Abstract

Abstract: While coherency between an RF-tone and OFDM signals in RoF systems at 60 GHz is de-correlated by fiber dispersion, both phase rotation term (PRT) on each subcarrier and inter-carrier interference (ICI) between subcarriers are induced at a receiver. We analytically calculate the powers of PRT and ICI under different parameters, such as subcarrier number, modulation format, laser linewidth and transmission distance. Moreover, dispersion-induced ICI is shown to be non-Gaussian distributed by its kurtosis, and its distribution depends on system parameters. Therefore, using only the power of ICI cannot predict accurate bit error rate (BER) and corresponding power penalty. We propose to use t-distribution to fit the distribution of ICI, and it can be used to compute BER precisely.

© 2010 OSA

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References

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    [CrossRef]
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    [CrossRef]
  3. Y. X. Guo, B. Luo, C. S. Park, L. C. Ong, M.-T. Zhou, and S. Kato, “60 GHz radio-over-fiber for Gbps transmission,” in Proc. Global Symp. Millimeter Waves (GSMM), 41–43 (2008).
  4. H.-C. Chien, A. Chowdhury, Z. Jia, Y.-T. Hsueh, and G.-K. Chang, “60 GHz millimeter-wave gigabit wireless services over long-reach passive optical network using remote signal regeneration and upconversion,” Opt. Express 17, 3016–3024 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe- 17-5-3016.
  5. C. T. Lin, E. Z. Wong, W. J. Jiang, P. T. Shih, J. Chen, and S. Chi, “28‐Gb/s 16‐QAM OFDM radio‐over‐fiber system within 7‐GHz license‐free band at 60 GHz employing all-optical up-conversion,” in Proc. CLEO 2009, Maryland, Baltimore, CPDA8 (2009).
  6. Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
    [CrossRef]
  7. C.-T. Lin, J. Chen, P.-T. Shih, W.-J. Jiang, and S. Chi, “Ultra-high data-rate 60 GHz radio-over-fiber systems employing optical frequency multiplication and OFDM formats,” J. Lightwave Technol. 28(16), 2296–2306 (2010).
    [CrossRef]
  8. J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009).
    [CrossRef]
  9. Z. Zan, M. Premaratne, and A. J. Lowery, “Laser RIN and linewidth requirements for direct detection optical OFDM,” in Proc. CLEO’08, San Jose, CWN2 (2008).
  10. W.-R. Peng, J. Chen, and S. Chi, “On the phase noise impact in direct-detection optical OFDM transmission,” IEEE Photon. Technol. Lett. 22(9), 649–651 (2010).
    [CrossRef]
  11. C.-T. Lin, P.-T. Shih, J. Chen, W.-Q. Xue, P.-C. Peng, and S. Chi, “Optical millimeter-wave signal generation using frequency quadrupling technique and no optical filtering,” IEEE Photon. Technol. Lett. 20(12), 1027–1029 (2008).
    [CrossRef]
  12. P.-T. Shih, J. Chen, C.-T. Lin, W.-J. Jiang, H.-S. Huang, P.-C. Peng, and S. Chi, “Optical millimeter-wave signal generation via frequency 12-tupling,” J. Lightwave Technol. 28(1), 71–78 (2010).
    [CrossRef]
  13. K. Higuma, S. Oikawa, Y. Hashimoto, H. Nagata, and M. Izutsu, “X-cut lithium niobate optical single-sideband modulator,” Electron. Lett. 37(8), 515–516 (2001).
    [CrossRef]
  14. M. S. El-Tanany, Y. Wu, and L. Hazy, “Analytical modeling and simulation of phase noise interference in OFDM-based digital television terrestrial broadcasting systems,” IEEE Trans. Broadcast 47(1), 20–31 (2001).
    [CrossRef]
  15. X. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM systems,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
    [CrossRef]
  16. D. Petrovic, W. Rave, and G. Fettweis, “Properties of the intercarrier interference due to phase noise in OFDM,” in Proc. ICC’05, 2605–2610 (2005)
  17. E. Costa and S. Pupolin, “M-QAM-OFDM system performance in the presence of a nonlinear amplifier and phase noise,” IEEE Trans. Commun. 50(3), 462–472 (2002).
    [CrossRef]
  18. K. Pearson, “Contributions to the mathematical theory of evolution. II. skew variation in homogeneous material,” Philos. Trans. Roy. Soc. London Ser. A 186(0), 343–414 (1895).
    [CrossRef]

