Abstract

In this work, we propose and investigate a novel technique for the generation of millimeter-wave (mm-wave), i.e. frequency sixuplexing technique. The proposed technique is comprised of two cascaded Mach-Zehnder modulators (MZMs). The first MZM, biased at maximum transmission, is only used for even-order optical harmonic generation, and then a second MZM, biased at minimum transmission, is used for both optical carrier suppression modulation and data signal modulation. As an example, we consider an RF at 10 GHz, which carries the data signal and drives the MZMs; and an mm-wave signal at 60 GHz, i.e. a frequency sixupler, is obtained. It is found that our proposed sixupler leads to an 8-dB higher RF power at 60 GHz and a 6-dB improvement in receiver sensitivity with comparison to the conventional technique, i.e. optical carrier suppression modulation. The generated mm-wave signal is robust to fiber chromatic dispersion. The proposed technique is verified by experiments.

© 2008 Optical Society of America

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  1. B. Masella and X. Zhang, “A novel single wavelength balanced system for radio over fiber links,” IEEE Photon. Technol. Lett. 18, 301–303 (2006).
    [Crossref]
  2. C. Wu and X. Zhang, “Impact of nonlinear distortion in radio over fiber systems with single-side band and tandem single-side band subcarrier modulations,” J. Lightwave Technol. 24, 2076–2090 (2006).
    [Crossref]
  3. G. Qi, J. Yao, J. Seregelyi, C. Bélisle, and S. Paquet, “Generation and distribution of a wideband continuously tunable mm-wave signal with an optical external modulation technique,” IEEE Trans. Microwave Theory Tech. 53, 3090–3097 (2005).
    [Crossref]
  4. J. Yu, Z. Jia, L. Yi, Y. Su, and G. Chang, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18, 256–267 (2006).
  5. T. Nakasyotani, H. Toda, T. Kuri, and K. Kitayama, “Wavelength-division-multiplexed Millimeter-waveband radio-on-fiber system using a supercontinuum light source,” J. Lightwave Technol. 24, 404–410 (2006).
    [Crossref]
  6. C. Lin, W. Peng, P. Peng, J. Chen, C. Peng, B. Chiou, and S. Chi, “Simultaneous generation of baseband and radio signals using only one single-electrode Mach-Zehnder modulator with enhanced linearity,” IEEE Photon. Technol. Lett. 18, 2481–2483 (2006).
    [Crossref]
  7. M. Mohamed, X. Zhang, B. Hraimel, and K. Wu “Efficient photonic generation of millimeter-waves using optical frequency multiplication in radio-over-fiber systems,” Proceeding of IEEE Topic meeting on Microwave Photonics 2007, paper Th.-4.20, Victoria, Canada.
  8. M. Larrode, A. Koonen, J. Vegas, and A. NgOma, “Bidirectional radio-over-fiber link employing optical frequency multiplication,” IEEE Photon. Technol. Lett. 18, 241–243 (2006).
    [Crossref]
  9. J. Zhang, H. Chen, M. Chen, T. Wang, and S. Xie “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression” IEEE Photon. Technol. Lett. 19, 1057–1059 (2007).
    [Crossref]
  10. B. Hraimel, M. Twati, and Ke Wu “Closed-form dynamic range expression of dual-electrode Mach-Zehnder modulator in radio-over-fiber WDM system” J. Lightwave Technol. 24, 2380–2387 (2006).
    [Crossref]
  11. T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Optics Lett. 32, 515–1517 (2007).
    [Crossref]

2007 (2)

J. Zhang, H. Chen, M. Chen, T. Wang, and S. Xie “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression” IEEE Photon. Technol. Lett. 19, 1057–1059 (2007).
[Crossref]

T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Optics Lett. 32, 515–1517 (2007).
[Crossref]

2006 (7)

B. Hraimel, M. Twati, and Ke Wu “Closed-form dynamic range expression of dual-electrode Mach-Zehnder modulator in radio-over-fiber WDM system” J. Lightwave Technol. 24, 2380–2387 (2006).
[Crossref]

