Abstract

We explore photonic ADC architectures based on encoding voltage-under-test into phase. The first step is to identify two basic optical building blocks: the optical phase comparator (1-bit ADC), based on interferometric comparison of phases in the well-known balanced photo-detection configuration, and the optical 1-bit DAC, namely electro-optic modulation with a bipolar electrical pulse. Equipped with these fundamental building blocks, we proceed to systematically port and adapt known ADC quantization architectures to photonic ADC, conceiving a hybrid between the Successive Approximation Register (SAR) and the Pipeline classic ADC architectures, referred to here as Spatially Distributed SAR (SDSAR). This novel photonic ADC, constructed out of B 1-bit ADCs and B-2 1-bit DACs, with B the number of bits, is not equivalent to any of the previous photonic ADCs in the literature, but appears superior to prior schemes in both optical power efficiency and electro-optic modulation complexity. We derive upper bounds on resolution, Effective Number of Bits (ENOB) performance as a function of average optical power for the new SDSAR device, developing analytic and numeric Monte-Carlo statistical models, comprising quantization, shot, thermal and DAC voltage noise sources. Our findings indicate that SDSAR is limited to ~11.5 ENOBs, assuming state-of-the-art mode-locked-lasers providing ~250 mW of average power (assuming ~7 dB excess losses). However, this upper bound is not tight, due to various physical impairments. In particular, the mode locked laser jitter is shown to have negligible impact on overall performance for RMS jitter < 20 fsec.

© 2012 OSA

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2011 (6)

L. Ma, P. Ghelfi, M. Yao, F. Berizzi, and A. Bogoni, “Demonstration of optical sample parallelisation for high-speed photonic assisted ADCs,” Electron. Lett. 47(5), 333–335 (2011).
[CrossRef]

R. Llorente, M. Morant, N. Amiot, and B. Uguen, “Novel photonic analog-to-digital converter architecture for precise localization of ultra-wide band radio transmitters,” IEEE J. Sel. Areas Comm. 29(6), 1321–1327 (2011).
[CrossRef]

P. Ghelfi, F. Scotti, A. T. Nguyen, G. Serafino, and A. Bogoni, “Novel architecture for a photonics-assisted radar transceiver based on a single mode-locking laser,” IEEE Photon. Technol. Lett. 23(10), 639–641 (2011).
[CrossRef]

S. Khan, M. A. Baghban, and S. Fathpour, “Electronically tunable silicon photonic delay lines,” Opt. Express 19(12), 11780–11785 (2011).
[CrossRef] [PubMed]

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Two-dimensional free-space beam steering with an optical phased array on silicon-on-insulator,” Opt. Express 19(22), 21595–21604 (2011).
[CrossRef] [PubMed]

J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Planar waveguides with less than 0.1 dB/m propagation loss fabricated with wafer bonding,” Opt. Express 19(24), 24090–24101 (2011).
[CrossRef] [PubMed]

2010 (3)

G. A. Sefler, J. Chou, J. A. Conway, and G. C. Valley, “Distortion correction in a high-resolution time-stretch ADC scalable to continuous time,” J. Lightwave Technol. 28(10), 1468–1476 (2010).
[CrossRef]

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O'Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay Lines in silicon photonics: Coupled resonators and photonic crystals, a comparison,” IEEE Photon. J. 2(2), 181–194 (2010).
[CrossRef]

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral Filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron. 16(1), 33–44 (2010).
[CrossRef]

2009 (4)

2008 (3)

2007 (2)

W. Li, H. Zhang, Q. Wu, Z. Zhang, and M. Yao, “All-optical analog-to-digital conversion based on polarization-differential interference and phase modulation,” IEEE Photon. Technol. Lett. 19(8), 625–627 (2007).
[CrossRef]

G. C. Valley, “Photonic analog-to-digital converters,” Opt. Express 15(5), 1955–1982 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (3)

2003 (3)

U. Keller, “Recent developments in compact ultrafast lasers,” Nature 424(6950), 831–838 (2003).
[CrossRef] [PubMed]

L. Y. Nathawad, R. Urata, B. A. Wooley, and D. A. B. Miller, “A 40-GHz-bandwidth, 4-bit, time-interleaved A/D converter using photoconductive sampling,” IEEE J. Solid-state Circuits 38(12), 2021–2030 (2003).
[CrossRef]

P. W. Juodawlkis, J. J. Hargreaves, R. D. Younger, G. W. Titi, and J. C. Twichell, “Optical down-sampling of wide-band microwave signals,” J. Lightwave Technol. 21(12), 3116–3124 (2003).
[CrossRef]

2001 (2)

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O’Donnell, K. G. Ray, and R. C. Williamson, “Optically sampled analog-to-digital converters,” IEEE Trans. Microw. Theory Tech. 49(10), 1840–1853 (2001).
[CrossRef]

J. C. Twichell, J. L. Wasserman, P. W. Juodawlkis, G. E. Betts, and R. C. Williamson, “High-linearity 208-MS/s photonic analog-to-digital converter using 1-to-4 optical time-division demultiplexers,” IEEE Photon. Technol. Lett. 13(7), 714–716 (2001).
[CrossRef]

2000 (1)

M. Currie, T. R. Clark, and P. J. Matthews, “Photonic analog-to-digital conversion by distributed phase modulation,” IEEE Photon. Technol. Lett. 12(12), 1689–1691 (2000).
[CrossRef]

1984 (1)

R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72(7), 802–819 (1984).
[CrossRef]

1975 (1)

H. F. Taylor, “An electrooptic analog-to-digital converter,” Proc. IEEE 63(10), 1524–1525 (1975).
[CrossRef]

Abdul, J.

Amiot, N.

R. Llorente, M. Morant, N. Amiot, and B. Uguen, “Novel photonic analog-to-digital converter architecture for precise localization of ultra-wide band radio transmitters,” IEEE J. Sel. Areas Comm. 29(6), 1321–1327 (2011).
[CrossRef]

Baets, R.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral Filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron. 16(1), 33–44 (2010).
[CrossRef]

Baghban, M. A.

Barton, J. S.

Bauters, J. F.

Becker, R. A.

R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72(7), 802–819 (1984).
[CrossRef]

Berizzi, F.

L. Ma, P. Ghelfi, M. Yao, F. Berizzi, and A. Bogoni, “Demonstration of optical sample parallelisation for high-speed photonic assisted ADCs,” Electron. Lett. 47(5), 333–335 (2011).
[CrossRef]

Betts, G. E.

