Abstract

This paper deals with the performance evaluation of a passive optical network (PON) using wavelength division multiplexing (WDM) by employing an arrayed-waveguide grating (AWG) in the remote node (RN). In such WDM-based PONs (WDMPONs) the RN demultiplexes the downstream WDM signal via distinct output ports through a static wavelength-routing mechanism. An analytical model is developed for evaluating the bit error rate (BER) performance of the demultiplexed channels as a function of the output port location in the AWG by incorporating a novel spectral-to-spatial domain transformation technique. The proposed model captures the impact of finite laser linewidth as well as the AWG characteristics in determining the received signal power at the individual optical network unit (ONU) attached to the output port (due to spectral-to-spatial mapping). Furthermore, the BER model accounts for several transmission impairments, including beat noise due to interchannel crosstalk in the AWG. Results indicate that the signal power captured by the output ports of AWGs and hence their BER values are not only determined by the Gaussian focal-field pattern of the AWG but also by the power spillover from adjacent ports for broader linewidths and higher data rates. A significant interplay is observed between data rate, source linewidth, and relevant device parameters while examining the loss and BER characteristics of a given AWG-based WDMPON.

© 2010 Optical Society of America

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2009

J. Kani, F. Bourgart, A. Cui, A. Raphel, F. Rodrigues, “Next generation PONs—Part I: technology roadmap and general requirements,” IEEE Commun. Mag., vol. 47, no. 11, pp. 43–49, Nov. 2009.
[CrossRef]

2008

2007

2006

T. Koonen, “Fiber to the home/fiber to the premises: what, where, and when?” Proc. IEEE, vol. 94, no. 5, pp. 911–934, May 2006.
[CrossRef]

2005

2004

2003

2002

1999

B. Ramamurthy, D. Datta, H. Feng, J. P. Heritage, B. Mukherjee, “Impact of transmission impairments on the teletraffic performance of wavelength-routed optical networks,” J. Lightwave Technol., vol. 17, no. 10, pp. 1713–1723, Oct. 1999.
[CrossRef]

Y. S. Jang, C.-H. Lee, Y. C. Chung, “Effects of crosstalk in WDM systems using spectrum-sliced light sources,” Photon. Technol. Lett., vol. 11, no. 6, pp. 715–717, June 1999.
[CrossRef]

1998

K. A. McGreer, “Arrayed waveguide gratings for wavelength routing,” IEEE Commun. Mag., vol. 36, no. 12, pp. 62–68, Dec. 1998.
[CrossRef]

1996

H. Takahashi, K. Oda, H. Toba, “Impact of crosstalk in an arrayed-waveguide multiplexer on N×N optical interconnection,” J. Lightwave Technol., vol. 14, no. 6, pp. 1097–1105, June 1996.
[CrossRef]

M. K. Smit, C. V. Dam, “PHASAR-based WDM-devices: principles, design and applications,” IEEE J. Sel. Top. Quantum Electron., vol. 2, no. 2, pp. 237–250, June 1996.
[CrossRef]

1995

H. Takahashi, K. Oda, H. Toba, Y. Inoue, “Transmission characteristics of arrayed waveguide N×N wavelength multiplexer,” J. Lightwave Technol., vol. 13, no. 3, pp. 447–445, Mar. 1995.
[CrossRef]

1990

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-waveguide grating for wavelength division multi/demultiplexer with nanometer resolution,” Electron. Lett., vol. 26, no. 2, pp. 87–88, Jan. 1990.
[CrossRef]

1988

M. K. Smit, “New focusing and dispersive planar component based on an optical phased array,” Electron. Lett., vol. 24, no. 7, pp. 385–386, Feb. 1988.
[CrossRef]

1986

J. Salz, “Modulation and detection for coherent communication,” IEEE Commun. Mag., vol. 24, no. 6, pp. 38–49, June 1986.
[CrossRef]

Ahn, J.-G.

Anderson, T. B.

E. Wong, K. L. Lee, T. B. Anderson, “Directly modulated self-seeding reflective semiconductor optical amplifiers as colorless transmitters in wavelength division multiplexed passive optical network,” J. Lightwave Technol., vol. 15, no. 1, pp. 67–74, Jan. 2007.
[CrossRef]

Anis, H.

Banerjee, A.

Bourgart, F.

