Abstract

An approach to measuring frequency response of high-speed electro-optic phase modulators via a single scan is proposed and experimentally demonstrated. This method employs low-speed photonic sampling to transfer the response information at any high frequency to a fixed low-frequency duplicate in the first Nyquist frequency range by setting the microwave frequency sweeping step equal to the repetition rate of the mode-locked laser. Through low-frequency detection and analysis in the electrical domain, the relative frequency response can be directly calculated from the relative intensity between the low-frequency duplicate and the direct current component without calibration of the photodetector response. In the experiment, the relative frequency responses (i.e., the S21 curve) of two 20 Gb/s LiNbO3-based electro-optic phase modulators and two 40 Gb/s samples are measured with a resolution of 96.9 MHz by using the proposed method, where the measurement results fit in with those obtained by using the conventional optical spectrum analysis method. The frequency measurement resolution can be further improved by using a mode-locked laser with a lower repetition rate.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

Y. Q. Heng, M. Xue, W. Chen, S. L. Han, J. Q. Liu, and S. L. Pan, “Large-dynamic frequency response measurement for broadband electro-optic phase modulators,” IEEE Photonics Technol. Lett. 31(4), 291–294 (2019).
[Crossref]

2018 (4)

M. Xue, S. F. Liu, and S. L. Pan, “High-resolution optical vector analysis based on symmetric double sideband modulation,” IEEE Photonics Technol. Lett. 30(5), 491–494 (2018).
[Crossref]

X. Yi, Q. F. Yang, K. Y. Yang, and K. Vahala, “Imaging soliton dynamics in optical microcavities,” Nat. Commun. 9(1), 3565 (2018).
[Crossref]

P. Imany, O. D. Odele, J. A. Jaramillo-Villegas, D. E. Leaird, and A. M. Weiner, “Characterization of coherent quantum frequency combs using electro-optic phase modulation,” Phys. Rev. A 97(1), 013813 (2018).
[Crossref]

X. L. Yan, X. H. Zou, W. Pan, L. S. Yan, and J. Azaña, “Fully digital programmable optical frequency comb generation and application,” Opt. Lett. 43(2), 283–286 (2018).
[Crossref]

2017 (1)

2015 (1)

2014 (2)

2013 (1)

2012 (3)

S. J. Zhang, X. X. Zhang, and Y. Liu, “Swept frequency measurement of electrooptic phase modulators using dispersive fibers,” Chin. Phys. Lett. 29(8), 084217 (2012).
[Crossref]

S. J. Zhang, X. X. Zhang, S. Liu, and Y. Liu, “Measurement of modulation index and half-wave voltage of an electro-optical phase modulator with a dispersion-based phase filter,” Opt. Commun. 285(24), 5089–5093 (2012).
[Crossref]

X. M. Wu, J. W. Mani, L. Xie, Y. Liu, X. Q. Qi, L. X. Wang, J. G. Liu, and N. H. Zhu, “Novel method for frequency response measurement of optoelectronic devices,” IEEE Photonics Technol. Lett. 24(7), 575–577 (2012).
[Crossref]

2011 (1)

B. M. Haas and T. E. Murphy, “Linearized down converting microwave photonic link using dual-wavelength phase modulation and optical filtering,” IEEE Photonics J. 3(1), 1–12 (2011).
[Crossref]

2009 (1)

2008 (1)

2005 (1)

S. Bhandare, S. K. Ibrahim, D. Sandel, H. B. Zhang, F. Wüst, and R. Noé, “Novel nonmagnetic 30-dB traveling-wave single-sideband optical isolator integrated in III/V material,” IEEE J. Select. Topics Quantum Electron. 11(2), 417–421 (2005).
[Crossref]

2003 (3)

P. D. Hale and D. F. Williams, “Calibrated measurement of optoelectronic frequency response,” IEEE Trans. Microwave Theory Tech. 51(4), 1422–1429 (2003).
[Crossref]

S. Oikawa, T. Kawanishi, and M. Izutsu, “Measurement of chirp parameters and halfwave voltages of Mach-Zehnder-type optical modulators by using a small signal operation,” IEEE Photonics Technol. Lett. 15(5), 682–684 (2003).
[Crossref]

Y. Q. Shi, L. S. Yan, and A. E. Willner, “High-speed electrooptic modulator characterization using optical spectrum analysis,” J. Lightwave Technol. 21(10), 2358–2367 (2003).
[Crossref]

1995 (1)

M. Levesque, M. Têtu, P. Tremblay, and M. Chamberland, “A novel technique to measure the dynamic response of an optical phase modulator,” IEEE Trans. Instrum. Meas. 44(5), 952–957 (1995).
[Crossref]

Azaña, J.