2010 (3)

2009 (1)

2008 (4)

A. M. J. Koonen and L. M. Garcia, “Radio-over-MMF techniques – part II: microwave to millimeter-wave systems,” J. Lightwave Technol. 26(15), 2396–2408 (2008).
[CrossRef]

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

C.-T. Lin, P.-T. Shih, J. Chen, W.-Q. Xue, P.-C. Peng, and S. Chi, “Optical millimeter-wave signal generation using frequency quadrupling technique and no optical filtering,” IEEE Photon. Technol. Lett. 20(12), 1027–1029 (2008).
[CrossRef]

X. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM systems,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
[CrossRef]

2007 (1)

2002 (1)

E. Costa and S. Pupolin, “M-QAM-OFDM system performance in the presence of a nonlinear amplifier and phase noise,” IEEE Trans. Commun. 50(3), 462–472 (2002).
[CrossRef]

2001 (2)

K. Higuma, S. Oikawa, Y. Hashimoto, H. Nagata, and M. Izutsu, “X-cut lithium niobate optical single-sideband modulator,” Electron. Lett. 37(8), 515–516 (2001).
[CrossRef]

M. S. El-Tanany, Y. Wu, and L. Hazy, “Analytical modeling and simulation of phase noise interference in OFDM-based digital television terrestrial broadcasting systems,” IEEE Trans. Broadcast 47(1), 20–31 (2001).
[CrossRef]

1895 (1)

K. Pearson, “Contributions to the mathematical theory of evolution. II. skew variation in homogeneous material,” Philos. Trans. Roy. Soc. London Ser. A 186(0), 343–414 (1895).
[CrossRef]

Armstrong, J.

Buck, J. A.

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

Chang, G.-K.

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

Chen, J.

C.-T. Lin, J. Chen, P.-T. Shih, W.-J. Jiang, and S. Chi, “Ultra-high data-rate 60 GHz radio-over-fiber systems employing optical frequency multiplication and OFDM formats,” J. Lightwave Technol. 28(16), 2296–2306 (2010).
[CrossRef]

W.-R. Peng, J. Chen, and S. Chi, “On the phase noise impact in direct-detection optical OFDM transmission,” IEEE Photon. Technol. Lett. 22(9), 649–651 (2010).
[CrossRef]

P.-T. Shih, J. Chen, C.-T. Lin, W.-J. Jiang, H.-S. Huang, P.-C. Peng, and S. Chi, “Optical millimeter-wave signal generation via frequency 12-tupling,” J. Lightwave Technol. 28(1), 71–78 (2010).
[CrossRef]

C.-T. Lin, P.-T. Shih, J. Chen, W.-Q. Xue, P.-C. Peng, and S. Chi, “Optical millimeter-wave signal generation using frequency quadrupling technique and no optical filtering,” IEEE Photon. Technol. Lett. 20(12), 1027–1029 (2008).
[CrossRef]

Chi, S.

C.-T. Lin, J. Chen, P.-T. Shih, W.-J. Jiang, and S. Chi, “Ultra-high data-rate 60 GHz radio-over-fiber systems employing optical frequency multiplication and OFDM formats,” J. Lightwave Technol. 28(16), 2296–2306 (2010).
[CrossRef]

W.-R. Peng, J. Chen, and S. Chi, “On the phase noise impact in direct-detection optical OFDM transmission,” IEEE Photon. Technol. Lett. 22(9), 649–651 (2010).
[CrossRef]

P.-T. Shih, J. Chen, C.-T. Lin, W.-J. Jiang, H.-S. Huang, P.-C. Peng, and S. Chi, “Optical millimeter-wave signal generation via frequency 12-tupling,” J. Lightwave Technol. 28(1), 71–78 (2010).
[CrossRef]

C.-T. Lin, P.-T. Shih, J. Chen, W.-Q. Xue, P.-C. Peng, and S. Chi, “Optical millimeter-wave signal generation using frequency quadrupling technique and no optical filtering,” IEEE Photon. Technol. Lett. 20(12), 1027–1029 (2008).
[CrossRef]

Chien, H.-C.

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

Chowdhury, A.

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

Costa, E.

E. Costa and S. Pupolin, “M-QAM-OFDM system performance in the presence of a nonlinear amplifier and phase noise,” IEEE Trans. Commun. 50(3), 462–472 (2002).
[CrossRef]

El-Tanany, M. S.

