Abstract

In this work we propose a pre-distortion technique for the mitigation of the nonlinear distortion present in directly modulated/detected OOFDM systems and explore the system performance achieved under varying system parameters. Simulation results show that the proposed pre-distortion technique efficiently mitigates the nonlinear distortion, achieving transmission information rates around 40Gbits/s and 18.5Gbits/s over 40km and 100km of single mode fiber links, respectively, under optimum operating conditions. Moreover, the proposed pre-distortion technique can potentially provide higher system performance to that obtained with nonlinear equalization at the receiver.

© 2014 Optical Society of America

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  1. W. Shieh, I. Djordjevic, OFDM for Optical Communications (Elsevier/Academic Press, 2009).
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    [CrossRef] [PubMed]
  3. T. Alves, J. Morgado, A. Cartaxo, “Linearity improvement of directly modulated PONs by digital predistortion of coexisting OFDM-based signals,” in Proceedings of Advanced Photonics Congress, (Optical Society of America, 2012), AW4A.2.
  4. Y. Bao, Z. Li, J. Li, X. Feng, B. Guan, G. Li, “Nonlinearity mitigation for high-speed optical OFDM transmitters using digital pre-distortion,” Opt. Express 21(6), 7354–7361 (2013).
    [CrossRef] [PubMed]
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    [CrossRef]
  6. C-C. Wei, “Analysis and iterative equalization of transient and adiabatic chirp effects in DML-based OFDM transmission sytems,” Opt. Express 20(23), 25774–25789 (2012).
    [CrossRef] [PubMed]
  7. D-Z. Hsu, C-C. Wei, H-Y. Chen, Y-C. Lu, C-Y. Song, C-C. Yang, J. Chen, “SSII cancellation in an EAM-based OFDM-IMDD transmission system employing a novel dynamic chirp model,” Opt. Express 21(1), 533–543 (2013).
    [CrossRef] [PubMed]
  8. N. S. André, K. Habel, H. Louchet, A. Richter, “Adaptive nonlinear volterra equalizer for mitigation of chirp-induced distortions in cost effective IMDD OFDM systems,” Opt. Express 21(18), 20999–21009 (2013).
    [CrossRef]
  9. W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, T. Drenski, “100 Gb/s optical IM-DD transmission with 10G-class devices enabled by 65 G samples/s CMOS DAC core,” in Proceedings of OFC/NFOEC2013 (OM3H1).
  10. C. Sánchez, J. L. Wei, B. Ortega, J. Capmany, “Comprehensive impairment and performance description of directly modulated/detected OOFDM systems,” J. Lightw. Technol. 31(20), 3277–3288 (2013).
    [CrossRef]
  11. G. P. Agrawal, N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, 1993).
  12. M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory 29(3), 439–441 (1983).
    [CrossRef]
  13. C. Sánchez, B. Ortega, J. L. Wei, J. Tang, J. Capmany, “Analytical formulation of directly modulated OOFDM signals transmitted over an IM/DD dispersive link,” Opt. Express 21(6), 7651–7666 (2013).
    [CrossRef] [PubMed]
  14. W-P. Peng, B. Zhang, K-M. Feng, X. Wu, A. E. Willner, S. Chi, “Spectrally efficient direct-detected OFDM transmission incorporating a tunable frequency gap and an iterative detection techniques,” J.Lightw. Technol. 31(20), 3277—3288 (2013).
  15. W. Yan, B. Liu, L. Li, Z. Tao, T. Takahara, J. C. Rasmussen, “Nonlinear distortion and DSP-based cmpensation in metro and access networks using discrete multi-tone,” at ECOC 2012, Mo.1.B.2.
  16. A. V. Oppenheim, R. W. Schaffer, J. R. Buck, Discrete-time signal processing (Prentice-Hall, 1999).
  17. G. P. Agrawal, Fiber-optic Communications Systems (Wiley, 1997).

2013 (7)

D. Lam, A. M. Fard, B. Buckley, B. Jalali, “Digital broadband linearization of optical links,” Opt. Letters 38(4), 446–448 (2013).
[CrossRef]

C. Sánchez, J. L. Wei, B. Ortega, J. Capmany, “Comprehensive impairment and performance description of directly modulated/detected OOFDM systems,” J. Lightw. Technol. 31(20), 3277–3288 (2013).
[CrossRef]

W-P. Peng, B. Zhang, K-M. Feng, X. Wu, A. E. Willner, S. Chi, “Spectrally efficient direct-detected OFDM transmission incorporating a tunable frequency gap and an iterative detection techniques,” J.Lightw. Technol. 31(20), 3277—3288 (2013).

