## Generalized OFDM (GOFDM) for ultra-high-speed optical transmission |

Optics Express, Vol. 19, Issue 7, pp. 6969-6979 (2011)

http://dx.doi.org/10.1364/OE.19.006969

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### Abstract

We propose a coded *N*-dimensional modulation scheme suitable for ultra-high-speed serial optical transport. The proposed scheme can be considered as a generalization of OFDM, and hence, we call it as generalized OFDM (GOFDM). In this scheme, the orthogonal subcarriers are used as basis functions and the signal constellation points are defined over this *N*-dimensional linear space. To facilitate implementation, we propose using *N*-dimensional pulse-amplitude modulation (ND-PAM) as the signal constellation diagram, which is obtained as the *N*-ary Cartesian product of one-dimensional PAM. In conventional OFDM, QAM/PSK signal constellation points are transmitted over orthogonal subcarriers and then they are multiplexed together in an OFDM stream. Individual subcarriers, therefore, carry *N* parallel QAM/PSK streams. In the proposed GOFDM scheme instead, an *N*-dimensional signal constellation point is transmitted over all *N* subcarriers simultaneously. When some of the subcarriers are severely affected by channel impairments, the constellation points carried by those subcarriers may be lost in the conventional OFDM. In comparison, under such conditions, the overall signal constellation point will face only small distortion in GOFDM and it can be recovered successfully using the information on the other high fidelity subcarriers. Furthermore, because the channel capacity is a logarithmic function of signal-to-noise ratio but a linear function of the number of dimensions, the spectral efficiency of optical transmission systems can be improved with GOFDM.

© 2011 OSA

## 1. Introduction

*N*-dimensional pulse-amplitude modulation (PAM), and hence, it can also be called as

*N*D-PAM. In GOFDM scheme,

*N*orthogonal subcarriers are used as basis functions to define an

*N*-dimensional linear function space. Although any

*N*-dimensional signal constellation can be used in the proposed scheme, we choose to employ

*N*D-PAM signal constellation, which is obtained by the

*N*-ary Cartesian product of one-dimensional (1D) PAM. The benefit we gain from using

*N*D-PAM as a signal constellation diagram instead of using a set of constellation points randomly chosen over the

*N*-dimensional space is the simplicity in implementation due to the ease of generating 1D-PAM and its

*N*-ary Cartesian product, i.e.,

*N*D-PAM. In conventional OFDM, QAM/PSK signal constellation points are transmitted over orthogonal subcarriers and then multiplexed together in an OFDM stream. Individual subcarriers therefore carry

*N*parallel QAM/PSK streams. Thus, when some of the subcarriers are severely degraded, the information carrier over those subcarriers will most probably be unrecoverable. In contrast, in GOFDM, an

*N*-dimensional signal constellation point is simultaneously transmitted over all

*N*subcarriers, which serve as the individual basis functions as we mentioned above. Even if some of the subcarriers are severely affected by channel distortion, the

*N*-dimensional signal constellation point will face only a partial distortion, which can be further reduced by using strong, channel-capacity-achieving channel codes. Furthermore, because channel capacity is a logarithmic function of signal-to-noise ratio (SNR) but a linear function of the number of dimensions (

*N*), the spectral efficiency of optical transmission systems can be dramatically improved by proposed

*N*-dimensional GOFDM scheme. Notice, however, that the complexity of symbol log-likelihood ratio (LLR) calculation operation increases with the number of dimensions, and it is clear that, in practice, three to seven dimensions should be used. As a workaround, we also propose a frequency-interleaved scheme that meticulously combines subsystems with reasonable number of (three to five) dimensions into a system with multi-Tb/s serial aggregate data rate.

## 2. Description of proposed GOFDM system

*N*D-PAM signal constellation is obtained by the

*N*-ary Cartesian product of the 1D-PAM signal constellation. The 1D-PAM signal constellation is described by the following set of signal amplitudes

*d*is the Euclidean distance between two neighboring signal amplitudes. The

*N*D-PAM signal constellation is therefore obtained by

*N*D-PAM constellation is given by

*M*-QAM-OFDM and GOFDM) the spectral efficiency for GOFDM is

*N*>2) times larger, which will become evident in Sec. 5, were several illustrative examples are given . In order to make a clear distinction between several

*N*D-PAM schemes with the same number of dimensions

*N*, we use

*L*-

^{N}*N*D-PAM notation, which unambiguously defines the signal constellation as the one with

*N*dimensions and with

*L*1D-PAM constellation points per dimension. For example, for

^{3}-3D-PAM. Notice that square-QAM can also be described as 2D Cartesian product of PAM.

