Abstract

White light emitting diodes (LEDs) have been widely utilized for illumination owing to their desired properties of inherent bright output, high efficiency, low power consumption and long life-time. They are also increasingly applied in optical wireless communications for realizing high data rate transmission. This paper presents an improved scheme relying on the insertion of a simple predistortion module before the decoder at the receiver of optical wireless communication systems that use white LEDs. The proposed predistortion scheme exploits the inherent nature of mixing the three unequal optical-power primary colours in generating white light to enhance the system’s performance. Specifically, we design this predistortion module by minimizing the upper bound of the error probability in conjunction with a soft-decision decoder. Our simulation results demonstrate that the detection performance is considerably improved with the aid of the proposed predistortion module.

© 2013 Optical Society of America

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References

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  1. IEEE Std. 802.15.7-2011, Part 15.7: Short-Range Wireless Optical Communication Using Visible Light (2011).
  2. R. Wang, J. Y. Duan, A. C. Shi, Y. J. Wang, and Y. L. Liu, “Indoor optical wireless communication system utilizing white LED lights,” in APCC (2009), pp. 617–621.
  3. D. W. K. Wong and G. Chen, “Optical design and multipath analysis for broadband optical wireless in an aircraft passenger cabin application,” IEEE Trans. Veh. Technol. 57(6), 3598–3606 (2008).
    [Crossref]
  4. M. D. Higgins, R. J. Green, and M. S. Leeson, “Optical wireless for intravehicle communications: a channel viability analysis,” IEEE Trans. Veh. Technol. 61(1), 123–129 (2012).
    [Crossref]
  5. A. M. Khalid, G. Cossu, R. Corsini, P. Choudhury, and E. Ciaramella, “1-Gb/s transmission over a phosphorescent white LED by using rate-adaptive discrete multitone modulation,” IEEE Photonics J. 4(5), 1465–1473 (2012).
    [Crossref]
  6. G. Cossu, A. M. Khalid, P. Choudhury, R. Corsini, and E. Ciaramella, “3.4 Gbit/s visible optical wireless transmission based on RGB LED,” Opt. Express 20(26), B501–B506 (2012).
    [Crossref] [PubMed]
  7. F. M. Wu, C. T. Lin, C. C. Wei, C. W. Chen, Z. Y. Chen, and K. Huang, “3.22-Gb/s WDM visible light communication of a single RGB LED employing carrier-less amplitude and phase modulation,” inOFC/NFOEC (2013), OTh1G.4.
  8. N. Fujimoto and H. Mochizuki, “477 Mbit/s visible light transmission based on OOK-NRZ modulation using a single commercially available visible LED and a practical LED driver with a pre-emphasis circuit,” in OFC/NFOEC(2013), JTh2A.73.
  9. Y. Tanaka, T. Komine, S. Haruyama, and M. Nakagawa, “Indoor visible light data transmission system utilizing white LED lights,” IEICE Trans. Commun. E86-B, (8)2440–2454 (2003).
  10. J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw-Hill, 2008).
  11. J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE85(2), 265–298 (1997).
    [Crossref]
  12. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
    [Crossref]
  13. A. J. Viterbi, “Convolutional codes and their performance in communication systems,” IEEE Trans. Commun. Technol. 19(5), 751–772 (1971).
    [Crossref]
  14. T. Komine, J. H. Lee, S. Haruyama, and M. Nakagawa, “Adaptive equalization system for visible light wireless communication utilizing multiple white LED lighting equipment,” IEEE Trans. Wireless Commun. 8(6), 2892–2900 (2009).
    [Crossref]
  15. D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory 45(2), 399–431 (1999).
    [Crossref]
  16. L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory 20(2), 284–287(1974).
    [Crossref]
  17. IEEE Std. 802.11-2012, Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications (2012).
  18. 3GPP2 C.S0002-F v1.0, Physical Layer Standard for cdma2000 Spread Spectrum Systems (2012).

2012 (3)

M. D. Higgins, R. J. Green, and M. S. Leeson, “Optical wireless for intravehicle communications: a channel viability analysis,” IEEE Trans. Veh. Technol. 61(1), 123–129 (2012).
[Crossref]

A. M. Khalid, G. Cossu, R. Corsini, P. Choudhury, and E. Ciaramella, “1-Gb/s transmission over a phosphorescent white LED by using rate-adaptive discrete multitone modulation,” IEEE Photonics J. 4(5), 1465–1473 (2012).
[Crossref]

G. Cossu, A. M. Khalid, P. Choudhury, R. Corsini, and E. Ciaramella, “3.4 Gbit/s visible optical wireless transmission based on RGB LED,” Opt. Express 20(26), B501–B506 (2012).
[Crossref] [PubMed]

