Abstract

The possibility of polarization-mode dispersion (PMD) compensation in fiber-optic communication systems with direct detection using a simple channel estimation technique and low-density parity-check (LDPC)-coded orthogonal frequency division multiplexing (OFDM) is demonstrated. It is shown that even for differential group delay (DGD) of 4/BW (BW is the OFDM signal bandwidth), the degradation due to the first-order PMD can be completely compensated for. Two classes of LDPC codes designed based on two different combinatorial objects (difference systems and product of combinatorial designs) suitable for use in PMD compensation are introduced.

© 2007 Optical Society of America

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  1. H. F. Haunstein,W. Sauer-Greff, A. Dittrich, K. Sticht, and R. Urbansky, "Principles for electronic equalization of polarization-mode dispersion," J. Lightwave Technol. 22, 1169-1182 (2004).
    [CrossRef]
  2. M. Jäger, T. Rankl, J. Spiedel, H. Bulöw, and F. Buchali, "Performance of turbo equalizers for optical PMD channels," J. Lightwave Technol. 24, 1226-1236 (2006).
    [CrossRef]
  3. I. B. Djordjevic, and B. Vasic, "Nonlinear BCJR equalizer for suppression of intrachannel nonlinearities in 40 Gb/s optical communications systems," Opt. Express 14, 4625-4635 (2006).
    [CrossRef] [PubMed]
  4. L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
    [CrossRef]
  5. C. Douillard, M. Jézéquel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, "Iterative correction of intersymbol interference: turbo equalization," Eur. Trans. Telecommun. 6, 507-511 (1995).
    [CrossRef]
  6. 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., submitted for publication.
  7. R. van Nee, and R. Prasad, OFDM Wireless Multimedia Communications (Artech House, Boston 2000).
  8. A. J. Lowery, L. Du, and J. Armstrong, "Orthogonal frequency division multiplexing for adaptive dispersion compensation in long haul WDM systems," in Proc. OFC Postdeadline Papers, Paper no. PDP39, 2006.
  9. I. B. Djordjevic and B. Vasic, "Orthogonal frequency-division multiplexing for high-speed optical transmission," Opt. Express 14, 3767-3775 (2006).
    [CrossRef] [PubMed]
  10. I. B. Djordjevic, and B. Vasic, "100 Gb/s transmission using orthogonal frequency-division multiplexing," IEEE Photon. Technol. Lett. 18, 1576-1578 (2006).
    [CrossRef]
  11. W. Shieh, W. Chen and R. S. Tucker, "Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems," Electron. Lett. 42, 996-997 (2006).
    [CrossRef]
  12. W. Shieh and C. Athaudage, "Coherent optical frequency division multiplexing," Electron. Lett. 42, 587-589 (2006).
    [CrossRef]
  13. W. Shieh, "PMD-supported coherent optical OFDM systems," IEEE Photonics Technol. Lett. 19, 134-136 (2006).
    [CrossRef]
  14. A. J. Lowery and J. Armstrong, "10 Gb/s multimode fiber link using power-efficient orthogonal-frequency-division multiplexing," Opt. Express 13, 10003-10009 (2005).
    [CrossRef] [PubMed]
  15. M. Cvijetic, Coherent and Nonlinear Lightwave Communications (Artech House, Boston 1996).
  16. A. Garcia Armada and M. Calvo, "Phase noise and sub-carrier effects on the performance of an OFDM communication system," IEEE Commun. Lett. 2, 11-13 (1998).
    [CrossRef]
  17. I. B. Djordjevic, S. Sankaranarayanan, S. K. 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]
  18. I. Anderson, Combinatorial Designs and Tournaments (Oxford University Press, 1997).
  19. D. Raghavarao, Constructions and Combinatorial Problems in Design of Experiments (Dover Publications, New York, 1988).
  20. C.-C. Lin, K.-L. Lin, H.-Ch. Chang, and C.-Y. Lee, "A 3.33Gb/s (1200,720) low-density parity check code decoder," in Proc. ESSCIRC 2005, 211-214 (2005), Grenoble, France.
  21. M. Morelli and U. Mengali, "A comparison of pilot-aided channel estimation methods for OFDM systems," IEEE Trans. Signal Process. 49, 3065-3073 (2001).
    [CrossRef]
  22. M. N. Vartak, "On application of Kronecker product of matrices to statistical designs," Ann. Math. Stat. 26, 420-438 (1955).
    [CrossRef]
  23. 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]
  24. 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]
  25. D. J. C. MacKay, "Encyclopedia of sparse graph codes," available at http://www.inference.phy.cam.ac.uk/mackay/codes/data.html

