Abstract

We propose a simple non-data-aided (or unsupervised) and universal cycle slip detection and correction (CS-DC) technique based on locating the minimum of the sliding average of twice estimated phase noise. The CS-DC can be appended to any carrier phase estimation(CPE) technique and is modulation format independent. We analytically derive the probability density function of the CS detection metric and study how the sliding window length and detection threshold affects CS detection performance. Simulation results reveal significant cycle slips reduction for various modulation formats with a residual CS probability of 2 × 10−7 for single carrier system even in unrealistic highly nonlinear system setups. In addition, we show that a second stage of CS-DC with a different sliding window length can further reduce the cycle slip probability by at least an order of magnitude. We also show that CS-DC is tolerant to inter-channel nonlinearities and residue frequency offset effects. Overall, the proposed CS-DC technique can be used in conjunction to other CS reduction techniques to maximize the ability of CS mitigation in next generation optical transceivers.

© 2014 Optical Society of America

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References

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  1. M. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009).
    [Crossref]
  2. E. Ibragimov, B. Zhang, T. J. Schmidt, C. Malouin, N. Fediakine, and H. Jiang, “Cycle slip probability in 100G PM-QPSK systems” in Proc. Opt. Fiber Commun. (OFC), San Diego, CA, Mar. 2010, Paper OWE2.
  3. C.R.S Fludger, D. Nuss, and T. Kupfer, “Cycle-slips in 100G DP-QPSK tranmission systems” in Proc. Opt. Fiber. Commun. (OFC), Los Angeles, CA, Mar. 2012, Paper OTu2G. 1.
  4. A. Bisplinghoff, S. Langenbach, T. Kupfer, and B. Schmauss, “Turbo differential decoding failure for a coherent phase slip channel” in Proc.Eur. Conf. Exhib. Opt. Commun. (ECOC), Amsterdam, Netherlands, Sep. 2012, Paper Mo.1.A.5.
    [Crossref]
  5. C. Xie and G. Raybon, “Digital PLL based frequency offset compensation and carrier phase estimation for 16-QAM coherent optical communication systems” in Proc.Eur. Conf. Exhib. Opt. Commun. (ECOC), Amsterdam, Netherlands, Sep. 2012, Paper Mo.1.A.2.
    [Crossref]
  6. S. Zhang, X. Li, P. Y. Kam, C. Yu, and J. Chen, “Pilot-assisted, decision-aided, maximum likelihood phase estimation in coherent optical phase-modulated systems with nonlinear phase noise,” IEEE Photon. Technol. Lett. 22(6), 380–382 (2010).
    [Crossref]
  7. T. Yoshida, T. Sugihara, K. Ishida, and T. Mizuochi, “Cycle slip compensation with polarization block coding for coherent optical transmission: two-dimensional phases constellation corresponds to a slip stage,” IEEE Signal Process. Mag. 31(2), 57–69 (2014).
    [Crossref]
  8. H. Zhang, Y. Cai, D. G. Foursa, and A. N. Pilipetskii, “Cycle slip mitigation in POLMUX-QPSK modulation” in Proc. Opt. Fiber Commun. (OFC), Los Angeles, CA, Mar. 2011, Paper OWE2.
  9. Y. Gao, A. P. T. Lau, and C. Lu, “Cycle-slip resilient carrier phase estimation for polarization multiplexed 16-QAM systems” in Proc. OptoElectron. Commun. Conf. (OECC), Busan, Korea, Jul. 2012, Paper 4B2–4.
    [Crossref]
  10. H. Cheng, Y. Li, M. Yu, J. Zang, J. Wu, and J. Lin, “Experimental demonstration of pilot-symbol-aided cycle slip mitigation for QPSK modulation format” in Proc. Opt. Fiber Commun. (OFC), San Francisco, CA, Mar. 2014, Paper Th4D.1.
  11. A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
    [Crossref]
  12. Y. Gao, A. P. T. Lau, C. Lu, Y. Dai, and X. Xu, “Blind cycle-slip detection and correction for coherent communication systems” in Proc.Eur. Conf. Exhib. Opt. Commun. (ECOC), London, U. K., Sep. 2013, Paper P.3.16.
  13. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
    [Crossref] [PubMed]
  14. A. J. Viterbi and A. N. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
    [Crossref]
  15. X. Zhou, J. Yu, and P. Magill, “Cascaded two-modulus algorithm for blind polarization de-multiplexing of 114-Gb/s PDM-8-QAM optical signals” in Proc. Opt. Fiber Commun. (OFC), San Diego, CA, Mar. 2009, Paper OWG3.
  16. Y. Gao, A. P. T. Lau, S. Yan, and C. Lu, “Low-complexity and phase noise tolerant carrier phase estimation for dual-polarization 16-QAM systems,” Opt. Express 19(22), 21717–21729 (2011).
    [Crossref] [PubMed]
  17. T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feed forward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
    [Crossref]
  18. A. Bisplinghoff, C. Vogel, T. Kupfer, S. Langebach, and B. Schmauss, “Slip-reduced carrier phase estimation for coherent transmisssion in the presence of non-linear phase noise” in Proc. Opt. Fiber Commun. (OFC), Anaheim, CA, Mar. 2013, Paper OTu3I.1.
  19. M. Selmi, Y. Jaouen, and P. Ciblat, “Accurate digital frequency offset estimator for coherent PolMux QAM transmission systems,” in Proc. Eur. Conf. Exhib. Opt. Commun. (ECOC), Vienna, Austria, Sep. 2009, Paper P3.08.

