Abstract

We demonstrate an efficient noise dithering procedure for measuring the power spectrum of a weak spectral doublet with a Fourier-transform spectrometer in which the subthreshold interferogram is measured by a 1-bit analog-to-digital converter without oversampling. In the absence of noise, no information is obtained regarding the doublet spectrum because the modulation term s(x) of its interferogram is below the instrumental detection limit B, i.e., |s(x)| < B, for all path difference x values. Extensive numerical experiments are carried out concerning the recovery of the doublet power spectrum that is represented by s(x) = (s0/2)exp(-π2x2/β)[cos(2πf1x) + cos(2πf2x)], where s0 is a constant, β is the linewidth factor, and 〈f〉 = (f1 + f2)/2. Different values of 〈f〉, s0, and β are considered to evaluate thoroughly the accuracy of the procedure to determine the unknown values of f1 and f2, the spectral linewidth, and the peak values of the spectral profiles. Our experiments show that, even for short observation times, the resonant frequencies of s(x) could be located with high accuracy over a wide range of 〈f〉 and β values. Signal-to-noise ratios as high as 50 are also gained for the recovered power spectra. The performance of the procedure is also analyzed with respect to another method that recovers the amplitude values of s(x) directly.

© 2001 Optical Society of America

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    [CrossRef]
  3. K. Wiesenfeld, F. Jaramillo, “Minireview of stochastic resonance,” Chaos 8, 539–548 (1998).
    [CrossRef]
  4. N. Jayant, P. Noll, Digital Coding of Waveforms (Prentice-Hall, Englewood Cliffs, N.J., 1984).
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    [CrossRef]
  6. F. Chapeau-Blondeau, X. Godivier, “Stochastic resonance in nonlinear transmission of spike signals: an exact model and an application to the neuron,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 2069–2076 (1996).
    [CrossRef]
  7. F. Vaudelle, J. Gazengel, G. Rivioire, X. Godivier, F. Chapeau-Blondeau, “Stochastic resonance and noise-enhanced transmission of spatial signals in optics: the case of scattering,” J. Opt. Soc. Am. B 15, 2674–2680 (1998).
    [CrossRef]
  8. A. Palonpon, J. Amistoso, J. Holdsworth, W. Garcia, C. Saloma, “Measurement of weak transmittances by stochastic resonance,” Opt. Lett. 23, 1480–1482 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. M. Lim, C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
    [CrossRef]
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    [CrossRef]
  17. C. Ho, C. Kuo, “Oversampling sigma-delta modulator stabilized by local nonlinear feedback loop technique,” IEEE Trans. Circuits Syst. II 47, 941–948 (2000).
    [CrossRef]
  18. M. Kozak, M. Karaman, I. Kale, “Efficient architectures for time-interleaved oversampling delta-sigma converters,” IEEE Trans. Circuits Syst. II 47, 802–809 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2000 (5)

C. Ho, C. Kuo, “Oversampling sigma-delta modulator stabilized by local nonlinear feedback loop technique,” IEEE Trans. Circuits Syst. II 47, 941–948 (2000).
[CrossRef]

M. Kozak, M. Karaman, I. Kale, “Efficient architectures for time-interleaved oversampling delta-sigma converters,” IEEE Trans. Circuits Syst. II 47, 802–809 (2000).
[CrossRef]

C. Zierhofer, “Adaptive sigma-delta modulation with one-bit quantization,” IEEE Trans. Circuits Syst. II 47, 408–415 (2000).
[CrossRef]

C. Dick, F. Harris, “FPGA signal processing using sigma-delta modulation,” IEEE Signal Process. Mag. 17, 20–35 (2000).
[CrossRef]

V. Daria, C. Saloma, “High-accuracy Fourier transform interferometry, without oversampling, with a 1-bit analog-to-digital converter,” Appl. Opt. 39, 108–113 (2000).
[CrossRef]

1999 (2)

D. Russell, L. Wilken, F. Moss, “Use of behavioural stochastic resonance by paddle fish for feeding,” Nature (London) 402, 291–294 (1999).
[CrossRef]

