## Maximum likelihood detection with beat noise estimation for minimizing bit error rate in OCDM-based system

Optics Express, Vol. 17, Issue 15, pp. 12433-12443 (2009)

http://dx.doi.org/10.1364/OE.17.012433

Acrobat PDF (561 KB)

### Abstract

We propose a maximum likelihood detection (MLD) technique that incorporates beat noise estimation (BNE). MLD can minimize a bit error rate theoretically because a bit pattern with the maximum posteriori probability is selected as the detected signals. Also, BNE can extract a specific beat noise from mixed multiple signals using a correlation. By combining these techniques, the influence of beat noise is reduced and the bit error rate becomes lower in an OCDM-based system. This paper describes the MLD algorithm and the BNE design. And numerical simulation results confirm the validity and performance of this technique.

© 2009 Optical Society of America

## 1. Introduction

9. W. Lee, H. Izadpanah, R. Menendez, S. Etemad, and P. J. Delfyett, “Synchronized Mode-Locked Semiconductor Lasers and Applications in Coherent Communications,” IEEE J. Lightwave Technol. **26**, 908–921 (2008).
[CrossRef]

11. A. Färbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J. -P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” European Conference and Exhibition on Optical Communication (ECOC) PD-Th. 4.1.5 (2004).

## 2. Maximum likelihood detection with beat noise estimation

### 2.1. MLD algorithm

*a*(

_{i}*t*) is the transmitted signal {0} or {1} with probability 1/2,

*A*is the electrical intensity of the carrier wave,

_{i}*c*is an orthogonal code assigned to the

_{ik}*i*th user’s

*k*th wavelength,

*f*is frequency,

_{ik}*ϕ*is the optical initial phase, and

_{ik}*M*is the number of wavelengths. In this case, an orthogonal code such as the Hadamard [5] code is employed. After demultiplexing the signals into each wavelength and direct detection, which denotes square-law detection at PDs, the received signals sk,rx can be expressed as shown by Eq. (2).

*N*is the number of users, the first term is signal information intensity, and

*b*in the second term is the beat noise as shown in Eq. (3).

_{i jk}*i*th and

*j*th users sharing the same

*k*th wavelength. And third term

*x*is thermal noise and denotes Gaussian noise generated during photo-electric conversion. The probability density function (PDF) of this thermal noise depends on the Gaussian function

_{k}*N*(0,

*σ*) as shown in Eq. (4).

*σ*is a standard deviation of the Gaussian distribution.

*a*can be expressed by Eq. (5).

_{i}*b*̂

_{i jk}is the estimated beat noise. And by removing the common term, Eq. (5) becomes Eq. (6).

*b*̂

_{i jk}equals the generated beat noise

*b*, the posteriori probability of the bit pattern has the maximum value. So, we apply BNE to MLD as our proposed technique.

_{i jk}*s*

_{k,rx}are the received signals, black and white circles (∙,∘) denote the signal points of each transmitted bit pattern such as (0, 0), and

*θ*is the beat noise phase when

*N*=2,

*M*=3. Because Eq. (6) means that a signal point with the minimum Euclidean distance from the received signal sk,rx is selected as a detected result, the threshold is set at a point that is equidistant from two signal points. If the beat noise phase is changed, that is to say the signal point (1, 1) is moved as in Fig. 4, the distance between the signal points is changed. In this way, the threshold is optimally set at a point equidistant from two signal points. The proposed method is superior to the hard decision technique in this respect, and it enables us to realize minimum BER.

### 2.2. BNE design

*b*̂

_{ijk}(

*t*) can be expressed

*τ*is the time constant of the LPF. It is assumed that

*τ*is sufficiently short for

*b*(

_{ijk}*t*) to remain unaltered during

*τ*, this means

*b*(

_{ijk}*t*)≒

*b*(

_{ijk}*t*+

*τ*), and is sufficiently longer than 1 bit time. Under the condition of

*τ, u*(

_{ij}*t*), which is a coefficient multiplied by the noise signals to estimate the specific beat noise

*b*(

_{ijk}*t*), is obtained by solving a simultaneous equation derived from the identity formula so that

*b*̂

_{ijk}≒

*b*.

_{ijk}*u*(

_{ij}*t*) for the condition

*b*̂

_{ijk}≒

*b*

_{ijk}. Here, there are three users on the OCDM system. In this case, transmitted signals have eight bit patterns as shown in Table 1. The received signals including the beat and Gaussian noises generated during photo-electric conversion and the coefficients for transmitted bit patterns are expressed as shown in Table 1, where

*θ*

_{1}=

*ϕ*

_{2}-

*ϕ*

_{1},

*θ*

_{2}=

*ϕ*

_{3}-

*ϕ*

_{1},

*θ*

_{3}=

*ϕ*

_{3}-

*ϕ*

_{2}used in Eq. (1),

*x*is Gaussian noise, and

_{k}*b*is the beat noise generated by the

_{ij}*i*th and

*j*th users sharing the same wavelength. We explain how to estimate the beat noise

*b*

_{12}. In this case, we classified the

*u*(

_{ij}*t*) by strength of correlation, for example, the coefficient of a bit pattern with a specific beat noise (P7) is α, another with a strong correlation with a specific beat noise (P8) is

*β*, and the others with a weak

*b*

_{12}and considering the correlation between each bit pattern, simultaneous Eqs. are derived as shown by Eq. (9).

