Abstract

The MSDD carrier phase estimation technique is derived here for optically coherent QPSK transmission, introducing the principle of operation while providing intuitive insight in terms of a multi-symbol extension of naïve delay-detection. We derive here for the first time Wiener-optimized and LMS-adapted versions of MSDD, introduce simplified hardware realizations, and evaluate complexity and numerical performance tradeoffs of this highly robust and low-complexity carrier phase recovery method. A multiplier-free carrier phase recovery version of the MSDD provides nearly optimal performance for linewidths up to ~0.5 MHz, whereas for wider linewidths, the Wiener or LMS versions provide optimal performance at about 9 taps, using 1 or 2 complex multipliers per tap.

© 2012 OSA

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References

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  1. A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
    [CrossRef]
  2. E. Ip and J. M. Kahn, “Carrier synchronization for 3- and 4-bit-per-symbol optical transmission,” J. Lightwave Technol. 23(12), 4110–4124 (2005).
    [CrossRef]
  3. R. Noé, “PLL-free synchronous QPSK polarization multiplex / diversity receiver concept with digital I & Q baseband processing,” IEEE Photon. Technol. Lett. 17(4), 887–889 (2005).
    [CrossRef]
  4. M. G. Taylor, “Accurate digital phase estimation process for coherent detection using a parallel digital processor,” in ECOC’05 European Conf. of Optical Communication, Tu 4.2.6 (2005).
  5. E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” J. Lightwave Technol. 25(9), 2675–2692 (2007).
    [CrossRef]
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    [CrossRef]
  7. M. G. Taylor, “Detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009).
    [CrossRef]
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    [CrossRef]
  10. K. Piyawanno, M. Kuschnerov, B. Spinnler, and B. Lankl, “Low complexity carrier recovery for coherent QAM using superscalar parallelization,” in ECOC’10 European Conf. of Optical Communication, We.7.A.3 (2010).
  11. D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun. 38(3), 300–308 (1990).
    [CrossRef]
  12. F. Edbauer, “Bit error rate of binary and quaternary DPSK signals with multiple differential feedback detection,” IEEE Trans. Commun. 40(3), 457–460 (1992).
    [CrossRef]
  13. M. Adachi and F. Sawahashi, “Decision feedback multiple-symbol differential detection for M-ary DPSK,” Electron. Lett. 29(15), 1385–1387 (1993).
    [CrossRef]
  14. F. Adachi and M. Sawahashi, “Decision feedback differential phase detection of M-ary DPSK signals,” IEEE Trans. Vehicular Technol. 44(2), 203–210 (1995).
    [CrossRef]
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    [CrossRef]
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  19. M. Nazarathy and Y. Yadin, “Approaching coherent homodyne performance with direct detection low-complexity advanced modulation formats,” in COTA’06 Coherent Optical Technologies and Applications (2006).
  20. X. Liu, “Data-aided multi-symbol phase estimation for receiver sensitivity enhancement in optical DQPSK, CThB4,” in COTA’06 Coherent Optical Techniques and Applications (2006).
  21. M. Nazarathy and Y. Atzmon, “Approaching coherent homodyne performance with direct detection low-complexity advanced modulation formats,” in COTA’08 Coherent Optical Techniques and Applications (2008).
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    [CrossRef] [PubMed]
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    [CrossRef]
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  25. J. Li, R. Schmogrow, D. Hillerkuss, M. Lauermann, M. Winter, K. Worms, C. Schubert, C. Koos, W. Freude, and W. J. Leuthold, “Self-coherent receiver for PolMUX coherent signals,” in OFC’11 Conference on Optical Fiber Communication, OWV5 (2011).
  26. N. Kikuchi and S. Sasaki, “Highly sensitive optical multilevel transmission of arbitrary quadrature-amplitude modulation (QAM) signals with direct detection,” J. Lightwave Technol. 28(1), 123–130 (2010).
    [CrossRef]
  27. N. Kikuchi, “Chromatic dispersion-tolerant higher-order multilevel transmission with optical delay detection,” in SPPCom’11 Signal Processing in Photonic Communications - OSA Technical Digest (2011).
  28. 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” in ICTON’11 13th International Conference on Transparent Optical Networks (2011).
  29. S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, (Springer, 2011).
  30. N. Sigron, I. Tselniker, M. Nazarathy, A. Gorshtein, D. Sadot, and I. Zelniker, “Ultimate single-carrier recovery for coherent detection,” in OFC’11 Conference on Optical Fiber Communication, OMJ2 (2011).
  31. M. Nazarathy, N. Sigron, and I. Tselniker, “Integrated carrier phase and frequency estimation for coherent detection based on multi-symbol differential detection (MSDD),” in SPPCom’11 Signal Processing in Photonic Communications - OSA Technical Digest, Invited paper SPMC1 (2011).
  32. N. Kikuchi, S. Sasaki, and T. Uda, “Phase-noise tolerant coherent polarization-multiplexed 16QAM Transmission with digital delay-detection, in ECOC’11 European Conference of Optical Communication (ECOC), Tu.3.A (2011).
  33. T. Adali and S. Haykin, Adaptive Signal Processing—Next Generation Solutions (John Wiley, 2010).
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    [CrossRef] [PubMed]
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    [CrossRef]

