Abstract

We examine how the accuracy in describing the exact location of a signal crossing affects the quality of the spectrum or the reconstructed two-dimensional image that is computed by a representation of sampled zero crossings. The position of a zero crossing within a Nyquist interval is described by the ratio between the number of clock pulses that have elapsed before the crossing occurred and the total number of clock pulses that could fit within the interval. The pulses scale the Nyquist interval, and the greater their total number, the more accurate the description of the crossing location. In a real zero-crossing detector the ability to increase the total number of square pulses contained within the Nyquist interval is limited by the finite response time of its circuit components [Opt. Lett. 18, 1468 (1993)].

© 1994 Optical Society of America

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References

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  1. F. E. Bond, C. R. Cahn, “On sampling the zeros of bandwidth limited signals,” IRE Trans. Inf. Theory IT-4, 110–113 (1958).
    [CrossRef]
  2. H. B. Voelker, “Toward a unified theory of modulation. Part II. Zero manipulation,” Proc. IEEE 54, 735–755 (1966).
    [CrossRef]
  3. B. F. Logan, “Information in zero crossing of bandpass signals,” Bell Syst. Tech. J.56, 487–510 (1977).
  4. A. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. IEEE 68, 308–328 (1980).
    [CrossRef]
  5. S. Curtis, A. Oppenheim, “Reconstruction of multidimensional signals from zero crossings,” J. Opt. Soc. Am. A4, 221–231 (1987).
  6. K. Piwnicki, “Modulation methods related to sine-wave crossings,” IEEE Trans. Commun. COM-31, 503–508 (1983).
    [CrossRef]
  7. S. Kay, R. Sudhaker, “A zero-crossing-based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96–104 (1986).
    [CrossRef]
  8. Y. Zeevi, A. Gavriely, S. Shamai, “Image representation by zero and sine wave crossings,” J. Opt. Soc. Am. A 4, 2045–2060 (1987).
    [CrossRef]
  9. C. Saloma, P. Haeberli, “Optical spectrum analysis from zero crossings,” Opt. Lett. 16, 1535–1537 (1991).
    [CrossRef] [PubMed]
  10. C. Saloma, V. Daria, “Performance of zero-crossing optical spectrum analyzer,” Opt. Lett. 18, 1468–1470 (1993).
    [CrossRef] [PubMed]
  11. National Semiconductor, General Purpose Linear Devices Databook (National Semiconductor, Santa Clara, Calif., 1989), pp. 1–1640;Texas Instruments, The TTL Data Book (Texas Instruments, Dallas, Tx., 1985), Vol. 1, pp. 16–1572;The TTL Data Book Vol. 2, 28–1392.
  12. A. Zakhor, A. Oppenheim, “Reconstruction of two-dimensional signals from level crossings,” Proc. IEEE 78, 31–55 (1990).
    [CrossRef]
  13. D. Skoog, Principles of Instrumental Analysis, 3rd ed. (Saunders, Philadelphia, Pa., 1985), pp. 14–17.
  14. C. Saloma, “Computational complexity and the observation of physical signals,” J. Appl. Phys. 74, 5314–5319 (1993).
    [CrossRef]
  15. Borland-Osborne, Turbo Pascal Reference Guide, Version 6.0 (McGraw–Hill, NewYork, 1991), pp. 91–94.
  16. C. Saloma, P. Haeberli, “Two-dimensional image reconstruction from Fourier coefficients that are directly computed from zero crossings,” Appl. Opt. 32, 3092–3093 (1993).
    [CrossRef] [PubMed]

1993

1991

1990

A. Zakhor, A. Oppenheim, “Reconstruction of two-dimensional signals from level crossings,” Proc. IEEE 78, 31–55 (1990).
[CrossRef]

1987

S. Curtis, A. Oppenheim, “Reconstruction of multidimensional signals from zero crossings,” J. Opt. Soc. Am. A4, 221–231 (1987).

Y. Zeevi, A. Gavriely, S. Shamai, “Image representation by zero and sine wave crossings,” J. Opt. Soc. Am. A 4, 2045–2060 (1987).
[CrossRef]

1986

S. Kay, R. Sudhaker, “A zero-crossing-based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96–104 (1986).
[CrossRef]

1983

K. Piwnicki, “Modulation methods related to sine-wave crossings,” IEEE Trans. Commun. COM-31, 503–508 (1983).
[CrossRef]

1980

A. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. IEEE 68, 308–328 (1980).
[CrossRef]

1977

B. F. Logan, “Information in zero crossing of bandpass signals,” Bell Syst. Tech. J.56, 487–510 (1977).

1966

H. B. Voelker, “Toward a unified theory of modulation. Part II. Zero manipulation,” Proc. IEEE 54, 735–755 (1966).
[CrossRef]

1958

F. E. Bond, C. R. Cahn, “On sampling the zeros of bandwidth limited signals,” IRE Trans. Inf. Theory IT-4, 110–113 (1958).
[CrossRef]

Bond, F. E.

F. E. Bond, C. R. Cahn, “On sampling the zeros of bandwidth limited signals,” IRE Trans. Inf. Theory IT-4, 110–113 (1958).
[CrossRef]

Cahn, C. R.

F. E. Bond, C. R. Cahn, “On sampling the zeros of bandwidth limited signals,” IRE Trans. Inf. Theory IT-4, 110–113 (1958).
[CrossRef]

Curtis, S.

S. Curtis, A. Oppenheim, “Reconstruction of multidimensional signals from zero crossings,” J. Opt. Soc. Am. A4, 221–231 (1987).

