Abstract

A novel and simple approach to optical wavelength measurement is presented in this paper. The working principle is demonstrated using a tunable waveguide micro ring resonator and single photodiode. The initial calibration is done with a set of known wavelengths and resonator tunings. The combined spectral sensitivity function of the resonator and photodiode at each tuning voltage was modeled by a neural network. For determining the unknown wavelengths, the resonator was tuned with a set of heating voltages and the corresponding photodiode signals were collected. The unknown wavelength was estimated, based on the collected photodiode signals, the calibrated neural networks, and an optimization algorithm. The wavelength estimate method provides a high spectral precision of about 8 pm (5·10−6 at 1550 nm) in the wavelength range between 1549 nm to 1553 nm. A higher precision of 5 pm (3·10−6) is achieved in the range between 1550.3 nm to 1550.8 nm, which is a factor of five improved compared to a simple lookup of data. The importance of our approach is that it strongly simplifies the optical system and enables optical integration. The approach is also of general importance, because it may be applicable to all wavelength monitoring devices which show an adjustable wavelength response.

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2012

2011

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett.89, 201113 (2006).

2008

K. Richard, P. Manson, and P. Ewart, “A widely tunable, high power, single-mode laser for linear and nonlinear spectroscopy,” Meas. Sci. Tech.19, 015603 (2008).
[CrossRef]

2007

2006

2005

G. Wysocki, R. F. Curl, F. K. Tittel, R. Maulini, J. M. Bulliard, and J. Faist, “Widely tunable mode-hop free external cavity quantum cascade laser for high resolution spectroscopic applications,” Appl. Phys. B-Lasers O.81(6), 769–777 (2005).
[CrossRef]

I. Lindsay, B. Adhimoolam, P. Gross, M. Klein, and K. Boller, “110GHz rapid, continuous tuning from an optical parametric oscillator pumped by a fiber-amplified DBR diode laser,” Opt. Express13(4), 1234–1239 (2005).
[CrossRef] [PubMed]

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

2003

C. Sookdhis, T. Mei, H. S. Djie, and J. Arokiaraj, “Passive wavelength monitor based on multimode interference waveguide,” Opt. Eng.42(12), 3421–3422 (2003).
[CrossRef]

2002

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature416(6877), 233–237 (2002).
[CrossRef] [PubMed]

2001

A. Banerjee, U. D. Rapol, A. Wasan, and V. Natarajan, “High-accuracy wavemeter based on a stabilized diode laser,” Appl. Phys. Lett.79, 2139–2141 (2001).
[CrossRef]

1998

X. J. Gu, “Wavelength-division multiplexing isolation fiber filter and light source using cascaded long-period fiber gratings,” Opt. Lett.23(7), 509–510 (1998).
[CrossRef]

B. Mason, S. P. DenBaars, and L. A. Coldren, “Tunable sampled-grating DBR lasers with integrated wavelength monitors,” IEEE Photonic. Tech. L.10(8), 1085–1087 (1998).
[CrossRef]

1995

J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, “Non-linear black-box modeling in system identification: a unified overview,” Automatica31(12), 1691–1724 (1995).
[CrossRef]

J. B. Cooper, P. E. Flecher, S. Albin, T. M. Vess, and W. T. Welch, “Elimination of mode hopping and frequency hysteresis in diode laser raman spectroscopy: the advantages of a distributed bragg reflector diode laser for raman excitation,” Appl. Spectrosc.49, 1692–1698 (1995).
[CrossRef]

1992

J. F. de Boer, M. P. van Albada, and A. Lagendijk, “Transmission and intensity correlations in wave propagation through random media,” Phys. Rev. B45, 658–666 (1992).
[CrossRef]

1987

R. Lippmann, “An introduction to computing with neural nets,” IEEE ASSP Magazine4(2), 4–22 (1987).
[CrossRef]

Adhimoolam, B.

Albin, S.

Arokiaraj, J.

C. Sookdhis, T. Mei, H. S. Djie, and J. Arokiaraj, “Passive wavelength monitor based on multimode interference waveguide,” Opt. Eng.42(12), 3421–3422 (2003).
[CrossRef]

Banerjee, A.