J. C. Twichell, J. L. Wasserman, P. W. Juodawlkis, G. E. Betts, and R. C. Williamson, “High-linearity 208-MS/s photonic analog-to-digital converter using 1-to-4 optical time-division demultiplexers,” IEEE Photon. Technol. Lett. 13(7), 714–716 (2001).
[CrossRef]

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O’Donnell, K. G. Ray, and R. C. Williamson, “Optically sampled analog-to-digital converters,” IEEE Trans. Microw. Theory Tech. 49(10), 1840–1853 (2001).
[CrossRef]

Blumenthal, D. J.

J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Planar waveguides with less than 0.1 dB/m propagation loss fabricated with wafer bonding,” Opt. Express 19(24), 24090–24101 (2011).
[CrossRef] [PubMed]

M. J. R. Heck, G. Kurczveil, E. F. Burmeister, H. Park, J. P. Mack, D. J. Blumenthal, and J. E. Bowers, ““Integrated recirculating optical buffers,” in Proc,” Proc. SPIE 16, 11124–11131 (2008).

Bogaerts, W.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral Filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron. 16(1), 33–44 (2010).
[CrossRef]

Bogoni, A.

P. Ghelfi, F. Scotti, A. T. Nguyen, G. Serafino, and A. Bogoni, “Novel architecture for a photonics-assisted radar transceiver based on a single mode-locking laser,” IEEE Photon. Technol. Lett. 23(10), 639–641 (2011).
[CrossRef]

L. Ma, P. Ghelfi, M. Yao, F. Berizzi, and A. Bogoni, “Demonstration of optical sample parallelisation for high-speed photonic assisted ADCs,” Electron. Lett. 47(5), 333–335 (2011).
[CrossRef]

Bovington, J. T.

Bowers, J. E.

Brouckaert, J.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral Filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron. 16(1), 33–44 (2010).
[CrossRef]

Bruinink, C. M.

Burmeister, E. F.

M. J. R. Heck, G. Kurczveil, E. F. Burmeister, H. Park, J. P. Mack, D. J. Blumenthal, and J. E. Bowers, ““Integrated recirculating optical buffers,” in Proc,” Proc. SPIE 16, 11124–11131 (2008).

Canciamilla, A.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O'Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay Lines in silicon photonics: Coupled resonators and photonic crystals, a comparison,” IEEE Photon. J. 2(2), 181–194 (2010).
[CrossRef]

Chi, H.

Chou, J.

Clark, T. R.

M. Currie, T. R. Clark, and P. J. Matthews, “Photonic analog-to-digital conversion by distributed phase modulation,” IEEE Photon. Technol. Lett. 12(12), 1689–1691 (2000).
[CrossRef]

Coldren, L. A.

Conway, J. A.

Currie, M.

M. Currie, “Optical quantization of microwave signals via distributed phase modulation,” J. Lightwave Technol. 23(2), 827–833 (2005).
[CrossRef]

M. Currie, T. R. Clark, and P. J. Matthews, “Photonic analog-to-digital conversion by distributed phase modulation,” IEEE Photon. Technol. Lett. 12(12), 1689–1691 (2000).
[CrossRef]

De La Rue, R.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O'Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay Lines in silicon photonics: Coupled resonators and photonic crystals, a comparison,” IEEE Photon. J. 2(2), 181–194 (2010).
[CrossRef]

De Vos, K.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral Filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron. 16(1), 33–44 (2010).
[CrossRef]

Doylend, J. K.

Dumon, P.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral Filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron. 16(1), 33–44 (2010).
[CrossRef]

Fathpour, S.

Feldster, A.

Ferrari, C.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O'Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay Lines in silicon photonics: Coupled resonators and photonic crystals, a comparison,” IEEE Photon. J. 2(2), 181–194 (2010).
[CrossRef]

Fu, X.

Q. Wu, H. Zhang, Y. Peng, X. Fu, and M. Yao, “40GS/s Optical analog-to-digital conversion system and its improvement,” Opt. Express 17(11), 9252–9257 (2009).
[CrossRef] [PubMed]

Q. Wu, H. Zhang, X. Fu, and M. Yao, “Spectral encoded photonic analog-to-digital converter based on cascaded unbalanced MZMs,” IEEE Photon. Technol. Lett. 21(4), 224–226 (2009).
[CrossRef]

Galt, S.

J. Stigwall and S. Galt, “Demonstration and analysis of a 40-gigasample/s interferometric analog-to-digital converter,” J. Lightwave Technol. 24(3), 1247–1256 (2006).
[CrossRef]

J. Stigwall and S. Galt, “Interferometric analog-to-digital conversion scheme,” IEEE Photon. Technol. Lett. 17(2), 468–470 (2005).
[CrossRef]

Ghelfi, P.

P. Ghelfi, F. Scotti, A. T. Nguyen, G. Serafino, and A. Bogoni, “Novel architecture for a photonics-assisted radar transceiver based on a single mode-locking laser,” IEEE Photon. Technol. Lett. 23(10), 639–641 (2011).
[CrossRef]

L. Ma, P. Ghelfi, M. Yao, F. Berizzi, and A. Bogoni, “Demonstration of optical sample parallelisation for high-speed photonic assisted ADCs,” Electron. Lett. 47(5), 333–335 (2011).
[CrossRef]

Hargreaves, J. J.

P. W. Juodawlkis, J. J. Hargreaves, R. D. Younger, G. W. Titi, and J. C. Twichell, “Optical down-sampling of wide-band microwave signals,” J. Lightwave Technol. 21(12), 3116–3124 (2003).
[CrossRef]

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O’Donnell, K. G. Ray, and R. C. Williamson, “Optically sampled analog-to-digital converters,” IEEE Trans. Microw. Theory Tech. 49(10), 1840–1853 (2001).
[CrossRef]

Heck, M. J. R.

Heideman, R. G.

Horowitz, M.

Ikeda, K.

Jalali, B.

John, D. D.

Juodawlkis, P. W.