Daria, V.

Gavriely, A.

Haeberli, P.

Kay, S.

S. Kay, R. Sudhaker, “A zero-crossing-based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96–104 (1986).
[CrossRef]

Logan, B. F.

B. F. Logan, “Information in zero crossing of bandpass signals,” Bell Syst. Tech. J.56, 487–510 (1977).

Oppenheim, A.

A. Zakhor, A. Oppenheim, “Reconstruction of two-dimensional signals from level crossings,” Proc. IEEE 78, 31–55 (1990).
[CrossRef]

S. Curtis, A. Oppenheim, “Reconstruction of multidimensional signals from zero crossings,” J. Opt. Soc. Am. A4, 221–231 (1987).

Piwnicki, K.

K. Piwnicki, “Modulation methods related to sine-wave crossings,” IEEE Trans. Commun. COM-31, 503–508 (1983).
[CrossRef]

Requicha, A.

A. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. IEEE 68, 308–328 (1980).
[CrossRef]

Saloma, C.

Shamai, S.

Skoog, D.

D. Skoog, Principles of Instrumental Analysis, 3rd ed. (Saunders, Philadelphia, Pa., 1985), pp. 14–17.

Sudhaker, R.

S. Kay, R. Sudhaker, “A zero-crossing-based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96–104 (1986).
[CrossRef]

Voelker, H. B.

H. B. Voelker, “Toward a unified theory of modulation. Part II. Zero manipulation,” Proc. IEEE 54, 735–755 (1966).
[CrossRef]

Zakhor, A.

A. Zakhor, A. Oppenheim, “Reconstruction of two-dimensional signals from level crossings,” Proc. IEEE 78, 31–55 (1990).
[CrossRef]

Zeevi, Y.

Appl. Opt.

Bell Syst. Tech. J.

B. F. Logan, “Information in zero crossing of bandpass signals,” Bell Syst. Tech. J.56, 487–510 (1977).

IEEE Trans. Acoust. Speech Signal Process

S. Kay, R. Sudhaker, “A zero-crossing-based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96–104 (1986).
[CrossRef]

IEEE Trans. Commun.

K. Piwnicki, “Modulation methods related to sine-wave crossings,” IEEE Trans. Commun. COM-31, 503–508 (1983).
[CrossRef]

IRE Trans. Inf. Theory

F. E. Bond, C. R. Cahn, “On sampling the zeros of bandwidth limited signals,” IRE Trans. Inf. Theory IT-4, 110–113 (1958).
[CrossRef]

J. Appl. Phys.

C. Saloma, “Computational complexity and the observation of physical signals,” J. Appl. Phys. 74, 5314–5319 (1993).
[CrossRef]

J. Opt. Soc. Am. A

S. Curtis, A. Oppenheim, “Reconstruction of multidimensional signals from zero crossings,” J. Opt. Soc. Am. A4, 221–231 (1987).

Y. Zeevi, A. Gavriely, S. Shamai, “Image representation by zero and sine wave crossings,” J. Opt. Soc. Am. A 4, 2045–2060 (1987).
[CrossRef]

Opt. Lett.

Proc. IEEE

H. B. Voelker, “Toward a unified theory of modulation. Part II. Zero manipulation,” Proc. IEEE 54, 735–755 (1966).
[CrossRef]

A. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. IEEE 68, 308–328 (1980).
[CrossRef]

A. Zakhor, A. Oppenheim, “Reconstruction of two-dimensional signals from level crossings,” Proc. IEEE 78, 31–55 (1990).
[CrossRef]

Other

D. Skoog, Principles of Instrumental Analysis, 3rd ed. (Saunders, Philadelphia, Pa., 1985), pp. 14–17.

Borland-Osborne, Turbo Pascal Reference Guide, Version 6.0 (McGraw–Hill, NewYork, 1991), pp. 91–94.

National Semiconductor, General Purpose Linear Devices Databook (National Semiconductor, Santa Clara, Calif., 1989), pp. 1–1640;Texas Instruments, The TTL Data Book (Texas Instruments, Dallas, Tx., 1985), Vol. 1, pp. 16–1572;The TTL Data Book Vol. 2, 28–1392.

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Figures (3)

Fig. 1
Fig. 1

(a) Interferogram s(x) representing a doublet, (b) its corresponding spectrum computed by Fourier transforming 128 equally sampled data points of s(x).

Fig. 2
Fig. 2

Computed spectra corresponding to different values of B: (a) B = 1, (b) B = 2, (c) B = 3, and (d) B = 4. Detection resolution inside the Nyquist interval is given by δ = 10B.

Fig. 3
Fig. 3

Effect of detection resolution δ = 10B on the visual quality of an image reconstruction: (a) 256 × 256 point equally sampled image of human chromosomes, (b) δ = 10−1, (c) δ = 10−2, and (d) δ = 10−3.

Equations (4)

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

s ( Z ) = m = M M c m Z m ,
s ( Z ) = c M Z M Π i = 1 2 M ( Z Z i ) = c M Z M ( a 0 + a 1 Z + a 2 Z 2 + , , + a 2 M Z 2 M ) ,
a m = 1 m q = 1 m s q a m q ,
a ( i ) = ± ( i ) 1 / 2 [ ( a i 1 s ( 1 ) ) 2 + ( s 1 a ( i 1 ) ) 2 + ( s 2 a ( i 2 ) ) 2 + ( a i 2 s ( 2 ) ) 2 + + s ( i ) 2 ] 1 / 2 .

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