## An analytical study of the improved nonlinear tolerance of DFT-spread OFDM and its unitary-spread OFDM generalization |

Optics Express, Vol. 20, Issue 23, pp. 25884-25901 (2012)

http://dx.doi.org/10.1364/OE.20.025884

Acrobat PDF (1672 KB)

### Abstract

DFT-spread (DFT-S) coherent optical OFDM was numerically and experimentally shown to provide improved nonlinear tolerance over an optically amplified dispersion uncompensated fiber link, relative to both conventional coherent OFDM and single-carrier transmission. Here we provide an analytic model rigorously accounting for this numerical result and precisely predicting the optimal bandwidth per DFT-S sub-band (or equivalently the optimal number of sub-bands per optical channel) required in order to maximize the link non-linear tolerance (NLT). The NLT advantage of DFT-S OFDM is traced to the particular statistical dependency introduced among the OFDM sub-carriers by means of the DFT spreading operation. We further extend DFT-S to a unitary-spread generalized modulation format which includes as special cases the DFT-S scheme as well as a new format which we refer to as wavelet-spread (WAV-S) OFDM, replacing the spreading DFTs by Hadamard matrices which have elements +/−1 hence are multiplier-free. The extra complexity incurred in the spreading operation is almost negligible, however the performance improvement with WAV-S relative to plain OFDM is more modest than that achieved by DFT-S, which remains the preferred format for nonlinear tolerance improvement, outperforming both plain OFDM and single-carrier schemes.

© 2012 OSA

## 1. Introduction

2. W. Shieh and Yan Tang, “Ultrahigh-Speed Signal Transmission Over Nonlinear and Dispersive Fiber Optic Channel: The Multicarrier Advantage,” IEEE Photonics J. **2**(3), 276–283 (2010). [CrossRef]

*Single-Carrier Frequency Division Multiple Access*(SC-FDMA) [11] and this scheme has been adopted in the fourth generation wireless cellular networks.

*Peak to Average Power Ratio*(PAPR) of the OFDM signal. The DFT-S OFDM format has been gaining traction [12] as it is a multicarrier format [2

2. W. Shieh and Yan Tang, “Ultrahigh-Speed Signal Transmission Over Nonlinear and Dispersive Fiber Optic Channel: The Multicarrier Advantage,” IEEE Photonics J. **2**(3), 276–283 (2010). [CrossRef]

2. W. Shieh and Yan Tang, “Ultrahigh-Speed Signal Transmission Over Nonlinear and Dispersive Fiber Optic Channel: The Multicarrier Advantage,” IEEE Photonics J. **2**(3), 276–283 (2010). [CrossRef]

*nonlinear tolerance*(NLT) advantage in terms of the reduced PAPR, the rationale being that each DFT-S sub-band amounts to a conventional single carrier of relatively low PAPR. As there are far fewer sub-bands per channel than there are OFDM sub-carriers, the PAPR is reduced.

*N*-pnt Hadamard matrices. We refer to the resulting system as

*Wavelet Spread*(WAV-S) OFDM. This OFDM variant was previously proposed in wireless transmission under the name

*Walsh-Hadamard OFDM*[15,16] but for a single sub-band. Both WAV-S and DFT-S OFDM are viewed here as special cases of our newly introduced generic U-S OFDM transmission scheme. The advantage of the WAV-S OFDM is its reduced complexity relative to the DFT-S OFDM, as the Hadamard unitary spreading matrix has elements +/−1, hence matrix multiplication reduces to signed addition, much simpler than DFT-spreading; the WAV-S pre/post processing is then multiplier-free and its complexity just marginally exceeds that of plain OFDM. However, the WAV-S nonlinear tolerance turns out to be inferior to that of DFT-S OFDM but still exceeds that of plain OFDM. Thus, WAV-S OFDM practically provides a new complexity-penalty-free option for slightly improving OFDM NL performance.

