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

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References

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  1. W. Shieh, Y. Tang, and B. S. Krongold, “DFT-Spread OFDM for Optical Communications,” in 9th International Conference on Optical Internet (COIN), (2010).
  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]
  3. F. Wang and X. Wang, “Coherent Optical DFT-Spread OFDM,” in Advances in Optical Technologies (Hindawi Publishing Corporation, 2011).
  4. Y. Tang, W. Shieh, and B. S. Krongold, “DFT-Spread OFDM for Fiber Nonlinearity Mitigation,” IEEE Photon. Technol. Lett.22(16), 1250–1252 (2010).
    [CrossRef]
  5. 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).
  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.24, 1704–1707 (2012).
  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 European Conference of Optical Communication (ECOC), (2011).
  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. Express20(3), 2379–2385 (2012).
    [CrossRef] [PubMed]
  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. Express19(27), 26198–26207 (2011).
    [CrossRef] [PubMed]
  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 OFC/NFOEC, paper OW4C.1, (2012).
  11. C. Ciochina and H. Sari, “A review of OFDMA and single-carrier FDMA,” in Wireless Conference (EW), 706– 710, (2010).
  12. X. Liu and S. Chandrasekhar, “High Spectral-Efficiency Transmission Techniques for Beyond 100-Gb/s Systems, SPMA1,” in SPPCom - Signal Processing in Photonic Communications - OSA Technical Digest, 1–36, (2011).
  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. Express16(20), 15777–15810 (2008).
    [CrossRef] [PubMed]
  14. S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, Ch. 3 (Springer, 2011).
  15. M. P. H. Jun and J. Cho, “PAPR Reduction in OFDM transmission using Hadamard Transform,” in IEEE International Conference on Communications1, 430–433, (2000).
  16. Y. Wu, C. K. Ho, and S. Sun, “On some properties of Walsh-Hadamard transformed OFDM,” in Proceedings IEEE 56th Vehicular Technology Conference4, 2096–2100, (2002).
  17. B. Porat, A Course in Digital Signal Processing (John Wiley and Sons, 1997).
  18. H. Myung, J. Lim, and D. Godman, “Peak-To-Average Power Ratio of Single Carrier FDMA Signals with Pulse Shapingý,” in IEEE 17th International Symposium on Personal, Indoor and Mobile Radio Communications, (2006)ý, pp. 1–5.
  19. 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).
  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. Express19(18), 17713–17728 (2011).
    [CrossRef] [PubMed]
  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]
  23. 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).
  24. 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]
  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]

2012 (3)

2011 (2)

2010 (3)

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]

Y. Tang, W. Shieh, and B. S. Krongold, “DFT-Spread OFDM for Fiber Nonlinearity Mitigation,” IEEE Photon. Technol. Lett.22(16), 1250–1252 (2010).
[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).

2008 (2)

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]

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. Express16(20), 15777–15810 (2008).
[CrossRef] [PubMed]

2001 (1)

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]

1997 (1)

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

Aung, A.

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]

Benlachtar, Y.

Berger, C. R.

Brandt-Pearce, M.

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

Cai, Y.

Chen, X.

Cho, P.

Cui, K.

Gao, G.

Hauske, F. N.

He, Z.

Q. Yang, Z. He, Z. Yang, S. Yu, X. Yi, and W. Shieh, “Coherent optical DFT-spread OFDM transmission using orthogonal band multiplexing,” Opt. Express20(3), 2379–2385 (2012).
[CrossRef] [PubMed]

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).

Huang, Y.

Jiang, T.

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).

Karagodsky, V.

Khurgin, J.

Killey, R. I.

Krongold, B. S.

Y. Tang, W. Shieh, and B. S. Krongold, “DFT-Spread OFDM for Fiber Nonlinearity Mitigation,” IEEE Photon. Technol. Lett.22(16), 1250–1252 (2010).
[CrossRef]

Lee, M. H.

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]

Li, A.

Li, C.

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).

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).

Li, L.

Liu, L.

Luo, M.

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).