J. Kani, F. Bourgart, A. Cui, A. Raphel, F. Rodrigues, “Next generation PONs—Part I: technology roadmap and general requirements,” IEEE Commun. Mag., vol. 47, no. 11, pp. 43–49, Nov. 2009.
[CrossRef]

Capmany, J.

Cheng, N.

Chung, Y. C.

Y. S. Jang, C.-H. Lee, Y. C. Chung, “Effects of crosstalk in WDM systems using spectrum-sliced light sources,” Photon. Technol. Lett., vol. 11, no. 6, pp. 715–717, June 1999.
[CrossRef]

Clarke, F.

Cui, A.

J. Kani, F. Bourgart, A. Cui, A. Raphel, F. Rodrigues, “Next generation PONs—Part I: technology roadmap and general requirements,” IEEE Commun. Mag., vol. 47, no. 11, pp. 43–49, Nov. 2009.
[CrossRef]

Dam, C. V.

M. K. Smit, C. V. Dam, “PHASAR-based WDM-devices: principles, design and applications,” IEEE J. Sel. Top. Quantum Electron., vol. 2, no. 2, pp. 237–250, June 1996.
[CrossRef]

Datta, D.

Fan, G.

Feng, H.

Gutierrez, D.

Heo, D.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Heritage, J. P.

Hong, X.

Hu, H. W.

Huang, W. P.

Hwang, S.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Inoue, Y.

H. Takahashi, K. Oda, H. Toba, Y. Inoue, “Transmission characteristics of arrayed waveguide N×N wavelength multiplexer,” J. Lightwave Technol., vol. 13, no. 3, pp. 447–445, Mar. 1995.
[CrossRef]

Jang, D. H.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Jang, Y. S.

Y. S. Jang, C.-H. Lee, Y. C. Chung, “Effects of crosstalk in WDM systems using spectrum-sliced light sources,” Photon. Technol. Lett., vol. 11, no. 6, pp. 715–717, June 1999.
[CrossRef]

Jeong, K.-T.

Jung, D. K.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Kang, J. K.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Kani, J.

J. Kani, F. Bourgart, A. Cui, A. Raphel, F. Rodrigues, “Next generation PONs—Part I: technology roadmap and general requirements,” IEEE Commun. Mag., vol. 47, no. 11, pp. 43–49, Nov. 2009.
[CrossRef]

Kato, K.

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-waveguide grating for wavelength division multi/demultiplexer with nanometer resolution,” Electron. Lett., vol. 26, no. 2, pp. 87–88, Jan. 1990.
[CrossRef]

Kazovsky, L. G.

Keh, Y. C.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Keiser, G. E.

G. E. Keiser, Optical Fiber Communications. McGraw Hill Inc., 1991.

Kim, B. W.

Kim, H. S.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Kim, K.

Kim, S. W.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Koonen, T.

T. Koonen, “Fiber to the home/fiber to the premises: what, where, and when?” Proc. IEEE, vol. 94, no. 5, pp. 911–934, May 2006.
[CrossRef]

Kramer, G.

Kwon, J. W.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Lee, C.-H.

S.-J. Park, C.-H. Lee, K.-T. Jeong, H.-J. Park, J.-G. Ahn, K.-H. Song, “Fiber-to-the-home services based on wavelength-division-multiplexing passive optical network,” J. Lightwave Technol., vol. 22, no. 11, pp. 2582–2590, Nov. 2004.
[CrossRef]

Y. S. Jang, C.-H. Lee, Y. C. Chung, “Effects of crosstalk in WDM systems using spectrum-sliced light sources,” Photon. Technol. Lett., vol. 11, no. 6, pp. 715–717, June 1999.
[CrossRef]

Lee, E. H.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Lee, J. K.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Lee, J. S.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Lee, K. L.

E. Wong, K. L. Lee, T. B. Anderson, “Directly modulated self-seeding reflective semiconductor optical amplifiers as colorless transmitters in wavelength division multiplexed passive optical network,” J. Lightwave Technol., vol. 15, no. 1, pp. 67–74, Jan. 2007.
[CrossRef]

Li, X.

Liang, W.

McGreer, K. A.

K. A. McGreer, “Arrayed waveguide gratings for wavelength routing,” IEEE Commun. Mag., vol. 36, no. 12, pp. 62–68, Dec. 1998.
[CrossRef]

Mukherjee, B.

