F.2..

[6] J. Li, L. Li, Z. Tao, T. Hoshida and J. C. Rasmussen, "Laser-Linewidth-Tolerant Feed-Forward Carrier Phase Estimator With Reduced Complexity for QAM," Journal of Lightwave Technology, vol. 29, no. 16, pp. 2358-2364, August 2011.

[7] Y. Wang and E. Serpedin, "A Class of Blind Phase Recovery Techniques for Higher Order QAM Modulations: Estimators and Bounds," IEEE Signal Processing Letters, vol. 9, no. 10, pp. 301-304, October 2002.

[8] Y. Gao, A. Lao, C. Lu, J. Wu, Y. Li, K. Xu, W. Li and J. Li, "Low-Complexity Two-Stage Carrier Phase Estimation for 16-QAM Systems using QPSK Partitioning and Maximum Likelihood Detection," in Optical Fiber Communication Conference and Exposition (OFC/NFOEC), 2011 and the National Fiber Optic Engineers Conference, 2011.


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