Abstract

A microwave photonic link (MPL) with simultaneous suppression of the even-order and third-order distortions using a polarization modulator (PolM), an optical bandpass filter (OBPF), and a balanced photodetector (BPD) is proposed and experimentally demonstrated. The even-order distortions are suppressed by utilizing orthogonal polarization modulation based on the PolM and balanced differential detection based on the BPD. The third-order distortions (IMD3) are suppressed by optimizing the spectral response of the OBPF with an optimal power ratio between the optical carrier and the sidebands of the phase-modulated signals from the PolM. Since the suppression of the IMD3 is achieved when the MPL is optimized for even-order distortion suppression, the proposed MPL can operate with simultaneous suppression of the even-order and third-order distortions. The proposed MPL is analyzed theoretically and is verified by an experiment. For a two-tone RF signal of f1 = 10 GHz and f2 = 19.95 GHz, the spurious-free dynamic range (SFDR2) is enhanced by 23.4 dB for the second harmonic (2f1), and 29.1 and 27.6 dB for the second intermodulation (f2-f1 and f1 + f2), as compared with a conventional MPL. For a two-tone RF signal of f1 = 9.95 GHz and f2 = 10 GHz, the SFDR3 is increased by 13.1 dB as compared with a conventional MPL.

© 2016 Optical Society of America

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References

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2016 (1)

X. Y. Han, E. M. Xu, and J. P. Yao, “Tunable single bandpass microwave photonic filter with an improved dynamic range,” IEEE Photonics Technol. Lett. 28(1), 11–14 (2016).
[Crossref]

2015 (1)

X. Chen, W. Li, and J. P. Yao, “Dynamic range enhancement for a microwave photonic link based on a polarization modulator,” IEEE Trans. Microw. Theory Tech. 63(7), 2384–2389 (2015).
[Crossref]

2014 (2)

2013 (2)

X. Chen, W. Li, and J. P. Yao, “Microwave photonic link with improved dynamic range using a polarization modulator,” IEEE Photonics Technol. Lett. 25(14), 1373–1376 (2013).
[Crossref]

W. Li and J. Yao, “Dynamic range improvement of a microwave photonic link based on bi-directional use of a polarization modulator in a Sagnac loop,” Opt. Express 21(13), 15692–15697 (2013).
[Crossref] [PubMed]

2012 (3)

2009 (2)

2007 (1)

J. Capmany and D. Novak, “Microwave photonics combines two world,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

2006 (2)

A. Seeds and K. J. Williams, “Microwave photonics,” J. Lightwave Technol. 24(12), 4628–4641 (2006).
[Crossref]

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[Crossref]

2005 (1)

Q. Lin, Z. Z. Ying, and G. Wei, “Design of a feedback predistortion linear power amplifier,” Microw. J. 48(5), 232–241 (2005).

2004 (1)

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577(1), 133–143 (2004).
[Crossref]

1999 (1)

E. Ackerman, “Broad-band linearization of a Mach-Zehnder electrooptic modulator,” IEEE Trans. Microw. Theory Tech. 47(12), 2271–2279 (1999).
[Crossref]

1996 (1)

W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photonics Technol. Lett. 8(1), 130–132 (1996).
[Crossref]

1993 (1)

M. L. Farwell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 5(7), 779–782 (1993).
[Crossref]

1990 (1)

S. K. Korotky and R. M. DeRidder, “Dual parallel modulation schemes for low-distortion analog optical transmission,” IEEE J. Sel. Areas Comm. 8(7), 1377–1381 (1990).
[Crossref]

Ackerman, E.

E. Ackerman, “Broad-band linearization of a Mach-Zehnder electrooptic modulator,” IEEE Trans. Microw. Theory Tech. 47(12), 2271–2279 (1999).
[Crossref]

Attygalle, M.

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[Crossref]

Baynes, F. N.

Beling, A.

Boller, K.-J.

Bull, J. D.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577(1), 133–143 (2004).
[Crossref]

Burns, W. K.

W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photonics Technol. Lett. 8(1), 130–132 (1996).
[Crossref]

Campbell, J. C.

Capmany, J.

J. Capmany and D. Novak, “Microwave photonics combines two world,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Chang, W. S. C.

M. L. Farwell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 5(7), 779–782 (1993).
[Crossref]

Chen, X.