P. W. Juodawlkis, J. J. Hargreaves, R. D. Younger, G. W. Titi, and J. C. Twichell, “Optical down-sampling of wide-band microwave signals,” J. Lightwave Technol. 21(12), 3116–3124 (2003).
[CrossRef]

J. C. Twichell, J. L. Wasserman, P. W. Juodawlkis, G. E. Betts, and R. C. Williamson, “High-linearity 208-MS/s photonic analog-to-digital converter using 1-to-4 optical time-division demultiplexers,” IEEE Photon. Technol. Lett. 13(7), 714–716 (2001).
[CrossRef]

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O’Donnell, K. G. Ray, and R. C. Williamson, “Optically sampled analog-to-digital converters,” IEEE Trans. Microw. Theory Tech. 49(10), 1840–1853 (2001).
[CrossRef]

Kärtner, F. X.

Keller, U.

U. Keller, “Recent developments in compact ultrafast lasers,” Nature 424(6950), 831–838 (2003).
[CrossRef] [PubMed]

Khan, S.

Kim, J.

Kitayama, K.-I.

Krauss, T. F.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O'Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay Lines in silicon photonics: Coupled resonators and photonic crystals, a comparison,” IEEE Photon. J. 2(2), 181–194 (2010).
[CrossRef]

Kurczveil, G.

M. J. R. Heck, G. Kurczveil, E. F. Burmeister, H. Park, J. P. Mack, D. J. Blumenthal, and J. E. Bowers, ““Integrated recirculating optical buffers,” in Proc,” Proc. SPIE 16, 11124–11131 (2008).

Leinse, A.

Leonberger, F. J.

R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72(7), 802–819 (1984).
[CrossRef]

Li, W.

W. Li, H. Zhang, Q. Wu, Z. Zhang, and M. Yao, “All-optical analog-to-digital conversion based on polarization-differential interference and phase modulation,” IEEE Photon. Technol. Lett. 19(8), 625–627 (2007).
[CrossRef]

Llorente, R.

R. Llorente, M. Morant, N. Amiot, and B. Uguen, “Novel photonic analog-to-digital converter architecture for precise localization of ultra-wide band radio transmitters,” IEEE J. Sel. Areas Comm. 29(6), 1321–1327 (2011).
[CrossRef]

Ma, L.

L. Ma, P. Ghelfi, M. Yao, F. Berizzi, and A. Bogoni, “Demonstration of optical sample parallelisation for high-speed photonic assisted ADCs,” Electron. Lett. 47(5), 333–335 (2011).
[CrossRef]

Mack, J. P.

M. J. R. Heck, G. Kurczveil, E. F. Burmeister, H. Park, J. P. Mack, D. J. Blumenthal, and J. E. Bowers, ““Integrated recirculating optical buffers,” in Proc,” Proc. SPIE 16, 11124–11131 (2008).

Matthews, P. J.

M. Currie, T. R. Clark, and P. J. Matthews, “Photonic analog-to-digital conversion by distributed phase modulation,” IEEE Photon. Technol. Lett. 12(12), 1689–1691 (2000).
[CrossRef]

Melloni, A.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O'Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay Lines in silicon photonics: Coupled resonators and photonic crystals, a comparison,” IEEE Photon. J. 2(2), 181–194 (2010).
[CrossRef]

Miller, D. A. B.

L. Y. Nathawad, R. Urata, B. A. Wooley, and D. A. B. Miller, “A 40-GHz-bandwidth, 4-bit, time-interleaved A/D converter using photoconductive sampling,” IEEE J. Solid-state Circuits 38(12), 2021–2030 (2003).
[CrossRef]

Morant, M.

R. Llorente, M. Morant, N. Amiot, and B. Uguen, “Novel photonic analog-to-digital converter architecture for precise localization of ultra-wide band radio transmitters,” IEEE J. Sel. Areas Comm. 29(6), 1321–1327 (2011).
[CrossRef]

Morichetti, F.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O'Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay Lines in silicon photonics: Coupled resonators and photonic crystals, a comparison,” IEEE Photon. J. 2(2), 181–194 (2010).
[CrossRef]

Namiki, S.

Nathawad, L. Y.

L. Y. Nathawad, R. Urata, B. A. Wooley, and D. A. B. Miller, “A 40-GHz-bandwidth, 4-bit, time-interleaved A/D converter using photoconductive sampling,” IEEE J. Solid-state Circuits 38(12), 2021–2030 (2003).
[CrossRef]

Nguyen, A. T.

P. Ghelfi, F. Scotti, A. T. Nguyen, G. Serafino, and A. Bogoni, “Novel architecture for a photonics-assisted radar transceiver based on a single mode-locking laser,” IEEE Photon. Technol. Lett. 23(10), 639–641 (2011).
[CrossRef]

O’Donnell, F. J.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O’Donnell, K. G. Ray, and R. C. Williamson, “Optically sampled analog-to-digital converters,” IEEE Trans. Microw. Theory Tech. 49(10), 1840–1853 (2001).
[CrossRef]

O'Faolain, L.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O'Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay Lines in silicon photonics: Coupled resonators and photonic crystals, a comparison,” IEEE Photon. J. 2(2), 181–194 (2010).
[CrossRef]

Park, H.

M. J. R. Heck, G. Kurczveil, E. F. Burmeister, H. Park, J. P. Mack, D. J. Blumenthal, and J. E. Bowers, ““Integrated recirculating optical buffers,” in Proc,” Proc. SPIE 16, 11124–11131 (2008).

Park, M. J.

Peng, Y.

Perrott, M. H.

Peters, J. D.

Ray, K. G.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O’Donnell, K. G. Ray, and R. C. Williamson, “Optically sampled analog-to-digital converters,” IEEE Trans. Microw. Theory Tech. 49(10), 1840–1853 (2001).
[CrossRef]

Rosenthal, A.

Samarelli, A.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O'Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay Lines in silicon photonics: Coupled resonators and photonic crystals, a comparison,” IEEE Photon. J. 2(2), 181–194 (2010).
[CrossRef]

Scotti, F.

P. Ghelfi, F. Scotti, A. T. Nguyen, G. Serafino, and A. Bogoni, “Novel architecture for a photonics-assisted radar transceiver based on a single mode-locking laser,” IEEE Photon. Technol. Lett. 23(10), 639–641 (2011).
[CrossRef]

Sefler, G. A.

Selvaraja, S. K.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral Filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron. 16(1), 33–44 (2010).
[CrossRef]

Serafino, G.

P. Ghelfi, F. Scotti, A. T. Nguyen, G. Serafino, and A. Bogoni, “Novel architecture for a photonics-assisted radar transceiver based on a single mode-locking laser,” IEEE Photon. Technol. Lett. 23(10), 639–641 (2011).
[CrossRef]

Shapira, Y. P.

