Abstract

It is well-known that the closed-loop operation in optical gyros offers wider dynamic range and better linearity. By adding a stair-like digital serrodyne wave to a phase modulator can be used as a frequency shifter. The width of one stair in this stair-like digital serrodyne wave should be set equal to the optical transmission time in the resonator, which is relaxed in the hybrid digital phase modulation (HDPM) scheme. The physical mechanism for this relaxation is firstly indicated in this paper. Detailed theoretical and experimental investigations are presented for the HDPM. Simulation and experimental results show that the width of one stair is not restricted by the optical transmission time, however, it should be optimized according to the rise time of the output of the digital-to-analogue converter. Based on the optimum parameters of the HDPM, a bias stability of 0.05°/s for the integration time of 400 seconds in 1 h has been carried out in an RMOG with a waveguide ring resonator with a length of 7.9 cm and a diameter of 2.5 cm.

© 2015 Optical Society of America

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References

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  1. S. Ezekiel and S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett. 30(9), 478–480 (1977).
    [Crossref]
  2. C. Ciminelli, F. Dell Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
    [Crossref]
  3. C. Monovoukas, A. K. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).
  4. C. Caterina, P. Francesco, and M. N. Armenise, “A new integrated optical angular velocity sensor,” Proc. SPIE 5728, 93–100 (2005).
  5. K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator micro optic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000).
    [Crossref]
  6. H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
    [Crossref]
  7. H. Ma, Z. He, and K. Hotate, “Reduction of backscattering induced noise by carrier suppression in waveguide-type optical ring resonator gyro,” J. Lightwave Technol. 29(1), 85–90 (2011).
    [Crossref]
  8. C. Ciminelli, F. Dell’Olio, M. N. Armenise, F. M. Soares, and W. Passenberg, “High performance InP ring resonator for new generation monolithically integrated optical Gyroscopes,” Opt. Express 21(1), 556–564 (2013).
    [Crossref] [PubMed]
  9. H. K. Hsiao and K. A. Winick, “Planar glass waveguide ring resonators with gain,” Opt. Express 15(26), 17783–17797 (2007).
    [Crossref] [PubMed]
  10. H. Mao, H. Ma, and Z. Jin, “Polarization maintaining silica waveguide resonator optic gyro using double phase modulation technique,” Opt. Express 19(5), 4632–4643 (2011).
    [PubMed]
  11. C. Ciminelli, F. Dell’Olio, and M. N. Armenise, “High-Q spiral resonator for optical gyroscope applications: numerical and experimental investigation,” IEEE Photonics J. 4(5), 1844–1854 (2012).
    [Crossref]
  12. F. Dell’Olio, C. Ciminelli, and M. N. Armenise, “Theoretical investigation of indium phosphide buried ring resonators for new angular velocity sensors,” Opt. Eng. 52(2), 024601 (2013).
    [Crossref]
  13. F. Dell’Olio, F. Indiveri, F. Innone, P. Dello Russo, C. Ciminelli, and M. N. Armenise, “System test of an optoelectronic gyroscope based on a high Q-factor InP ring resonator,” Opt. Eng. 53(12), 127104 (2014).
    [Crossref]
  14. L. Feng, J. Wang, Y. Zhi, Y. Tang, Q. Wang, H. Li, and W. Wang, “Transmissive resonator optic gyro based on silica waveguide ring resonator,” Opt. Express 22(22), 27565–27575 (2014).
    [Crossref] [PubMed]
  15. J. Wang, L. Feng, Y. Tang, and Y. Zhi, “Resonator integrated optic gyro employing trapezoidal phase modulation technique,” Opt. Lett. 40(2), 155–158 (2015).
    [Crossref] [PubMed]
  16. K. Hotate and M. Harumoto, “Resonator fiber optic gyro using digital serrodyne modulation,” J. Lightwave Technol. 15(3), 466–473 (1997).
    [Crossref]
  17. K. Hotate and G. Hayashi, “Resonator fiber optic gyro using digital serrodyne modulation method to reduce the noise induced by the backscattering and closed-loop operation using digital signal processing,” Proc. SPIE 3746, 104 (1999).
  18. X. Wang, M. Kishi, Z. He, and K. Hotate, “Closed loop resonator fiber optic gyro with precisely controlled bipolar digital serrodyne modulation,” Proc. SPIE 8351, 83513G (2012).
  19. Z. Jin, X. Yu, and H. Ma, “Closed-loop resonant fiber optic gyro with an improved digital serrodyne modulation,” Opt. Express 21(22), 26578–26588 (2013).
    [Crossref] [PubMed]
  20. X. L. Zhang, H. I. Ma, Z. H. Jin, and C. Ding, “Open-loop operation experiments in a resonator fiber-optic gyro using the phase modulation spectroscopy technique,” Appl. Opt. 45(31), 7961–7965 (2006).
    [Crossref] [PubMed]
  21. R. E. Meyer, S. Ezekiel, D. W. Stowe, and V. J. Tekippe, “Passive fiber-optic ring resonator for rotation sensing,” Opt. Lett. 8(12), 644–646 (1983).
    [Crossref] [PubMed]