Meiman, Y.

Member, S.

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]

Milder, P. A.

Nazarathy, M.

Ng, B. P.

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]

Noe, R.

Park, J. Y.

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]

Peddanarappagari, K.

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

Rahardja, S.

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]

Rajan, B. S.

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]

Shieh, W.

Shpantzer, I.

Tang, Y.

Y. Tang, W. Shieh, and B. S. Krongold, “DFT-Spread OFDM for Fiber Nonlinearity Mitigation,” IEEE Photon. Technol. Lett.22(16), 1250–1252 (2010).
[CrossRef]

Tucker, R. S.

Wavelet, A.

Weidenfeld, R.

Xiao, X.

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).

Xie, C.

Xiong, Q.

Xue, D.

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).

Yan Tang,

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]

Yang, Q.

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).

Q. Yang, Z. He, Z. Yang, S. Yu, X. Yi, and W. Shieh, “Coherent optical DFT-spread OFDM transmission using orthogonal band multiplexing,” Opt. Express20(3), 2379–2385 (2012).
[CrossRef] [PubMed]

Yang, Z.

Yi, X.

Q. Yang, Z. He, Z. Yang, S. Yu, X. Yi, and W. Shieh, “Coherent optical DFT-spread OFDM transmission using orthogonal band multiplexing,” Opt. Express20(3), 2379–2385 (2012).
[CrossRef] [PubMed]

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).

Yu, S.

IEEE Photon. Technol. Lett. (2)

Y. Tang, W. Shieh, and B. S. Krongold, “DFT-Spread OFDM for Fiber Nonlinearity Mitigation,” IEEE Photon. Technol. Lett.22(16), 1250–1252 (2010).
[CrossRef]

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).

IEEE Photonics J. (1)

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]

IEEE Trans. Circ. Syst. II (1)

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]

IEEE Trans. Sig. Process. (1)

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]

J. Lightwave Technol. (3)

Opt. Express (4)

Other (13)

S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, Ch. 3 (Springer, 2011).

M. P. H. Jun and J. Cho, “PAPR Reduction in OFDM transmission using Hadamard Transform,” in IEEE International Conference on Communications1, 430–433, (2000).

Y. Wu, C. K. Ho, and S. Sun, “On some properties of Walsh-Hadamard transformed OFDM,” in Proceedings IEEE 56th Vehicular Technology Conference4, 2096–2100, (2002).

B. Porat, A Course in Digital Signal Processing (John Wiley and Sons, 1997).

H. Myung, J. Lim, and D. Godman, “Peak-To-Average Power Ratio of Single Carrier FDMA Signals with Pulse Shapingý,” in IEEE 17th International Symposium on Personal, Indoor and Mobile Radio Communications, (2006)ý, pp. 1–5.

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).

W. Shieh, Y. Tang, and B. S. Krongold, “DFT-Spread OFDM for Optical Communications,” in 9th International Conference on Optical Internet (COIN), (2010).

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 European Conference of Optical Communication (ECOC), (2011).

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).

C. Ciochina and H. Sari, “A review of OFDMA and single-carrier FDMA,” in Wireless Conference (EW), 706– 710, (2010).

X. Liu and S. Chandrasekhar, “High Spectral-Efficiency Transmission Techniques for Beyond 100-Gb/s Systems, SPMA1,” in SPPCom - Signal Processing in Photonic Communications - OSA Technical Digest, 1–36, (2011).

F. Wang and X. Wang, “Coherent Optical DFT-Spread OFDM,” in Advances in Optical Technologies (Hindawi Publishing Corporation, 2011).

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).

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Figures (8)

Fig. 1
Fig. 1

(a) DFT-S OFDM link with (de)spreading pre(post)-processing. M sub-single-carriers are transmitted over M FDM sub-bands per channel (b). Tx signal flow for a single sub-band.

Fig. 2
Fig. 2

Cyclically extended Single-Carrier Tx (a) obtained as a special case of DFT-S with M = 1 SSC sub-band. (b) resulting CE-SC block diagram. The Tx is just a single-carrier one with added CP; the Rx drops the CP and performs DFT-based FD one-tap per tone equalization.