2014 (2)

T. Yoshida, T. Sugihara, K. Ishida, and T. Mizuochi, “Cycle slip compensation with polarization block coding for coherent optical transmission: two-dimensional phases constellation corresponds to a slip stage,” IEEE Signal Process. Mag. 31(2), 57–69 (2014).
[Crossref]

A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
[Crossref]

2011 (1)

2010 (1)

S. Zhang, X. Li, P. Y. Kam, C. Yu, and J. Chen, “Pilot-assisted, decision-aided, maximum likelihood phase estimation in coherent optical phase-modulated systems with nonlinear phase noise,” IEEE Photon. Technol. Lett. 22(6), 380–382 (2010).
[Crossref]

2009 (2)

2008 (1)

1983 (1)

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

Chagnon, M.

A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
[Crossref]

Chen, J.

S. Zhang, X. Li, P. Y. Kam, C. Yu, and J. Chen, “Pilot-assisted, decision-aided, maximum likelihood phase estimation in coherent optical phase-modulated systems with nonlinear phase noise,” IEEE Photon. Technol. Lett. 22(6), 380–382 (2010).
[Crossref]

Gao, Y.

A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
[Crossref]

Y. Gao, A. P. T. Lau, S. Yan, and C. Lu, “Low-complexity and phase noise tolerant carrier phase estimation for dual-polarization 16-QAM systems,” Opt. Express 19(22), 21717–21729 (2011).
[Crossref] [PubMed]

Hoffmann, S.

Ishida, K.

T. Yoshida, T. Sugihara, K. Ishida, and T. Mizuochi, “Cycle slip compensation with polarization block coding for coherent optical transmission: two-dimensional phases constellation corresponds to a slip stage,” IEEE Signal Process. Mag. 31(2), 57–69 (2014).
[Crossref]

Kam, P. Y.

S. Zhang, X. Li, P. Y. Kam, C. Yu, and J. Chen, “Pilot-assisted, decision-aided, maximum likelihood phase estimation in coherent optical phase-modulated systems with nonlinear phase noise,” IEEE Photon. Technol. Lett. 22(6), 380–382 (2010).
[Crossref]

Lau, A. P. T.

A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
[Crossref]

Y. Gao, A. P. T. Lau, S. Yan, and C. Lu, “Low-complexity and phase noise tolerant carrier phase estimation for dual-polarization 16-QAM systems,” Opt. Express 19(22), 21717–21729 (2011).
[Crossref] [PubMed]

Li, X.

S. Zhang, X. Li, P. Y. Kam, C. Yu, and J. Chen, “Pilot-assisted, decision-aided, maximum likelihood phase estimation in coherent optical phase-modulated systems with nonlinear phase noise,” IEEE Photon. Technol. Lett. 22(6), 380–382 (2010).
[Crossref]

Lu, C.

