Abstract

Modulation format independence is one of the key challenges in digital signal processing (DSP) techniques for future elastic optical transmissions. We proposed a modulation-format-independent blind phase search (MFI-BPS) algorithm for square M-ary quadrature amplitude modulation (M-QAM) systems, in which modulation format recognition (MFR) and carrier phase estimation (CPE), are included and implemented both in a feed-forward manner. Comprehensive simulation and the experimental studies on 224 Gbit/s polarization multiplexing 16-QAM (PM-16QAM) systems demonstrate the feasibility and the effectiveness of the proposed MFI-BPS algorithm.

© 2014 Optical Society of America

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References

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2014 (2)

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]

X. Zhou, C. Lu, A. P. T. Lau, and K. P. Long, “Low-complexity carrier phase recovery for square M-QAM based on S-BPS algorithm,” IEEE Photon. Technol. Lett. 26(18), 1863–1866 (2014).
[Crossref]

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. L. Gao, A. P. T. Lau, and C. Lu, “Modulation-format-independent carrier phase estimation for square M-QAM systems,” IEEE Photon. Technol. Lett. 25(11), 1073–1076 (2013).
[Crossref]

2012 (2)

2011 (1)

2010 (3)

X. Zhou, “An improved feed-forward carrier recovery algorithm for coherent receivers with M-QAM modulation format,” IEEE Photon. Technol. Lett. 22(14), 1051–1053 (2010).
[Crossref]

I. Fatadin, D. Ives, and S. J. Savory, “Laser linewidth tolerance for 16-QAM coherent optical systems using QPSK partitioning,” IEEE Photon. Technol. Lett. 22(9), 631–633 (2010).
[Crossref]

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag. 48(8), 138–145 (2010).
[Crossref]

2009 (1)

2008 (1)

I. Fatadin, S. J. Savory, and D. Ives, “Compensation of quadrature imbalance in an optical QPSK coherent receiver,” IEEE Photon. Technol. Lett. 20(20), 1733–1735 (2008).
[Crossref]

1985 (1)

M. Tur, B. Moslehi, and J. W. Goodman, “Theory of laser phase noise in recirculating fiber-optic delay lines,” J. Lightwave Technol. 3(1), 20–31 (1985).
[Crossref]

1983 (1)

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

1980 (1)

D. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. 28(11), 1867–1875 (1980).
[Crossref]

1978 (1)

W. Weber, “Differential encoding for multiple amplitude and phase shift keying systems,” IEEE Trans. Commun. 26(3), 385–391 (1978).
[Crossref]

Cartledge, J. C.

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]

Q. Zhuge, M. Morsy-Osman, X. Xu, M. E. Mousa-Pasandi, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Pilot-aided carrier phase recovery for M-QAM using superscalar parallelization based PLL,” Opt. Express 20(17), 19599–19609 (2012).
[Crossref] [PubMed]

El-Sahn, Z. A.

Fatadin, I.

I. Fatadin, D. Ives, and S. J. Savory, “Laser linewidth tolerance for 16-QAM coherent optical systems using QPSK partitioning,” IEEE Photon. Technol. Lett. 22(9), 631–633 (2010).
[Crossref]

I. Fatadin, S. J. Savory, and D. Ives, “Compensation of quadrature imbalance in an optical QPSK coherent receiver,” IEEE Photon. Technol. Lett. 20(20), 1733–1735 (2008).
[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]

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]

Gao, Y. L.

Y. L. Gao, A. P. T. Lau, and C. Lu, “Modulation-format-independent carrier phase estimation for square M-QAM systems,” IEEE Photon. Technol. Lett. 25(11), 1073–1076 (2013).
[Crossref]

Y. L. Gao, A. P. T. Lau, S. Y. 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]

Gerstel, O.

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50(2), 12–20 (2012).
[Crossref]

Godard, D.

D. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. 28(11), 1867–1875 (1980).
[Crossref]

Goodman, J. W.

M. Tur, B. Moslehi, and J. W. Goodman, “Theory of laser phase noise in recirculating fiber-optic delay lines,” J. Lightwave Technol. 3(1), 20–31 (1985).
[Crossref]

Hirano, A.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag. 48(8), 138–145 (2010).
[Crossref]

Hoffmann, S.

Ives, D.

I. Fatadin, D. Ives, and S. J. Savory, “Laser linewidth tolerance for 16-QAM coherent optical systems using QPSK partitioning,” IEEE Photon. Technol. Lett. 22(9), 631–633 (2010).
[Crossref]

I. Fatadin, S. J. Savory, and D. Ives, “Compensation of quadrature imbalance in an optical QPSK coherent receiver,” IEEE Photon. Technol. Lett. 20(20), 1733–1735 (2008).
[Crossref]

Jinno, M.

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50(2), 12–20 (2012).
[Crossref]

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag. 48(8), 138–145 (2010).
[Crossref]

Ke, J. H.