B. Masella and X. Zhang, “A novel single wavelength balanced system for radio over fiber links,” IEEE Photon. Technol. Lett. 18, 301–303 (2006).
[Crossref]

C. Wu and X. Zhang, “Impact of nonlinear distortion in radio over fiber systems with single-side band and tandem single-side band subcarrier modulations,” J. Lightwave Technol. 24, 2076–2090 (2006).
[Crossref]

J. Yu, Z. Jia, L. Yi, Y. Su, and G. Chang, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18, 256–267 (2006).

T. Nakasyotani, H. Toda, T. Kuri, and K. Kitayama, “Wavelength-division-multiplexed Millimeter-waveband radio-on-fiber system using a supercontinuum light source,” J. Lightwave Technol. 24, 404–410 (2006).
[Crossref]

C. Lin, W. Peng, P. Peng, J. Chen, C. Peng, B. Chiou, and S. Chi, “Simultaneous generation of baseband and radio signals using only one single-electrode Mach-Zehnder modulator with enhanced linearity,” IEEE Photon. Technol. Lett. 18, 2481–2483 (2006).
[Crossref]

M. Larrode, A. Koonen, J. Vegas, and A. NgOma, “Bidirectional radio-over-fiber link employing optical frequency multiplication,” IEEE Photon. Technol. Lett. 18, 241–243 (2006).
[Crossref]

2005 (1)

G. Qi, J. Yao, J. Seregelyi, C. Bélisle, and S. Paquet, “Generation and distribution of a wideband continuously tunable mm-wave signal with an optical external modulation technique,” IEEE Trans. Microwave Theory Tech. 53, 3090–3097 (2005).
[Crossref]

Bélisle, C.

G. Qi, J. Yao, J. Seregelyi, C. Bélisle, and S. Paquet, “Generation and distribution of a wideband continuously tunable mm-wave signal with an optical external modulation technique,” IEEE Trans. Microwave Theory Tech. 53, 3090–3097 (2005).
[Crossref]

Chang, G.

J. Yu, Z. Jia, L. Yi, Y. Su, and G. Chang, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18, 256–267 (2006).

Chen, H.

J. Zhang, H. Chen, M. Chen, T. Wang, and S. Xie “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression” IEEE Photon. Technol. Lett. 19, 1057–1059 (2007).
[Crossref]

Chen, J.

C. Lin, W. Peng, P. Peng, J. Chen, C. Peng, B. Chiou, and S. Chi, “Simultaneous generation of baseband and radio signals using only one single-electrode Mach-Zehnder modulator with enhanced linearity,” IEEE Photon. Technol. Lett. 18, 2481–2483 (2006).
[Crossref]

Chen, M.

J. Zhang, H. Chen, M. Chen, T. Wang, and S. Xie “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression” IEEE Photon. Technol. Lett. 19, 1057–1059 (2007).
[Crossref]

Chi, S.

C. Lin, W. Peng, P. Peng, J. Chen, C. Peng, B. Chiou, and S. Chi, “Simultaneous generation of baseband and radio signals using only one single-electrode Mach-Zehnder modulator with enhanced linearity,” IEEE Photon. Technol. Lett. 18, 2481–2483 (2006).
[Crossref]

Chiou, B.

C. Lin, W. Peng, P. Peng, J. Chen, C. Peng, B. Chiou, and S. Chi, “Simultaneous generation of baseband and radio signals using only one single-electrode Mach-Zehnder modulator with enhanced linearity,” IEEE Photon. Technol. Lett. 18, 2481–2483 (2006).
[Crossref]

Hraimel, B.

B. Hraimel, M. Twati, and Ke Wu “Closed-form dynamic range expression of dual-electrode Mach-Zehnder modulator in radio-over-fiber WDM system” J. Lightwave Technol. 24, 2380–2387 (2006).
[Crossref]

M. Mohamed, X. Zhang, B. Hraimel, and K. Wu “Efficient photonic generation of millimeter-waves using optical frequency multiplication in radio-over-fiber systems,” Proceeding of IEEE Topic meeting on Microwave Photonics 2007, paper Th.-4.20, Victoria, Canada.

