## A complex-weighted, decision-aided, maximum-likelihood carrier phase and frequency-offset estimation algorithm for coherent optical detection |

Optics Express, Vol. 20, Issue 18, pp. 20102-20114 (2012)

http://dx.doi.org/10.1364/OE.20.020102

Acrobat PDF (1661 KB)

### Abstract

We present a symbol-by-symbol coherent optical receiver, which employs a novel, complex-weighted, decision-aided, maximum-likelihood (CW-DA-ML) carrier phase and frequency offset estimator. The CW-DA-ML carrier estimator uses a CW transversal filter to generate a carrier reference phasor, and the filter weights are automatically adapted on-line by linear regression on the observed signals. A complete modulo-*R* reduced frequency offset estimation (FOE) range of *R*/2 is achieved, independent of modulation format, where *R* is the symbol rate. Carrier phase and frequency tracking is achieved rapidly. The acquisition speed of frequency offset in quaternary phase-shift keying (4-PSK) signals is more than 5 times faster than that of differential FOE. A constant penalty of approximately 1 dB at bit-error rate of 10^{−4} is demonstrated for all frequency offsets in 4-PSK signals with laser-linewidth-symbol-duration product of 8^{−5}.

© 2012 OSA

## 1. Introduction

1. F. Derr, “Coherent optical QPSK intradyne system: concept and digital receiver realization,” J. Lightwave Technol. **10**(9), 1290–1296 (1992). [CrossRef]

*M*th power carrier estimation (CE) was proposed in [2

2. D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol. **24**(1), 12–21 (2006). [CrossRef]

*M*-ary phase-shift keying (MPSK) formats only and requires nonlinear operations which increase system latency. A computationally linear, decision-aided, maximum-likelihood (DA-ML) CE with comparable performance to block

*M*th power CE was derived in [3

3. P. Y. Kam, “Maximum likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Trans. Commun. **34**(6), 522–527 (1986). [CrossRef]

4. S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express **17**(2), 703–715 (2009). [CrossRef] [PubMed]

*f*, between LO and transmitter laser can be as large as

*M*th power and DA-ML CE were designed to estimate the carrier phase without considering any frequency offset. Block

*M*th power and DA-ML CE have limited frequency offset tolerance of 10

^{−3}times the symbol rate at laser-linewidth-symbol-duration product of 6.35

^{−5}[2

2. D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol. **24**(1), 12–21 (2006). [CrossRef]

*M*th power CE in [7

7. A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,” IEEE Photon. Technol. Lett. **19**(6), 366–368 (2007). [CrossRef]

*M*th power to remove data modulation. Consequently, it is only applicable to MPSK formats and the FOE range is limited to

*R*/2

*M*, where

*R*is the symbol rate. With higher order modulation formats, the FOE range shrinks while the complexity of raising to the

*M*th power increases. This differential FOE has an

*R*/

*M*frequency estimate ambiguity, requiring differential encoding (DE) which leads to a DE-induced performance penalty [8

8. W. J. Weber, “Differential encoding for multiple amplitude and phase shift keying systems,” IEEE Trans. Commun. **26**(3), 385–391 (1978). [CrossRef]

3. P. Y. Kam, “Maximum likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Trans. Commun. **34**(6), 522–527 (1986). [CrossRef]

4. S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express **17**(2), 703–715 (2009). [CrossRef] [PubMed]

*R*reduced frequency offset range of

*R*/2 can be estimated for arbitrary modulation formats. CW-DA-ML CE achieves near-ideal FOE with a low training overhead cost and no

*a priori*knowledge of the system statistics. Carrier phase and frequency are acquired rapidly. The FOE for 4-PSK signals is more than 5 times faster than that of the differential FOE of [7

7. A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,” IEEE Photon. Technol. Lett. **19**(6), 366–368 (2007). [CrossRef]

*T*, and

*H*denote conjugate, transpose, and complex conjugate transpose, respectively. All phase quantities are treated modulo-2

*π*to account for their circular nature and frequency quantities are treated modulo-

*R*, which is justified later in Subsection 4.3.

