Abstract

An improved self-coherent digital-signal-processing-based optical receiver that utilizes polarization diversity is presented and analyzed. It demonstrates that coherent performance can be approached by employing field reconstruction on differential-phase detection and judicious digital signal processing. Performance is improved compared to known techniques by addressing two shortcomings that characterize self-coherent detection: 1. the phase drift caused during field reconstruction; and 2. the loss of synchronization induced by a zero-intensity sample with undefined phase. These will be met with phase estimation and periodic correction, and with polarization-diversity and a non-linear quantization scheme, respectively. It is demonstrated by means of simulations that the improved receiver allows achieving detection of an optical orthogonal frequency division multiplexing at uncoded bit-error-rate of 3*10-5, with 12 bit non-uniform quantization, over an 18 dB signal-to-noise ratio channel.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. X. Liu, S. Chandrasekhar, and A. Leven, “Digital self-coherent detection,” Opt. Express 16(2), 792–803 (2008).
    [Crossref]
  2. J. Li, R. Schmogrow, D. Hillerkuss, P. C. Schindler, M. Nazarthy, C. Schmidt-Langhorst, S.-B. Ezra, I. Tselniker, C. Koos, W. Freude, and J. Leuthold, “A self-coherent receiver for detection of PolMUX coherent signals,” Opt. Express 20(19), 21413 (2012).
    [Crossref]
  3. I. Tselniker, M. Nazarathy, S.-B. Ezra, J. Li, and J. Leuthold, “Self-coherent complex field reconstruction with in-phase and quadrature delay detection without a direct-detection branch,” Opt. Express 20(14), 15452 (2012).
    [Crossref]
  4. S. Adhikari, S. L. Jansen, M. Alfiad, B. Inan, V. a J. M. Sleiffer, a. Lobato, P. Leoni, and W. Rosenkranz, “Self-coherent optical OFDM: An interesting alternative to direct or coherent detection,” Int. Conf. Transparent Opt. Networks1–4 (2011).
  5. A. Magen and O. Amrani, “Approaching coherent performance in differential detection via diversity,” Opt. Express 23(4), 4529 (2015).
    [Crossref]
  6. R. Noe, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photonics Technol. Lett. 17(4), 887–889 (2005).
    [Crossref]
  7. W. Shieh, “Maximum-likelihood phase and channel estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 20(8), 605–607 (2008).
    [Crossref]
  8. K. P. Zhong, J. H. Ke, Y. Gao, and J. C. Cartledge, “Linewidth-Tolerant and Low-Complexity Two-Stage Carrier Phase Estimation Based on Modified QPSK Partitioning for Dual-Polarization 16-QAM Systems,” J. Lightwave Technol. 31(1), 50–57 (2013).
    [Crossref]
  9. I. Fatadin, D. Ives, and S. J. Savory, “Laser Linewidth Tolerance for 16-QAM Coherent Optical Systems Using QPSK Partitioning,” IEEE Photonics Technol. Lett. 22(9), 631–633 (2010).
    [Crossref]
  10. S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser Linewidth Tolerance of Decision-Aided Maximum Likelihood Phase Estimation in Coherent Optical $M$-ary PSK and QAM Systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
    [Crossref]
  11. S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express 17(2), 703 (2009).
    [Crossref]
  12. X. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 19(12), 919–921 (2007).
    [Crossref]
  13. D. van den Borne, S. Calabro, S. L. Jansen, E. Gottwald, G. D. Khoe, and H. de Waardt, “Differential quadrature phase shift keying with close to homodyne performance based on multi-symbol phase estimation,” OFC/NFOEC - Conf. Opt. Fiber Commun. Natl. Fiber Opt. Eng. Conf. (2005).
  14. E. W. Weisstein, “Modified Bessel Function of the Second Kind,” http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html .
  15. Y. Wang, L. H. Wang, J. H. Ge, and B. Ai, “An efficient nonlinear companding transform for reducing PAPR of OFDM signals,” IEEE Trans. Broadcast. 58(4), 677–684 (2012).
    [Crossref]
  16. Y. Wang, J. Ge, L. Wang, J. Li, and B. Ai, “Nonlinear companding transform for reduction of peak-to-average power ratio in OFDM systems,” IEEE Trans. Broadcast. 59(2), 369–375 (2013).
    [Crossref]
  17. J. Hou, J. Ge, D. Zhai, and J. Li, “Peak-to-average power ratio reduction of OFDM signals with nonlinear companding scheme,” IEEE Trans. Broadcast. 56(2), 258–262 (2010).
    [Crossref]

