Abstract

Optoelectronic oscillators (OEOs) are promising sources of low phase noise radio frequency (RF) signals. However, at X-band frequencies, the long optical fiber delay line required for a high oscillator Q also leads to spurious modes (spurs) spaced too narrowly to be filtered by RF filters. The dual injection-locked OEO (DIL-OEO) has been proposed as a solution to this problem. In this work, we describe in detail the construction of a DIL-OEO. We also present experimental data from our systematic study of injection-locking in DIL-OEOs. With this data, we optimize the DIL-OEO, achieving both low phase noise and low spurs. Finally, we present data demonstrating a 60 dB suppression of the nearest-neighbor spur without increasing the phase noise within 1 kHz of the 10 GHz central oscillating mode.

© 2011 Optical Society of America

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References

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  1. X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13, 1725–1735 (1996).
    [Crossref]
  2. X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
    [Crossref]
  3. D. A. Howe and A. Hati, “Low-noise X-band oscillator and amplifier technologies: Comparison and status,” in “Proceedings of the Joint IEEE International Frequency Control Symposium and Precise Time and Time Interval (PTTI) Systems and Applications Meeting,” (Vancouver, Canada, 2005), pp. 481–487.
  4. D. Green, C. McNeilage, and J. H. Searls, “A low phase noise microwave sapphire loop oscillator,” in “Proceedings of the IEEE International Frequency Control Symposium,” (Miami, FL, 2006), pp. 852–860.
  5. X. S. Yao and L. Maleki, “Ultra-low phase noise dual-loop optoelectronic oscillator,” in “Technical Digest of the Optical Fiber Communication Conference and Exhibit (OFC ’98),” (San Jose, CA, 1998), pp. 353–354.
  6. X. S. Yao, L. Davis, and L. Maleki, “Coupled optoelectronic oscillators for generating both RF signal and optical pulses,” J. Lightwave Technol. 18, 73–78 (2000).
    [Crossref]
  7. W. Zhou and G. Blasche, “Injection-locked dual opto-electronic oscillator with ultra-low phase noise and ultra-low spurious level,” IEEE Trans. Microwave Theory Tech. 53, 929–933 (2005).
    [Crossref]
  8. O. Okusaga, W. Zhou, E. C. Levy, M. Horowitz, G. M. Carter, and C. R. Menyuk, “Experimental and simulation study of dual injection-locked OEOs,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 875–879.
  9. C. R. Menyuk, E. C. Levy, O. Okusaga, M. Horowitz, G. M. Carter, and W. Zhou, “An analytical model of the dual-injection-locked opto-electronic oscillator (DIL-OEO),” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 870–874.
  10. E. C. Levy, M. Horowitz, and C. R. Menyuk, “Modeling optoelectronic oscillators,” J. Opt. Soc. Am. B 26, 148–159 (2009).
    [Crossref]
  11. E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
    [Crossref] [PubMed]
  12. E. Rubiola, Phase Noise and Frequency Stability in Oscillators, The Cambridge RF and Microwave Engineering Series (Cambridge University Press, Cambridge, England, 2008).
    [Crossref]
  13. E. Rubiola, E. Salik, S. Huang, N. Yu, and L. Maleki, “Photonic-delay technique for phase-noise measurement of microwave oscillators,” J. Opt. Soc. Am. B 22, 987–997 (2005).
    [Crossref]
  14. E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Study of dual-loop optoelectronic oscillators,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 505–507.

2010 (1)

2009 (1)

2005 (2)

E. Rubiola, E. Salik, S. Huang, N. Yu, and L. Maleki, “Photonic-delay technique for phase-noise measurement of microwave oscillators,” J. Opt. Soc. Am. B 22, 987–997 (2005).
[Crossref]

W. Zhou and G. Blasche, “Injection-locked dual opto-electronic oscillator with ultra-low phase noise and ultra-low spurious level,” IEEE Trans. Microwave Theory Tech. 53, 929–933 (2005).
[Crossref]

2000 (1)

1996 (2)

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13, 1725–1735 (1996).
[Crossref]

Blasche, G.

W. Zhou and G. Blasche, “Injection-locked dual opto-electronic oscillator with ultra-low phase noise and ultra-low spurious level,” IEEE Trans. Microwave Theory Tech. 53, 929–933 (2005).
[Crossref]

Carter, G. M.