Izutsu, M.

T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Optics Lett. 32, 515–1517 (2007).
[Crossref]

Jia, Z.

J. Yu, Z. Jia, L. Yi, Y. Su, and G. Chang, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18, 256–267 (2006).

Kawanishi, T.

T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Optics Lett. 32, 515–1517 (2007).
[Crossref]

Kitayama, K.

Koonen, A.

M. Larrode, A. Koonen, J. Vegas, and A. NgOma, “Bidirectional radio-over-fiber link employing optical frequency multiplication,” IEEE Photon. Technol. Lett. 18, 241–243 (2006).
[Crossref]

Kuri, T.

Larrode, M.

M. Larrode, A. Koonen, J. Vegas, and A. NgOma, “Bidirectional radio-over-fiber link employing optical frequency multiplication,” IEEE Photon. Technol. Lett. 18, 241–243 (2006).
[Crossref]

Lin, C.

C. Lin, W. Peng, P. Peng, J. Chen, C. Peng, B. Chiou, and S. Chi, “Simultaneous generation of baseband and radio signals using only one single-electrode Mach-Zehnder modulator with enhanced linearity,” IEEE Photon. Technol. Lett. 18, 2481–2483 (2006).
[Crossref]

Masella, B.

B. Masella and X. Zhang, “A novel single wavelength balanced system for radio over fiber links,” IEEE Photon. Technol. Lett. 18, 301–303 (2006).
[Crossref]

Mohamed, M.

M. Mohamed, X. Zhang, B. Hraimel, and K. Wu “Efficient photonic generation of millimeter-waves using optical frequency multiplication in radio-over-fiber systems,” Proceeding of IEEE Topic meeting on Microwave Photonics 2007, paper Th.-4.20, Victoria, Canada.

Nakasyotani, T.

NgOma, A.

M. Larrode, A. Koonen, J. Vegas, and A. NgOma, “Bidirectional radio-over-fiber link employing optical frequency multiplication,” IEEE Photon. Technol. Lett. 18, 241–243 (2006).
[Crossref]

Paquet, S.

G. Qi, J. Yao, J. Seregelyi, C. Bélisle, and S. Paquet, “Generation and distribution of a wideband continuously tunable mm-wave signal with an optical external modulation technique,” IEEE Trans. Microwave Theory Tech. 53, 3090–3097 (2005).
[Crossref]

Peng, C.

C. Lin, W. Peng, P. Peng, J. Chen, C. Peng, B. Chiou, and S. Chi, “Simultaneous generation of baseband and radio signals using only one single-electrode Mach-Zehnder modulator with enhanced linearity,” IEEE Photon. Technol. Lett. 18, 2481–2483 (2006).
[Crossref]

Peng, P.

C. Lin, W. Peng, P. Peng, J. Chen, C. Peng, B. Chiou, and S. Chi, “Simultaneous generation of baseband and radio signals using only one single-electrode Mach-Zehnder modulator with enhanced linearity,” IEEE Photon. Technol. Lett. 18, 2481–2483 (2006).
[Crossref]

Peng, W.

C. Lin, W. Peng, P. Peng, J. Chen, C. Peng, B. Chiou, and S. Chi, “Simultaneous generation of baseband and radio signals using only one single-electrode Mach-Zehnder modulator with enhanced linearity,” IEEE Photon. Technol. Lett. 18, 2481–2483 (2006).
[Crossref]

Qi, G.

G. Qi, J. Yao, J. Seregelyi, C. Bélisle, and S. Paquet, “Generation and distribution of a wideband continuously tunable mm-wave signal with an optical external modulation technique,” IEEE Trans. Microwave Theory Tech. 53, 3090–3097 (2005).
[Crossref]

Sakamoto, T.

T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Optics Lett. 32, 515–1517 (2007).
[Crossref]

Seregelyi, J.

