Pilot-aided carrier phase recovery for M-QAM using superscalar parallelization based PLL

Qunbi Zhuge; Mohamed Morsy-Osman; Xian Xu; Mohammad E. Mousa-Pasandi; Mathieu Chagnon; Ziad A. El-Sahn; David V. Plant

doi:10.1364/OE.20.019599

Abstract

In this paper, we present a carrier phase recovery (CPR) algorithm using a modified superscalar parallelization based phase locked loop (M-SSP-PLL) combined with a maximum-likelihood (ML) phase estimation. Compared to the original SSP-PLL, M-SSP-PLL + ML reduces the required buffer size using a novel superscalar structure. In addition, by removing the differential coding/decoding and employing ML phase recovery it also improves the performance. In simulation, we show that the laser linewidth tolerance of M-SSP-PLL + ML is comparable to blind phase search (BPS) algorithm, which is known to be one of the best CPR algorithms in terms of performance for arbitrary QAM formats. In 28 Gbaud QPSK (112 Gb/s) and 16-QAM (224 Gb/s), and 7 Gbaud 64-QAM (84 Gb/s) experiments, it is also demonstrated that M-SSP-PLL + ML can increase the transmission distance by at least 12% compared to BPS for each of them. Finally, the computational complexity is discussed and a significant reduction is shown for our algorithm with respect to BPS.

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References

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Publication

P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010).
[Crossref]
A. H. Gnauck, P. J. Winzer, A. Konczykowska, F. Jorge, J. Dupuy, M. Riet, G. Charlet, B. Zhu, and D. W. Peckham, “Generation and transmission of 21.4-Gbaud PDM 64-QAM using a novel high-power DAC driving a single I/Q modulator,” J. Lightwave Technol. 30(4), 532–536 (2012).
[Crossref]
M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009).
[Crossref]
T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
[Crossref]
E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” J. Lightwave Technol. 25(9), 2675–2692 (2007).
[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]
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]
Q. Zhuge, C. Chen, and D. V. Plant, “Low computation complexity two-stage feedforward carrier recovery algorithm for M-QAM,” in Proc. OFC'11, Paper. OMJ5.
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]
X. Zhou and Y. Sun, “Low-complexity, blind phase recovery for coherent receivers using QAM modulation,” in Proc. OFC'11, Paper. OMJ3.
K. Piyawanno, M. Kuschnerov, B. Spinnler, and B. Lankl, “Low complexity carrier recovery for coherent QAM using superscalar parallelization,” in Proc. ECOC'10, Paper. We.7.A.3.
Q. Zhuge, M. E. Pasandi, X. Xu, B. Châtelain, Z. Pan, M. Osman, and D. V. Plant, “Linewidth-tolerant low complexity pilot-aided carrier phase recovery for M-QAM using superscalar parallelization,” in Proc. OFC'12, Paper. OTu2G.2.
I. Fatadin, D. Ives, and S. J. Savory, “Blind equalization and carrier phase recovery in a 16-QAM optical coherent system,” J. Lightwave Technol. 27(15), 3042–3049 (2009).
[Crossref]
T. Nakagawa, M. Matsui, T. Kobayashi, K. Ishihara, R. Kudo, M. Mizoguchi, and Y. Miyamoto, “Non-data-aided wide-range frequency offset estimator for QAM optical coherent receivers,” in Proc. OFC'11, Paper. OMJ1.
X. Chen, A. Al Amin, and W. Shieh, “Characterization and monitoring of laser linewidths in coherent systems,” J. Lightwave Technol. 29(17), 2533–2537 (2011).
[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]

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]

2009 (3)

M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009).
[Crossref]

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
[Crossref]

I. Fatadin, D. Ives, and S. J. Savory, “Blind equalization and carrier phase recovery in a 16-QAM optical coherent system,” J. Lightwave Technol. 27(15), 3042–3049 (2009).
[Crossref]

2007 (1)

E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” J. Lightwave Technol. 25(9), 2675–2692 (2007).
[Crossref]

Al Amin, A.

X. Chen, A. Al Amin, and W. Shieh, “Characterization and monitoring of laser linewidths in coherent systems,” J. Lightwave Technol. 29(17), 2533–2537 (2011).
[Crossref]

Buhl, L. L.

P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010).
[Crossref]

Charlet, G.

