Abstract

We present a comprehensive study of nonlinear distortions from an optical OFDM transmitter. Nonlinearities are introduced by the combination of effects from the digital-to-analog converter (DAC), electrical power amplifier (PA) and optical modulator in the presence of high peak-to-average power ratio (PAPR). We introduce parameters to quantify the transmitter nonlinearity. High input backoff avoids OFDM signal compression from the PA, but incurs high penalties in power efficiency. At low input backoff, common PAPR reduction techniques are not effective in suppressing the PA nonlinear distortion. A bit error distribution investigation shows a technique combining nonlinear predistortion with PAPR mitigation could achieve good power efficiency by allowing low input backoff. We use training symbols to extract the transmitter nonlinear function. We show that piecewise linear interpolation (PLI) leads to an accurate transmitter nonlinearity characterization. We derive a semi-analytical solution for bit error rate (BER) that validates the PLI approximation accurately captures transmitter nonlinearity. The inverse of the PLI estimate of the nonlinear function is used as a predistorter to suppress transmitter nonlinearity. We investigate performance of the proposed scheme by Monte Carlo simulations. Our simulations show that when DAC resolution is more than 4 bits, BER below forward error correction limit of 3.8 × 10−3 can be achieved by using predistortion with very low input power backoff for electrical PA and optical modulator.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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2014 (1)

2013 (4)

2012 (2)

D. Qian, M.-F. Huang, E. Ip, Y.-K. Huang, Y. Shao, J. Hu, and T. Wang, “High capacity/spectral efficiency 101.7-Tb/s WDM transmission using PDM-128QAM-OFDM over 165-km SSMF within C- and L-bands,” J. Lightwave Technol. 30(10), 1540–1548 (2012).
[Crossref]

J. A. Fredenburg and M. P. Flynn, “Statistical analysis of ENOB and yield in binary weighted ADCs and DACs with random element mismatch,” IEEE Trans. Commun. 59(7), 1396–1408 (2012).

2011 (2)

2009 (2)

2008 (4)

Y. Tang, K.-P. Ho, and W. Shieh, “Coherent optical OFDM transmitter design employing predistortion,” IEEE Photon. Technol. Lett. 20(11), 954–956 (2008).
[Crossref]

W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008).
[Crossref] [PubMed]

S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008).
[Crossref]

T. Jiang and Y. Wu, “An overview: peak-to-average power ratio reduction techniques for OFDM signals,” IEEE Trans. Broadcast. 54(2), 257–268 (2008).
[Crossref]

2007 (1)

Y. Tang, W. Shieh, X. Yi, and R. Evans, “Optimum design for RF-to-optical up-converter in coherent optical OFDM systems,” IEEE Photon. Technol. Lett. 19(7), 483–485 (2007).
[Crossref]

2005 (1)

S. H. Han and J. H. Lee, “An overview of peak-to-average power ratio reduction techniques for multicarrier transmission,” IEEE Wireless Commun. 12(2), 56–65 (2005).
[Crossref]

2002 (1)

I. Koffman and V. Roman, “Broadband wireless access solutions based on OFDM access in IEEE 802.16,” IEEE Commun. Mag. 40(4), 96–103 (2002).
[Crossref]

2001 (1)

A. N. D’Andrea, V. Lottici, and R. Reggiannini, “Nonlinear predistortion of OFDM signals over frequency-selective fading channels,” IEEE Trans. Commun. 49(5), 837–843 (2001).
[Crossref]

2000 (1)

D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun. 48(10), 1755–1764 (2000).
[Crossref]

1999 (1)

H. W. Kang, Y.-S. Cho, and D.-H. Youn, “On compensating nonlinear distortions of an OFDM system using an efficient adaptive predistorter,” IEEE Trans. Commun. 47(4), 522–526 (1999).
[Crossref]

1995 (1)

H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag. 33(2), 100–109 (1995).
[Crossref]

1982 (1)

H. E. Rowe, “Memoryless nonlinearities with Gaussian inputs: elementary results,” Bell Syst. Tech. J. 61(7), 1519–1525 (1982).
[Crossref]

Amiralizadeh, S.