X. Chen, W. Li, and J. P. Yao, “Dynamic range enhancement for a microwave photonic link based on a polarization modulator,” IEEE Trans. Microw. Theory Tech. 63(7), 2384–2389 (2015).
[Crossref]

X. Chen, W. Li, and J. P. Yao, “Microwave photonic link with improved dynamic range using a polarization modulator,” IEEE Photonics Technol. Lett. 25(14), 1373–1376 (2013).
[Crossref]

Cundiff, S. T.

DeRidder, R. M.

S. K. Korotky and R. M. DeRidder, “Dual parallel modulation schemes for low-distortion analog optical transmission,” IEEE J. Sel. Areas Comm. 8(7), 1377–1381 (1990).
[Crossref]

Devgan, P. S.

Diddams, S. A.

Diehl, J. F.

Fairburn, M.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577(1), 133–143 (2004).
[Crossref]

Farwell, M. L.

M. L. Farwell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 5(7), 779–782 (1993).
[Crossref]

Fortier, T. M.

Fu, J.

Ghanipour, P.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577(1), 133–143 (2004).
[Crossref]

Gopalakrishnan, G. K.

W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photonics Technol. Lett. 8(1), 130–132 (1996).
[Crossref]

Han, X. Y.

X. Y. Han, E. M. Xu, and J. P. Yao, “Tunable single bandpass microwave photonic filter with an improved dynamic range,” IEEE Photonics Technol. Lett. 28(1), 11–14 (2016).
[Crossref]

Heideman, R. G.

Hoekman, M.

Hraimel, B.

B. Hraimel and X. Zhang, “Low-cost broadband predistortion-linearized single-drive x-cut Mach-Zehnder modulator for radio-over-fiber system,” IEEE Photonics Technol. Lett. 24(18), 1571–1573 (2012).
[Crossref]

Huang, M.

Huber, D. R.

M. L. Farwell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 5(7), 779–782 (1993).
[Crossref]

Hulzinga, A.

Jaeger, N. A. F.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577(1), 133–143 (2004).
[Crossref]

Kato, H.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577(1), 133–143 (2004).
[Crossref]

Korotky, S. K.

S. K. Korotky and R. M. DeRidder, “Dual parallel modulation schemes for low-distortion analog optical transmission,” IEEE J. Sel. Areas Comm. 8(7), 1377–1381 (1990).
[Crossref]

Leinse, A.

Li, K. J.

Li, W.

X. Chen, W. Li, and J. P. Yao, “Dynamic range enhancement for a microwave photonic link based on a polarization modulator,” IEEE Trans. Microw. Theory Tech. 63(7), 2384–2389 (2015).
[Crossref]

X. Chen, W. Li, and J. P. Yao, “Microwave photonic link with improved dynamic range using a polarization modulator,” IEEE Photonics Technol. Lett. 25(14), 1373–1376 (2013).
[Crossref]

W. Li and J. Yao, “Dynamic range improvement of a microwave photonic link based on bi-directional use of a polarization modulator in a Sagnac loop,” Opt. Express 21(13), 15692–15697 (2013).
[Crossref] [PubMed]

Lim, C.

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[Crossref]

Lin, Q.

Q. Lin, Z. Z. Ying, and G. Wei, “Design of a feedback predistortion linear power amplifier,” Microw. J. 48(5), 232–241 (2005).

Moeller, R. P.

W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photonics Technol. Lett. 8(1), 130–132 (1996).
[Crossref]

Nirmalathas, A.

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[Crossref]

Novak, D.

J. Capmany and D. Novak, “Microwave photonics combines two world,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[Crossref]

Oldenbeuving, R. M.

Pan, S.

Quinlan, F.

Reid, A.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577(1), 133–143 (2004).
[Crossref]

Roeloffzen, C. G. H.

Rouvalis, E.

Seeds, A.

Steffan, A. G.

Sunderman, C. E.

Taddei, C.

Urick, V. J.

van Dijk, P. W.

Verpoorte, J.

Waterhouse, R.

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[Crossref]

Wei, G.

Q. Lin, Z. Z. Ying, and G. Wei, “Design of a feedback predistortion linear power amplifier,” Microw. J. 48(5), 232–241 (2005).

Weiner, A. M.

Williams, K. J.

Willits, J. T.

Xie, X. J.

Xu, E. M.

X. Y. Han, E. M. Xu, and J. P. Yao, “Tunable single bandpass microwave photonic filter with an improved dynamic range,” IEEE Photonics Technol. Lett. 28(1), 11–14 (2016).
[Crossref]

Yao, J.