Bhandare, S.

S. Bhandare, S. K. Ibrahim, D. Sandel, H. B. Zhang, F. Wüst, and R. Noé, “Novel nonmagnetic 30-dB traveling-wave single-sideband optical isolator integrated in III/V material,” IEEE J. Select. Topics Quantum Electron. 11(2), 417–421 (2005).
[Crossref]

Chamberland, M.

M. Levesque, M. Têtu, P. Tremblay, and M. Chamberland, “A novel technique to measure the dynamic response of an optical phase modulator,” IEEE Trans. Instrum. Meas. 44(5), 952–957 (1995).
[Crossref]

Chan, E. H. W.

Chen, W.

Y. Q. Heng, M. Xue, W. Chen, S. L. Han, J. Q. Liu, and S. L. Pan, “Large-dynamic frequency response measurement for broadband electro-optic phase modulators,” IEEE Photonics Technol. Lett. 31(4), 291–294 (2019).
[Crossref]

Chi, H.

Chong, Y. H.

Dong, P.

Haas, B. M.

B. M. Haas and T. E. Murphy, “Linearized down converting microwave photonic link using dual-wavelength phase modulation and optical filtering,” IEEE Photonics J. 3(1), 1–12 (2011).
[Crossref]

Hale, P. D.

P. D. Hale and D. F. Williams, “Calibrated measurement of optoelectronic frequency response,” IEEE Trans. Microwave Theory Tech. 51(4), 1422–1429 (2003).
[Crossref]

Han, S. L.

Y. Q. Heng, M. Xue, W. Chen, S. L. Han, J. Q. Liu, and S. L. Pan, “Large-dynamic frequency response measurement for broadband electro-optic phase modulators,” IEEE Photonics Technol. Lett. 31(4), 291–294 (2019).
[Crossref]

Heng, Y. Q.

Y. Q. Heng, M. Xue, W. Chen, S. L. Han, J. Q. Liu, and S. L. Pan, “Large-dynamic frequency response measurement for broadband electro-optic phase modulators,” IEEE Photonics Technol. Lett. 31(4), 291–294 (2019).
[Crossref]

Ibrahim, S. K.

S. Bhandare, S. K. Ibrahim, D. Sandel, H. B. Zhang, F. Wüst, and R. Noé, “Novel nonmagnetic 30-dB traveling-wave single-sideband optical isolator integrated in III/V material,” IEEE J. Select. Topics Quantum Electron. 11(2), 417–421 (2005).
[Crossref]

Imany, P.

P. Imany, O. D. Odele, J. A. Jaramillo-Villegas, D. E. Leaird, and A. M. Weiner, “Characterization of coherent quantum frequency combs using electro-optic phase modulation,” Phys. Rev. A 97(1), 013813 (2018).
[Crossref]

Izutsu, M.

S. Oikawa, T. Kawanishi, and M. Izutsu, “Measurement of chirp parameters and halfwave voltages of Mach-Zehnder-type optical modulators by using a small signal operation,” IEEE Photonics Technol. Lett. 15(5), 682–684 (2003).
[Crossref]

Jaramillo-Villegas, J. A.

P. Imany, O. D. Odele, J. A. Jaramillo-Villegas, D. E. Leaird, and A. M. Weiner, “Characterization of coherent quantum frequency combs using electro-optic phase modulation,” Phys. Rev. A 97(1), 013813 (2018).
[Crossref]

Kawanishi, T.

S. Oikawa, T. Kawanishi, and M. Izutsu, “Measurement of chirp parameters and halfwave voltages of Mach-Zehnder-type optical modulators by using a small signal operation,” IEEE Photonics Technol. Lett. 15(5), 682–684 (2003).
[Crossref]

Leaird, D. E.

P. Imany, O. D. Odele, J. A. Jaramillo-Villegas, D. E. Leaird, and A. M. Weiner, “Characterization of coherent quantum frequency combs using electro-optic phase modulation,” Phys. Rev. A 97(1), 013813 (2018).
[Crossref]

Levesque, M.

M. Levesque, M. Têtu, P. Tremblay, and M. Chamberland, “A novel technique to measure the dynamic response of an optical phase modulator,” IEEE Trans. Instrum. Meas. 44(5), 952–957 (1995).
[Crossref]

Li, X. L.

