Abstract

The microwave reflection coefficient is commonly used to characterize the impedance of high-speed optoelectronic devices. Error and uncertainty in equivalent circuit parameters measured using this data are systematically evaluated. The commonly used nonlinear least-squares method for estimating uncertainty is shown to give unsatisfactory and incorrect results due to the nonlinear relationship between the circuit parameters and the measured data. Markov chain Monte Carlo methods are shown to provide superior results, both for individual devices and for assessing within-die variation.

© 2016 Optical Society of America

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References

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  1. C. R. Doerr, “Silicon photonic integration in telecommunications,” Front. Phys. 3, 37 (2015).
    [Crossref]
  2. S. Bowers, B. Abiri, F. Aflatouni, and A. Hajimiri, “A compact optically driven travelling-wave radiating source,” in Optical Fiber Communication Conference, (OSA, 2014), p. Tu2A.3.
  3. C.-M. Chang, J. H. Sinsky, P. Dong, G. de Valicourt, and Y.-K. Chen, “High-power dual-fed traveling wave photodetector circuits in silicon photonics,” Opt. Express 23, 22857 (2015).
    [Crossref] [PubMed]
  4. M. S. Hai, M. Ménard, and O. Liboiron-Ladouceur, “A 20 Gb/s SiGe photoreceiver based on optical time sampling,” in European Conference on Optical Communications, (2015), p. Tu.1.3.5.
  5. K. Minoglou, E. D. Kyriakis-Bitzaros, D. Syvridis, and G. Halkias, “A compact nonlinear equivalent circuit model and parameter extraction method for packaged high-speed VCSELs,” J. Lightwave Technol. 22, 2823–2827 (2004).
    [Crossref]
  6. P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.
  7. S. M. Sze, Physics of Semiconductor Devices, 2nd ed. (John Wiley & Sons, 1981).
  8. G. P. Agrawal, Optical Receivers, 3rd ed. (John Wiley & Sons, 2002), vol. 6, chap. 4.
  9. G. Wang, T. Tokumitsu, I. Hanawa, K. Sato, and M. Kobayashi, “Analysis of high speed p-i-n photodiode S-parameters by a novel small-signal equivalent circuit model,” IEEE Microwave and Wireless Components Lett. 12, 378–380 (2002).
    [Crossref]
  10. R. Lewén, S. Irmscher, and U. Eriksson, “Microwave cad circuit modeling of a traveling-wave electroabsorption modulator,” IEEE Trans. Microwave Theory Tech. 51, 1117–1128 (2003).
    [Crossref]
  11. Cascade Microtech, Calibration Tools: Consistent Parameter Extraction for Advanced RF Devices (Cascade Microtech, 2012).
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    [Crossref]
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    [Crossref] [PubMed]
  17. D. Melati, E. Lovati, and A. Melloni, “Statistical process design kits: analysis of fabrication tolerances in integrated photonic circuits,” in Advanced Photonics 2015, (OSA, 2015), p. IT4A.5.
    [Crossref]
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    [Crossref]
  19. Y. Yang, Y. Ma, H. Guan, Y. Liu, S. Danziger, S. Ocheltree, K. Bergman, T. Baehr-Jones, and M. Hochberg, “Phase coherence length in silicon photonic platform,” Opt. Express 23, 16890–16902 (2015).
    [Crossref] [PubMed]
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  21. X. Zhang, A. Hosseini, H. Subbaraman, S. Wang, Q. Zhan, J. Luo, A. K.-Y. Jen, and R. T. Chen, “Integrated photonic electromagnetic field sensor based on broadband bowtie antenna coupled silicon organic hybrid modulator,” J. Lightwave Technol. 32, 3774–3784 (2014).
    [Crossref]
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    [Crossref]
  23. W. Loh, S. Yegnanarayanan, J. J. Plant, F. J. O’Donnell, M. E. Grein, J. Klamkin, S. M. Duff, and P. W. Juodawlkis, “Low-noise RF-amplifier-free slab-coupled optical waveguide coupled optoelectronic oscillators: physics and operation,” Opt. Express 20, 19420–19430 (2012).
    [Crossref] [PubMed]

