Abstract

In a visible light communication (VLC) system, the nonlinear characteristic of the light emitting diode (LED) in transmitter is a limiting factor of system performance. Modern modulation signals with large peak-to-power-ratio (PAPR) suffers uneven distortion. The nonlinear response directly impacts the intensity modulation and direct detection VLC system with pulse-amplitude modulation (PAM). The amplitude of the PAM signal is distorted unevenly and large signal is vulnerable to noise. Orthogonal linear transformations, such as discrete multi-tone (DMT) modulation, can spread the nonlinear effects evenly to each data symbol, thus perform better than PAM signals. In this paper, we provide theoretical analysis on the benefit of DMT modulation in nonlinear VLC system. We show that the DMT modulation is a better choice than the PAM modulation for the VLC system as the DMT modulation is more robust against nonlinearity. We also show that the post-distortion nonlinear elimination method, which is applied at the receiver, can be a reliable solution to the nonlinear VLC system. Simulation results show that the post-distortion greatly improves the system performance for the DMT modulation.

© 2015 Optical Society of America

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References

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2014 (2)

Y. Wang, X. Huang, J. Zhang, Y. Wang, and N. Chi, “Enhanced performance of visible light communication employing 512-QAM N-SC-FDE and DD-LMS,” Opt. Express 22(13) 15328–15334 (2014).
[Crossref] [PubMed]

H. Qian, S. Yao, S. Cai, and T. Zhou, “Adaptive postdistortion for nonlinear LEDs in visible light communications,” IEEE Photon. J. 6(4), 1–8 (2014).
[Crossref]

2013 (3)

2012 (3)

D. J. Barros, S. K. Wilson, and J. M. Kahn, “Comparison of orthogonal frequency-division multiplexing and pulse-amplitude modulation in indoor optical wireless links,” IEEE T. Commun. 60(1), 153–163 (2012).
[Crossref]

D. Lee, K. Choi, K.-D. Kim, and Y. Park, “Visible light wireless communications based on predistorted OFDM,” Opt. Commun. 285(7), 1767–1770 (2012).
[Crossref]

R. Mesleh, H. Elgala, and H. Haas, “LED nonlinearity mitigation techniques in optical wireless OFDM communication systems,” IEEE/OSA J. Opt. Commun. Network. 4(11), 865–875 (2012).
[Crossref]

2011 (2)

R. Mesleh, H. Elgala, and H. Haas, “On the performance of different OFDM based optical wireless communication systems,” IEEE/OSA J. Opt. Commun. Network. 3(8), 620–628 (2011).
[Crossref]

H. Elgala, R. Mesleh, and H. Haas, “Indoor optical wireless communication: potential and state-of-the-art,” IEEE Commun. Mag. 49(9), 56–62 (2011).
[Crossref]

2009 (2)

2008 (1)

J. Armstrong and B. Schmidt, “Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN,” IEEE Commun. Lett. 12(5), 343–345 (2008).
[Crossref]

2006 (1)

J. Armstrong and A. Lowery, “Power efficient optical OFDM,” IET Electr. Lett. 42(6), 370–372 (2006).
[Crossref]

2004 (2)

P. Schniter, “Low-complexity equalization of OFDM in doubly selective channels,” IEEE T. Signal Process. 52 (4), 1002–1011 (2004).
[Crossref]

R. Raich and G. T. Zhou, “Orthogonal polynomials for complex Gaussian processes,” IEEE T. Signal Process. 52, 2788–2797 (2004).
[Crossref]

2003 (2)

J. Tellado, L. M. Hoo, and J. M. Cioffi, “Maximum-likelihood detection of nonlinearly distorted multicarrier symbols by iterative decoding,” IEEE T. Commun 51(2), 218–228 (2003).
[Crossref]

T. Komine and M. Nakagawa, “Integrated system of white LED visible-light communication and power-line communication,” IEEE T. Consum. Electr. 49(1), 71–79 (2003).
[Crossref]

1978 (1)

K. Asatani and T. Kimura, “Analyses of LED nonlinear distortions,” IEEE J. Solid-st. Circ 13(1), 125–133 (1978).
[Crossref]

1974 (1)

N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE T. Comput. 100(1), 90–93 (1974).
[Crossref]

1967 (1)

Å. Bjöorck, “Solving linear least squares problems by gram-schmidt orthogonalization,” BIT Numer. Math. 7(1), 1–21 (1967).
[Crossref]

Ahmed, N.