2010

2009

2008

2007

2005

E. Ip and J. M. Kahn, “Carrier synchronization for 3- and 4-bit-per-symbol optical transmission,” J. Lightwave Technol. 23(12), 4110–4124 (2005).
[CrossRef]

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

1995

F. Adachi and M. Sawahashi, “Decision feedback differential phase detection of M-ary DPSK signals,” IEEE Trans. Vehicular Technol. 44(2), 203–210 (1995).
[CrossRef]

1993

M. Adachi and F. Sawahashi, “Decision feedback multiple-symbol differential detection for M-ary DPSK,” Electron. Lett. 29(15), 1385–1387 (1993).
[CrossRef]

1992

F. Edbauer, “Bit error rate of binary and quaternary DPSK signals with multiple differential feedback detection,” IEEE Trans. Commun. 40(3), 457–460 (1992).
[CrossRef]

1990

D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun. 38(3), 300–308 (1990).
[CrossRef]

1983

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[CrossRef]

Adachi, F.

F. Adachi and M. Sawahashi, “Decision feedback differential phase detection of M-ary DPSK signals,” IEEE Trans. Vehicular Technol. 44(2), 203–210 (1995).
[CrossRef]

Adachi, M.

M. Adachi and F. Sawahashi, “Decision feedback multiple-symbol differential detection for M-ary DPSK,” Electron. Lett. 29(15), 1385–1387 (1993).
[CrossRef]

Adamczyk, O.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Atzmon, Y.

Bhandare, S.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Chandrasekhar, S.

Chen, J.

Choi, H. Y.

Christen, L.

Chung, Y. C.

Divsalar, D.

D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun. 38(3), 300–308 (1990).
[CrossRef]

Edbauer, F.

F. Edbauer, “Bit error rate of binary and quaternary DPSK signals with multiple differential feedback detection,” IEEE Trans. Commun. 40(3), 457–460 (1992).
[CrossRef]

Ho, K. P.

Hoffmann, S.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Ip, E.

Kahn, J. M.

Kam, P. Y.

Kikuchi, N.

Leven, A.

Liu, X.

Lize, Y. K.

Nazarathy, M.

Noe, R.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Noé, R.

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

Peveling, R.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Pfau, T.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Porrmann, M.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Sasaki, S.

Sawahashi, F.

M. Adachi and F. Sawahashi, “Decision feedback multiple-symbol differential detection for M-ary DPSK,” Electron. Lett. 29(15), 1385–1387 (1993).
[CrossRef]

Sawahashi, M.

F. Adachi and M. Sawahashi, “Decision feedback differential phase detection of M-ary DPSK signals,” IEEE Trans. Vehicular Technol. 44(2), 203–210 (1995).
[CrossRef]

Shieh, W.

Simon, M. K.

D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun. 38(3), 300–308 (1990).
[CrossRef]

Takushima, Y.

Taylor, M. G.

Viterbi, A.

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[CrossRef]

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[CrossRef]

Willner, A. E.

Wordehoff, C.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

Yu, C.

-yuen Kam, P.

Zhang, S.

Electron. Lett.

M. Adachi and F. Sawahashi, “Decision feedback multiple-symbol differential detection for M-ary DPSK,” Electron. Lett. 29(15), 1385–1387 (1993).
[CrossRef]

IEEE Photon. Technol. Lett.

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

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008).
[CrossRef]

IEEE Trans. Commun.