2015 (4)

2014 (2)

2012 (1)

2004 (1)

2003 (1)

R. Lewén, S. Irmscher, and U. Eriksson, “Microwave cad circuit modeling of a traveling-wave electroabsorption modulator,” IEEE Trans. Microwave Theory Tech. 51, 1117–1128 (2003).
[Crossref]

2002 (1)

G. Wang, T. Tokumitsu, I. Hanawa, K. Sato, and M. Kobayashi, “Analysis of high speed p-i-n photodiode S-parameters by a novel small-signal equivalent circuit model,” IEEE Microwave and Wireless Components Lett. 12, 378–380 (2002).
[Crossref]

1997 (1)

1996 (1)

A. Gelman, X.-L. Meng, and H. Stern, “Posterior predictive assessment of model fitness via realized discrepancies,” Statistica Sinica 6, 733–807 (1996).

1992 (1)

C. Michael and M. Ismail, “Statistical modeling of device mismatch for analog MOS integrated circuits,” IEEE J. Solid-State Circuits 27, 154–166 (1992).
[Crossref]

Abiri, B.

S. Bowers, B. Abiri, F. Aflatouni, and A. Hajimiri, “A compact optically driven travelling-wave radiating source,” in Optical Fiber Communication Conference, (OSA, 2014), p. Tu2A.3.

Absil, P. P.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

Aflatouni, F.

S. Bowers, B. Abiri, F. Aflatouni, and A. Hajimiri, “A compact optically driven travelling-wave radiating source,” in Optical Fiber Communication Conference, (OSA, 2014), p. Tu2A.3.

Agrawal, G. P.

G. P. Agrawal, Optical Receivers, 3rd ed. (John Wiley & Sons, 2002), vol. 6, chap. 4.

Baehr-Jones, T.

Baets, R.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

Bergman, K.

Bishop, C. M.

C. M. Bishop, Pattern Recognition and Machine Learning, 1st ed. (Springer Science+Business, 2006).

Bogaerts, W.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

Bowers, J. E.

Bowers, S.

S. Bowers, B. Abiri, F. Aflatouni, and A. Hajimiri, “A compact optically driven travelling-wave radiating source,” in Optical Fiber Communication Conference, (OSA, 2014), p. Tu2A.3.

Chang, C.-M.

Chen, H.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

Chen, R. T.

Chen, Y.-K.

Daniel, L.

Danziger, S.

De Coster, J.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

De Heyn, P.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

de Valicourt, G.

Doerr, C. R.

C. R. Doerr, “Silicon photonic integration in telecommunications,” Front. Phys. 3, 37 (2015).
[Crossref]

Dong, P.

Duff, S. M.

Dumon, P.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

Eriksson, U.

R. Lewén, S. Irmscher, and U. Eriksson, “Microwave cad circuit modeling of a traveling-wave electroabsorption modulator,” IEEE Trans. Microwave Theory Tech. 51, 1117–1128 (2003).
[Crossref]

Gelman, A.

A. Gelman, X.-L. Meng, and H. Stern, “Posterior predictive assessment of model fitness via realized discrepancies,” Statistica Sinica 6, 733–807 (1996).

Gilks, W. R.

W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Markov Chain Monte Carlo in Practice (Chapman & Hall, 1996).

Grein, M. E.

Guan, H.

Hai, M. S.

M. S. Hai, M. Ménard, and O. Liboiron-Ladouceur, “A 20 Gb/s SiGe photoreceiver based on optical time sampling,” in European Conference on Optical Communications, (2015), p. Tu.1.3.5.

Hajimiri, A.

S. Bowers, B. Abiri, F. Aflatouni, and A. Hajimiri, “A compact optically driven travelling-wave radiating source,” in Optical Fiber Communication Conference, (OSA, 2014), p. Tu2A.3.

Halkias, G.

Hanawa, I.

G. Wang, T. Tokumitsu, I. Hanawa, K. Sato, and M. Kobayashi, “Analysis of high speed p-i-n photodiode S-parameters by a novel small-signal equivalent circuit model,” IEEE Microwave and Wireless Components Lett. 12, 378–380 (2002).
[Crossref]

Hochberg, M.