M. S. El-Tanany, Y. Wu, and L. Hazy, “Analytical modeling and simulation of phase noise interference in OFDM-based digital television terrestrial broadcasting systems,” IEEE Trans. Broadcast 47(1), 20–31 (2001).
[CrossRef]

Garcia, L. M.

George, J.

Hashimoto, Y.

K. Higuma, S. Oikawa, Y. Hashimoto, H. Nagata, and M. Izutsu, “X-cut lithium niobate optical single-sideband modulator,” Electron. Lett. 37(8), 515–516 (2001).
[CrossRef]

Hazy, L.

M. S. El-Tanany, Y. Wu, and L. Hazy, “Analytical modeling and simulation of phase noise interference in OFDM-based digital television terrestrial broadcasting systems,” IEEE Trans. Broadcast 47(1), 20–31 (2001).
[CrossRef]

Higuma, K.

K. Higuma, S. Oikawa, Y. Hashimoto, H. Nagata, and M. Izutsu, “X-cut lithium niobate optical single-sideband modulator,” Electron. Lett. 37(8), 515–516 (2001).
[CrossRef]

Hsueh, Y.-T.

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

Huang, H.-S.

Izutsu, M.

K. Higuma, S. Oikawa, Y. Hashimoto, H. Nagata, and M. Izutsu, “X-cut lithium niobate optical single-sideband modulator,” Electron. Lett. 37(8), 515–516 (2001).
[CrossRef]

Jia, Z.

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

Jiang, W.-J.

Kobyakov, A.

Koonen, A. M. J.

Lin, C.-T.

Ma, Y.

Nagata, H.

K. Higuma, S. Oikawa, Y. Hashimoto, H. Nagata, and M. Izutsu, “X-cut lithium niobate optical single-sideband modulator,” Electron. Lett. 37(8), 515–516 (2001).
[CrossRef]

Oikawa, S.

K. Higuma, S. Oikawa, Y. Hashimoto, H. Nagata, and M. Izutsu, “X-cut lithium niobate optical single-sideband modulator,” Electron. Lett. 37(8), 515–516 (2001).
[CrossRef]

Pearson, K.

K. Pearson, “Contributions to the mathematical theory of evolution. II. skew variation in homogeneous material,” Philos. Trans. Roy. Soc. London Ser. A 186(0), 343–414 (1895).
[CrossRef]

Peng, P.-C.

P.-T. Shih, J. Chen, C.-T. Lin, W.-J. Jiang, H.-S. Huang, P.-C. Peng, and S. Chi, “Optical millimeter-wave signal generation via frequency 12-tupling,” J. Lightwave Technol. 28(1), 71–78 (2010).
[CrossRef]

C.-T. Lin, P.-T. Shih, J. Chen, W.-Q. Xue, P.-C. Peng, and S. Chi, “Optical millimeter-wave signal generation using frequency quadrupling technique and no optical filtering,” IEEE Photon. Technol. Lett. 20(12), 1027–1029 (2008).
[CrossRef]

Peng, W.-R.

W.-R. Peng, J. Chen, and S. Chi, “On the phase noise impact in direct-detection optical OFDM transmission,” IEEE Photon. Technol. Lett. 22(9), 649–651 (2010).
[CrossRef]

Pupolin, S.

E. Costa and S. Pupolin, “M-QAM-OFDM system performance in the presence of a nonlinear amplifier and phase noise,” IEEE Trans. Commun. 50(3), 462–472 (2002).
[CrossRef]

Sauer, M.

Shieh, W.

Shih, P.-T.

Wu, Y.

M. S. El-Tanany, Y. Wu, and L. Hazy, “Analytical modeling and simulation of phase noise interference in OFDM-based digital television terrestrial broadcasting systems,” IEEE Trans. Broadcast 47(1), 20–31 (2001).
[CrossRef]

Xue, W.-Q.

C.-T. Lin, P.-T. Shih, J. Chen, W.-Q. Xue, P.-C. Peng, and S. Chi, “Optical millimeter-wave signal generation using frequency quadrupling technique and no optical filtering,” IEEE Photon. Technol. Lett. 20(12), 1027–1029 (2008).
[CrossRef]

Yi, X.

Yu, J.

