Abstract

We propose and demonstrate the asymmetric direct detection (ADD) of polarization division multiplexed single-sideband (PDM-SSB) signals with orthogonal offset carriers. ADD exploits the photocurrent difference to eliminate the Y-Pol interference in the X-Pol, and the X-Pol signal intensity to eliminate the X-Pol interference in the Y-Pol without resorting to iterative algorithms. This enables not only low-complexity signal linearization but also a simplified receiver front-end composed of a single optical filter, two single-ended photodiodes and two analog-to-digital converters (ADC). In the experiment, we first perform a parametric study of the proposed scheme at 40 Gbaud in the back-to-back configuration (B2B) to evaluate the performance impact of different system parameters including the carrier to signal power ratio (CSPR), the matched filter roll-off, and the filter guard band. Next, we demonstrate the transmission of 416 Gbit/s PDM 16-QAM signal over 80 km single-mode fiber (SMF) below the soft-decision forward error correction (SD-FEC) threshold of 2×10−2. We also numerically study the effectiveness of a 2×2 multiple-input-multiple-output MIMO equalizer in alleviating the inter-polarization linear crosstalk resulting from the non-orthogonal PDM-SSB signals due to polarization-dependent loss (PDL), which is not negligible for potential on-chip implementation of ADD.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. Cisco Visual Networking Index: Forecast and Trends, 2017–2022 White Paper, https://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/white-paper-c11-741490.html .
  2. K. Zhong, X. Zhou, J. Huo, C. Yu, C. Lu, and A. P. T. Lau, “Digital Signal Processing for Short-Reach Optical Communications: A Review of Current Technologies and Future Trends,” J. Lightwave Technol. 36(2), 377–400 (2018).
    [Crossref]
  3. M. Morsy-Osman, M. Chagnon, and D. Plant, “Four Dimensional Modulation and Stokes Direct Detection of Polarization Division Multiplexed Intensities, Inter Polarization Phase and Inter Polarization Differential Phase,” J. Lightwave Technol. 34(7), 1585–1592 (2016).
    [Crossref]
  4. M. Chagnon, “Optical Communications for Short Reach,” J. Lightwave Technol. 37(8), 1779–1797 (2019).
    [Crossref]
  5. L. Zhang, T. Zuo, Y. Mao, Q. Zhang, E. Zhou, G. N. Liu, and X. Xu, “Beyond 100-Gb/s Transmission Over 80-km SMF Using Direct-Detection SSB-DMT at C-Band,” J. Lightwave Technol. 34(2), 723–729 (2016).
    [Crossref]
  6. Q. Zhang, N. Stojanovic, C. Xie, C. Prodaniuc, and P. Laskowski, “Transmission of single lane 128 Gbit/s PAM-4 signals over an 80 km SSMF link, enabled by DDMZM aided dispersion pre-compensation,” Opt. Express 24(21), 24580–24591 (2016).
    [Crossref]
  7. Z. Xing, A. Samani, M. Xiang, E. El-Fiky, T. M. Hoang, D. Patel, R. Li, M. Qiu, M. G. Saber, M. Morsy-Osman, and D. V. Plant, “100 Gb/s PAM4 transmission system for datacenter interconnects using a SiP ME-MZM based DAC-less transmitter and a VSB self-coherent receiver,” Opt. Express 26(18), 23969–23979 (2018).
    [Crossref]
  8. Z. Li, M. S. Erkılınç, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “SSBI Mitigation and the Kramers–Kronig Scheme in Single-Sideband Direct-Detection Transmission With Receiver-Based Electronic Dispersion Compensation,” J. Lightwave Technol. 35(10), 1887–1893 (2017).
    [Crossref]
  9. H.-Y. Chen, N. Kaneda, J. Lee, J. Chen, and Y.-K. Chen, “Optical filter requirements in an EML-based single-sideband PAM4 intensity-modulation and direct-detection transmission system,” Opt. Express 25(6), 5852–5860 (2017).
    [Crossref]
  10. C. Antonelli, A. Mecozzi, M. Shtaif, X. Chen, S. Chandrasekhar, and P. J. Winzer, “Polarization Multiplexing With the Kramers-Kronig Receiver,” J. Lightwave Technol. 35(24), 5418–5424 (2017).
    [Crossref]
  11. D. Che, C. Sun, and W. Shieh, “Single-Channel 480-Gb/s Direct Detection of POL-MUX IQ Signal Using Single-Sideband Stokes Vector Receiver,” in Optical Fiber Communication Conference (OFC), OSA Technical Digest (online) (Optical Society of America, Tu2C.7 (2018).
  12. T. M. Hoang, M. Y. S. Sowailem, Q. Zhuge, Z. Xing, M. Morsy-Osman, E. El-Fiky, S. Fan, M. Xiang, and D. V. Plant, “Single wavelength 480 Gb/s direct detection over 80 km SSMF enabled by Stokes vector Kramers Kronig transceiver,” Opt. Express 25(26), 33534–33542 (2017).
    [Crossref]
  13. D. Che, C. Sun, and W. Shieh, “Maximizing the spectral efficiency of Stokes vector receiver with optical field recovery,” Opt. Express 26(22), 28976–28981 (2018).
    [Crossref]
  14. A. Mecozzi, C. Antonelli, and M. Shtaif, “Kramers–Kronig coherent receiver,” Optica 3(11), 1220–1227 (2016).
    [Crossref]
  15. X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer, “Kramers–Kronig Receivers for 100-km Datacenter Interconnects,” J. Lightwave Technol. 36(1), 79–89 (2018).
    [Crossref]
  16. X. Li, J. Xiao, and J. Yu, “Heterodyne detection and transmission of 60-Gbaud PDM-QPSK signal with SE of 4b/s/Hz,” Opt. Express 22(8), 9307–9313 (2014).
    [Crossref]
  17. Y. Zhu, M. Jiang, and F. Zhang, “Direct detection of polarization multiplexed single sideband signals with orthogonal offset carriers,” Opt. Express 26(12), 15887–15898 (2018).
    [Crossref]
  18. D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Sci. Appl. 1(3), e1 (2012).
    [Crossref]
  19. H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and S.-i. Itabashi, “Silicon photonic circuit with polarization diversity,” Opt. Express 16(7), 4872–4880 (2008).
    [Crossref]
  20. F. Y. Gardes, D. J. Thomson, N. G. Emerson, and G. T. Reed, “40 Gb/s silicon photonics modulator for TE and TM polarisations,” Opt. Express 19(12), 11804–11814 (2011).
    [Crossref]
  21. S. T. Le, K. Schuh, M. Chagnon, F. Buchali, and H. Buelow, “1.6Tbps WDM Direct Detection Transmission with Virtual-Carrier over 1200 km,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2018), Tu2D.5.
  22. S. T. Le, K. Schuh, M. Chagnon, F. Buchali, R. Dischler, V. Aref, H. Buelow, and K. M. Engenhardt, “1.72-Tb/s Virtual-Carrier-Assisted Direct-Detection Transmission Over 200 km,” J. Lightwave Technol. 36(6), 1347–1353 (2018).
    [Crossref]
  23. M. Selmi, Y. Jaouen, and P. Ciblat, “Accurate digital frequency offset estimator for coherent PolMux QAM transmission systems,” in 2009 35th European Conference on Optical Communication, 2009), 1–2.
  24. L. Charles, B. Villeneuve, Z. Zhang, D. McGhan, H. Sun, and M. O’Sullivan, “WDM performance and PMD tolerance of a coherent 40-Gbit/s dual-polarization QPSK transceiver,” J. Lightwave Technol. 26(1), 168–175 (2008).
    [Crossref]