## 2. Phase and polarization diversity intradyne coherent receiver

*x-*and

*y*-polarized, in-phase,

*I*, and quadrature-phase,

*Q*, photocurrents by mixing

*R*which amounts to sampling the complex signal at 2

*R*, therefore satisfying the Nyquist theorem for fully preserving the optical signal and noise statistics.

*k*th symbol interval

*T*= symbol duration) are clock-synchronized with one complex sample per symbol. They are assumed to be free of intersymbol interference (ISI), polarization crosstalk, and nonlinear distortions. The

*k*th interval, respectively. Here,

1. F. Derr, “Coherent optical QPSK intradyne system: concept and digital receiver realization,” J. Lightwave Technol. **10**(9), 1290–1296 (1992). [CrossRef]

*k*th data symbol. The LO-ASE beat noise and LO shot noise are modeled as additive white Gaussian noise (AWGN), and are combined to form

*f*as seen in Eq. (2). Optical frequency bands around

## 3. Carrier estimation and data detection

### 3.1 Complex-weighted DA-ML carrier estimation

3. P. Y. Kam, “Maximum likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Trans. Commun. **34**(6), 522–527 (1986). [CrossRef]

4. S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express **17**(2), 703–715 (2009). [CrossRef] [PubMed]

*LT*, where

*L*is the estimator filter length. The RP

**34**(6), 522–527 (1986). [CrossRef]

**17**(2), 703–715 (2009). [CrossRef] [PubMed]

*L*received signals aswhere

*k*th symbol from DD and

*ω*is unknown in practice. Thus, we propose to choose the complex weights automatically and adaptively at each time

*k*based on the observations

11. P. Y. Kam, K. H. Chua, and X. Yu, “Adaptive symbol-by-symbol reception of MPSK on the Gaussian channel with unknown carrier phase characteristics,” IEEE Trans. Commun. **46**(10), 1275–1279 (1998). [CrossRef]

*L*-by-1 filter-weight vector and

*L*-by-1 filter-input vector at time

*k*. The error

*l*

*L*-by-

*L*time-average autocorrelation matrix and

*L*-by-1 time-average cross-correlation vector. The

*k*can be obtained recursively using the matrix inversion lemma [12]. Therefore, CW-DA-ML CE avoids any matrix inversion. The optimum

### 3.2 Symbol-by-symbol data detection

*S*, and rearranging them to yield

_{i}*N*known symbols is used to enable

## 4. Numerical results and discussion

*L*= 5 is maintained for all CE algorithms tested. The modulo-2

*π*reduced ∆

*ω*is assumed to have a probability density function given by

*p*(∆

*ω*) = 1/2

*π*for ∆

*ω*

*π*,

*π*), where ∆

*ω*is random but time invariant. The models of the carrier phase,

*ω*, are not known to the receiver.

### 4.1 Operation of CW-DA-ML CE

*N*= 50 and a laser linewidth of 2 MHz.

*l*∆

*ω*. Hence,

^{4}runs at frequency offsets of 0, 0.03, and 3 GHz are plotted for SNR values of 7, 10, and 13 dB in Fig. 3 . Ideal decision feedback is assumed, i.e.,

*k*. Using Eq. (2), the MSE

*∆f*,

*k*= 100 to

*k*= 200 the average of

^{4}runs at each time point

*k*. A

^{−2}demonstrates the good tracking of

*N*≈2

*L*, is sufficient to aid the complex weight

7. A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,” IEEE Photon. Technol. Lett. **19**(6), 366–368 (2007). [CrossRef]

**19**(6), 366–368 (2007). [CrossRef]

### 4.2 Bit-error rate performance with CW-DA-ML CE

*M*th power [2

2. D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol. **24**(1), 12–21 (2006). [CrossRef]

**34**(6), 522–527 (1986). [CrossRef]

**17**(2), 703–715 (2009). [CrossRef] [PubMed]

*N*= 10 known symbols is used for DA-ML and CW-DA-ML CE. DE according to [8

8. W. J. Weber, “Differential encoding for multiple amplitude and phase shift keying systems,” IEEE Trans. Commun. **26**(3), 385–391 (1978). [CrossRef]

*M*th power CE, and to counter runaways due to decision errors in DA-ML and CW-DA-ML CE.

