## Novel configuration of finite-impulse-response filters tolerant to carrier-phase fluctuations in digital coherent optical receivers for higher-order quadrature amplitude modulation signals |

Optics Express, Vol. 20, Issue 24, pp. 26236-26251 (2012)

http://dx.doi.org/10.1364/OE.20.026236

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### Abstract

We propose a novel configuration of the finite-impulse-response (FIR) filter adapted by the phase-dependent decision-directed least-mean-square (DD-LMS) algorithm in digital coherent optical receivers. Since fast carrier-phase fluctuations are removed from the error signal which updates tap coefficients of the FIR filter, we can achieve stable adaptation of filter-tap coefficients for higher-order quadrature-amplitude modulation (QAM) signals. Computer simulations show that our proposed scheme is much more tolerant to the phase noise and the frequency offset than the conventional DD-LMS scheme. Such theoretical predictions are also validated experimentally by using a 10-Gsymbol/s dual-polarization 16-QAM signal.

© 2012 OSA

## 1. Introduction

1. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express **16**(2), 804–817 (2008). [CrossRef] [PubMed]

2. K. Kikuchi, “Clock recovering characteristics of adaptive finite-impulse-response filters in digital coherent optical receivers,” Opt. Express **19**(6), 5611–5619 (2011). [CrossRef] [PubMed]

3. D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. **28**(11), 1867–1875 (1980). [CrossRef]

5. P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. **28**(4), 547–556 (2010). [CrossRef]

6. K. Kikuchi, “Performance analyses of polarization demultiplexing based on constant-modulus algorithm in digital coherent optical receivers,” Opt. Express **19**(10), 9868–9880 (2011). [CrossRef] [PubMed]

7. S. U. H. Qureshi, “Adaptive equalization,” Proc. IEEE **73**(9), 1349–1387 (1985). [CrossRef]

## 2. Adaptation algorithms of FIR filters

*E*(

*n*), where

*n*denotes the number of the sampled sequence. The column vector

**E**(

*n*) input to the FIR filter is then defined aswhere

*M*is the filter order (

*i.e*., the number of taps is given as

*M*+ 1) and

*T*stands for the transpose of a matrix. The delay-tap coefficient vector

**p**(

*n*) of the FIR filter is given byIn this case, the FIR-filter output

*E'*(

*n*) can be calculated asThe tap vector of the FIR filter is adapted by the LMS algorithm [8] aswhere

*μ*is the step-size parameter and the superscript * denotes complex conjugation. The definition of the error signal

_{p}*e*(

_{p}*n*) depends on tap-adaptation algorithms.

*k*-fold oversampled data (

*k*: integer), we need to update tap coefficients every

*k*samples. In addition, the tap adaptation rate should be decreased much below the symbol rate when the feedback delay cannot be ignored in real digital circuits.

### 2.1 CMA, MMA, and phase-independent DD-LMS algorithm

*e*(

_{CMA}*n*) in CMA is given [3

3. D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. **28**(11), 1867–1875 (1980). [CrossRef]

*r*is the reference radius on the complex IQ plane. Since Eq. (5) does not include the phase error, the FIR filter never tracks phase fluctuations. As a result, we can individually achieve FIR-filter adaptation and phase estimation without any mutual coupling. Although CMA works well for constant-envelope signals such as phase-shift keying (PSK) signals, the demodulation performance is degraded for higher-order QAM signals accompanying amplitude-modulation (AM) components. Such degradation can be reduced by using MMA, in which the reference radius is determined by using radius-decision grids in a symbol-by-symbol manner. For example, the standard square 16-QAM signal has the three-level reference radius. The error signal

*e*(

_{MMA}*n*) in MMA is given [4, 5

5. P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. **28**(4), 547–556 (2010). [CrossRef]

*r*(

*n*) is the reference radius closest to |

*E'*(

*n*)| on the complex plane.

*e*(

_{PI-LMS}*n*) is given [5

5. P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. **28**(4), 547–556 (2010). [CrossRef]

*d*(

*n*) is the decoded signal; namely, |

*d*(

*n*)| stands for the reference radius. Since Euclidean distances among symbols is generally longer than the radius difference among them, the symbol-decision criterion provides better filtering performance compared with the radius-decision criterion used in CMA/MMA. Equation (7) also excludes the phase error.