2019 (1)

2018 (6)

2017 (4)

2016 (4)

2014 (1)

2012 (1)

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Sci. Appl. 1(3), e1 (2012).
[Crossref]

2011 (1)

2008 (2)

Antonelli, C.

Aref, V.

Bauters, J.

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Sci. Appl. 1(3), e1 (2012).
[Crossref]

Bayvel, P.

Bowers, J. E.

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Sci. Appl. 1(3), e1 (2012).
[Crossref]

Buchali, F.

S. T. Le, K. Schuh, M. Chagnon, F. Buchali, R. Dischler, V. Aref, H. Buelow, and K. M. Engenhardt, “1.72-Tb/s Virtual-Carrier-Assisted Direct-Detection Transmission Over 200 km,” J. Lightwave Technol. 36(6), 1347–1353 (2018).
[Crossref]

S. T. Le, K. Schuh, M. Chagnon, F. Buchali, and H. Buelow, “1.6Tbps WDM Direct Detection Transmission with Virtual-Carrier over 1200 km,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2018), Tu2D.5.

Buelow, H.

S. T. Le, K. Schuh, M. Chagnon, F. Buchali, R. Dischler, V. Aref, H. Buelow, and K. M. Engenhardt, “1.72-Tb/s Virtual-Carrier-Assisted Direct-Detection Transmission Over 200 km,” J. Lightwave Technol. 36(6), 1347–1353 (2018).
[Crossref]

S. T. Le, K. Schuh, M. Chagnon, F. Buchali, and H. Buelow, “1.6Tbps WDM Direct Detection Transmission with Virtual-Carrier over 1200 km,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2018), Tu2D.5.