*b*

_{12}below and pattern diagrams in Fig. 6.

*b*

_{12}. For estimating a specific beat noise

*b*

_{12}, the obtained coefficient is outputted: Fig. 6(b). As previously indicated, the coefficient of bit pattern P7, which has only a specific beat noise

*b*

_{12}is

*b*

_{12}is

*b*

_{12}: Fig. 6(d).

### 2.3. Optimum time constant of LPF

*τ*, which is also known as the cutoff frequency

*f*. This relation is defined as shown by Eq. (11).

_{c}*τ*is thought of as having an optimum value for the two reasons given below. One is that if the averaging time, which is defined as about double the

*τ*value, is short, the number of samples for bit patterns becomes small. And this increases the dispersion of the occurrence rate for each bit pattern. As a result, many estimation errors are generated during BNE. On the other hand, if the averaging time is long, the BNE output exhibits a time lag. In particular, there are many estimation errors when there is a large rate of change in the beat noise over a constant time. In addition, if

*τ*is longer, the cutoff frequency

*f*of the LPF decreases from Eq. (11). In this case, because the high frequency content of the beat noise is eliminated, the estimation errors increase. For example, the beat noise includes LD phase noise, so the cutoff frequency should be far higher than the spectrum of the phase noise for outputting the phase noise directly through BNE. Figure 7 is a diagram of the LPF frequency characteristics for

_{c}*τ*and the LD phase noise spectrum of as previously described. In this way, because the estimation errors become greater if

*τ*is too long or too short, it is expected that there is an optimum time constant

*τ*.

## 3. Simulation results

*τ*of the LPF. Figure 9 shows the time constant characteristic. In this Fig., the horizontal axis is the time constant

*τ*, and the vertical axis is the BER. Also there are three users, and the SNR is 18 dB. From the result, the time constant was at its optimum value, in this case

*τ*=2.78 ns, as previously described in Sec. 2.3, and this is confirmed.

*µ*m. The OSC parameter was 50 giga samples per second (50 GS/s) for an observation time of 10

*µ*s.

*τ*of the LPF is 2.78 ns. The dot-dashed line denotes the hard decision characteristic with and without beat noise, and the dashed line indicates the proposed method without estimate error, instead substituting the well-known beat noise as the estimated beat noise. The solid line shows the proposed method, and the diamonds (⋄) also show the proposed method when using empirical data as the beat noise value. In addition, the Hadamard codes, {1,1,0,0}, {1,0,1,0}, and {1,0,0,1}, were used. And here, the result using the length of the 3 codes is shown as another dashed line. When the number of users increased from two to three, and the number of wavelengths (the length of the codes) remained 3, an error floor was observed. In this case, the Hadamard codes, {1,1,0,0}, {1,0,1,0}, and {0,1,1,0}, were assigned. This is because a signal point for a given bit pattern overlapped another signal point when the beat noise phase was changed as described in Sec. 2.1. This confirms the need to select codes to avoid the signal points overlapping.

## 4. Conclusion

## References and links

1. | A. Stok and E. H. Sargent, “The Role of Optical CDMA in Access Networks,” IEEE Commun. Mag. |

2. | D. Zaccarin and M. Kavehrad, “An Optical CDMA System Based on Spectral Encoding of LED,” IEEE Photon. Technol. Lett. |

3. | V. J. Hernandez, W. Cong, J. Hu, C. Yang, N. K. Fontaine, R. P. Scott, Z. Ding, B. H. Kolner, J. P. Heritage, and S. J. B. Yoo, “A 320-Gb/s Capacity SPECTS O-CDMA Network TestbedWith Enhanced Spectral Efficiency Through Forward Error Correction,” IEEE J. Lightwave Technol. |

4. | S. Huang, K. Kitayama, K. Baba, and M. Murata, “Impact of MAI Noise Cycle Attack on OCDMA-based Optical Networks and its Diagnostic/Mitigation Algorithm,” in |

5. | S. Kaneko, H. Suzuki, N. Miki, H. Kimura, and M. Tsubokawa, “1-bit/s/Hz Spectral Efficiency OCDM Technique Based on Multi-frequency Homodyne Detection and Optical OFDM,” European Conference and Exhibition on Optical Communication (ECOC) |