D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun. 38(3), 300–308 (1990).
[CrossRef]

F. Edbauer, “Bit error rate of binary and quaternary DPSK signals with multiple differential feedback detection,” IEEE Trans. Commun. 40(3), 457–460 (1992).
[CrossRef]

IEEE Trans. Inf. Theory

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[CrossRef]

IEEE Trans. Vehicular Technol.

F. Adachi and M. Sawahashi, “Decision feedback differential phase detection of M-ary DPSK signals,” IEEE Trans. Vehicular Technol. 44(2), 203–210 (1995).
[CrossRef]

J. Lightwave Technol.

Opt. Express

Other

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,” in OFC’05 Conference on Optical Fiber Communication (2005).

M. Nazarathy and Y. Yadin, “Approaching coherent homodyne performance with direct detection low-complexity advanced modulation formats,” in COTA’06 Coherent Optical Technologies and Applications (2006).

X. Liu, “Data-aided multi-symbol phase estimation for receiver sensitivity enhancement in optical DQPSK, CThB4,” in COTA’06 Coherent Optical Techniques and Applications (2006).

M. Nazarathy and Y. Atzmon, “Approaching coherent homodyne performance with direct detection low-complexity advanced modulation formats,” in COTA’08 Coherent Optical Techniques and Applications (2008).

K. Piyawanno, M. Kuschnerov, B. Spinnler, and B. Lankl, “Low complexity carrier recovery for coherent QAM using superscalar parallelization,” in ECOC’10 European Conf. of Optical Communication, We.7.A.3 (2010).

M. G. Taylor, “Algorithms for coherent detection what can we learn from other fields?” in OFC/NFOEC’10, Conf. on Optical Fiber Communication, OThL4 (2010).

M. G. Taylor, “Accurate digital phase estimation process for coherent detection using a parallel digital processor,” in ECOC’05 European Conf. of Optical Communication, Tu 4.2.6 (2005).

J. Li, R. Schmogrow, D. Hillerkuss, M. Lauermann, M. Winter, K. Worms, C. Schubert, C. Koos, W. Freude, and W. J. Leuthold, “Self-coherent receiver for PolMUX coherent signals,” in OFC’11 Conference on Optical Fiber Communication, OWV5 (2011).

N. Kikuchi, “Chromatic dispersion-tolerant higher-order multilevel transmission with optical delay detection,” in SPPCom’11 Signal Processing in Photonic Communications - OSA Technical Digest (2011).

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” in ICTON’11 13th International Conference on Transparent Optical Networks (2011).

S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, (Springer, 2011).

N. Sigron, I. Tselniker, M. Nazarathy, A. Gorshtein, D. Sadot, and I. Zelniker, “Ultimate single-carrier recovery for coherent detection,” in OFC’11 Conference on Optical Fiber Communication, OMJ2 (2011).

M. Nazarathy, N. Sigron, and I. Tselniker, “Integrated carrier phase and frequency estimation for coherent detection based on multi-symbol differential detection (MSDD),” in SPPCom’11 Signal Processing in Photonic Communications - OSA Technical Digest, Invited paper SPMC1 (2011).

N. Kikuchi, S. Sasaki, and T. Uda, “Phase-noise tolerant coherent polarization-multiplexed 16QAM Transmission with digital delay-detection, in ECOC’11 European Conference of Optical Communication (ECOC), Tu.3.A (2011).

T. Adali and S. Haykin, Adaptive Signal Processing—Next Generation Solutions (John Wiley, 2010).

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Figures (14)

Fig. 1
Fig. 1

QPSK/QAM transmitter scheme with (modulus-preserving) differential precoder, compatible with a MSDD-based receiver.

Fig. 2
Fig. 2

Link model including the Carrier Recovery System.

Fig. 3
Fig. 3

(a): Carrier Recovery (CR) system operation. (b): Demodulator as complex conjugate multiplier.

Fig. 4
Fig. 4

(a): MSDD CR system operating on a window of L past samples in order to generate an improved reference for demodulation. (b): The MSDD CPE may be alternatively realized by acting on each past sample at a time, rotating the respective past sample in order to bring it in approximate alignment with the last sample (the conventional DD reference). Either of these rotated-into-alignment past samples may be selected to serve as a “partial” reference. The improved reference will be obtained by taking a linear combination of all of the partial reference (Fig. 5). The multipliers vertically arrayed on the left indicate that, for each past noisy sample, rotating the sample before it by a corresponding transmission symbol yields a result approximately aligned with the respective past sample.