G. Qi, J. Yao, J. Seregelyi, C. Bélisle, and S. Paquet, “Generation and distribution of a wideband continuously tunable mm-wave signal with an optical external modulation technique,” IEEE Trans. Microwave Theory Tech. 53, 3090–3097 (2005).
[Crossref]

Su, Y.

J. Yu, Z. Jia, L. Yi, Y. Su, and G. Chang, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18, 256–267 (2006).

Toda, H.

Twati, M.

Vegas, J.

M. Larrode, A. Koonen, J. Vegas, and A. NgOma, “Bidirectional radio-over-fiber link employing optical frequency multiplication,” IEEE Photon. Technol. Lett. 18, 241–243 (2006).
[Crossref]

Wang, T.

J. Zhang, H. Chen, M. Chen, T. Wang, and S. Xie “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression” IEEE Photon. Technol. Lett. 19, 1057–1059 (2007).
[Crossref]

Wu, C.

Wu, K.

M. Mohamed, X. Zhang, B. Hraimel, and K. Wu “Efficient photonic generation of millimeter-waves using optical frequency multiplication in radio-over-fiber systems,” Proceeding of IEEE Topic meeting on Microwave Photonics 2007, paper Th.-4.20, Victoria, Canada.

Wu, Ke

Xie, S.

J. Zhang, H. Chen, M. Chen, T. Wang, and S. Xie “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression” IEEE Photon. Technol. Lett. 19, 1057–1059 (2007).
[Crossref]

Yao, J.

G. Qi, J. Yao, J. Seregelyi, C. Bélisle, and S. Paquet, “Generation and distribution of a wideband continuously tunable mm-wave signal with an optical external modulation technique,” IEEE Trans. Microwave Theory Tech. 53, 3090–3097 (2005).
[Crossref]

Yi, L.

J. Yu, Z. Jia, L. Yi, Y. Su, and G. Chang, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18, 256–267 (2006).

Yu, J.

J. Yu, Z. Jia, L. Yi, Y. Su, and G. Chang, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18, 256–267 (2006).

Zhang, J.

J. Zhang, H. Chen, M. Chen, T. Wang, and S. Xie “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression” IEEE Photon. Technol. Lett. 19, 1057–1059 (2007).
[Crossref]

Zhang, X.

B. Masella and X. Zhang, “A novel single wavelength balanced system for radio over fiber links,” IEEE Photon. Technol. Lett. 18, 301–303 (2006).
[Crossref]

C. Wu and X. Zhang, “Impact of nonlinear distortion in radio over fiber systems with single-side band and tandem single-side band subcarrier modulations,” J. Lightwave Technol. 24, 2076–2090 (2006).
[Crossref]

M. Mohamed, X. Zhang, B. Hraimel, and K. Wu “Efficient photonic generation of millimeter-waves using optical frequency multiplication in radio-over-fiber systems,” Proceeding of IEEE Topic meeting on Microwave Photonics 2007, paper Th.-4.20, Victoria, Canada.

IEEE Photon. Technol. Lett. (5)

B. Masella and X. Zhang, “A novel single wavelength balanced system for radio over fiber links,” IEEE Photon. Technol. Lett. 18, 301–303 (2006).
[Crossref]

J. Yu, Z. Jia, L. Yi, Y. Su, and G. Chang, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18, 256–267 (2006).

M. Larrode, A. Koonen, J. Vegas, and A. NgOma, “Bidirectional radio-over-fiber link employing optical frequency multiplication,” IEEE Photon. Technol. Lett. 18, 241–243 (2006).
[Crossref]

J. Zhang, H. Chen, M. Chen, T. Wang, and S. Xie “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression” IEEE Photon. Technol. Lett. 19, 1057–1059 (2007).
[Crossref]

C. Lin, W. Peng, P. Peng, J. Chen, C. Peng, B. Chiou, and S. Chi, “Simultaneous generation of baseband and radio signals using only one single-electrode Mach-Zehnder modulator with enhanced linearity,” IEEE Photon. Technol. Lett. 18, 2481–2483 (2006).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