S. Freeman, M. Quick, M. Morin, R. Anderson, C. Desilets, T. Linnenbrink, M. O’Donnell, “Delta-sigma oversampled ultrasound beamformer with dynamic delays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 320–332 (1999).
[CrossRef]

1998 (8)

S. Gal, G. Cathebras, Y. Bertrand, “Measurement of small resistance variations using sigma delta technique,” Electron. Lett. 34, 1578–1579 (1998).
[CrossRef]

L. Gammaitoni, P. Hanggi, P. Jung, F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–288 (1998).
[CrossRef]

M. Litong, C. Saloma, “Detection of sub-threshold oscillations by sinusoid-crossing sampling,” Phys. Rev. E 57, 3579–3588 (1998).
[CrossRef]

K. Wiesenfeld, F. Jaramillo, “Minireview of stochastic resonance,” Chaos 8, 539–548 (1998).
[CrossRef]

M. Lim, C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
[CrossRef]

F. Vaudelle, J. Gazengel, G. Rivioire, X. Godivier, F. Chapeau-Blondeau, “Stochastic resonance and noise-enhanced transmission of spatial signals in optics: the case of scattering,” J. Opt. Soc. Am. B 15, 2674–2680 (1998).
[CrossRef]

A. Palonpon, J. Amistoso, J. Holdsworth, W. Garcia, C. Saloma, “Measurement of weak transmittances by stochastic resonance,” Opt. Lett. 23, 1480–1482 (1998).
[CrossRef]

M. Nazario, C. Saloma, “Signal recovery in sinusoid crossing sampling using the minimum negativity constraint,” Appl. Opt. 37, 2953–2963 (1998).
[CrossRef]

1997 (2)

S. Kurikov, D. Prilutskii, S. Selishchev, “Use of a sigma-delta analog-to-digital converter in multichannel electrocardiographs,” Biomed. Eng. (USSR) 31(4), 190–198 (1997).
[CrossRef]

V. Daria, C. Saloma, “Bandwidth and detection limit in crossing-based spectrum analyzer,” Rev. Sci. Instrum. 68, 240–242 (1997).
[CrossRef]

1996 (4)

F. Chapeau-Blondeau, X. Godivier, “Stochastic resonance in nonlinear transmission of spike signals: an exact model and an application to the neuron,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 2069–2076 (1996).
[CrossRef]

A. Bulsara, L. Gammaitoni, “Tuning in to noise,” Phys. Today 49(3), 39–47 (1996).
[CrossRef]

O. Norman, “A band-pass delta-sigma modulator for ultrasound imaging at 160 MHz clock rate,” IEEE J. Solid-State Circuits 31, 2036–2041 (1996).
[CrossRef]

H. Verhoeven, J. Huijsing, “Design of integrated thermal flow sensors using thermal sigma-delta modulation,” Sens. Actuators A 52, 198–202 (1996).
[CrossRef]

1995 (1)

K. Wiesenfeld, F. Moss, “Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDS,” Nature (London) 373, 33–36 (1995).
[CrossRef]

1993 (4)

F. Riedijk, J. Huijsing, “A smart balanced thermal pyranometer using a sigma-delta A-to-D converter for direct communication with microcontrollers,” Sens. Actuators A 37/38, 16–25 (1993).
[CrossRef]

F. Riedijk, G. Rademaker, J. Huijsing, “A dual-bit low-offset sigma delta analog-to-digital converter for integrated smart sensors,” Sens. Actuators A 36, 157–166 (1993).
[CrossRef]

K. Minami, S. Kawata, “Dynamic range enhancement of Fourier transform infrared spectrum measurement using delta sigma modulation,” Appl. Opt. 32, 4822–4827 (1993).
[CrossRef] [PubMed]

C. Saloma, V. Daria, “Performance of a zero-crossing optical spectrum analyzer,” Opt. Lett. 18, 1468–1470 (1993).
[CrossRef] [PubMed]

1991 (1)

1985 (1)

Amistoso, J.