N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE T. Comput. 100(1), 90–93 (1974).
[Crossref]

Armstrong, J.

J. Armstrong and B. Schmidt, “Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN,” IEEE Commun. Lett. 12(5), 343–345 (2008).
[Crossref]

J. Armstrong and A. Lowery, “Power efficient optical OFDM,” IET Electr. Lett. 42(6), 370–372 (2006).
[Crossref]

Asatani, K.

K. Asatani and T. Kimura, “Analyses of LED nonlinear distortions,” IEEE J. Solid-st. Circ 13(1), 125–133 (1978).
[Crossref]

Barros, D. J.

D. J. Barros, S. K. Wilson, and J. M. Kahn, “Comparison of orthogonal frequency-division multiplexing and pulse-amplitude modulation in indoor optical wireless links,” IEEE T. Commun. 60(1), 153–163 (2012).
[Crossref]

Benedetto, S.

S. Benedetto and E. Biglieri, Principles of Digital Transmission with Wireless Applications (Kluwer Academic Publishers, July1999).

Biglieri, E.

S. Benedetto and E. Biglieri, Principles of Digital Transmission with Wireless Applications (Kluwer Academic Publishers, July1999).

Bjöorck, Å.

Å. Bjöorck, “Solving linear least squares problems by gram-schmidt orthogonalization,” BIT Numer. Math. 7(1), 1–21 (1967).
[Crossref]

Brillinger, D. R.

D. R. Brillinger, Time Series: Data Analysis and Theory (Holden-day Inc., San Francisco: 1981).

Cai, S.

H. Qian, S. Yao, S. Cai, and T. Zhou, “Adaptive postdistortion for nonlinear LEDs in visible light communications,” IEEE Photon. J. 6(4), 1–8 (2014).
[Crossref]

Chen, J.

H. Qian, J. Chen, S. Yao, and Z. Yu, “One-bit sigma-delta modulator for nonlinear visible light communication systems,” IEEE Photon. Technol. Lett. doc. ID 10.1109/LPT.2014.2376971 (posted 04 December 2014, in press).

Chi, N.

Choi, K.

D. Lee, K. Choi, K.-D. Kim, and Y. Park, “Visible light wireless communications based on predistorted OFDM,” Opt. Commun. 285(7), 1767–1770 (2012).
[Crossref]

Cioffi, J. M.

J. Tellado, L. M. Hoo, and J. M. Cioffi, “Maximum-likelihood detection of nonlinearly distorted multicarrier symbols by iterative decoding,” IEEE T. Commun 51(2), 218–228 (2003).
[Crossref]

Dimitrov, S.

Elgala, H.

R. Mesleh, H. Elgala, and H. Haas, “LED nonlinearity mitigation techniques in optical wireless OFDM communication systems,” IEEE/OSA J. Opt. Commun. Network. 4(11), 865–875 (2012).
[Crossref]

H. Elgala, R. Mesleh, and H. Haas, “Indoor optical wireless communication: potential and state-of-the-art,” IEEE Commun. Mag. 49(9), 56–62 (2011).
[Crossref]

R. Mesleh, H. Elgala, and H. Haas, “On the performance of different OFDM based optical wireless communication systems,” IEEE/OSA J. Opt. Commun. Network. 3(8), 620–628 (2011).
[Crossref]

H. Elgala, R. Mesleh, and H. Haas, “Indoor broadcasting via white LEDs and OFDM,” IEEE T. Consum. Electr. 55 (3), 1127–1134 (2009).
[Crossref]

Grobe, L.

L. Grobe, A. Paraskevopoulos, J. Hilt, D. Schulz, F. Lassak, F. Hartlieb, C. Kottke, V. Jungnickel, and K. Langer, “High-speed visible light communication systems,” IEEE Commun. Mag. 51(12), 60–66 (2013).
[Crossref]

Haas, H.