### 2.2 Phase-dependent DD-LMS algorithm

*e*(

_{PD-LMS}*n*) is expressed [7

7. S. U. H. Qureshi, “Adaptive equalization,” Proc. IEEE **73**(9), 1349–1387 (1985). [CrossRef]

## 3. Novel FIR-filtering scheme adapted by the DD-LMS algorithm

### 3.1 Proposal of the novel FIR-filter configuration

*e*(

_{p}*n*) in the proposed configuration is then given bywhere

*d*(

*n*) represents either the desired signal in the training mode or the decoded signal in the tracking mode, and

*f*(

*n*) denotes a complex number obtained by the decision-directed phase estimator as shown below. Note that in the conventional DD-LMS schemes shown by Figs. 1 (c) and 1(d), the error signal given by Eq. (8) does not include

*f*(

*n*).

*μ*is the step-size parameter and

_{f}*e*(

_{f}*n*) the error signal controlling the tap coefficient. Using Eq. (10),

*f*(

*n*) is updated in a symbol-by-symbol manner so that

*e*(

_{f}*n*) approaches zero. The small positive offset

*ε*prevents the denominator of the second term of the right-hand side of Eq. (10) becoming zero for any input signal.

9. Y. Mori, C. Zhang, K. Igarashi, K. Katoh, and K. Kikuchi, “Unrepeated 200-km transmission of 40-Gbit/s 16-QAM signals using digital coherent receiver,” Opt. Express **17**(3), 1435–1441 (2009). [CrossRef] [PubMed]

*d*(

*n*){

*f*(

*n*)/|

*f*(

*n*)|}

^{−1}, and the phase estimator rotates the phase of the FIR-filter output by

*f*(

*n*)/|

*f*(

*n*)|; thus, the final output approaches

*d*(

*n*). In such a case, Eq. (9), where fast phase fluctuations are eliminated, gives us a proper error signal for the FIR filter. On the other hand, in the conventional scheme, the error signal given by Eq. (8) is not proper, because the FIR-filter output cannot approach

*d*(

*n*) due to fast phase fluctuations that the FIR filter cannot track. Thus, in our proposed scheme, the FIR filter only compensates for slowly time-varying impairments, whereas the phase estimator tracks fast phase fluctuations. Such interruption-free operation of the FIR filter and the phase estimator improves the filtering performance drastically.

*μ*′

*=*

_{p}*μ*|

_{p}*E*(

*n*)|

^{2}. On the other hand, Eqs. (10) and (11) yieldwhere the offset

*ε*in Eq. (10) is neglected. In Eqs. (12) and (13), the first terms of the right-hand sides represent the contribution of the current tap weight to the next one, which depends on the step-size parameters. On the other hand, the second terms stand for updating values of the tap coefficients. We find that

*p*(

*n*) of the FIR filter tracks the phase left after

*f*(

*n*) of the phase estimator does it, whereas

*f*(

*n*) tracks the phase left after

*p*(

*n*) does it. Even when the adaptation speed for

*p*(

*n*) of the FIR filter is slow owing to the small step-size parameter,

*f*(

*n*) of the phase estimator complements the phase-estimation process. Thus, the FIR filter and the phase estimator work for phase estimation in a mutually complementary manner.

### 3.2 Numerical analyses of the proposed configuration

*μ*= 1 and

_{p}/μ_{f}*M*+ 1 = 1 in left Figs. 3(a) and 3(d),

*μ*= 1 and

_{p}/μ_{f}*M*+ 1 = 4 in middle Figs. 3(b) and 3(e), and

*μ*= 1/4 and

_{p}/μ_{f}*M*+ 1 = 1 in right Figs. 3(c) and 3(f). The tracking speed of the FIR filter in middle and right figures is slower than that in left figures.

*φ*(

_{p}*n*) defined as the phase difference of the signal between the output and input ports of the FIR filter. The blue broken curves represent the unwrapped phase

*φ*(

_{f}*n*) defined as the phase difference of the signal between the output and input ports of the phase estimator. The green broken curves show the total phase

*φ*(

_{p}*n*) +

*φ*(

_{f}*n*). The actual value of the phase fluctuation is denoted as

*φ*(

_{n}*n*), and the black curves represent -

*φ*(

_{n}*n*).