D-Z. Hsu, C-C. Wei, H-Y. Chen, Y-C. Lu, C-Y. Song, C-C. Yang, J. Chen, “SSII cancellation in an EAM-based OFDM-IMDD transmission system employing a novel dynamic chirp model,” Opt. Express 21(1), 533–543 (2013).
[CrossRef] [PubMed]

Y. Bao, Z. Li, J. Li, X. Feng, B. Guan, G. Li, “Nonlinearity mitigation for high-speed optical OFDM transmitters using digital pre-distortion,” Opt. Express 21(6), 7354–7361 (2013).
[CrossRef] [PubMed]

C. Sánchez, B. Ortega, J. L. Wei, J. Tang, J. Capmany, “Analytical formulation of directly modulated OOFDM signals transmitted over an IM/DD dispersive link,” Opt. Express 21(6), 7651–7666 (2013).
[CrossRef] [PubMed]

N. S. André, K. Habel, H. Louchet, A. Richter, “Adaptive nonlinear volterra equalizer for mitigation of chirp-induced distortions in cost effective IMDD OFDM systems,” Opt. Express 21(18), 20999–21009 (2013).
[CrossRef]

2012 (1)

2011 (1)

1983 (1)

M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory 29(3), 439–441 (1983).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, 1993).

G. P. Agrawal, Fiber-optic Communications Systems (Wiley, 1997).

Alves, T.

T. Alves, J. Morgado, A. Cartaxo, “Linearity improvement of directly modulated PONs by digital predistortion of coexisting OFDM-based signals,” in Proceedings of Advanced Photonics Congress, (Optical Society of America, 2012), AW4A.2.

André, N. S.

Bao, Y.

Buck, J. R.

A. V. Oppenheim, R. W. Schaffer, J. R. Buck, Discrete-time signal processing (Prentice-Hall, 1999).

Buckley, B.

D. Lam, A. M. Fard, B. Buckley, B. Jalali, “Digital broadband linearization of optical links,” Opt. Letters 38(4), 446–448 (2013).
[CrossRef]

Capmany, J.

C. Sánchez, J. L. Wei, B. Ortega, J. Capmany, “Comprehensive impairment and performance description of directly modulated/detected OOFDM systems,” J. Lightw. Technol. 31(20), 3277–3288 (2013).
[CrossRef]

C. Sánchez, B. Ortega, J. L. Wei, J. Tang, J. Capmany, “Analytical formulation of directly modulated OOFDM signals transmitted over an IM/DD dispersive link,” Opt. Express 21(6), 7651–7666 (2013).
[CrossRef] [PubMed]

Cartaxo, A.

T. Alves, J. Morgado, A. Cartaxo, “Linearity improvement of directly modulated PONs by digital predistortion of coexisting OFDM-based signals,” in Proceedings of Advanced Photonics Congress, (Optical Society of America, 2012), AW4A.2.

Carvalho, N. B.

Chen, H-Y.

Chen, J.

Chi, S.

W-P. Peng, B. Zhang, K-M. Feng, X. Wu, A. E. Willner, S. Chi, “Spectrally efficient direct-detected OFDM transmission incorporating a tunable frequency gap and an iterative detection techniques,” J.Lightw. Technol. 31(20), 3277—3288 (2013).

Costa, M.

M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory 29(3), 439–441 (1983).
[CrossRef]

Djordjevic, I.

W. Shieh, I. Djordjevic, OFDM for Optical Communications (Elsevier/Academic Press, 2009).

Drenski, T.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, T. Drenski, “100 Gb/s optical IM-DD transmission with 10G-class devices enabled by 65 G samples/s CMOS DAC core,” in Proceedings of OFC/NFOEC2013 (OM3H1).

Dutta, N. K.

G. P. Agrawal, N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, 1993).

Fard, A. M.

D. Lam, A. M. Fard, B. Buckley, B. Jalali, “Digital broadband linearization of optical links,” Opt. Letters 38(4), 446–448 (2013).
[CrossRef]

Feng, K-M.