D. Tsonev, S. Sinanovic, and H. Haas, “Complete modeling of nonlinear distortion in OFDM-based optical wireless communication,” J. Lightwave Technol. 31(18), 3064–3076 (2013).
[Crossref]

R. Mesleh, H. Elgala, and H. Haas, “LED nonlinearity mitigation techniques in optical wireless OFDM communication systems,” IEEE/OSA J. Opt. Commun. Network. 4(11), 865–875 (2012).
[Crossref]

H. Elgala, R. Mesleh, and H. Haas, “Indoor optical wireless communication: potential and state-of-the-art,” IEEE Commun. Mag. 49(9), 56–62 (2011).
[Crossref]

R. Mesleh, H. Elgala, and H. Haas, “On the performance of different OFDM based optical wireless communication systems,” IEEE/OSA J. Opt. Commun. Network. 3(8), 620–628 (2011).
[Crossref]

H. Elgala, R. Mesleh, and H. Haas, “Indoor broadcasting via white LEDs and OFDM,” IEEE T. Consum. Electr. 55 (3), 1127–1134 (2009).
[Crossref]

Han, D.

Y. Xiang, M. Zhang, X. Tang, J. Wu, and D. Han, “A post-processing channel estimation method for DCO-OFDM visible light communication,” IEEE 8th International Symposium on Communication Systems, Networks & Digital Signal Processing (CSNDSP) (IEEE, 2012), pp.1–4.

Hartlieb, F.

L. Grobe, A. Paraskevopoulos, J. Hilt, D. Schulz, F. Lassak, F. Hartlieb, C. Kottke, V. Jungnickel, and K. Langer, “High-speed visible light communication systems,” IEEE Commun. Mag. 51(12), 60–66 (2013).
[Crossref]

Hass, H.

Hilt, J.

L. Grobe, A. Paraskevopoulos, J. Hilt, D. Schulz, F. Lassak, F. Hartlieb, C. Kottke, V. Jungnickel, and K. Langer, “High-speed visible light communication systems,” IEEE Commun. Mag. 51(12), 60–66 (2013).
[Crossref]

Hoo, L. M.

J. Tellado, L. M. Hoo, and J. M. Cioffi, “Maximum-likelihood detection of nonlinearly distorted multicarrier symbols by iterative decoding,” IEEE T. Commun 51(2), 218–228 (2003).
[Crossref]

Huang, X.

Inan, B.

Jungnickel, V.

L. Grobe, A. Paraskevopoulos, J. Hilt, D. Schulz, F. Lassak, F. Hartlieb, C. Kottke, V. Jungnickel, and K. Langer, “High-speed visible light communication systems,” IEEE Commun. Mag. 51(12), 60–66 (2013).
[Crossref]

Kahn, J. M.

D. J. Barros, S. K. Wilson, and J. M. Kahn, “Comparison of orthogonal frequency-division multiplexing and pulse-amplitude modulation in indoor optical wireless links,” IEEE T. Commun. 60(1), 153–163 (2012).
[Crossref]

Kamalakis, T.

Kim, K.-D.

D. Lee, K. Choi, K.-D. Kim, and Y. Park, “Visible light wireless communications based on predistorted OFDM,” Opt. Commun. 285(7), 1767–1770 (2012).
[Crossref]

Kimura, T.

K. Asatani and T. Kimura, “Analyses of LED nonlinear distortions,” IEEE J. Solid-st. Circ 13(1), 125–133 (1978).
[Crossref]

Komine, T.

T. Komine and M. Nakagawa, “Integrated system of white LED visible-light communication and power-line communication,” IEEE T. Consum. Electr. 49(1), 71–79 (2003).
[Crossref]

Kottke, C.

L. Grobe, A. Paraskevopoulos, J. Hilt, D. Schulz, F. Lassak, F. Hartlieb, C. Kottke, V. Jungnickel, and K. Langer, “High-speed visible light communication systems,” IEEE Commun. Mag. 51(12), 60–66 (2013).
[Crossref]

Langer, K.

L. Grobe, A. Paraskevopoulos, J. Hilt, D. Schulz, F. Lassak, F. Hartlieb, C. Kottke, V. Jungnickel, and K. Langer, “High-speed visible light communication systems,” IEEE Commun. Mag. 51(12), 60–66 (2013).
[Crossref]

Lassak, F.

L. Grobe, A. Paraskevopoulos, J. Hilt, D. Schulz, F. Lassak, F. Hartlieb, C. Kottke, V. Jungnickel, and K. Langer, “High-speed visible light communication systems,” IEEE Commun. Mag. 51(12), 60–66 (2013).
[Crossref]

Lee, D.

