## Evaluation of large girth LDPC codes for PMD compensation by turbo equalization

Optics Express, Vol. 16, Issue 17, pp. 13450-13455 (2008)

http://dx.doi.org/10.1364/OE.16.013450

Acrobat PDF (142 KB)

### Abstract

Large-girth quasi-cyclic LDPC codes have been experimentally evaluated for use in PMD compensation by turbo equalization for a 10 Gb/s NRZ optical transmission system, and observing one sample per bit. Net effective coding gain improvement for girth-10, rate 0.906 code of length 11936 over maximum a posteriori probability (MAP) detector for differential group delay of 125 ps is 6.25 dB at BER of 10^{-6}. Girth-10 LDPC code of rate 0.8 outperforms the girth-10 code of rate 0.906 by 2.75 dB, and provides the net effective coding gain improvement of 9 dB at the same BER. It is experimentally determined that girth-10 LDPC codes of length around 15000 approach channel capacity limit within 1.25 dB.

© 2008 Optical Society of America

## 1. Introduction

1. T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. **10**, 376–386 (2004). [CrossRef]

2. I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC coding based turbo equalization,” IEEE Photon. Technol. Lett. **19**, 1163–1165 (2007). [CrossRef]

3. B. Vasic, I. B. Djordjevic, and R. Kostuk, “Low-density parity check codes and iterative decoding for long-haul optical communication systems,” J. Lightwave Technol. **21**, 438–446 (2003). [CrossRef]

2. I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC coding based turbo equalization,” IEEE Photon. Technol. Lett. **19**, 1163–1165 (2007). [CrossRef]

*g*=10) quasi-cyclic (also known as array or block-circulant) LDPC codes. The length of the code is a limiting factor to the large girth as it has been shown in [5

5. M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory **50**, 1788–1793 (2004). [CrossRef]

5. M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory **50**, 1788–1793 (2004). [CrossRef]

## 2. Large girth quasi-cyclic LDPC codes

*I*is

*pxp*(

*p*is a prime number) identity matrix,

*D*is

*pxp*permutation matrix (

*d*

_{i,i+1}=

*d*

_{p,1}=1,

*i*=1,2,…,

*p*-1; other elements of

*D*are zeros), while

*r*and c represent the number of rows and columns in (1), respectively. The set of integers

*S*are to be carefully chosen from the set {0,1,…,

*p*-1} so that the cycles of short length, in the corresponding Tanner graph representation of (1), are avoided. We have shown in

**Error! Reference source not found.**that large girth,

*g*=10, LDPC codes provide excellent improvement in coding gain over corresponding turbo-product codes (TPCs). At the same time the complexity of LDPC codes is lower than that of TPCs, selecting them as excellent candidates for application to systems for beyond 40 Gb/s transmission. Namely, the minimum distance for an LDPC code is given by the Tanner bound [8

8. R. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inf. Theory **IT-27**, 533–547 (1981). [CrossRef]

*g*and

*w*are respectively the girth of the LDPC code and the column weight of the parity check matrix. The operator ⌊ ⌋ indicates the largest integer that is smaller or equal to the enclosed number. Equation (2) shows that the linear increase in the girth results in exponential increase of the minimum distance. Notice that this bound is tight only for short codes (in the order of hundreds), nevertheless it provides a guideline of how to design the LDPC codes of large minimum distance. For example, by selecting

*p*=1123 and

*S*={2, 5, 13, 20, 37, 58, 91, 135, 160, 220, 292, 354, 712, 830} an LDPC code of rate 0.8, girth

*g*=10, column weight 3 and length

*N*=16845 is obtained.

## 3. Experimental setup and PMD compensator

9. L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. **19**, 1852–1854 (2007). [CrossRef]

*2m*+1, meaning that that every bit

*x*from the transmitted sequence

_{i}**was affected by the preceding neighboring**

*X**m*(

*x*

_{i-m},

*x*

_{i-m+1},‥,

*x*

_{i-1}) bits as well as the following neighboring

*m*(

*x*

_{i+1},…,

*x*

_{i+m}) bits in the sequence. This subsequence (also known as bit-configuration) defines the state of the trellis as

**=(**

*s**x*

_{i-m},‥,

*x*,

_{i}*x*

_{i+1},…,

*x*

_{i+m}). The Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm based equalizer provides soft reliabilities for the channel output

**as it operates on the channel trellis. The channel is uniquely defined in any discrete moment in time in terms of the triple {previous state, channel output, next state}.**

*Y**s*that corresponds to the transmitted bit

*x*is defined as the conditional probability

_{i}*p(y*|

_{i}**), where**

*s**y*is the received sample corresponding to

_{i}*x*. Figure 2 illustrates the trellis concept. Figure 2(a) shows the trellis states for memory

_{i}*2m*+

*1*=

*5*and Fig. 2(b) shows the DGD effect on the conditional PDF distribution for two states:

**=‘11011’ and ‘00100’. With increase of the DGD the PDF mean for the first state shifts to the right and the curve becomes wider. This leads to degradation in BER performance, as shown later. Notice that the trellis description shown in Fig. 2(a) is description of the channel used by turbo equalizer. If the memory of the trellis is the same as the channel memory, turbo equalizer should be able completely to eliminate the channel distortions.**

*s*## 4. Experimental results

9. L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. **19**, 1852–1854 (2007). [CrossRef]

^{-4}and outperforms it at lower BER values. QC LDPC2 code of

*R*=0.906 outperforms the random code by 0.75 dB at BER of 10

^{-6}. QC LDPC3 code of

*R*=0.906 outperforms LDPC2 code by 1 dB at BER=10

^{-6}. QC LDPC4 code of

*R*=0.8 outperforms QC LDPC3 code of

*R*=0.906 by 2 dB at BER=10

^{-6}.