*NonLinear Interference*(NLI) statistical properties of U-S and DFT-S OFDM. Section 8 presents a key result of the paper, deriving the compact analytic model for NL performance of U-S OFDM and its DFT-S and WAV-S special cases. Section 9 concludes the paper. Two appendices detail various mathematical derivations. The abbreviations used in this paper are stated in Appendix C.

## 2. Review of DFT-S OFDM and its interpretation as FDM of sub-single-carriers

*MN*is preceded (followed) by an array of (I)DFT modules each of size

*N*. Here

*M*is also the number of sub-bands, which is also the number of (I)DFTs forming the pre/post-processing arrays while

*N*is the number of OFDM tones per sub-band. A mathematical description of the DFT-S transmitter pre-processing expresses the vector

*N*-pnt consecutive sub-blocks of the

*L*=

*MN*-pnt vector of input samples,

*M*sub-bands is pre-processed at the transmitter by an

*N*-pnt DFT, which is referred to as the

*spreading*DFT. The

*M*outputs of the spreading DFT array are concatenated to form an

*MN*-pnt vector,

*time-domain*(TD) samples, which is CP-extended and applied to the Tx DAC.

*M*equal sub-blocks, each of which is passed through an

*N*-pnt IDFT. The sub-band IDFTs in the Rx form the

*de-spreading*IDFT array. The cascade of the main

*MN*-pnt DFT and the de-spreading array acts to demultiplex and decimate each of the sub-bands, undoing the action of the DFT-S Tx. The de-spreading DFT-S post-processing corresponds to inverting Eq. (1) by performing the exchanges

*M*identical IDFT blocks on the main diagonal.

*tones*, and we also make the convention of referring to the DFT-S sub-bands as

*sub-single-carriers*(SSC), as the signal within each sub-band is effectively indistinguishable from a time-domain single-carrier signal, albeit of reduced bandwidth

*B*/

*M*where

*B*is the overall channel bandwidth, as shown next.

### 2.1 DFT-S OFDM as frequency division (de)multiplexer with interpolation (decimation)

*frequency-division-multiplex*(FDM) carrying

*M*SSCs side-by-side in frequency. To this end consider exciting a single SSC input of the DFT-S OFDM Tx (turning off the other

*M*-1 sub-band signals). The block diagram now reduces to a single

*N*-pnt DFT feeding an

*MN*-pnt IDFT (Fig. 1(b)) with zero-padded frequency inputs on either side of the active

*N*inputs fed by the single

*N*-pnt DFT singled out of the DFT-S spreading array. It is well-known in DSP theory [17] that zero-padding input of an IDFT corresponds to time-interpolation of the IDFT output. It follows that the DFT-IDFT cascade generates a time→frequency→time map composition, compounding to a time→time mapping generating TD interpolation. Thus, the DFT-S transmitter time-domain interpolates its center SSC. Moreover, whenever the SSC is shifted off center, as the inputs into the main IDFT are effectively in the

*frequency-domain*(FD), then this is a frequency shift amounting to modulation (multiplication by an harmonic signal) in the TD. We conclude that each of the SSCs may be viewed as a TD single-carrier signal which is both interpolated and modulated onto a sub-carrier at a center frequency corresponding to the sub-band center position. When all the SSCs are excited together by simultaneously applying

*M*TD inputs to the DFT spreading array, we then obtain multiple interpolated SSCs modulated onto a regular grid of center frequencies. A composite FDM signal is then formed. The DFT-S transmitter provides sub-carrier multiplexing of multiple (

*M*) narrowband single carrier QAM transmissions. This intuitive account of DFT-S Tx operation is mathematically formalized in Appendix A, rigorously proving that the DFT-S Tx indeed functions as FDM + interpolator.

### 2.2 OFDM and cyclically extended SC as special cases of DFT-S OFDM

*M*= 1 (but

*Single Carrier*(SC) transmission (with cyclic extension) as shown in Fig. 2(b), referred to in the wireless literature as SC-FDMA. A more appropriate name adopted here in the optical context is

*Cyclically Extended Single Carrier*(CE-SC), where CE alludes to equipping the conventional SC transmission with a

*cyclic prefix*(CP). The CE-SC Tx (Fig. 2(b)) is akin to a single carrier conventional QAM Tx, except that each block of

*N*consecutive SC symbols is prepended or appended a cyclic extension (cyclic prefix or suffix), e.g., a CP consisting of the last

*N*-pnt block is replicated ahead of the

*N*-pnt block to form an

*M*= 1).