*L*-

^{N}*N*D-PAM signal constellation is depicted in Fig. 1(a) . As seen in the figure,

*b*independent binary data streams are encoded in parallel using

*b*identical binary

*n*and

*k*denote the codeword length and the information word length, respectively. (We should note here that different LDPC codes can be used to protect information streams on each one of the

*b*branches. Here, without loss of generality, we used identical LDPC code to simplify the notation.) The codewords are written row-wise into a

*N*D-mapper reads a

*b*-bit symbol from the bit interleaver column-wise and outputs the constellation point corresponding to that symbol. The

*N*D-mapper is implemented as a look-up table (LUT) with

*b*input bits serving as a memory address that select the

*N*-coordinates of the corresponding ND-PAM signal constellation point. For example, the LUT for

^{3}-3D-PAM) is shown in Table 1 . Upon mapping, the inverse fast Fourier transform (IFFT) is applied to perform modulation. We should stress that not all the subcarriers need to be used for modulation. In fact, we propose using

*N*out of

*N*

_{sc}available subcarriers

*b*binary LDPC encoders and the interleaver (see shaded block in Fig. 1(a)) are replaced by a single nonbinary LDPC encoder that outputs

*b*-bit symbols ready to be mapped to constellation points. Thus, the need for the bit-to-symbol interface is avoided. Starting with the

*N*D-mapper block the same configuration applies to both binary- and nonbinary-LDPC-coded transmission scenarios.

*N*outputs of the FFT block, corresponding to

*N*-dimensions of the

*N*D-PAM signal constellation, are used as inputs to an

*a posteriori*probability (APP) demapper.

**r**denotes the received

*N*-tuple at a given symbol interval,

*N*-tuple (i.e.,

*N*coordinates of the constellation point) at the output of the

*N*D-mapper corresponding to the

*b*-bit symbol

*b*-bit all-zero symbol. The term

*a priori*probability of the symbol

*j*th binary LDPC decoder as the bit LLR estimate at the current symbol interval. As shown in Fig. 1, the bit LLRs are forwarded to LDPC decoders, which enhance the bit LLR estimates through decoding, and feed the extrinsic bit LLRs back to the demapper for the demapper to use them as prior estimates on

*b*binary LDPC decoders (see shaded block in Fig. 1(b)) and the bit LLR calculation block are replaced by a single nonbinary LDPC decoder (depicted below the shaded block in Fig. 1(b)). Moreover, the use of a nonbinary LDPC decoder eliminates the need for the iterative information exchange between the dempaping and the decoding blocks, which in addition to decreasing computational complexity, reduces the latency. Finally, nonbinary LDPC-coded modulation schemes provide much larger coding gains compared to their corresponding bit-interleaved LDPC-coded modulation schemes, as we have shown in [10

10. M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “Non-binary quasi-cyclic LDPC based coded modulation for beyond 100 Gb/s transmission,” IEEE Photon. Technol. Lett. **22**(6), 434–436 (2010). [CrossRef]

11. M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “Polarization-multiplexed rate-adaptive non-binary-quasi-cyclic-LDPC-coded multilevel modulation with coherent detection for optical transport networks,” Opt. Express **18**(3), 1820–1832 (2010). [CrossRef] [PubMed]

*rbR*

_{s}bits per second, where

*R*

_{s}is the symbol rate. For example, by setting

*N*-dimensional signal constellations while in OFDM only 2D signal constellations are used. Since, for the same symbol energy, the Euclidean distance between signal constellation points is much larger in

*N*-dimensional space

## 3. Description of frequency-interleaved GOFDM

*N*. To keep the complexity of the APP demaper reasonably low, we can employ the following approach. We first split the total number of subcarriers