*R*=0.906 for DGD of 0ps, 50ps and125ps. The OSNR penalty for DGD of 125ps is 3dB at BER=10

^{-6}. Coding gain improvement over BCJR equalizer (with memory 2

*m*+1=5) for DGD=125ps is 6.25dB at BER=10

^{-6}. Larger coding gains are expected at lower BERs. Polynomial fit of 4

^{th}order was used to obtain the smoothened version of the measurement curve.

## 5. Conclusions

9. L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. **19**, 1852–1854 (2007). [CrossRef]

*R*=0.8 and girth 10. It provides the net effective coding gain improvement over, BCJR equalizer of 9dB at BER=10

^{-6}in the presence of 125ps of DGD. All tested codes were within 1.25dB away from capacity limit. Notice that girth-10 LDPC codes do not exhibit error floor phenomena down to 10

^{-15}[11]. That allows extrapolation of results down to BER of 10

^{-15}. Such an extrapolation indicates that the net effective coding gain improvement over BCJR equalizer for LDPC4 code is about 12.5 dB at BER of 10

^{-15}. Previous simulation results [2

2. I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC coding based turbo equalization,” IEEE Photon. Technol. Lett. **19**, 1163–1165 (2007). [CrossRef]

## References and links

1. | T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. |

2. | I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC coding based turbo equalization,” IEEE Photon. Technol. Lett. |

3. | B. Vasic, I. B. Djordjevic, and R. Kostuk, “Low-density parity check codes and iterative decoding for long-haul optical communication systems,” J. Lightwave Technol. |

4. | I. B. Djordjevic, S. Sankaranarayanan, S. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40-Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. |

5. | M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory |

6. | O. Milenkovic, I. B. Djordjevic, and B. Vasic, “Block-circulant low-density parity-check codes for optical communication systems,” IEEE J. Sel. Top. Quantum Electron. |

7. | I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. JWA53. |

8. | R. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inf. Theory |

9. | L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. |

10. | H. Xiao-Yu et al., “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE Globecom |

11. | Y. Miyata, W. Matsumoto, H. Yoshida, and T. Mizuochi, “Efficient FEC for optical communications using concatenated codes to combat error-floor,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. OTuE4. |

12. | M. Arabaci, I. B. Djordjevic, and R. Saunders, “FPGA implementation of LDPC decoders for large girth LDPC codes,” in preparation. |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.2300) Fiber optics and optical communications : Fiber measurements

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 17, 2008

Revised Manuscript: August 8, 2008

Manuscript Accepted: August 11, 2008

Published: August 15, 2008

**Citation**

Lyubomir L. Minkov, Ivan B. Djordjevic, Lei Xu, Ting Wang, and Franko Kueppers, "Evaluation of large girth LDPC codes for PMD compensation by turbo equalization," Opt. Express **16**, 13450-13455 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13450

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### References

- T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004). [CrossRef]
- I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, "PMD compensation by LDPC coding based turbo equalization," IEEE Photon. Technol. Lett. 19, 1163-1165 (2007). [CrossRef]
- B. Vasic, I. B. Djordjevic, and R. Kostuk, "Low-density parity check codes and iterative decoding for long-haul optical communication systems," J. Lightwave Technol. 21, 438-446 (2003). [CrossRef]
- I. B. Djordjevic S. Sankaranarayanan, S. Chilappagari, and B. Vasic, "Low-density parity-check codes for 40-Gb/s optical transmission systems," IEEE J. Sel. Top. Quantum Electron. 12, 555-562 (2006). [CrossRef]
- M. P. C. Fossorier, "Quasi-cyclic low-density parity-check codes from circulant permutation matrices," IEEE Trans. Inf. Theory 50, 1788-1793 (2004). [CrossRef]
- O. Milenkovic, I. B. Djordjevic, and B. Vasic, "Block-circulant low-density parity-check codes for optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 294-299 (2004). [CrossRef]
- I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, "Large girth low-density parity-check codes for long-haul high-speed optical communications," in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24-28 2008, Paper no. JWA53.
- R. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inf. Theory IT-27, 533-547 (1981). [CrossRef]
- L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, "Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization," IEEE Photon. Technol. Lett. 19, 1852-1854 (2007). [CrossRef]
- H. Xiao-Yu et al., "Efficient implementations of the sum-product algorithm for decoding of LDPC codes," in Proc. IEEE Globecom 2, 1036-1036E (2001).
- Y. Miyata, W. Matsumoto, H. Yoshida, and T. Mizuochi, "Efficient FEC for optical communications using concatenated codes to combat error-floor," in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24-28 2008, Paper no. OTuE4.
- M. Arabaci, I. B. Djordjevic, and R. Saunders, "FPGA implementation of LDPC decoders for large girth LDPC codes," in preparation.

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