2015 (1)

2013 (2)

K. P. Zhong, J. H. Ke, Y. Gao, and J. C. Cartledge, “Linewidth-Tolerant and Low-Complexity Two-Stage Carrier Phase Estimation Based on Modified QPSK Partitioning for Dual-Polarization 16-QAM Systems,” J. Lightwave Technol. 31(1), 50–57 (2013).
[Crossref]

Y. Wang, J. Ge, L. Wang, J. Li, and B. Ai, “Nonlinear companding transform for reduction of peak-to-average power ratio in OFDM systems,” IEEE Trans. Broadcast. 59(2), 369–375 (2013).
[Crossref]

2012 (3)

2010 (2)

J. Hou, J. Ge, D. Zhai, and J. Li, “Peak-to-average power ratio reduction of OFDM signals with nonlinear companding scheme,” IEEE Trans. Broadcast. 56(2), 258–262 (2010).
[Crossref]

I. Fatadin, D. Ives, and S. J. Savory, “Laser Linewidth Tolerance for 16-QAM Coherent Optical Systems Using QPSK Partitioning,” IEEE Photonics Technol. Lett. 22(9), 631–633 (2010).
[Crossref]

2009 (2)

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser Linewidth Tolerance of Decision-Aided Maximum Likelihood Phase Estimation in Coherent Optical $M$-ary PSK and QAM Systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
[Crossref]

S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express 17(2), 703 (2009).
[Crossref]

2008 (2)

X. Liu, S. Chandrasekhar, and A. Leven, “Digital self-coherent detection,” Opt. Express 16(2), 792–803 (2008).
[Crossref]

W. Shieh, “Maximum-likelihood phase and channel estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 20(8), 605–607 (2008).
[Crossref]

2007 (1)

X. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

2005 (1)

R. Noe, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photonics Technol. Lett. 17(4), 887–889 (2005).
[Crossref]

Adhikari, S.

S. Adhikari, S. L. Jansen, M. Alfiad, B. Inan, V. a J. M. Sleiffer, a. Lobato, P. Leoni, and W. Rosenkranz, “Self-coherent optical OFDM: An interesting alternative to direct or coherent detection,” Int. Conf. Transparent Opt. Networks1–4 (2011).

Ai, B.

Y. Wang, J. Ge, L. Wang, J. Li, and B. Ai, “Nonlinear companding transform for reduction of peak-to-average power ratio in OFDM systems,” IEEE Trans. Broadcast. 59(2), 369–375 (2013).
[Crossref]

Y. Wang, L. H. Wang, J. H. Ge, and B. Ai, “An efficient nonlinear companding transform for reducing PAPR of OFDM signals,” IEEE Trans. Broadcast. 58(4), 677–684 (2012).
[Crossref]

Alfiad, M.

S. Adhikari, S. L. Jansen, M. Alfiad, B. Inan, V. a J. M. Sleiffer, a. Lobato, P. Leoni, and W. Rosenkranz, “Self-coherent optical OFDM: An interesting alternative to direct or coherent detection,” Int. Conf. Transparent Opt. Networks1–4 (2011).

Amrani, O.

Calabro, S.

D. van den Borne, S. Calabro, S. L. Jansen, E. Gottwald, G. D. Khoe, and H. de Waardt, “Differential quadrature phase shift keying with close to homodyne performance based on multi-symbol phase estimation,” OFC/NFOEC - Conf. Opt. Fiber Commun. Natl. Fiber Opt. Eng. Conf. (2005).

Cartledge, J. C.

Chandrasekhar, S.

Chen, J.

S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express 17(2), 703 (2009).
[Crossref]

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser Linewidth Tolerance of Decision-Aided Maximum Likelihood Phase Estimation in Coherent Optical $M$-ary PSK and QAM Systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
[Crossref]

de Waardt, H.