Munoz, P.

Nishi, I.

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-waveguide grating for wavelength division multi/demultiplexer with nanometer resolution,” Electron. Lett., vol. 26, no. 2, pp. 87–88, Jan. 1990.
[CrossRef]

Oda, K.

H. Takahashi, K. Oda, H. Toba, “Impact of crosstalk in an arrayed-waveguide multiplexer on N×N optical interconnection,” J. Lightwave Technol., vol. 14, no. 6, pp. 1097–1105, June 1996.
[CrossRef]

H. Takahashi, K. Oda, H. Toba, Y. Inoue, “Transmission characteristics of arrayed waveguide N×N wavelength multiplexer,” J. Lightwave Technol., vol. 13, no. 3, pp. 447–445, Mar. 1995.
[CrossRef]

Oh, Y. J.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Oh, Y. K.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Park, H.-J.

Park, J. W.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Park, M. K.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Park, S. B.

D. J. Shin, Y. C. Keh, J. W. Kwon, E. H. Lee, J. K. Lee, M. K. Park, J. W. Park, J. K. Kang, Y. K. Oh, S. W. Kim, I. K. Yun, H. C. Shin, D. Heo, J. S. Lee, H. S. Shin, H. S. Kim, S. B. Park, D. K. Jung, S. Hwang, Y. J. Oh, D. H. Jang, C. S. Shim, “C/S-band WDM-PON employing colorless bidirectional transceivers and SOA-based broadband light sources,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., San Diego, 2005, paper PDP36.

Park, S.-J.

Park, Y.

Pastor, D.

Peebles, P. Z.

P. Z. Peebles, Probability, Random Variables and Random Signal Principles. India: Tata McGraw Hill Ltd., 2002.

Qiu, S.

Ramamurthy, B.

Raphel, A.

J. Kani, F. Bourgart, A. Cui, A. Raphel, F. Rodrigues, “Next generation PONs—Part I: technology roadmap and general requirements,” IEEE Commun. Mag., vol. 47, no. 11, pp. 43–49, Nov. 2009.
[CrossRef]

Rodrigues, F.

J. Kani, F. Bourgart, A. Cui, A. Raphel, F. Rodrigues, “Next generation PONs—Part I: technology roadmap and general requirements,” IEEE Commun. Mag., vol. 47, no. 11, pp. 43–49, Nov. 2009.
[CrossRef]

Salz, J.

J. Salz, “Modulation and detection for coherent communication,” IEEE Commun. Mag., vol. 24, no. 6, pp. 38–49, June 1986.
[CrossRef]

Shaw, W. T.

Shim, C. S.

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Figures (11)

Fig. 1
Fig. 1

Block schematic of an AWG-based WDMPON configuration.

Fig. 2
Fig. 2

Routing in a 1 × 2 AWG.

Fig. 3
Fig. 3

Far-field pattern of demultiplexed optical channel routing.

Fig. 4
Fig. 4

Demultiplexed optical channels of AWG-based RN far-field intensity profile at the image plane.

Fig. 5
Fig. 5

Loss characteristics of AWG for different laser linewidths: R b = 1.25 Gbps , P OLT = 6.0 dBm , AWG insertion loss = 6.5 dB .

Fig. 6
Fig. 6

Interchannel crosstalk characteristics for 1.25 Gbps channels, P OLT = 6.0 dBm .

Fig. 7
Fig. 7

Comparative crosstalk characteristics for different data rates and port locations.

Fig. 8
Fig. 8

BER characteristics of 1.25 Gbps channels for different laser linewidths without including beat noise: P OLT = 6.0 dBm , Δ θ ch = 100 GHz .

Fig. 9
Fig. 9

BER characteristics of 10.0 Gbps channels for different laser linewidths: P OLT = + 3.0 dBm , Δ θ ch = 100 GHz .

Fig. 10
Fig. 10

BER versus laser linewidth for different data rates at port #0, B L = 500 MHz .

Fig. 11
Fig. 11

BER performance of 1.25 Gbps channels including beat noise for different port counts: P OLT = 6.0 dBm , B L = 500 MHz , Δ θ ch = 100 GHz .