D. Lee, K. Choi, K.-D. Kim, and Y. Park, “Visible light wireless communications based on predistorted OFDM,” Opt. Commun. 285(7), 1767–1770 (2012).
[Crossref]

Lowery, A.

J. Armstrong and A. Lowery, “Power efficient optical OFDM,” IET Electr. Lett. 42(6), 370–372 (2006).
[Crossref]

Mesleh, R.

R. Mesleh, H. Elgala, and H. Haas, “LED nonlinearity mitigation techniques in optical wireless OFDM communication systems,” IEEE/OSA J. Opt. Commun. Network. 4(11), 865–875 (2012).
[Crossref]

R. Mesleh, H. Elgala, and H. Haas, “On the performance of different OFDM based optical wireless communication systems,” IEEE/OSA J. Opt. Commun. Network. 3(8), 620–628 (2011).
[Crossref]

H. Elgala, R. Mesleh, and H. Haas, “Indoor optical wireless communication: potential and state-of-the-art,” IEEE Commun. Mag. 49(9), 56–62 (2011).
[Crossref]

H. Elgala, R. Mesleh, and H. Haas, “Indoor broadcasting via white LEDs and OFDM,” IEEE T. Consum. Electr. 55 (3), 1127–1134 (2009).
[Crossref]

Nakagawa, M.

T. Komine and M. Nakagawa, “Integrated system of white LED visible-light communication and power-line communication,” IEEE T. Consum. Electr. 49(1), 71–79 (2003).
[Crossref]

Natarajan, T.

N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE T. Comput. 100(1), 90–93 (1974).
[Crossref]

Neokosmidis, I.

Paraskevopoulos, A.

L. Grobe, A. Paraskevopoulos, J. Hilt, D. Schulz, F. Lassak, F. Hartlieb, C. Kottke, V. Jungnickel, and K. Langer, “High-speed visible light communication systems,” IEEE Commun. Mag. 51(12), 60–66 (2013).
[Crossref]

Park, Y.

D. Lee, K. Choi, K.-D. Kim, and Y. Park, “Visible light wireless communications based on predistorted OFDM,” Opt. Commun. 285(7), 1767–1770 (2012).
[Crossref]

Proakis, J. G.

J. G. Proakis, Digital Communications (McGraw-Hill, 3rd ed., 1995).

Qian, H.

H. Qian, S. Yao, S. Cai, and T. Zhou, “Adaptive postdistortion for nonlinear LEDs in visible light communications,” IEEE Photon. J. 6(4), 1–8 (2014).
[Crossref]

H. Qian, J. Chen, S. Yao, and Z. Yu, “One-bit sigma-delta modulator for nonlinear visible light communication systems,” IEEE Photon. Technol. Lett. doc. ID 10.1109/LPT.2014.2376971 (posted 04 December 2014, in press).

Raich, R.

R. Raich and G. T. Zhou, “Orthogonal polynomials for complex Gaussian processes,” IEEE T. Signal Process. 52, 2788–2797 (2004).
[Crossref]

Rao, K. R.

N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE T. Comput. 100(1), 90–93 (1974).
[Crossref]

Schmidt, B.

J. Armstrong and B. Schmidt, “Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN,” IEEE Commun. Lett. 12(5), 343–345 (2008).
[Crossref]

Schniter, P.

P. Schniter, “Low-complexity equalization of OFDM in doubly selective channels,” IEEE T. Signal Process. 52 (4), 1002–1011 (2004).
[Crossref]

Schulz, D.

L. Grobe, A. Paraskevopoulos, J. Hilt, D. Schulz, F. Lassak, F. Hartlieb, C. Kottke, V. Jungnickel, and K. Langer, “High-speed visible light communication systems,” IEEE Commun. Mag. 51(12), 60–66 (2013).
[Crossref]

Sinanovic, S.

Sphicopoulos, T.

Tang, X.

Y. Xiang, M. Zhang, X. Tang, J. Wu, and D. Han, “A post-processing channel estimation method for DCO-OFDM visible light communication,” IEEE 8th International Symposium on Communication Systems, Networks & Digital Signal Processing (CSNDSP) (IEEE, 2012), pp.1–4.

Tellado, J.