S. Amiralizadeh and L. A. Rusch, “Transmitter sensitivity to high PAPR in coherent optical OFDM systems,” in CLEO: Science and Innovations, (Optical Society of America, 2014), paper SW1J.5.

Armstrong, J.

J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009).
[Crossref]

A. J. Lowery, L. Du, and J. Armstrong, “Orthogonal frequency division multiplexing for adaptive dispersion compensation in long haul WDM systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2006), paper PDP39.

Bao, H.

Bao, Y.

Barros, D. J. F.

Benlachtar, Y.

Berger, C.

Bulakçi, Ö.

Ö. Bulakçi, M. Schuster, C.-A. Bunge, and B. Spinnler, “Precoding based peak-to-average power ratio reduction for optical OFDM demonstrated on compatible single-sideband modulation with direct detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2008), paper JThA56.

Bunge, C.-A.

Ö. Bulakçi, M. Schuster, C.-A. Bunge, and B. Spinnler, “Precoding based peak-to-average power ratio reduction for optical OFDM demonstrated on compatible single-sideband modulation with direct detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2008), paper JThA56.

Capmany, J.

Chandrasekhar, S.

S. Chandrasekhar, X. Liu, P. J. Winzer, T. Lotz, C. J. Youn, Y. H. Kwon, and E. S. Nam, “Field demonstration of 3×341-Gb/s PDM-OFDM-256 iterative polar modulation signals with a record 11.0-b/s/Hz intrachannel spectral efficiency,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5A.8.

Cho, Y.-S.

H. W. Kang, Y.-S. Cho, and D.-H. Youn, “On compensating nonlinear distortions of an OFDM system using an efficient adaptive predistorter,” IEEE Trans. Commun. 47(4), 522–526 (1999).
[Crossref]

Coelho, L. D.

B. Goebel, B. Fesl, L. D. Coelho, and N. Hanik, “On the effect of FWM in coherent optical OFDM systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2008), paper JWA58.

D’Andrea, A. N.

A. N. D’Andrea, V. Lottici, and R. Reggiannini, “Nonlinear predistortion of OFDM signals over frequency-selective fading channels,” IEEE Trans. Commun. 49(5), 837–843 (2001).
[Crossref]

Dardari, D.

D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun. 48(10), 1755–1764 (2000).
[Crossref]

Du, L.

A. J. Lowery, L. Du, and J. Armstrong, “Orthogonal frequency division multiplexing for adaptive dispersion compensation in long haul WDM systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2006), paper PDP39.

Evans, R.

Y. Tang, W. Shieh, X. Yi, and R. Evans, “Optimum design for RF-to-optical up-converter in coherent optical OFDM systems,” IEEE Photon. Technol. Lett. 19(7), 483–485 (2007).
[Crossref]

Feng, X.

Fesl, B.

B. Goebel, B. Fesl, L. D. Coelho, and N. Hanik, “On the effect of FWM in coherent optical OFDM systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2008), paper JWA58.

Flynn, M. P.

J. A. Fredenburg and M. P. Flynn, “Statistical analysis of ENOB and yield in binary weighted ADCs and DACs with random element mismatch,” IEEE Trans. Commun. 59(7), 1396–1408 (2012).

Fredenburg, J. A.

J. A. Fredenburg and M. P. Flynn, “Statistical analysis of ENOB and yield in binary weighted ADCs and DACs with random element mismatch,” IEEE Trans. Commun. 59(7), 1396–1408 (2012).

Goebel, B.

B. Goebel, B. Fesl, L. D. Coelho, and N. Hanik, “On the effect of FWM in coherent optical OFDM systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2008), paper JWA58.

S. Hellerbrand, B. Goebel, and N. Hanik, “Trellis shaping for reduction of the peak-to-average power ratio in coherent optical OFDM systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2009), paper JThA48.

Gregorio, F. H.

F. H. Gregorio, “Analysis and compensation of nonlinear power amplifier effects in multi-antenna OFDM systems,” Ph.D. dissertation, Helsinki University of Technology, Finland (2007).

Guan, B.

Haas, H.

Han, S. H.

S. H. Han and J. H. Lee, “An overview of peak-to-average power ratio reduction techniques for multicarrier transmission,” IEEE Wireless Commun. 12(2), 56–65 (2005).
[Crossref]

Hanik, N.