Tables (2)

Tables Icon

Table 1 AWG Device Parameters

Tables Icon

Table 2 Description of the System Parameters

Equations (28)

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N FPR d a sin θ in + N g Δ L + N FPR d a sin θ op = m λ ,
β FPR d a sin θ in + β g Δ L + β FPR d a sin θ op = 2 π m ,
Δ L = m c f c N c .
θ N g Δ L N FPR d a = N g m c N FPR d a f c N c .
D = Δ x d f = R a d θ d f .
d θ d f = 1 f c 2 N g m c N FPR d a N c .
P o i ( θ ) = P o e 2 θ 2 θ w 2 ,
S i ( f ) = A 2 4 π 2 N 0 [ 1 1 + ( f + f o i π N 0 ) 2 + 1 1 + ( f f o i π N 0 ) 2 ] ,
P sig i ( θ , f ) = e 2 θ 2 θ w 2 { A 2 2 π B L [ 1 1 + ( f + f o i B L 2 ) 2 + 1 1 + ( f f o i B L 2 ) 2 ] } .
( f f o ) = R a θ D .
f o i = f o + i Δ f ch .
f f o i = R a θ D i R a Δ θ ch D .
P sig i ( θ ) = A 2 π B L e 2 ( i Δ θ ch ) 2 θ w 2 [ 1 { 1 + ( 2 R a θ B L D i 2 R a Δ θ ch B L D ) 2 } ] .
P sig i = A 2 π e 2 ( i Δ θ ch ) 2 θ w 2 tan 1 ( R a Δ θ wg B L D ) .
P sig 0 = A 2 π tan 1 ( R a Δ θ wg B L D ) .
P xt i = i Δ θ ch Δ θ wg 2 i Δ θ ch + Δ θ wg 2 A 2 π B L ( e 2 ( i + 1 ¯ Δ θ ch ) 2 θ w 2 ) [ 1 1 + ( 2 R a θ B L D i 2 R a ( i + 1 ) Δ θ ch B L D ) 2 ] d θ i Δ θ ch + Δ θ wg 2 i Δ θ ch Δ θ wg 2 A 2 π B L ( e 2 ( i 1 ¯ Δ θ ch ) 2 θ w 2 ) [ 1 1 + ( 2 R a θ B L D i 2 R a ( i + 1 ) Δ θ ch B L D ) 2 ] d θ .
P xt i = A 2 2 π e 2 ( i + 1 ¯ Δ θ ch ) 2 θ w 2 [ tan 1 { 2 R a B L D ( Δ θ ch + Δ θ wg 2 ) } tan 1 { 2 R a B L D ( Δ θ ch Δ θ wg 2 ) } ] A 2 2 π e 2 ( i 1 ¯ Δ θ ch ) 2 θ w 2 [ tan 1 { 2 R a B L D ( Δ θ ch Δ θ wg 2 ) } tan 1 { 2 R a B L D ( Δ θ ch + Δ θ wg 2 ) } ] .
P xt 0 = A 2 π e 2 Δ θ ch 2 θ w 2 [ tan 1 { 2 R a B L D ( Δ θ ch + Δ θ wg 2 ) } tan 1 { 2 R a B L D ( Δ θ ch Δ θ wg 2 ) } ] .
σ th 2 = 4 K T B e R L ,
σ sh 0 2 = 2 q ϵ R λ P sig i B e , σ sh 1 2 = 2 q R λ P sig i B e ,
σ rin 0 2 = ( ϵ R λ P sig i ) 2 RIN B e , σ rin 1 2 = ( R λ P sig i ) 2 RIN B e ,
σ xt 2 = 2 q R λ P xt i n adj B e p r on .
σ sg _ xt 2 = 2 ξ pol R λ 2 P sig i P xt i p r on
σ xt _ xt 2 = 2 ξ pol R λ 2 P adj + 1 i P adj 1 i p r on 2 ( n adj 1 ) .
σ 0 = σ th 2 + σ sh 0 2 + σ rin 0 2 + σ xt 2 + σ xt _ xt 2 ,
σ 1 = σ th 2 + σ sh 1 2 + σ rin 1 2 + σ xt 2 + σ xt _ xt 2 + σ sig _ xt 2 .
I th = R λ P sig i σ 0 + ϵ R λ P sig i σ 1 σ 1 + σ 0 ,
P e = 1 4 { erfc [ R λ P sig 1 I th 2 σ 1 ] + erfc [ I th ϵ R λ P sig 0 2 σ 0 ] } .