Yao, J. P.

X. Y. Han, E. M. Xu, and J. P. Yao, “Tunable single bandpass microwave photonic filter with an improved dynamic range,” IEEE Photonics Technol. Lett. 28(1), 11–14 (2016).
[Crossref]

X. Chen, W. Li, and J. P. Yao, “Dynamic range enhancement for a microwave photonic link based on a polarization modulator,” IEEE Trans. Microw. Theory Tech. 63(7), 2384–2389 (2015).
[Crossref]

X. Chen, W. Li, and J. P. Yao, “Microwave photonic link with improved dynamic range using a polarization modulator,” IEEE Photonics Technol. Lett. 25(14), 1373–1376 (2013).
[Crossref]

Ying, Z. Z.

Q. Lin, Z. Z. Ying, and G. Wei, “Design of a feedback predistortion linear power amplifier,” Microw. J. 48(5), 232–241 (2005).

Zhang, X.

B. Hraimel and X. Zhang, “Low-cost broadband predistortion-linearized single-drive x-cut Mach-Zehnder modulator for radio-over-fiber system,” IEEE Photonics Technol. Lett. 24(18), 1571–1573 (2012).
[Crossref]

Zhou, Q. G.

Zhuang, L.

IEEE J. Sel. Areas Comm. (1)

S. K. Korotky and R. M. DeRidder, “Dual parallel modulation schemes for low-distortion analog optical transmission,” IEEE J. Sel. Areas Comm. 8(7), 1377–1381 (1990).
[Crossref]

IEEE Photonics Technol. Lett. (5)

B. Hraimel and X. Zhang, “Low-cost broadband predistortion-linearized single-drive x-cut Mach-Zehnder modulator for radio-over-fiber system,” IEEE Photonics Technol. Lett. 24(18), 1571–1573 (2012).
[Crossref]

M. L. Farwell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 5(7), 779–782 (1993).
[Crossref]

X. Chen, W. Li, and J. P. Yao, “Microwave photonic link with improved dynamic range using a polarization modulator,” IEEE Photonics Technol. Lett. 25(14), 1373–1376 (2013).
[Crossref]

W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photonics Technol. Lett. 8(1), 130–132 (1996).
[Crossref]

X. Y. Han, E. M. Xu, and J. P. Yao, “Tunable single bandpass microwave photonic filter with an improved dynamic range,” IEEE Photonics Technol. Lett. 28(1), 11–14 (2016).
[Crossref]

IEEE Trans. Microw. Theory Tech. (3)

X. Chen, W. Li, and J. P. Yao, “Dynamic range enhancement for a microwave photonic link based on a polarization modulator,” IEEE Trans. Microw. Theory Tech. 63(7), 2384–2389 (2015).
[Crossref]

E. Ackerman, “Broad-band linearization of a Mach-Zehnder electrooptic modulator,” IEEE Trans. Microw. Theory Tech. 47(12), 2271–2279 (1999).
[Crossref]

C. Lim, M. Attygalle, A. Nirmalathas, D. Novak, and R. Waterhouse, “Analysis of optical carrier-to-sideband ratio for improving transmission performance in fiber-radio links,” IEEE Trans. Microw. Theory Tech. 54(5), 2181–2187 (2006).
[Crossref]

J. Lightwave Technol. (3)

Microw. J. (1)

Q. Lin, Z. Z. Ying, and G. Wei, “Design of a feedback predistortion linear power amplifier,” Microw. J. 48(5), 232–241 (2005).

Nat. Photonics (1)

J. Capmany and D. Novak, “Microwave photonics combines two world,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Proc. SPIE (1)

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577(1), 133–143 (2004).
[Crossref]

Other (5)

J. Dai, K. Xu, R. M. Duan, Y. Cui, J. Wu, and J. T. Lin, “Optical linearization for intensity-modulated analog links employing equivalent incoherent combination technique,” 2011 the IEEE International Topical Meeting on Microwave Photonics (MWP 11’), Singapore, 230–233 (2011).
[Crossref]

Y. Cui, K. Xu, Y. T. Dai, and J. L. Lin, “Suppression of second-order harmonic distortion in ROF links utilizing dual-output MZM and balanced detection,” 2012 the IEEE International Topical Meeting on Microwave Photonics (MWP 12’), Noordwijk, Netherland, 103–106 (2012).