Singer, L.

Sorel, M.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O'Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay Lines in silicon photonics: Coupled resonators and photonic crystals, a comparison,” IEEE Photon. J. 2(2), 181–194 (2010).
[CrossRef]

Stigwall, J.

J. Stigwall and S. Galt, “Demonstration and analysis of a 40-gigasample/s interferometric analog-to-digital converter,” J. Lightwave Technol. 24(3), 1247–1256 (2006).
[CrossRef]

J. Stigwall and S. Galt, “Interferometric analog-to-digital conversion scheme,” IEEE Photon. Technol. Lett. 17(2), 468–470 (2005).
[CrossRef]

Taylor, H. F.

H. F. Taylor, “An electrooptic analog-to-digital converter,” Proc. IEEE 63(10), 1524–1525 (1975).
[CrossRef]

Titi, G. W.

Twichell, J. C.

P. W. Juodawlkis, J. J. Hargreaves, R. D. Younger, G. W. Titi, and J. C. Twichell, “Optical down-sampling of wide-band microwave signals,” J. Lightwave Technol. 21(12), 3116–3124 (2003).
[CrossRef]

J. C. Twichell, J. L. Wasserman, P. W. Juodawlkis, G. E. Betts, and R. C. Williamson, “High-linearity 208-MS/s photonic analog-to-digital converter using 1-to-4 optical time-division demultiplexers,” IEEE Photon. Technol. Lett. 13(7), 714–716 (2001).
[CrossRef]

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O’Donnell, K. G. Ray, and R. C. Williamson, “Optically sampled analog-to-digital converters,” IEEE Trans. Microw. Theory Tech. 49(10), 1840–1853 (2001).
[CrossRef]

Uguen, B.

R. Llorente, M. Morant, N. Amiot, and B. Uguen, “Novel photonic analog-to-digital converter architecture for precise localization of ultra-wide band radio transmitters,” IEEE J. Sel. Areas Comm. 29(6), 1321–1327 (2011).
[CrossRef]

Urata, R.

L. Y. Nathawad, R. Urata, B. A. Wooley, and D. A. B. Miller, “A 40-GHz-bandwidth, 4-bit, time-interleaved A/D converter using photoconductive sampling,” IEEE J. Solid-state Circuits 38(12), 2021–2030 (2003).
[CrossRef]

Valley, G. C.

Van Thourhout, D.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral Filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron. 16(1), 33–44 (2010).
[CrossRef]

Wasserman, J. L.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O’Donnell, K. G. Ray, and R. C. Williamson, “Optically sampled analog-to-digital converters,” IEEE Trans. Microw. Theory Tech. 49(10), 1840–1853 (2001).
[CrossRef]

J. C. Twichell, J. L. Wasserman, P. W. Juodawlkis, G. E. Betts, and R. C. Williamson, “High-linearity 208-MS/s photonic analog-to-digital converter using 1-to-4 optical time-division demultiplexers,” IEEE Photon. Technol. Lett. 13(7), 714–716 (2001).
[CrossRef]

Williamson, R. C.

J. C. Twichell, J. L. Wasserman, P. W. Juodawlkis, G. E. Betts, and R. C. Williamson, “High-linearity 208-MS/s photonic analog-to-digital converter using 1-to-4 optical time-division demultiplexers,” IEEE Photon. Technol. Lett. 13(7), 714–716 (2001).
[CrossRef]

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O’Donnell, K. G. Ray, and R. C. Williamson, “Optically sampled analog-to-digital converters,” IEEE Trans. Microw. Theory Tech. 49(10), 1840–1853 (2001).
[CrossRef]

R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72(7), 802–819 (1984).
[CrossRef]

Woodward, C. E.

R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72(7), 802–819 (1984).
[CrossRef]

Wooley, B. A.

L. Y. Nathawad, R. Urata, B. A. Wooley, and D. A. B. Miller, “A 40-GHz-bandwidth, 4-bit, time-interleaved A/D converter using photoconductive sampling,” IEEE J. Solid-state Circuits 38(12), 2021–2030 (2003).
[CrossRef]

Wu, Q.

Q. Wu, H. Zhang, Y. Peng, X. Fu, and M. Yao, “40GS/s Optical analog-to-digital conversion system and its improvement,” Opt. Express 17(11), 9252–9257 (2009).
[CrossRef] [PubMed]

Q. Wu, H. Zhang, X. Fu, and M. Yao, “Spectral encoded photonic analog-to-digital converter based on cascaded unbalanced MZMs,” IEEE Photon. Technol. Lett. 21(4), 224–226 (2009).
[CrossRef]

W. Li, H. Zhang, Q. Wu, Z. Zhang, and M. Yao, “All-optical analog-to-digital conversion based on polarization-differential interference and phase modulation,” IEEE Photon. Technol. Lett. 19(8), 625–627 (2007).
[CrossRef]

Yao, J.

Yao, M.

L. Ma, P. Ghelfi, M. Yao, F. Berizzi, and A. Bogoni, “Demonstration of optical sample parallelisation for high-speed photonic assisted ADCs,” Electron. Lett. 47(5), 333–335 (2011).
[CrossRef]

Q. Wu, H. Zhang, Y. Peng, X. Fu, and M. Yao, “40GS/s Optical analog-to-digital conversion system and its improvement,” Opt. Express 17(11), 9252–9257 (2009).
[CrossRef] [PubMed]

Q. Wu, H. Zhang, X. Fu, and M. Yao, “Spectral encoded photonic analog-to-digital converter based on cascaded unbalanced MZMs,” IEEE Photon. Technol. Lett. 21(4), 224–226 (2009).
[CrossRef]

W. Li, H. Zhang, Q. Wu, Z. Zhang, and M. Yao, “All-optical analog-to-digital conversion based on polarization-differential interference and phase modulation,” IEEE Photon. Technol. Lett. 19(8), 625–627 (2007).
[CrossRef]

Younger, R. D.

P. W. Juodawlkis, J. J. Hargreaves, R. D. Younger, G. W. Titi, and J. C. Twichell, “Optical down-sampling of wide-band microwave signals,” J. Lightwave Technol. 21(12), 3116–3124 (2003).
[CrossRef]

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O’Donnell, K. G. Ray, and R. C. Williamson, “Optically sampled analog-to-digital converters,” IEEE Trans. Microw. Theory Tech. 49(10), 1840–1853 (2001).
[CrossRef]

Zach, S.

Zhang, H.

