Abstract

An approach to microwave frequency measurement with a high resolution and a broad bandwidth is proposed based on three parallel low-speed photonic analog-to-digital converters (ADCs) architecture. Through simultaneously bandpass sampling the input microwave signal with the three photonic ADCs, the input frequency can be calculated from the Fourier frequencies of the photonic ADCs using the proposed frequency recovery algorithm. Theoretical analysis and simulation results indicate that the proposed method is applicable for both single-tone and multi-tone microwave signals. By employing three ~1 GS/s@8 bits photonic ADCs, 0-100 GHz frequency measurement with an error of ± 0.5 MHz and a spur-free dynamic range of 94 dB-Hz2/3 over the full band has been numerically demonstrated. Additionally, a proof-of-concept experiment is carried out to demonstrate the effectiveness of the proposed method, where a frequency measurement range of 0-20 GHz with a measurement error of ± 8 kHz is realized by utilizing three photonic ADCs with sampling rates of 27.690 MS/s, 27.710 MS/s, and 27.730 MS/s. Larger frequency measurement range can be achieved by using an optical modulator with a larger bandwidth.

© 2017 Optical Society of America

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References

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2014 (2)

T. A. Nguyen, E. H. W. Chan, and R. A. Minasian, “Instantaneous high-resolution multiple-frequency measurement system based on frequency-to-time mapping technique,” Opt. Lett. 39(8), 2419–2422 (2014).
[Crossref] [PubMed]

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

2013 (4)

L. X. Wang, N. H. Zhu, W. Li, H. Wang, J. Y. Zheng, and J. G. Liu, “Polarization division multiplexed photonic radio-frequency channelizer using an optical comb,” Opt. Commun. 286, 282–287 (2013).
[Crossref]

C. Wang and J. Yao, “Ultrahigh-resolution photonic-assisted microwave frequency identification based on temporal channelization,” IEEE T. Microw. Theory 61(12), 4275–4282 (2013).
[Crossref]

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory 61(9), 3470–3478 (2013).
[Crossref]

B. Lu, W. Pan, X. Zou, B. Luo, L. Yan, X. Liu, and S. Xiang, “Photonic frequency measurement and signal separation for pulsed/CW microwave signals,” IEEE Photonics Technol. Lett. 25(5), 500–503 (2013).
[Crossref]

2012 (4)

2010 (1)

2009 (1)

L. V. T. Nguyen, “Microwave photonic technique for frequency measurement of simultaneous signals,” IEEE Photonics Technol. Lett. 21(10), 642–644 (2009).
[Crossref]

2008 (1)

H. Chi, X. Zou, and J. Yao, “An approach to the measurement of microwave frequency based on optical power monitoring,” IEEE Photonics Technol. Lett. 20(14), 1249–1251 (2008).
[Crossref]

2007 (1)

2006 (1)

A. Lippman, “The new age of wireless,” Sci. Am. 295(4), 40 (2006).
[Crossref] [PubMed]

2000 (1)

J. C. Twichell and R. Helkey, “Phase-encoded optical sampling for analog-to-digital converters,” IEEE Photonics Technol. Lett. 12(9), 1237–1239 (2000).
[Crossref]

Berizzi, F.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Bogoni, A.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Byun, H.

Capria, A.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Chan, E. H. W.

Chen, J.

Chi, H.

H. Chi, X. Zou, and J. Yao, “An approach to the measurement of microwave frequency based on optical power monitoring,” IEEE Photonics Technol. Lett. 20(14), 1249–1251 (2008).
[Crossref]

Dahlem, M. S.

Dai, Y.

X. Xie, Y. Dai, Y. Ji, K. Xu, Y. Li, J. Wu, and J. Lin, “Broadband photonic radio-frequency channelization based on a 39-GHz optical frequency comb,” IEEE Photonics Technol. Lett. 24(8), 661–663 (2012).
[Crossref]

DeVore, P. T.

C. M. Gee, G. Sefler, P. T. DeVore, and G. C. Valley, “Spurious-Free dynamic range of a high-resolution photonic time-stretch analog-to-digital converter system,” Microw. Opt. Technol. Lett. 54(11), 2558–2563 (2012).
[Crossref]

DiLello, N. A.

Fu, J.

Gee, C. M.

C. M. Gee, G. Sefler, P. T. DeVore, and G. C. Valley, “Spurious-Free dynamic range of a high-resolution photonic time-stretch analog-to-digital converter system,” Microw. Opt. Technol. Lett. 54(11), 2558–2563 (2012).
[Crossref]

Geis, M. W.

Ghelfi, P.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Grein, M. E.

Gupta, S.

Helkey, R.