2015 (1)

2014 (2)

F. Dell’Olio, F. Indiveri, F. Innone, P. Dello Russo, C. Ciminelli, and M. N. Armenise, “System test of an optoelectronic gyroscope based on a high Q-factor InP ring resonator,” Opt. Eng. 53(12), 127104 (2014).
[Crossref]

L. Feng, J. Wang, Y. Zhi, Y. Tang, Q. Wang, H. Li, and W. Wang, “Transmissive resonator optic gyro based on silica waveguide ring resonator,” Opt. Express 22(22), 27565–27575 (2014).
[Crossref] [PubMed]

2013 (3)

2012 (2)

X. Wang, M. Kishi, Z. He, and K. Hotate, “Closed loop resonator fiber optic gyro with precisely controlled bipolar digital serrodyne modulation,” Proc. SPIE 8351, 83513G (2012).

C. Ciminelli, F. Dell’Olio, and M. N. Armenise, “High-Q spiral resonator for optical gyroscope applications: numerical and experimental investigation,” IEEE Photonics J. 4(5), 1844–1854 (2012).
[Crossref]

2011 (2)

2010 (1)

C. Ciminelli, F. Dell Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
[Crossref]

2007 (1)

2006 (2)

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[Crossref]

X. L. Zhang, H. I. Ma, Z. H. Jin, and C. Ding, “Open-loop operation experiments in a resonator fiber-optic gyro using the phase modulation spectroscopy technique,” Appl. Opt. 45(31), 7961–7965 (2006).
[Crossref] [PubMed]

2005 (1)

C. Caterina, P. Francesco, and M. N. Armenise, “A new integrated optical angular velocity sensor,” Proc. SPIE 5728, 93–100 (2005).

2000 (2)

K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator micro optic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000).
[Crossref]

C. Monovoukas, A. K. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).

1999 (1)

K. Hotate and G. Hayashi, “Resonator fiber optic gyro using digital serrodyne modulation method to reduce the noise induced by the backscattering and closed-loop operation using digital signal processing,” Proc. SPIE 3746, 104 (1999).

1997 (1)

K. Hotate and M. Harumoto, “Resonator fiber optic gyro using digital serrodyne modulation,” J. Lightwave Technol. 15(3), 466–473 (1997).
[Crossref]

1983 (1)

1977 (1)

S. Ezekiel and S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett. 30(9), 478–480 (1977).
[Crossref]

Armenise, M. N.

F. Dell’Olio, F. Indiveri, F. Innone, P. Dello Russo, C. Ciminelli, and M. N. Armenise, “System test of an optoelectronic gyroscope based on a high Q-factor InP ring resonator,” Opt. Eng. 53(12), 127104 (2014).
[Crossref]

F. Dell’Olio, C. Ciminelli, and M. N. Armenise, “Theoretical investigation of indium phosphide buried ring resonators for new angular velocity sensors,” Opt. Eng. 52(2), 024601 (2013).
[Crossref]

C. Ciminelli, F. Dell’Olio, M. N. Armenise, F. M. Soares, and W. Passenberg, “High performance InP ring resonator for new generation monolithically integrated optical Gyroscopes,” Opt. Express 21(1), 556–564 (2013).
[Crossref] [PubMed]

C. Ciminelli, F. Dell’Olio, and M. N. Armenise, “High-Q spiral resonator for optical gyroscope applications: numerical and experimental investigation,” IEEE Photonics J. 4(5), 1844–1854 (2012).
[Crossref]

C. Ciminelli, F. Dell Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
[Crossref]

C. Caterina, P. Francesco, and M. N. Armenise, “A new integrated optical angular velocity sensor,” Proc. SPIE 5728, 93–100 (2005).