Fig. 5
Fig. 5

(a): MSDD CR system generating an improved demodulation reference by a linear combination of past samples rotated into approximate alignment with the last sample. (b): Phasor diagram description of the process of rotation-into-alignment of past samples. The improved reference is obtained here as a sum (special case of general linear combination) of the aligned past samples. The summing process yields an averaging effect, improving the signal to noise ratio of the resulting reference, relative to just using the last sample as delay-demodulation reference.

Fig. 6
Fig. 6

QPSK receiver block diagram detailing the “U-not U” MSDD efficient hardware structure, with Wiener-optimal or adaptive coefficients (the adaptive coefficients control is detailed in the next figure). Notice the presence of two Uop modules, however the one within the slicer does not incur extra complexity, as it may be implemented as a look-up table. The delay line with multipliers at the top of the figure incurs negligible complexity, as multiplication with QPSK constellation elements are trivial.

Fig. 7
Fig. 7

Ultra-low complexity MSDD CR for QPSK reception with multiplier-free CPE, requiring just a single full-fledged complex multiplier, the one used for demodulation – all the other multipliers marked green, are trivial.

Fig. 8
Fig. 8

QPSK/QAM receiver with “U-notU” adaptive MSDD, including full detail on the coefficients adaptation mechanism. This scheme, intended here for QPSK, is “QAM-ready”. For QPSK, the Uop-2 module may be discarded (and Uop-3 is always a trivial lookup-table) thus just Uop-1 is required to implement the QPSK U-notU version.

Fig. 9
Fig. 9

(a): Tx implementation for the MSDD polyphase receiver implementation. (b): Polyphase implementation of the MSDD CR. Processing is partitioned into P parallel sub-modules, each acting on a received polyphase. Notice that the clock rates of the DP modules in the Tx and MSDD modules in the Rx are reduced by a factor of P.

Fig. 10
Fig. 10

(a): Tx implementation for the MSDD polyphase receiver implementation. (b): Polyphase implementation of the MSDD CR. Processing is partitioned into P parallel sub-modules, each acting on a received polyphase. Notice that the clock rates of the DP modules in the Tx and MSDD modules in the Rx are reduced by a factor of P.

Fig. 11
Fig. 11

Viterbi&Viterbi M-power QPSK CR, against which the MSDD CR is compared.

Fig. 12
Fig. 12

Viterbi&Viterbi M-power QPSK CR, against which the MSDD CR is compared, for an idealized situation whereby LW = 0, i.e. there is no LPN, just white ASE noise.

Fig. 13
Fig. 13

Adaptive MSDD performance. (left): Optimizing MSDD over the number of adaptive combining taps for various linewidths (LW). (right): Converged adaptive coefficients (red) vs. the optimal Wiener coefficients (blue).

Fig. 14
Fig. 14

MSDD CR vs. Viterbi&Viterbi M-power CR BER vs. OSNR performance. The bottom two curves represent theoretical limits corresponding to a purely white noise idealized channel without and with hard differential detection. The top curve corresponds to the Viterbi&Viterbi M-pwr algorithm (worst performance). The MSDD performance is generally better, but successively degrades as linewidth is increased (0.1, 0.5, 1 MHz). At 0.1 MHz, the curves for Wiener-Optimal, LMS and Uniform coefficients coincide. It is then apparent that for systems based on coherent-grade 100 KHz lasers, MSDD OSNR is 1.9 dB better than that of the MPWR system at BER = 10E-3. Moreover, this performance may be attained with the multiplier-free CPE (Uniform coefficients), and there is no need to use the more complex version with optimized coefficients. At 0.5 MHz linewidth, the OSNR gap between uniform and optimized coefficients is just 0.15 dB indicating that even in this case it is still worth adopting the CPE-multipliers-free hardware simplified MSDD. However, at 1 MHz the gap between uniform and optimized coefficients widens to almost 1 dB indicating that in this case the more complex MSDD system with LMS or Wiener optimized coefficients is required for best performance. In all cases the Wiener and LMS coefficients track each other almost perfectly, mutually validating the mathematical analyses leading to the Wiener solution and the LMS update equations.