A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
[Crossref]

Y. Gao, A. P. T. Lau, S. Yan, and C. Lu, “Low-complexity and phase noise tolerant carrier phase estimation for dual-polarization 16-QAM systems,” Opt. Express 19(22), 21717–21729 (2011).
[Crossref] [PubMed]

Mizuochi, T.

T. Yoshida, T. Sugihara, K. Ishida, and T. Mizuochi, “Cycle slip compensation with polarization block coding for coherent optical transmission: two-dimensional phases constellation corresponds to a slip stage,” IEEE Signal Process. Mag. 31(2), 57–69 (2014).
[Crossref]

Morsy-Osman, M.

A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
[Crossref]

Noe, R.

Pfau, T.

Plant, D. V.

A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
[Crossref]

Savory, S. J.

Sugihara, T.

T. Yoshida, T. Sugihara, K. Ishida, and T. Mizuochi, “Cycle slip compensation with polarization block coding for coherent optical transmission: two-dimensional phases constellation corresponds to a slip stage,” IEEE Signal Process. Mag. 31(2), 57–69 (2014).
[Crossref]

Sui, Q.

A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
[Crossref]

Taylor, M.

Viterbi, A. J.

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

Viterbi, A. N.

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

Wang, D.

A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
[Crossref]

Xu, X.

A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
[Crossref]

Yan, S.

Yoshida, T.

T. Yoshida, T. Sugihara, K. Ishida, and T. Mizuochi, “Cycle slip compensation with polarization block coding for coherent optical transmission: two-dimensional phases constellation corresponds to a slip stage,” IEEE Signal Process. Mag. 31(2), 57–69 (2014).
[Crossref]

Yu, C.

S. Zhang, X. Li, P. Y. Kam, C. Yu, and J. Chen, “Pilot-assisted, decision-aided, maximum likelihood phase estimation in coherent optical phase-modulated systems with nonlinear phase noise,” IEEE Photon. Technol. Lett. 22(6), 380–382 (2010).
[Crossref]

Zhang, S.

S. Zhang, X. Li, P. Y. Kam, C. Yu, and J. Chen, “Pilot-assisted, decision-aided, maximum likelihood phase estimation in coherent optical phase-modulated systems with nonlinear phase noise,” IEEE Photon. Technol. Lett. 22(6), 380–382 (2010).
[Crossref]

Zhuge, Q.

A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
[Crossref]

IEEE Photon. Technol. Lett. (1)

S. Zhang, X. Li, P. Y. Kam, C. Yu, and J. Chen, “Pilot-assisted, decision-aided, maximum likelihood phase estimation in coherent optical phase-modulated systems with nonlinear phase noise,” IEEE Photon. Technol. Lett. 22(6), 380–382 (2010).
[Crossref]

IEEE Signal Process. Mag. (2)

T. Yoshida, T. Sugihara, K. Ishida, and T. Mizuochi, “Cycle slip compensation with polarization block coding for coherent optical transmission: two-dimensional phases constellation corresponds to a slip stage,” IEEE Signal Process. Mag. 31(2), 57–69 (2014).
[Crossref]

A. P. T. Lau, Y. Gao, Q. Sui, D. Wang, Q. Zhuge, M. Morsy-Osman, M. Chagnon, X. Xu, C. Lu, and D. V. Plant, “Advanced DSP techniques enabling high spectral efficiency and flexible transmissions: toward elastic optical networks,” IEEE Signal Process. Mag. 31(2), 82–92 (2014).
[Crossref]

IEEE Trans. Inf. Theory (1)

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

J. Lightwave Technol. (2)

Opt. Express (2)

Other (11)

Y. Gao, A. P. T. Lau, C. Lu, Y. Dai, and X. Xu, “Blind cycle-slip detection and correction for coherent communication systems” in Proc.Eur. Conf. Exhib. Opt. Commun. (ECOC), London, U. K., Sep. 2013, Paper P.3.16.

X. Zhou, J. Yu, and P. Magill, “Cascaded two-modulus algorithm for blind polarization de-multiplexing of 114-Gb/s PDM-8-QAM optical signals” in Proc. Opt. Fiber Commun. (OFC), San Diego, CA, Mar. 2009, Paper OWG3.