D. van den Borne, S. Calabro, S. L. Jansen, E. Gottwald, G. D. Khoe, and H. de Waardt, “Differential quadrature phase shift keying with close to homodyne performance based on multi-symbol phase estimation,” OFC/NFOEC - Conf. Opt. Fiber Commun. Natl. Fiber Opt. Eng. Conf. (2005).

Ezra, S.-B.

Fatadin, I.

I. Fatadin, D. Ives, and S. J. Savory, “Laser Linewidth Tolerance for 16-QAM Coherent Optical Systems Using QPSK Partitioning,” IEEE Photonics Technol. Lett. 22(9), 631–633 (2010).
[Crossref]

Freude, W.

Gao, Y.

Ge, J.

Y. Wang, J. Ge, L. Wang, J. Li, and B. Ai, “Nonlinear companding transform for reduction of peak-to-average power ratio in OFDM systems,” IEEE Trans. Broadcast. 59(2), 369–375 (2013).
[Crossref]

J. Hou, J. Ge, D. Zhai, and J. Li, “Peak-to-average power ratio reduction of OFDM signals with nonlinear companding scheme,” IEEE Trans. Broadcast. 56(2), 258–262 (2010).
[Crossref]

Ge, J. H.

Y. Wang, L. H. Wang, J. H. Ge, and B. Ai, “An efficient nonlinear companding transform for reducing PAPR of OFDM signals,” IEEE Trans. Broadcast. 58(4), 677–684 (2012).
[Crossref]

Gottwald, E.

D. van den Borne, S. Calabro, S. L. Jansen, E. Gottwald, G. D. Khoe, and H. de Waardt, “Differential quadrature phase shift keying with close to homodyne performance based on multi-symbol phase estimation,” OFC/NFOEC - Conf. Opt. Fiber Commun. Natl. Fiber Opt. Eng. Conf. (2005).

Hillerkuss, D.

Hou, J.

J. Hou, J. Ge, D. Zhai, and J. Li, “Peak-to-average power ratio reduction of OFDM signals with nonlinear companding scheme,” IEEE Trans. Broadcast. 56(2), 258–262 (2010).
[Crossref]

Inan, B.

S. Adhikari, S. L. Jansen, M. Alfiad, B. Inan, V. a J. M. Sleiffer, a. Lobato, P. Leoni, and W. Rosenkranz, “Self-coherent optical OFDM: An interesting alternative to direct or coherent detection,” Int. Conf. Transparent Opt. Networks1–4 (2011).

Ives, D.

I. Fatadin, D. Ives, and S. J. Savory, “Laser Linewidth Tolerance for 16-QAM Coherent Optical Systems Using QPSK Partitioning,” IEEE Photonics Technol. Lett. 22(9), 631–633 (2010).
[Crossref]

Jansen, S. L.

S. Adhikari, S. L. Jansen, M. Alfiad, B. Inan, V. a J. M. Sleiffer, a. Lobato, P. Leoni, and W. Rosenkranz, “Self-coherent optical OFDM: An interesting alternative to direct or coherent detection,” Int. Conf. Transparent Opt. Networks1–4 (2011).

D. van den Borne, S. Calabro, S. L. Jansen, E. Gottwald, G. D. Khoe, and H. de Waardt, “Differential quadrature phase shift keying with close to homodyne performance based on multi-symbol phase estimation,” OFC/NFOEC - Conf. Opt. Fiber Commun. Natl. Fiber Opt. Eng. Conf. (2005).

Kam, P. Y.

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser Linewidth Tolerance of Decision-Aided Maximum Likelihood Phase Estimation in Coherent Optical $M$-ary PSK and QAM Systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
[Crossref]

S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express 17(2), 703 (2009).
[Crossref]

Ke, J. H.

Khoe, G. D.

D. van den Borne, S. Calabro, S. L. Jansen, E. Gottwald, G. D. Khoe, and H. de Waardt, “Differential quadrature phase shift keying with close to homodyne performance based on multi-symbol phase estimation,” OFC/NFOEC - Conf. Opt. Fiber Commun. Natl. Fiber Opt. Eng. Conf. (2005).

Koos, C.

Leoni, P.