Hosseini, A.

Irmscher, S.

R. Lewén, S. Irmscher, and U. Eriksson, “Microwave cad circuit modeling of a traveling-wave electroabsorption modulator,” IEEE Trans. Microwave Theory Tech. 51, 1117–1128 (2003).
[Crossref]

Ismail, M.

C. Michael and M. Ismail, “Statistical modeling of device mismatch for analog MOS integrated circuits,” IEEE J. Solid-State Circuits 27, 154–166 (1992).
[Crossref]

Jen, A. K.-Y.

Juodawlkis, P. W.

Klamkin, J.

Kobayashi, M.

G. Wang, T. Tokumitsu, I. Hanawa, K. Sato, and M. Kobayashi, “Analysis of high speed p-i-n photodiode S-parameters by a novel small-signal equivalent circuit model,” IEEE Microwave and Wireless Components Lett. 12, 378–380 (2002).
[Crossref]

Kyriakis-Bitzaros, E. D.

Lepage, G.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

Lewén, R.

R. Lewén, S. Irmscher, and U. Eriksson, “Microwave cad circuit modeling of a traveling-wave electroabsorption modulator,” IEEE Trans. Microwave Theory Tech. 51, 1117–1128 (2003).
[Crossref]

Liboiron-Ladouceur, O.

M. S. Hai, M. Ménard, and O. Liboiron-Ladouceur, “A 20 Gb/s SiGe photoreceiver based on optical time sampling,” in European Conference on Optical Communications, (2015), p. Tu.1.3.5.

Liu, Y.

Loh, W.

Lovati, E.

D. Melati, E. Lovati, and A. Melloni, “Statistical process design kits: analysis of fabrication tolerances in integrated photonic circuits,” in Advanced Photonics 2015, (OSA, 2015), p. IT4A.5.
[Crossref]

Luo, J.

Ma, Y.

Maleki, L.

Marzouk, Y.

Masood, A.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

Melati, D.

D. Melati, E. Lovati, and A. Melloni, “Statistical process design kits: analysis of fabrication tolerances in integrated photonic circuits,” in Advanced Photonics 2015, (OSA, 2015), p. IT4A.5.
[Crossref]

Melloni, A.

T.-W. Weng, Z. Zhang, Z. Su, Y. Marzouk, A. Melloni, and L. Daniel, “Uncertainty quantification of silicon photonic devices with correlated and non-Gaussian random parameters,” Opt. Express 23, 4242–4254 (2015).
[Crossref] [PubMed]

D. Melati, E. Lovati, and A. Melloni, “Statistical process design kits: analysis of fabrication tolerances in integrated photonic circuits,” in Advanced Photonics 2015, (OSA, 2015), p. IT4A.5.
[Crossref]

Ménard, M.

M. S. Hai, M. Ménard, and O. Liboiron-Ladouceur, “A 20 Gb/s SiGe photoreceiver based on optical time sampling,” in European Conference on Optical Communications, (2015), p. Tu.1.3.5.

Meng, X.-L.

A. Gelman, X.-L. Meng, and H. Stern, “Posterior predictive assessment of model fitness via realized discrepancies,” Statistica Sinica 6, 733–807 (1996).

Michael, C.

C. Michael and M. Ismail, “Statistical modeling of device mismatch for analog MOS integrated circuits,” IEEE J. Solid-State Circuits 27, 154–166 (1992).
[Crossref]

Minoglou, K.

O’Donnell, F. J.

Ocheltree, S.

Pantouvaki, M.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

Piels, M.

Plant, J. J.

Richardson, S.

W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Markov Chain Monte Carlo in Practice (Chapman & Hall, 1996).

Sato, K.

G. Wang, T. Tokumitsu, I. Hanawa, K. Sato, and M. Kobayashi, “Analysis of high speed p-i-n photodiode S-parameters by a novel small-signal equivalent circuit model,” IEEE Microwave and Wireless Components Lett. 12, 378–380 (2002).
[Crossref]

Sinsky, J. H.

Spiegelhalter, D. J.

W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Markov Chain Monte Carlo in Practice (Chapman & Hall, 1996).