Fig. 3
Fig. 3

Numeric simulation of the received FWM NLI MER over a 16-QAM DFT-S OFDM optically amplified fiber link, for an SSF transmission experiment with the following parameters: MN=128;BW=25GHz;α=0.2dB/Km;γ=1.3 (WKm) 1 ;D=17ps/(nmKm);L=6×100Km;P=0dBm. 38400 data-symbols were used to gather the statistics (and we have verified that the MER converged to steady values).

Fig. 4
Fig. 4

PAPR of the transmitted DFT-S OFDM signal, parameterized by the number of SSC sub-bands. Other parameters: RRC Tx filter with parameter alpha = 0.1; 4x upsampling; QAM16 constellation; main FFT size MN = 128. The abbreviations anlg = analog and dig = digital refer to the PAPR type.

Fig. 5
Fig. 5

Numeric simulation of the received FWM NLI MER over a 16-QAM DFT-S OFDM optically amplified fiber link. The two curves present Monte Carlo simulations respectively based on the SSF and third-order Volterra series models with the following parameters: MN=128;BW=25GHz;α=0.2dB/Km;γ=1.3 (WKm) 1 ;D=17ps/(nmKm);L=6×100Km;P=0dBm. Notice that the worst case deviation of the Volterra-based trilinear model is bounded by 0.4 dB. 38400 data-symbols were used to gather the statistics (and we have verified that the MER converged to steady values).

Fig. 6
Fig. 6

Numeric Monte Carlo simulation of the PDF of the modulus (magnitude) and angle of the output samples of a 16-QAM DFT-S transmitter, parameterized by the number N (64,128,256) of tones per sub-band. The empirical histograms track the respective theoretical PDFs which are Rayleigh for the modulus and uniform over 2π for the phase. The plot pertains to the tone indexed i = 1.

Fig. 7
Fig. 7

(a) Averaged (over all tones) NLI MER vs. the number of sub-bands for a 16-PSK DFT-S OFDM channel over an optically amplified fiber link with the following parameters: Three curves are shown. The analytic curve accurately tracks the Monte-Carlo trilinear Volterra series simulation while the SSF slightly deviates away for the high plotted range of M values. (b) Averaged NLI MER vs. sub-bands, for 16-QAM vs. 16-PSK, over a link with:

Fig. 8
Fig. 8

NLI MER averaged (over all tones) vs. the number of unitary-spread sub-bands over an optically amplified fiber link with the following parameters: BW=25GHz;α=0.2dB/Km;γ=1.3 (WKm) 1 ;D=17ps/(nmKm);P=0dBm. (a) 16-QAM MC-SSF simulation over L=10×100Km comparing WAV-S vs. DTF-S OFDM NL performance for MN = 128 tones (also vs. CE-SC and plain OFDM discrete-points). Plain OFDM NL performance is exceeded with both DFT-S and WAV-S systems, but only DFT-S exceeds the single-carrier (CE-SC) performance. (b) 16-PSK channel over L=2×100Km with MN = 32 tones simulated analytically and also with MC trilinear Volterra series. The analytic WAV-S curve accurately tracks the MC-Trilinear simulation and indicates improved performance relative to plain OFDM, but worse performance relative to single-carrier. The DFT spread performance at its optimized peak (occurring for M = 2 sub-bands) exceeds the performance of all other systems (but requires higher complexity)