Liu, J. G.

X. M. Wu, J. W. Mani, L. Xie, Y. Liu, X. Q. Qi, L. X. Wang, J. G. Liu, and N. H. Zhu, “Novel method for frequency response measurement of optoelectronic devices,” IEEE Photonics Technol. Lett. 24(7), 575–577 (2012).
[Crossref]

Liu, J. Q.

Y. Q. Heng, M. Xue, W. Chen, S. L. Han, J. Q. Liu, and S. L. Pan, “Large-dynamic frequency response measurement for broadband electro-optic phase modulators,” IEEE Photonics Technol. Lett. 31(4), 291–294 (2019).
[Crossref]

Liu, S.

S. J. Zhang, X. X. Zhang, S. Liu, and Y. Liu, “Measurement of modulation index and half-wave voltage of an electro-optical phase modulator with a dispersion-based phase filter,” Opt. Commun. 285(24), 5089–5093 (2012).
[Crossref]

Liu, S. F.

M. Xue, S. F. Liu, and S. L. Pan, “High-resolution optical vector analysis based on symmetric double sideband modulation,” IEEE Photonics Technol. Lett. 30(5), 491–494 (2018).
[Crossref]

Liu, Y.

S. J. Zhang, H. Wang, X. H. Zou, Y. L. Zhang, R. G. Lu, and Y. Liu, “Self-calibrating measurement of high-speed electro-optic phase modulators based on two-tone modulation,” Opt. Lett. 39(12), 3504–3507 (2014).
[Crossref]

S. J. Zhang, X. X. Zhang, S. Liu, and Y. Liu, “Measurement of modulation index and half-wave voltage of an electro-optical phase modulator with a dispersion-based phase filter,” Opt. Commun. 285(24), 5089–5093 (2012).
[Crossref]

X. M. Wu, J. W. Mani, L. Xie, Y. Liu, X. Q. Qi, L. X. Wang, J. G. Liu, and N. H. Zhu, “Novel method for frequency response measurement of optoelectronic devices,” IEEE Photonics Technol. Lett. 24(7), 575–577 (2012).
[Crossref]

S. J. Zhang, X. X. Zhang, and Y. Liu, “Swept frequency measurement of electrooptic phase modulators using dispersive fibers,” Chin. Phys. Lett. 29(8), 084217 (2012).
[Crossref]

Lu, R. G.

Mani, J. W.

X. M. Wu, J. W. Mani, L. Xie, Y. Liu, X. Q. Qi, L. X. Wang, J. G. Liu, and N. H. Zhu, “Novel method for frequency response measurement of optoelectronic devices,” IEEE Photonics Technol. Lett. 24(7), 575–577 (2012).
[Crossref]

Minasian, R. A.

Murphy, T. E.

B. M. Haas and T. E. Murphy, “Linearized down converting microwave photonic link using dual-wavelength phase modulation and optical filtering,” IEEE Photonics J. 3(1), 1–12 (2011).
[Crossref]

Noé, R.

S. Bhandare, S. K. Ibrahim, D. Sandel, H. B. Zhang, F. Wüst, and R. Noé, “Novel nonmagnetic 30-dB traveling-wave single-sideband optical isolator integrated in III/V material,” IEEE J. Select. Topics Quantum Electron. 11(2), 417–421 (2005).
[Crossref]

Odele, O. D.

P. Imany, O. D. Odele, J. A. Jaramillo-Villegas, D. E. Leaird, and A. M. Weiner, “Characterization of coherent quantum frequency combs using electro-optic phase modulation,” Phys. Rev. A 97(1), 013813 (2018).
[Crossref]

Oikawa, S.

S. Oikawa, T. Kawanishi, and M. Izutsu, “Measurement of chirp parameters and halfwave voltages of Mach-Zehnder-type optical modulators by using a small signal operation,” IEEE Photonics Technol. Lett. 15(5), 682–684 (2003).
[Crossref]

Pan, S. L.

Y. Q. Heng, M. Xue, W. Chen, S. L. Han, J. Q. Liu, and S. L. Pan, “Large-dynamic frequency response measurement for broadband electro-optic phase modulators,” IEEE Photonics Technol. Lett. 31(4), 291–294 (2019).
[Crossref]

M. Xue, S. F. Liu, and S. L. Pan, “High-resolution optical vector analysis based on symmetric double sideband modulation,” IEEE Photonics Technol. Lett. 30(5), 491–494 (2018).
[Crossref]

S. L. Pan and J. P. Yao, “Photonics-based broadband microwave measurement,” J. Lightwave Technol. 35(16), 3498–3513 (2017).
[Crossref]

Pan, W.