*N*subgroups of

*N*subcarriers. Next, the

*k*th group of subcarriers

*N*-dimensional signal constellation is formed by taking the

*k*th element of all subgroups. Finally, we perform encoding, modulation, transmission, demodulation, decoding on all groups as shown in Fig. 1. In such a way, if several subcarriers (coordinates) are affected by channel distortion, they will belong to different constellation points. Hence, via frequency interleaving/deinterleaving the immunity of GOFDM to channel distortion will be even more enhanced compared to the conventional OFDM. By using sufficiently high dimensionality of signal constellations

## 4. Binary and nonbinary quasi-cyclic LDPC codes

6. I. B. Djordjevic, M. Arabaci, and L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightwave Technol. **27**(16), 3518–3530 (2009). [CrossRef]

12. M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory **50**(8), 1788–1793 (2004). [CrossRef]

**I**is the

*B*is a prime number),

*r*and

*c*represent the number of block-rows and block-columns in Eq. (6), respectively. (A (

*w*

_{c},

*w*)-

_{r}*regular*LDPC(

*n*,

*k*) code is a linear block code whose parity-check matrix

**H**contains exactly

*w*

_{c}1s in each column and exactly

6. I. B. Djordjevic, M. Arabaci, and L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightwave Technol. **27**(16), 3518–3530 (2009). [CrossRef]

*S*should be carefully chosen from the set

*n*,

*k*) code is drawn according to the following rule: check node

*c*is connected to variable node

*v*whenever the element

*h*in the parity-check matrix

_{cv}*q*), i.e., the Galois field of

*q*elements, can be obtained by properly assigning nonzero elements from GF(

*q*) to the 1s in the parity-check matrix of the corresponding binary QC-LDPC code, as we described in [13

13. M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “High-rate non-binary regular quasi-cyclic LDPC codes for optical communications,” J. Lightwave Technol. **27**(23), 5261–5267 (2009). [CrossRef]

*q*-ary LDPC codes and was given the name

*q*-ary SPA (QSPA) by Davey and MacKay [14]. They also proposed an efficient implementation of QSPA based on the fast Fourier transform (FFT), which we refer to as FFT-QSPA. FFT-QSPA is particularly efficient over the nonbinary fields whose order is a power of 2 since the complex arithmetic due to FFT can be avoided. In this paper, we use nonbinary LDPC codes over the extension fields of the binary field, i.e.

*q*= 2

*, for some positive integer*

^{b}*b*, in order to benefit from this nice property. Further details on FFT-QSPA and a reduced complexity variant of it can be found in [10

10. M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “Non-binary quasi-cyclic LDPC based coded modulation for beyond 100 Gb/s transmission,” IEEE Photon. Technol. Lett. **22**(6), 434–436 (2010). [CrossRef]

11. M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “Polarization-multiplexed rate-adaptive non-binary-quasi-cyclic-LDPC-coded multilevel modulation with coherent detection for optical transport networks,” Opt. Express **18**(3), 1820–1832 (2010). [CrossRef] [PubMed]

## 5. Performance analysis

*L*-ary 1D-PAM:where erfc(⋅) is the complementary error function. The uncoded symbol error probability of

*N*D-PAM is thenThe average symbol energy

*d*is the Euclidean distance between two neighboring signal amplitudes (the subscript

_{N}*N*is used to denote the number of dimensions). The corresponding expression for

*M*-QAM is given by

*b*= log

_{2}

*M*= log

_{2}(

*L*)] and the same symbol rate we can establish the following connection between

^{N}*N*D-PAM Euclidean distance and that of QAM:

*N*D-PAM Euclidean distance squared increases exponentially with number of dimensions (for

*N*>2) compared to

*E*

_{b}as follows

*P*

_{s}can be evaluated against bit energy-to-power spectral density ratio

*E*

_{b}/

*N*

_{0}. In Fig. 2(a) , we provide symbol error probabilities obtained by Eq. (11) and compare them against those obtained via Monte Carlo simulations. An excellent agreement is observed. We can see also that an increase in the number of dimensions results in small performance degradation as long as the orhtogonality among subcarriers is maintained. In order to be consistent with existent optical and digital communication literature [18