Anderson, R.

S. Freeman, M. Quick, M. Morin, R. Anderson, C. Desilets, T. Linnenbrink, M. O’Donnell, “Delta-sigma oversampled ultrasound beamformer with dynamic delays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 320–332 (1999).
[CrossRef]

Bertrand, Y.

S. Gal, G. Cathebras, Y. Bertrand, “Measurement of small resistance variations using sigma delta technique,” Electron. Lett. 34, 1578–1579 (1998).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, New York, 1998).

Bulsara, A.

A. Bulsara, L. Gammaitoni, “Tuning in to noise,” Phys. Today 49(3), 39–47 (1996).
[CrossRef]

Cathebras, G.

S. Gal, G. Cathebras, Y. Bertrand, “Measurement of small resistance variations using sigma delta technique,” Electron. Lett. 34, 1578–1579 (1998).
[CrossRef]

Chapeau-Blondeau, F.

F. Vaudelle, J. Gazengel, G. Rivioire, X. Godivier, F. Chapeau-Blondeau, “Stochastic resonance and noise-enhanced transmission of spatial signals in optics: the case of scattering,” J. Opt. Soc. Am. B 15, 2674–2680 (1998).
[CrossRef]

F. Chapeau-Blondeau, X. Godivier, “Stochastic resonance in nonlinear transmission of spike signals: an exact model and an application to the neuron,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 2069–2076 (1996).
[CrossRef]

Cramer, H.

H. Cramer, M. Leadbetter, Stationary and Related Stochastic Processes (Wiley, New York, 1967).

Daria, V.

Desilets, C.

S. Freeman, M. Quick, M. Morin, R. Anderson, C. Desilets, T. Linnenbrink, M. O’Donnell, “Delta-sigma oversampled ultrasound beamformer with dynamic delays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 320–332 (1999).
[CrossRef]

Dick, C.

C. Dick, F. Harris, “FPGA signal processing using sigma-delta modulation,” IEEE Signal Process. Mag. 17, 20–35 (2000).
[CrossRef]

Fales, C.

Freeman, S.

S. Freeman, M. Quick, M. Morin, R. Anderson, C. Desilets, T. Linnenbrink, M. O’Donnell, “Delta-sigma oversampled ultrasound beamformer with dynamic delays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 320–332 (1999).
[CrossRef]

Gal, S.

S. Gal, G. Cathebras, Y. Bertrand, “Measurement of small resistance variations using sigma delta technique,” Electron. Lett. 34, 1578–1579 (1998).
[CrossRef]

Gammaitoni, L.

L. Gammaitoni, P. Hanggi, P. Jung, F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–288 (1998).
[CrossRef]

A. Bulsara, L. Gammaitoni, “Tuning in to noise,” Phys. Today 49(3), 39–47 (1996).
[CrossRef]

Garcia, W.

Gazengel, J.

Godivier, X.

F. Vaudelle, J. Gazengel, G. Rivioire, X. Godivier, F. Chapeau-Blondeau, “Stochastic resonance and noise-enhanced transmission of spatial signals in optics: the case of scattering,” J. Opt. Soc. Am. B 15, 2674–2680 (1998).
[CrossRef]

F. Chapeau-Blondeau, X. Godivier, “Stochastic resonance in nonlinear transmission of spike signals: an exact model and an application to the neuron,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 2069–2076 (1996).
[CrossRef]

Griffiths, P.

P. Griffiths, Chemical Infrared Fourier Transform Spectroscopy (Wiley, New York, 1975).

Haeberli, P.

Hanggi, P.

L. Gammaitoni, P. Hanggi, P. Jung, F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–288 (1998).
[CrossRef]

Harris, F.

C. Dick, F. Harris, “FPGA signal processing using sigma-delta modulation,” IEEE Signal Process. Mag. 17, 20–35 (2000).
[CrossRef]

Haylo, N.

Ho, C.

