Abstract

The performances of radio-over-fiber (RoF) link with fixed incident optical power on photodetector (PD) are improved through carrier suppression method. Firstly, a precise analytical model is proposed to quantify the relationship between the improvement of link gain, noise figure (NF), spur-free dynamic range (SFDR) and the carrier suppression ratio x, in which, the modulation index m is fully considered for the first time to our knowledge. Then the optimum optical carrier-to-sideband ratio (CSR) for RoF link performances in both double-sideband and single-sideband modulation is obtained from the optimum x for the link performances. Finally the experiments with the carrier subtraction method realized by Stimulated Brillouin scattering (SBS) are carried out and the experimental results show good agreement with the simulation ones.

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References

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  1. E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
    [CrossRef]
  2. R. C. Williamson and R. D. Esman, “RF photonics,” J. Lightwave Technol. 26(9), 1145–1153 (2008).
    [CrossRef]
  3. C. H. Cox III, E. I. Ackerman, and J. L. Prince, “What do we need to get great link performance?” Microwave Photonics, International topical meeting (Germany, 1997), pp. 215–218.
  4. C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
    [CrossRef]
  5. A. Karim and J. Devenport, “Noise Figure Reduction in Externally Modulated Analog Fiber-Optic Links,” IEEE Photon. Technol. Lett. 19(5), 312–314 (2007).
    [CrossRef]
  6. M. L. Farewell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach-Zehnder modulator,” IEEE Photon. Technol. Lett. 5(7), 779–782 (1993).
    [CrossRef]
  7. J. Devenport and A. Karim, “Optimization of an externally modulated RF photonic link,” Fiber Integr. Opt. 27(1), 7–14 (2008).
  8. W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photon. Technol. Lett. 8(1), 130–132 (1996).
    [CrossRef]
  9. M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
    [CrossRef]
  10. K. J. Williams and R. D. Esman, “Stimulated Brillouin scattering for improvement of microwave fiber-optic link efficiency,” Electron. Lett. 30(23), 1965–1966 (1994).
    [CrossRef]
  11. Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005).
    [CrossRef]
  12. 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]
  13. R. D. Esman and K. J. Williams, “Wideband efficiency improvement of fiber optic systems by carrier subtraction,” IEEE Photon. Technol. Lett. 7(2), 218–220 (1995).
    [CrossRef]
  14. C. H. Cox, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990).
    [CrossRef]
  15. C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
    [CrossRef]

2008 (2)

J. Devenport and A. Karim, “Optimization of an externally modulated RF photonic link,” Fiber Integr. Opt. 27(1), 7–14 (2008).

R. C. Williamson and R. D. Esman, “RF photonics,” J. Lightwave Technol. 26(9), 1145–1153 (2008).
[CrossRef]

2007 (1)

A. Karim and J. Devenport, “Noise Figure Reduction in Externally Modulated Analog Fiber-Optic Links,” IEEE Photon. Technol. Lett. 19(5), 312–314 (2007).
[CrossRef]

2006 (2)

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (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)

Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005).
[CrossRef]

1997 (1)

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

1996 (1)

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

1995 (1)

R. D. Esman and K. J. Williams, “Wideband efficiency improvement of fiber optic systems by carrier subtraction,” IEEE Photon. Technol. Lett. 7(2), 218–220 (1995).
[CrossRef]

1994 (2)

M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
[CrossRef]

K. J. Williams and R. D. Esman, “Stimulated Brillouin scattering for improvement of microwave fiber-optic link efficiency,” Electron. Lett. 30(23), 1965–1966 (1994).
[CrossRef]

1993 (2)

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

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

1990 (1)

C. H. Cox, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990).
[CrossRef]

Ackerman, E.

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

Ackerman, E. I.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[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]

Betts, G. E.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

C. H. Cox, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990).
[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 Photon. Technol. Lett. 8(1), 130–132 (1996).
[CrossRef]

Chang, W. S. C.

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

Charezenko, W.

M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
[CrossRef]

Chen, K. S.

Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005).
[CrossRef]

Cox, C.

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

Cox, C. H.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

C. H. Cox, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990).
[CrossRef]

Daryoush, A. S.

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

Devenport, J.