Chagnon, M.

Chandrasekhar, S.

Charles, L.

Che, D.

D. Che, C. Sun, and W. Shieh, “Maximizing the spectral efficiency of Stokes vector receiver with optical field recovery,” Opt. Express 26(22), 28976–28981 (2018).
[Crossref]

D. Che, C. Sun, and W. Shieh, “Single-Channel 480-Gb/s Direct Detection of POL-MUX IQ Signal Using Single-Sideband Stokes Vector Receiver,” in Optical Fiber Communication Conference (OFC), OSA Technical Digest (online) (Optical Society of America, Tu2C.7 (2018).

Chen, H.-Y.

Chen, J.

Chen, X.

Chen, Y.-K.

Ciblat, P.

M. Selmi, Y. Jaouen, and P. Ciblat, “Accurate digital frequency offset estimator for coherent PolMux QAM transmission systems,” in 2009 35th European Conference on Optical Communication, 2009), 1–2.

Dai, D.

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Sci. Appl. 1(3), e1 (2012).
[Crossref]

Dischler, R.

El-Fiky, E.

Emerson, N. G.

Engenhardt, K. M.

Erkilinç, M. S.

Fan, S.

Fukuda, H.

Galdino, L.

Gardes, F. Y.

Hoang, T. M.

Huo, J.

Itabashi, S.-i.

Jaouen, Y.

M. Selmi, Y. Jaouen, and P. Ciblat, “Accurate digital frequency offset estimator for coherent PolMux QAM transmission systems,” in 2009 35th European Conference on Optical Communication, 2009), 1–2.

Jiang, M.

Kaneda, N.

Killey, R. I.

Laskowski, P.

Lau, A. P. T.

Le, S. T.

S. T. Le, K. Schuh, M. Chagnon, F. Buchali, R. Dischler, V. Aref, H. Buelow, and K. M. Engenhardt, “1.72-Tb/s Virtual-Carrier-Assisted Direct-Detection Transmission Over 200 km,” J. Lightwave Technol. 36(6), 1347–1353 (2018).
[Crossref]

S. T. Le, K. Schuh, M. Chagnon, F. Buchali, and H. Buelow, “1.6Tbps WDM Direct Detection Transmission with Virtual-Carrier over 1200 km,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2018), Tu2D.5.

Lee, J.

Li, R.

Li, X.

Li, Z.

Liu, G. N.

Lu, C.

Mao, Y.

McGhan, D.

Mecozzi, A.

Morsy-Osman, M.

O’Sullivan, M.

Patel, D.

Plant, D.

Plant, D. V.

Prodaniuc, C.

Qiu, M.

Raybon, G.

Reed, G. T.

Saber, M. G.

Samani, A.

Schuh, K.

S. T. Le, K. Schuh, M. Chagnon, F. Buchali, R. Dischler, V. Aref, H. Buelow, and K. M. Engenhardt, “1.72-Tb/s Virtual-Carrier-Assisted Direct-Detection Transmission Over 200 km,” J. Lightwave Technol. 36(6), 1347–1353 (2018).
[Crossref]

S. T. Le, K. Schuh, M. Chagnon, F. Buchali, and H. Buelow, “1.6Tbps WDM Direct Detection Transmission with Virtual-Carrier over 1200 km,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2018), Tu2D.5.

Selmi, M.

M. Selmi, Y. Jaouen, and P. Ciblat, “Accurate digital frequency offset estimator for coherent PolMux QAM transmission systems,” in 2009 35th European Conference on Optical Communication, 2009), 1–2.

Shi, K.

Shieh, W.

D. Che, C. Sun, and W. Shieh, “Maximizing the spectral efficiency of Stokes vector receiver with optical field recovery,” Opt. Express 26(22), 28976–28981 (2018).
[Crossref]

D. Che, C. Sun, and W. Shieh, “Single-Channel 480-Gb/s Direct Detection of POL-MUX IQ Signal Using Single-Sideband Stokes Vector Receiver,” in Optical Fiber Communication Conference (OFC), OSA Technical Digest (online) (Optical Society of America, Tu2C.7 (2018).

Shinojima, H.

Shtaif, M.

Sillekens, E.

Sowailem, M. Y. S.

Stojanovic, N.

Sun, C.