A. Bisplinghoff, C. Vogel, T. Kupfer, S. Langebach, and B. Schmauss, “Slip-reduced carrier phase estimation for coherent transmisssion in the presence of non-linear phase noise” in Proc. Opt. Fiber Commun. (OFC), Anaheim, CA, Mar. 2013, Paper OTu3I.1.

M. Selmi, Y. Jaouen, and P. Ciblat, “Accurate digital frequency offset estimator for coherent PolMux QAM transmission systems,” in Proc. Eur. Conf. Exhib. Opt. Commun. (ECOC), Vienna, Austria, Sep. 2009, Paper P3.08.

H. Zhang, Y. Cai, D. G. Foursa, and A. N. Pilipetskii, “Cycle slip mitigation in POLMUX-QPSK modulation” in Proc. Opt. Fiber Commun. (OFC), Los Angeles, CA, Mar. 2011, Paper OWE2.

Y. Gao, A. P. T. Lau, and C. Lu, “Cycle-slip resilient carrier phase estimation for polarization multiplexed 16-QAM systems” in Proc. OptoElectron. Commun. Conf. (OECC), Busan, Korea, Jul. 2012, Paper 4B2–4.
[Crossref]

H. Cheng, Y. Li, M. Yu, J. Zang, J. Wu, and J. Lin, “Experimental demonstration of pilot-symbol-aided cycle slip mitigation for QPSK modulation format” in Proc. Opt. Fiber Commun. (OFC), San Francisco, CA, Mar. 2014, Paper Th4D.1.

E. Ibragimov, B. Zhang, T. J. Schmidt, C. Malouin, N. Fediakine, and H. Jiang, “Cycle slip probability in 100G PM-QPSK systems” in Proc. Opt. Fiber Commun. (OFC), San Diego, CA, Mar. 2010, Paper OWE2.

C.R.S Fludger, D. Nuss, and T. Kupfer, “Cycle-slips in 100G DP-QPSK tranmission systems” in Proc. Opt. Fiber. Commun. (OFC), Los Angeles, CA, Mar. 2012, Paper OTu2G. 1.

A. Bisplinghoff, S. Langenbach, T. Kupfer, and B. Schmauss, “Turbo differential decoding failure for a coherent phase slip channel” in Proc.Eur. Conf. Exhib. Opt. Commun. (ECOC), Amsterdam, Netherlands, Sep. 2012, Paper Mo.1.A.5.
[Crossref]

C. Xie and G. Raybon, “Digital PLL based frequency offset compensation and carrier phase estimation for 16-QAM coherent optical communication systems” in Proc.Eur. Conf. Exhib. Opt. Commun. (ECOC), Amsterdam, Netherlands, Sep. 2012, Paper Mo.1.A.2.
[Crossref]

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Figures (11)