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

A( A (1) A (2) ... A (M) )=( DFT N DFT N ... DFT N )( B (1) B (2) ... B (M) )
MER[k]= average-power mean-square-fluctuations = 1 MN i=1 MN | B i [k] | 2 1 MN i=1 MN | R i [k] B i [k] | 2 = P B i=1 MN | R i FWM [k] | 2
r i FWM = ( j,k )S[ i ] H i;jk A j A k A j+ki *
A i = t=1 MN W i,t B t A=WB
A i 1 A i 2 * = t 1 t 2 W i 1 , t 1 W i 2 , t 2 * B t 1 B t 2 * = σ 2 t 1 W i 1 , t 1 W i 2 , t 1 * = σ 2 δ i 1 , i 2
W MN×MN =diag{ U N×N , U N×N ,..., U N×N }
W MN×MN WAV-S =diag{ HA D N ,HA D N ,...,HA D N }
HA D 2N =( HA D N HA D N HA D N HA D N );HA D 2 =( 1 1 1 1 ).
| r i FWM | 2 = ( j 1 , k 1 ),( j 2 , k 2 )S[ i ] A j 1 A j 2 * A k 1 A k 2 * A j 1 + k 1 i * A j 2 + k 2 i H i; j 1 k 1 H i; j 2 k 2 *
r i FWM = ( j,k )S[ i ] H i;jk t 1 t 2 t 3 B t 1 W j, t 1 B t 2 W k, t 2 ( B t 3 W (j+ki), t 3 ) * = t 1 t 2 t 3 B t 1 B t 2 B t 3 * ( j,k )S[ i ] H i;jk W j, t 1 W k, t 2 W (j+ki), t 3 * = t 1 t 2 t 3 h i; t 1 t 2 t 3 B t 1 B t 2 B t 3 *
h i; t 1 t 2 t 3 ( j,k )S[ i ] H i;jk W j, t 1 W k, t 2 W (j+ki), t 3 *
t 1 t 2 t 3 | h i; t 1 t 2 t 3 | 2 = ( j,k )S[ i ] | H i;j,k | 2
h i; t 1 t 2 t 3 = h i; t 2 t 1 t 3
h i; t 1 t 2 t 3 ( j,k )S[ i ] H i;jk W j, t 1 W k, t 2 W (j+ki), t 3 * = ( j,k )S[ i ] H i;jk e j 2π N j t 1 e j 2π N k t 2 e j 2π N ( j+ki ) t 3 = e j 2π N i t 3 ( j,k )S[ i ] H i;jk e j 2π N [ j( t 1 t 3 )+k( t 2 t 3 ) ] = e j 2π N i t 3 ( j,k ) 1 S[ i ] [j,k] H i;jk e j 2π N [ j( t 1 t 3 )+k( t 2 t 3 ) ] = = (u,v) DF T j,k { 1 S[ i ] [j,k] H i;jk } | (u,v)=( t 1 t 3 , t 2 t 3 ) e j 2π N i t 3
| r i FWM | 2 = t 1 , t 2 , t 3 , t 1 , t 2 , t 3 B t 1 B t 1 * B t 2 B t 2 * B t 3 * B t 3 h i; t 1 t 2 t 3 h i; t 1 t 2 t 3 *
| r i FWM | 2 =2 P 3 t 3 t 1 t 2 t 3 | h i; t 1 t 2 t 3 | 2 + P 3 t 1 t 2 | h i; t 1 t 1 t 2 | 2 + P 3 t 1 | h i; t 1 t 1 t 1 | 2 = P 3 [ 2 t 1 , t 2 , t 3 | h i; t 1 t 2 t 3 | 2 t 1 t 2 | h i; t 1 t 1 t 2 | 2 4 t 1 t 2 | h i; t 1 t 2 t 2 | 2 t 1 | h i; t 1 t 1 t 1 | 2 ]
| r i FWM | 2 =2 P 3 ( j,k )S[ i ] | H i;j,k | 2 P 3 ( j 1 , k 1 )S[ i ] ( j 2 , k 2 )S[ i ] H i; j 1 , k 1 H i; j 2 , k 2 * Q[ j 1 , j 2 , k 1 , k 2 ]