C. Ho, C. Kuo, “Oversampling sigma-delta modulator stabilized by local nonlinear feedback loop technique,” IEEE Trans. Circuits Syst. II 47, 941–948 (2000).
[CrossRef]

Holdsworth, J.

Huck, F.

Huijsing, J.

H. Verhoeven, J. Huijsing, “Design of integrated thermal flow sensors using thermal sigma-delta modulation,” Sens. Actuators A 52, 198–202 (1996).
[CrossRef]

F. Riedijk, J. Huijsing, “A smart balanced thermal pyranometer using a sigma-delta A-to-D converter for direct communication with microcontrollers,” Sens. Actuators A 37/38, 16–25 (1993).
[CrossRef]

F. Riedijk, G. Rademaker, J. Huijsing, “A dual-bit low-offset sigma delta analog-to-digital converter for integrated smart sensors,” Sens. Actuators A 36, 157–166 (1993).
[CrossRef]

Jaramillo, F.

K. Wiesenfeld, F. Jaramillo, “Minireview of stochastic resonance,” Chaos 8, 539–548 (1998).
[CrossRef]

Jayant, N.

N. Jayant, P. Noll, Digital Coding of Waveforms (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Jung, P.

L. Gammaitoni, P. Hanggi, P. Jung, F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–288 (1998).
[CrossRef]

Kale, I.

M. Kozak, M. Karaman, I. Kale, “Efficient architectures for time-interleaved oversampling delta-sigma converters,” IEEE Trans. Circuits Syst. II 47, 802–809 (2000).
[CrossRef]

Karaman, M.

M. Kozak, M. Karaman, I. Kale, “Efficient architectures for time-interleaved oversampling delta-sigma converters,” IEEE Trans. Circuits Syst. II 47, 802–809 (2000).
[CrossRef]

Kawata, S.

Kozak, M.

M. Kozak, M. Karaman, I. Kale, “Efficient architectures for time-interleaved oversampling delta-sigma converters,” IEEE Trans. Circuits Syst. II 47, 802–809 (2000).
[CrossRef]

Kuo, C.

C. Ho, C. Kuo, “Oversampling sigma-delta modulator stabilized by local nonlinear feedback loop technique,” IEEE Trans. Circuits Syst. II 47, 941–948 (2000).
[CrossRef]

Kurikov, S.

S. Kurikov, D. Prilutskii, S. Selishchev, “Use of a sigma-delta analog-to-digital converter in multichannel electrocardiographs,” Biomed. Eng. (USSR) 31(4), 190–198 (1997).
[CrossRef]

Leadbetter, M.

H. Cramer, M. Leadbetter, Stationary and Related Stochastic Processes (Wiley, New York, 1967).

Lim, M.

M. Lim, C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
[CrossRef]

Linnenbrink, T.

S. Freeman, M. Quick, M. Morin, R. Anderson, C. Desilets, T. Linnenbrink, M. O’Donnell, “Delta-sigma oversampled ultrasound beamformer with dynamic delays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 320–332 (1999).
[CrossRef]

Litong, M.

M. Litong, C. Saloma, “Detection of sub-threshold oscillations by sinusoid-crossing sampling,” Phys. Rev. E 57, 3579–3588 (1998).
[CrossRef]

Manolakis, D.

J. Proakis, D. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1989).

Marchesoni, F.

L. Gammaitoni, P. Hanggi, P. Jung, F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–288 (1998).
[CrossRef]

Minami, K.

Morin, M.

S. Freeman, M. Quick, M. Morin, R. Anderson, C. Desilets, T. Linnenbrink, M. O’Donnell, “Delta-sigma oversampled ultrasound beamformer with dynamic delays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 320–332 (1999).
[CrossRef]

Moss, F.

D. Russell, L. Wilken, F. Moss, “Use of behavioural stochastic resonance by paddle fish for feeding,” Nature (London) 402, 291–294 (1999).
[CrossRef]

K. Wiesenfeld, F. Moss, “Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDS,” Nature (London) 373, 33–36 (1995).
[CrossRef]

Nazario, M.

Noll, P.