J. Tellado, L. M. Hoo, and J. M. Cioffi, “Maximum-likelihood detection of nonlinearly distorted multicarrier symbols by iterative decoding,” IEEE T. Commun 51(2), 218–228 (2003).
[Crossref]

Tsonev, D.

Walewski, J. W.

Wang, Y.

Wilson, S. K.

D. J. Barros, S. K. Wilson, and J. M. Kahn, “Comparison of orthogonal frequency-division multiplexing and pulse-amplitude modulation in indoor optical wireless links,” IEEE T. Commun. 60(1), 153–163 (2012).
[Crossref]

Wu, J.

Y. Xiang, M. Zhang, X. Tang, J. Wu, and D. Han, “A post-processing channel estimation method for DCO-OFDM visible light communication,” IEEE 8th International Symposium on Communication Systems, Networks & Digital Signal Processing (CSNDSP) (IEEE, 2012), pp.1–4.

Xiang, Y.

Y. Xiang, M. Zhang, X. Tang, J. Wu, and D. Han, “A post-processing channel estimation method for DCO-OFDM visible light communication,” IEEE 8th International Symposium on Communication Systems, Networks & Digital Signal Processing (CSNDSP) (IEEE, 2012), pp.1–4.

Yao, S.

H. Qian, S. Yao, S. Cai, and T. Zhou, “Adaptive postdistortion for nonlinear LEDs in visible light communications,” IEEE Photon. J. 6(4), 1–8 (2014).
[Crossref]

H. Qian, J. Chen, S. Yao, and Z. Yu, “One-bit sigma-delta modulator for nonlinear visible light communication systems,” IEEE Photon. Technol. Lett. doc. ID 10.1109/LPT.2014.2376971 (posted 04 December 2014, in press).

Yu, Z.

H. Qian, J. Chen, S. Yao, and Z. Yu, “One-bit sigma-delta modulator for nonlinear visible light communication systems,” IEEE Photon. Technol. Lett. doc. ID 10.1109/LPT.2014.2376971 (posted 04 December 2014, in press).

Zhang, J.

Zhang, M.

Y. Xiang, M. Zhang, X. Tang, J. Wu, and D. Han, “A post-processing channel estimation method for DCO-OFDM visible light communication,” IEEE 8th International Symposium on Communication Systems, Networks & Digital Signal Processing (CSNDSP) (IEEE, 2012), pp.1–4.

Zhou, G. T.

R. Raich and G. T. Zhou, “Orthogonal polynomials for complex Gaussian processes,” IEEE T. Signal Process. 52, 2788–2797 (2004).
[Crossref]

Zhou, T.

H. Qian, S. Yao, S. Cai, and T. Zhou, “Adaptive postdistortion for nonlinear LEDs in visible light communications,” IEEE Photon. J. 6(4), 1–8 (2014).
[Crossref]

BIT Numer. Math. (1)

Å. Bjöorck, “Solving linear least squares problems by gram-schmidt orthogonalization,” BIT Numer. Math. 7(1), 1–21 (1967).
[Crossref]

IEEE Commun. Lett. (1)

J. Armstrong and B. Schmidt, “Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN,” IEEE Commun. Lett. 12(5), 343–345 (2008).
[Crossref]

IEEE Commun. Mag. (2)

H. Elgala, R. Mesleh, and H. Haas, “Indoor optical wireless communication: potential and state-of-the-art,” IEEE Commun. Mag. 49(9), 56–62 (2011).
[Crossref]

L. Grobe, A. Paraskevopoulos, J. Hilt, D. Schulz, F. Lassak, F. Hartlieb, C. Kottke, V. Jungnickel, and K. Langer, “High-speed visible light communication systems,” IEEE Commun. Mag. 51(12), 60–66 (2013).
[Crossref]

IEEE J. Solid-st. Circ (1)

K. Asatani and T. Kimura, “Analyses of LED nonlinear distortions,” IEEE J. Solid-st. Circ 13(1), 125–133 (1978).
[Crossref]

IEEE Photon. J. (1)

H. Qian, S. Yao, S. Cai, and T. Zhou, “Adaptive postdistortion for nonlinear LEDs in visible light communications,” IEEE Photon. J. 6(4), 1–8 (2014).
[Crossref]

IEEE T. Commun (1)