S. Adhikari, S. L. Jansen, M. Alfiad, B. Inan, V. a J. M. Sleiffer, a. Lobato, P. Leoni, and W. Rosenkranz, “Self-coherent optical OFDM: An interesting alternative to direct or coherent detection,” Int. Conf. Transparent Opt. Networks1–4 (2011).

Leuthold, J.

Leven, A.

Li, J.

Y. Wang, J. Ge, L. Wang, J. Li, and B. Ai, “Nonlinear companding transform for reduction of peak-to-average power ratio in OFDM systems,” IEEE Trans. Broadcast. 59(2), 369–375 (2013).
[Crossref]

I. Tselniker, M. Nazarathy, S.-B. Ezra, J. Li, and J. Leuthold, “Self-coherent complex field reconstruction with in-phase and quadrature delay detection without a direct-detection branch,” Opt. Express 20(14), 15452 (2012).
[Crossref]

J. Li, R. Schmogrow, D. Hillerkuss, P. C. Schindler, M. Nazarthy, C. Schmidt-Langhorst, S.-B. Ezra, I. Tselniker, C. Koos, W. Freude, and J. Leuthold, “A self-coherent receiver for detection of PolMUX coherent signals,” Opt. Express 20(19), 21413 (2012).
[Crossref]

J. Hou, J. Ge, D. Zhai, and J. Li, “Peak-to-average power ratio reduction of OFDM signals with nonlinear companding scheme,” IEEE Trans. Broadcast. 56(2), 258–262 (2010).
[Crossref]

Liu, X.

Lobato, a.

S. Adhikari, S. L. Jansen, M. Alfiad, B. Inan, V. a J. M. Sleiffer, a. Lobato, P. Leoni, and W. Rosenkranz, “Self-coherent optical OFDM: An interesting alternative to direct or coherent detection,” Int. Conf. Transparent Opt. Networks1–4 (2011).

Magen, A.

Nazarathy, M.

Nazarthy, M.

Noe, R.

R. Noe, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photonics Technol. Lett. 17(4), 887–889 (2005).
[Crossref]

Rosenkranz, W.

S. Adhikari, S. L. Jansen, M. Alfiad, B. Inan, V. a J. M. Sleiffer, a. Lobato, P. Leoni, and W. Rosenkranz, “Self-coherent optical OFDM: An interesting alternative to direct or coherent detection,” Int. Conf. Transparent Opt. Networks1–4 (2011).

Savory, S. J.

I. Fatadin, D. Ives, and S. J. Savory, “Laser Linewidth Tolerance for 16-QAM Coherent Optical Systems Using QPSK Partitioning,” IEEE Photonics Technol. Lett. 22(9), 631–633 (2010).
[Crossref]

Schindler, P. C.

Schmidt-Langhorst, C.

Schmogrow, R.

Shieh, W.

W. Shieh, “Maximum-likelihood phase and channel estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 20(8), 605–607 (2008).
[Crossref]

X. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

Sleiffer, V. a J. M.

S. Adhikari, S. L. Jansen, M. Alfiad, B. Inan, V. a J. M. Sleiffer, a. Lobato, P. Leoni, and W. Rosenkranz, “Self-coherent optical OFDM: An interesting alternative to direct or coherent detection,” Int. Conf. Transparent Opt. Networks1–4 (2011).

Tang, Y.

X. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

Tselniker, I.

van den Borne, D.

D. van den Borne, S. Calabro, S. L. Jansen, E. Gottwald, G. D. Khoe, and H. de Waardt, “Differential quadrature phase shift keying with close to homodyne performance based on multi-symbol phase estimation,” OFC/NFOEC - Conf. Opt. Fiber Commun. Natl. Fiber Opt. Eng. Conf. (2005).

Wang, L.

Y. Wang, J. Ge, L. Wang, J. Li, and B. Ai, “Nonlinear companding transform for reduction of peak-to-average power ratio in OFDM systems,” IEEE Trans. Broadcast. 59(2), 369–375 (2013).
[Crossref]

Wang, L. H.

Y. Wang, L. H. Wang, J. H. Ge, and B. Ai, “An efficient nonlinear companding transform for reducing PAPR of OFDM signals,” IEEE Trans. Broadcast. 58(4), 677–684 (2012).
[Crossref]

Wang, Y.