Qi, X. Q.

X. M. Wu, J. W. Mani, L. Xie, Y. Liu, X. Q. Qi, L. X. Wang, J. G. Liu, and N. H. Zhu, “Novel method for frequency response measurement of optoelectronic devices,” IEEE Photonics Technol. Lett. 24(7), 575–577 (2012).
[Crossref]

Sandel, D.

S. Bhandare, S. K. Ibrahim, D. Sandel, H. B. Zhang, F. Wüst, and R. Noé, “Novel nonmagnetic 30-dB traveling-wave single-sideband optical isolator integrated in III/V material,” IEEE J. Select. Topics Quantum Electron. 11(2), 417–421 (2005).
[Crossref]

Shi, Y. Q.

Têtu, M.

M. Levesque, M. Têtu, P. Tremblay, and M. Chamberland, “A novel technique to measure the dynamic response of an optical phase modulator,” IEEE Trans. Instrum. Meas. 44(5), 952–957 (1995).
[Crossref]

Tremblay, P.

M. Levesque, M. Têtu, P. Tremblay, and M. Chamberland, “A novel technique to measure the dynamic response of an optical phase modulator,” IEEE Trans. Instrum. Meas. 44(5), 952–957 (1995).
[Crossref]

Vahala, K.

X. Yi, Q. F. Yang, K. Y. Yang, and K. Vahala, “Imaging soliton dynamics in optical microcavities,” Nat. Commun. 9(1), 3565 (2018).
[Crossref]

Wang, H.

Wang, L. X.

X. M. Wu, J. W. Mani, L. Xie, Y. Liu, X. Q. Qi, L. X. Wang, J. G. Liu, and N. H. Zhu, “Novel method for frequency response measurement of optoelectronic devices,” IEEE Photonics Technol. Lett. 24(7), 575–577 (2012).
[Crossref]

Weiner, A. M.

P. Imany, O. D. Odele, J. A. Jaramillo-Villegas, D. E. Leaird, and A. M. Weiner, “Characterization of coherent quantum frequency combs using electro-optic phase modulation,” Phys. Rev. A 97(1), 013813 (2018).
[Crossref]

Williams, D. F.

P. D. Hale and D. F. Williams, “Calibrated measurement of optoelectronic frequency response,” IEEE Trans. Microwave Theory Tech. 51(4), 1422–1429 (2003).
[Crossref]

Willner, A. E.

Wu, X. M.

X. M. Wu, J. W. Mani, L. Xie, Y. Liu, X. Q. Qi, L. X. Wang, J. G. Liu, and N. H. Zhu, “Novel method for frequency response measurement of optoelectronic devices,” IEEE Photonics Technol. Lett. 24(7), 575–577 (2012).
[Crossref]

Wüst, F.

S. Bhandare, S. K. Ibrahim, D. Sandel, H. B. Zhang, F. Wüst, and R. Noé, “Novel nonmagnetic 30-dB traveling-wave single-sideband optical isolator integrated in III/V material,” IEEE J. Select. Topics Quantum Electron. 11(2), 417–421 (2005).
[Crossref]

Xie, L.

X. M. Wu, J. W. Mani, L. Xie, Y. Liu, X. Q. Qi, L. X. Wang, J. G. Liu, and N. H. Zhu, “Novel method for frequency response measurement of optoelectronic devices,” IEEE Photonics Technol. Lett. 24(7), 575–577 (2012).
[Crossref]

Xue, M.

Y. Q. Heng, M. Xue, W. Chen, S. L. Han, J. Q. Liu, and S. L. Pan, “Large-dynamic frequency response measurement for broadband electro-optic phase modulators,” IEEE Photonics Technol. Lett. 31(4), 291–294 (2019).
[Crossref]

M. Xue, S. F. Liu, and S. L. Pan, “High-resolution optical vector analysis based on symmetric double sideband modulation,” IEEE Photonics Technol. Lett. 30(5), 491–494 (2018).
[Crossref]

Yan, L. S.

Yan, X. L.

Yang, C.

Yang, K. Y.