S. Hellerbrand, B. Goebel, and N. Hanik, “Trellis shaping for reduction of the peak-to-average power ratio in coherent optical OFDM systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2009), paper JThA48.

B. Goebel, B. Fesl, L. D. Coelho, and N. Hanik, “On the effect of FWM in coherent optical OFDM systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2008), paper JWA58.

Hellerbrand, S.

S. Hellerbrand, B. Goebel, and N. Hanik, “Trellis shaping for reduction of the peak-to-average power ratio in coherent optical OFDM systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2009), paper JThA48.

Ho, K.-P.

Y. Tang, K.-P. Ho, and W. Shieh, “Coherent optical OFDM transmitter design employing predistortion,” IEEE Photon. Technol. Lett. 20(11), 954–956 (2008).
[Crossref]

Hu, J.

Huang, M.-F.

Huang, Y.-K.

Ip, E.

Jansen, S. L.

Jeanclaude, I.

H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag. 33(2), 100–109 (1995).
[Crossref]

Jiang, T.

T. Jiang and Y. Wu, “An overview: peak-to-average power ratio reduction techniques for OFDM signals,” IEEE Trans. Broadcast. 54(2), 257–268 (2008).
[Crossref]

Kahn, J. M.

Kang, H. W.

H. W. Kang, Y.-S. Cho, and D.-H. Youn, “On compensating nonlinear distortions of an OFDM system using an efficient adaptive predistorter,” IEEE Trans. Commun. 47(4), 522–526 (1999).
[Crossref]

Karam, G.

H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag. 33(2), 100–109 (1995).
[Crossref]

Killey, R.

Koffman, I.

I. Koffman and V. Roman, “Broadband wireless access solutions based on OFDM access in IEEE 802.16,” IEEE Commun. Mag. 40(4), 96–103 (2002).
[Crossref]

Kwon, Y. H.

S. Chandrasekhar, X. Liu, P. J. Winzer, T. Lotz, C. J. Youn, Y. H. Kwon, and E. S. Nam, “Field demonstration of 3×341-Gb/s PDM-OFDM-256 iterative polar modulation signals with a record 11.0-b/s/Hz intrachannel spectral efficiency,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5A.8.

Lee, J. H.

S. H. Han and J. H. Lee, “An overview of peak-to-average power ratio reduction techniques for multicarrier transmission,” IEEE Wireless Commun. 12(2), 56–65 (2005).
[Crossref]

Li, G.

Li, J.

Li, Z.

Liu, X.

S. Chandrasekhar, X. Liu, P. J. Winzer, T. Lotz, C. J. Youn, Y. H. Kwon, and E. S. Nam, “Field demonstration of 3×341-Gb/s PDM-OFDM-256 iterative polar modulation signals with a record 11.0-b/s/Hz intrachannel spectral efficiency,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5A.8.

London, Y.

Lottici, V.

A. N. D’Andrea, V. Lottici, and R. Reggiannini, “Nonlinear predistortion of OFDM signals over frequency-selective fading channels,” IEEE Trans. Commun. 49(5), 837–843 (2001).
[Crossref]

Lotz, T.

S. Chandrasekhar, X. Liu, P. J. Winzer, T. Lotz, C. J. Youn, Y. H. Kwon, and E. S. Nam, “Field demonstration of 3×341-Gb/s PDM-OFDM-256 iterative polar modulation signals with a record 11.0-b/s/Hz intrachannel spectral efficiency,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5A.8.

Lowery, A. J.

A. J. Lowery, L. Du, and J. Armstrong, “Orthogonal frequency division multiplexing for adaptive dispersion compensation in long haul WDM systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2006), paper PDP39.

Milder, P.

Morita, I.

Morsy-Osman, M. H.

Nam, E. S.

S. Chandrasekhar, X. Liu, P. J. Winzer, T. Lotz, C. J. Youn, Y. H. Kwon, and E. S. Nam, “Field demonstration of 3×341-Gb/s PDM-OFDM-256 iterative polar modulation signals with a record 11.0-b/s/Hz intrachannel spectral efficiency,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5A.8.

Nazarathy, M.

Ortega, B.

Plant, D. V.

Pochmara, J.