*φ*(

_{p}*n*) =

*φ*(

_{f}*n*) and

*φ*(

_{p}*n*) +

*φ*(

_{f}*n*) = -

*φ*(

_{n}*n*). On the other hand, in Figs. 3(b), 3(e), 3(c), and 3(f), the tracking speed of the FIR filter is slower than that of the phase estimator. In such cases, the phase estimator mainly tracks the phase fluctuation, that is, |

*φ*(

_{p}*n*+ 1)-

*φ*(

_{p}*n*)|<<|

*φ*(

_{f}*n*+ 1)-

*φ*(

_{f}*n*)|; however, the total phase always tracks the actual phase fluctuation:

*φ*(

_{p}*n*) +

*φ*(

_{f}*n*) = -

*φ*(

_{n}*n*).

*α*aswhere the ratio of differential phases between symbols is averaged over

*N*symbols. Smaller

*α*values mean that the contribution of the FIR filter for phase tracking is less than that of the phase estimator.

*α*as a function of the number of taps

*M*+ 1. In Fig. 3(a), we include the 100-kHz laser linewidth and ignore the frequency offset, whereas in Fig. 4(b), the 10-MHz frequency offset is taken into account and the laser linewidth is neglected. Squares, dots, and plus marks correspond to

*μ*= 1, 1/4, and 1/16, respectively. As shown by solid curves,

_{p}/μ_{f}*α*for each

*μ*obeys the relation given byNoting that the tracking speed of the FIR filter is proportional to the step size parameter and the reciprocal of the number of delay taps [8], we find from Eq. (15) that the ratio of contributions of the FIR filter and the phase estimator is determined from the ratio of their effective tracking speeds. When the FIR filter cannot track the fast phase fluctuation owing to the small step-size parameter and/or the long delay tap, the one-tap phase estimator eliminates the phase fluctuation residual after the FIR filter. Then, we can achieve stable FIR filtering with the long delay tap even under fast phase fluctuations.

_{p}/μ_{f}### 3.3 FIR-filter configuration enabling frequency-offset cancelation

*s*(

*n*) of the second-stage phase estimator is updated by where

*μ*is the step-size parameter for the second-stage phase estimator,

_{s}*e*(

_{s}*n*) is the error signal, and

*ε*stands for the small positive offset. In this case, the error signal for the FIR filter given by Eq. (9) should be modified asThe output from the FIR filter is phase-controlled by

*f*(

*n*) and

*s*(

*n*), and we finally obtain the decoded symbol. With this configuration, the frequency-offset tolerance of the receiver is improved.

### 3.4 FIR-filter configuration for dual-polarization systems

*E*(

_{x,y}*n*) be the sampled complex amplitude from the

*x*- or

*y*-port of the phase- and polarization-diversity homodyne receiver. The column vector

**E**

*(*

_{x,y}*n*) given byis incident on the

*x*or

*y*input port of the butterfly-structured FIR filters. The delay-tap coefficient vector

**p**

*(*

_{k,l}*n*) of the butterfly-structured FIR filter, where

*k*and

*l*are either

*x*or

*y*, is given byEach tap vector is updated in accordance with

*μ*is the step-size parameter. The error signal

_{p}*e*(

_{px,y}*n*) is defined by

*d*(

_{x,y}*n*) denotes either the desired signal in the training mode or the decoded signal in the tracking mode. Complex numbers

*f*(

_{x,y}*n*) and

*s*(

_{x,y}*n*) are estimated by the first-stage and the second-stage phase estimator, respectively. The output of the butterfly-structured FIR filters is then given asThese outputs are phase-controlled by the dual-stage decision-directed phase estimators.

*f*(

_{x,y}*n*) is updated by where

*μ*is the step-size parameter,

_{f}*e*(

_{fx,y}*n*) the error signal controlling the tap coefficient of the first-stage phase estimator, and

*ε*the small positive offset. On the other hand, the tap coefficient of the second-stage phase estimator

*s*(

_{x,y}*n*) is updated in accordance with where

*μ*is the step-size parameter. Generally,

_{s}*μ*should be much smaller than

_{s}*μ*, because the laser frequency changes very slowly. The error signal

_{f}*e*(

_{sx,y}*n*) adapts the tap coefficient of the second-stage phase estimator.