*N*-pnt DFT followed by an

*N*-pnt IDFT (which corresponds to a degenerate DFT-S array with a single de-spreading IDFT transformation). However, in the Rx, the DFT and IDFT no longer cancel out, as in between them there are inserted one-tap complex multiplications used to equalize the cyclically extended fiber channel.

## 3. Numeric simulations of the nonlinear tolerance of DFT-S OFDM

**2**(3), 276–283 (2010). [CrossRef]

**2**(3), 276–283 (2010). [CrossRef]

*Split-Step-Fourier*(SSF) method for a single polarization scalar channel and DFT-S transceivers as described in Fig. 1(a).

*Modulation Error Ratio*(MER) of the NLI accompanying the received signal due to the Four-Wave-Mixing (FWM) between the tones, after compensating the

*Chromatic Dispersion*(CD) as well as compensating the

*Self-Phase-Modulation*(SPM) of each tone as well as the

*Cross-Phase-Modulation*(XPM) among the tones [13

13. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express **16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*k*for an ergodic process) of the following ‘instantaneous MER’ expression (here

*i*is the SSC index,

*k*is discrete-time, and

**R**-vector at the Rx de-spreading array output as defined in Fig. 1(a)).

## 4. PAPR is not an accurate predictor of the nonlinear tolerance of DFT-S OFDM

20. C. R. Berger, Y. Benlachtar, R. I. Killey, and P. A. Milder, “Theoretical and experimental evaluation of clipping and quantization noise for optical OFDM,” Opt. Express **19**(18), 17713–17728 (2011). [CrossRef] [PubMed]

*Complementary Cumulative Distribution Function*(CCDF) of the DFT-S Tx signal launched into the fiber link is plotted in Fig. 4 , showing both the digital-domain and analog-domain PAPR distributions per OFDM symbol for a

*Root-Raised-Cosine*(RRC) Tx filter and 16-QAM transmission. The plots are parameterized by the number

*M*of SSC sub-bands. The PAPR of conventional OFDM is retrieved for

*M = MN*(when each sub-band is one-tone wide, and the link simplifies to conventional OFDM, as previously discussed), whereas the single-carrier CE-SC case corresponds to DFT-S with

*M*= 1. It is apparent that OFDM exhibits the worst PAPR. The PAPR monotonically improves with decreasing

*M*, finally reaching a minimum for the CE-SC case (

*M*= 1).

## 5. Monte Carlo Volterra series NL simulation of the DFT-S link

13. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express **16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*Volterra Series Transfer Function*(VSTF) formalism, e.g [21

21. K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. **15**(12), 2232–2241 (1997). [CrossRef]

22. L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Intrachannel Nonlinearity Compensation by Inverse Volterra Series Transfer Function,” J. Lightwave Technol. **30**(3), 310–316 (2012). [CrossRef]

*i*-th tone.

13. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express **16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*j,k*) index pairs corresponding to valid FWM frequency triplets,

*i*-th in-band OFDM tone. The

*trilinear transform*of Eq. (3), with frequency-dependent weighting given by the VSTF, corresponds to the third-order term of a Volterra Series expansion. Truncating the series to its third-order turns out to provide a highly accurate model (the accuracy of which is numerically verified as explained below) which may be used as a more efficient substitute for the well-known SSF method of fiber non-linear propagation.

## 6. Unitary-Spread OFDM as generalization of DFT-S and its FWM generation model

*Unitary-Spread*OFDM. We proceed to derive the nonlinear tolerance of this more generic format, finally restricting the general U-S NLT results to the special DFT-S case, as well as to other special cases.