W-P. Peng, B. Zhang, K-M. Feng, X. Wu, A. E. Willner, S. Chi, “Spectrally efficient direct-detected OFDM transmission incorporating a tunable frequency gap and an iterative detection techniques,” J.Lightw. Technol. 31(20), 3277—3288 (2013).

Feng, X.

Guan, B.

Habel, K.

Hsu, D-Z.

Jalali, B.

D. Lam, A. M. Fard, B. Buckley, B. Jalali, “Digital broadband linearization of optical links,” Opt. Letters 38(4), 446–448 (2013).
[CrossRef]

Lam, D.

D. Lam, A. M. Fard, B. Buckley, B. Jalali, “Digital broadband linearization of optical links,” Opt. Letters 38(4), 446–448 (2013).
[CrossRef]

Li, G.

Li, J.

Li, L.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, T. Drenski, “100 Gb/s optical IM-DD transmission with 10G-class devices enabled by 65 G samples/s CMOS DAC core,” in Proceedings of OFC/NFOEC2013 (OM3H1).

W. Yan, B. Liu, L. Li, Z. Tao, T. Takahara, J. C. Rasmussen, “Nonlinear distortion and DSP-based cmpensation in metro and access networks using discrete multi-tone,” at ECOC 2012, Mo.1.B.2.

Li, Z.

Liu, B.

W. Yan, B. Liu, L. Li, Z. Tao, T. Takahara, J. C. Rasmussen, “Nonlinear distortion and DSP-based cmpensation in metro and access networks using discrete multi-tone,” at ECOC 2012, Mo.1.B.2.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, T. Drenski, “100 Gb/s optical IM-DD transmission with 10G-class devices enabled by 65 G samples/s CMOS DAC core,” in Proceedings of OFC/NFOEC2013 (OM3H1).

Liu, Z.

Louchet, H.

Lu, Y-C.

Morgado, J.

T. Alves, J. Morgado, A. Cartaxo, “Linearity improvement of directly modulated PONs by digital predistortion of coexisting OFDM-based signals,” in Proceedings of Advanced Photonics Congress, (Optical Society of America, 2012), AW4A.2.

Nishihara, M.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, T. Drenski, “100 Gb/s optical IM-DD transmission with 10G-class devices enabled by 65 G samples/s CMOS DAC core,” in Proceedings of OFC/NFOEC2013 (OM3H1).

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schaffer, J. R. Buck, Discrete-time signal processing (Prentice-Hall, 1999).

Ortega, B.

C. Sánchez, J. L. Wei, B. Ortega, J. Capmany, “Comprehensive impairment and performance description of directly modulated/detected OOFDM systems,” J. Lightw. Technol. 31(20), 3277–3288 (2013).
[CrossRef]

C. Sánchez, B. Ortega, J. L. Wei, J. Tang, J. Capmany, “Analytical formulation of directly modulated OOFDM signals transmitted over an IM/DD dispersive link,” Opt. Express 21(6), 7651–7666 (2013).
[CrossRef] [PubMed]

Peng, W-P.

W-P. Peng, B. Zhang, K-M. Feng, X. Wu, A. E. Willner, S. Chi, “Spectrally efficient direct-detected OFDM transmission incorporating a tunable frequency gap and an iterative detection techniques,” J.Lightw. Technol. 31(20), 3277—3288 (2013).

Rasmussen, J. C.

W. Yan, B. Liu, L. Li, Z. Tao, T. Takahara, J. C. Rasmussen, “Nonlinear distortion and DSP-based cmpensation in metro and access networks using discrete multi-tone,” at ECOC 2012, Mo.1.B.2.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, T. Drenski, “100 Gb/s optical IM-DD transmission with 10G-class devices enabled by 65 G samples/s CMOS DAC core,” in Proceedings of OFC/NFOEC2013 (OM3H1).

Richter, A.

Sánchez, C.

C. Sánchez, B. Ortega, J. L. Wei, J. Tang, J. Capmany, “Analytical formulation of directly modulated OOFDM signals transmitted over an IM/DD dispersive link,” Opt. Express 21(6), 7651–7666 (2013).
[CrossRef] [PubMed]

C. Sánchez, J. L. Wei, B. Ortega, J. Capmany, “Comprehensive impairment and performance description of directly modulated/detected OOFDM systems,” J. Lightw. Technol. 31(20), 3277–3288 (2013).
[CrossRef]

Schaffer, R. W.