*f*(

_{x,y}*n*) can be averaged, by using the carrier-phase correlation between the polarization tributaries, asUsing

*f*(

_{ave}*n*) instead of

*f*(

_{x,y}*n*) improves the tolerance against the phase noise and the frequency offset, because the signal-to-noise ratio (SNR) is enhanced by averaging the tap coefficients of both of the polarization tributaries. On the other hand, we should independently use the carrier phase estimated from each polarization tributary, when carrier phases have no correlation between the two polarization tributaries.

### 3.5 Advantage of our scheme for practical implementation

## 4. Simulations of the tolerance against the phase noise and the frequency offset in our proposed scheme

*δf*stands for the linewidth of lasers for the transmitter and LO. The state of polarization of the incoming signal and the frequency offset

*Δf*are assumed to be constant within the calculated symbol-sequence interval. Although the phase singularity of the received signal is avoided by the training sequence, differential encoding is employed to mitigate the impact of cycle slips in the phase-estimation process [11]. After the spectrum of the modulated signal is shaped by the Nyquist filter with the roll-off factor of 1, the signal is incident on the coherent receiver.

2. K. Kikuchi, “Clock recovering characteristics of adaptive finite-impulse-response filters in digital coherent optical receivers,” Opt. Express **19**(6), 5611–5619 (2011). [CrossRef] [PubMed]

*E*/

_{b}*N*

_{0,}when the proposed scheme is employed. Phase estimation is done individually for each polarization tributary. Squares, circles, and triangles denote BERs when FIR-filter orders

*M*are 4, 16, and 64, respectively. Red and green curves illustrate BERs when

*Δf*= 0 Hz, and

*δf*= 0 Hz and 100 kHz, respectively. Blue curves are BERs calculated when

*Δf*= 100 MHz and

*δf*= 100 kHz. For comparison, Figs. 7(b) and 7(c) depict BER characteristics when the conventional phase-dependent DD-LMS algorithm is used for filter-tap adaptation. Figure 7(b) corresponds to the scheme shown in Fig. 1(c), and Fig. 7(c) to that shown in Fig. 1(d).

*μ*in Eq. (24) is optimized in the range of 1/2

_{f}*(*

^{m}*m =*0, 1, 2,⋅⋅⋅, 6) so that each bit-error ratio (BER) is minimized. Since we ignore fluctuations of the state of polarization and the laser frequency,

*μ*in Eq. (21) and

_{p}*μ*in Eq. (26) are fixed to the smallest value of 1/2

_{s}^{6}. In the scheme shown in Fig. 1(c), the step-size parameter for FIR-filter adaptation is optimized in the range of 1/2

*. When the scheme shown in Fig. 1(d) is employed, we search for an optimum combination of the three step-size parameters used for adaptation of the FIR filter and the phase estimator based on the dual-stage phase rotator. In this case, the step-size parameter of the FIR filter is selected from 1/2*

^{m}*. On the other hand, the step-size parameter of the first-stage phase estimator takes any one of 1/2*

^{m}*or zero, and that of the second-stage phase estimator is set to 1/2*

^{m}^{6}.

^{−3}calculated as a function of the laser linewidth normalized to the symbol rate. As a reference, the laser linewidth in the 10-Gsymbol/s system is shown on the top axis. Red, green, and blue curves correspond to 4-QAM, 16-QAM, and 64-QAM formats, respectively, when

*M*= 64. Solid and broken curves respectively represent power penalties with and without using the phase correlation between the polarization tributaries.

^{−3}when the frequency offset is changed and the laser linewidth is ignored. The frequency offset normalized to the symbol rate is shown in the bottom axis, while that in the 10-Gsymbol/s system is shown on the top axis. Definitions of the curves are the same as those of Fig. 8.

^{−3}in 10-Gsymbol/s QAM systems. For example, when we employ the phase correlation between the polarization tributaries, acceptable values of the laser linewidth and the frequency offset can go up to 400 kHz and 400 MHz, respectively, in the 10-Gsymbol/s 16-QAM systems regardless of the FIR-filter order. Such values increase in proportion to the symbol rate. Since these power penalties are independent of the filter order, we find that these are determined only from the phase noise and the frequency offset.