### 6.1 Extending DFT-S OFDM to the generalized U-S OFDM modulation format

*N*-pnt (I)DFTs. Evidently, taking

*N*-pnt DFT matrix (with its (

*m,n*) element given by

### 6.2 Wavelet-Spread OFDM

### 6.3 Exploring additional U-S OFDM special cases

24. M. H. Lee, S. Member, B. S. Rajan, and J. Y. Park, “A Generalized Reverse Jacket Transform,” IEEE Trans. Circ. Syst. II **48**(7), 684–690 (2001). [CrossRef]

25. A. Aung, B. P. Ng, and S. Rahardja, “Sequency-Ordered Complex Hadamard Transform: Properties, Computational Complexity and Applications,” IEEE Trans. Sig. Process. **56**(8), 3562–3571 (2008). [CrossRef]

## 7. Statistics of the unitary-spread samples

*MN*becomes sufficiently large. The asymptotically Gaussian distribution was also verified by simulation, though the results are omitted.

## 8. Nonlinear performance of U-S OFDM

*analytic*model (further to the SSF and tri-linear

*numerical*models) for the FWM generation. This is actually a novel analytic version of the trilinear model - it predicts the FWM NLI of DFT-S by means of a closed-form formula (further generalized to additional OFDM variants based on alternative spreading transforms). This third model still requires numerical evaluation of a very large analytic formula involving summations, but it no longer resorts to Monte-Carlo simulation. This new trilinear analytic model precisely assesses the numerical FWM tolerance of DFT-S and more generalized OFDM variants.

**W**-matrix (irrespective of the specific nature of the spreading transform), a useful Parseval-like formula may be readily derived between the VSTF and the VSK:An important symmetry property of the VSK readily stems from Eq. (11) which turns out to be symmetric in its first and second indices (the proof that

### 8.1 Evaluating the VSK in the case of cyclically-extended single-carrier transmission

*N*(to which CP is added) the spreading DFT and main IDFT of DFT-S now both have the same size,

*N*, hence cancel out, yet it is still useful to think of the

*N*-pnt DFT-S and

*N*-pnt IDFT. The generic unitary spreading

**W**-matrix now reduces to the DFT matrix with elements

*i*) by the 2-D DFT-like transform acting on the 2-D VSTF sequence

*(j,k)*plane to the set of points

### 8.2 NLI power of U-S OFDM

*MN*of the total power

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

### 8.3 DFT-S OFDM NL performance

*MN*= 32 and

*M*= 1,2,4,8,16,32, due to the heavy numerical combinatorial complexity of the analytic evaluation averaging over all

*i*-s. It is apparent that the analytic version tracks the Monte-Carlo trilinear Volterra based version very well, however the SSF simulations do deviate from the two tracking (analytic and MC-trilinear) curves. It turns out that the match is much better for higher

*MN*values (e.g. as shown in Fig. 3), but for those values the evaluation of the two agreeing trilinear and analytic Volterra based expressions would just take too long due to the large number of combinations. In Fig. 7(b) we further compare the nonlinear performance of DFT-S for 16-QAM and 16-PSK constellations. It is apparent that in both cases there is a similar trend, however 16-QAM generally incurs worse NLI MER.

### 8.4 WAV-S OFDM NL performance

## 9. Conclusions

*N*log(

*N*)]complexity in par with that of the DFT. Our treatment should be further extended to address dual polarization operation. Other types of unitary spreading matrixes (e.g. complex hadamard generalizations) should be further explored.

## Appendix A. DFT-S as a superposition of single carrier sub-bands

*MN*data symbols

*MN*-pnt IDFT yielding the Tx output. Due to the linearity of the IDFT operation it is possible to analyze its output as a superposition of responses

*m*-th sub-band. The

*N*-pnt data segment

*mN*-1] and [(

*m*+ 1)

*N*,

*MN*-1], then the

*MN*-pnt IDFT is applied:

*m*-th SSC is given by an interpolated single-carrier signal, frequency shifted (in the digital domain) by modulation with an harmonic phase factor given by