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
[Crossref] [PubMed]

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Study of dual-loop optoelectronic oscillators,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 505–507.

O. Okusaga, W. Zhou, E. C. Levy, M. Horowitz, G. M. Carter, and C. R. Menyuk, “Experimental and simulation study of dual injection-locked OEOs,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 875–879.

C. R. Menyuk, E. C. Levy, O. Okusaga, M. Horowitz, G. M. Carter, and W. Zhou, “An analytical model of the dual-injection-locked opto-electronic oscillator (DIL-OEO),” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 870–874.

Davis, L.

Green, D.

D. Green, C. McNeilage, and J. H. Searls, “A low phase noise microwave sapphire loop oscillator,” in “Proceedings of the IEEE International Frequency Control Symposium,” (Miami, FL, 2006), pp. 852–860.

Hati, A.

D. A. Howe and A. Hati, “Low-noise X-band oscillator and amplifier technologies: Comparison and status,” in “Proceedings of the Joint IEEE International Frequency Control Symposium and Precise Time and Time Interval (PTTI) Systems and Applications Meeting,” (Vancouver, Canada, 2005), pp. 481–487.

Horowitz, M.

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
[Crossref] [PubMed]

E. C. Levy, M. Horowitz, and C. R. Menyuk, “Modeling optoelectronic oscillators,” J. Opt. Soc. Am. B 26, 148–159 (2009).
[Crossref]

O. Okusaga, W. Zhou, E. C. Levy, M. Horowitz, G. M. Carter, and C. R. Menyuk, “Experimental and simulation study of dual injection-locked OEOs,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 875–879.

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Study of dual-loop optoelectronic oscillators,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 505–507.

C. R. Menyuk, E. C. Levy, O. Okusaga, M. Horowitz, G. M. Carter, and W. Zhou, “An analytical model of the dual-injection-locked opto-electronic oscillator (DIL-OEO),” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 870–874.

Howe, D. A.

D. A. Howe and A. Hati, “Low-noise X-band oscillator and amplifier technologies: Comparison and status,” in “Proceedings of the Joint IEEE International Frequency Control Symposium and Precise Time and Time Interval (PTTI) Systems and Applications Meeting,” (Vancouver, Canada, 2005), pp. 481–487.

Huang, S.

Levy, E. C.

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
[Crossref] [PubMed]

E. C. Levy, M. Horowitz, and C. R. Menyuk, “Modeling optoelectronic oscillators,” J. Opt. Soc. Am. B 26, 148–159 (2009).
[Crossref]

O. Okusaga, W. Zhou, E. C. Levy, M. Horowitz, G. M. Carter, and C. R. Menyuk, “Experimental and simulation study of dual injection-locked OEOs,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 875–879.

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Study of dual-loop optoelectronic oscillators,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 505–507.

C. R. Menyuk, E. C. Levy, O. Okusaga, M. Horowitz, G. M. Carter, and W. Zhou, “An analytical model of the dual-injection-locked opto-electronic oscillator (DIL-OEO),” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 870–874.

Maleki, L.

E. Rubiola, E. Salik, S. Huang, N. Yu, and L. Maleki, “Photonic-delay technique for phase-noise measurement of microwave oscillators,” J. Opt. Soc. Am. B 22, 987–997 (2005).
[Crossref]

X. S. Yao, L. Davis, and L. Maleki, “Coupled optoelectronic oscillators for generating both RF signal and optical pulses,” J. Lightwave Technol. 18, 73–78 (2000).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13, 1725–1735 (1996).
[Crossref]

X. S. Yao and L. Maleki, “Ultra-low phase noise dual-loop optoelectronic oscillator,” in “Technical Digest of the Optical Fiber Communication Conference and Exhibit (OFC ’98),” (San Jose, CA, 1998), pp. 353–354.

McNeilage, C.

D. Green, C. McNeilage, and J. H. Searls, “A low phase noise microwave sapphire loop oscillator,” in “Proceedings of the IEEE International Frequency Control Symposium,” (Miami, FL, 2006), pp. 852–860.

Menyuk, C. R.