K. Wesolowski and J. Pochmara, “Efficient algorithm for adjustment of adaptive predistorter in OFDM transmitter,” in Proceedings of IEEE Vehicular Technology Conference, pp. 2491–2496 (2008).

Prasad, R.

R. Prasad, OFDM for Wireless Communications Systems (Artech House, 2004).

Qian, D.

Reggiannini, R.

A. N. D’Andrea, V. Lottici, and R. Reggiannini, “Nonlinear predistortion of OFDM signals over frequency-selective fading channels,” IEEE Trans. Commun. 49(5), 837–843 (2001).
[Crossref]

Roman, V.

I. Koffman and V. Roman, “Broadband wireless access solutions based on OFDM access in IEEE 802.16,” IEEE Commun. Mag. 40(4), 96–103 (2002).
[Crossref]

Rowe, H. E.

H. E. Rowe, “Memoryless nonlinearities with Gaussian inputs: elementary results,” Bell Syst. Tech. J. 61(7), 1519–1525 (1982).
[Crossref]

Rusch, L. A.

S. Amiralizadeh and L. A. Rusch, “Transmitter sensitivity to high PAPR in coherent optical OFDM systems,” in CLEO: Science and Innovations, (Optical Society of America, 2014), paper SW1J.5.

Sadot, D.

Sánchez, C.

Sari, H.

H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag. 33(2), 100–109 (1995).
[Crossref]

Schenk, T. C. W.

Schuster, M.

Ö. Bulakçi, M. Schuster, C.-A. Bunge, and B. Spinnler, “Precoding based peak-to-average power ratio reduction for optical OFDM demonstrated on compatible single-sideband modulation with direct detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2008), paper JThA56.

Shao, Y.

Shieh, W.

Y. Tang, K.-P. Ho, and W. Shieh, “Coherent optical OFDM transmitter design employing predistortion,” IEEE Photon. Technol. Lett. 20(11), 954–956 (2008).
[Crossref]

W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008).
[Crossref] [PubMed]

Y. Tang, W. Shieh, X. Yi, and R. Evans, “Optimum design for RF-to-optical up-converter in coherent optical OFDM systems,” IEEE Photon. Technol. Lett. 19(7), 483–485 (2007).
[Crossref]

Shulkind, G.

Sinanovic, S.

Spinnler, B.

Ö. Bulakçi, M. Schuster, C.-A. Bunge, and B. Spinnler, “Precoding based peak-to-average power ratio reduction for optical OFDM demonstrated on compatible single-sideband modulation with direct detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2008), paper JThA56.

Takeda, N.

Tanaka, H.

Tang, Y.

W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008).
[Crossref] [PubMed]

Y. Tang, K.-P. Ho, and W. Shieh, “Coherent optical OFDM transmitter design employing predistortion,” IEEE Photon. Technol. Lett. 20(11), 954–956 (2008).
[Crossref]

Y. Tang, W. Shieh, X. Yi, and R. Evans, “Optimum design for RF-to-optical up-converter in coherent optical OFDM systems,” IEEE Photon. Technol. Lett. 19(7), 483–485 (2007).
[Crossref]

Thompson, S. C.

S. C. Thompson, “Constant envelope OFDM phase modulation,” Ph.D. dissertation, Univ. of California, San Diego, CA (2005).

Tralli, V.

D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun. 48(10), 1755–1764 (2000).
[Crossref]

Tsonev, D.

Vaccari, A.

D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun. 48(10), 1755–1764 (2000).
[Crossref]

Wang, T.

Wesolowski, K.

K. Wesolowski and J. Pochmara, “Efficient algorithm for adjustment of adaptive predistorter in OFDM transmitter,” in Proceedings of IEEE Vehicular Technology Conference, pp. 2491–2496 (2008).

Winzer, P. J.

S. Chandrasekhar, X. Liu, P. J. Winzer, T. Lotz, C. J. Youn, Y. H. Kwon, and E. S. Nam, “Field demonstration of 3×341-Gb/s PDM-OFDM-256 iterative polar modulation signals with a record 11.0-b/s/Hz intrachannel spectral efficiency,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5A.8.

Wu, Y.