2009 (1)

T. Komine, J. H. Lee, S. Haruyama, and M. Nakagawa, “Adaptive equalization system for visible light wireless communication utilizing multiple white LED lighting equipment,” IEEE Trans. Wireless Commun. 8(6), 2892–2900 (2009).
[Crossref]

2008 (1)

D. W. K. Wong and G. Chen, “Optical design and multipath analysis for broadband optical wireless in an aircraft passenger cabin application,” IEEE Trans. Veh. Technol. 57(6), 3598–3606 (2008).
[Crossref]

2003 (1)

Y. Tanaka, T. Komine, S. Haruyama, and M. Nakagawa, “Indoor visible light data transmission system utilizing white LED lights,” IEICE Trans. Commun. E86-B, (8)2440–2454 (2003).

2002 (1)

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

1999 (1)

D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory 45(2), 399–431 (1999).
[Crossref]

1974 (1)

L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory 20(2), 284–287(1974).
[Crossref]

1971 (1)

A. J. Viterbi, “Convolutional codes and their performance in communication systems,” IEEE Trans. Commun. Technol. 19(5), 751–772 (1971).
[Crossref]

Bahl, L.

L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory 20(2), 284–287(1974).
[Crossref]

Barry, J. R.

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE85(2), 265–298 (1997).
[Crossref]

Chen, C. W.

F. M. Wu, C. T. Lin, C. C. Wei, C. W. Chen, Z. Y. Chen, and K. Huang, “3.22-Gb/s WDM visible light communication of a single RGB LED employing carrier-less amplitude and phase modulation,” inOFC/NFOEC (2013), OTh1G.4.

Chen, G.

D. W. K. Wong and G. Chen, “Optical design and multipath analysis for broadband optical wireless in an aircraft passenger cabin application,” IEEE Trans. Veh. Technol. 57(6), 3598–3606 (2008).
[Crossref]

Chen, Z. Y.

F. M. Wu, C. T. Lin, C. C. Wei, C. W. Chen, Z. Y. Chen, and K. Huang, “3.22-Gb/s WDM visible light communication of a single RGB LED employing carrier-less amplitude and phase modulation,” inOFC/NFOEC (2013), OTh1G.4.

Choudhury, P.

A. M. Khalid, G. Cossu, R. Corsini, P. Choudhury, and E. Ciaramella, “1-Gb/s transmission over a phosphorescent white LED by using rate-adaptive discrete multitone modulation,” IEEE Photonics J. 4(5), 1465–1473 (2012).
[Crossref]

G. Cossu, A. M. Khalid, P. Choudhury, R. Corsini, and E. Ciaramella, “3.4 Gbit/s visible optical wireless transmission based on RGB LED,” Opt. Express 20(26), B501–B506 (2012).
[Crossref] [PubMed]

Ciaramella, E.

G. Cossu, A. M. Khalid, P. Choudhury, R. Corsini, and E. Ciaramella, “3.4 Gbit/s visible optical wireless transmission based on RGB LED,” Opt. Express 20(26), B501–B506 (2012).
[Crossref] [PubMed]

A. M. Khalid, G. Cossu, R. Corsini, P. Choudhury, and E. Ciaramella, “1-Gb/s transmission over a phosphorescent white LED by using rate-adaptive discrete multitone modulation,” IEEE Photonics J. 4(5), 1465–1473 (2012).
[Crossref]

Cocke, J.

L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory 20(2), 284–287(1974).
[Crossref]

Corsini, R.

G. Cossu, A. M. Khalid, P. Choudhury, R. Corsini, and E. Ciaramella, “3.4 Gbit/s visible optical wireless transmission based on RGB LED,” Opt. Express 20(26), B501–B506 (2012).
[Crossref] [PubMed]

A. M. Khalid, G. Cossu, R. Corsini, P. Choudhury, and E. Ciaramella, “1-Gb/s transmission over a phosphorescent white LED by using rate-adaptive discrete multitone modulation,” IEEE Photonics J. 4(5), 1465–1473 (2012).
[Crossref]

Cossu, G.

A. M. Khalid, G. Cossu, R. Corsini, P. Choudhury, and E. Ciaramella, “1-Gb/s transmission over a phosphorescent white LED by using rate-adaptive discrete multitone modulation,” IEEE Photonics J. 4(5), 1465–1473 (2012).
[Crossref]

G. Cossu, A. M. Khalid, P. Choudhury, R. Corsini, and E. Ciaramella, “3.4 Gbit/s visible optical wireless transmission based on RGB LED,” Opt. Express 20(26), B501–B506 (2012).
[Crossref] [PubMed]

Duan, J. Y.