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
[Crossref] [PubMed]

E. C. Levy, M. Horowitz, and C. R. Menyuk, “Modeling optoelectronic oscillators,” J. Opt. Soc. Am. B 26, 148–159 (2009).
[Crossref]

O. Okusaga, W. Zhou, E. C. Levy, M. Horowitz, G. M. Carter, and C. R. Menyuk, “Experimental and simulation study of dual injection-locked OEOs,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 875–879.

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Study of dual-loop optoelectronic oscillators,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 505–507.

C. R. Menyuk, E. C. Levy, O. Okusaga, M. Horowitz, G. M. Carter, and W. Zhou, “An analytical model of the dual-injection-locked opto-electronic oscillator (DIL-OEO),” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 870–874.

Okusaga, O.

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
[Crossref] [PubMed]

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Study of dual-loop optoelectronic oscillators,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 505–507.

O. Okusaga, W. Zhou, E. C. Levy, M. Horowitz, G. M. Carter, and C. R. Menyuk, “Experimental and simulation study of dual injection-locked OEOs,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 875–879.

C. R. Menyuk, E. C. Levy, O. Okusaga, M. Horowitz, G. M. Carter, and W. Zhou, “An analytical model of the dual-injection-locked opto-electronic oscillator (DIL-OEO),” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 870–874.

Rubiola, E.

E. Rubiola, E. Salik, S. Huang, N. Yu, and L. Maleki, “Photonic-delay technique for phase-noise measurement of microwave oscillators,” J. Opt. Soc. Am. B 22, 987–997 (2005).
[Crossref]

E. Rubiola, Phase Noise and Frequency Stability in Oscillators, The Cambridge RF and Microwave Engineering Series (Cambridge University Press, Cambridge, England, 2008).
[Crossref]

Salik, E.

Searls, J. H.

D. Green, C. McNeilage, and J. H. Searls, “A low phase noise microwave sapphire loop oscillator,” in “Proceedings of the IEEE International Frequency Control Symposium,” (Miami, FL, 2006), pp. 852–860.

Yao, X. S.

X. S. Yao, L. Davis, and L. Maleki, “Coupled optoelectronic oscillators for generating both RF signal and optical pulses,” J. Lightwave Technol. 18, 73–78 (2000).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13, 1725–1735 (1996).
[Crossref]

X. S. Yao and L. Maleki, “Ultra-low phase noise dual-loop optoelectronic oscillator,” in “Technical Digest of the Optical Fiber Communication Conference and Exhibit (OFC ’98),” (San Jose, CA, 1998), pp. 353–354.

Yu, N.

Zhou, W.

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
[Crossref] [PubMed]

W. Zhou and G. Blasche, “Injection-locked dual opto-electronic oscillator with ultra-low phase noise and ultra-low spurious level,” IEEE Trans. Microwave Theory Tech. 53, 929–933 (2005).
[Crossref]

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Study of dual-loop optoelectronic oscillators,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 505–507.

O. Okusaga, W. Zhou, E. C. Levy, M. Horowitz, G. M. Carter, and C. R. Menyuk, “Experimental and simulation study of dual injection-locked OEOs,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 875–879.

C. R. Menyuk, E. C. Levy, O. Okusaga, M. Horowitz, G. M. Carter, and W. Zhou, “An analytical model of the dual-injection-locked opto-electronic oscillator (DIL-OEO),” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 870–874.

IEEE J. Quantum Electron. (1)

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

W. Zhou and G. Blasche, “Injection-locked dual opto-electronic oscillator with ultra-low phase noise and ultra-low spurious level,” IEEE Trans. Microwave Theory Tech. 53, 929–933 (2005).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (3)

Opt. Express (1)

Other (7)

O. Okusaga, W. Zhou, E. C. Levy, M. Horowitz, G. M. Carter, and C. R. Menyuk, “Experimental and simulation study of dual injection-locked OEOs,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 875–879.

C. R. Menyuk, E. C. Levy, O. Okusaga, M. Horowitz, G. M. Carter, and W. Zhou, “An analytical model of the dual-injection-locked opto-electronic oscillator (DIL-OEO),” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 870–874.

E. Rubiola, Phase Noise and Frequency Stability in Oscillators, The Cambridge RF and Microwave Engineering Series (Cambridge University Press, Cambridge, England, 2008).
[Crossref]

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Study of dual-loop optoelectronic oscillators,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 505–507.