Equations (48)

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

| A ˜ k |=| A ˜ k1 |=A; A ˜ k = s ˜ k + A ˜ k1 s ˜ k = A ˜ k A ˜ k1
z ˜ U{ z ˜ } z ˜ /| z ˜ |= e j z ˜ = z ˜ / z ˜ *
A ˜ k = s ˜ k A ˜ k1 | A ˜ k |=| s ˜ k |and A ˜ k = s ˜ k + A ˜ k1 .
r ˜ k = A ˜ k e j ϕ k LPN e jθk + n ˜ k
r ˜ k = A ˜ k e j ϕ k LPN + n ˜ k = A ˜ k p ˜ k ; p ˜ k ( 1+ η ˜ k ) e j ϕ k LPN ; η ˜ k e j ϕ k LPN n ˜ k / A ˜ k
ϕ k LPN = m=0 k Ω m ; Ω m ~N[0,2πΔνT]; Ω m Ω n =2πΔ ν ^ δ mn
p ˜ k ( 1+ η ˜ k ) e j ϕ k LPN =| 1+ η ˜ k | e j ϕ k ASE e j ϕ k LPN e j ϕ k ; ϕ k = ϕ k LPN + ϕ k ASE ; ϕ k ASE =( 1+ η ˜ k )
s ˜ k = r ˜ k r ˜ k1 * =| r ˜ k || r ˜ k1 | e j( A ˜ k A ˜ k1 + ϕ k ϕ k1 ) =| r ˜ k || r ˜ k1 | e j( s ˜ k + ϕ k ϕ k1 )
R ˜ k1 (i) s ˜ k1 s ˜ k2 ... s ˜ ki+1 r ˜ ki i=1,2,...,
R ˜ k1 (i) s ˜ ^ k1 s ˜ ^ k2 ... s ˜ ^ ki+1 r ˜ ki ,i=1,2,...,L
R ˜ k1 = c 1 r ˜ k1 R ˜ k1 (1) + c 2 s ˜ ^ k1 r ˜ k2 R ˜ k1 (2) + c 3 s ˜ ^ k1 s ˜ ^ k2 r ˜ k3 R ˜ k1 (3) + c 4 s ˜ ^ k1 s ˜ ^ k2 s ˜ ^ k3 r ˜ k4 R ˜ k1 (4) +...
s ˜ k = r ˜ k [ i=1 L c i R ˜ k1 (i) ] * = i=1 L c i r ˜ k R ˜ k1 (i)* = i=1 L c i s ˜ k (i)
s ˜ k (i) = r ˜ k R ˜ k1 (i)* = [ s ˜ k1 s ˜ k2 ... s ˜ ki+1 ] * r ˜ k r ˜ ki *
R ˜ k1 (i) =U{ r ˜ ki s ˜ ki+1 s ˜ ki+2 ... s ˜ k1 }= r ˜ ki s ˜ ki+1 s ˜ ki+2 ... s ˜ k1
R ˜ k1 = i=1 L c i R ˜ k1 (i)
s ˜ k (i) r ˜ k R ˜ k1 (i)* = r ˜ k U{ R ˜ k1 (i)* }= r ˜ k U { r ˜ ki s ˜ ki+1 s ˜ ki+2 ... s ˜ k1 } * = r ˜ k r ˜ ki * s ˜ ki+1 * s ˜ ki+2 * ... s ˜ k1 * =( A ˜ k p ˜ k )( A ˜ ki * p ˜ ki * ) s ˜ ki+1 * s ˜ ki+2 * ... s ˜ k1 * = A ˜ k A ˜ ki * s ˜ ki+1 * s ˜ ki+2 * ... s ˜ k1 * p ˜ k p ˜ ki * = s ˜ k p ˜ k p ˜ ki *
s ˜ k = i=1 L c ¯ i r ˜ k R ˜ k1 (i)* = i=1 L c ¯ i s ˜ k (i) = c ¯ T s ˜ k = c s ˜ k s ˜ k [ s ˜ k (1) , s ˜ k (2) ,..., s ˜ k (L) ] T = [ r ˜ k R ˜ k1 (1)* , r ˜ k R ˜ k1 (2)* ,..., r ˜ k R ˜ k1 (L)* ] T = r ˜ k R ˜ k1 * ; R ˜ k1 [ R ˜ k1 (1) , R ˜ k (2) ,..., R ˜ k (L) ] T ;c [ c 1 , c 2 ,..., c L ] T