whereQ[ j 1 , j 2 , k 1 , k 2 ] t W j 1 ,t W j 2 ,t * W k 1 ,t W k 2 ,t * δ j 1 + k 1 , j 2 + k 2 + +4 W k 1 ,t W k 2 ,t * W j 1 + k 1 i,t * W j 2 + k 2 i,t δ j 1 , j 2 4 W j 1 ,t W j 2 ,t * W k 1 ,t W k 2 ,t * W j 1 + k 1 i,t * W j 2 + k 2 i,t
A={ DFT { B t } t=0 N1 ,DFT { B t } t=N 2N1 ,...,DFT { B t } t=( M1 )N MN1 }
s n (m) = ν=mN ( m+1 )N1 ( t=0 N1 B mN+t e j 2π N ( νmN )t ) e +j 2π MN νn = t=0 N1 B mN+t ν=mN ( m+1 )N1 e j 2π N νt e +j 2π MN νn = t=0 N1 B mN+t ν=mN ( m+1 )N1 e +j 2π N ν( n M t ) = ν =νmN t=0 N1 B mN+t ν =0 N1 e +j 2π N ( ν +mN )( n M t ) = t=0 N1 B mN+t ν =0 N1 e +j 2π N ν ( n M t ) e +j 2π M nm =N e +j 2π M nm t=0 N1 B mN+t e jπ( n M t ) N1 N din c N [n/Mt] wheredin c N [u] sin( πu ) Nsin( πu/N )
ν =0 N1 e +j 2π N ν ( n M t ) = e jπ( n/Mt )( N1 )/N sin[ π( n/Mt ) ] sin[ π N ( n/Mt ) ] =N e jπ( n/Mt )( N1 )/N dinc N [n/Mt]
Q[ j 1 , j 2 , k 1 , k 2 ] t W j 1 ,t W j 2 ,t * W k 1 ,t W k 2 ,t * δ j 1 + k 1 , j 2 + k 2 + +4 W k 1 ,t W k 2 ,t * W j 1 + k 1 i,t * W j 2 + k 2 i,t δ j 1 , j 2 4 W j 1 ,t W j 2 ,t * W k 1 ,t W k 2 ,t * W j 1 + k 1 i,t * W j 2 + k 2 i,t = 1 {( j 1 , j 2 , k 1 , k 2 )| j 1 = j 2 = k 1 = k 2 } ( j 1 , j 2 , k 1 , k 2 )+4 1 {( j 1 , j 2 , k 1 , k 2 )| k 1 = k 2 = j 1 + k 1 i= j 2 + k 2 i, j 1 = j 2 } ( j 1 , j 2 , k 1 , k 2 ) 4 1 {( j 1 , j 2 , k 1 , k 2 )| j 1 = j 2 = k 1 = k 2 = j 1 + k 1 i= j 2 + k 2 i} ( j 1 , j 2 , k 1 , k 2 )
| r i NL | 2 = P 3 [ 2 ( j,k )S[ i ] | H i;j,k | 2 ( j,j )S[ i ] | H i;j,j | 2 ]
Q[ j 1 , j 2 , k 1 , k 2 ]= t [ e j 2π N t( ( j 1 + k 1 )( j 2 + k 2 ) ) δ j 1 + k 1 , j 2 + k 2 +4 e j 2π N t( ( k 1 + j 2 + k 2 i )( k 2 + j 1 + k 1 i ) ) δ j 1 , j 2 2 e j 2π N t( ( j 1 + k 1 + j 2 + k 2 i )( j 2 + k 2 + j 1 + k 1 i ) ) ] = t [ 1 {( j 1 , j 2 , k 1 , k 2 )| j 1 + k 1 = j 2 + k 2 } ( j 1 , j 2 , k 1 , k 2 )+4 1 {( j 1 , j 2 , k 1 , k 2 )| j 1 = j 2 } ( j 1 , j 2 , k 1 , k 2 )4 ]
| r i NL | 2 =2 P 3 ( j,k )S[ i ] | H i;j,k | 2 +4N P 3 ( j 1 , k 1 )S[ i ] ( j 2 j 1 , k 2 )S[ i ] H i; j 1 , k 1 H i; j 2 , k 2 * N P 3 ( j 1 , k 1 )S[ i ] ( j 2 , k 2 )S[ i ] j 1 + k 1 = j 2 + k 2 H i; j 1 , k 1 H i; j 2 , k 2 * .

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