J. C. Twichell and R. Helkey, “Phase-encoded optical sampling for analog-to-digital converters,” IEEE Photonics Technol. Lett. 12(9), 1237–1239 (2000).
[Crossref]

Holzwarth, C. W.

Hoyt, J. L.

Ippen, E. P.

Jalali, B.

Ji, Y.

X. Xie, Y. Dai, Y. Ji, K. Xu, Y. Li, J. Wu, and J. Lin, “Broadband photonic radio-frequency channelization based on a 39-GHz optical frequency comb,” IEEE Photonics Technol. Lett. 24(8), 661–663 (2012).
[Crossref]

Kärtner, F. X.

Khilo, A.

Laghezza, F.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Lazzeri, E.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Li, W.

L. X. Wang, N. H. Zhu, W. Li, H. Wang, J. Y. Zheng, and J. G. Liu, “Polarization division multiplexed photonic radio-frequency channelizer using an optical comb,” Opt. Commun. 286, 282–287 (2013).
[Crossref]

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory 61(9), 3470–3478 (2013).
[Crossref]

Li, Y.

X. Xie, Y. Dai, Y. Ji, K. Xu, Y. Li, J. Wu, and J. Lin, “Broadband photonic radio-frequency channelization based on a 39-GHz optical frequency comb,” IEEE Photonics Technol. Lett. 24(8), 661–663 (2012).
[Crossref]

Lin, J.

X. Xie, Y. Dai, Y. Ji, K. Xu, Y. Li, J. Wu, and J. Lin, “Broadband photonic radio-frequency channelization based on a 39-GHz optical frequency comb,” IEEE Photonics Technol. Lett. 24(8), 661–663 (2012).
[Crossref]

Lippman, A.

A. Lippman, “The new age of wireless,” Sci. Am. 295(4), 40 (2006).
[Crossref] [PubMed]

Liu, J. G.

L. X. Wang, N. H. Zhu, W. Li, H. Wang, J. Y. Zheng, and J. G. Liu, “Polarization division multiplexed photonic radio-frequency channelizer using an optical comb,” Opt. Commun. 286, 282–287 (2013).
[Crossref]

Liu, X.

B. Lu, W. Pan, X. Zou, B. Luo, L. Yan, X. Liu, and S. Xiang, “Photonic frequency measurement and signal separation for pulsed/CW microwave signals,” IEEE Photonics Technol. Lett. 25(5), 500–503 (2013).
[Crossref]

Lu, B.

B. Lu, W. Pan, X. Zou, B. Luo, L. Yan, X. Liu, and S. Xiang, “Photonic frequency measurement and signal separation for pulsed/CW microwave signals,” IEEE Photonics Technol. Lett. 25(5), 500–503 (2013).
[Crossref]

Luo, B.

B. Lu, W. Pan, X. Zou, B. Luo, L. Yan, X. Liu, and S. Xiang, “Photonic frequency measurement and signal separation for pulsed/CW microwave signals,” IEEE Photonics Technol. Lett. 25(5), 500–503 (2013).
[Crossref]

X. Zou, W. Pan, B. Luo, and L. Yan, “Full-scale phase demodulation approach for photonic instantaneous frequency measurement,” Opt. Lett. 35(16), 2747–2749 (2010).
[Crossref] [PubMed]

Lyszczarz, T. M.

Malacarne, A.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Minasian, R. A.

Motamedi, A.

Nejadmalayeri, A. H.

Nguyen, L. V. T.

L. V. T. Nguyen, “Microwave photonic technique for frequency measurement of simultaneous signals,” IEEE Photonics Technol. Lett. 21(10), 642–644 (2009).
[Crossref]

Nguyen, T. A.

Onori, D.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Orcutt, J. S.

Pan, S.

Pan, W.

B. Lu, W. Pan, X. Zou, B. Luo, L. Yan, X. Liu, and S. Xiang, “Photonic frequency measurement and signal separation for pulsed/CW microwave signals,” IEEE Photonics Technol. Lett. 25(5), 500–503 (2013).
[Crossref]

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory 61(9), 3470–3478 (2013).
[Crossref]

X. Zou, W. Pan, B. Luo, and L. Yan, “Full-scale phase demodulation approach for photonic instantaneous frequency measurement,” Opt. Lett. 35(16), 2747–2749 (2010).
[Crossref] [PubMed]

Peng, M. Y.

Perrott, M.

Pinna, S.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Popovic, M. A.

Porzi, C.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Ram, R. J.

Sander, M. Y.

Scaffardi, M.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Scotti, F.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Sefler, G.

C. M. Gee, G. Sefler, P. T. DeVore, and G. C. Valley, “Spurious-Free dynamic range of a high-resolution photonic time-stretch analog-to-digital converter system,” Microw. Opt. Technol. Lett. 54(11), 2558–2563 (2012).
[Crossref]

Serafino, G.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Smith, H. I.