J. Devenport and A. Karim, “Optimization of an externally modulated RF photonic link,” Fiber Integr. Opt. 27(1), 7–14 (2008).

A. Karim and J. Devenport, “Noise Figure Reduction in Externally Modulated Analog Fiber-Optic Links,” IEEE Photon. Technol. Lett. 19(5), 312–314 (2007).
[CrossRef]

Esman, R. D.

R. C. Williamson and R. D. Esman, “RF photonics,” J. Lightwave Technol. 26(9), 1145–1153 (2008).
[CrossRef]

R. D. Esman and K. J. Williams, “Wideband efficiency improvement of fiber optic systems by carrier subtraction,” IEEE Photon. Technol. Lett. 7(2), 218–220 (1995).
[CrossRef]

K. J. Williams and R. D. Esman, “Stimulated Brillouin scattering for improvement of microwave fiber-optic link efficiency,” Electron. Lett. 30(23), 1965–1966 (1994).
[CrossRef]

Farewell, M. L.

M. L. Farewell, W. S. C. Chang, and D. R. Huber, “Increased linear dynamic range by low biasing the Mach-Zehnder modulator,” IEEE Photon. Technol. Lett. 5(7), 779–782 (1993).
[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 Photon. Technol. Lett. 8(1), 130–132 (1996).
[CrossRef]

Hamilton, M. C.

M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
[CrossRef]

Helkey, R.

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

Huber, D. R.

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

Johnson, L. M.

C. H. Cox, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990).
[CrossRef]

Karim, A.

J. Devenport and A. Karim, “Optimization of an externally modulated RF photonic link,” Fiber Integr. Opt. 27(1), 7–14 (2008).

A. Karim and J. Devenport, “Noise Figure Reduction in Externally Modulated Analog Fiber-Optic Links,” IEEE Photon. Technol. Lett. 19(5), 312–314 (2007).
[CrossRef]

Kasemset, D.

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

LaGasse, M. J.

M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
[CrossRef]

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]

Moeller, R. P.

W. K. Burns, G. K. Gopalakrishnan, and R. P. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photon. 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.

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]

Prince, J. L.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

Samant, N. R.

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

Shen, Y. C.

Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005).
[CrossRef]

Thaniyavarn, S.

M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
[CrossRef]

Wanuga, S.

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[CrossRef]

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]

Williams, K. J.

R. D. Esman and K. J. Williams, “Wideband efficiency improvement of fiber optic systems by carrier subtraction,” IEEE Photon. Technol. Lett. 7(2), 218–220 (1995).
[CrossRef]

K. J. Williams and R. D. Esman, “Stimulated Brillouin scattering for improvement of microwave fiber-optic link efficiency,” Electron. Lett. 30(23), 1965–1966 (1994).
[CrossRef]

Williamson, R. C.

Zhang, X. M.

Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005).
[CrossRef]

Electron. Lett. (2)

M. J. LaGasse, W. Charezenko, M. C. Hamilton, and S. Thaniyavarn, “Optical carrier filtering for high dynamic range fibre optic links,” Electron. Lett. 30(25), 2157–2158 (1994).
[CrossRef]

K. J. Williams and R. D. Esman, “Stimulated Brillouin scattering for improvement of microwave fiber-optic link efficiency,” Electron. Lett. 30(23), 1965–1966 (1994).
[CrossRef]

Fiber Integr. Opt. (1)

J. Devenport and A. Karim, “Optimization of an externally modulated RF photonic link,” Fiber Integr. Opt. 27(1), 7–14 (2008).

IEEE Photon. Technol. Lett. (4)

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

A. Karim and J. Devenport, “Noise Figure Reduction in Externally Modulated Analog Fiber-Optic Links,” IEEE Photon. Technol. Lett. 19(5), 312–314 (2007).
[CrossRef]

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

R. D. Esman and K. J. Williams, “Wideband efficiency improvement of fiber optic systems by carrier subtraction,” IEEE Photon. Technol. Lett. 7(2), 218–220 (1995).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (5)

C. H. Cox, G. E. Betts, and L. M. Johnson, “An analytic and experimental comparison of direct and external modulation in analog fiber-optic links,” IEEE Trans. Microw. Theory Tech. 38(5), 501–509 (1990).
[CrossRef]

