Abstract

A practical method of computing the spectral components directly from measured zero crossings of interferograms is presented. The method requires the sampling of only one zero crossing per Nyquist interval and yields results with a normalized mean-square error that is better than 10−6 with respect to the fast Fourier transformation when the zero crossing is located within the Nyquist interval with an accuracy of one part in 106. The method is also robust against error frequencies that may arise owing to the finite range in the floating-point representation of numbers in a computer. The error frequencies appear whenever a large number of crossings is processed. This type of error is not related to the accuracy of locating the zero crossings and limits the operational bandwidth of a zero-crossing-based optical spectrum analyzer.

© 1991 Optical Society of America

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References

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  1. I. N. Herstein, Topics in Algebra, 2nd ed. (Wiley, New York, 1975), p. 242.
  2. A. Montowski, M. Stark, Introduction to Higher Algebra (Pergamon, Oxford, 1964), p. 360.
  3. B. Logan, Bell Syst. Tech. J. 56, 510 (1977).
  4. Y. Zeevi, A. Gavriely, S. Shimai, J. Opt. Soc. Am. A 4, 2045 (1987).
    [CrossRef]
  5. W. Press, B. Fleming, S. Teukolsky, W. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986), p. 246.
  6. T. Okamoto, S. Kawata, S. Minami, Appl. Opt. 23, 269 (1984).
    [CrossRef] [PubMed]
  7. S. Kay, R. Sudhakar, IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96 (1986).
    [CrossRef]
  8. C. Saloma, P. Haeberli, “Image reconstruction using Fourier coefficients that are computed from zero crossing,” submitted to J. Opt. Soc. Am. A.

1987 (1)

1986 (1)

S. Kay, R. Sudhakar, IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96 (1986).
[CrossRef]

1984 (1)

1977 (1)

B. Logan, Bell Syst. Tech. J. 56, 510 (1977).

Fleming, B.

W. Press, B. Fleming, S. Teukolsky, W. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986), p. 246.

Gavriely, A.

Haeberli, P.

C. Saloma, P. Haeberli, “Image reconstruction using Fourier coefficients that are computed from zero crossing,” submitted to J. Opt. Soc. Am. A.

Herstein, I. N.

I. N. Herstein, Topics in Algebra, 2nd ed. (Wiley, New York, 1975), p. 242.

Kawata, S.

Kay, S.

S. Kay, R. Sudhakar, IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96 (1986).
[CrossRef]

Logan, B.

B. Logan, Bell Syst. Tech. J. 56, 510 (1977).

Minami, S.

Montowski, A.

A. Montowski, M. Stark, Introduction to Higher Algebra (Pergamon, Oxford, 1964), p. 360.

Okamoto, T.

Press, W.

W. Press, B. Fleming, S. Teukolsky, W. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986), p. 246.

Saloma, C.

C. Saloma, P. Haeberli, “Image reconstruction using Fourier coefficients that are computed from zero crossing,” submitted to J. Opt. Soc. Am. A.

Shimai, S.

Stark, M.

A. Montowski, M. Stark, Introduction to Higher Algebra (Pergamon, Oxford, 1964), p. 360.

Sudhakar, R.

S. Kay, R. Sudhakar, IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96 (1986).
[CrossRef]

Teukolsky, S.

W. Press, B. Fleming, S. Teukolsky, W. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986), p. 246.

Vetterling, W.

W. Press, B. Fleming, S. Teukolsky, W. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986), p. 246.

Zeevi, Y.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

B. Logan, Bell Syst. Tech. J. 56, 510 (1977).

IEEE Trans. Acoust. Speech Signal Process (1)

S. Kay, R. Sudhakar, IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96 (1986).
[CrossRef]

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

Other (4)

W. Press, B. Fleming, S. Teukolsky, W. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986), p. 246.

I. N. Herstein, Topics in Algebra, 2nd ed. (Wiley, New York, 1975), p. 242.

A. Montowski, M. Stark, Introduction to Higher Algebra (Pergamon, Oxford, 1964), p. 360.

C. Saloma, P. Haeberli, “Image reconstruction using Fourier coefficients that are computed from zero crossing,” submitted to J. Opt. Soc. Am. A.

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Figures (5)

Fig. 1
Fig. 1

Spectrum of an analytic doublet computed with the algorithm based on Newton's formula for N = 1000 zero crossings.

Fig. 2
Fig. 2

Normalized mean-square error (NMSE) versus accuracy in the zero-crossing location within the Nyquist interval (N = 64). The reference was computed by using a fast-Fourier-transform algorithm.

Fig. 3
Fig. 3

Spectra computed by using the algorithm based on Newton's formula, with a He–Ne laser and N = 500 zero crossings.

Fig. 4
Fig. 4

Illustration of low-frequency errors arising from finite floating-point representation, with a doublet by sequential computation, N = 80, and 10-byte representation.

Fig. 5
Fig. 5

Block diagram of circuit implementing zero-crossing detection and transport to the computer.

Equations (8)

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s ( x ) = m = M M c m e im ω x ,
s ( Z ) = m = M M c m Z m = c M Z M + c M + 1 Z M + 1 + + c M Z M ,
s ( z ) = c M Z M ( Z Z 1 ) ( Z Z 2 ) ( Z Z 2 M ) = c M Z M Π i = 1 2 M ( Z Z i ) .
a 1 = Z 1 + Z 2 + Z 3 + + Z N , a 2 = Z 1 Z 2 + + Z 1 Z N + Z 2 Z 3 + + Z 2 Z N + + Z N 1 Z N , a N 1 = Z 1 Z 2 Z 3 Z N 1 + Z 1 Z 2 Z N 3 Z N + + Z 2 Z 3 Z N , a N = Z 1 Z 2 Z 3 Z 4 Z N ,
P N ( Z ) = a 0 Z N + a 1 Z N 1 + a 2 Z N 2 + + a N = a 0 ( Z Z 1 ) ( Z Z 2 ) ( Z Z 3 ) ( Z Z N ) ,
s k = Z 1 k + Z 2 k + Z 3 k + + Z N k .
1 a 1 + a 0 = 0 2 a 2 + s 1 a 1 + s 2 a 0 = 0 , 3 a 3 + s 1 a 2 + s 2 a 1 + s 3 a 0 = 0 , N a N + s 1 a N 1 + s 2 a N 2 + s 3 a N 2 + s 3 a N 3 + + s N a 0 = 0.
a k = 1 k n = 1 k s n a k n .

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