Sorace-Agaskar, C. M.

Spector, S. J.

Sun, J.

Twichell, J. C.

J. C. Twichell and R. Helkey, “Phase-encoded optical sampling for analog-to-digital converters,” IEEE Photonics Technol. Lett. 12(9), 1237–1239 (2000).
[Crossref]

Valley, G. C.

C. M. Gee, G. Sefler, P. T. DeVore, and G. C. Valley, “Spurious-Free dynamic range of a high-resolution photonic time-stretch analog-to-digital converter system,” Microw. Opt. Technol. Lett. 54(11), 2558–2563 (2012).
[Crossref]

S. Gupta, G. C. Valley, and B. Jalali, “Distortion cancellation in time-stretch analog-to-digital converter,” J. Lightwave Technol. 25(12), 3716–3721 (2007).
[Crossref]

Vercesi, V.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Wang, C.

C. Wang and J. Yao, “Ultrahigh-resolution photonic-assisted microwave frequency identification based on temporal channelization,” IEEE T. Microw. Theory 61(12), 4275–4282 (2013).
[Crossref]

Wang, H.

L. X. Wang, N. H. Zhu, W. Li, H. Wang, J. Y. Zheng, and J. G. Liu, “Polarization division multiplexed photonic radio-frequency channelizer using an optical comb,” Opt. Commun. 286, 282–287 (2013).
[Crossref]

Wang, J. P.

Wang, L. X.

L. X. Wang, N. H. Zhu, W. Li, H. Wang, J. Y. Zheng, and J. G. Liu, “Polarization division multiplexed photonic radio-frequency channelizer using an optical comb,” Opt. Commun. 286, 282–287 (2013).
[Crossref]

Wu, J.

X. Xie, Y. Dai, Y. Ji, K. Xu, Y. Li, J. Wu, and J. Lin, “Broadband photonic radio-frequency channelization based on a 39-GHz optical frequency comb,” IEEE Photonics Technol. Lett. 24(8), 661–663 (2012).
[Crossref]

Xiang, S.

B. Lu, W. Pan, X. Zou, B. Luo, L. Yan, X. Liu, and S. Xiang, “Photonic frequency measurement and signal separation for pulsed/CW microwave signals,” IEEE Photonics Technol. Lett. 25(5), 500–503 (2013).
[Crossref]

Xie, X.

X. Xie, Y. Dai, Y. Ji, K. Xu, Y. Li, J. Wu, and J. Lin, “Broadband photonic radio-frequency channelization based on a 39-GHz optical frequency comb,” IEEE Photonics Technol. Lett. 24(8), 661–663 (2012).
[Crossref]

Xu, K.

X. Xie, Y. Dai, Y. Ji, K. Xu, Y. Li, J. Wu, and J. Lin, “Broadband photonic radio-frequency channelization based on a 39-GHz optical frequency comb,” IEEE Photonics Technol. Lett. 24(8), 661–663 (2012).
[Crossref]

Yan, L.

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory 61(9), 3470–3478 (2013).
[Crossref]

B. Lu, W. Pan, X. Zou, B. Luo, L. Yan, X. Liu, and S. Xiang, “Photonic frequency measurement and signal separation for pulsed/CW microwave signals,” IEEE Photonics Technol. Lett. 25(5), 500–503 (2013).
[Crossref]

X. Zou, W. Pan, B. Luo, and L. Yan, “Full-scale phase demodulation approach for photonic instantaneous frequency measurement,” Opt. Lett. 35(16), 2747–2749 (2010).
[Crossref] [PubMed]

Yao, J.

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory 61(9), 3470–3478 (2013).
[Crossref]

C. Wang and J. Yao, “Ultrahigh-resolution photonic-assisted microwave frequency identification based on temporal channelization,” IEEE T. Microw. Theory 61(12), 4275–4282 (2013).
[Crossref]

S. Pan, J. Fu, and J. Yao, “Photonic approach to the simultaneous measurement of the frequency, amplitude, pulse width, and time of arrival of a microwave signal,” Opt. Lett. 37(1), 7–9 (2012).
[Crossref] [PubMed]

H. Chi, X. Zou, and J. Yao, “An approach to the measurement of microwave frequency based on optical power monitoring,” IEEE Photonics Technol. Lett. 20(14), 1249–1251 (2008).
[Crossref]

Yoon, J. U.

Zheng, J. Y.

L. X. Wang, N. H. Zhu, W. Li, H. Wang, J. Y. Zheng, and J. G. Liu, “Polarization division multiplexed photonic radio-frequency channelizer using an optical comb,” Opt. Commun. 286, 282–287 (2013).
[Crossref]

Zhou, G. R.