A. H. Gnauck, P. J. Winzer, A. Konczykowska, F. Jorge, J. Dupuy, M. Riet, G. Charlet, B. Zhu, and D. W. Peckham, “Generation and transmission of 21.4-Gbaud PDM 64-QAM using a novel high-power DAC driving a single I/Q modulator,” J. Lightwave Technol. 30(4), 532–536 (2012).
[Crossref]

Chen, X.

X. Chen, A. Al Amin, and W. Shieh, “Characterization and monitoring of laser linewidths in coherent systems,” J. Lightwave Technol. 29(17), 2533–2537 (2011).
[Crossref]

Doerr, C. R.

P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010).
[Crossref]

Dupuy, J.

A. H. Gnauck, P. J. Winzer, A. Konczykowska, F. Jorge, J. Dupuy, M. Riet, G. Charlet, B. Zhu, and D. W. Peckham, “Generation and transmission of 21.4-Gbaud PDM 64-QAM using a novel high-power DAC driving a single I/Q modulator,” J. Lightwave Technol. 30(4), 532–536 (2012).
[Crossref]

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, D. Ives, and S. J. Savory, “Blind equalization and carrier phase recovery in a 16-QAM optical coherent system,” J. Lightwave Technol. 27(15), 3042–3049 (2009).
[Crossref]

Gao, Y.

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]

Gnauck, A. H.

A. H. Gnauck, P. J. Winzer, A. Konczykowska, F. Jorge, J. Dupuy, M. Riet, G. Charlet, B. Zhu, and D. W. Peckham, “Generation and transmission of 21.4-Gbaud PDM 64-QAM using a novel high-power DAC driving a single I/Q modulator,” J. Lightwave Technol. 30(4), 532–536 (2012).
[Crossref]

P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010).
[Crossref]

Hoffmann, S.

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
[Crossref]

Ip, E.

E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” J. Lightwave Technol. 25(9), 2675–2692 (2007).
[Crossref]

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, D. Ives, and S. J. Savory, “Blind equalization and carrier phase recovery in a 16-QAM optical coherent system,” J. Lightwave Technol. 27(15), 3042–3049 (2009).
[Crossref]

Jorge, F.

A. H. Gnauck, P. J. Winzer, A. Konczykowska, F. Jorge, J. Dupuy, M. Riet, G. Charlet, B. Zhu, and D. W. Peckham, “Generation and transmission of 21.4-Gbaud PDM 64-QAM using a novel high-power DAC driving a single I/Q modulator,” J. Lightwave Technol. 30(4), 532–536 (2012).
[Crossref]

Kahn, J. M.

E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” J. Lightwave Technol. 25(9), 2675–2692 (2007).
[Crossref]

Konczykowska, A.

A. H. Gnauck, P. J. Winzer, A. Konczykowska, F. Jorge, J. Dupuy, M. Riet, G. Charlet, B. Zhu, and D. W. Peckham, “Generation and transmission of 21.4-Gbaud PDM 64-QAM using a novel high-power DAC driving a single I/Q modulator,” J. Lightwave Technol. 30(4), 532–536 (2012).
[Crossref]

Lau, A. P. T.

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]

Lu, C.

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]

Magarini, M.

P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010).
[Crossref]

Noe, R.

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
[Crossref]

Peckham, D. W.

A. H. Gnauck, P. J. Winzer, A. Konczykowska, F. Jorge, J. Dupuy, M. Riet, G. Charlet, B. Zhu, and D. W. Peckham, “Generation and transmission of 21.4-Gbaud PDM 64-QAM using a novel high-power DAC driving a single I/Q modulator,” J. Lightwave Technol. 30(4), 532–536 (2012).
[Crossref]

Pfau, T.

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
[Crossref]

Riet, M.

A. H. Gnauck, P. J. Winzer, A. Konczykowska, F. Jorge, J. Dupuy, M. Riet, G. Charlet, B. Zhu, and D. W. Peckham, “Generation and transmission of 21.4-Gbaud PDM 64-QAM using a novel high-power DAC driving a single I/Q modulator,” J. Lightwave Technol. 30(4), 532–536 (2012).
[Crossref]

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, D. Ives, and S. J. Savory, “Blind equalization and carrier phase recovery in a 16-QAM optical coherent system,” J. Lightwave Technol. 27(15), 3042–3049 (2009).
[Crossref]

Shieh, W.