N. Jayant, P. Noll, Digital Coding of Waveforms (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Norman, O.

O. Norman, “A band-pass delta-sigma modulator for ultrasound imaging at 160 MHz clock rate,” IEEE J. Solid-State Circuits 31, 2036–2041 (1996).
[CrossRef]

O’Donnell, M.

S. Freeman, M. Quick, M. Morin, R. Anderson, C. Desilets, T. Linnenbrink, M. O’Donnell, “Delta-sigma oversampled ultrasound beamformer with dynamic delays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 320–332 (1999).
[CrossRef]

Palonpon, A.

Prilutskii, D.

S. Kurikov, D. Prilutskii, S. Selishchev, “Use of a sigma-delta analog-to-digital converter in multichannel electrocardiographs,” Biomed. Eng. (USSR) 31(4), 190–198 (1997).
[CrossRef]

Proakis, J.

J. Proakis, D. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1989).

Quick, M.

S. Freeman, M. Quick, M. Morin, R. Anderson, C. Desilets, T. Linnenbrink, M. O’Donnell, “Delta-sigma oversampled ultrasound beamformer with dynamic delays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 320–332 (1999).
[CrossRef]

Rademaker, G.

F. Riedijk, G. Rademaker, J. Huijsing, “A dual-bit low-offset sigma delta analog-to-digital converter for integrated smart sensors,” Sens. Actuators A 36, 157–166 (1993).
[CrossRef]

Riedijk, F.

F. Riedijk, G. Rademaker, J. Huijsing, “A dual-bit low-offset sigma delta analog-to-digital converter for integrated smart sensors,” Sens. Actuators A 36, 157–166 (1993).
[CrossRef]

F. Riedijk, J. Huijsing, “A smart balanced thermal pyranometer using a sigma-delta A-to-D converter for direct communication with microcontrollers,” Sens. Actuators A 37/38, 16–25 (1993).
[CrossRef]

Rivioire, G.

Russell, D.

D. Russell, L. Wilken, F. Moss, “Use of behavioural stochastic resonance by paddle fish for feeding,” Nature (London) 402, 291–294 (1999).
[CrossRef]

Saloma, C.

Samms, R.

Selishchev, S.

S. Kurikov, D. Prilutskii, S. Selishchev, “Use of a sigma-delta analog-to-digital converter in multichannel electrocardiographs,” Biomed. Eng. (USSR) 31(4), 190–198 (1997).
[CrossRef]

Stacey, K.

Vaudelle, F.

Verhoeven, H.

H. Verhoeven, J. Huijsing, “Design of integrated thermal flow sensors using thermal sigma-delta modulation,” Sens. Actuators A 52, 198–202 (1996).
[CrossRef]

Wiesenfeld, K.

K. Wiesenfeld, F. Jaramillo, “Minireview of stochastic resonance,” Chaos 8, 539–548 (1998).
[CrossRef]

K. Wiesenfeld, F. Moss, “Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDS,” Nature (London) 373, 33–36 (1995).
[CrossRef]

Wilken, L.

D. Russell, L. Wilken, F. Moss, “Use of behavioural stochastic resonance by paddle fish for feeding,” Nature (London) 402, 291–294 (1999).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, New York, 1998).

Zierhofer, C.

C. Zierhofer, “Adaptive sigma-delta modulation with one-bit quantization,” IEEE Trans. Circuits Syst. II 47, 408–415 (2000).
[CrossRef]

Appl. Opt. (3)

Biomed. Eng. (USSR) (1)

S. Kurikov, D. Prilutskii, S. Selishchev, “Use of a sigma-delta analog-to-digital converter in multichannel electrocardiographs,” Biomed. Eng. (USSR) 31(4), 190–198 (1997).
[CrossRef]

Chaos (1)

K. Wiesenfeld, F. Jaramillo, “Minireview of stochastic resonance,” Chaos 8, 539–548 (1998).
[CrossRef]

Electron. Lett. (1)

S. Gal, G. Cathebras, Y. Bertrand, “Measurement of small resistance variations using sigma delta technique,” Electron. Lett. 34, 1578–1579 (1998).
[CrossRef]