2006

I. B. Djordjevic, and B. Vasic, "100 Gb/s transmission using orthogonal frequency-division multiplexing," IEEE Photon. Technol. Lett. 18, 1576-1578 (2006).
[CrossRef]

W. Shieh, W. Chen and R. S. Tucker, "Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems," Electron. Lett. 42, 996-997 (2006).
[CrossRef]

W. Shieh and C. Athaudage, "Coherent optical frequency division multiplexing," Electron. Lett. 42, 587-589 (2006).
[CrossRef]

W. Shieh, "PMD-supported coherent optical OFDM systems," IEEE Photonics Technol. Lett. 19, 134-136 (2006).
[CrossRef]

I. B. Djordjevic, S. Sankaranarayanan, S. K. 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]

I. B. Djordjevic and B. Vasic, "Orthogonal frequency-division multiplexing for high-speed optical transmission," Opt. Express 14, 3767-3775 (2006).
[CrossRef] [PubMed]

M. Jäger, T. Rankl, J. Spiedel, H. Bulöw, and F. Buchali, "Performance of turbo equalizers for optical PMD channels," J. Lightwave Technol. 24, 1226-1236 (2006).
[CrossRef]

I. B. Djordjevic, and B. Vasic, "Nonlinear BCJR equalizer for suppression of intrachannel nonlinearities in 40 Gb/s optical communications systems," Opt. Express 14, 4625-4635 (2006).
[CrossRef] [PubMed]

2005

2004

H. F. Haunstein,W. Sauer-Greff, A. Dittrich, K. Sticht, and R. Urbansky, "Principles for electronic equalization of polarization-mode dispersion," J. Lightwave Technol. 22, 1169-1182 (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]

2003

2001

M. Morelli and U. Mengali, "A comparison of pilot-aided channel estimation methods for OFDM systems," IEEE Trans. Signal Process. 49, 3065-3073 (2001).
[CrossRef]

1998

A. Garcia Armada and M. Calvo, "Phase noise and sub-carrier effects on the performance of an OFDM communication system," IEEE Commun. Lett. 2, 11-13 (1998).
[CrossRef]

1995

C. Douillard, M. Jézéquel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, "Iterative correction of intersymbol interference: turbo equalization," Eur. Trans. Telecommun. 6, 507-511 (1995).
[CrossRef]

1974

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
[CrossRef]

1955

M. N. Vartak, "On application of Kronecker product of matrices to statistical designs," Ann. Math. Stat. 26, 420-438 (1955).
[CrossRef]

Armstrong, J.

A. J. Lowery and J. Armstrong, "10 Gb/s multimode fiber link using power-efficient orthogonal-frequency-division multiplexing," Opt. Express 13, 10003-10009 (2005).
[CrossRef] [PubMed]

A. J. Lowery, L. Du, and J. Armstrong, "Orthogonal frequency division multiplexing for adaptive dispersion compensation in long haul WDM systems," in Proc. OFC Postdeadline Papers, Paper no. PDP39, 2006.

Athaudage, C.

W. Shieh and C. Athaudage, "Coherent optical frequency division multiplexing," Electron. Lett. 42, 587-589 (2006).
[CrossRef]

Bahl, L. R.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
[CrossRef]

Batshon, H. G.

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., submitted for publication.

Berrou, C.