D. Che, C. Sun, and W. Shieh, “Maximizing the spectral efficiency of Stokes vector receiver with optical field recovery,” Opt. Express 26(22), 28976–28981 (2018).
[Crossref]

D. Che, C. Sun, and W. Shieh, “Single-Channel 480-Gb/s Direct Detection of POL-MUX IQ Signal Using Single-Sideband Stokes Vector Receiver,” in Optical Fiber Communication Conference (OFC), OSA Technical Digest (online) (Optical Society of America, Tu2C.7 (2018).

Sun, H.

Thomsen, B. C.

Thomson, D. J.

Tsuchizawa, T.

Villeneuve, B.

Watanabe, T.

Winzer, P.

Winzer, P. J.

Xiang, M.

Xiao, J.

Xie, C.

Xing, Z.

Xu, X.

Yamada, K.

Yu, C.

Yu, J.

Zhang, F.

Zhang, L.

Zhang, Q.

Zhang, Z.

Zhong, K.

Zhou, E.

Zhou, X.

Zhu, Y.

Zhuge, Q.

Zuo, T.

J. Lightwave Technol. (9)

Z. Li, M. S. Erkılınç, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “SSBI Mitigation and the Kramers–Kronig Scheme in Single-Sideband Direct-Detection Transmission With Receiver-Based Electronic Dispersion Compensation,” J. Lightwave Technol. 35(10), 1887–1893 (2017).
[Crossref]

S. T. Le, K. Schuh, M. Chagnon, F. Buchali, R. Dischler, V. Aref, H. Buelow, and K. M. Engenhardt, “1.72-Tb/s Virtual-Carrier-Assisted Direct-Detection Transmission Over 200 km,” J. Lightwave Technol. 36(6), 1347–1353 (2018).
[Crossref]

L. Zhang, T. Zuo, Y. Mao, Q. Zhang, E. Zhou, G. N. Liu, and X. Xu, “Beyond 100-Gb/s Transmission Over 80-km SMF Using Direct-Detection SSB-DMT at C-Band,” J. Lightwave Technol. 34(2), 723–729 (2016).
[Crossref]

M. Morsy-Osman, M. Chagnon, and D. Plant, “Four Dimensional Modulation and Stokes Direct Detection of Polarization Division Multiplexed Intensities, Inter Polarization Phase and Inter Polarization Differential Phase,” J. Lightwave Technol. 34(7), 1585–1592 (2016).
[Crossref]

L. Charles, B. Villeneuve, Z. Zhang, D. McGhan, H. Sun, and M. O’Sullivan, “WDM performance and PMD tolerance of a coherent 40-Gbit/s dual-polarization QPSK transceiver,” J. Lightwave Technol. 26(1), 168–175 (2008).
[Crossref]

C. Antonelli, A. Mecozzi, M. Shtaif, X. Chen, S. Chandrasekhar, and P. J. Winzer, “Polarization Multiplexing With the Kramers-Kronig Receiver,” J. Lightwave Technol. 35(24), 5418–5424 (2017).
[Crossref]

K. Zhong, X. Zhou, J. Huo, C. Yu, C. Lu, and A. P. T. Lau, “Digital Signal Processing for Short-Reach Optical Communications: A Review of Current Technologies and Future Trends,” J. Lightwave Technol. 36(2), 377–400 (2018).
[Crossref]

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer, “Kramers–Kronig Receivers for 100-km Datacenter Interconnects,” J. Lightwave Technol. 36(1), 79–89 (2018).
[Crossref]

M. Chagnon, “Optical Communications for Short Reach,” J. Lightwave Technol. 37(8), 1779–1797 (2019).
[Crossref]

Light: Sci. Appl. (1)

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Sci. Appl. 1(3), e1 (2012).
[Crossref]

Opt. Express (9)

Z. Xing, A. Samani, M. Xiang, E. El-Fiky, T. M. Hoang, D. Patel, R. Li, M. Qiu, M. G. Saber, M. Morsy-Osman, and D. V. Plant, “100 Gb/s PAM4 transmission system for datacenter interconnects using a SiP ME-MZM based DAC-less transmitter and a VSB self-coherent receiver,” Opt. Express 26(18), 23969–23979 (2018).
[Crossref]

H.-Y. Chen, N. Kaneda, J. Lee, J. Chen, and Y.-K. Chen, “Optical filter requirements in an EML-based single-sideband PAM4 intensity-modulation and direct-detection transmission system,” Opt. Express 25(6), 5852–5860 (2017).
[Crossref]

T. M. Hoang, M. Y. S. Sowailem, Q. Zhuge, Z. Xing, M. Morsy-Osman, E. El-Fiky, S. Fan, M. Xiang, and D. V. Plant, “Single wavelength 480 Gb/s direct detection over 80 km SSMF enabled by Stokes vector Kramers Kronig transceiver,” Opt. Express 25(26), 33534–33542 (2017).
[Crossref]