Y. Cui, Y. T. Dai, K. Xu, F. F. Yin, and J. T. Lin, “Multi-octave operation of analogy optical link using parallel intensity modulators,” 2013 12th International Conference on Optical Communications & Networks (ICOCN 13’), Chengdu, China, 1–4 (2013).

V. J. Urick, K. J. Williams, and J. D. McKinney, Fundamentals of Microwave Photonics (John Wiley & Sons, Inc., Hoboken, 2015).

D. Marpaung, “High dynamic range analog photonic links design and implementation,” PhD thesis, University of Twente, Netherland (2009).

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Figures (8)

Fig. 1
Fig. 1 The schematic of the proposed MPL. TLS: tunable laser source, PolM: polarization modulator, OBPF: optical bandpass filter, EDFA: erbium-doped fiber amplifier, PC: polarization controller, PBS: polarization beam splitter, PD: photodetector, BPD: balanced photodetector.
Fig. 2
Fig. 2 The spectral components of the orthogonally polarized phase-modulated signals at the output from the OBPF. (a) x polarization and (b) y polarization. The solid and dash arc lines denote the generation of the positive and negative IMD3 components, respectively, during the optical to electrical conversion at the PDs in the BPD.
Fig. 3
Fig. 3 Electrical spectra of the detected RF signals when a two-tone RF signal at 10 and 19.95 GHz is applied. (a) Conventional MPL; (b) proposed MPL; (c) the comparison of the IMD2 (f1 + f2)
Fig. 4
Fig. 4 Zoom-in view of the electrical spectra of the detected RF signals when a two-tone RF signal f1 = 10 GHz and f2 = 19.95 GHz is applied. (a) At 10 GHz and (b) at 19.95 GHz for the conventional MPL; (c) at 10 GHz and (d) at 19.95 GHz for the proposed MPL.
Fig. 5
Fig. 5 The measured powers of the fundamental and the even-order components when a two-tone RF signal of f1 = 10 GHz and f2 = 19.95 GHz is applied. (a) SHD (2f1), (b) IMD2 (f2-f1) and (c) IMD2 (f1 + f2) for the conventional MPL; (d) SHD (2f1), (e) IMD2 (f2-f1) and (f) IMD2 (f1 + f2) for the proposed MPL.
Fig. 6
Fig. 6 Measured optical spectra of a phase-modulated modulated signal (a) before and (b) after the OBPF.
Fig. 7
Fig. 7 Electrical spectra of the detected microwave signals when a two-tone RF signal of f1 = 9.95 GHz and f2 = 10 GHz is applied to the PolM. (a) A conventional MPL, and (b) an optimized MPL.
Fig. 8
Fig. 8 Measured microwave powers of the fundamental and the IMD3 terms when a two-tone signal of f1 = 9.95 GHz and f2 = 10 GHz is applied.

Tables (2)

Tables Icon

Table 1 Fundamental components ω1 and ω2 of the output microwave signals from the BPD

Tables Icon

Table 2 IMD3 components (2ω1-ω2) and (2ω2-ω1) of the output microwave signals from the BPD

Equations (16)