G. Qi, J. Yao, J. Seregelyi, C. Bélisle, and S. Paquet, “Generation and distribution of a wideband continuously tunable mm-wave signal with an optical external modulation technique,” IEEE Trans. Microwave Theory Tech. 53, 3090–3097 (2005).
[Crossref]

J. Lightwave Technol. (3)

Optics Lett. (1)

T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Optics Lett. 32, 515–1517 (2007).
[Crossref]

Other (1)

M. Mohamed, X. Zhang, B. Hraimel, and K. Wu “Efficient photonic generation of millimeter-waves using optical frequency multiplication in radio-over-fiber systems,” Proceeding of IEEE Topic meeting on Microwave Photonics 2007, paper Th.-4.20, Victoria, Canada.

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Figures (9)

Fig. 1.
Fig. 1.

System layout where the proposed modulation technique, i.e. a frequency sixupler, is used for the mm-wave generation using two dual-electrode-MZMs. EDFA: erbium-doped fiber amplifier, ESA: electrical spectrum analyzer, SSMF standard single- mode fiber, MZM: dual-electrode MZM, OBF: optical bandpass filter, EBPF: electrical bandpass filter, ELPF: electrical low passband filter, PD: photodiode. Insets (i), (ii) and (iii) indicate optical spectra.

Fig. 2.
Fig. 2.

Simulated Q-factor vs. fiber length. mRF =70% is used.

Fig. 3.
Fig. 3.

(a) simulated Q-factor vs. RF modulation index and (b) simulated mm-wave power vs. RF modulation index.

Fig. 4.
Fig. 4.

Electrical spectra of the generated mm-wave signal at 60 GHz using the sixupler at the transmission distance of (a) 0-km and (b) 50-km. mRF=60% is used.

Fig. 5.
Fig. 5.

BER vs. received optical power using the both proposed and conventional techniques for the generation of optical mm-wave signals at 60 GHz.

Fig. 6.
Fig. 6.

(a) simulated Q-factor vs. local oscillator RF and (b) simulated mm-wave power vs. local oscillator RF. mRF =70% is used. A fiber length of 0 and 25 km is considered, respectively.

Fig. 7.
Fig. 7.

Simulated Q-factor versus fiber length for the six combinations of the bias conditions, i.e. (a) QTB/MITB, QTB/MATB and MITB/MATB, and (b) for MATB/MATB, MITB/MITB, and QTB/QTB.

Fig. 8.
Fig. 8.

Measured optical spectra for using (a) proposed technique and (b) conventional technique.

Fig. 9.
Fig. 9.

Simulated optical spectra for using (a) proposed technique and (b) conventional technique. VRF=1.26 V is used.

Equations (18)