Q. Wu, H. Zhang, Y. Peng, X. Fu, and M. Yao, “40GS/s Optical analog-to-digital conversion system and its improvement,” Opt. Express 17(11), 9252–9257 (2009).
[CrossRef] [PubMed]

Q. Wu, H. Zhang, X. Fu, and M. Yao, “Spectral encoded photonic analog-to-digital converter based on cascaded unbalanced MZMs,” IEEE Photon. Technol. Lett. 21(4), 224–226 (2009).
[CrossRef]

W. Li, H. Zhang, Q. Wu, Z. Zhang, and M. Yao, “All-optical analog-to-digital conversion based on polarization-differential interference and phase modulation,” IEEE Photon. Technol. Lett. 19(8), 625–627 (2007).
[CrossRef]

Zhang, Z.

W. Li, H. Zhang, Q. Wu, Z. Zhang, and M. Yao, “All-optical analog-to-digital conversion based on polarization-differential interference and phase modulation,” IEEE Photon. Technol. Lett. 19(8), 625–627 (2007).
[CrossRef]

Electron. Lett. (1)

L. Ma, P. Ghelfi, M. Yao, F. Berizzi, and A. Bogoni, “Demonstration of optical sample parallelisation for high-speed photonic assisted ADCs,” Electron. Lett. 47(5), 333–335 (2011).
[CrossRef]

IEEE J. Sel. Areas Comm. (1)

R. Llorente, M. Morant, N. Amiot, and B. Uguen, “Novel photonic analog-to-digital converter architecture for precise localization of ultra-wide band radio transmitters,” IEEE J. Sel. Areas Comm. 29(6), 1321–1327 (2011).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral Filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron. 16(1), 33–44 (2010).
[CrossRef]

IEEE J. Solid-state Circuits (1)

L. Y. Nathawad, R. Urata, B. A. Wooley, and D. A. B. Miller, “A 40-GHz-bandwidth, 4-bit, time-interleaved A/D converter using photoconductive sampling,” IEEE J. Solid-state Circuits 38(12), 2021–2030 (2003).
[CrossRef]

IEEE Photon. J. (1)

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O'Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay Lines in silicon photonics: Coupled resonators and photonic crystals, a comparison,” IEEE Photon. J. 2(2), 181–194 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (6)

J. C. Twichell, J. L. Wasserman, P. W. Juodawlkis, G. E. Betts, and R. C. Williamson, “High-linearity 208-MS/s photonic analog-to-digital converter using 1-to-4 optical time-division demultiplexers,” IEEE Photon. Technol. Lett. 13(7), 714–716 (2001).
[CrossRef]

M. Currie, T. R. Clark, and P. J. Matthews, “Photonic analog-to-digital conversion by distributed phase modulation,” IEEE Photon. Technol. Lett. 12(12), 1689–1691 (2000).
[CrossRef]

P. Ghelfi, F. Scotti, A. T. Nguyen, G. Serafino, and A. Bogoni, “Novel architecture for a photonics-assisted radar transceiver based on a single mode-locking laser,” IEEE Photon. Technol. Lett. 23(10), 639–641 (2011).
[CrossRef]

Q. Wu, H. Zhang, X. Fu, and M. Yao, “Spectral encoded photonic analog-to-digital converter based on cascaded unbalanced MZMs,” IEEE Photon. Technol. Lett. 21(4), 224–226 (2009).
[CrossRef]

J. Stigwall and S. Galt, “Interferometric analog-to-digital conversion scheme,” IEEE Photon. Technol. Lett. 17(2), 468–470 (2005).
[CrossRef]

W. Li, H. Zhang, Q. Wu, Z. Zhang, and M. Yao, “All-optical analog-to-digital conversion based on polarization-differential interference and phase modulation,” IEEE Photon. Technol. Lett. 19(8), 625–627 (2007).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (1)

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O’Donnell, K. G. Ray, and R. C. Williamson, “Optically sampled analog-to-digital converters,” IEEE Trans. Microw. Theory Tech. 49(10), 1840–1853 (2001).
[CrossRef]

J. Lightwave Technol. (6)

Nature (1)

U. Keller, “Recent developments in compact ultrafast lasers,” Nature 424(6950), 831–838 (2003).
[CrossRef] [PubMed]

Opt. Express (8)

G. C. Valley, “Photonic analog-to-digital converters,” Opt. Express 15(5), 1955–1982 (2007).
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H. Chi and J. Yao, “A photonic analog-to-digital conversion scheme using Mach-Zehnder modulators with identical half-wave voltages,” Opt. Express 16(2), 567–572 (2008).
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J. Kim, M. J. Park, M. H. Perrott, and F. X. Kärtner, “Photonic subsampling analog-to-digital conversion of microwave signals at 40-GHz with higher than 7-ENOB resolution,” Opt. Express 16(21), 16509–16515 (2008).
[CrossRef] [PubMed]

Q. Wu, H. Zhang, Y. Peng, X. Fu, and M. Yao, “40GS/s Optical analog-to-digital conversion system and its improvement,” Opt. Express 17(11), 9252–9257 (2009).
[CrossRef] [PubMed]

S. Khan, M. A. Baghban, and S. Fathpour, “Electronically tunable silicon photonic delay lines,” Opt. Express 19(12), 11780–11785 (2011).
[CrossRef] [PubMed]

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Two-dimensional free-space beam steering with an optical phased array on silicon-on-insulator,” Opt. Express 19(22), 21595–21604 (2011).
[CrossRef] [PubMed]

J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Planar waveguides with less than 0.1 dB/m propagation loss fabricated with wafer bonding,” Opt. Express 19(24), 24090–24101 (2011).
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K. Ikeda, J. Abdul, S. Namiki, and K.-I. Kitayama, “Optical quantizing and coding for ultrafast A/D conversion using nonlinear fiber-optic switches based on Sagnac interferometer,” Opt. Express 13(11), 4296–4302 (2005).
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Proc. IEEE (2)

H. F. Taylor, “An electrooptic analog-to-digital converter,” Proc. IEEE 63(10), 1524–1525 (1975).
[CrossRef]

R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72(7), 802–819 (1984).
[CrossRef]

Proc. SPIE (1)

M. J. R. Heck, G. Kurczveil, E. F. Burmeister, H. Park, J. P. Mack, D. J. Blumenthal, and J. E. Bowers, ““Integrated recirculating optical buffers,” in Proc,” Proc. SPIE 16, 11124–11131 (2008).