## Appendix B. Analytic NLT evaluation for U-S special cases: OFDM, SC, DFT-S

### Conventional OFDM

**-**

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

### Cyclically extended Single Carrier (CE-SC) transmission

## Appendix C - Glossary

ADC = Analog to Digital Converter | MC = Monte Carlo | Rx = Receiver |

CE-SC = Cyclically Extended Single Carrier | MER = Modulation Error Rate | RRC = Root Raised Cosine |

CCDF = Complementary Cumulative Distribution Function | NL = Nonlinear | SSC = Sub Single Carrier |

CLT = Central Limit Theorem | NLC = Nonlinear Compensator | SPM = Self Phase Modulation |

CP = Cyclic Prefix | NLT = Nonlinear Tolerance | SSF = Split Step Fourier |

CD = Chromatic Dispersion | NLI = Nonlinear Interference | Tx = Transmitter |

DAC = Digital to Analog Converter | PAPR = Peak to Average Power Ratio | TD = Time Domain |

FWM = Four Wave Mixing | PDF = Probability Distribution Function | VSK = Volterra Series Kernel |

FDM = Frequency Domain Multiplexer | PSK = Phase Shift Keying | VSTF = Volterra Series Transfer Function |

IID = Independent Identically Distributed | QAM = Quadrature Amplitude Modulation | XPM = Cross Phase Modulation |

## Acknowledgments

## References and links

1. | W. Shieh, Y. Tang, and B. S. Krongold, “DFT-Spread OFDM for Optical Communications,” in |

2. | W. Shieh and Yan Tang, “Ultrahigh-Speed Signal Transmission Over Nonlinear and Dispersive Fiber Optic Channel: The Multicarrier Advantage,” IEEE Photonics J. |

3. | F. Wang and X. Wang, “Coherent Optical DFT-Spread OFDM,” in |

4. | Y. Tang, W. Shieh, and B. S. Krongold, “DFT-Spread OFDM for Fiber Nonlinearity Mitigation,” IEEE Photon. Technol. Lett. |

5. | Y. Tang, W. Shieh, and B. S. Krongold, “Fiber Nonlinearity Mitigation in 428-Gb / s Multiband Coherent Optical OFDM Systems,” in |

6. | C. Li, Q. Yang, T. Jiang, Z. He, M. Luo, C. Li, X. Xiao, D. Xue, and X. Yi, “Investigation of Coherent Optical Multi-band DFT-S OFDM in Long Haul Transmission,” IEEE Photon. Technol. Lett. |

7. | Q. Yang, Z. He, Z. Yang, S. Yu, X. Yi, A. A. Amin, and W. Shieh, “Coherent optical DFT-Spread OFDM in Band-Multiplexed Transmissions, We.8.A.6,” in |

8. | Q. Yang, Z. He, Z. Yang, S. Yu, X. Yi, and W. Shieh, “Coherent optical DFT-spread OFDM transmission using orthogonal band multiplexing,” Opt. Express |

9. | X. Chen, A. Li, G. Gao, and W. Shieh, “Experimental demonstration of improved fiber nonlinearity tolerance for unique-word DFT-spread OFDM systems,” Opt. Express |

10. | A. Li and G. Chen, Xi, A. Guanjun, A. Amin, W. Shieh, William, B. S. Krongold, “Transmission of 1. 63-Tb/s PDM-16QAM Unique-word DFT-Spread OFDM Signal over 1, 010-km SSMF,” in |

11. | C. Ciochina and H. Sari, “A review of OFDMA and single-carrier FDMA,” in |

12. | X. Liu and S. Chandrasekhar, “High Spectral-Efficiency Transmission Techniques for Beyond 100-Gb/s Systems, SPMA1,” in |

13. | M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express |

14. | S. Kumar, |

15. | M. P. H. Jun and J. Cho, “PAPR Reduction in OFDM transmission using Hadamard Transform,” in |

16. | Y. Wu, C. K. Ho, and S. Sun, “On some properties of Walsh-Hadamard transformed OFDM,” in |

17. | B. Porat, |

18. | H. Myung, J. Lim, and D. Godman, “Peak-To-Average Power Ratio of Single Carrier FDMA Signals with Pulse Shapingý,” in |