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

E o u t , P o l M , x ( t ) = E 0 2 e j ω c t e j ( m 1 cos ω 1 t + m 2 cos ω 2 t )
E o u t , P o l M , y ( t ) = E 0 2 e j ω c t e j ( m 1 cos ω 1 t + m 2 cos ω 2 t )
E o u t , P o l M , x ( t ) = E 0 2 e j ω c t k = + l = + J k ( m 1 ) J l ( m 2 ) e j [ k ω 1 t + l ω 2 t + ( k + l ) π 2 ]
E o u t , P o l M , y ( t ) = E 0 2 e j ω c t k = + l = + J k ( m 1 ) J l ( m 2 ) e j [ k ω 1 t + l ω 2 t + ( k + l ) 3 π 2 ]
E o u t , O B P F , x ( t ) = E 0 2 e j ω c t k = + l = + J k ( m 1 ) J l ( m 2 ) R ( ω c + k ω 1 + l ω 2 ) e j [ k ω 1 t + l ω 2 t + ( k + l ) π 2 ]
E o u t , O B P F , y ( t ) = E 0 2 e j ω c t k = + l = + J k ( m 1 ) J l ( m 2 ) R ( ω c + k ω 1 + l ω 2 ) e j [ k ω 1 t + l ω 2 t + ( k + l ) 3 π 2 ]
E o u t , P B S , x ( t ) = E 0 2 e j ω c t k = + l = + J k ( m 1 ) J l ( m 2 ) R ( ω c + k ω 1 + l ω 2 ) G e j [ k ω 1 t + l ω 2 t + ( k + l ) π 2 ]
E o u t , P B S , y ( t ) = E 0 2 e j ω c t k = + l = + J k ( m 1 ) J l ( m 2 ) R ( ω c + k ω 1 + l ω 2 ) G e j [ k ω 1 t + l ω 2 t + ( k + l ) 3 π 2 ]
i 1 ( t ) = E o u t , P B S , x ( t ) E * o u t , P B S , x ( t ) = | E 0 | 2 2 k = + l = + p = + q = + { J k ( m 1 ) J l ( m 2 ) J p ( m 1 ) J q ( m 2 ) R ( ω c + k ω 1 + l ω 2 ) R ( ω c + p ω 1 + q ω 2 ) G e j [ ( k p ) ω 1 t + ( l q ) ω 2 t + ( k + l p q ) π 2 ] } = | E 0 | 2 2 k = + l = + p = + q = + { J k ( m 1 ) J l ( m 2 ) J p ( m 1 ) J q ( m 2 ) R ( ω c + k ω 1 + l ω 2 ) R ( ω c + p ω 1 + q ω 2 ) G e j [ ( k p ) ω 1 t + ( l q ) ω 2 t ] { ( 1 ) N , k p + l q = 2 N ( 1 ) N e j π 2 , k p + l q = 2 N + 1 } k + l 0 , p + q 0 , N = 0 ± 1 , ± 2 ,
i 2 ( t ) = E o u t , P B S , y ( t ) E * o u t , P B S , y ( t ) = | E 0 | 2 2 k = + l = + p = + q = + { J k ( m 1 ) J l ( m 2 ) J p ( m 1 ) J q ( m 2 ) R ( ω c + k ω 1 + l ω 2 ) R ( ω c + p ω 1 + q ω 2 ) G e j [ ( k p ) ω 1 t + ( l q ) ω 2 t + ( k + l p q ) 3 π 2 ] } = | E 0 | 2 2 k = + l = + p = + q = + { J k ( m 1 ) J l ( m 2 ) J p ( m 1 ) J q ( m 2 ) R ( ω c + k ω 1 + l ω 2 ) R ( ω c + p ω 1 + q ω 2 ) G e j [ ( k p ) ω 1 t + ( l q ) ω 2 t ] { ( 1 ) N , k p + l q = 2 N ( 1 ) N e j π 2 , k p + l q = 2 N + 1 } k + l 0 , p + q 0 , N = 0 ± 1 , ± 2 ,
i ( t ) = i 1 ( t ) i 2 ( t ) = { 0 , k p + l q = 2 N | E 0 | 2 k = + l = + p = + q = + { J k ( m 1 ) J l ( m 2 ) J p ( m 1 ) J q ( m 2 ) R ( ω c + k ω 1 + l ω 2 ) R ( ω c + p ω 1 + q ω 2 ) G e j [ ( k p ) ω 1 t + ( l q ) ω 2 t ] ( 1 ) N e j π 2 } , k p + l q = 2 N + 1 k + l 0 , p + q 0 , N = 0 ± 1 , ± 2 ,
k p + l q = 2 N + 1 , N = 0 , ± 1 , ± 2
C 2 ω 1 ω 2 = 2 J 12 J 02 R 1 [ R 2 J 21 J 01 R 0 J 21 J 01 R 0 J 11 J 11 ] + 2 R 2 J 12 J 22 [ R 3 J 21 J 01 R 1 J 11 J 11 ] + 2 R 1 R 0 J 01 J 21 J 12 J 22
C 2 ω 2 ω 1 = 2 J 11 J 01 R 1 [ R 2 J 22 J 02 R 0 J 22 J 02 R 0 J 12 J 12 ] + 2 R 2 J 11 J 21 [ R 3 J 22 J 02 R 1 J 12 J 12 ] + 2 R 1 R 0 J 11 J 21 J 02 J 22
C 2 ω 1 ω 2 = C 2 ω 2 ω 1 = m 3 8 R 1 [ R 2 ( 1 m 2 4 + m 2 8 R 3 R 1 ) R 0 ( 3 m 2 8 ) ]
α = R 2 R 0 3 2 = 9 9.54 dB

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