R. Wang, J. Y. Duan, A. C. Shi, Y. J. Wang, and Y. L. Liu, “Indoor optical wireless communication system utilizing white LED lights,” in APCC (2009), pp. 617–621.

Fujimoto, N.

N. Fujimoto and H. Mochizuki, “477 Mbit/s visible light transmission based on OOK-NRZ modulation using a single commercially available visible LED and a practical LED driver with a pre-emphasis circuit,” in OFC/NFOEC(2013), JTh2A.73.

Green, R. J.

M. D. Higgins, R. J. Green, and M. S. Leeson, “Optical wireless for intravehicle communications: a channel viability analysis,” IEEE Trans. Veh. Technol. 61(1), 123–129 (2012).
[Crossref]

Haruyama, S.

T. Komine, J. H. Lee, S. Haruyama, and M. Nakagawa, “Adaptive equalization system for visible light wireless communication utilizing multiple white LED lighting equipment,” IEEE Trans. Wireless Commun. 8(6), 2892–2900 (2009).
[Crossref]

Y. Tanaka, T. Komine, S. Haruyama, and M. Nakagawa, “Indoor visible light data transmission system utilizing white LED lights,” IEICE Trans. Commun. E86-B, (8)2440–2454 (2003).

Higgins, M. D.

M. D. Higgins, R. J. Green, and M. S. Leeson, “Optical wireless for intravehicle communications: a channel viability analysis,” IEEE Trans. Veh. Technol. 61(1), 123–129 (2012).
[Crossref]

Huang, K.

F. M. Wu, C. T. Lin, C. C. Wei, C. W. Chen, Z. Y. Chen, and K. Huang, “3.22-Gb/s WDM visible light communication of a single RGB LED employing carrier-less amplitude and phase modulation,” inOFC/NFOEC (2013), OTh1G.4.

Jelinek, F.

L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory 20(2), 284–287(1974).
[Crossref]

Kahn, J. M.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE85(2), 265–298 (1997).
[Crossref]

Khalid, A. M.

A. M. Khalid, G. Cossu, R. Corsini, P. Choudhury, and E. Ciaramella, “1-Gb/s transmission over a phosphorescent white LED by using rate-adaptive discrete multitone modulation,” IEEE Photonics J. 4(5), 1465–1473 (2012).
[Crossref]

G. Cossu, A. M. Khalid, P. Choudhury, R. Corsini, and E. Ciaramella, “3.4 Gbit/s visible optical wireless transmission based on RGB LED,” Opt. Express 20(26), B501–B506 (2012).
[Crossref] [PubMed]

Komine, T.

T. Komine, J. H. Lee, S. Haruyama, and M. Nakagawa, “Adaptive equalization system for visible light wireless communication utilizing multiple white LED lighting equipment,” IEEE Trans. Wireless Commun. 8(6), 2892–2900 (2009).
[Crossref]

Y. Tanaka, T. Komine, S. Haruyama, and M. Nakagawa, “Indoor visible light data transmission system utilizing white LED lights,” IEICE Trans. Commun. E86-B, (8)2440–2454 (2003).

Lee, J. H.

T. Komine, J. H. Lee, S. Haruyama, and M. Nakagawa, “Adaptive equalization system for visible light wireless communication utilizing multiple white LED lighting equipment,” IEEE Trans. Wireless Commun. 8(6), 2892–2900 (2009).
[Crossref]

Leeson, M. S.

M. D. Higgins, R. J. Green, and M. S. Leeson, “Optical wireless for intravehicle communications: a channel viability analysis,” IEEE Trans. Veh. Technol. 61(1), 123–129 (2012).
[Crossref]

Lin, C. T.

F. M. Wu, C. T. Lin, C. C. Wei, C. W. Chen, Z. Y. Chen, and K. Huang, “3.22-Gb/s WDM visible light communication of a single RGB LED employing carrier-less amplitude and phase modulation,” inOFC/NFOEC (2013), OTh1G.4.

Liu, Y. L.

R. Wang, J. Y. Duan, A. C. Shi, Y. J. Wang, and Y. L. Liu, “Indoor optical wireless communication system utilizing white LED lights,” in APCC (2009), pp. 617–621.

MacKay, D. J. C.

D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory 45(2), 399–431 (1999).
[Crossref]

Mochizuki, H.

N. Fujimoto and H. Mochizuki, “477 Mbit/s visible light transmission based on OOK-NRZ modulation using a single commercially available visible LED and a practical LED driver with a pre-emphasis circuit,” in OFC/NFOEC(2013), JTh2A.73.