Balsamo, S. R.

S. Ezekiel and S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett. 30(9), 478–480 (1977).
[Crossref]

Campanella, C. E.

C. Ciminelli, F. Dell Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
[Crossref]

Caterina, C.

C. Caterina, P. Francesco, and M. N. Armenise, “A new integrated optical angular velocity sensor,” Proc. SPIE 5728, 93–100 (2005).

Ciminelli, C.

F. Dell’Olio, F. Indiveri, F. Innone, P. Dello Russo, C. Ciminelli, and M. N. Armenise, “System test of an optoelectronic gyroscope based on a high Q-factor InP ring resonator,” Opt. Eng. 53(12), 127104 (2014).
[Crossref]

F. Dell’Olio, C. Ciminelli, and M. N. Armenise, “Theoretical investigation of indium phosphide buried ring resonators for new angular velocity sensors,” Opt. Eng. 52(2), 024601 (2013).
[Crossref]

C. Ciminelli, F. Dell’Olio, M. N. Armenise, F. M. Soares, and W. Passenberg, “High performance InP ring resonator for new generation monolithically integrated optical Gyroscopes,” Opt. Express 21(1), 556–564 (2013).
[Crossref] [PubMed]

C. Ciminelli, F. Dell’Olio, and M. N. Armenise, “High-Q spiral resonator for optical gyroscope applications: numerical and experimental investigation,” IEEE Photonics J. 4(5), 1844–1854 (2012).
[Crossref]

C. Ciminelli, F. Dell Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
[Crossref]

Dell Olio, F.

C. Ciminelli, F. Dell Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
[Crossref]

Dell’Olio, F.

F. Dell’Olio, F. Indiveri, F. Innone, P. Dello Russo, C. Ciminelli, and M. N. Armenise, “System test of an optoelectronic gyroscope based on a high Q-factor InP ring resonator,” Opt. Eng. 53(12), 127104 (2014).
[Crossref]

F. Dell’Olio, C. Ciminelli, and M. N. Armenise, “Theoretical investigation of indium phosphide buried ring resonators for new angular velocity sensors,” Opt. Eng. 52(2), 024601 (2013).
[Crossref]

C. Ciminelli, F. Dell’Olio, M. N. Armenise, F. M. Soares, and W. Passenberg, “High performance InP ring resonator for new generation monolithically integrated optical Gyroscopes,” Opt. Express 21(1), 556–564 (2013).
[Crossref] [PubMed]

C. Ciminelli, F. Dell’Olio, and M. N. Armenise, “High-Q spiral resonator for optical gyroscope applications: numerical and experimental investigation,” IEEE Photonics J. 4(5), 1844–1854 (2012).
[Crossref]

Dello Russo, P.

F. Dell’Olio, F. Indiveri, F. Innone, P. Dello Russo, C. Ciminelli, and M. N. Armenise, “System test of an optoelectronic gyroscope based on a high Q-factor InP ring resonator,” Opt. Eng. 53(12), 127104 (2014).
[Crossref]

Ding, C.

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[Crossref]

X. L. Zhang, H. I. Ma, Z. H. Jin, and C. Ding, “Open-loop operation experiments in a resonator fiber-optic gyro using the phase modulation spectroscopy technique,” Appl. Opt. 45(31), 7961–7965 (2006).
[Crossref] [PubMed]

Ezekiel, S.

Feng, L.

Francesco, P.

C. Caterina, P. Francesco, and M. N. Armenise, “A new integrated optical angular velocity sensor,” Proc. SPIE 5728, 93–100 (2005).

Harumoto, M.

K. Hotate and M. Harumoto, “Resonator fiber optic gyro using digital serrodyne modulation,” J. Lightwave Technol. 15(3), 466–473 (1997).
[Crossref]

Hayashi, G.

K. Hotate and G. Hayashi, “Resonator fiber optic gyro using digital serrodyne modulation method to reduce the noise induced by the backscattering and closed-loop operation using digital signal processing,” Proc. SPIE 3746, 104 (1999).

He, Z.