X. Chen, A. Al Amin, and W. Shieh, “Characterization and monitoring of laser linewidths in coherent systems,” J. Lightwave Technol. 29(17), 2533–2537 (2011).
[Crossref]

Taylor, M. G.

M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009).
[Crossref]

Winzer, P. J.

A. H. Gnauck, P. J. Winzer, A. Konczykowska, F. Jorge, J. Dupuy, M. Riet, G. Charlet, B. Zhu, and D. W. Peckham, “Generation and transmission of 21.4-Gbaud PDM 64-QAM using a novel high-power DAC driving a single I/Q modulator,” J. Lightwave Technol. 30(4), 532–536 (2012).
[Crossref]

P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010).
[Crossref]

Yan, S.

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]

Zhou, X.

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]

Zhu, B.

A. H. Gnauck, P. J. Winzer, A. Konczykowska, F. Jorge, J. Dupuy, M. Riet, G. Charlet, B. Zhu, and D. W. Peckham, “Generation and transmission of 21.4-Gbaud PDM 64-QAM using a novel high-power DAC driving a single I/Q modulator,” J. Lightwave Technol. 30(4), 532–536 (2012).
[Crossref]

IEEE Photon. Technol. Lett. (2)

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]

J. Lightwave Technol. (7)

X. Chen, A. Al Amin, and W. Shieh, “Characterization and monitoring of laser linewidths in coherent systems,” J. Lightwave Technol. 29(17), 2533–2537 (2011).
[Crossref]

A. H. Gnauck, P. J. Winzer, A. Konczykowska, F. Jorge, J. Dupuy, M. Riet, G. Charlet, B. Zhu, and D. W. Peckham, “Generation and transmission of 21.4-Gbaud PDM 64-QAM using a novel high-power DAC driving a single I/Q modulator,” J. Lightwave Technol. 30(4), 532–536 (2012).
[Crossref]

E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” J. Lightwave Technol. 25(9), 2675–2692 (2007).
[Crossref]

M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009).
[Crossref]

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
[Crossref]

I. Fatadin, D. Ives, and S. J. Savory, “Blind equalization and carrier phase recovery in a 16-QAM optical coherent system,” J. Lightwave Technol. 27(15), 3042–3049 (2009).
[Crossref]

P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010).
[Crossref]

Opt. Express (1)

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]

Other (5)

X. Zhou and Y. Sun, “Low-complexity, blind phase recovery for coherent receivers using QAM modulation,” in Proc. OFC'11, Paper. OMJ3.

K. Piyawanno, M. Kuschnerov, B. Spinnler, and B. Lankl, “Low complexity carrier recovery for coherent QAM using superscalar parallelization,” in Proc. ECOC'10, Paper. We.7.A.3.

Q. Zhuge, M. E. Pasandi, X. Xu, B. Châtelain, Z. Pan, M. Osman, and D. V. Plant, “Linewidth-tolerant low complexity pilot-aided carrier phase recovery for M-QAM using superscalar parallelization,” in Proc. OFC'12, Paper. OTu2G.2.

T. Nakagawa, M. Matsui, T. Kobayashi, K. Ishihara, R. Kudo, M. Mizoguchi, and Y. Miyamoto, “Non-data-aided wide-range frequency offset estimator for QAM optical coherent receivers,” in Proc. OFC'11, Paper. OMJ1.

Q. Zhuge, C. Chen, and D. V. Plant, “Low computation complexity two-stage feedforward carrier recovery algorithm for M-QAM,” in Proc. OFC'11, Paper. OMJ5.

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Fig. 1 (a) The block diagram of a first-order PLL. (b) The interleaving implementation of PLL in parallelized processing.

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Fig. 2 (a) The original superscalar buffer structure in [11]. (b) The proposed superscalar buffer structure. CH: channel.

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Fig. 3 The combination of the modified SSP-PLL algorithm with a ML carrier recovery.

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Fig. 4 (a) The simulation setup. Simulated linewidth tolerance of various algorithms for (b) QPSK, (c) 16-QAM and (d) 64-QAM.

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Fig. 5 Experimental setup. ODL: optical delay line, PBS/PBC: polarization beam splitter/combiner, PC: polarization controller, SW: switch.

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Fig. 6 (a) BER vs. block length with 2 pilot symbols for QPSK and 4 pilot symbols for both 16-QAM and 64-QAM. (b) BER vs. number of pilot symbols with S = 100 for QPSK and S = 200 for both 16-QAM and 64-QAM.