| ε ˜ k | 2 = | s ˜ k s ˜ k | 2 = | s ˜ k c s ˜ k | 2
Γ s ˜ k c= Γ s ˜ k , s ˜ k j=1 L Φ ij c j = v i ,i=1,2,...,L
v i [ Γ s ˜ k , s ˜ k ] i ; Φ ij [ Γ s ˜ ] ij
j=1 L ( 1+ SNR k Tx 1 ( 1+ δ ij ) ) e πΔ ν ^ | ij | c j = e πΔ ν ^ i ,i=1,2,...,L
SNR k Tx 1 = σ n ˜ 2 / | s ˜ k | 2 = σ n ˜ 2 / 1 m α=1 m | s ˜ k (α) | 2
ε ˜ k s ˜ k =0 ε ˜ k * s ˜ k = ( s ˜ k * c T s ˜ k * ) s ˜ k =0
U[ k ] ε ˜ k * s ˜ k = ε ˜ k * r ˜ k R ˜ k1 * ; U i [ k ] ε ˜ k * s ˜ k (i) = ε ˜ k * r ˜ k R ˜ k1 (i)* = ε ˜ k * r ˜ k r ˜ ki s ˜ ki+1 s ˜ ki+2 ... s ˜ k1
| ε ˜ k | 2 = ε ˜ k ε ˜ ¯ k =( s ˜ k c ¯ T s ˜ k )( s ˜ k * c T s ˜ k * )
c | ε ˜ k | 2 = ε ˜ k ε ˜ ¯ k =2 c ¯ { ( s ˜ k c ¯ T s ˜ k )( s ˜ k * c T s ˜ k * ) }=2( s ˜ k * c T s ˜ k * ) c ¯ ( s ˜ k c ¯ T s ˜ k ) =2( s ˜ k * c T s ˜ k * ) c ¯ ( c ¯ T s ˜ k )=2( s ˜ k * c T s ˜ k * ) s ˜ k =2 ε ˜ k * s ˜ k =2U[ k ]
c[k+1]=c[k] μ 2 c | ε ˜ k | 2 | c=c[k] c[k+1]=c[k]+μU[ k ] c i [k+1]= c i [k]+μ U i [ k ]
c i [k+1]= c i [k]+μ ε ˜ k * r ˜ k R ˜ k1 (i)*
ε ˜ k = s ˜ k s ˜ k = s ˜ k r ˜ k R ˜ k1 *
( )/| |=( )( 1/ | | 2 )=[ Re( ),Im( ) ][ 1/ [ Re( ) ] 2 + [ Im( ) ] 2 ]
A ˜ ki s ˜ ki+1 = A ˜ ki+1 A ˜ ki+1 s ˜ ki+2 = A ˜ ki+2 .... A ˜ k2 s ˜ k1 = A ˜ k1
A ˜ ki s ˜ ki+1 s ˜ ki+2 .. s ˜ k1 = A ˜ k1
A ˜ k A ˜ k1 * =( s ˜ k A ˜ k1 ) A ˜ k1 * = s ˜ k ( A ˜ k1 A ˜ k1 * )= s ˜ k
s ˜ k = A ˜ k A ˜ ki * s ˜ ki+1 * s ˜ ki+2 * ... s ˜ k1 *
A ˜ k A ˜ ki * s ˜ ki+1 * s ˜ ki+2 * ... s ˜ k1 * = A ˜ k ( A ˜ ki s ˜ ki+1 s ˜ ki+2 ... s ˜ k1 ) * = A ˜ k A ˜ k1 * = s ˜ k
0=[ ε ˜ k s ˜ k (1)* , ε ˜ k s ˜ k (2)* ,..., ε ˜ k s ˜ k (2)* ]= ε ˜ k [ s ˜ k (1)* , s ˜ k (2)* ,..., s ˜ k (L)* ] = ε ˜ k s ˜ k
0= ε ˜ k s ˜ k = ( s ˜ k c s ˜ k ) s ˜ k = s ˜ k s ˜ k c s ˜ k s ˜ k c s ˜ k s ˜ k = s ˜ k s ˜ k s ˜ k s ˜ k c= s ˜ k s ˜ k Γ s ˜ k c= Γ s ˜ k , s ˜ k
v i [ Γ s ˜ k , s ˜ k ] i = s ˜ k (i) s ˜ k * = s ˜ k p ˜ k p ˜ ki * s ˜ k * = | s ˜ k | 2 p ˜ k p ˜ ki *