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

E. Ackerman, S. Wanuga, D. Kasemset, A. S. Daryoush, and N. R. Samant, “Maximum dynamic range operation of a microwave external modulation fiber-optic link,” IEEE Trans. Microw. Theory Tech. 41(8), 1299–1306 (1993).
[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]

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

J. Lightwave Technol. (1)

Opt. Eng. (1)

Y. C. Shen, X. M. Zhang, and K. S. Chen, “Stimulated Brillouin scattering for efficient improvement of radio-over-fiber systems,” Opt. Eng. 44(10), 105003 (2005).
[CrossRef]

Other (1)

C. H. Cox III, E. I. Ackerman, and J. L. Prince, “What do we need to get great link performance?” Microwave Photonics, International topical meeting (Germany, 1997), pp. 215–218.

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Figures (8)

Fig. 1
Fig. 1

The simulation results of normalized RF link gain with regards to the carrier suppression ratio x under different modulation index m. a: the curve of (1-x)−2; b: m = 0.04; c: m = 0.1; d: m = 0.3; e: m = 0.5; f: m = 0.7.

Fig. 2
Fig. 2

The simulation results of NF with respected to the carrier suppression ratio x under different modulation index m. a: The solid line with m = 0.3 and incident optical power on PD of 2 mW; b: The solid line with m = 0.1 and incident optical power on PD of 2 mW; c: The dashed line with m = 0.3 and incident optical power on PD of 5 mW; d: The dashed line with m = 0.1 and incident optical power on PD of 5 mW.

Fig. 3
Fig. 3

The experimental configuration of RoF link with SBS as a carrier filter.

Fig. 4
Fig. 4

The simulation (solid lines) and experimental results (dots) of link gain and NF with respected to carrier suppression ratio x, in which RF signal is 18 GHz and m = 0.1.

Fig. 5
Fig. 5

The simulation (solid lines) and experimental results (dots) of link gain and NF with regards to carrier suppression ratio x, in which RF signal is 18 GHz and m = 0.3.

Fig. 6
Fig. 6

The simulation (solid lines) and experimental results (dots) of link gain and NF with regards to carrier suppression ratio x, in which RF input signal is 9 GHz and m = 0.5.

Fig. 7
Fig. 7

The results of the second-order harmonic relative to fundamental component with regards to carrier suppression ratio x. Solid line: theoretical curve of Eq. (21) with m = 0.3. Diamonds: experimental results.

Fig. 8
Fig. 8

RF output power versus RF input power for RoF link with carrier subtraction method. The modulation frequency is 9 GHz.

Tables (1)

Tables Icon

Table 1 Comparison of SFDR in RoF link with and without carrier subtraction

Equations (21)