H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and S.-i. Itabashi, “Silicon photonic circuit with polarization diversity,” Opt. Express 16(7), 4872–4880 (2008).
[Crossref]

Q. Zhang, N. Stojanovic, C. Xie, C. Prodaniuc, and P. Laskowski, “Transmission of single lane 128 Gbit/s PAM-4 signals over an 80 km SSMF link, enabled by DDMZM aided dispersion pre-compensation,” Opt. Express 24(21), 24580–24591 (2016).
[Crossref]

Y. Zhu, M. Jiang, and F. Zhang, “Direct detection of polarization multiplexed single sideband signals with orthogonal offset carriers,” Opt. Express 26(12), 15887–15898 (2018).
[Crossref]

D. Che, C. Sun, and W. Shieh, “Maximizing the spectral efficiency of Stokes vector receiver with optical field recovery,” Opt. Express 26(22), 28976–28981 (2018).
[Crossref]

F. Y. Gardes, D. J. Thomson, N. G. Emerson, and G. T. Reed, “40 Gb/s silicon photonics modulator for TE and TM polarisations,” Opt. Express 19(12), 11804–11814 (2011).
[Crossref]

X. Li, J. Xiao, and J. Yu, “Heterodyne detection and transmission of 60-Gbaud PDM-QPSK signal with SE of 4b/s/Hz,” Opt. Express 22(8), 9307–9313 (2014).
[Crossref]

Optica (1)

Other (4)

D. Che, C. Sun, and W. Shieh, “Single-Channel 480-Gb/s Direct Detection of POL-MUX IQ Signal Using Single-Sideband Stokes Vector Receiver,” in Optical Fiber Communication Conference (OFC), OSA Technical Digest (online) (Optical Society of America, Tu2C.7 (2018).

S. T. Le, K. Schuh, M. Chagnon, F. Buchali, and H. Buelow, “1.6Tbps WDM Direct Detection Transmission with Virtual-Carrier over 1200 km,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2018), Tu2D.5.

M. Selmi, Y. Jaouen, and P. Ciblat, “Accurate digital frequency offset estimator for coherent PolMux QAM transmission systems,” in 2009 35th European Conference on Optical Communication, 2009), 1–2.

Cisco Visual Networking Index: Forecast and Trends, 2017–2022 White Paper, https://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/white-paper-c11-741490.html .

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Figures (8)

Fig. 1.
Fig. 1. The transmitted signal spectrum and the Rx signal spectrum evolution in the linearization DSP.
Fig. 2.
Fig. 2. Experimental set-up and DSP of the proposed scheme.
Fig. 3.
Fig. 3. (a) signal spectrum before and after optical filtering, (b) (c) (d) BER as a function of the Y-Pol CSPR when X-Pol CSPR equals to 11.85 dB, 14.24 dB, and 16,39 dB, respectively.
Fig. 4.
Fig. 4. BER as a function of the RRC filter roll-off factor.
Fig. 5.
Fig. 5. Average BER versus the guard band size when $\alpha $ is set to either zero or an optimized value.
Fig. 6.
Fig. 6. (a) BER as a function of IOP in B2B; (b) BER as a function of launch power after 80 km.
Fig. 7.
Fig. 7. BER versus symbol rate.
Fig. 8.
Fig. 8. (a) Angle deviation from 90o versus PDL. (b) Aggregate SNR versus X-Y angle deviation from 90o.

Equations (6)

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

E T = ( T X e j w X t + E X T Y e j ( w Y t ) + E Y ) + ( n X n Y )
I 1 = | T X | 2 + | E X + n X | 2 + 2 T X Re ( ( E X + n X ) e j w X t ) + | T Y | 2 + | E Y + n Y | 2 + 2 T Y Re ( ( E Y + n Y ) e j w Y t ) + n T h 1
I 2 = | α T X | 2 + | E X + n X | 2 + 2 α T X Re ( ( E X + n X ) e j w X t ) + | T Y | 2 + | E Y + n Y | 2 + 2 T Y Re ( ( E Y + n Y ) e j w Y t ) + n T h 2
I 1 I 2 = ( 1 α 2 ) | T X | 2 + 2 ( 1 α ) T X Re ( ( E X + n X ) e j w X t ) + n T h 3
E T = ( T X e j w X t + E X + cos ( θ ) ( T Y e j w Y t + E Y ) sin ( θ ) ( T Y e j w Y t + E Y ) ) + ( n X n Y )
E T = ( T X e j w X t + E X sin ( θ ) ( T Y e j ( w Y t ) + E Y ) ) + ( n X n Y )

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