A. Banerjee, U. D. Rapol, A. Wasan, and V. Natarajan, “High-accuracy wavemeter based on a stabilized diode laser,” Appl. Phys. Lett.79, 2139–2141 (2001).
[CrossRef]

Beale, M.

H. Demuth, M. Beale, and M. Hagan, MATLAB Neural Network Toolbox 5 User’s Guide (The MathWorks, Inc., 2007).

Benisty, H.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Benveniste, A.

J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, “Non-linear black-box modeling in system identification: a unified overview,” Automatica31(12), 1691–1724 (1995).
[CrossRef]

Bergman, K.

Boller, K.

Borreman, A.

Bulliard, J. M.

G. Wysocki, R. F. Curl, F. K. Tittel, R. Maulini, J. M. Bulliard, and J. Faist, “Widely tunable mode-hop free external cavity quantum cascade laser for high resolution spectroscopic applications,” Appl. Phys. B-Lasers O.81(6), 769–777 (2005).
[CrossRef]

Cao, H.

Coldren, L. A.

B. Mason, S. P. DenBaars, and L. A. Coldren, “Tunable sampled-grating DBR lasers with integrated wavelength monitors,” IEEE Photonic. Tech. L.10(8), 1085–1087 (1998).
[CrossRef]

Cooper, J. B.

Cuisin, C.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Curl, R. F.

G. Wysocki, R. F. Curl, F. K. Tittel, R. Maulini, J. M. Bulliard, and J. Faist, “Widely tunable mode-hop free external cavity quantum cascade laser for high resolution spectroscopic applications,” Appl. Phys. B-Lasers O.81(6), 769–777 (2005).
[CrossRef]

de Boer, J. F.

J. F. de Boer, M. P. van Albada, and A. Lagendijk, “Transmission and intensity correlations in wave propagation through random media,” Phys. Rev. B45, 658–666 (1992).
[CrossRef]

Delyon, B.

J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, “Non-linear black-box modeling in system identification: a unified overview,” Automatica31(12), 1691–1724 (1995).
[CrossRef]

Demuth, H.

H. Demuth, M. Beale, and M. Hagan, MATLAB Neural Network Toolbox 5 User’s Guide (The MathWorks, Inc., 2007).

DenBaars, S. P.

B. Mason, S. P. DenBaars, and L. A. Coldren, “Tunable sampled-grating DBR lasers with integrated wavelength monitors,” IEEE Photonic. Tech. L.10(8), 1085–1087 (1998).
[CrossRef]

Derouin, E.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Djie, H. S.

C. Sookdhis, T. Mei, H. S. Djie, and J. Arokiaraj, “Passive wavelength monitor based on multimode interference waveguide,” Opt. Eng.42(12), 3421–3422 (2003).
[CrossRef]

Drisse, O.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Duan, G-H

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Ewart, P.

K. Richard, P. Manson, and P. Ewart, “A widely tunable, high power, single-mode laser for linear and nonlinear spectroscopy,” Meas. Sci. Tech.19, 015603 (2008).
[CrossRef]

Faist, J.

G. Wysocki, R. F. Curl, F. K. Tittel, R. Maulini, J. M. Bulliard, and J. Faist, “Widely tunable mode-hop free external cavity quantum cascade laser for high resolution spectroscopic applications,” Appl. Phys. B-Lasers O.81(6), 769–777 (2005).
[CrossRef]

Farrell, G.

Flecher, P. E.

Freir, T.

Gaeta, A. L.

Geuzebroek, D. H.

Glorennec, P.

J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, “Non-linear black-box modeling in system identification: a unified overview,” Automatica31(12), 1691–1724 (1995).
[CrossRef]

Golka, S.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Gross, P.

Gu, X. J.

Hagan, M.

H. Demuth, M. Beale, and M. Hagan, MATLAB Neural Network Toolbox 5 User’s Guide (The MathWorks, Inc., 2007).

Hänsch, T. W.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature416(6877), 233–237 (2002).
[CrossRef] [PubMed]

Haykin, S.

S. Haykin, Neural Networks: a Comprehensive Foundation (Macmillan, 1994).

Heideman, R. G.

Heidrich, H.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Hensel, H. J.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Hjalmarsson, H.