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

m = π 2 V π V mod
E b e f o r e ( t ) = L cos ( π V D C 2 V π + m cos ω r f t ) E 0 ( t ) = L [ cos π V D C 2 V π cos ( m cos ω r f t ) sin π V D C 2 V π sin ( m cos ω r f t ) ] E 0 ( t ) = 2 L 2 [ cos ( m cos ω r f t ) sin ( m cos ω r f t ) ] E 0 ( t ) = 2 L 2 [ J 0 ( m ) + 2 k = 1 + ( 1 ) k J 2 k ( m ) cos ( 2 k ω r f t ) 2 k = 0 + ( 1 ) k J 2 k + 1 ( m ) cos [ ( 2 k + 1 ) ω r f t ] ] E 0 ( t )
P b e f o r e ( t ) = P 0 L 2 [ 1 + cos ( π V D C V π + 2 m cos ω r f t ) ] = P 0 L 2 { 1 + cos π V D C V π [ J 0 ( 2 m ) + 2 k = 1 + ( 1 ) k J 2 k ( 2 m ) cos ( 2 k ω r f t ) ] sin π V D C V π [ 2 k = 0 + ( 1 ) k J 2 k + 1 ( 2 m ) cos [ ( 2 k + 1 ) ω r f t ] ] }
P a v e r a g e _ b e f o r e = P 0 L 2 [ 1 + cos π V D C V π J 0 ( 2 m ) ] = P 0 L 2
E a f t e r ( t ) = 2 L 2 [ J 0 ( m ) + 2 k = 1 + ( 1 ) k J 2 k ( m ) cos ( 2 k ω r f t ) 2 k = 0 + ( 1 ) k J 2 k + 1 ( m ) cos [ ( 2 k + 1 ) ω r f t ] x J 0 ( m ) ] E 1 ( t ) = L cos ( π V D C 2 V π + m cos ω r f t ) E 1 ( t ) 2 L 2 x J 0 ( m ) E 1 ( t )
P a f t e r ( t ) = E a f t e r ( t ) E a f t e r * ( t ) = P 1 L 2 { 1 + cos π V D C V π [ J 0 ( 2 m ) + 2 k = 1 + ( 1 ) k J 2 k ( 2 m ) cos ( 2 k ω r f t ) ] sin π V D C V π [ 2 k = 0 + ( 1 ) k J 2 k + 1 ( 2 m ) cos [ ( 2 k + 1 ) ω r f t ] ] } P 1 L 2 2 x J 0 ( m ) [ J 0 ( m ) + 2 k = 1 + ( 1 ) k J 2 k ( m ) cos ( 2 k ω r f t ) 2 k = 0 + ( 1 ) k J 2 k + 1 ( m ) cos [ ( 2 k + 1 ) ω r f t ] ] + P 1 L 2 x 2 J 0 ( m ) 2
P a v e r a g e _ a f t e r = P 1 L 2 [ 1 + J 0 ( m ) 2 ( x 2 2 x ) ]
P 1 P 0 = 1 1 + J 0 ( m ) 2 ( x 2 2 x )
G a f t e r = { P 1 L 2 [ 2 J 1 ( 2 m ) 4 x J 0 ( m ) J 1 ( m ) ] } 2 G b e f o r e = [ P 0 L 2 2 J 1 ( 2 m ) ] 2
G a f t e r G b e f o r e = [ 1 1 + J 0 ( m ) 2 ( x 2 2 x ) J 1 ( 2 m ) 2 x J 0 ( m ) J 1 ( m ) J 1 ( 2 m ) ] 2
E b e f o r e ( t ) = 2 P 0 L 2 { J 0 ( m ) cos ( ω 0 t ) J 1 ( m ) [ cos ( ω 0 t ω mod t ) + cos ( ω 0 t + ω mod t ) ] J 2 ( m ) [ cos ( ω 0 t 2 ω mod t ) + cos ( ω 0 t + 2 ω mod t ) ] }
P a v e r a g e _ b e f o r e = P 0 L 2 { [ J 0 ( m ) ] 2 + 2 [ J 1 ( m ) ] 2 + 2 [ J 2 ( m ) ] 2 } P 0 L 2 ( 1 + m 2 2 )
E a f t e r ( t ) = 2 P 1 L 2 { ( 1 x ) J 0 ( m ) cos ( ω 0 t + φ 0 ) J 1 ( m ) [ cos ( ω 0 t ω mod t ) + cos ( ω 0 t + ω mod t ) ] J 2 ( m ) [ cos ( ω 0 t 2 ω mod t ) + cos ( ω 0 t + 2 ω mod t ) ] }
P a v e r a g e _ a f t e r = P 1 L 2 { [ ( 1 x ) J 0 ( m ) ] 2 + 2 [ J 1 ( m ) ] 2 + 2 [ J 2 ( m ) ] 2 } P 1 L 2 [ ( 1 x ) 2 + m 2 2 ]
C S R = ( 1 x ) 2 m 2 4
G a f t e r = [ ( 1 x ) ( 1 + m 2 2 ) ( 1 x ) 2 + m 2 2 ] 2 G b e f o r e
x G = 1 m 2 2
x G _ s i n g l e b a n d = 1 m 2 4
F = N o i s e _ o u t G f k T 0
S F D R = 2 3 ( I P 3 + 174 N F ) = 2 3 ( I P 3 N o i s e + G ) [dB Hz 2 3 ]
G 2 f _ a f t e r G f _ a f t e r = [ x J 1 ( m ) 2 ( 1 x ) J 0 ( m ) ] 2

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