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P. W. Juodawlkis, J. J. Hargreaves, and J. C. Twichell, “Impact of photodetector nonlinearities on photonic analog-to-digital converters,” in Lasers and Electro-Optics, 2002. CLEO ’02, 11–12 (2002).

M. Gustavsson, J. J. Wikner, and N. N. Tan, CMOS Data Converters for Communications (Kluwer Academic, 2002).

B. Murmann, “‘ADC Performance Survey 1997-2011,’ [Online]. Available: http://www.stanford.edu/~murmann/adcsurvey.html ,” (2011).

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Figures (10)

Fig. 1
Fig. 1

Fundamental PADC building block – the Phase Comparator (P-CMP). (a) The P-CMP block diagram and opto-electronic realization of the Phase Detector as a directional coupler terminated in a balanced photo-diode pair. A sign-detector (slicer) acting on the balanced photo-diodes photocurrent completes the P-CMP function; the two P-CMP effective inputs are the optical phases ϕ, θ and the output is a bit indicating whether or not ϕ>θ. (b) Transfer characteristic of the phase detector (normalized photo-current vs. phase ϕ). The slicer following the Phase-Detector determines which of the two half-circle angular decision regions, delineated by the diameter at angle θ, the phase under test falls in.

Fig. 2
Fig. 2

SDSAR PADC system. P-CMP are phase comparators (with internal structure detailed in Fig. 1), biased by quasi-static phase controls (little filled circles). Following an optical sampling front-end, consisting of an OCG feeding dual phase modulators driven by the voltage under test, cascaded interferometric ‘bit extractor’ measurement stages converge onto the phase-under-test by successively subtracting a binary sequence of phase values from the accumulated phases of the optically sampled pulses along the optical transmission line. The phase subtractions are effected by means of phase rotator modulators with lengths forming a geometric sequence with ratio ½ . The subtracted phase values, as generated by 1-bit DAC voltages, are determined by the phase measurements of the prior stage(s) (XORs of certain prior 1-bit ADC decisions). The 1-bit ADC outputs, cn, are processed by simple combinational logic in order to generate the B ADC codeword bits bn. Photonic and electronic matching delays are required to synchronize the successive stages in this feed-forward approach.

Fig. 3
Fig. 3

Phase-domain rotations occurring in the SDSAR PADC of Fig. 2, and supports of the phase probability density distributions at the QPSK 2-bit extractor input (a), and at the output of the phase rotator modulator in the first 1-bit extractor (b).

Fig. 4
Fig. 4

(a): Block diagram of the Monte-Carlo simulation for the SDSAR PADC. (b,c): Detailed block diagrams of the two module types composing the SDSAR (a), with physical gain factors and additive gaussian noise sources.

Fig. 5
Fig. 5

ENOB vs. net Optical Source Power for SDSAR (a) and flash (b). Theoretical performance (solid lines) and simulation results (X markers).

Fig. 6
Fig. 6

SDSAR simulation of ENOB vs. net Optical Source Power with sinusoidal input distribution (solid lines), as well as with uniform input distribution (X markers).

Fig. 7
Fig. 7

Partial contributions of the various noise sources to SDSAR simulated performance. ENOB vs. Optical Source Power (log scale) for 8-bits (a), 10-bits (b), and 12-bits (c). Notice that the ENOB vs. optical power for all noise sources (black solid) essentially coincides with that due to shot + thermal (blue), implying that the DAC noise contribution is negligible. The shot-noise-only (red) rapidly approaches the all-sources performance for optical source power >10 mW, indicating shot-noise limited operation.

Fig. 8
Fig. 8

ENOB vs. Photon Count per Pulse (log scale) at each balanced photo-diodes pair for SDSAR (a) and flash (b). Theoretical performance (solid lines) and simulation results (X markers).

Fig. 9
Fig. 9

(a): ENOB penalty vs. RMS jitter either in fsec for our 10 GS/s PADC (top scale) or normalized by sampling interval (bottom scale) parameterized by the ADC number of bits. (b): 11-bits SDSAR PADC: ENOB vs. optical power lower bound: sinusoidal input model curves parameterized by jitter (2,10,20,50 fs) as well as broadband ADC input signal simulations (discrete data points) corresponding to the same jitter values.

Fig. 10
Fig. 10

Canonical block diagram of SDSAR with unity inter-stage gains. The structure is formally equivalent to an s-stages pipeline with unity inter-stage gains and one bit per stage for stages n = 2,3,..., s (stage 1 generates two bits).

Tables (2)

Tables Icon

Table 1 System parameters for the flash and/or SDSAR PADCs a

Tables Icon

Table 2 The 12 terms in the leftmost column are the abbreviations specific to this paper – the other two columns contain abbreviations in general use.

Equations (64)