J. Tellado, L. M. Hoo, and J. M. Cioffi, “Maximum-likelihood detection of nonlinearly distorted multicarrier symbols by iterative decoding,” IEEE T. Commun 51(2), 218–228 (2003).
[Crossref]

IEEE T. Commun. (1)

D. J. Barros, S. K. Wilson, and J. M. Kahn, “Comparison of orthogonal frequency-division multiplexing and pulse-amplitude modulation in indoor optical wireless links,” IEEE T. Commun. 60(1), 153–163 (2012).
[Crossref]

IEEE T. Comput. (1)

N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE T. Comput. 100(1), 90–93 (1974).
[Crossref]

IEEE T. Consum. Electr. (2)

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Figures (9)

Fig. 1
Fig. 1 Block diagrams of a VLC transceiver.
Fig. 2
Fig. 2 Nonlinear characteristics of LED and corresponding input signal distribution.
Fig. 3
Fig. 3 Normalized nonlinear offset for single carrier modulation. The solid line shows the offset generated by the nonlinearity. When the signal amplitude is within [−0.125, 0.135], the nonlinear distortion is less than −30 dB and is not shown in the figure.
Fig. 4
Fig. 4 Normalized constellation diagram of 64-QAM for single carrier modulation. The blue dots are the original 64-QAM constellation and the red dots are the distorted nonlinear signal with AWGN noise. The LED nonlinearity is a 7th-order polynomial model whose coefficients are given by Table 1. The SNR at the receiver is 25 dB. A total of 64000 symbols are shown.
Fig. 5
Fig. 5 Block diagrams of a multi-carrier VLC system with post-distortion nonlinear elimination algorithm.
Fig. 6
Fig. 6 The normalized power for DCO-OFDM system with 64, 1024 data subcarriers, respectively. The blue solid line shows the signal power of input signal; the black dash line shows the signal power of output distorted signal; and the red solid line shows the power of distortion term. The results are obtained by averaging 10000 independent realizations.
Fig. 7
Fig. 7 Normalized constellation diagram of 64-QAM for DCO-OFDM modulation. The blue dots are the original 64-QAM constellation and the red dots are the distorted signal with AWGN noise. The SNR at the receiver is 25 dB. A total of 1000 independent DCO-OFDM symbols are shown.
Fig. 8
Fig. 8 Channel response and the BER performance of the proposed schemes.
Fig. 9
Fig. 9 VLC system performance with nonlinear LED. In both cases, from bottom to up, the black solid line shows the performance of a linear system; the blue dashed line shows the performance of the DMT system with post-distortion; the blue solid line shows the performance of the DMT system without post-distortion; the red dashed line shows the performance of the PAM system with post-distortion; the red solid line shows the performance of the PAM system without post-distortion.

Tables (2)

Tables Icon

Table 1 Normalized model coefficients of the LED.

Tables Icon

Table 2 Iterative post-distortion nonlinear elimination algorithm.

Equations (33)