Kozicki, B.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag. 48(8), 138–145 (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]

X. Zhou, C. Lu, A. P. T. Lau, and K. P. Long, “Low-complexity carrier phase recovery for square M-QAM based on S-BPS algorithm,” IEEE Photon. Technol. Lett. 26(18), 1863–1866 (2014).
[Crossref]

Y. L. Gao, A. P. T. Lau, and C. Lu, “Modulation-format-independent carrier phase estimation for square M-QAM systems,” IEEE Photon. Technol. Lett. 25(11), 1073–1076 (2013).
[Crossref]

Y. L. Gao, A. P. T. Lau, S. Y. 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]

Long, K. P.

X. Zhou, C. Lu, A. P. T. Lau, and K. P. Long, “Low-complexity carrier phase recovery for square M-QAM based on S-BPS algorithm,” IEEE Photon. Technol. Lett. 26(18), 1863–1866 (2014).
[Crossref]

Lord, A.

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50(2), 12–20 (2012).
[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]

X. Zhou, C. Lu, A. P. T. Lau, and K. P. Long, “Low-complexity carrier phase recovery for square M-QAM based on S-BPS algorithm,” IEEE Photon. Technol. Lett. 26(18), 1863–1866 (2014).
[Crossref]

Y. L. Gao, A. P. T. Lau, and C. Lu, “Modulation-format-independent carrier phase estimation for square M-QAM systems,” IEEE Photon. Technol. Lett. 25(11), 1073–1076 (2013).
[Crossref]

Y. L. Gao, A. P. T. Lau, S. Y. 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]

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]

Q. Zhuge, M. Morsy-Osman, X. Xu, M. E. Mousa-Pasandi, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Pilot-aided carrier phase recovery for M-QAM using superscalar parallelization based PLL,” Opt. Express 20(17), 19599–19609 (2012).
[Crossref] [PubMed]

Moslehi, B.

M. Tur, B. Moslehi, and J. W. Goodman, “Theory of laser phase noise in recirculating fiber-optic delay lines,” J. Lightwave Technol. 3(1), 20–31 (1985).
[Crossref]

Mousa-Pasandi, M. E.

Noé, 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]

Q. Zhuge, M. Morsy-Osman, X. Xu, M. E. Mousa-Pasandi, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Pilot-aided carrier phase recovery for M-QAM using superscalar parallelization based PLL,” Opt. Express 20(17), 19599–19609 (2012).
[Crossref] [PubMed]

Savory, S. J.

I. Fatadin, D. Ives, and S. J. Savory, “Laser linewidth tolerance for 16-QAM coherent optical systems using QPSK partitioning,” IEEE Photon. Technol. Lett. 22(9), 631–633 (2010).
[Crossref]

I. Fatadin, S. J. Savory, and D. Ives, “Compensation of quadrature imbalance in an optical QPSK coherent receiver,” IEEE Photon. Technol. Lett. 20(20), 1733–1735 (2008).
[Crossref]

Sone, Y.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag. 48(8), 138–145 (2010).
[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]

Takara, H.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag. 48(8), 138–145 (2010).
[Crossref]

Tanaka, T.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag. 48(8), 138–145 (2010).
[Crossref]

Tur, M.

M. Tur, B. Moslehi, and J. W. Goodman, “Theory of laser phase noise in recirculating fiber-optic delay lines,” J. Lightwave Technol. 3(1), 20–31 (1985).
[Crossref]

Viterbi, A. J.

A. J. Viterbi and A. M. 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. M.

A. J. Viterbi and A. M. 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]

Watanabe, A.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag. 48(8), 138–145 (2010).
[Crossref]

Weber, W.

W. Weber, “Differential encoding for multiple amplitude and phase shift keying systems,” IEEE Trans. Commun. 26(3), 385–391 (1978).
[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]

Q. Zhuge, M. Morsy-Osman, X. Xu, M. E. Mousa-Pasandi, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Pilot-aided carrier phase recovery for M-QAM using superscalar parallelization based PLL,” Opt. Express 20(17), 19599–19609 (2012).
[Crossref] [PubMed]

Yan, S. Y.

Yoo, S. J. B.

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50(2), 12–20 (2012).
[Crossref]

Zhong, K. P.

Zhou, X.

X. Zhou, C. Lu, A. P. T. Lau, and K. P. Long, “Low-complexity carrier phase recovery for square M-QAM based on S-BPS algorithm,” IEEE Photon. Technol. Lett. 26(18), 1863–1866 (2014).
[Crossref]

X. Zhou, “An improved feed-forward carrier recovery algorithm for coherent receivers with M-QAM modulation format,” IEEE Photon. Technol. Lett. 22(14), 1051–1053 (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]

Q. Zhuge, M. Morsy-Osman, X. Xu, M. E. Mousa-Pasandi, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Pilot-aided carrier phase recovery for M-QAM using superscalar parallelization based PLL,” Opt. Express 20(17), 19599–19609 (2012).
[Crossref] [PubMed]

IEEE Commun. Mag. (2)