A. V. Oppenheim, R. W. Schaffer, J. R. Buck, Discrete-time signal processing (Prentice-Hall, 1999).

Shieh, W.

W. Shieh, I. Djordjevic, OFDM for Optical Communications (Elsevier/Academic Press, 2009).

Song, C-Y.

Takahara, T.

W. Yan, B. Liu, L. Li, Z. Tao, T. Takahara, J. C. Rasmussen, “Nonlinear distortion and DSP-based cmpensation in metro and access networks using discrete multi-tone,” at ECOC 2012, Mo.1.B.2.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, T. Drenski, “100 Gb/s optical IM-DD transmission with 10G-class devices enabled by 65 G samples/s CMOS DAC core,” in Proceedings of OFC/NFOEC2013 (OM3H1).

Tanaka, T.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, T. Drenski, “100 Gb/s optical IM-DD transmission with 10G-class devices enabled by 65 G samples/s CMOS DAC core,” in Proceedings of OFC/NFOEC2013 (OM3H1).

Tang, J.

Tao, Z.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, T. Drenski, “100 Gb/s optical IM-DD transmission with 10G-class devices enabled by 65 G samples/s CMOS DAC core,” in Proceedings of OFC/NFOEC2013 (OM3H1).

W. Yan, B. Liu, L. Li, Z. Tao, T. Takahara, J. C. Rasmussen, “Nonlinear distortion and DSP-based cmpensation in metro and access networks using discrete multi-tone,” at ECOC 2012, Mo.1.B.2.

Violas, M. A.

Wei, C-C.

Wei, J. L.

C. Sánchez, B. Ortega, J. L. Wei, J. Tang, J. Capmany, “Analytical formulation of directly modulated OOFDM signals transmitted over an IM/DD dispersive link,” Opt. Express 21(6), 7651–7666 (2013).
[CrossRef] [PubMed]

C. Sánchez, J. L. Wei, B. Ortega, J. Capmany, “Comprehensive impairment and performance description of directly modulated/detected OOFDM systems,” J. Lightw. Technol. 31(20), 3277–3288 (2013).
[CrossRef]

Willner, A. E.

W-P. Peng, B. Zhang, K-M. Feng, X. Wu, A. E. Willner, S. Chi, “Spectrally efficient direct-detected OFDM transmission incorporating a tunable frequency gap and an iterative detection techniques,” J.Lightw. Technol. 31(20), 3277—3288 (2013).

Wu, X.

W-P. Peng, B. Zhang, K-M. Feng, X. Wu, A. E. Willner, S. Chi, “Spectrally efficient direct-detected OFDM transmission incorporating a tunable frequency gap and an iterative detection techniques,” J.Lightw. Technol. 31(20), 3277—3288 (2013).

Yan, W.

W. Yan, B. Liu, L. Li, Z. Tao, T. Takahara, J. C. Rasmussen, “Nonlinear distortion and DSP-based cmpensation in metro and access networks using discrete multi-tone,” at ECOC 2012, Mo.1.B.2.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, T. Drenski, “100 Gb/s optical IM-DD transmission with 10G-class devices enabled by 65 G samples/s CMOS DAC core,” in Proceedings of OFC/NFOEC2013 (OM3H1).

Yang, C-C.

Zhang, B.

W-P. Peng, B. Zhang, K-M. Feng, X. Wu, A. E. Willner, S. Chi, “Spectrally efficient direct-detected OFDM transmission incorporating a tunable frequency gap and an iterative detection techniques,” J.Lightw. Technol. 31(20), 3277—3288 (2013).

IEEE Trans. Inform. Theory (1)

M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory 29(3), 439–441 (1983).
[CrossRef]

J. Lightw. Technol. (1)

C. Sánchez, J. L. Wei, B. Ortega, J. Capmany, “Comprehensive impairment and performance description of directly modulated/detected OOFDM systems,” J. Lightw. Technol. 31(20), 3277–3288 (2013).
[CrossRef]

J.Lightw. Technol. (1)

W-P. Peng, B. Zhang, K-M. Feng, X. Wu, A. E. Willner, S. Chi, “Spectrally efficient direct-detected OFDM transmission incorporating a tunable frequency gap and an iterative detection techniques,” J.Lightw. Technol. 31(20), 3277—3288 (2013).