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E s ( t ) = t ff 1 P 2 ( 1 + γ 1 2 ) n = ( 1 ) n + 1 j n J n ( π m RF 2 ) ( γ 1 + ( 1 ) n ) e j ( n ω RF + ω 0 ) t
E o ( t ) = t ff 1 t ff 2 P 4 ( 1 + γ 1 2 ) ( 1 + γ 2 2 )
× n = 2 2 j n J n ( ψ ) ( 1 ) n + 1 ( γ 1 + ( 1 ) n ) e j ( ω 0 + n ω RF ) t k = j k J k ( ζ ) ( γ 2 + ( 1 ) k + 1 ) e j k ω RF t
as E out ( t ) = j 2 d ( t ) t ff 1 t ff 2 P ( 1 + γ 1 2 ) ( 1 + γ 2 2 ) × [ A 1 e j ( ω RF t + ϕ RF ) + A 3 e j 3 ( ω RF t + ϕ RF ) + c . c . ] e j ω 0 t
A 1 = H F ( ω RF ) × { ( γ 1 + 1 ) ( γ 2 + 1 ) J 2 ( ψ ) [ J 1 ( ψ ) J 3 ( ψ ) ] ( γ 1 1 ) ( γ 2 1 ) J 1 ( ψ ) J 2 ( ψ ) ( γ 2 + 1 ) J 1 ( ψ ) [ ( γ 1 + 1 ) J 0 ( ψ ) + ( γ 1 1 ) J 1 ( ψ ) ] }
A 3 = { ( γ 1 + 1 ) ( γ 2 + 1 ) [ J 0 ( ψ ) J 3 ( ψ ) + J 2 ( ψ ) J 1 ( ψ ) ] ( γ 1 1 ) ( γ 2 1 ) J 1 ( ψ ) J 2 ( ψ ) } H F ( 3 ω RF )
S ( t ) = j 2 t ff 1 t ff 2 PG e α L ( 1 + γ 1 2 ) ( 1 + γ 2 2 ) e j ω 0 t
× { ( A 1 d ( t + 1 6 Δ τ ) e j [ ω RF ( t + β 1 L ) + ϕ RF ] + A 1 * d ( t 1 6 Δ τ ) e j [ ω RF ( t + β 1 L ) + ϕ RF ] ) e j 1 2 β 2 L ω RF 2 + ( A 3 d ( t + 1 2 Δ τ ) e j 3 [ ω RF ( t + β 1 L ) + ϕ RF ] + A 3 * d ( t 1 2 Δ τ ) e j 3 [ ω RF ( t + β 1 L ) + ϕ RF ] ) e j 1 2 β 2 L ( 3 ω RF ) 2 }
I 6 ω RF ( d ( t ) ) t ff 1 t ff 2 P Ge α L 2 ( 1 + γ 1 2 ) ( 1 + γ 2 2 ) A 3 2 d ( t 1 2 Δ τ ) d ( t + 1 2 Δ τ ) cos ( 6 ω RF ( t + Δ τ ) )
I 6 ω RF ( d ( t ) ) t ff 1 t ff 2 P G e α L 2 ( 1 + γ 1 2 ) ( 1 + γ 2 2 ) A 3 d ( t ) 2 cos ( 6 ω RF t )
P ¯ 3 P ¯ 1 = ( H F ( 3 ω RF ) H F ( ω RF ) ) 2
× [ ε J 0 ( ψ ) J 3 ( ψ ) + ( ε 1 ) J 1 ( ψ ) J 2 ( ψ ) ] 2 [ J 2 ( ψ ) ( ( ε 1 ) J 1 ( ψ ) ε J 3 ( ψ ) ) ε J 0 ( ψ ) J 1 ( ψ ) + ε J 1 2 ( ψ ) ] 2 35.7 dB
E s ( t ) = 1 2 t ff 1 P ( 1 + γ 1 2 ) e j ω 0 t n = j n ( γ 1 + ( 1 ) n + 1 ) J n ( π d ( t ) m RF ) e j n ω RF t
S ( t ) = j ( 1 + γ 1 ) P G t ff 1 e α L 2 ( 1 + γ 1 2 ) e j ω 0 t
× { J 1 ( π m RF ) ( d ( t + 1 6 Δ τ ) H F ( ω RF ) e j [ ω RF ( t + β 1 L ) + ϕ RF ] + d ( t 1 6 Δ τ ) H F * ( ω RF ) e j [ ω RF ( t + β 1 L ) + ϕ RF ] ) e j 1 2 β 2 L ( ω RF ) 2 J 3 ( π m RF ) ( d ( t + 1 2 Δ τ ) H F ( 3 ω RF ) e j 3 [ ω RF ( t + β 1 L ) + ϕ RF ] + d ( t 1 2 Δ τ ) H F * ( 3 ω RF ) e j 3 [ ω RF ( t + β 1 L ) + ϕ RF ] ) e j 1 2 β 2 L ( 3 ω RF ) 2 }
I 6 ω RF ( d ( t ) ) ( 1 + γ 1 ) 2 ( 1 + γ 1 2 ) t ff 1 G P e α L d ( t 1 2 Δ τ ) d ( t + 1 2 Δ τ ) J 3 2 ( π m RF )
× H F ( 3 ω RF ) 2 cos [ 6 ω RF ( t + Δ τ ) ]
P ¯ 3 P ¯ 1 J 3 2 ( π m RF ) J 1 2 ( π m RF ) 5.2 dB

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