*M*th power and DA-ML CE are intolerant to frequency offsets. For instance, in block

*M*th power CE of 4-PSK signals, as the frequency offset increases from 0 to 100 and 200 MHz, the

^{−4}. Moreover, block

*M*th power and DA-ML CE become increasingly sensitive to frequency offset with increasing modulation order. For a fixed frequency offset of 100 MHz, as modulation format increases from 4- to 8-PSK, block

*M*th power CE incurs an increasing

*M*th power and DA-ML CE originates from the violation of their assumption that

^{−4}. Regardless of ∆

*f*, BER of CW-DA-ML CE over all tested SNR values closely replicate that of DA-ML CE at zero frequency offset. Hence, CW-DA-ML CE approaches ideal FOE, without requiring any statistical knowledge of AWGN, carrier phase, or frequency offset.

*f*= 100 MHz with ideal decision feedback, i.e.,

*k*. In the range of SNR shown, the performance difference between actual and ideal decision feedback is not noticeable for 4-PSK, and is slightly more pronounced at the lower SNR region for 8-PSK. The effect of decision errors on the receiver performance becomes negligible for BER's below 10

^{−3}.

### 4.3 FOE range of CW-DA-ML CE

*M*th power CE [2

**24**(1), 12–21 (2006). [CrossRef]

**34**(6), 522–527 (1986). [CrossRef]

**17**(2), 703–715 (2009). [CrossRef] [PubMed]

**19**(6), 366–368 (2007). [CrossRef]

*R*/2, in the presence of different laser linewidths, ∆

*ν*. Any integer-multiple-of-

*R*frequency offset, ∆

*f*, translates into the same modulo-2

*π*reduced ∆

*ω*, thus having the same rotational effect on

*R*reduced frequency offset range

*R*/2. The differential FOE utilizes a subsequent block

*M*th power CE to correct any residual carrier phase, DE to overcome its

*R*/

*M*frequency estimate ambiguity, and a sample size of 500 to acquire the frequency offset, following that of [7

**19**(6), 366–368 (2007). [CrossRef]

*M*th power and differential FOE, which erase data modulation by raising the received signal to the

*M*th power, and extract the phase using the arctan(

*M*th power CE has a maximum allowable frequency offset of

**19**(6), 366–368 (2007). [CrossRef]

*R*reduced, modulation-format-independent, FOE range of

*R*/2 as witnessed in Fig. 5. This is attributed to the use of the RP,

*π*). The use of the RP also eliminates the need for phase unwrapping in carrier phase and frequency offset estimation. The circular-2

*π*nature of the RP results in frequency-estimate duplicity of

*R*, but this duplicity does not call for pilot assistance or DE to ensure correct DD. From Fig. 5(a), it is notable that a penalty of only approximately 1-dB is incurred at BER = 10

^{−4}for a 50 Gbit/s 4-PSK signal having a laser linewidth of 2 MHz, regardless of the frequency offset present.

### 4.4 Complexity and parallel implementation of CW-DA-ML CE

*M*th power CE) requires

*L*+ 1)/

*L*accesses to a read-only-memory to map arctan(

*L*factor arises from sharing of the common arctan(

*L*symbols during the block

*M*th power processing. The higher complexity of CW-DA-ML CE compared to differential FOE is traded off with CW-DA-ML CE’s wider and faster frequency estimation, and its applicability to arbitrary modulation formats. The update of

*k*, i.e., skip steps 4 to 8 in Table 1. Then, the complexity of CW-DA-ML CE becomes simply 4

*L*+ 6 and 4

*L*real multiplications and additions, respectively, per symbol.

## 5. Conclusions

*M*th power and decision-aided, maximum-likelihood carrier estimation (DA-ML CE) can cope with small frequency offsets. To combat larger frequency offsets, an FOE is necessary. Hence, we proposed a symbol-by-symbol receiver employing a novel complex-weighted (CW) DA-ML CE for optical channels with both carrier phase and frequency offset uncertainty. A near-ideal FOE over a complete modulo-

*R*reduced frequency range of

*R*/2 (

*R*= symbol rate) is demonstrated such that the bit-error rate performance is only limited by the carrier phase variance. CW-DA-ML CE is applicable to both

*M*-ary phase-shift keying (MPSK) and quadrature amplitude modulation formats. The FOE for 4-PSK signals is more than 5 times faster than that of the differential FOE at signal-to-noise ratio values above 7 dB.