Zhu, N. H.

L. X. Wang, N. H. Zhu, W. Li, H. Wang, J. Y. Zheng, and J. G. Liu, “Polarization division multiplexed photonic radio-frequency channelizer using an optical comb,” Opt. Commun. 286, 282–287 (2013).
[Crossref]

Zou, X.

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory 61(9), 3470–3478 (2013).
[Crossref]

B. Lu, W. Pan, X. Zou, B. Luo, L. Yan, X. Liu, and S. Xiang, “Photonic frequency measurement and signal separation for pulsed/CW microwave signals,” IEEE Photonics Technol. Lett. 25(5), 500–503 (2013).
[Crossref]

X. Zou, W. Pan, B. Luo, and L. Yan, “Full-scale phase demodulation approach for photonic instantaneous frequency measurement,” Opt. Lett. 35(16), 2747–2749 (2010).
[Crossref] [PubMed]

H. Chi, X. Zou, and J. Yao, “An approach to the measurement of microwave frequency based on optical power monitoring,” IEEE Photonics Technol. Lett. 20(14), 1249–1251 (2008).
[Crossref]

IEEE Photonics Technol. Lett. (5)

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Figures (9)

Fig. 1
Fig. 1 Schematic diagram of the proposed photonic-assisted microwave frequency measurement. MLL: mode-locked laser, MZM: Mach-Zehnder modulator, PD: photodetector, AM: amplifier, ADC: analog to digital converter, DSP: digital signal processor, RF: radio frequency.
Fig. 2
Fig. 2 The flow chart of frequency recovery algorithm.
Fig. 3
Fig. 3 Simulation results for a single-tone input microwave signal. (a) FFT spectrum of photonic ADC1. (b) FFT spectrum of photonic ADC2. (c) FFT spectrum of photonic ADC3.
Fig. 4
Fig. 4 Frequency measurement error of simulation for a single-tone input microwave signal with different frequencies.
Fig. 5
Fig. 5 Simulation result of the spur-free dynamic range measurement.
Fig. 6
Fig. 6 Simulation results for a three-tone input microwave signal. (a) FFT spectrum of photonic ADC1. (b) FFT spectrum of photonic ADC2. (c) FFT spectrum of photonic ADC3.
Fig. 7
Fig. 7 Experimental measured spectra of a single-tone RF signal with a frequency of 20 GHz under sampling rate of (a) 27.690 MS/s, (b) 27.710 MS/s and (c) 27.730 MS/s.
Fig. 8
Fig. 8 Experimental results of measured frequency and measurement error for a single-tone input microwave signal with different frequencies
Fig. 9
Fig. 9 Simulated spectra of a single-tone RF signal with a frequency of 20 GHz under a sampling rate of (a) 27.690 MS/s, (b) 27.710 MS/s and (c) 27.730 MS/s.

Tables (3)

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Table 1 Frequency measuring results for a multi-tone input microwave signal

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Table 2 Theoretical frequency measuring result of a special multi-tone microwave signal

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Table 3 Simulation results of a single-tone RF signal under different sampling rates drifts

Equations (9)

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S M L L 1 ( f ) = n = N N p n δ ( f ± f c ± n f 1 )
S s i g ( f ) = δ ( f + f s ) + δ ( f f s )
S S A M 1 ( f ) = 1 2 S M L L 1 ( f ) [ δ ( f ) + m S s i g ( f ) ] = 1 2 n = N N p n [ δ ( f ± f c ± n f 1 ) + m π δ ( f ± f c ± n f 1 ± f s ) ]
S P D 1 ( f ) = n = 2 N 2 N [ D n δ ( f n f 1 ) + E n δ ( f n f 1 + f s ) + E n δ ( f n f 1 f s ) + ]
f F 1 = { f s n 1 f 1 rem ( f s / f 1 ) f 1 / 2 ( n 1 + 1 ) f 1 f s rem ( f s / f 1 ) > f 1 / 2
f s 1 = { n 1 f 1 + f F 1 , rem ( f s 1 / f 1 ) f 1 / 2 ( n 1 + 1 ) f 1 f F 1 , rem ( f s 1 / f 1 ) > f 1 / 2
f s 2 = { n 2 f 2 + f F 2 rem ( f s 2 / f 2 ) f 2 / 2 ( n 2 + 1 ) f 2 f F 2 rem ( f s 2 / f 2 ) > f 2 / 2
f s 3 = { n 3 f 3 + f F 3 rem ( f s 3 / f 3 ) f 3 / 2 ( n 3 + 1 ) f 3 f F 3 rem ( f s 3 / f 3 ) > f 3 / 2
f s f = f s 1 = f s 2 = f s 3

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