D. A. Howe and A. Hati, “Low-noise X-band oscillator and amplifier technologies: Comparison and status,” in “Proceedings of the Joint IEEE International Frequency Control Symposium and Precise Time and Time Interval (PTTI) Systems and Applications Meeting,” (Vancouver, Canada, 2005), pp. 481–487.

D. Green, C. McNeilage, and J. H. Searls, “A low phase noise microwave sapphire loop oscillator,” in “Proceedings of the IEEE International Frequency Control Symposium,” (Miami, FL, 2006), pp. 852–860.

X. S. Yao and L. Maleki, “Ultra-low phase noise dual-loop optoelectronic oscillator,” in “Technical Digest of the Optical Fiber Communication Conference and Exhibit (OFC ’98),” (San Jose, CA, 1998), pp. 353–354.

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Figures (10)

Fig. 1
Fig. 1 A schematic illustration of the dual injection-locked OEO.

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Fig. 2
Fig. 2 A schematic illustration of the DIL-OEO showing the field coupling ratios γij.

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Fig. 3
Fig. 3 Effect of increasing the injection ratio on (a) the phase noise at 1 kHz, (b) the phase noise at 10 kHz, and (c) the first spur level of the master loop. Theoretical data derived from Eq. (12) are included for comparison.

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Fig. 4
Fig. 4 Effect of increasing the injection ratio on (a) the phase noise at 1 kHz, (b) the phase noise at 10 kHz, and (c) the first spur level of the slave loop. Theoretical data derived from Eq. (12) are included for comparison.

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Fig. 5
Fig. 5 Phase noise data for free-running 44 m and 547 m slave loops. The 547 m slave has a spur at 375 kHz while the 44 m slave has no spurs over the entire measurement range.

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Fig. 6
Fig. 6 Phase noise and spurious mode data from free-running and injection-locked OEOs.

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Fig. 7
Fig. 7 A plot of the phase noise at 1 kHz versus the coupling coefficient. The data were collected from the injection-locked master and slave loops. Phase noise at 1 kHz from the free-running, master, and slave loops are all included for comparison.

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Fig. 8
Fig. 8 A plot of the phase noise at 10 kHz versus the coupling coefficient. The data were collected from the injection-locked master and slave loops. Phase noise at 10 kHz from the free-running, master, and slave loops are all included for comparison.

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Fig. 9
Fig. 9 A plot of spur level versus coupling coefficient. The data was collected from the injection-locked master and slave loops.

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Fig. 10
Fig. 10 A plot of the phase noise data from a DIL-OEO with −6 dB coupling between master and slave loops. The phase noise was measured from both master and slave loops. Phase noise data from the free-running master and slave loops are included for comparison.

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Equations (25)