18. M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express **17**(13), 10814–10819 (2009). [CrossRef] [PubMed]

*L*results in negligible BER performance degradation. The comparison of corresponding curves with

^{3}-3D-PAM-based-GOFDM outperforms corresponding 64-QAM-OFDM with an impressive 4.28 dB additional coding gain at the BER of 10

^{−8}. The polarization-multiplexed (PolMUX) 4

^{4}-3D-PAM performs just slightly worse than PolMux 16-QAM-OFDM, but provides an aggregate information bit rate of

^{3}-3D-PAM, the nonbinary coded modulation scheme outperforms the binary scheme by 0.87 dB. On the other hand, for GOFDM based on 8

^{3}-3D-PAM, the nonbinary coded modulation scheme outperforms the binary scheme by even a larger margin of 1.29 dB. What is also interesting to notice from Fig. 3 is that the proposed

*N*D-PAM signal constellation performs close to the optimum signal constellation based on sphere-packing method [15,16

16. N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, “Minimal-energy clusters of hard spheres,” Discrete Comput. Geom. **14**(1), 237–259 (1995). [CrossRef]

*N*D-PAM is much easier to implement than the optimum signal constellation, which is comprised of real number amplitudes up to 12 digits after the decimal point [16

16. N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, “Minimal-energy clusters of hard spheres,” Discrete Comput. Geom. **14**(1), 237–259 (1995). [CrossRef]

## 6. Conclusion

*N*-dimensional Cartesian product of PAM, called ND-PAM, as the signal constellation. In this scheme, the orthogonal subcarriers are used as bases functions. Because in proposed GOFDM scheme the

*N*-dimensional signal constellation point is transmitted over all

*N*subcarriers, even if during long-haul transmission some of the subcarriers are severely affected by channel distortion, the overall signal constellation point will face only small distortion. In addition, because the channel capacity is a linear function of number of dimensions, the spectral efficiency of optical transmission systems can be improved with GOFDM. This scheme is the next generations, both 400 Gb/s and 1 Tb/s, Ethernet enabling technology. The binary LDPC-coded GOFDM scheme significantly (>4dB) outperforms corresponding conventional QAM-OFDM counterpart. The use of nonbinary LDPC codes provides the additional improvement of 1.29 dB for GOFDM based on 8

^{3}-3D-PAM. Notice that this paper was concerned with theoretical OSNR limits and coding gain improvements for proposed scheme, to clearly identify high potentials of proposed GOFDM scheme in terms of OSNR sensitivity and spectral efficiency.

## Acknowledgments

## References and links

1. | J. Hong, T. Schmidt, M. Traverso, and E. Yoshikazu, “40G and 100G modules enable next generation networks,” Proc. SPIE |

2. | W. Shieh, and I. Djordjevic, |

3. | Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access,” Opt. Express |

4. | Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. |

5. | J. McDonough, “Moving standards to 100 GbE and beyond,” IEEE Commun. Mag. |

6. | I. B. Djordjevic, M. Arabaci, and L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightwave Technol. |

7. | H. G. Batshon, I. B. Djordjevic, L. Xu, and T. Wang, “Modified hybrid subcarrier/amplitude/ phase/polarization LDPC-coded modulation for 400 Gb/s optical transmission and beyond,” Opt. Express |

8. | H. G. Batshon, I. B. Djordjevic, and T. Schmidt, “Ultra high speed optical transmission using subcarrier-multiplexed four-dimensional LDPC-coded modulation,” Opt. Express |

9. | H. G. Batshon, I. B. Djordjevic, L. Xu, and T. Wang, “Multidimensional LDPC-coded modulation for beyond 400 Gb/s per wavelength transmission,” IEEE Photon. Technol. Lett. |

10. | M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “Non-binary quasi-cyclic LDPC based coded modulation for beyond 100 Gb/s transmission,” IEEE Photon. Technol. Lett. |

11. | M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “Polarization-multiplexed rate-adaptive non-binary-quasi-cyclic-LDPC-coded multilevel modulation with coherent detection for optical transport networks,” Opt. Express |

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

13. | M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “High-rate non-binary regular quasi-cyclic LDPC codes for optical communications,” J. Lightwave Technol. |