C. Douillard, M. Jézéquel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, "Iterative correction of intersymbol interference: turbo equalization," Eur. Trans. Telecommun. 6, 507-511 (1995).
[CrossRef]

Buchali, F.

Bulöw, H.

Calvo, M.

A. Garcia Armada and M. Calvo, "Phase noise and sub-carrier effects on the performance of an OFDM communication system," IEEE Commun. Lett. 2, 11-13 (1998).
[CrossRef]

Chen, W.

W. Shieh, W. Chen and R. S. Tucker, "Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems," Electron. Lett. 42, 996-997 (2006).
[CrossRef]

Chilappagari, S. K.

I. B. Djordjevic, S. Sankaranarayanan, S. K. 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]

Cocke, J.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
[CrossRef]

Cvijetic, M.

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., submitted for publication.

Didier, P.

C. Douillard, M. Jézéquel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, "Iterative correction of intersymbol interference: turbo equalization," Eur. Trans. Telecommun. 6, 507-511 (1995).
[CrossRef]

Dittrich, A.

Djordjevic, I. B.

I. B. Djordjevic, and B. Vasic, "Nonlinear BCJR equalizer for suppression of intrachannel nonlinearities in 40 Gb/s optical communications systems," Opt. Express 14, 4625-4635 (2006).
[CrossRef] [PubMed]

I. B. Djordjevic and B. Vasic, "Orthogonal frequency-division multiplexing for high-speed optical transmission," Opt. Express 14, 3767-3775 (2006).
[CrossRef] [PubMed]

I. B. Djordjevic, S. Sankaranarayanan, S. K. 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]

I. B. Djordjevic, and B. Vasic, "100 Gb/s transmission using orthogonal frequency-division multiplexing," IEEE Photon. Technol. Lett. 18, 1576-1578 (2006).
[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]

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, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, "PMD compensation by LDPC coding based turbo equalization," IEEE Photon. Technol. Lett., submitted for publication.

Douillard, C.

C. Douillard, M. Jézéquel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, "Iterative correction of intersymbol interference: turbo equalization," Eur. Trans. Telecommun. 6, 507-511 (1995).
[CrossRef]

Du, L.

A. J. Lowery, L. Du, and J. Armstrong, "Orthogonal frequency division multiplexing for adaptive dispersion compensation in long haul WDM systems," in Proc. OFC Postdeadline Papers, Paper no. PDP39, 2006.

Garcia Armada, A.

A. Garcia Armada and M. Calvo, "Phase noise and sub-carrier effects on the performance of an OFDM communication system," IEEE Commun. Lett. 2, 11-13 (1998).
[CrossRef]

Glavieux, A.

C. Douillard, M. Jézéquel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, "Iterative correction of intersymbol interference: turbo equalization," Eur. Trans. Telecommun. 6, 507-511 (1995).
[CrossRef]

Haunstein, H. F.

Jäger, M.

Jelinek, F.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
[CrossRef]

Jézéquel, M.

C. Douillard, M. Jézéquel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, "Iterative correction of intersymbol interference: turbo equalization," Eur. Trans. Telecommun. 6, 507-511 (1995).
[CrossRef]

Kostuk, R.

Lowery, A. J.

A. J. Lowery and J. Armstrong, "10 Gb/s multimode fiber link using power-efficient orthogonal-frequency-division multiplexing," Opt. Express 13, 10003-10009 (2005).
[CrossRef] [PubMed]

A. J. Lowery, L. Du, and J. Armstrong, "Orthogonal frequency division multiplexing for adaptive dispersion compensation in long haul WDM systems," in Proc. OFC Postdeadline Papers, Paper no. PDP39, 2006.

Mengali, U.

M. Morelli and U. Mengali, "A comparison of pilot-aided channel estimation methods for OFDM systems," IEEE Trans. Signal Process. 49, 3065-3073 (2001).
[CrossRef]

Milenkovic, O.

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]

Morelli, M.

M. Morelli and U. Mengali, "A comparison of pilot-aided channel estimation methods for OFDM systems," IEEE Trans. Signal Process. 49, 3065-3073 (2001).
[CrossRef]

Picart, A.