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag. 48(8), 138–145 (2010).
[Crossref]

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50(2), 12–20 (2012).
[Crossref]

IEEE Photon. Technol. Lett. (5)

I. Fatadin, D. Ives, and S. J. Savory, “Laser linewidth tolerance for 16-QAM coherent optical systems using QPSK partitioning,” IEEE Photon. Technol. Lett. 22(9), 631–633 (2010).
[Crossref]

Y. L. Gao, A. P. T. Lau, and C. Lu, “Modulation-format-independent carrier phase estimation for square M-QAM systems,” IEEE Photon. Technol. Lett. 25(11), 1073–1076 (2013).
[Crossref]

X. Zhou, C. Lu, A. P. T. Lau, and K. P. Long, “Low-complexity carrier phase recovery for square M-QAM based on S-BPS algorithm,” IEEE Photon. Technol. Lett. 26(18), 1863–1866 (2014).
[Crossref]

I. Fatadin, S. J. Savory, and D. Ives, “Compensation of quadrature imbalance in an optical QPSK coherent receiver,” IEEE Photon. Technol. Lett. 20(20), 1733–1735 (2008).
[Crossref]

X. Zhou, “An improved feed-forward carrier recovery algorithm for coherent receivers with M-QAM modulation format,” IEEE Photon. Technol. Lett. 22(14), 1051–1053 (2010).
[Crossref]

IEEE Signal Process. Mag. (1)

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. Commun. (2)

W. Weber, “Differential encoding for multiple amplitude and phase shift keying systems,” IEEE Trans. Commun. 26(3), 385–391 (1978).
[Crossref]

D. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. 28(11), 1867–1875 (1980).
[Crossref]

IEEE Trans. Inf. Theory (1)

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Figures (13)

Fig. 1
Fig. 1 Block diagram of the S-BPS algorithm.
Fig. 2
Fig. 2 Normalized S(k,b) as a function of phase compensation error (NCA = n, n = log2(M)/2).
Fig. 3
Fig. 3 Signal distribution changes with constellation aggregation for 4-QAM, 16-QAM and 64-QAM modulation formats.
Fig. 4
Fig. 4 The block diagram of full-order detection (FOD).
Fig. 5
Fig. 5 Normalized SFOD as a function of carrier phase error at NCA = 1, 2 and 3 respectively for (a) 4-QAM, (b) 16-QAM and (c) 64-QAM signals.
Fig. 6
Fig. 6 Diff of each step based on different input signals: 4-QAM, 16-QAM and 64-QAM respectively.
Fig. 7
Fig. 7 Correct probability of MFR as a function of the summing window length 2L1 with different linewidth and Eb/N0 (a) for 4-QAM and16-QAM, (b) for 64-QAM and 256-QAM.
Fig. 8
Fig. 8 Correct probability of MFR as a function of the summing window length 2L1 with the sum of different number of Diff, (a) for 16-QAM, (b) for 64-QAM.
Fig. 9
Fig. 9 Correct probability of MFR as a function of the number of test phase angles B1.
Fig. 10
Fig. 10 BER as a function of Eb/N0 for 4-QAM, 16-QAM, 64-QAM and 256-QAM by using the proposed MFI-BPS algorithm.
Fig. 11
Fig. 11 (a) Experimental setup for 224Gbit/s PM-16-QAM system, (b) offline DSP. EDFA: erbium doped fiber amplifier; OBPF: optical bandpass filter; PC: polarization controller; PBS: polarization beam splitter; PBC: polarization beam combiner.
Fig. 12
Fig. 12 Correct probability of MFR as a function of 2L1 and m at OSNR = 22.3 dB.
Fig. 13
Fig. 13 BER performance as a function of OSNR with MFI-BPS and standard BPS algorithm.

Equations (10)

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z(k,b)=r(k)exp(j φ b ), [ φ b = b B π 2 π 4 , b=1,2,...,B ]
{ I(k,b,i)=abs( I(k,b,i1) ) D 2 i1 Q(k,b,i)=abs( Q(k,b,i1) ) D 2 i1 ,i=1,..., N CA
S(k,b)= p=k-L+1 k+L ( abs( I(p,b, N CA ) )+abs( Q(p,b, N CA ) ) )
θ ˜ (k)=unwrap( φ b ^ )
b ^ = argmin b=1,2,,B ( S(k,b) )
S FOD (k,b, N CA )= p=k- L 1 +1 k+ L 1 ( abs( I(p,b, N CA ) )+abs( Q(p,b, N CA ) ) )
Diff( N CA )= max( S FOD (k,b, N CA ) ) b=1,2,..., B 1 / min( S FOD (k,b, N CA ) ) b=1,2,..., B 1
N ^ CA = argmax N CA =1,, log 2 ( M max )/2 Diff( N CA )
N ^ CA = argmax N CA =1,, log 2 ( M max )/2 ( m Diff( N CA ) )
BER= 2 log 2 M ( 1 1 M )erfc[ 3 log 2 M 2( M1 ) E b N 0 ]

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