Stern, H.

A. Gelman, X.-L. Meng, and H. Stern, “Posterior predictive assessment of model fitness via realized discrepancies,” Statistica Sinica 6, 733–807 (1996).

Su, Z.

Subbaraman, H.

Syvridis, D.

Sze, S. M.

S. M. Sze, Physics of Semiconductor Devices, 2nd ed. (John Wiley & Sons, 1981).

Tarantola, A.

A. Tarantola, Inverse Problem Theory (SIAM, 2005).

Tokumitsu, T.

G. Wang, T. Tokumitsu, I. Hanawa, K. Sato, and M. Kobayashi, “Analysis of high speed p-i-n photodiode S-parameters by a novel small-signal equivalent circuit model,” IEEE Microwave and Wireless Components Lett. 12, 378–380 (2002).
[Crossref]

Van Campenhout, J.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

Van Thourhout, D.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

Verheyen, P.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

Wang, G.

G. Wang, T. Tokumitsu, I. Hanawa, K. Sato, and M. Kobayashi, “Analysis of high speed p-i-n photodiode S-parameters by a novel small-signal equivalent circuit model,” IEEE Microwave and Wireless Components Lett. 12, 378–380 (2002).
[Crossref]

Wang, S.

Weng, T.-W.

Yang, Y.

Yao, X. S.

Yegnanarayanan, S.

Zhan, Q.

Zhang, X.

Zhang, Z.

Front. Phys. (1)

C. R. Doerr, “Silicon photonic integration in telecommunications,” Front. Phys. 3, 37 (2015).
[Crossref]

IEEE J. Solid-State Circuits (1)

C. Michael and M. Ismail, “Statistical modeling of device mismatch for analog MOS integrated circuits,” IEEE J. Solid-State Circuits 27, 154–166 (1992).
[Crossref]

IEEE Microwave and Wireless Components Lett. (1)

G. Wang, T. Tokumitsu, I. Hanawa, K. Sato, and M. Kobayashi, “Analysis of high speed p-i-n photodiode S-parameters by a novel small-signal equivalent circuit model,” IEEE Microwave and Wireless Components Lett. 12, 378–380 (2002).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

R. Lewén, S. Irmscher, and U. Eriksson, “Microwave cad circuit modeling of a traveling-wave electroabsorption modulator,” IEEE Trans. Microwave Theory Tech. 51, 1117–1128 (2003).
[Crossref]

J. Lightwave Technol. (3)

Opt. Express (4)

Opt. Lett. (1)

Statistica Sinica (1)

A. Gelman, X.-L. Meng, and H. Stern, “Posterior predictive assessment of model fitness via realized discrepancies,” Statistica Sinica 6, 733–807 (1996).

Other (10)

D. Melati, E. Lovati, and A. Melloni, “Statistical process design kits: analysis of fabrication tolerances in integrated photonic circuits,” in Advanced Photonics 2015, (OSA, 2015), p. IT4A.5.
[Crossref]

Cascade Microtech, Calibration Tools: Consistent Parameter Extraction for Advanced RF Devices (Cascade Microtech, 2012).

W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Markov Chain Monte Carlo in Practice (Chapman & Hall, 1996).

A. Tarantola, Inverse Problem Theory (SIAM, 2005).

C. M. Bishop, Pattern Recognition and Machine Learning, 1st ed. (Springer Science+Business, 2006).

M. S. Hai, M. Ménard, and O. Liboiron-Ladouceur, “A 20 Gb/s SiGe photoreceiver based on optical time sampling,” in European Conference on Optical Communications, (2015), p. Tu.1.3.5.

S. Bowers, B. Abiri, F. Aflatouni, and A. Hajimiri, “A compact optically driven travelling-wave radiating source,” in Optical Fiber Communication Conference, (OSA, 2014), p. Tu2A.3.

P. Verheyen, M. Pantouvaki, J. Van Campenhout, P. P. Absil, H. Chen, P. De Heyn, G. Lepage, J. De Coster, P. Dumon, A. Masood, D. Van Thourhout, R. Baets, and W. Bogaerts, “Highly uniform 25 Gb/s Si photonics platform for high-density, low-power WDM optical interconnects,” in Advanced Photonics for Communications, (OSA, 2014), p. IW3A.4.