X. Wang, M. Kishi, Z. He, and K. Hotate, “Closed loop resonator fiber optic gyro with precisely controlled bipolar digital serrodyne modulation,” Proc. SPIE 8351, 83513G (2012).

H. Ma, Z. He, and K. Hotate, “Reduction of backscattering induced noise by carrier suppression in waveguide-type optical ring resonator gyro,” J. Lightwave Technol. 29(1), 85–90 (2011).
[Crossref]

Hotate, K.

X. Wang, M. Kishi, Z. He, and K. Hotate, “Closed loop resonator fiber optic gyro with precisely controlled bipolar digital serrodyne modulation,” Proc. SPIE 8351, 83513G (2012).

H. Ma, Z. He, and K. Hotate, “Reduction of backscattering induced noise by carrier suppression in waveguide-type optical ring resonator gyro,” J. Lightwave Technol. 29(1), 85–90 (2011).
[Crossref]

K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator micro optic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000).
[Crossref]

K. Hotate and G. Hayashi, “Resonator fiber optic gyro using digital serrodyne modulation method to reduce the noise induced by the backscattering and closed-loop operation using digital signal processing,” Proc. SPIE 3746, 104 (1999).

K. Hotate and M. Harumoto, “Resonator fiber optic gyro using digital serrodyne modulation,” J. Lightwave Technol. 15(3), 466–473 (1997).
[Crossref]

Hsiao, H. K.

Indiveri, F.

F. Dell’Olio, F. Indiveri, F. Innone, P. Dello Russo, C. Ciminelli, and M. N. Armenise, “System test of an optoelectronic gyroscope based on a high Q-factor InP ring resonator,” Opt. Eng. 53(12), 127104 (2014).
[Crossref]

Innone, F.

F. Dell’Olio, F. Indiveri, F. Innone, P. Dello Russo, C. Ciminelli, and M. N. Armenise, “System test of an optoelectronic gyroscope based on a high Q-factor InP ring resonator,” Opt. Eng. 53(12), 127104 (2014).
[Crossref]

Jin, Z.

Jin, Z. H.

Kishi, M.

X. Wang, M. Kishi, Z. He, and K. Hotate, “Closed loop resonator fiber optic gyro with precisely controlled bipolar digital serrodyne modulation,” Proc. SPIE 8351, 83513G (2012).

Li, H.

Ma, H.

Ma, H. I.

Mao, H.

Maseeh, F.

C. Monovoukas, A. K. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).

Meyer, R. E.

Monovoukas, C.

C. Monovoukas, A. K. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).

Passenberg, W.

Soares, F. M.

Stowe, D. W.

Suzuki, K.

Swiecki, A. K.

C. Monovoukas, A. K. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).

Takiguchi, K.

Tang, Y.

Tekippe, V. J.

Wang, J.

Wang, Q.

Wang, W.

Wang, X.

X. Wang, M. Kishi, Z. He, and K. Hotate, “Closed loop resonator fiber optic gyro with precisely controlled bipolar digital serrodyne modulation,” Proc. SPIE 8351, 83513G (2012).

Winick, K. A.

Yu, X.

Zhang, X.

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[Crossref]

Zhang, X. L.

Zhi, Y.

Adv. Opt. Photonics (1)

C. Ciminelli, F. Dell Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. Ezekiel and S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett. 30(9), 478–480 (1977).
[Crossref]

IEEE Photonics J. (1)

C. Ciminelli, F. Dell’Olio, and M. N. Armenise, “High-Q spiral resonator for optical gyroscope applications: numerical and experimental investigation,” IEEE Photonics J. 4(5), 1844–1854 (2012).
[Crossref]

J. Lightwave Technol. (3)

Opt. Eng. (3)

F. Dell’Olio, C. Ciminelli, and M. N. Armenise, “Theoretical investigation of indium phosphide buried ring resonators for new angular velocity sensors,” Opt. Eng. 52(2), 024601 (2013).
[Crossref]

F. Dell’Olio, F. Indiveri, F. Innone, P. Dello Russo, C. Ciminelli, and M. N. Armenise, “System test of an optoelectronic gyroscope based on a high Q-factor InP ring resonator,” Opt. Eng. 53(12), 127104 (2014).
[Crossref]

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Proc. SPIE (4)

C. Monovoukas, A. K. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).

C. Caterina, P. Francesco, and M. N. Armenise, “A new integrated optical angular velocity sensor,” Proc. SPIE 5728, 93–100 (2005).