Nakagawa, M.

T. Komine, J. H. Lee, S. Haruyama, and M. Nakagawa, “Adaptive equalization system for visible light wireless communication utilizing multiple white LED lighting equipment,” IEEE Trans. Wireless Commun. 8(6), 2892–2900 (2009).
[Crossref]

Y. Tanaka, T. Komine, S. Haruyama, and M. Nakagawa, “Indoor visible light data transmission system utilizing white LED lights,” IEICE Trans. Commun. E86-B, (8)2440–2454 (2003).

Proakis, J. G.

J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw-Hill, 2008).

Raviv, J.

L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory 20(2), 284–287(1974).
[Crossref]

Salehi, M.

J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw-Hill, 2008).

Shi, A. C.

R. Wang, J. Y. Duan, A. C. Shi, Y. J. Wang, and Y. L. Liu, “Indoor optical wireless communication system utilizing white LED lights,” in APCC (2009), pp. 617–621.

Tanaka, Y.

Y. Tanaka, T. Komine, S. Haruyama, and M. Nakagawa, “Indoor visible light data transmission system utilizing white LED lights,” IEICE Trans. Commun. E86-B, (8)2440–2454 (2003).

Viterbi, A. J.

A. J. Viterbi, “Convolutional codes and their performance in communication systems,” IEEE Trans. Commun. Technol. 19(5), 751–772 (1971).
[Crossref]

Wang, R.

R. Wang, J. Y. Duan, A. C. Shi, Y. J. Wang, and Y. L. Liu, “Indoor optical wireless communication system utilizing white LED lights,” in APCC (2009), pp. 617–621.

Wang, Y. J.

R. Wang, J. Y. Duan, A. C. Shi, Y. J. Wang, and Y. L. Liu, “Indoor optical wireless communication system utilizing white LED lights,” in APCC (2009), pp. 617–621.

Wei, C. C.

F. M. Wu, C. T. Lin, C. C. Wei, C. W. Chen, Z. Y. Chen, and K. Huang, “3.22-Gb/s WDM visible light communication of a single RGB LED employing carrier-less amplitude and phase modulation,” inOFC/NFOEC (2013), OTh1G.4.

Wong, D. W. K.

D. W. K. Wong and G. Chen, “Optical design and multipath analysis for broadband optical wireless in an aircraft passenger cabin application,” IEEE Trans. Veh. Technol. 57(6), 3598–3606 (2008).
[Crossref]

Wu, F. M.

F. M. Wu, C. T. Lin, C. C. Wei, C. W. Chen, Z. Y. Chen, and K. Huang, “3.22-Gb/s WDM visible light communication of a single RGB LED employing carrier-less amplitude and phase modulation,” inOFC/NFOEC (2013), OTh1G.4.

Zhu, X.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

IEEE Photonics J. (1)

A. M. Khalid, G. Cossu, R. Corsini, P. Choudhury, and E. Ciaramella, “1-Gb/s transmission over a phosphorescent white LED by using rate-adaptive discrete multitone modulation,” IEEE Photonics J. 4(5), 1465–1473 (2012).
[Crossref]

IEEE Trans. Commun. (1)

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

IEEE Trans. Commun. Technol. (1)

A. J. Viterbi, “Convolutional codes and their performance in communication systems,” IEEE Trans. Commun. Technol. 19(5), 751–772 (1971).
[Crossref]

IEEE Trans. Inf. Theory (2)

D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory 45(2), 399–431 (1999).
[Crossref]

L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory 20(2), 284–287(1974).
[Crossref]

IEEE Trans. Veh. Technol. (2)

D. W. K. Wong and G. Chen, “Optical design and multipath analysis for broadband optical wireless in an aircraft passenger cabin application,” IEEE Trans. Veh. Technol. 57(6), 3598–3606 (2008).
[Crossref]

M. D. Higgins, R. J. Green, and M. S. Leeson, “Optical wireless for intravehicle communications: a channel viability analysis,” IEEE Trans. Veh. Technol. 61(1), 123–129 (2012).
[Crossref]

IEEE Trans. Wireless Commun. (1)

T. Komine, J. H. Lee, S. Haruyama, and M. Nakagawa, “Adaptive equalization system for visible light wireless communication utilizing multiple white LED lighting equipment,” IEEE Trans. Wireless Commun. 8(6), 2892–2900 (2009).
[Crossref]

IEICE Trans. Commun. (1)

Y. Tanaka, T. Komine, S. Haruyama, and M. Nakagawa, “Indoor visible light data transmission system utilizing white LED lights,” IEICE Trans. Commun. E86-B, (8)2440–2454 (2003).