X. Yi, Q. F. Yang, K. Y. Yang, and K. Vahala, “Imaging soliton dynamics in optical microcavities,” Nat. Commun. 9(1), 3565 (2018).
[Crossref]

Yang, Q. F.

X. Yi, Q. F. Yang, K. Y. Yang, and K. Vahala, “Imaging soliton dynamics in optical microcavities,” Nat. Commun. 9(1), 3565 (2018).
[Crossref]

Yao, J. P.

Yi, X.

X. Yi, Q. F. Yang, K. Y. Yang, and K. Vahala, “Imaging soliton dynamics in optical microcavities,” Nat. Commun. 9(1), 3565 (2018).
[Crossref]

R. A. Minasian, E. H. W. Chan, and X. Yi, “Microwave photonic signal processing,” Opt. Express 21(19), 22918–22936 (2013).
[Crossref]

Zhang, H. B.

S. Bhandare, S. K. Ibrahim, D. Sandel, H. B. Zhang, F. Wüst, and R. Noé, “Novel nonmagnetic 30-dB traveling-wave single-sideband optical isolator integrated in III/V material,” IEEE J. Select. Topics Quantum Electron. 11(2), 417–421 (2005).
[Crossref]

Zhang, S. J.

S. J. Zhang, H. Wang, X. H. Zou, Y. L. Zhang, R. G. Lu, and Y. Liu, “Self-calibrating measurement of high-speed electro-optic phase modulators based on two-tone modulation,” Opt. Lett. 39(12), 3504–3507 (2014).
[Crossref]

S. J. Zhang, X. X. Zhang, and Y. Liu, “Swept frequency measurement of electrooptic phase modulators using dispersive fibers,” Chin. Phys. Lett. 29(8), 084217 (2012).
[Crossref]

S. J. Zhang, X. X. Zhang, S. Liu, and Y. Liu, “Measurement of modulation index and half-wave voltage of an electro-optical phase modulator with a dispersion-based phase filter,” Opt. Commun. 285(24), 5089–5093 (2012).
[Crossref]

Zhang, X. X.

S. J. Zhang, X. X. Zhang, S. Liu, and Y. Liu, “Measurement of modulation index and half-wave voltage of an electro-optical phase modulator with a dispersion-based phase filter,” Opt. Commun. 285(24), 5089–5093 (2012).
[Crossref]

S. J. Zhang, X. X. Zhang, and Y. Liu, “Swept frequency measurement of electrooptic phase modulators using dispersive fibers,” Chin. Phys. Lett. 29(8), 084217 (2012).
[Crossref]

Zhang, Y. L.

Zhou, Z. H.

Zhu, N. H.

X. M. Wu, J. W. Mani, L. Xie, Y. Liu, X. Q. Qi, L. X. Wang, J. G. Liu, and N. H. Zhu, “Novel method for frequency response measurement of optoelectronic devices,” IEEE Photonics Technol. Lett. 24(7), 575–577 (2012).
[Crossref]

Zou, X. H.

Chin. Phys. Lett. (1)

S. J. Zhang, X. X. Zhang, and Y. Liu, “Swept frequency measurement of electrooptic phase modulators using dispersive fibers,” Chin. Phys. Lett. 29(8), 084217 (2012).
[Crossref]

IEEE J. Select. Topics Quantum Electron. (1)

S. Bhandare, S. K. Ibrahim, D. Sandel, H. B. Zhang, F. Wüst, and R. Noé, “Novel nonmagnetic 30-dB traveling-wave single-sideband optical isolator integrated in III/V material,” IEEE J. Select. Topics Quantum Electron. 11(2), 417–421 (2005).
[Crossref]

IEEE Photonics J. (1)

B. M. Haas and T. E. Murphy, “Linearized down converting microwave photonic link using dual-wavelength phase modulation and optical filtering,” IEEE Photonics J. 3(1), 1–12 (2011).
[Crossref]

IEEE Photonics Technol. Lett. (4)

X. M. Wu, J. W. Mani, L. Xie, Y. Liu, X. Q. Qi, L. X. Wang, J. G. Liu, and N. H. Zhu, “Novel method for frequency response measurement of optoelectronic devices,” IEEE Photonics Technol. Lett. 24(7), 575–577 (2012).
[Crossref]

M. Xue, S. F. Liu, and S. L. Pan, “High-resolution optical vector analysis based on symmetric double sideband modulation,” IEEE Photonics Technol. Lett. 30(5), 491–494 (2018).
[Crossref]