IEEE J. Solid-State Circuits (1)

O. Norman, “A band-pass delta-sigma modulator for ultrasound imaging at 160 MHz clock rate,” IEEE J. Solid-State Circuits 31, 2036–2041 (1996).
[CrossRef]

IEEE Signal Process. Mag. (1)

C. Dick, F. Harris, “FPGA signal processing using sigma-delta modulation,” IEEE Signal Process. Mag. 17, 20–35 (2000).
[CrossRef]

IEEE Trans. Circuits Syst. II (3)

C. Ho, C. Kuo, “Oversampling sigma-delta modulator stabilized by local nonlinear feedback loop technique,” IEEE Trans. Circuits Syst. II 47, 941–948 (2000).
[CrossRef]

M. Kozak, M. Karaman, I. Kale, “Efficient architectures for time-interleaved oversampling delta-sigma converters,” IEEE Trans. Circuits Syst. II 47, 802–809 (2000).
[CrossRef]

C. Zierhofer, “Adaptive sigma-delta modulation with one-bit quantization,” IEEE Trans. Circuits Syst. II 47, 408–415 (2000).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

S. Freeman, M. Quick, M. Morin, R. Anderson, C. Desilets, T. Linnenbrink, M. O’Donnell, “Delta-sigma oversampled ultrasound beamformer with dynamic delays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 320–332 (1999).
[CrossRef]

Int. J. Bifurcation Chaos Appl. Sci. Eng. (1)

F. Chapeau-Blondeau, X. Godivier, “Stochastic resonance in nonlinear transmission of spike signals: an exact model and an application to the neuron,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 2069–2076 (1996).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Nature (London) (2)

D. Russell, L. Wilken, F. Moss, “Use of behavioural stochastic resonance by paddle fish for feeding,” Nature (London) 402, 291–294 (1999).
[CrossRef]

K. Wiesenfeld, F. Moss, “Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDS,” Nature (London) 373, 33–36 (1995).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. E (2)

M. Lim, C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
[CrossRef]

M. Litong, C. Saloma, “Detection of sub-threshold oscillations by sinusoid-crossing sampling,” Phys. Rev. E 57, 3579–3588 (1998).
[CrossRef]

Phys. Today (1)

A. Bulsara, L. Gammaitoni, “Tuning in to noise,” Phys. Today 49(3), 39–47 (1996).
[CrossRef]

Rev. Mod. Phys. (1)

L. Gammaitoni, P. Hanggi, P. Jung, F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–288 (1998).
[CrossRef]

Rev. Sci. Instrum. (1)

V. Daria, C. Saloma, “Bandwidth and detection limit in crossing-based spectrum analyzer,” Rev. Sci. Instrum. 68, 240–242 (1997).
[CrossRef]

Sens. Actuators A (3)

F. Riedijk, G. Rademaker, J. Huijsing, “A dual-bit low-offset sigma delta analog-to-digital converter for integrated smart sensors,” Sens. Actuators A 36, 157–166 (1993).
[CrossRef]

H. Verhoeven, J. Huijsing, “Design of integrated thermal flow sensors using thermal sigma-delta modulation,” Sens. Actuators A 52, 198–202 (1996).
[CrossRef]

F. Riedijk, J. Huijsing, “A smart balanced thermal pyranometer using a sigma-delta A-to-D converter for direct communication with microcontrollers,” Sens. Actuators A 37/38, 16–25 (1993).
[CrossRef]

Other (5)

N. Jayant, P. Noll, Digital Coding of Waveforms (Prentice-Hall, Englewood Cliffs, N.J., 1984).

P. Griffiths, Chemical Infrared Fourier Transform Spectroscopy (Wiley, New York, 1975).

J. Proakis, D. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1989).

H. Cramer, M. Leadbetter, Stationary and Related Stochastic Processes (Wiley, New York, 1967).

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, New York, 1998).