Opt. Express (6)

Opt. Letters (1)

D. Lam, A. M. Fard, B. Buckley, B. Jalali, “Digital broadband linearization of optical links,” Opt. Letters 38(4), 446–448 (2013).
[CrossRef]

Other (7)

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, T. Drenski, “100 Gb/s optical IM-DD transmission with 10G-class devices enabled by 65 G samples/s CMOS DAC core,” in Proceedings of OFC/NFOEC2013 (OM3H1).

W. Shieh, I. Djordjevic, OFDM for Optical Communications (Elsevier/Academic Press, 2009).

T. Alves, J. Morgado, A. Cartaxo, “Linearity improvement of directly modulated PONs by digital predistortion of coexisting OFDM-based signals,” in Proceedings of Advanced Photonics Congress, (Optical Society of America, 2012), AW4A.2.

W. Yan, B. Liu, L. Li, Z. Tao, T. Takahara, J. C. Rasmussen, “Nonlinear distortion and DSP-based cmpensation in metro and access networks using discrete multi-tone,” at ECOC 2012, Mo.1.B.2.

A. V. Oppenheim, R. W. Schaffer, J. R. Buck, Discrete-time signal processing (Prentice-Hall, 1999).

G. P. Agrawal, Fiber-optic Communications Systems (Wiley, 1997).

G. P. Agrawal, N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, 1993).

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Figures (8)

Fig. 1
Fig. 1

OOFDM system. Blocks in red: parts due to the proposed pre-distortion technique.

Fig. 2
Fig. 2

Block diagram of the proposed pre-distortion technique.

Fig. 3
Fig. 3

Nonlinear distortion cancellation ratio ηcanc for a) i0 = 45mA and b) i0 = 85mA. Optical fiber length equals to 40km.

Fig. 4
Fig. 4

Transmission information rate as a function of CR and Δi for L = 40km a) i0 = 45mA, b) i0 = 65mA, c) i0 = 85mA, and L = 100km e) i0 = 45mA, f) i0 = 65mA, g) i0 = 85mA.

Fig. 5
Fig. 5

a) Information transmission rate achieved in function of the number of samples for the cyclic extensions, b) BER of each subcarrier and corresponding modulation format. Blue: optical fiber length equals to 40km. Red: optical fiber lengths equals to 100km.

Fig. 6
Fig. 6

a) System transfer function of the conventional system, b) SNR of the conventional system, c) System transfer function of the system with pre-distortion technique, d) SNR of the system with pre-distortion technique (SNRprd).

Fig. 7
Fig. 7

Obtained BER values obtained through brute force simulations.

Fig. 8
Fig. 8

Constellation diagrams of the 62th to 110th subcarriers for L = 40km. a) Conventional DM/DD OOFDM system (128-QAM), b) DM/DD OOFDM system without pre-distortion technique (512-QAM), and c) DM/DD OOFDM system with pre-distortion technique (512-QAM).

Tables (1)

Tables Icon

Table 1 Laser parameters

Equations (16)

