## Signal Recovery in Sinusoid-Crossing Sampling by use of the Minimum-Negativity Constraint

Applied Optics, Vol. 37, Issue 14, pp. 2953-2963 (1998)

http://dx.doi.org/10.1364/AO.37.002953

Acrobat PDF (253 KB)

### Abstract

High-frequency components that are lost when a signal *s*(*x*) of bandwidth *W* is low-pass filtered in sinusoid-crossing sampling are recovered by use of the minimum-negativity constraint. The lost high-frequency components are recovered from the information that is available in the Fourier spectrum, which is computed directly from locations of intersections {*x*_{i}} between *s*(*x*) and the reference sinusoid *r*(*x*) = *A* cos(2π*f*_{r}*x*), where the index *i* = 1, 2, …, 2*M* = 2*Tf*_{r}, and *T* is the sampling period. Low-pass filtering occurs when *f*_{r} < *W*/2. If ‖*s*(*x*)‖ ≤ *A* for all values of *x* within *T*, then acrossing exists within each period Δ = 1/2*f*_{r}. The recovery procedure is investigated for the practical case of when *W* is not known *a priori* and *s*(*x*) is corrupted by additive Gaussian noise.

© 1998 Optical Society of America

**OCIS Codes**

(070.1170) Fourier optics and signal processing : Analog optical signal processing

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

**Citation**

Mary Ann Nazario and Caesar Saloma, "Signal Recovery in Sinusoid-Crossing Sampling by use of the Minimum-Negativity Constraint," Appl. Opt. **37**, 2953-2963 (1998)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-37-14-2953

Sort: Year | Journal | Reset

### References

- C. Saloma and V. R. Daria, “Performance of a zero-crossing optical spectrum analyzer,” Opt. Lett. 18, 1468–1470 (1993).
- V. Daria and C. Saloma, “Bandwidth and detection limit in a crossing-based spectrum analyzer,” Rev. Sci. Instrum. 68, 240–242 (1997).
- C. Saloma and P. Haeberli, “Optical spectrum analysis by zero crossings,” Opt. Lett. 16, 1535–1537 (1991).
- F. Bond and C. Cahn, “On sampling the xeros of bandwidth limited signals,” IRE Trans. Inf. Theory IT-4, 110–113 (1958).
- H. Voelker, “Toward a unified theory of modulation. Part II. Zero manipulation,” Proc. IEEE 54, 735–755 (1996).
- B. Logan, “Information in zero crossings of bandpass signals,” Bell Sys. Tech. J. 56, 487–510 (1977).
- A. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. IEEE 68, 308–328 (1980).
- Y. Zeevi, A. Gavriely, and S. Shamai, “Image representation by zero and sine wave crossings.” J. Opt. Soc. Am. A 4, 2045–2060 (1987).
- S. Kay and R. Sudhakar, “A zero-crossing based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. 34, 96–104 (1987).
- A. Zakhor and A. Oppenheim, “Reconstruction of two-dimensional signals from level crossings,” Proc. IEEE 78, 31–55 (1990).
- A. Zakhor and G. Alustad, “Two-dimensional polynomical interpolation from nonuniform samples,” IEEE Trans. Signal Process. 40, 169–175 (1992).
- Y. Zeevi and E. Shlomot, “Nonuniform sampling and antialiasing in image representation,” IEEE Trans. Signal Process. 41, 1223–1229 (1993).
- K. Minami, S. Kawata, and S. Minami, “Zero-crossing sampling of Fourier transform interferograms and spectrum reconstruction using real-zero interpolation,” Appl. Opt. 31, 6322–6327 (1992).
- C. Saloma, “Computational complexity and observation of physical signals,” J. Appl. Phys. 74, 5314–5319 (1993).
- A. Montowski and A. Stark, Introduction to Higher Algebra (Pergamon, Oxford, 1964), pp. 364–369.
- J. Proakis and D. Manolakis, Digital Signal Processing: Principles, Algorithm, and Applications, 2nd ed. (Macmillan, New York, 1992), pp. 943–944.
- J. Hopfield, “Pattern recognition computation using action potential timing for stimulus representation,” Nature 376, 33–36 (1995).
- C. Koch, “Computation and the single neuron,” Nature 385, 207–210 (1997).
- C. M. Blanca, V. Daria, and C. Saloma, “Spectral recovery in crossing-based spectral analysis by analytic continuation,” Appl. Opt. 35, 6417–6423 (1996).
- M. Escobido and C. Saloma, “Detection accuracy in zero-crossing based spectrum analysis and image reconstruction,” Appl. Opt. 35, 6417–6423 (1994).
- S. J. Howard, “Continuation of discrete Fourier spectra using a minimum-negativity constraint,” J. Opt. Soc. Am. 7, 819–824 (1981).
- S. J. Howard, “Fast algorithm for implementing the minimum-negativity constraint for Fourier spectrum extrapolation,” Appl. Opt. 25, 1670–1675 (1986).
- W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986), pp. 24–29.
- G. Johhson, “Constructions of particular random processes,” Proc. IEEE 82, 270–285 (1994).
- J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).
- L. Gammaitoni, F. Marchesoni, E. Menicella-Saetta, and S. Santucci, “Multiplicative stochastic resonance,” Phys. Rev. E 49, 4878–4881 (1994).
- F. O. Huck, C. Fales, N. Haylo, R. W. Samms, and K. Stacey, “Image gathering and processing: information and fidelity,” J. Opt. Soc. Am. A 2, 1644–1666 (1985).
- E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, New York, 1963).
- B. McNamara and K. Weisenfeld, “Theory of stochastic resonance,” Phys. Rev. A 39, 4854–4869 (1989).
- K. Weisenfeld and F. Moss, “Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDS,” Nature 373, 33–36 (1995).
- L. Gammaitoni, “Stochastic resonance and the dithering effect in threshold physical systems,” Phys. Rev. E 52, 4691–4698 (1995).
- A. Bulsara and L. Gammaitoni, “Tuning in to noise,” Phys. Today 39–45 (March 1996).
- M. Litong and C. Saloma, “Detection of subthreshold oscillations in sinusoid-crossing sampling,” Phys. Rev. E 57, 3579–3588 (1998).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.