Fig. 1
Fig. 1 Block diagram of the proposed CS-DC technique. The received signal r i and decided symbol d i are used to form y i and the magnitude of its sliding average z i . When CS occurs at ics, z i undergo an abrupt drop and this very feature can be used to identify the presence of CS. To correct the CS, one can then evaluate the evolution of estimated phase φ ^ i around ics to determine if φ ^ i should be rotated by π/2 or π/2 .
Fig. 2
Fig. 2 (a) Estimated phase evolution indicating the presence of cycle-slips and (b) evolution of the corresponding parameter zi for cycle-slip detection and correction (CS-DC).
Fig. 3
Fig. 3 Probability density function of (a) x61 without CS and (b) x61 with CS obtained from theory and Monte Carlo simulations. The linewidth duration product is Δv T s =6× 10 4 .
Fig. 4
Fig. 4 Probability density function of z 61|CS and z 61|noCS . The linewidth duration product is Δv T s =6× 10 4 .
Fig. 5
Fig. 5 PPost versus Z th with different PPre for (a) K + 1 = 41, (b) K + 1 = 61 and (c) K + 1 = 81.
Fig. 6
Fig. 6 CSP with and without the proposed CS-DC technique for a single carrier 112Gb/s PM-QPSK system with various OSNR and CPE lengths over a (a) 2400 km and (b) 7200 km link. The signal launched power is 4 dBm and the laser linewidths are 100kHz. Without CS-DC, the amount of CS for each data point ranges from 10s to more than 1200. With two-stage CS-DC, the CS probability is driven down to 0 most of the time and at most 10−6 under highly unrealistic system conditions.
Fig. 7
Fig. 7 CSP with and without the proposed CS-DC technique for a single carrier 224 Gb/s PM-16-QAM system with various OSNR and CPE lengths over a (a) 1200 km and (b) 2400 km link. The launched power is 4 dBm and the laser linewidths are 100kHz. Without CS-DC, the amount of CS for each data point ranges from 10s to more than 1700. With the proposed two-stage CS-DC, the CS probability is driven down to 0 most of the time and at most 3 × 10−7 under highly unrealistic conditions.
Fig. 8
Fig. 8 CSP without CS-DC, with 1-stage CS-DC and with 2-stage CS-DC techniques for (a) 5 × 112Gbit/s PM-QPSK Nyquist-WDM system over 2400km SMF link and (b) 5 × 224Gbit/s PM-16QAM Nyquist-WDM system over 1200km SMF link with various OSNR and CPE lengths. The signal launched power is 4dBm per channel and the laser linewidths are 100kHz. Without CS-DC, the amount of CS for each data point ranges from 10s to more than 2400. With two-stage CS-DC, the CSP is driven down to 0 most of the time and at most 8 × 10−6 and 7 × 10−6 respectively for QPSK and 16QAM signals under highly unrealistic system conditions.
Fig. 9
Fig. 9 Block diagram of two-stage CS-DC with different window lengths K1 + 1 and K2 + 1. The structure can help detect and correct multiple cycle-slips that occurred close to each other such that a single CS-DC may fail to identity all the cycle slips correctly.
Fig. 10
Fig. 10 CSP and required OSNR at BER of 0.04 for VVPE, SR-CPE and SR-CPE + CS-DC. N1 and N2 are half-filter lengths of short and long filers respectively in SR-CPE.
Fig. 11
Fig. 11 CSP without CS-DC, with 1-stage CS-DC and with 2-stage CS-DC for (a) 112Gbit/s QPSK transmission system over 7200km SMF link and (b) 224Gbit/s 16QAM transmission systems over 2400km SMF link with various residue FO MSE. The CPE half-filter lengths are set to be 10 and 15 for QPSK and 16QAM respectively. The launch power is 4dBm. The OSNRs are set to be 16 dB and 18dB respectively for QPSK and 16QAM. K denotes the average length of CS-DC. The best average lengths are 100 and 200 for 1-stage and 2-stage CS-DC respectively.

Equations (13)

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r i = s i e j φ i + n i
y i = [ r i d i * / | r i d i * | ] M { e jM φ i no CS at i e jM( φ i ±π/M) = e jM φ i CS occured at i .
z i = | k=iK/2 i+K/2 y k | / ( K+1 ) = | k=iK/2 i+K/2 e j2 φ ^ k | / ( K+1 ) ,
x K+1 = k=iK/2 i+K/2 y k = k=iK/2 i+K/2 e j2 φ k
x K+1 = e j2 φ iK/2 ( ...( 1+ e j ϕ 2 ( 1+ e j ϕ 1 ) ) )
x k+1 = e j ϕ k+1 ( 1+ x k )
f x k+1 (r,θ)= f x k ( 1+ r 2 2rcosθ , tan 1 ( rsinθ rcosθ1 ) ) f Φ ( θ )
f z K+1|no CS (r)=(K+1) 0 2π f x K+1 ( (K+1)r,θ ) dθ.
z K+1|CS =| k=iK/2 i1 e j2 φ k +1+ k=i+1 i+K/2 e j2( φ k ±π/2) |/(K+1) =| k=iK/2 i1 e j2 φ k +1 k=i+1 i+K/2 e j2 φ k |/(K+1).
P post = P pre P miss +( 1 P pre ) P FA
P FA = 0 Z th f z K+1|no CS (ς)dς
P miss = Z th f z K+1|CS (ς)dς
MSE=E[ | Δf T s | 2 ]

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