19. | B. Goebel, S. Hellerbrand, N. Haufe, and N. Hanik, “PAPR reduction techniques for coherent optical OFDM transmission,” in |

20. | C. R. Berger, Y. Benlachtar, R. I. Killey, and P. A. Milder, “Theoretical and experimental evaluation of clipping and quantization noise for optical OFDM,” Opt. Express |

21. | K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. |

22. | L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Intrachannel Nonlinearity Compensation by Inverse Volterra Series Transfer Function,” J. Lightwave Technol. |

23. | A. Li, W. Shieh, R. S. Tucker, and A. Wavelet, “Wavelet Packet Transform-Based OFDM for Optical Communications,” J. Lightwave Technol. |

24. | M. H. Lee, S. Member, B. S. Rajan, and J. Y. Park, “A Generalized Reverse Jacket Transform,” IEEE Trans. Circ. Syst. II |

25. | A. Aung, B. P. Ng, and S. Rahardja, “Sequency-Ordered Complex Hadamard Transform: Properties, Computational Complexity and Applications,” IEEE Trans. Sig. Process. |

**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 4, 2012

Revised Manuscript: October 19, 2012

Manuscript Accepted: October 22, 2012

Published: November 1, 2012

**Citation**

Gal Shulkind and Moshe Nazarathy, "An analytical study of the improved nonlinear tolerance of DFT-spread OFDM and its unitary-spread OFDM generalization," Opt. Express **20**, 25884-25901 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-25884

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### References

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- W. Shieh and Yan Tang, “Ultrahigh-Speed Signal Transmission Over Nonlinear and Dispersive Fiber Optic Channel: The Multicarrier Advantage,” IEEE Photonics J.2(3), 276–283 (2010). [CrossRef]
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- Y. Tang, W. Shieh, and B. S. Krongold, “Fiber Nonlinearity Mitigation in 428-Gb / s Multiband Coherent Optical OFDM Systems,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2010).
- C. Li, Q. Yang, T. Jiang, Z. He, M. Luo, C. Li, X. Xiao, D. Xue, and X. Yi, “Investigation of Coherent Optical Multi-band DFT-S OFDM in Long Haul Transmission,” IEEE Photon. Technol. Lett.24, 1704–1707 (2012).
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- A. Li and G. Chen, Xi, A. Guanjun, A. Amin, W. Shieh, William, B. S. Krongold, “Transmission of 1. 63-Tb/s PDM-16QAM Unique-word DFT-Spread OFDM Signal over 1, 010-km SSMF,” in OFC/NFOEC, paper OW4C.1, (2012).
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- B. Goebel, S. Hellerbrand, N. Haufe, and N. Hanik, “PAPR reduction techniques for coherent optical OFDM transmission,” in 2009 11th International Conference on Transparent Optical Networks1, 1–4, (2009).
- C. R. Berger, Y. Benlachtar, R. I. Killey, and P. A. Milder, “Theoretical and experimental evaluation of clipping and quantization noise for optical OFDM,” Opt. Express19(18), 17713–17728 (2011). [CrossRef] [PubMed]
- K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol.15(12), 2232–2241 (1997). [CrossRef]
- L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Intrachannel Nonlinearity Compensation by Inverse Volterra Series Transfer Function,” J. Lightwave Technol.30(3), 310–316 (2012). [CrossRef]
- A. Li, W. Shieh, R. S. Tucker, and A. Wavelet, “Wavelet Packet Transform-Based OFDM for Optical Communications,” J. Lightwave Technol.28, 3519–3528 (2010).
- M. H. Lee, S. Member, B. S. Rajan, and J. Y. Park, “A Generalized Reverse Jacket Transform,” IEEE Trans. Circ. Syst. II48(7), 684–690 (2001). [CrossRef]
- A. Aung, B. P. Ng, and S. Rahardja, “Sequency-Ordered Complex Hadamard Transform: Properties, Computational Complexity and Applications,” IEEE Trans. Sig. Process.56(8), 3562–3571 (2008). [CrossRef]

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