Y. Wang, J. Ge, L. Wang, J. Li, and B. Ai, “Nonlinear companding transform for reduction of peak-to-average power ratio in OFDM systems,” IEEE Trans. Broadcast. 59(2), 369–375 (2013).
[Crossref]

Y. Wang, L. H. Wang, J. H. Ge, and B. Ai, “An efficient nonlinear companding transform for reducing PAPR of OFDM signals,” IEEE Trans. Broadcast. 58(4), 677–684 (2012).
[Crossref]

Weisstein, E. W.

E. W. Weisstein, “Modified Bessel Function of the Second Kind,” http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html .

Yi, X.

X. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

Yu, C.

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser Linewidth Tolerance of Decision-Aided Maximum Likelihood Phase Estimation in Coherent Optical $M$-ary PSK and QAM Systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
[Crossref]

S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express 17(2), 703 (2009).
[Crossref]

Zhai, D.

J. Hou, J. Ge, D. Zhai, and J. Li, “Peak-to-average power ratio reduction of OFDM signals with nonlinear companding scheme,” IEEE Trans. Broadcast. 56(2), 258–262 (2010).
[Crossref]

Zhang, S.

S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express 17(2), 703 (2009).
[Crossref]

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser Linewidth Tolerance of Decision-Aided Maximum Likelihood Phase Estimation in Coherent Optical $M$-ary PSK and QAM Systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
[Crossref]

Zhong, K. P.

IEEE Photonics Technol. Lett. (5)

R. Noe, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photonics Technol. Lett. 17(4), 887–889 (2005).
[Crossref]

W. Shieh, “Maximum-likelihood phase and channel estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 20(8), 605–607 (2008).
[Crossref]

I. Fatadin, D. Ives, and S. J. Savory, “Laser Linewidth Tolerance for 16-QAM Coherent Optical Systems Using QPSK Partitioning,” IEEE Photonics Technol. Lett. 22(9), 631–633 (2010).
[Crossref]

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser Linewidth Tolerance of Decision-Aided Maximum Likelihood Phase Estimation in Coherent Optical $M$-ary PSK and QAM Systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
[Crossref]

X. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

IEEE Trans. Broadcast. (3)

Y. Wang, L. H. Wang, J. H. Ge, and B. Ai, “An efficient nonlinear companding transform for reducing PAPR of OFDM signals,” IEEE Trans. Broadcast. 58(4), 677–684 (2012).
[Crossref]

Y. Wang, J. Ge, L. Wang, J. Li, and B. Ai, “Nonlinear companding transform for reduction of peak-to-average power ratio in OFDM systems,” IEEE Trans. Broadcast. 59(2), 369–375 (2013).
[Crossref]

J. Hou, J. Ge, D. Zhai, and J. Li, “Peak-to-average power ratio reduction of OFDM signals with nonlinear companding scheme,” IEEE Trans. Broadcast. 56(2), 258–262 (2010).
[Crossref]

J. Lightwave Technol. (1)

Opt. Express (5)

Other (3)

D. van den Borne, S. Calabro, S. L. Jansen, E. Gottwald, G. D. Khoe, and H. de Waardt, “Differential quadrature phase shift keying with close to homodyne performance based on multi-symbol phase estimation,” OFC/NFOEC - Conf. Opt. Fiber Commun. Natl. Fiber Opt. Eng. Conf. (2005).

E. W. Weisstein, “Modified Bessel Function of the Second Kind,” http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html .

S. Adhikari, S. L. Jansen, M. Alfiad, B. Inan, V. a J. M. Sleiffer, a. Lobato, P. Leoni, and W. Rosenkranz, “Self-coherent optical OFDM: An interesting alternative to direct or coherent detection,” Int. Conf. Transparent Opt. Networks1–4 (2011).

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Figures (6)

Fig. 1.
Fig. 1. Detailed block diagram of the drift estimation and correction process.
Fig. 2.
Fig. 2. AP noise of a QPSK signal with 9 bit quantizer and 12 dB SNR, noise and drift estimation (left), and noise after correction of drift (right).
Fig. 3.
Fig. 3. BER Performance vs. quantization depth; variety of receivers with a uniform-scale quantizer. 16QAM (left) and 16QAM-OFDM (right) signals at SNR = 18 dB.
Fig. 4.
Fig. 4. BER Performance vs. quantization depth; variety of receivers with a logarithmic scale quantizer.
Fig. 5.
Fig. 5. Performance vs. SNR. Different receiver schemes, for 12 quantization bits, compared in terms of BER for a 16QAM-OFDM signal.
Fig. 6.
Fig. 6. The analyzed, polarization-diversity, X-DSCD receiver scheme.