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Figures (6)

Fig. 1
Fig. 1

Correspondence between Dm+/D and s(m) values that are in units of rn for D = 1000. The value of σ is in units of rn. The corresponding Dm-/D plots (not shown) are mirror images of the Dm+/D plots about s(m) = 0. The Dm+ value represents that number of times in D attempts that s(m) + nσ(m) overshoots a predetermined +rn threshold level in the sampling interval Δm. The plots can be used to determine the unknown s(m) value from the measured Dm+/D for any given σ ≥ 2rn (theory is shown by solid curves).

Fig. 2
Fig. 2

Schematic of the dithering procedure to calculate the power spectrum {Pσ(k)} of {s(m) + nσ(m)}. The set {〈xm〉} contains the average SC locations in each of the 2M Δm’s within T. The equally sampled representation {r(m)} is derived from {rxm〉} with the ideal interpolation formulation. The spectrum is calculated as {Pσ(k)} = ℱ[{r(m)} ⊗ {r(m)}]. The SC location 〈xm〉 in Δm is averaged from ten independent crossing measurements. When s(x) is detectable, its {Pσ(k)} is measured with nσ(x) = 0, and the averaging of the xm’s is unnecessary. DFT, discrete Fourier transform.

Fig. 3
Fig. 3

Recovered power spectrum {Pσ(k)} (solid circles), where 〈f〉 = 〈k〉/T and 〈k〉 = 0.5(k1 + k2): (a) 〈k〉 = 15, (b) 〈k〉 = 25, and (c) 〈k〉 = 55. Each data point represents the average of ten raw Pσ(k) values. Parameter values used are σ = B = 0.016, s0 = 0.96B, L = 0.3, D = 100, T = 1, 2M = 128, and N = 25. Dotted lines represent the true doublet spectra {Ps(k)}.

Fig. 4
Fig. 4

Linfoot’s criteria for recovered {Pσ(k)}: (a) F versus σ for different L values; (b) F, Q, and C as a function of L (σ = B = 0.016). Parameter values are s0 = 0.96B, D = 100, T = 1, 2M = 128, and N = 256.

Fig. 5
Fig. 5

Effect of spectral averaging in the recovery of unknown Ps(k1) and Ps(k2) values for L = 0 and L = 0.4: (a) Percentage error versus D showing ε1(L = 0, solid circles), ε2(L = 0, open circles), ε2 (L = 0.4, crosses), and ε2 (L = 0.4, open squares); (b) SNR versus D showing SNR1 (0, solid circles), SNR2 (0, open circles), SNR1 (0.4, crosses), and SNR2 (0.4, open squares); and (c) SNR1 and SNR2 versus s0 for D = 100, showing SNR1 (0, solid circles), SNR2 (0, open circles), SNR1 (0.4, crosses), SNR2 (0.4, open squares). Parameter values are σ = B = 0.016, T = 1, 2M = 128, and N = 256. The 〈{Pσ(k)〉 and {〈xm〉} are calculated from D/10 and ten trials, respectively.

Fig. 6
Fig. 6

SNR as a function of the number of trials D/10 used to calculate {〈xm〉} for L = 0 and L = 0.4. Parameter values are σ = B = 0.016, T = 1, 2M = 128, and N = 256. The 〈{Pσ(k)〉 is calculated from ten trials. The solid circles are described by a SNR equaling 81.326D/10 + 54.486.

Equations (9)

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Dm+=D1-Φσ|rn|-sm,
Dm-=DΦσ-|rn|-sm,
sm=|rn|-σ+2σDm+/D  if Dm+>Dm-,
sm=σ-|rn|-2σDm-/D  if Dm+<Dm-,
sm=0,  if Dm+=Dm-.
δs2=2σ/D2δDm+2=2σ/D2δDm-2,
Pσk=rmrm=smsm+nσmnσm+smnσm+nσmsm
smsm+nσmnσmPsk+Pnk.
Ek=rm*rm=sm+nσm*sm+nσm=Sk+NkSk+Nk*=|Sk|2+|Nk|2+SkN*k+NkS*k.

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