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Fig. 7 Constellations of (a) QPSK (4800 km), (b) 16-QAM (640 km) and (c) 64-QAM  (320 km).

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Fig. 8 (a) BER vs. OSNR at back-to-back transmission and (b) BER vs. distance for QPSK with one 2.6 MHz and one 0.6 MHz linewidth DFB laser at the two ends.

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Fig. 9 (a) BER vs. OSNR at back-to-back transmission and (b) BER vs. distance for 16-QAM with two ECLs.

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Fig. 10 (a) BER vs. OSNR at back-to-back transmission and (b) BER vs. distance for 16-QAM with one ECL and one 1.3 MHz linewidth DFB laser.

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Fig. 11 (a) BER vs. OSNR at back-to-back transmission and (b) BER vs. distance for 16-QAM with one ECL and one 2.6 MHz linewidth DFB laser.

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Fig. 12 (a) BER vs. OSNR at back-to-back transmission and (b) BER vs. distance for 64-QAM with two ECLs.

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Tables (2)

Table 1 Summary of Laser Linewidth Tolerance.

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Table 2 Complexity comparison between CPR algorithms

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Equations (4)

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

O v e r h e a d = \frac{N_{P S}}{S}

H_{k} = \sum_{n = k - N}^{k + N} r_{n} {[{\hat{r^{'}}}_{n}]}^{*}

ϕ_{k}^{M L} = \tan^{- 1} (Im [H_{k}] / Re [H_{k}])

O v e r h e a d = \frac{N_{P S} / 2}{S}

	BPS	ILP-PLL + ML	O-SSP-PLL	M-SSP-PLL + ML
QPSK	3 × 10⁻⁴	6.9 × 10⁻⁵	3.5 × 10⁻⁵	5.4 × 10⁻⁴
16-QAM	9.3 × 10⁻⁵	1.4 × 10⁻⁵	1.5 × 10⁻⁵	1.3 × 10⁻⁴
64-QAM	3.6 × 10⁻⁵	3.5 × 10⁻⁶	5.8 × 10⁻⁶	2.7 × 10⁻⁵

	Real Multiplier	Real Adder	Slicer	Buffer Units
M-SSP-PLL + ML	15	2L + 6 (48)	2	2S₁ + L (221/421)
BPS	6B (192/384)	(L + 6)B (864/1728)	B (32/64)	L·B (672/1344)
O-SSP-PLL	7	4	1	2S₂ (>1024)
ILP-SSP-PLL	15	2L + 6 (48)	2	L (21)

	BPS	ILP-PLL + ML	O-SSP-PLL	M-SSP-PLL + ML
QPSK	3 × 10⁻⁴	6.9 × 10⁻⁵	3.5 × 10⁻⁵	5.4 × 10⁻⁴
16-QAM	9.3 × 10⁻⁵	1.4 × 10⁻⁵	1.5 × 10⁻⁵	1.3 × 10⁻⁴
64-QAM	3.6 × 10⁻⁵	3.5 × 10⁻⁶	5.8 × 10⁻⁶	2.7 × 10⁻⁵

	Real Multiplier	Real Adder	Slicer	Buffer Units
M-SSP-PLL + ML	15	2L + 6 (48)	2	2S₁ + L (221/421)
BPS	6B (192/384)	(L + 6)B (864/1728)	B (32/64)	L·B (672/1344)
O-SSP-PLL	7	4	1	2S₂ (>1024)
ILP-SSP-PLL	15	2L + 6 (48)	2	L (21)

Abstract

References

2012 (1)

2011 (2)

2010 (3)

2009 (3)

2007 (1)

Al Amin, A.

Buhl, L. L.

Charlet, G.

Chen, X.

Doerr, C. R.

Dupuy, J.

Fatadin, I.

Gao, Y.

Gnauck, A. H.

Hoffmann, S.

Ip, E.

Ives, D.

Jorge, F.

Kahn, J. M.

Konczykowska, A.

Lau, A. P. T.

Lu, C.

Magarini, M.

Noe, R.

Peckham, D. W.

Pfau, T.

Riet, M.

Savory, S. J.

Shieh, W.

Taylor, M. G.

Winzer, P. J.

Yan, S.

Zhou, X.

Zhu, B.

IEEE Photon. Technol. Lett. (2)

J. Lightwave Technol. (7)

Opt. Express (1)

Other (5)

Cited By

Figures (12)

Tables (2)

Equations (4)

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