J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, “Non-linear black-box modeling in system identification: a unified overview,” Automatica31(12), 1691–1724 (1995).
[CrossRef]

Holzwarth, R.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature416(6877), 233–237 (2002).
[CrossRef] [PubMed]

Janiak, K.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Johnson, N.

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett.89, 201113 (2006).

Juditsky, A.

J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, “Non-linear black-box modeling in system identification: a unified overview,” Automatica31(12), 1691–1724 (1995).
[CrossRef]

Kiesel, P.

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett.89, 201113 (2006).

Klein, M.

Kung, S. Y.

S. Y. Kung, Digital Neural Networks (Prentice-Hall, 1993).

Lagendijk, A.

J. F. de Boer, M. P. van Albada, and A. Lagendijk, “Transmission and intensity correlations in wave propagation through random media,” Phys. Rev. B45, 658–666 (1992).
[CrossRef]

Lau, R. K. W.

Legouézigou, L.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Legouézigou, O.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Leinse, A.

Lindsay, I.

Lippmann, R.

R. Lippmann, “An introduction to computing with neural nets,” IEEE ASSP Magazine4(2), 4–22 (1987).
[CrossRef]

Lipson, M.

Ljung, L.

J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, “Non-linear black-box modeling in system identification: a unified overview,” Automatica31(12), 1691–1724 (1995).
[CrossRef]

Malzer, S.

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett.89, 201113 (2006).

Manson, P.

K. Richard, P. Manson, and P. Ewart, “A widely tunable, high power, single-mode laser for linear and nonlinear spectroscopy,” Meas. Sci. Tech.19, 015603 (2008).
[CrossRef]

Martinelli, M.

Mason, B.

B. Mason, S. P. DenBaars, and L. A. Coldren, “Tunable sampled-grating DBR lasers with integrated wavelength monitors,” IEEE Photonic. Tech. L.10(8), 1085–1087 (1998).
[CrossRef]

Maulini, R.

G. Wysocki, R. F. Curl, F. K. Tittel, R. Maulini, J. M. Bulliard, and J. Faist, “Widely tunable mode-hop free external cavity quantum cascade laser for high resolution spectroscopic applications,” Appl. Phys. B-Lasers O.81(6), 769–777 (2005).
[CrossRef]

Mei, T.

C. Sookdhis, T. Mei, H. S. Djie, and J. Arokiaraj, “Passive wavelength monitor based on multimode interference waveguide,” Opt. Eng.42(12), 3421–3422 (2003).
[CrossRef]

Melloni, A.

Menard, M.

Mohta, S.

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett.89, 201113 (2006).

Morichetti, F.

Natarajan, V.

A. Banerjee, U. D. Rapol, A. Wasan, and V. Natarajan, “High-accuracy wavemeter based on a stabilized diode laser,” Appl. Phys. Lett.79, 2139–2141 (2001).
[CrossRef]

Okawachi, Y.

Ophir, N.

Padmaraju, K.

Pommereau, F.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Rajan, G.

Rapol, U. D.

A. Banerjee, U. D. Rapol, A. Wasan, and V. Natarajan, “High-accuracy wavemeter based on a stabilized diode laser,” Appl. Phys. Lett.79, 2139–2141 (2001).
[CrossRef]

Redding, B.

Richard, K.

K. Richard, P. Manson, and P. Ewart, “A widely tunable, high power, single-mode laser for linear and nonlinear spectroscopy,” Meas. Sci. Tech.19, 015603 (2008).
[CrossRef]

Salem, R.

Schmidt, O.

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett.89, 201113 (2006).

Sjöberg, J.

J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, “Non-linear black-box modeling in system identification: a unified overview,” Automatica31(12), 1691–1724 (1995).
[CrossRef]

Sookdhis, C.

C. Sookdhis, T. Mei, H. S. Djie, and J. Arokiaraj, “Passive wavelength monitor based on multimode interference waveguide,” Opt. Eng.42(12), 3421–3422 (2003).
[CrossRef]

Tittel, F. K.

G. Wysocki, R. F. Curl, F. K. Tittel, R. Maulini, J. M. Bulliard, and J. Faist, “Widely tunable mode-hop free external cavity quantum cascade laser for high resolution spectroscopic applications,” Appl. Phys. B-Lasers O.81(6), 769–777 (2005).
[CrossRef]

Udem, T.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature416(6877), 233–237 (2002).
[CrossRef] [PubMed]

van Albada, M. P.