C. Douillard, M. Jézéquel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, "Iterative correction of intersymbol interference: turbo equalization," Eur. Trans. Telecommun. 6, 507-511 (1995).
[CrossRef]

Rankl, T.

Raviv, J.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
[CrossRef]

Sankaranarayanan, S.

I. B. Djordjevic, S. Sankaranarayanan, S. K. 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]

Sauer-Greff, W.

Shieh, W.

W. Shieh, "PMD-supported coherent optical OFDM systems," IEEE Photonics Technol. Lett. 19, 134-136 (2006).
[CrossRef]

W. Shieh and C. Athaudage, "Coherent optical frequency division multiplexing," Electron. Lett. 42, 587-589 (2006).
[CrossRef]

W. Shieh, W. Chen and R. S. Tucker, "Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems," Electron. Lett. 42, 996-997 (2006).
[CrossRef]

Spiedel, J.

Sticht, K.

Tucker, R. S.

W. Shieh, W. Chen and R. S. Tucker, "Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems," Electron. Lett. 42, 996-997 (2006).
[CrossRef]

Urbansky, R.

Vartak, M. N.

M. N. Vartak, "On application of Kronecker product of matrices to statistical designs," Ann. Math. Stat. 26, 420-438 (1955).
[CrossRef]

Vasic, B.

I. B. Djordjevic, and B. Vasic, "Nonlinear BCJR equalizer for suppression of intrachannel nonlinearities in 40 Gb/s optical communications systems," Opt. Express 14, 4625-4635 (2006).
[CrossRef] [PubMed]

I. B. Djordjevic and B. Vasic, "Orthogonal frequency-division multiplexing for high-speed optical transmission," Opt. Express 14, 3767-3775 (2006).
[CrossRef] [PubMed]

I. B. Djordjevic, and B. Vasic, "100 Gb/s transmission using orthogonal frequency-division multiplexing," IEEE Photon. Technol. Lett. 18, 1576-1578 (2006).
[CrossRef]

I. B. Djordjevic, S. Sankaranarayanan, S. K. 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]

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]

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]

Wang, T.

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., submitted for publication.

Xu, L.

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., submitted for publication.

Ann. Math. Stat.

M. N. Vartak, "On application of Kronecker product of matrices to statistical designs," Ann. Math. Stat. 26, 420-438 (1955).
[CrossRef]

Electron. Lett.

W. Shieh, W. Chen and R. S. Tucker, "Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems," Electron. Lett. 42, 996-997 (2006).
[CrossRef]

W. Shieh and C. Athaudage, "Coherent optical frequency division multiplexing," Electron. Lett. 42, 587-589 (2006).
[CrossRef]

Eur. Trans. Telecommun.

C. Douillard, M. Jézéquel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, "Iterative correction of intersymbol interference: turbo equalization," Eur. Trans. Telecommun. 6, 507-511 (1995).
[CrossRef]

IEEE Commun. Lett.

A. Garcia Armada and M. Calvo, "Phase noise and sub-carrier effects on the performance of an OFDM communication system," IEEE Commun. Lett. 2, 11-13 (1998).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

I. B. Djordjevic, S. Sankaranarayanan, S. K. 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]

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]

IEEE Photon. Technol. Lett.

I. B. Djordjevic, and B. Vasic, "100 Gb/s transmission using orthogonal frequency-division multiplexing," IEEE Photon. Technol. Lett. 18, 1576-1578 (2006).
[CrossRef]

IEEE Photonics Technol. Lett.

W. Shieh, "PMD-supported coherent optical OFDM systems," IEEE Photonics Technol. Lett. 19, 134-136 (2006).
[CrossRef]

IEEE Trans. Inf. Theory

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
[CrossRef]

IEEE Trans. Signal Process.