6. | E. D. J. Smith, P. T. Gough, and D. P. Taylor, “Noise limits of optical spectral-encoding CDMA systems,” Electron. Lett. |

7. | L. Tanc̆evski and L. A. Rusch, “Impact of the Beat Noise on the Performance of 2-D Optical CDMA Systems,” IEEE Commun. Lett. |

8. | M. Fujiwara, J. Kani, H. Suzuki, and K. Iwatsuki, “Impact of Backreflection on Upstream Transmission in WDM Single-Fiber Loopback Access Networks,” IEEE J. Lightwave Technol. |

9. | W. Lee, H. Izadpanah, R. Menendez, S. Etemad, and P. J. Delfyett, “Synchronized Mode-Locked Semiconductor Lasers and Applications in Coherent Communications,” IEEE J. Lightwave Technol. |

10. | J. H. Winters and S. Kasturia, “Constrained Maximum-Likelihood Detection for High-Speed Fiber-Optic Systems,” in |

11. | A. Färbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J. -P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” European Conference and Exhibition on Optical Communication (ECOC) PD-Th. 4.1.5 (2004). |

12. | P. G. Hoel, |

**OCIS Codes**

(060.4250) Fiber optics and optical communications : Networks

(060.4510) Fiber optics and optical communications : Optical communications

(070.4550) Fourier optics and signal processing : Correlators

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: January 7, 2009

Revised Manuscript: June 4, 2009

Manuscript Accepted: June 7, 2009

Published: July 7, 2009

**Citation**

Takahito Kirihara, Noriki Miki, Shin Kaneko, Hideaki Kimura, and Kiyomi Kumozaki, "Maximum likelihood detection with beat
noise estimation for minimizing bit error
rate in OCDM-based system," Opt. Express **17**, 12433-12443 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12433

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### References

- A. Stok and E. H. Sargent, "The Role of Optical CDMA in Access Networks," IEEE Commun. Mag. 40, 83-87 (2002). [CrossRef]
- D. Zaccarin and M. Kavehrad, "An Optical CDMA System Based on Spectral Encoding of LED," IEEE Photon. Technol. Lett. 4, 479-482 (1993). [CrossRef]
- V. J. Hernandez, W. Cong, J. Hu, C. Yang, N. K. Fontaine, R. P. Scott, Z. Ding, B. H. Kolner, J. P. Heritage, and S. J. B. Yoo, "A 320-Gb/s Capacity SPECTS O-CDMA Network TestbedWith Enhanced Spectral Efficiency Through Forward Error Correction," IEEE J. Lightwave Technol. 25, 79-86 (2007). [CrossRef]
- S. Huang, K. Kitayama, K. Baba, and M. Murata, "Impact of MAI Noise Cycle Attack on OCDMA-based Optical Networks and its Diagnostic/Mitigation Algorithm," in Proceedings of IEEE Global Communications Conference (GLOBECOM) (IEEE, 2007) pp. 2412-2416.
- S. Kaneko, H. Suzuki, N. Miki, H. Kimura, and M. Tsubokawa, "1-bit/s/Hz Spectral Efficiency OCDM Technique Based on Multi-frequency Homodyne Detection and Optical OFDM," European Conference and Exhibition on Optical Communication (ECOC) 5, PS- P088, 203-204 (2007).
- E. D. J. Smith, P. T. Gough, and D. P. Taylor, "Noise limits of optical spectral-encoding CDMA systems," Electron. Lett. 31, 1469-1470 (1995). [CrossRef]
- L. Tancevski and L. A. Rusch, "Impact of the Beat Noise on the Performance of 2-D Optical CDMA Systems," IEEE Commun. Lett. 4, 264-266 (2000). [CrossRef]
- M. Fujiwara, J. Kani, H. Suzuki, and K. Iwatsuki, "Impact of Backreflection on Upstream Transmission in WDM Single-Fiber Loopback Access Networks," IEEE J. Lightwave Technol. 24, 740-746 (2006). [CrossRef]
- W. Lee, H. Izadpanah, R. Menendez, S. Etemad, and P. J. Delfyett, "Synchronized Mode-Locked Semiconductor Lasers and Applications in Coherent Communications," IEEE J. Lightwave Technol. 26, 908-921 (2008). [CrossRef]
- J. H. Winters and S. Kasturia, "Constrained Maximum-Likelihood Detection for High-Speed Fiber-Optic Systems," in Proceedings of IEEE Global Communications Conference (GLOBECOM) (IEEE, 1991) pp. 1574-1579.
- A. F¨arbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J. -P. Elbers, H. Wernz, H. Griesser, and C. Glingener, "Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation," European Conference and Exhibition on Optical Communication (ECOC) PD-Th. 4.1.5 (2004).
- P. G. Hoel, Elementary Statistics, 2nd edition (John Wiley & Sons Inc., New York, 1966)

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