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V d K d = sin [ ϕ θ ] K r
b sgn { sin ( ϕ θ ) } = sgn { ϕ θ } = { + 1 i f ϕ θ 1 i f ϕ < θ f o r ϕ θ ( π , π )
i Σ / Δ ( t ) = ρ 2 P r ( t ) [ 1 ± sin ( ϕ ( t ) θ ) ] = 1 2 i r ( t ) [ 1 ± sin ( ϕ ( t ) θ ) ]
i Σ ( t ) + i Δ ( t ) = i r ( t ) ; i d ( t ) = i Σ i Δ = i r ( t ) sin [ ϕ ( t ) θ ]
P r ( t ) = P p e a k 1 [ 0 , τ p ] ( t ) .
i Σ / Δ = 1 2 i p e a k 1 [ 0 , τ p ] ( t ) [ 1 ± sin ( ϕ θ ) ] .
K d = sin [ ϕ θ ] 1 e i p e a k τ p = sin [ ϕ θ ] 1 e q r
K d = sin [ ϕ θ ] K r ; K r 1 e q r = 1 e i p e a k τ p
K d = sin [ ϕ θ ] K r = K r sin Δ ϕ K r Δ ϕ Δ ϕ = K d / K r .
var K d = var { K Σ K Δ } = var { K Σ } + var { K Δ } = K Σ + K Δ = K Σ + K Δ = K r .
K S H R M S = K r .
q T H R M S = V a r { q T H } = S i T H h I & D 2 = 1 2 i N E C 2 τ I = i N E C 1 2 τ I .
K T H R M S = q T H R M S / e = τ I / 2 ( i N E C / e ) .
K d = K Σ K Δ = sin [ ϕ θ ] K r + K d S H + T H ; K d S H + T H ~ N [ 0 , K r + 1 2 e 2 τ I i N E C 2 ] .
snr Q N R M S = μ p π / 2 ϕ Q N R M S = μ p π / 2 2 π 2 M / 12 = μ p M 6 = 2 μ p M 1.5 .
ϕ Q N R M S = V a r ϕ Q N = Δ ϕ / 12 = π / ( M 12 )
K Q N R M S K r ϕ Q N R M S = π M 12 K r .
K DAC(1) R M S = K r 1 2 k T R F D A C τ D A C 1 ( π / V π ( 1 ) ) .
s [ b ] = 1 + ( 1 2 ) 2 + ( 1 4 ) 2 + ( 1 8 ) 2 ... + ( 1 2 B 2 ) 2 = i = 0 B 2 ( 1 2 i ) 2 = 4 3 ( 1 4 ( B 1 ) ) B 4 3 = 1.3333.
( K DAC R M S ) 2 = 2 3 K r 2 ( π V π ( 1 ) ) 2 k T R F D A C τ D A C 1 ( 1 4 ( B 1 ) ) .
K Q N R M S K r ϕ Q N R M S = π M 12 K r ; K S H R M S = K r ; K T H R M S = q T H R M S / e = 1 2 τ I ( 1 e i N E C ) ; K DAC(1) R M S = π V π ( 1 ) K r 1 2 k T R F D A C τ D A C 1 .
Δ ϕ = K d / K r
ϕ Q N R M S = K Q N R M S / K r = π M 12 ϕ S H R M S = K S H R M S / K r = K r / K r = 1 / K r ϕ T H R M S = K T H R M S / K r = 1 2 τ I ( i N E C / e ) / K r ϕ DAC(1) R M S = K DAC(1) R M S / K r = π V π ( 1 ) 1 2 k T R F D A C τ D A C 1 .
ϕ A D C ( n ) [ k ] = ϕ Q N ( n ) [ k ] + ϕ e x s ( n ) [ k ] ; ϕ e x s ( n ) [ k ] ϕ T H ( n ) [ k ] + ϕ S H ( n ) [ k ] .
ϕ A D C ( B 1 ) [ k ] = ϕ Q N ( B 1 ) [ k ] + ϕ T H ( B 1 ) [ k ] + ϕ S H ( B 1 ) [ k ] .
ϕ D A C [ k ] = ϕ D A C ( 1 ) [ k ] + ϕ D A C ( 2 ) [ k ] / 2 + ϕ D A C ( 3 ) [ k ] / 2 2 + .... + + ϕ D A C ( B 2 ) [ k ] / 2 B 2 .
ϕ d [ k ] = ϕ I N [ k ] + ϕ A D C ( B 1 ) [ k ] + ϕ DAC [ k ] ϕ ε S D S A R [ k ] ,
[ ϕ DAC ( n ) R M S ] 2 = [ ϕ DAC(1) R M S ] 2 / 2 n 1 = ( π V π ( 1 ) ) 2 k T R F D A C τ D A C 1 / 2 n
var ϕ D A C = var ϕ D A C ( 1 ) [ k ] [ 1 + 1 4 + ( 1 4 ) 2 + ( 1 4 ) 3 + ... + ( 1 4 ) B 1 ] ; ϕ DAC R M S = ϕ DAC(1) R M S m = 1 n ( 1 4 ) m 1 = π V π ( 1 ) 1 2 k T R F D A C τ D A C 1 4 3 ( 1 4 ( B 1 ) ) = π V π ( 1 ) 2 3 k T R F D A C τ D A C 1 ( 1 4 ( B 1 ) ) .
ϕ DAC R M S = π V π ( 1 ) 2 3 k T R F D A C τ D A C 1 ( 1 4 ( B 1 ) ) .
( [ ϕ ε S D S A R ] R M S ) 2 = ( ϕ A D C ( B 1 ) R M S ) 2 + ( ϕ D A C R M S ) 2 = ( ϕ Q N R M S ) 2 + ( ϕ S H R M S ) 2 + ( ϕ T H R M S ) 2 + ( ϕ D A C R M S ) 2 = ( π M 12 ) 2 + ( 1 / K r ) 2 + ( 1 2 τ I ( 1 e i N E C ) / K r ) 2 + ( 1 2 τ I K N E C / K r ) 2 + ( π V π ( 1 ) 2 3 k T R F D A C τ D A C 1 ( 1 4 ( B 1 ) ) ) 2 = π 2 12 M 2 + K r 1 + 1 2 τ I ( 1 e i N E C ) 2 K r 2 + ( π V π ( 1 ) ) 2 2 3 k T R F D A C τ D A C 1 ( 1 4 ( B 1 ) ) .
F SDSAR / Q N = μ p 1 ( [ ϕ ε S D S A R ] R M S ) 2 / ( ϕ Q N R M S ) 2 = [ 1 + 12 M 2 π 2 K r 1 + 6 M 2 π 2 τ I ( 1 e i N E C ) 2 K r 2 + 8 ( M V π ( 1 ) ) 2 k T R F D A C τ D A C 1 ] μ p 2 .
K r S D S A R = K O C G L e x s B = η h ν 0 P ¯ O C G f s L e x s B .
B eff S D S A R { P ¯ O C G ; B } = B 1 2 log 2 { [ 1 + 12 M 2 π 2 K r 1 + 6 M 2 π 2 τ I ( 1 e i N E C ) 2 K r 2 + 8 ( M V π ( 1 ) ) 2 k T R F D A C τ D A C 1 ] μ p 2 } where K r = η h ν 0 P ¯ O C G f s L e x s B ; M = 2 B 1 .