## References and links

1. | F. Derr, “Coherent optical QPSK intradyne system: concept and digital receiver realization,” J. Lightwave Technol. |

2. | D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol. |

3. | P. Y. Kam, “Maximum likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Trans. Commun. |

4. | S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express |

5. | “Integrable Tunable Laser Assembly Multi Source Agreement,” OIF-ITLA-MSA-01.1 (Optical Internetworking Forum, Los Angeles, 2005). |

6. | A. Meiyappan, P. Y. Kam, and H. Kim, “Performance of decision-aided maximum-likelihood carrier phase estimation with frequency offset,” in Proc. OFC/NFOEC, Los Angeles, CA, 2012, paper OTu2G.6. |

7. | A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,” IEEE Photon. Technol. Lett. |

8. | W. J. Weber, “Differential encoding for multiple amplitude and phase shift keying systems,” IEEE Trans. Commun. |

9. | K.-P. Ho, |

10. | P. Y. Kam, S. S. Ng, and T. S. Ng, “Optimum symbol-by-symbol detection of uncoded digital data over the Gaussian channel with unknown carrier phase,” IEEE Trans. Commun. |

11. | P. Y. Kam, K. H. Chua, and X. Yu, “Adaptive symbol-by-symbol reception of MPSK on the Gaussian channel with unknown carrier phase characteristics,” IEEE Trans. Commun. |

12. | J. G. Proakis, |

13. | U. Mengali and A. N. D' Andrea, |

14. | H. Meyr, M. Moeneclaey, and S. Fechtel, |

15. | T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 29, 2012

Revised Manuscript: August 2, 2012

Manuscript Accepted: August 2, 2012

Published: August 17, 2012

**Citation**

Adaickalavan Meiyappan, Pooi-Yuen Kam, and Hoon Kim, "A complex-weighted, decision-aided, maximum-likelihood carrier phase and frequency-offset estimation algorithm for coherent optical detection," Opt. Express **20**, 20102-20114 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-20102

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### References

- F. Derr, “Coherent optical QPSK intradyne system: concept and digital receiver realization,” J. Lightwave Technol.10(9), 1290–1296 (1992). [CrossRef]
- D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol.24(1), 12–21 (2006). [CrossRef]
- P. Y. Kam, “Maximum likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Trans. Commun.34(6), 522–527 (1986). [CrossRef]
- S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express17(2), 703–715 (2009). [CrossRef] [PubMed]
- “Integrable Tunable Laser Assembly Multi Source Agreement,” OIF-ITLA-MSA-01.1 (Optical Internetworking Forum, Los Angeles, 2005).
- A. Meiyappan, P. Y. Kam, and H. Kim, “Performance of decision-aided maximum-likelihood carrier phase estimation with frequency offset,” in Proc. OFC/NFOEC, Los Angeles, CA, 2012, paper OTu2G.6.
- A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,” IEEE Photon. Technol. Lett.19(6), 366–368 (2007). [CrossRef]
- W. J. Weber, “Differential encoding for multiple amplitude and phase shift keying systems,” IEEE Trans. Commun.26(3), 385–391 (1978). [CrossRef]
- K.-P. Ho, Phase-modulated Optical Communication (Springer, 2005).
- P. Y. Kam, S. S. Ng, and T. S. Ng, “Optimum symbol-by-symbol detection of uncoded digital data over the Gaussian channel with unknown carrier phase,” IEEE Trans. Commun.42(8), 2543–2552 (1994). [CrossRef]
- P. Y. Kam, K. H. Chua, and X. Yu, “Adaptive symbol-by-symbol reception of MPSK on the Gaussian channel with unknown carrier phase characteristics,” IEEE Trans. Commun.46(10), 1275–1279 (1998). [CrossRef]
- J. G. Proakis, Digital Communications (McGraw-Hill, 2008).
- U. Mengali and A. N. D' Andrea, Synchronization Techniques for Digital Receivers (Plenum Press, 1997).
- H. Meyr, M. Moeneclaey, and S. Fechtel, Digital Communication Receivers (John Wiley, 1997).
- T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol.27(8), 989–999 (2009). [CrossRef]

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