J. F. de Boer, M. P. van Albada, and A. Lagendijk, “Transmission and intensity correlations in wave propagation through random media,” Phys. Rev. B45, 658–666 (1992).
[CrossRef]

Verdult, V.

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).
[CrossRef]

Verhaegen, M.

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).
[CrossRef]

Vess, T. M.

Viasnoff-Schwoob, E.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G-H Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral outcoupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett.86, 101107 (2005).
[CrossRef]

Wang, P.

Wang, Q.

Wasan, A.

A. Banerjee, U. D. Rapol, A. Wasan, and V. Natarajan, “High-accuracy wavemeter based on a stabilized diode laser,” Appl. Phys. Lett.79, 2139–2141 (2001).
[CrossRef]

Weisbuch, C.

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Figures (10)

Fig. 1
Fig. 1

General scheme of the wavemeter. Laser light with wavelength λx and power Px passes the tunable color filter, i.e., a micro-ring resonator (MRR) in this paper. The spectral transmission function of the MRR can be changed by applying a heater voltage vk. The transmitted light is collected by a photodiode, which yields a measurement signal yk.

Fig. 2
Fig. 2

Microscope picture of the waveguide chip (left) with micro-ring resonators (MRRs) and a schematic drawing (right) of the same chip. ”R” denotes the MRR with the heater on top (gray). ”E” denotes electrical contacts (gold colored) and the black lines represent waveguides. The fiber from the tunable laser was connected to the MRR from ”IN”. ”PD” denotes the position of the photodiode. Since only one MRR was used at the same time, there was no crosstalk between the MRR and the heaters of neighboring MRRs.

Fig. 3
Fig. 3

Spectral sensitivity function S(λ, v) at different v. ”+”: calibration data; lines: curves fitted by neural networks. For clarity, curves are shown only for five heating voltages v =0 V, 2.5 V, 5.0 V, 7.5 V and 10 V.

Fig. 4
Fig. 4

Variance accounted for (VAF) of the neural networks with respect to the number of neurons (Q). The VAF in the vertical axis is averaged over VAF of all 21 neural networks. As more neurons are used, the accuracy of the neural network increases until Q = 14.

Fig. 5
Fig. 5

Wavelength estimation error λ̂xλx in the test set.

Fig. 6
Fig. 6

Histogram of the wavelength estimation error. The maximal estimation error for the test set is 22 pm. 95% of the wavelength estimation errors are below 8 pm in the range between 1549 nm and 1553 nm.

Fig. 7
Fig. 7

Histogram of the relative power estimation error (xpx)/px in the test data set. The relative power estimation error is at maximum 1.6%, and in 95% of the cases less than 0.8%.

Fig. 8
Fig. 8

Comparison between LUT and the proposed NN+NLLS method. 95% of the errors fall below 27 pm for LUT and 5 pm for the proposed method (about one fifth of that for LUT).

Fig. 9
Fig. 9

Mean of the wavelength estimation error decreases as more data points are used in the estimation.

Fig. 10
Fig. 10

Influence of noise on the wavelength estimation error. 95% of the wavelength estimation error is still within 10 pm when the standard deviation of the noise is 22.5 mV

Equations (6)

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y k = P x f ( λ x , v k ) d ( λ x ) + η k = P x S ( λ x , v k ) + η k .
y k ( λ ) = P S ^ ( λ , v k ) = P i = 1 Q w 1 i k tanh ( w 2 i k λ + s 1 i k ) + s 2 i k .
Y [ y 1 y 2 y N ] = [ P x S ( λ x , v 1 ) + η 1 P x S ( λ x , v 2 ) + η 2 P x S ( λ x , v N ) + η N ] .
( λ ^ x , P ^ x ) = arg lim λ ^ x , P ^ x 1 N | Y Y ^ | 2 J ,
Y ^ [ y ^ 1 y ^ 2 y ^ N ] = [ P ^ x S ^ ( λ ^ x , v 1 ) P ^ x S ^ ( λ ^ x , v 2 ) P ^ x S ^ ( λ ^ x , v N ) ] .
A ( y ^ k , y k ) = ( 1 var ( y ^ k y k ) var ( y k ) ) × 100 % .

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