## 5. Experiments

*M*are 4, 16, and 64. Green and blue curves represent BER characteristics when

*Δf*= 0 Hz and 100 MHz, respectively. For comparison, BER performances of the conventional DD-LMS schemes are illustrated in Figs. 11(b) and 11(c). Figure 11(b) is obtained by the scheme shown in Fig. 1(c)), whereas Fig. 11 (c) is measured in the scheme shown in Fig. 1 (d).

## 6. Conclusion

## Acknowledgments

## References and links

1. | S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express |

2. | K. Kikuchi, “Clock recovering characteristics of adaptive finite-impulse-response filters in digital coherent optical receivers,” Opt. Express |

3. | D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. |

4. | H. Louchet, K. Kuzmin, and A. Richter, “Improved DSP algorithms for coherent 16-QAM transmission,” in Technical Digest of |

5. | P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. |

6. | K. Kikuchi, “Performance analyses of polarization demultiplexing based on constant-modulus algorithm in digital coherent optical receivers,” Opt. Express |

7. | S. U. H. Qureshi, “Adaptive equalization,” Proc. IEEE |

8. | S. Haykin, |

9. | Y. Mori, C. Zhang, K. Igarashi, K. Katoh, and K. Kikuchi, “Unrepeated 200-km transmission of 40-Gbit/s 16-QAM signals using digital coherent receiver,” Opt. Express |

10. | T. Tsukamoto, Y. Ishikawa, and K. Kikuchi, “Optical homodyne receiver comprising phase and polarization diversities with digital signal processing,” in Technical Digest of |

11. | C. R. S. Fludger, D. Nuss, and T. Kupfer, “Cycle-slips in 100G DP-QPSK transmission systems,” in 2012 OSA Technical Digest of |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.2330) Fiber optics and optical communications : Fiber optics communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: September 13, 2012

Revised Manuscript: October 25, 2012

Manuscript Accepted: October 25, 2012

Published: November 6, 2012

**Citation**

Yojiro Mori, Chao Zhang, and Kazuro Kikuchi, "Novel configuration of finite-impulse-response filters tolerant to carrier-phase fluctuations in digital coherent optical receivers for higher-order quadrature amplitude modulation signals," Opt. Express **20**, 26236-26251 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26236

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### References

- S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express16(2), 804–817 (2008). [CrossRef] [PubMed]
- K. Kikuchi, “Clock recovering characteristics of adaptive finite-impulse-response filters in digital coherent optical receivers,” Opt. Express19(6), 5611–5619 (2011). [CrossRef] [PubMed]
- D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun.28(11), 1867–1875 (1980). [CrossRef]
- H. Louchet, K. Kuzmin, and A. Richter, “Improved DSP algorithms for coherent 16-QAM transmission,” in Technical Digest of European Conference on Optical Communication (ECOC 2008), Tu.1.E.6.
- P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol.28(4), 547–556 (2010). [CrossRef]
- K. Kikuchi, “Performance analyses of polarization demultiplexing based on constant-modulus algorithm in digital coherent optical receivers,” Opt. Express19(10), 9868–9880 (2011). [CrossRef] [PubMed]
- S. U. H. Qureshi, “Adaptive equalization,” Proc. IEEE73(9), 1349–1387 (1985). [CrossRef]
- S. Haykin, Adaptive Filter Theory (Prentice Hall, 2001).
- Y. Mori, C. Zhang, K. Igarashi, K. Katoh, and K. Kikuchi, “Unrepeated 200-km transmission of 40-Gbit/s 16-QAM signals using digital coherent receiver,” Opt. Express17(3), 1435–1441 (2009). [CrossRef] [PubMed]
- T. Tsukamoto, Y. Ishikawa, and K. Kikuchi, “Optical homodyne receiver comprising phase and polarization diversities with digital signal processing,” in Technical Digest of European Conference on Optical Communication (ECOC 2006), Th3.5.2.
- C. R. S. Fludger, D. Nuss, and T. Kupfer, “Cycle-slips in 100G DP-QPSK transmission systems,” in 2012 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2012), OTu2G.1.

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