T. Jiang and Y. Wu, “An overview: peak-to-average power ratio reduction techniques for OFDM signals,” IEEE Trans. Broadcast. 54(2), 257–268 (2008).
[Crossref]

Yi, X.

Y. Tang, W. Shieh, X. Yi, and R. Evans, “Optimum design for RF-to-optical up-converter in coherent optical OFDM systems,” IEEE Photon. Technol. Lett. 19(7), 483–485 (2007).
[Crossref]

Youn, C. J.

S. Chandrasekhar, X. Liu, P. J. Winzer, T. Lotz, C. J. Youn, Y. H. Kwon, and E. S. Nam, “Field demonstration of 3×341-Gb/s PDM-OFDM-256 iterative polar modulation signals with a record 11.0-b/s/Hz intrachannel spectral efficiency,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5A.8.

Youn, D.-H.

H. W. Kang, Y.-S. Cho, and D.-H. Youn, “On compensating nonlinear distortions of an OFDM system using an efficient adaptive predistorter,” IEEE Trans. Commun. 47(4), 522–526 (1999).
[Crossref]

Zhuge, Q.

Bell Syst. Tech. J. (1)

H. E. Rowe, “Memoryless nonlinearities with Gaussian inputs: elementary results,” Bell Syst. Tech. J. 61(7), 1519–1525 (1982).
[Crossref]

IEEE Commun. Mag. (2)

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Supplementary Material (1)

NameDescription
» Visualization 1: MP4 (10899 KB)      This video shows impact of sweeping IBO on PAPR modal region.

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Figures (9)

Fig. 1
Fig. 1 M-QAM CO-OFDM system block diagram. SMF: single-mode fiber, S/P: serial-to-parallel, P/S: parallel-to-serial.
Fig. 2
Fig. 2 Transfer function of MZM biased at null point for CO-OFDM.
Fig. 3
Fig. 3 EVM versus electrical PA IBO for 16-QAM CO-OFDM system with (a) 4-, (b) 5-and (c) 6-bit DAC; dashed red line: without clipping, solid blue line: with clipping. BER of 3.8 × 10−3 is considered as forward error correction (FEC) limit.
Fig. 4
Fig. 4 (a) Bit error distribution for IBO = 6 dB, αd = 1 and 4-bit DAC as a function of PAPR for one OFDM frame (1000 OFDM symbols) with different clipping levels. (b) Normalized gain versus normalized input amplitude of the electrical PA. Rectangles show the range within one standard deviation of the mean of PAPR for different clipping levels. Star represents the PA input voltage amplitude equal to the average amplitude when IBO = 0 dB. A video is available for this plot when sweeping IBO ( Visualization 1).
Fig. 5
Fig. 5 Error distribution of one OFDM frame (1000 OFDM symbols) versus PAPR for different electrical PA IBOs when αd = 1 and 4-bit DAC is used. The number of errors is averaged over 200 OFDM frames.
Fig. 6
Fig. 6 Comparison of power backoff requirement for the transmitter with and without predistortion.
Fig. 7
Fig. 7 Block diagram of the models used for theoretical analysis. Each noise source has impact on total SNR and degrades signal quality.
Fig. 8
Fig. 8 (a) BER versus IBO for 16-QAM CO-OFDM. Solid lines refer to BER estimate with side information on nonlinearity, dashed lines with triangle markers refer to BER estimate with a PLI estimate of nonlinearity, and circle markers refer to BER estimate from Monte Carlo error counting. Black: ideal DAC, αd = 0.7; blue: ideal DAC, αd = 1; red: 4-bit DAC with CR = 1.9 and αd = 1. The ASE noise power is assumed to be zero. The DAC noise variance is calculated by Eq. (31) in [29] and is equal to 0.013 σ x 2 . (b) Power attenuation at I/Q modulator due to input signal backoff. Solid lines: κ2 obtained with side information on nonlinearity; markers: Monte Carlo simulations.
Fig. 9
Fig. 9 BER versus received SNR for 16-QAM CO-OFDM system with (a) 4-bit DAC and CR = 1.9; (b) 5-bit DAC and CR = 2.1. Circles: αd = 1, squares: αd = 0.7.