Opt. Express (1)

Other (8)

F. M. Wu, C. T. Lin, C. C. Wei, C. W. Chen, Z. Y. Chen, and K. Huang, “3.22-Gb/s WDM visible light communication of a single RGB LED employing carrier-less amplitude and phase modulation,” inOFC/NFOEC (2013), OTh1G.4.

N. Fujimoto and H. Mochizuki, “477 Mbit/s visible light transmission based on OOK-NRZ modulation using a single commercially available visible LED and a practical LED driver with a pre-emphasis circuit,” in OFC/NFOEC(2013), JTh2A.73.

IEEE Std. 802.15.7-2011, Part 15.7: Short-Range Wireless Optical Communication Using Visible Light (2011).

R. Wang, J. Y. Duan, A. C. Shi, Y. J. Wang, and Y. L. Liu, “Indoor optical wireless communication system utilizing white LED lights,” in APCC (2009), pp. 617–621.

J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw-Hill, 2008).

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE85(2), 265–298 (1997).
[Crossref]

IEEE Std. 802.11-2012, Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications (2012).

3GPP2 C.S0002-F v1.0, Physical Layer Standard for cdma2000 Spread Spectrum Systems (2012).

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Figures (6)

Fig. 1
Fig. 1 Optical wireless communication system using RGB-type white LEDs with the proposed predistortion module inserted before the decoder at receiver.
Fig. 2
Fig. 2 BER performance comparison of the RGB-type LED based optical wireless systems with and without the predistortion module. The convolutional code with the soft-decision Viterbi decoder is employed.
Fig. 3
Fig. 3 BER performance comparison of the RGB-type LED based optical wireless systems with and without the predistortion module. The LDPC code with the BP decoder is employed.
Fig. 4
Fig. 4 BER performance comparison of the RGB-type LED based optical wireless systems with and without the predistortion module. The turbo code with the BCJR decoder is employed.
Fig. 5
Fig. 5 BER performance comparison of the case-1 RGB-type LED based optical wireless systems with and without the predistortion module under the blue LED illumination perturbation. The convolutional code with the soft-decision Viterbi decoder is employed.
Fig. 6
Fig. 6 BER performance comparison of the case-2 RGB-type LED based optical wireless systems with and without the predistortion module under the blue LED illumination perturbation. The convolutional code with the soft-decision Viterbi decoder is employed.

Tables (1)

Tables Icon

Table 1 Combination of three primary colours and the corresponding optical power mixing ratio Mr : Mg : Mb for emitting white light [9].

Equations (14)

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

η = γ e λ h c ,
E r : E g : E b = M r η r : M g η g : M b η b .
r ( t ) = { r ( t ) F r , r ( t ) red light ; r ( t ) F g , r ( t ) green light ; r ( t ) F b , r ( t ) blue light .
P e d = d free n d P d ,
P d = Q ( d E s N 0 ) ,
Q ( x ) = 1 2 π x exp ( t 2 2 ) d t .
P e d = d free k = 1 n d Q ( ( r d , k E r + g d , k E g + b d , k E b ) 2 d N 0 ) ,
P e d = d free k = 1 n d Q ( ( r d , k F r E r + g d , k F g E g + b d , k F b E b ) 2 ( r d , k F r 2 + g d , k F g 2 + b d , k F b 2 ) N 0 ) .
d = d free k = 1 n d Q ( ( r d , k F r E r + g d , k F g E g + b d , k F b E b ) 2 ( r d , k F r 2 + g d , k F g 2 + b d , k F b 2 ) N 0 ) = d = d free k = 1 n d Q ( ( r d , k F r r d , k E r + g d , k F g g d , k E g + b d , k F b b d , k E b ) 2 ( r d , k F r 2 + g d , k F g 2 + b d , k F b 2 ) N 0 ) d = d free k = 1 n d Q ( ( r d , k F r 2 + g d , k F g 2 + b d , k F b 2 ) ( r d , k E r + g d , k E r + b d , k E r ) ( r d , k F r 2 + g d , k F g 2 + b d , k F b 2 ) N 0 ) = d = d free k = 1 n d Q ( ( r d , k E r + g d , k E r + b d , k E r ) N 0 ) ,
F r : F g : F b = E r : E g : E b .
F r 2 + F g 2 + F b 2 = 3 .
F r = 3 M r η r M r 2 η r 2 + M g 2 η g 2 + M b 2 η b 2 ,
F g = 3 M g η g M r 2 η r 2 + M g 2 η g 2 + M b 2 η b 2 ,
F b = 3 M b η b M r 2 η r 2 + M g 2 η g 2 + M b 2 η b 2 .

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