b 0 c 0 ; b 1 c 1 c 0 ¯ ; b n | n = 2 , 3 , ... , B 1 c n c 1 ; d 2 = c 0 c 1 ; d n | n = 3 , 4 , ... , B 1 = b n 1 .
ϕ ^ I N = π 2 n = 0 B 1 b n ± 2 n
c n ± | n = 0 , ... , B 1 = sgn { K d [ n ] } ; K d [ n ] = sin [ ϕ n ] K r + K d S H + T H [ n ] ; ϕ n = ϕ n 1 ( ρ n + δ n ) ; ρ n = d n ± π / 2 n , n = 2 , 4 , ... , B 1 ; δ n ~ N [ 0 , σ D A C 2 ] ; σ D A C 2 ( π / V π ( 1 ) ) 2 k T R F D A C τ D A C 1 / 2 n ; K r = η h ν 0 P ¯ O C G / ( f s L e x s B ) ; K d S H + T H [ n ] ~ N [ 0 , σ S H + T H 2 ] ; σ S H + T H 2 K r + 1 2 τ I ( i N E C / e ) 2
b n ± | n = 3 , ... , B 1 = sgn { sin [ ϕ n 1 [ c n 1 c 1 ] ± π / 2 n ρ n δ n ϕ n ] K r + K d S H + T H [ n ] } b 2 ± = sgn { sin [ ϕ 1 [ c 0 c 1 ] ± π / 2 2 δ 2 ] K r + K d S H + T H [ n ] } b 0 ± = c 0 ± = sgn { sin [ ϕ I N ] K r + K d S H + T H [ 0 ] } ; b 1 ± = c 1 ± = sgn { cos [ ϕ I N ] K r + K d S H + T H [ 1 ] }
b 0 = c 0 = 1 if ϕ I N ( 0 , π ) ; b 0 = c 0 = 0 otherwise b 1 = c 1 = 1 if ϕ I N ( π / 2 , 3 π / 2 ) ; b 1 = c 1 = 0 otherwise .
MSE = ( ϕ ^ I N ϕ I N ) 2 = t = 1 T ( ϕ ^ I N [ t ] ϕ I N [ t ] ) 2
B eff S D S A R -sim = B 1 2 log 2 { μ p 2 ( ϕ ^ I N ϕ I N ) 2 / σ Q N 2 } ; σ Q N 2 = ( π / ( 2 B 1 12 ) ) 2 .
K r p h o t / p u l s e SH=QN = K r η = 12 M 2 π 2 η = 12 2 2 ( B 1 ) π 2 η .
K r F L A S H = η h ν 0 P ¯ O C G / ( f s L e x s 2 B 1 ) .
E N O B p e n a l t y ( 1 ) + ( 2 ) = E N O B p e n a l t y ( 1 ) + 1 2 log 2 ( 1 + ( 2 2 E N O B p e n a l t y ( 2 ) 1 ) / 2 2 E N O B p e n a l t y ( 1 ) ) .
d = Q B [ a ] = a + Q B [ a ] a ε Q N = a + ε Q N .
d = Q [ a + n e x s ] = a + Q [ a + n e x s ] a ε = a + ε .
var n t o t ( B ) = var n Q ( B e f f ) = Δ 2 / 12 = ( FS / 2 B e f f ) 2 / 12
S N D R = 1.5 2 2 B eff ; B eff = 1 2 log 2 ( S N D R / 1.5 ) ; B eff = ( S N D R d B 1.76 d B ) / 6.02 d B .
S N D R QN only ideal = 1.5 2 2 B ; B = 1 2 log 2 ( S N D R QN only ideal / 1.5 ) .
S N D R QN only ideal = A r m s σ Q N = 1 2 ( F S / 2 ) Δ / 12 = F S Δ 12 2 2 = 2 B 1.5 .
F / Q N = S N D R QN only ideal S N D R = P s / P Q N μ P 2 P s / P t o t = μ p 2 P t o t P Q N .
F / Q N = μ p 2 P t o t P Q N = μ p 2 P Q N ( B ) + P e x s P Q N ( B ) = μ p 2 [ 1 + P e x s P Q N ] ; P Q N ( B ) = Δ 2 12 = ( FS / 2 B ) 2 12
E N O B p e n a l B B eff = 1 2 log 2 ( S N D R QN only ideal S N D R ) = 1 2 log 2 ( F / Q N ) .
B eff = B 1 2 log 2 ( F / Q N ) ; F / Q N = μ p 2 P Q N ( B ) + P e x s P Q N ( B ) .
B eff = B 1 2 log 2 ( F / Q N ) F / Q N = μ p 2 ε 2 / P Q N ( B ) = μ p 2 ( d a ) 2 M S E / P Q N ( B )
d = n = 1 s d ( n )
r ( s ) = a n = 1 s a ^ ( n ) = a n = 1 s ( d ( n ) + δ ( n ) ) = a n = 1 s d ( n ) n = 1 s δ ( n ) .
r ( 1 ) = a a ^ ( 1 ) = a ( a + ε ( 1 ) + δ ( 1 ) ) = ε ( 1 ) δ ( 1 ) r ( 2 ) = r ( 1 ) a ^ ( 2 ) = r ( 1 ) ( r ( 1 ) + ε ( 2 ) + δ ( 2 ) ) = ε ( 2 ) δ ( 2 ) r ( s 1 ) = r ( s 2 ) a ^ ( s 1 ) = r ( s 2 ) ( r ( s 2 ) + ε ( s 1 ) + δ ( s 1 ) ) = ε ( s 1 ) δ ( s 1 ) r ( s ) = r ( s 1 ) a ^ ( s ) = r ( s 1 ) ( r ( s 1 ) + ε ( s ) + δ ( s ) ) = ε ( s ) δ ( s ) .
d ( 1 ) = a + ε ( 1 ) ; d ( 2 ) = r ( 1 ) + ε ( 2 ) = ε ( 1 ) δ ( 1 ) + ε ( 2 ) d ( s 1 ) = r ( s 2 ) + ε ( s 1 ) = ε ( s 2 ) δ ( s 2 ) + ε ( s 1 ) d ( s ) = r ( s 1 ) + ε ( s ) = ε ( s 1 ) δ ( s 1 ) + ε ( s ) .
d = a + ε ( s ) n = 1 s δ ( n ) ε t o t .
s k | ϕ = d v ( t ) d t | t = t k = 2 π f 0 A sin ( 2 π f 0 t k + φ ) .
n τ 2 | ϕ = ( s k | φ τ ) 2 | φ = ( s k | φ ) 2 τ 2 | φ = ( s k | φ ) 2 τ 2 = ( 2 π f 0 A sin ( 2 π f 0 t k + φ ) ) 2 σ τ 2 .
n τ 2 = E φ n τ 2 | φ = E φ { ( 2 π f 0 A sin ( 2 π f 0 t k + φ ) ) 2 σ τ } = 1 2 ( 2 π f 0 A ) 2 σ τ 2 .
SNR j i t t e r E φ { v 2 ( t ) } n τ 2 = 1 2 A 2 1 2 ( 2 π f 0 A σ τ ) 2 = 1 ( 2 π f 0 σ τ ) 2 .

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