K. Hotate and G. Hayashi, “Resonator fiber optic gyro using digital serrodyne modulation method to reduce the noise induced by the backscattering and closed-loop operation using digital signal processing,” Proc. SPIE 3746, 104 (1999).

X. Wang, M. Kishi, Z. He, and K. Hotate, “Closed loop resonator fiber optic gyro with precisely controlled bipolar digital serrodyne modulation,” Proc. SPIE 8351, 83513G (2012).

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Figures (11)

Fig. 1
Fig. 1 Schematic diagram of thee closed-loop RMOG based on the hybrid digital phase modulation scheme. CIR1, CIR2: circulators; PI1, PI2: proportional integrators; LIA1, LIA2: lock-in amplifiers; PD1, PD2: photodetectors.
Fig. 2
Fig. 2 Basic model of the HDPM with two PMs. (a) Y-PM is used for sinusoidal modulation and PM1 is used for serrodyne modulation. (b) Analysis model of the WRR with the HDPM.
Fig. 3
Fig. 3 Simulation results of the sinusoidal demodulation.
Fig. 4
Fig. 4 Simulation results of the gyro output.
Fig. 5
Fig. 5 Implementation of the digital serrodyne waveform by using the FPGA for theory analysis. (a) An ideal digital serrodyne waveform setting on the FPGA. τ is the width of one stair. (b) The practical digital serrodyne waveform output from the DAC when τ>tr, tr is the rise time of the DAC. (c) Optimal digital serrodyne waveform output from the DAC whenτ = tr.
Fig. 6
Fig. 6 Measurements of the digital serrodyne waveform. (a), (b), (c) Measurements when τ = 3tr, τ = 2tr and τ = tr.
Fig. 7
Fig. 7 Influence of the sideband suppression level.
Fig. 8
Fig. 8 Simulation results of the sideband suppression.
Fig. 9
Fig. 9 Measurement results of the frequency shift performance with different widths of one stair. Serrodyne wave for τ = 8.3 ns in (a) and τ = 25 ns in (b). Measured spectrum for τ = 8.3 ns in (c) and τ = 25 ns in (d).
Fig. 10
Fig. 10 Measurement results of the two closed loops.
Fig. 11
Fig. 11 Measurement results of the double closed-loop RMOG.

Equations (13)

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

E Rin (t)= E 0 r= n= F r J n ( M 1 )expj( ω 0 +n ω 1 +r ω 3 )t
E Rout (t)= E 0 r= n= F r J n ( M )exp[ j( ω 0 +n ω 1 +r ω 3 )t ] | h nr | e j ϕ nr
V PD2out (t)= 1 2 c ε 0 E 0 2 R ν n= n = r= r = F r F r J n J n exp[ j[(n n ) ω 1 +(r r ) ω 3 ]t ]| h nr || h n r | e j( ϕ nr ϕ n r )
V LIA2 (t)=AsinφBcosφ,
A= 1 2 c ε 0 E 0 2 R ν r= F r 2 n=0 J n J n+1 [ | h nr || h (n+1)r |cos( ϕ (n+1)r ϕ nr ) | h nr || h (n+1)r |cos( ϕ nr ϕ (n+1)r ) ]
,
B= 1 2 c ε 0 E 0 2 R ν r= F r 2 n=0 J n J n+1 [ | h nr || h (n+1)r |sin( ϕ (n+1)r ϕ nr ) | h nr || h (n+1)r |sin( ϕ nr ϕ (n+1)r ) ]
,
V LIA2r = V LIA21 + r1 V LIA2r = V LIA21 + V LIA21 S = 1+S S V LIA21
,
S= F 1 2 r1 F r 2
e= 1 1+S
F r = 1 T 3 t T 3 2 t+ T 3 2 e s(t) e jr ω 3 t dt = sin(rπ ϕ 0 ) rπp ϕ 0 ( 1 p ϕ 0 (1 e j2p( ϕ 0 rπ) f 3 t r )( e j2(p1)rπ f 3 t r 1) 2rπ[ cos(2p( ϕ 0 rπ) f 3 t r )1 ] ) + (1X)sin( ϕ 0 rXπ) ϕ 0 +rπ(1X)

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