M. Morelli and U. Mengali, "A comparison of pilot-aided channel estimation methods for OFDM systems," IEEE Trans. Signal Process. 49, 3065-3073 (2001).
[CrossRef]

J. Lightwave Technol.

Opt. Express

Other

D. J. C. MacKay, "Encyclopedia of sparse graph codes," available at http://www.inference.phy.cam.ac.uk/mackay/codes/data.html

I. Anderson, Combinatorial Designs and Tournaments (Oxford University Press, 1997).

D. Raghavarao, Constructions and Combinatorial Problems in Design of Experiments (Dover Publications, New York, 1988).

C.-C. Lin, K.-L. Lin, H.-Ch. Chang, and C.-Y. Lee, "A 3.33Gb/s (1200,720) low-density parity check code decoder," in Proc. ESSCIRC 2005, 211-214 (2005), Grenoble, France.

M. Cvijetic, Coherent and Nonlinear Lightwave Communications (Artech House, Boston 1996).

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., submitted for publication.

R. van Nee, and R. Prasad, OFDM Wireless Multimedia Communications (Artech House, Boston 2000).

A. J. Lowery, L. Du, and J. Armstrong, "Orthogonal frequency division multiplexing for adaptive dispersion compensation in long haul WDM systems," in Proc. OFC Postdeadline Papers, Paper no. PDP39, 2006.

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Figures (5)

Fig. 1.
Fig. 1.

Transmitter configuration (a), receiver configuration (b), and OFDM symbol after cyclic extension (c). LDPCE-LDPC encoder, LDPCD-LDPC decoder, S/P-serial-to-parallel converter, MZM-Mach-Zehnder modulator.

Fig. 2.
Fig. 2.

Constellation diagrams before (a), and after (b) channel estimation for DGD equal to 4/BW (BW-the OFDM signal bandwidth), and k= ½ in the absence of noise.

Fig. 3.
Fig. 3.

BER performance of proposed LDPC codes on an AWGN channel

Fig. 4.
Fig. 4.

BER performance of LDPC(8547,6922)-coded OFDM for different DGD values, and: (a) QPSK, or (b) 16-QAM. The power-splitting ratio between two PSPs was set to k=1/2.

Fig. 5.
Fig. 5.

BER performance of LDPC-coded OFDM for different DGD values, and different LDPC codes of rate R≥0.93. The power-splitting ratio between two PSPs was set to k=1/2.

Equations (12)

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

s OFDM ( t ) = Re { k = w ( t kT ) i = ( N FFT 2 ) ( N FFT 2 ) 1 X i , k e j 2 π i T FFT ( t kT ) e j 2 π f RF t } kT ( T G 2 ) T win t kT + T FFT + ( T G 2 ) + T win
i ( t ) = R { k ( s OFDM ( t ) + b ) * h V ( t ) 2 + 1 k ( s OFDM ( t ) + b ) * h H ( t ) 2 } ,
i ( t ) 2 Rb [ k s OFDM ( t Δ τ 2 ) + ( 1 k ) s OFDM ( t + Δ τ 2 ) ] ,
r ( t ) = [ i ( t ) k RF cos ( ω RF t ) ] * h e ( τ ) + n ( t ) ,
Y i , k = h i e j θ k X i , k + n i , k ,
X ̂ i , k = ( h i * h i 2 ) e j θ k Y i , k .
E ( t ) e j [ ω DFB t + ϕ PN ( t ) ] [ s OFDM ( t ) + b ] ,
i ( t ) E ( t ) 2 e j [ ω DFB t + ϕ PN ( t ) ] [ s OFDM ( t ) + b ] 2 = e j [ ω DFB t + ϕ PN ( t ) ] 2 s OFDM ( t ) + b 2 = s OFDM ( t ) + b 2 ,
v balanced ( t ) e j [ ( ω DFB 1 ω DFB 2 ) t + ϕ P N 1 ( t ) ϕ PN 2 ( t ) ] s OFDM ( t ) ,
v = 20 t + 1 , k = 5 , λ = 1
H = [ H 1 H 2 H t ] ,
R 1 b 1 k 1 b P k P ( v 1 v 2 v P ) .

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