S. M. Sze, Physics of Semiconductor Devices, 2nd ed. (John Wiley & Sons, 1981).

G. P. Agrawal, Optical Receivers, 3rd ed. (John Wiley & Sons, 2002), vol. 6, chap. 4.

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Figures (6)

Fig. 1
Fig. 1 Small-circuit model of a photodiode. The intrinsic diode impedance is shown at left, while pad parasitics are on the right.
Fig. 2
Fig. 2 Microwave reflection data and error margins in the (a) S11, (b) Z, and (c) Y domains. The center curve in each figure is smoothed measured data, while the upper and lower bounds come from the worst case (in the mean-squared sense) error possible for that measurement. The directivity was 37 dB, the microwave propagation constant of the pads was 7.5e-9 m−1, and the misplacement is 25 μm
Fig. 3
Fig. 3 Influence of Markov chain design parameters on convergence and output. (a) Convergence and final acceptance rates for different proposal distribution widths. (b) Probability density function of the diode capacitance for different proposal distribution widths and problem formulations. Requiring a single probe placement error over the entire fitting range results in a narrower PDF.
Fig. 4
Fig. 4 (a) Joint probability density functions of series resistance and parallel capacitance when all parameters are fit simultaneously to the response of a single device. The joint PDF is shown on the bottom plane, with the MCMC result shown as a scatter plot and the NLLS results shown as contours. The back two planes show 1D PDFs of the parameters separately. (b, c) Original data and both nonlinear least squares maximum likelihood fits on the Smith chart.
Fig. 5
Fig. 5 Probability density functions for (a) series resistance · unit length, (b) diode capacitance per unit length, (c) pad capacitance, and (d) parallel resistance calculated using least-squares fitting (LS) and Markov chain Monte Carlo (MCMC). The means of each distribution are indicated with circular markers.
Fig. 6
Fig. 6 Cross-validation predictive densities of complex photodiode impedance at 20 GHz calculated by (a) least squares and (b) Markov chain Monte Carlo. The measured value (which is the same in both figures) is also shown.

Tables (1)

Tables Icon

Table 1 Pseudo-code for Gibbs sampler.

Equations (19)

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

S 11 ( ω ) = Z t o t ( ω ) Z 0 Z t o t ( ω ) + Z 0
Z P D ( ω ) = j ω C p R s + R s + R p j ω C p R p
S 11 , m e a n s ( ω ) = e 2 j ω β Δ S 11 ( ω ) ,
d = g ( m )
σ ( m ) = k ρ ( d | m ) ρ M ( m )
ρ ( d | m ) = k exp ( 1 2 ( d g ( m ) ) T C D 1 ( d g ( m ) ) ) .
r = d g ( m ¯ )
P = { k ρ ( d | w ) ρ M ( w ) q ( x n | w ) k ρ ( d | x n ) ρ M ( x n ) q ( w | x n ) , 1 }
B ϕ = N M 1 i = 1 M ( ϕ i ¯ ϕ ¯ ) 2 where ϕ i ¯ = m e a n ( ϕ i j ) and ϕ ¯ = m e a n ( ϕ i )
W ϕ = 1 M i = 1 M s i 2 where s i 2 = v a r ( ϕ i j ) .
σ ϕ 2 ^ = N 1 N W ϕ + 1 N B ϕ .
β = σ ϕ 2 ^ W ϕ .
C M ˜ = ( G C D 1 G T + C M , prior 1 ) 1
m ¯ = [ C p ¯ / C p 0 R p ¯ / R p 0 R s ¯ / R s 0 C p a d ¯ / C p a d 0 Δ ¯ / Δ 0 ]
m ¯ N ( m true ¯ , C ) .
ν n + 1 = ν n + N P D
Ξ n + 1 = Ξ n + 1 2 i = 1 N P D ( m i m ¯ ) 2
m ¯ n + 1 | m i , n N ( m ¯ , C n + 1 ) .
C n + 1 | m i , n W 1 ( Ξ n + 1 , ν n + 1 )

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