Y. Q. Heng, M. Xue, W. Chen, S. L. Han, J. Q. Liu, and S. L. Pan, “Large-dynamic frequency response measurement for broadband electro-optic phase modulators,” IEEE Photonics Technol. Lett. 31(4), 291–294 (2019).
[Crossref]

S. Oikawa, T. Kawanishi, and M. Izutsu, “Measurement of chirp parameters and halfwave voltages of Mach-Zehnder-type optical modulators by using a small signal operation,” IEEE Photonics Technol. Lett. 15(5), 682–684 (2003).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

M. Levesque, M. Têtu, P. Tremblay, and M. Chamberland, “A novel technique to measure the dynamic response of an optical phase modulator,” IEEE Trans. Instrum. Meas. 44(5), 952–957 (1995).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

P. D. Hale and D. F. Williams, “Calibrated measurement of optoelectronic frequency response,” IEEE Trans. Microwave Theory Tech. 51(4), 1422–1429 (2003).
[Crossref]

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X. Yi, Q. F. Yang, K. Y. Yang, and K. Vahala, “Imaging soliton dynamics in optical microcavities,” Nat. Commun. 9(1), 3565 (2018).
[Crossref]

Opt. Commun. (1)

S. J. Zhang, X. X. Zhang, S. Liu, and Y. Liu, “Measurement of modulation index and half-wave voltage of an electro-optical phase modulator with a dispersion-based phase filter,” Opt. Commun. 285(24), 5089–5093 (2012).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (1)

P. Imany, O. D. Odele, J. A. Jaramillo-Villegas, D. E. Leaird, and A. M. Weiner, “Characterization of coherent quantum frequency combs using electro-optic phase modulation,” Phys. Rev. A 97(1), 013813 (2018).
[Crossref]

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Figures (3)

Fig. 1.
Fig. 1. Schematic diagram of the proposed method. MLL: mode-locked lasers; PC: polarization controller; OC: optical coupler; PD: photodetector; ESA: electrical spectrum analyzer.
Fig. 2.
Fig. 2. Measured electrical spectra from the PD when the input microwave frequency is tuned to be 4.0029 GHz, 7.9758 GHz, 11.9487 GHz and 15.6309 GHz, respectively.
Fig. 3.
Fig. 3. Measured relative frequency response (i.e., the S21 curve) of (a) sample 1, (b) sample 2, (c) sample 3 and (d) sample 4.

Equations (13)

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S M L L ( t ) = F 0 l = + p ( t l / l f r f r ) = n  ={-}  N N { p n e j [ 2 π ( f 0 + n f r ) t ] + p n e j [ 2 π ( f 0 + n f r ) t ] } ,
S C W b e f o r e P M ( t ) = 2 2 S M L L ( t ) ,
S a n t i C W b e f o r e P M ( t ) = 2 2 S M L L ( t ) e j π 2 ,
S C W ( t ) = 1 2 S M L L ( t ) e j m ( f s ) cos ( 2 π f s t ) e j π / j π 2 2 ,
S a n t i C W ( t ) = 1 2 S M L L ( t ) e j π / j π 2 2 e j φ ,
S r ( t ) = S M L L ( t ) e j π / j π 2 2 e j m ( f s ) cos ( 2 π f s t ) / j m ( f s ) cos ( 2 π f s t ) 2 2 + j φ / φ 2 2 cos [ φ 2 m ( f s ) 2 cos ( 2 π f s t ) ] ,
I P D ( t ) R ( f ) S r ( t ) S r ( t ) = n = N N l = N N p n p l { R ( ( n l ) f r ) [ 1 + cos φ J 0 ( m ( f s ) ) ] cos ( 2 π ( n l ) f r t ) + 2 R ( ( n l ) f r ± f s ) sin φ J 1 ( m ( f s ) ) cos ( 2 π ( n f r l f r ± f s ) t ) + } ,
f F = { f s k f r r e m ( f s / f s f r f r ) f r / f r 2 2 k f r f s r e m ( f s / f s f r f r ) > f r / f r 2 2 ,
I F = 2 E k R ( f F ) sin φ J 1 ( m ( f s ) ) ,
I D C = E 0 R ( 0 ) [ 1 + cos φ J 0 ( m ( f s ) ) ] ,
E k = 2 n = N N p n p n + k ,
E 0 E 1 E 2 E n ,
J 1 ( m ( f s ) ) = R ( 0 ) 2 R ( f F ) I F I D C 1 + cos φ sin φ ,

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