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

Electron. Lett. (1)

K. Higuma, S. Oikawa, Y. Hashimoto, H. Nagata, and M. Izutsu, “X-cut lithium niobate optical single-sideband modulator,” Electron. Lett. 37(8), 515–516 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

W.-R. Peng, J. Chen, and S. Chi, “On the phase noise impact in direct-detection optical OFDM transmission,” IEEE Photon. Technol. Lett. 22(9), 649–651 (2010).
[CrossRef]

C.-T. Lin, P.-T. Shih, J. Chen, W.-Q. Xue, P.-C. Peng, and S. Chi, “Optical millimeter-wave signal generation using frequency quadrupling technique and no optical filtering,” IEEE Photon. Technol. Lett. 20(12), 1027–1029 (2008).
[CrossRef]

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

IEEE Trans. Broadcast (1)

M. S. El-Tanany, Y. Wu, and L. Hazy, “Analytical modeling and simulation of phase noise interference in OFDM-based digital television terrestrial broadcasting systems,” IEEE Trans. Broadcast 47(1), 20–31 (2001).
[CrossRef]

IEEE Trans. Commun. (1)

E. Costa and S. Pupolin, “M-QAM-OFDM system performance in the presence of a nonlinear amplifier and phase noise,” IEEE Trans. Commun. 50(3), 462–472 (2002).
[CrossRef]

J. Lightwave Technol. (6)

Philos. Trans. Roy. Soc. London Ser. A (1)

K. Pearson, “Contributions to the mathematical theory of evolution. II. skew variation in homogeneous material,” Philos. Trans. Roy. Soc. London Ser. A 186(0), 343–414 (1895).
[CrossRef]

Other (5)

D. Petrovic, W. Rave, and G. Fettweis, “Properties of the intercarrier interference due to phase noise in OFDM,” in Proc. ICC’05, 2605–2610 (2005)

Y. X. Guo, B. Luo, C. S. Park, L. C. Ong, M.-T. Zhou, and S. Kato, “60 GHz radio-over-fiber for Gbps transmission,” in Proc. Global Symp. Millimeter Waves (GSMM), 41–43 (2008).

H.-C. Chien, A. Chowdhury, Z. Jia, Y.-T. Hsueh, and G.-K. Chang, “60 GHz millimeter-wave gigabit wireless services over long-reach passive optical network using remote signal regeneration and upconversion,” Opt. Express 17, 3016–3024 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe- 17-5-3016.

C. T. Lin, E. Z. Wong, W. J. Jiang, P. T. Shih, J. Chen, and S. Chi, “28‐Gb/s 16‐QAM OFDM radio‐over‐fiber system within 7‐GHz license‐free band at 60 GHz employing all-optical up-conversion,” in Proc. CLEO 2009, Maryland, Baltimore, CPDA8 (2009).

Z. Zan, M. Premaratne, and A. J. Lowery, “Laser RIN and linewidth requirements for direct detection optical OFDM,” in Proc. CLEO’08, San Jose, CWN2 (2008).

Supplementary Material (1)

» Media 1: AVI (4096 KB)     

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Figures (11)

Fig. 1
Fig. 1

OFDM RoF transmission at 60-GHz band

Fig. 2
Fig. 2

Schematic plot of dispersion-induced phase difference between the RF-tone and subcarriers. (Media 1)

Fig. 3
Fig. 3

The ICI and PRT powers of OFDM signals over 32 subcarriers with 1-MHz laser linewidth after 100-km transmission.

Fig. 4
Fig. 4

Variances of (a) PRT and (b) normalized ICI for with 1-MHz linewidth

Fig. 5
Fig. 5

The kurtosis excess of the real part of ICI

Fig. 6
Fig. 6

Normalized CDF of (a) PRT and (b) the real part of ICI for 16-QAM OFDM signals

Fig. 7
Fig. 7

BER curves of OFDM signals over (a) 32 subcarriers and (b) 128 subcarriers after 1400 km (4-QAM), 350 km (16-QAM) and 87.5 km (64-QAM) SMF transmission with 1-MHz laser linewidth.