Φ ij [ Γ s ˜ ] ij = s ˜ k (i) s ˜ k (j)* = s ˜ k p ˜ k p ˜ ki * ( s ˜ k p ˜ k p ˜ kj * ) * = | s ˜ k | 2 p ˜ k p ˜ ki * ( p ˜ k p ˜ kj * ) *
p ˜ k 1 p ˜ k 2 * =( 1+ η ˜ k 1 ) e j ϕ k 1 LPN ( 1+ η ˜ k 2 * ) e j ϕ k 2 LPN =( 1+ η ˜ k 1 + η ˜ k 2 * + η ˜ k 1 η ˜ k 2 * ) e j ϕ k 1 k 2
ϕ k 1 k 2 LPN ϕ k 2 LPN ϕ k 1 LPN = k= k 1 +1 k 2 Ω k
| ϕ k 1 k 2 LPN | 2 =| k 2 k 1 | Ω l 2 =| k 2 k 1 |2πΔνT=| k 2 k 1 |2πΔ ν ^
exp [j ϕ k 1 k 2 ] = e πΔ ν ^ | k 1 k 2 |
p ˜ k 1 p ˜ k 2 * =( 1+ η ˜ k 1 0 + η ˜ k 2 * 0 + η ˜ k 1 η ˜ k 2 * σ η ˜ 2 δ k 1 k 2 ) e j( ϕ k 1 LPN ϕ k 2 LPN ) p ˜ k 1 p ˜ k 2 * = Γ p ˜ [ k 1 k 2 ]=( 1+ σ η ˜ 2 δ k 1 k 2 ) e πΔ ν ^ | k 1 k 2 | p ˜ k p ˜ ki * = Γ p ˜ [i]=( 1+ σ η ˜ 2 δ i ) e πΔ ν ^ | i |
p ˜ k 1 p ˜ k 2 * ( p ˜ k 1 p ˜ k 2 * ) * =( 1+ η ˜ k 1 + η ˜ k 2 * + η ˜ k 1 η ˜ k 2 * ) e j ϕ k 1 k 2 ( 1+ η ˜ k 1 * + η ˜ k 2 + η ˜ k 1 * η ˜ k 2 ) e j ϕ k 1 k 2 =(1+ η ˜ k 1 + η ˜ k 2 * + η ˜ k 1 * + η ˜ k 2 + η ˜ k 1 η ˜ k 1 * + η ˜ k 1 η ˜ k 2 + η ˜ k 2 * η ˜ k 1 * + η ˜ k 2 * η ˜ k 2 + η ˜ k 1 η ˜ k 1 * η ˜ k 2 + η ˜ k 2 * η ˜ k 1 * η ˜ k 2 + η ˜ k 1 η ˜ k 2 * η ˜ k 1 * + η ˜ k 1 η ˜ k 2 * η ˜ k 2 + η ˜ k 1 η ˜ k 2 * η ˜ k 1 * η ˜ k 2 ) e j( ϕ k 1 k 2 ϕ k 1 k 2 )
p ˜ k 1 p ˜ k 2 * p ˜ k 1 * p ˜ k 2 = p ˜ k 1 p ˜ k 2 * ( p ˜ k 1 p ˜ k 2 * ) * =( 1+ η ˜ k 1 η ˜ k 1 * + η ˜ k 2 * η ˜ k 2 ) e j( ϕ k 1 k 2 ϕ k 1 k 2 ) =[ 1+ σ η ˜ 2 ( δ k 1 k 1 + δ k 2 k 2 ) ] e j( ϕ k 1 k 2 ϕ k 1 k 2 )
p ˜ k p ˜ ki * ( p ˜ k p ˜ kj * ) * =[ 1+ σ η ˜ 2 ( δ kk + δ ij ) ] e j( ϕ kki ϕ kkj ) =[ 1+ σ η ˜ 2 ( 1+ δ ij ) ] e j( ϕ kik ϕ kjk ) =[ 1+ σ η ˜ 2 ( 1+ δ ij ) ] e πΔ ν ^ | ij |
v i [ Γ s ˜ k , s ˜ k ] i = s ˜ k (i) s ˜ k * = | s ˜ k | 2 ( 1+ σ η ˜ 2 δ i ) e πΔ ν ^ | i | Φ ij [ Γ s ˜ ] ij = s ˜ k (i) s ˜ k (j)* =[ 1+ σ η ˜ 2 ( 1+ δ ij ) ] e πΔ ν ^ | ij |

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