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x [ n ] = 2 Re { k = 1 N X k exp ( j 2 π k n F S ) }
s ( t ) k = 1 N | X k | cos ( Ω k t + φ X k ) * h t r x ( t ) + n clip ( t ) , 0 t T
i ( t ) = i 0 + i m ( t ) = i 0 + m ( k = 1 N | X k | cos ( Ω k t + φ X k ) * h t r x ( t ) + n t r x ( t ) ) = i 0 + k = 1 N 2 i k cos ( Ω k t + φ i k ) + n t r x ( t ) , 0 t T
d p ( t ) d t = [ Γ G ( n , p ) γ ] p ( t ) + R s p ( n ) d n ( t ) d t = i ( t ) e V γ e ( n ) G ( n , p ) p ( t ) d ϕ d t = 1 2 α Γ v g a g ( n ( t ) n t )
H fib ( ω ) = e α fib 2 L e j β ( ω ) L = e α fib 2 L e j ( β 0 + β 1 ( ω ω 0 ) + 1 2 β 2 ( ω ω 0 ) 2 ) L
SNR [ k ] = | H [ k ] | 2 | X [ k ] | 2 σ clip 2 [ k ] + σ ISI & ICI 2 [ k ] + σ s & t 2 + σ IMD 2 [ k ] , k = 1 , 2 , N
I [ k ] = I p , DML [ k ] + I ϕ , DML [ k ] + I p / ϕ , β 2 [ k ] + I ϕ , β 2 [ k ] + I p , β 2 [ k ] , k = 1 , 2 N
R ( bits / s ) = k = 1 N log 2 ( M [ k ] ) ( B W / ( F S / 2 ) ) 1 ( 1 + η pre + η pos )
BER T = k = 1 N log 2 ( M [ k ] ) BER [ k ] k = 1 N log 2 ( M [ k ] )
I p , DML [ k ] = H p 11 ( Ω k ) cos ( θ k ) ( l = 1 k / 2 1 H p 1 ( Ω l ) H p 1 ( Ω k l ) i l i k l e j ( φ i l + φ i k l ) + l = k + 1 N H p 1 ( Ω l ) H p 1 * ( Ω l k ) i l i l k e j ( φ i l φ i l k ) )
H ϕ 11 ( Ω l , Ω k l ) = ( H ϕ 1 ( Ω k ) H p 1 ( Ω k ) H p 11 ( Ω k ) 1 2 α Γ v g a g Γ G n ( n 0 , p 0 ) ( δ 1 n ( Ω l ) δ 1 p ( Ω l ) + δ 1 n ( Ω k l ) δ 1 p ( Ω k 1 ) ) ( Γ p 0 G n ( n 0 , p 0 ) + R s p ( n 0 ) + n 0 d R s p ( n ) d n | n = n 0 ) ) H p 1 ( Ω l ) H p 1 ( Ω k l )
I rec 0 [ k ] = l = 1 k / 2 1 ( H p / ϕ , β 2 { 1 , 1 } ( Ω l ) H p / ϕ , β 2 { 1 , 1 } ( Ω k l ) + H ϕ , β 2 { 1 , 1 } ( Ω l ) H ϕ , β 2 { 1 , 1 } ( Ω k l ) + H p , β 2 { 1 , 1 } ( Ω k ) H p , β 2 { 1 , 1 } ( Ω k l ) + H p , DML ( Ω k ) H p , DML { 1 , 1 } ( Ω l ) H p , DML { 1 , 1 } ( Ω k l ) + H ϕ , DML ( Ω k ) H ϕ , DML { 1 , 1 } ( Ω l ) H ϕ , DML { 1 , 1 } ( Ω k l ) ) X l 0 X k l 0 + l = k + 1 N ( H p / ϕ , β 2 { 1 , 1 } ( Ω l ) H p / ϕ , β 2 { 1 , 1 } ( Ω l k ) + H ϕ , β 2 { 1 , 1 } ( Ω l ) H ϕ , β 2 { 1 , 1 } ( Ω l k ) + H p , β 2 { 1 , 11 } ( Ω k ) H p , β 2 { 1 , 1 } ( Ω l k ) + H p , DML ( Ω k ) H p , DML { 1 , 1 } ( Ω l ) H p , DML { 1 , 1 } ( Ω l k ) + H ϕ , DML ( Ω k ) H ϕ , DML { 1 , 1 } ( Ω l ) H ϕ , DML ( Ω l k ) ) X l 0 ( X l k 0 ) * , k = 1 , 2 , N
X k 1 = X k 0 I rec 0 [ k ] / H [ k ] , k = 1 , 2 , N
Y [ k ] = H [ k ] X k 1 + I 1 [ k ] = H [ k ] X k 0 I rec 0 [ k ] + I 1 [ k ] , k = 1 , 2 , N
Y [ k ] = H [ k ] X k 0 I rec 0 [ k ] + I 0 [ k ] + χ 1 [ k ] H [ k ] X k 0 + χ 1 [ k ] , k = 1 , 2 , N
SNR prd [ k ] = κ 2 | H [ k ] | 2 | X [ k ] | 2 κ 2 ( σ clip 2 [ k ] + σ ISI & ICI 2 [ k ] + σ shot 2 ) + η canc [ k ] σ IMD 2 [ k ] + σ ther 2 , k = 1 , 2 , N

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