Fig. 8
Fig. 8

SNR penalty as a function of transmission distance for 16-QAM OFDM signals with 1-MHz laser linewidth

Fig. 11
Fig. 11

SNR penalty as a function of transmission distance for 64-QAM OFDM signals with 4-MHz laser linewidth

Fig. 9
Fig. 9

SNR penalty as a function of transmission distance for 64-QAM OFDM signals with 1-MHz laser linewidth

Fig. 10
Fig. 10

SNR penalty as a function of transmission distance for 16-QAM OFDM signals with 4-MHz laser linewidth

Equations (25)

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

R q = H q S q + W q ,
I q ( k ) = 1 N n = 0 N 1 e j φ q ( n ) e j 2 π N n k ,
R q = H q S q I q ( 0 ) + m = 0 m q N 1 H m S m I m ( q m ) + W q .
φ ¯ q = 1 N n = 0 N 1 φ q ( n ) .
σ PRT 2 ( q ) = 2 π β N 2 [ N t q + 2 Δ n = 1 n q ( N Δ n ) ( t q Δ n Δ t ) ] ,
σ ˜ ICI 2 ( q ) = σ ICI 2 ( q ) P S = m = 0 m q N 1 | I m ( q m ) | 2 .
| I m ( k ) | 2 = 1 N 2 Δ n = 1 N N 1 f m ( Δ n ) e j 2 π N Δ n k ,
f m ( Δ n ) = k = | Δ n | N 1 e sgn ( Δ n ) j ( φ m ( k ) φ m ( k | Δ n | ) ) .
| I m ( k ) | 2 = 1 N + 2 N 2 Δ n = 1 N 1 ( N Δ n ) cos ( 2 π N Δ n k ) e 2 π β min ( Δ n Δ t , t m ) .
σ PRT 2 ( q ) n q < < N 2 π β [ t q ( 2 n q + 1 ) n q Δ t ( n q + 1 ) ] t q n q Δ t 2 π β t q 2 N Δ t .
σ PRT 2 ( q ) = n q = N 2 π β [ t q N ( N 2 1 ) 3 N 2 Δ t ] N 2 > > 1 2 π β ( t q N Δ t 3 ) ,
σ ˜ ICI 2 ( q ) t m = t q m = 0 m q N 1 | I q ( q m ) | 2 = 1 | I q ( 0 ) | 2 .
σ ˜ ICI 2 ( q ) 1 1 N 2 N 2 [ Δ n = 1 n q ( 1 2 π β Δ n Δ t ) ( N Δ n ) + ( 1 2 π β t q ) Δ n = n q + 1 N 1 ( N Δ n ) ]
= 2 π β N 2 [ ( N n q 1 ) ( N n q ) t q + ( N 2 n q + 1 3 ) ( n q + 1 ) n q Δ t ] .
σ ˜ ICI 2 ( q ) t q n q Δ t 2 π β t q ( 1 t q N Δ t )
σ ˜ ICI 2 ( q ) N 2 > > 1 2 π β N Δ t 3 .
γ 2 , ICI ( q ) = 2 | S m | 4 σ ˜ ICI 4 ( q ) P S 2 m = 0 m q N 1 | I m ( q m ) | 4 + 4 σ ˜ ICI 4 ( q ) m = 0 m q N 1 m ' = 0 m ' q , m N 1 | I m ( q m ) I m ' ( q m ' ) | 2 4 .
BER = 8 N M log 2 M q = 0 N 1 π π e θ 2 2 σ PRT 2 ( q ) 2 π σ PRT ( q ) [ k , k ' = 1 M / 2 Q ( η Θ k , k ' ( θ ) ) +     k , k ' = 1 k M / 2 M / 2 Q ( η Θ ^ k , k ' ( θ ) ) ]   d θ ,
η = 3 M 1 ( ρ 1 + σ ˜ ICI 2 ( q ) ) 1 ,
Θ k , k ' ( θ ) = ( 2 k 1 ) cos θ ( 2 k ' 1 ) sin θ 2 ( k 1 ) ,
Θ ^ k , k ' ( θ ) = ( 2 k 1 ) cos θ + ( 2 k ' 1 ) sin θ + 2 k ,
p ν ( x ) = Γ ( ν + 1 2 ) π ( ν 2 ) Γ ( ν 2 ) ( 1 + x 2 ν 2 ) ν + 1 2 ,
T ( x , ν W ) = 1 2 x Γ ( ν W + 1 2 ) π ( ν W 2 ) Γ ( ν 2 ) F 2 1 ( 1 2 , ν W + 1 2 ; 3 2 ; x 2 ν W 2 ) ,
( a + b ) 4 ( a + b ) 2 2 3 = a 4 + b 4 + 6 a 2 b 2 a 2 2 + b 2 2 + 2 a 2 b 2 3 = γ 2 , a σ a 4 ( σ a 2 + σ b 2 ) 2 .
ν W = 4 + 6 γ 2 , ICI ( q ) ( 1 + 1 ρ σ ˜ ICI 2 ( q ) ) 2 .

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