14. | M. C. Davey, |

15. | T. M. Cover, and J. A. Thomas, |

16. | N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, “Minimal-energy clusters of hard spheres,” Discrete Comput. Geom. |

17. | I. B. Djordjevic, L. Xu, and T. Wang, “Coded multidimensional pulse amplitude modulation for ultra-high-speed optical transmission,” in Proc. |

18. | M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express |

19. | J. G. Proakis, |

**OCIS Codes**

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

(060.1660) Fiber optics and optical communications : Coherent communications

(060.4080) Fiber optics and optical communications : Modulation

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: February 17, 2011

Revised Manuscript: March 19, 2011

Manuscript Accepted: March 20, 2011

Published: March 25, 2011

**Citation**

Ivan Djordjevic, Murat Arabaci, Lei Xu, and Ting Wang, "Generalized OFDM (GOFDM) for ultra-high-speed optical transmission," Opt. Express **19**, 6969-6979 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6969

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### References

- J. Hong, T. Schmidt, M. Traverso, and E. Yoshikazu, “40G and 100G modules enable next generation networks,” Proc. SPIE 7631, 763115, 763115-7 (2009). [CrossRef]
- W. Shieh, and I. Djordjevic, OFDM for Optical Communications (Elsevier/Academic Press, 2009).
- Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access,” Opt. Express 17(11), 9421–9427 (2009). [CrossRef] [PubMed]
- Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. 27(16), 3511–3517 (2009). [CrossRef]
- J. McDonough, “Moving standards to 100 GbE and beyond,” IEEE Commun. Mag. 45(11), 6–9 (2007). [CrossRef]
- I. B. Djordjevic, M. Arabaci, and L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightwave Technol. 27(16), 3518–3530 (2009). [CrossRef]
- H. G. Batshon, I. B. Djordjevic, L. Xu, and T. Wang, “Modified hybrid subcarrier/amplitude/ phase/polarization LDPC-coded modulation for 400 Gb/s optical transmission and beyond,” Opt. Express 18(13), 14108–14113 (2010). [CrossRef] [PubMed]
- H. G. Batshon, I. B. Djordjevic, and T. Schmidt, “Ultra high speed optical transmission using subcarrier-multiplexed four-dimensional LDPC-coded modulation,” Opt. Express 18(19), 20546–20551 (2010). [CrossRef] [PubMed]
- H. G. Batshon, I. B. Djordjevic, L. Xu, and T. Wang, “Multidimensional LDPC-coded modulation for beyond 400 Gb/s per wavelength transmission,” IEEE Photon. Technol. Lett. 21(16), 1139–1141 (2009). [CrossRef]
- M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “Non-binary quasi-cyclic LDPC based coded modulation for beyond 100 Gb/s transmission,” IEEE Photon. Technol. Lett. 22(6), 434–436 (2010). [CrossRef]
- M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “Polarization-multiplexed rate-adaptive non-binary-quasi-cyclic-LDPC-coded multilevel modulation with coherent detection for optical transport networks,” Opt. Express 18(3), 1820–1832 (2010). [CrossRef] [PubMed]
- M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory 50(8), 1788–1793 (2004). [CrossRef]
- M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “High-rate non-binary regular quasi-cyclic LDPC codes for optical communications,” J. Lightwave Technol. 27(23), 5261–5267 (2009). [CrossRef]
- M. C. Davey, Error-Correction using Low-Density Parity-Check Codes, Ph.D. dissertation, (University of Cambridge, 1999).
- T. M. Cover, and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
- N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, “Minimal-energy clusters of hard spheres,” Discrete Comput. Geom. 14(1), 237–259 (1995). [CrossRef]
- I. B. Djordjevic, L. Xu, and T. Wang, “Coded multidimensional pulse amplitude modulation for ultra-high-speed optical transmission,” in Proc. OFC/NFOEC 2011, Paper No. JThA041, Los Angeles Convention Center, Los Angeles, CA, USA, March 6–10, 2011.
- M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express 17(13), 10814–10819 (2009). [CrossRef] [PubMed]
- J. G. Proakis, Digital Communications (McGraw-Hill, 2001).

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