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r ( n ) = h ( y ( n ) ) + v ( n ) ,
y ( n ) = f LED ( X ( n ) ) = p = 1 P a p x p ( n ) ,
P = i = 1 M P i = i = 1 M { p A i γ i f A i ( y ) d y } ,
f A i ( y ) = 1 2 π σ v e ( y f LED ( A i ) ) 2 / 2 σ v 2 ,
P = i = 2 M 2 M Q ( f LED ( A i ) f LED ( A i 1 ) 2 σ v ) ,
x ( n ) = k = 0 N 1 X k e j 2 π n k / N ,
X k = { X k k = 1 , , N / 2 1 X N k * k = N / 2 + 1 , , N 1 0 k = 0 , N / 2 ,
R k = k = 0 N 1 r ( n ) e j 2 π k n / N = n = 0 N 1 f LED ( X ( n ) ) e j 2 π k n / N + V k ,
R k = n = 0 N 1 p = 1 P a p x p ( n ) e j 2 π k n / N + V k a 1 X k + n = 0 N 1 n = 0 P a p x P ( n ) e j 2 π k n / N + V k .
c 2 y ( l ) = E { y ( n ) , y ( n + l ) } E { y ( n ) } E { y ( n + l ) } = cum { y ( n ) , y ( n + l ) } ,
c p x ( l 1 , l 2 , , l p 1 ) = cum { x ( n ) , x ( n + l 1 ) , , x ( n + l p 1 ) } .
c p x ( l 1 , l 2 , , l p 1 ) = 0 , p > 2 .
c 2 y ( l ) = cum { p = 1 P a p x p ( n ) , q = 1 P a q x q ( n + l ) } = p = 1 P q = 1 P a P a P cum { x p ( n ) , x q ( n + l ) } .
x ( n ) x ( n ) p x ( n + l ) x ( n + l ) q .
t 11 ( l ) = cum { x ( n ) , x ( n + l ) } = c 2 x ( l ) .
t 13 ( l ) = cum { x ( n ) , x 3 ( n + l ) } = 3 cum { x ( n ) , x ( n + l ) } cum { x ( n + l ) , x ( n + l ) } + cum { x ( n ) , x ( n + l ) , x ( n + l ) , x ( n + l ) } = 3 c 2 x ( 0 ) c 2 x ( l ) + c 4 x ( 0 , l , l , l ) = 3 σ x 2 c 2 x ( l ) ,
t 33 ( l ) = cum { x 3 ( n + l ) } = 6 c 2 x 3 ( l ) + 9 σ x 4 c 2 x ( l ) .
c 2 y ( l ) = cum { y ( n ) , y ( n + l ) } = p = 1 7 q = 1 7 a p a q t p q = 288 a 7 2 c 2 x 7 ( l ) + ( 120 a 5 2 + 288 σ x 2 a 7 2 + 3456 σ x 4 a 7 2 ) c 2 x 5 ( l ) + ( 6 a 3 2 + 60 σ x 2 a 3 a 5 + 600 σ x 4 a 5 2 + 144 σ x 4 a 3 a 7 + 864 σ x 6 a 5 a 7 + 5184 σ x 8 a 7 2 ) c 2 x 3 ( l ) + ( a 1 2 + 6 σ x 2 a 1 a 3 + 9 σ 3 4 a 3 2 + 30 σ x 4 a 1 a 5 + 45 σ x 6 a 3 a 5 + 24 σ x 6 a 1 a 7 + 225 σ x 8 a 5 2 + 76 σ x 8 a 3 a 7 + 288 σ x 1 0 a 5 a 7 + 1152 σ x 1 2 a 7 2 ) c 2 x ( l ) .
c 2 x ( l ) = { σ x 2 , l = 0 , 0 , l 0 .
S 2 y ( k ) = 10368 a 7 2 σ x 14 + 1092 a 5 a 7 σ x 12 + ( 945 a 5 2 + 220 a 3 a 7 ) σ x 10 + ( 210 a 3 a 5 + 24 a 1 a 7 ) σ x 8 + ( 15 a 3 2 + 30 a 1 a 5 ) σ x 6 + 6 a 1 a 3 σ x 4 + a 1 2 σ x 2 ,
E [ ψ p ( x ( n ) ) , ψ q ( x ( n ) ) ] = { 1 p = q , 0 p q ,
E [ ψ p ( x ( n ) ) , ψ q ( x ( n ) ) ] = ψ p ( x ( n ) ) , ψ q ( x ( n ) ) f p ( x ) d x ,
y ( n ) = p = 1 P σ p ψ p ( x ( n ) ) ,
c 2 y ( l ) = cum { y ( n ) , y ( n + l ) } = cum { p = 1 P α p ψ p ( x ( n ) ) , q = 1 P α p ψ q ( x ( n ) ) } = p = 1 P q = 1 P α p α q cum { ψ p ( x ( n ) ) , ψ q ( x ( n + l ) ) } .
cum { ψ p ( x ( n ) ) , ψ q ( x ( n + l ) ) } = { c 2 x p ( l ) p = q , 0 p q .
c 2 y ( l ) = p = 1 P α p 2 c 2 x p ( l ) .
S 2 y ( k ) = p = 0 P σ p 2 σ x 2 p .
x ( n ) = c ( n ) n = 0 N 1 X k cos [ ( k + 0.5 ) π N n ] ,
c ( n ) = { 1 N n = 0 , 2 N n 0 .
r ( m ) = IDFT ( X ^ ( m 1 ) ) .
e ( m ) = h LED ( r ( m ) ) a 1 r ( m ) .
E ( m ) = DFT ( e ( m ) ) .
X ^ ( m ) = ( R E ( m ) ) / a 1 .

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