Equations (27)

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x ( t ) = 1 N k = 0 N 1 X k exp ( j 2 π f k t ) , 0 t < T s ,
PAPR = max 0 t < T s [ | x ( t ) | 2 ] 1 T s 0 T s | x ( t ) | 2 d t .
g SSPA ( A ( t ) ) = g 0 A ( t ) [ 1 + ( A ( t ) A sat ) 2 p ] 1 2 p ,
g SSPA ( A ( t ) ) = g 0 P sat [ 1 + ( P sat P in ( t ) ) p ] 1 2 p ,
E out ( t ) = E in ( t ) sin ( π α d V d ( t ) 2 V π ) ,
g TX ( A ) = sin ( π α d 2 V π g SSPA ( A ) g SSPA ( A max ) ) ,
A max A sat = PAPR IBO .
z ( t ) = κ x ( t ) + y ( t ) ,
κ = E [ x ( t ) z ( t ) ] σ x 2 = 1 σ x 2 x g ( x ) 1 2 π σ x 2 exp ( x 2 2 σ x 2 ) d x ,
σ y 2 = E [ z ( t ) 2 ] κ 2 σ x 2 = g 2 ( x ) 1 2 x σ x 2 exp ( x 2 2 σ x 2 ) d x κ 2 σ x 2 ,
SNR = R os κ 2 σ x 2 κ 2 σ DAC 2 + σ y 2 + σ AWGN 2 ,
SNR R os SNR DAC SNR NL SNR AWGN SNR DAC + SNR DAC SNR NL + SNR DAC SNR AWGN + SNR NL SNR AWGN .
BER = 2 log 2 M M 1 M erfc ( 3 2 ( M 1 ) SNR ) .
g ^ ( x ) = { a i x + b i , x i x < x i + 1 g ( x m + 1 ) , x m + 1 x ,
κ = 2 σ x 2 i = 1 m [ a i ψ 2 ( x i , x i + 1 ) + b i ψ 1 ( x i , x i + 1 ) ] + 2 σ x 2 g ( x m + 1 ) ψ 1 ( x m + 1 , ) ,
σ y 2 = 2 i = 1 m [ a i 2 ψ 2 ( x i , x i + 1 ) + 2 a i b i ψ 1 ( x i , x i + 1 ) + b i 2 ψ 0 ( x i , x i + 1 ) ] + 2 g 2 ( x m + 1 ) ψ 0 ( x m + 1 , ) κ 2 σ x 2 ,
r [ n ] = g I ( x I [ n ] ) + j g Q ( x Q [ n ] ) ,
I i , I = { x I [ n ] | ( i 1 2 ) A I , max m x I [ n ] < ( i + 1 2 ) A I , max m } ,
I i , Q = { x Q [ n ] | ( i 1 2 ) A Q , max m x Q [ n ] < ( i + 1 2 ) A Q , max m } ,
c I ( i ) = x I [ n ] I i , I g I ( x I [ n ] ) x I [ n ] | I i , I | , c Q ( i ) = x Q [ n ] I i , Q g Q ( x Q [ n ] ) x Q [ n ] | I i , Q | ,
f LI , I ( x ) = { m [ c I ( i ) c I ( i 1 ) ] A I , max ( x i A I , max m ) + c I ( i ) , ( i 1 ) A I , max m x < i A I , max m , c I ( m ) , A I , max x ,
f LI , Q ( x ) = { m [ c Q ( i ) c Q ( i 1 ) ] A Q , max ( x i A Q , max m ) + c Q ( i ) , ( i 1 ) A Q , max m x < i A Q , max m , c Q ( m ) , A Q , max x ,
g ^ I ( x ) = x f LI , I ( x ) and g ^ Q ( x ) = x f LI , Q ( x ) .
ψ k ( v , w ) = 1 2 π σ x 2 v w x k exp ( x 2 2 σ x 2 ) d x .
ψ 0 ( v , w ) = Q ( v σ x ) Q ( w σ x ) ,
ψ 1 ( v , w ) = σ x 2 π [ exp ( v 2 2 σ x 2 ) exp ( w 2 2 σ x 2 ) ] ,
ψ 2 ( v , w ) = σ x 2 ψ 0 ( v , w ) + σ x 2 π [ v exp ( v 2 2 σ x 2 ) w exp ( w 2 2 σ x 2 ) ] ,

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