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

A ( t ) = G M [ A ( t − τ ) ] A ( t − τ ) + S ( t ) ,
M ( P ) = 2 ( P P sat ) 1 / 2 J 1 [ ( P sat P ) 1 / 2 ] ,
M ( P ) = 1 1 + P / P sat .
A ˜ ( ω ) = ( 1 − Δ ) A ˜ ( ω ) exp ( − j ω τ ) + S ˜ ( ω ) ,
A ˜ ( ω ) = S ˜ ( ω ) 1 − ( 1 − Δ ) exp ( − j ω τ ) .
P ( ω ) = N ( ω ) | 1 − ( 1 − Δ ) exp ( − j ω τ ) | 2 ,
P 0 = N 0 / | Δ | 2 .
P ( ω ) = N ( ω ) Δ 2 + ω 2 τ 2 .
A 1 ( t ) = γ 11 G 1 M 1 A 1 ( t − τ 1 ) + γ 12 A 2 ( t ) + S 1 ( t ) , A 2 ( t ) = γ 22 G 2 M 2 A 2 ( t − τ 2 ) + γ 21 A 1 ( t ) + S 2 ( t ) ,
A ˜ 1 ( ω ) = 1 D ( ω ) { [ 1 − ( 1 − Δ 2 ) exp ( − j ω τ 2 ) ] S ˜ 1 ( ω ) + γ 12 S ˜ 2 ( ω ) } , A ˜ 2 ( ω ) = 1 D ( ω ) { [ 1 − ( 1 − Δ 1 ) exp ( − j ω τ 1 ) ] S ˜ 2 ( ω ) + γ 21 S ˜ 1 ( ω ) } ,
D ( ω ) = [ 1 − ( 1 − Δ 1 ) ] exp ( − j ω τ 1 ) [ 1 − ( 1 − Δ 2 ) ] exp ( − j ω τ 2 ) − γ 12 γ 21 ,
P 1 ( ω ) = 1 | D ( ω ) | 2 [ | 1 − ( 1 − Δ 2 ) exp ( − j ω τ 2 ) | 2 N 1 ( ω ) + γ 12 2 N 2 ( ω ) ] , P 2 ( ω ) = 1 | D ( ω ) | 2 [ | 1 − ( 1 − Δ 1 ) exp ( − j ω τ 1 ) | 2 N 2 ( ω ) + γ 21 2 N 1 ( ω ) ] .
D 0 ≡ D ( ω ) | ω = 0 = Δ 1 Δ 2 − γ 12 γ 21 .
P 1 ( ω ) | ω = 0 = 1 | D 0 | 2 [ | Δ 2 | 2 N 1 ( ω ) | ω = 0 + | γ 12 | 2 N 2 ( ω ) | ω = 0 ] , P 2 ( ω ) | ω = 0 = 1 | D 0 | 2 [ | Δ 1 | 2 N 2 ( ω ) | ω = 0 + | γ 21 | 2 N 1 ( ω ) | ω = 0 ] .
P 10 ≥ | γ 12 | 2 N 20 | D 0 | 2 , ⇒ | D 0 | 2 ≤ | γ 12 | 2 N 20 / P 10 .
Δ 1 = α γ 21 + ɛ 1 ,                     Δ 2 = α − 1 γ 12 + ɛ 2 ,
W 1 = 1 4 τ 1 W 1 Δ 2 2 N 1 W + | γ 12 | 2 N 2 W Δ 1 Δ 2 − γ 12 γ 21 , W 2 = 1 4 τ 1 W 1 Δ 1 2 N 2 W + | γ 21 | 2 N 1 W Δ 1 Δ 2 − γ 12 γ 21 ,
ɛ 1 = N 1 W 4 τ 1 W 1 ( 1 + W 2 W 1 | γ 12 | 2 | γ 21 | 2 N 2 W N 1 W ) − 1 ,
| 1 − ( 1 − Δ 2 ) exp ( − j ω τ 2 ) | 2 ≈ Δ 2 2 = | γ 21 | 2 , | 1 − ( 1 − Δ 1 ) exp ( − j ω τ 1 ) | 2 ≈ Δ 1 2 + ω 2 τ 1 2 = | γ 12 | 2 + ω 2 τ 1 2 .
P 1 ( ω ) N 1 ( ω ) = 1 ɛ 1 2 + ω 2 τ 1 2 ,                 P 2 ( ω ) N 2 ( ω ) = 1 + ω 2 τ 1 2 / | γ 21 | 2 ɛ 1 2 + ω 2 τ 1 2 .
| 1 − ( 1 − Δ 2 ) exp ( − j ω τ 2 ) | 2 ≈ Δ 2 2 + ( 2 π n τ 2 / τ 1 ) 2 = | γ 21 | 2 + ( 2 π n τ 2 / τ 1 ) 2 , | 1 − ( 1 − Δ 1 ) exp ( − j ω τ 1 ) | 2 ≈ Δ 1 2 = | γ 12 | 2 ,
P 1 ( ω n ) N 1 ( ω n ) = 1 + ( 2 π n / | γ 21 | τ 1 ) 2 ɛ 1 2 + | γ 12 / γ 21 | 2 ( 2 π n τ 2 / τ 1 ) 2 , P 2 ( ω n ) N 2 ( ω n ) = 1 ɛ 1 2 + | γ 12 / γ 21 | 2 ( 2 π n τ 2 / τ 1 ) 2 .
Γ i j = 10 log 10 | γ i j | 2 .
Γ 12 = 10 log 10 | γ 12 | 2 .
C ≡ Γ 21 Γ 11 ≈ Γ 12 Γ 22 .

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