Equations (24)

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

u x I ( t ) Re { E ¯ x ( t ) E ¯ x ( t τ ) } = E x ( t ) E x ( t τ ) cos ( Δ ϕ τ , x ( t ) ) u x Q ( t ) Im { E ¯ x ( t ) E ¯ x ( t τ ) } = E x ( t ) E x ( t τ ) sin ( Δ ϕ τ , x ( t ) ) ,
ϕ x ( t ) ϕ x ( t τ ) Δ ϕ τ , x ( t ) = tan 1 ( u x Q ( t ) / u x I ( t ) ) ,
E ¯ x ( t = t 0 + k τ ) = E ¯ k , x = P k , x e j ϕ k , x = P k , x e j ( ϕ 0 , x + i = 1 k Δ ϕ i , x ) .
u k , x y I E k , x E k , y cos ( Δ ϕ k , x y ) u k , x y Q E k , x E k , y sin ( Δ ϕ k , x y ) ,
ϕ k , x ϕ k , y Δ ϕ k , x y = tan 1 ( u k , x y Q / u k , x y I ) ,
ϕ k , x = Δ ϕ k , x y ϕ k , y ,
n k Δ ϕ = Δ ϕ ^ k Δ ϕ k ,
E ^ k = ϕ ^ k = ϕ ^ 0 + i = 1 k Δ ϕ ^ i = ϕ 0 + n ϕ 0 + i = 1 k ( Δ ϕ i + n i Δ ϕ ) = ϕ k + n k ϕ .
n k Δ ϕ = tan 1 ( u ~ k Q u ~ k I ) Δ ϕ k ,
n k Δ ϕ = tan 1 ( u ~ k Q u ~ k I ) Δ ϕ k .
var ( n k ϕ ) = k var ( n 1 Δ ϕ ) .
E [ ϕ k ϕ ^ k c ] = 0 ,
ϕ ^ d r i f t = 1 L i = k L + 1 k ( s ^ i E ^ i ) .
{ ϕ ^ k d r i f t } k = m + 1 m + M = 1 L i = m L + 1 m ( s ^ i E ^ i ) .
E ^ k c E ^ k e j ϕ ^ k d r i f t .
{ ϕ ^ k d r i f t } k = m + 1 m + M = 1 i = m L + 1 m D i i = m L + 1 m ( s ^ i c ϕ ^ i ) D i , D i = { 1 s ^ i c h 0 s ^ i c h ,
E ^ q = P ^ q e j ( ϕ q + n q ϕ ) = P ^ q e j ϕ q + j ( n d r i f t + n q r a n d o m ) = e j n d r i f t P ^ q e j ( ϕ q + n q r a n d o m ) ,
{ D F T [ { E ^ q } m + 1 m + Q ] } k = m + 1 m + Q = q = m + 1 m + Q E ^ q e j q k 2 π Q = e j n d r i f t q = m m + Q P ^ q e j ( ϕ q + n q r a n d o m ) e j q k 2 π Q = e j n d r i f t { s ^ k e j n m } k = m + 1 m + Q ,
δ L S B { max x x | x = 0 } ,
ϕ ^ k , x = Δ ϕ k , x y + ϕ k , y + n k , x y Δ ϕ + n k , y ϕ .
u k { I , Q } log = M max F { exp [ n ln ( F + 1 ) ( 2 N 1 ) / 2 ] 1 }
n = f l o o r ( ln ( | u k { I , Q } F / M max | + 1 ) ln ( F + 1 ) / ln ( F + 1 ) [ ( 2 N 1 ) / 2 ] [ ( 2 N 1 ) / 2 ] ) ,
δ u n i f o r m = 2 M 2 ( N 1 ) ,
δ log = e log ( F ) / 2 ( N 1 ) M / M F F ,

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