Abstract

By the use of analytic continuation, the correct spectrum of an undersampled analog input signal fa(t) of a true bandwidth B is recovered from an aliased Fourier spectrum that is computed directly from a data set consisting of sinusoid-crossing locations {ti}, where the signal fa(t) intersects with a reference sinusoid r(t) with a frequency of W < B/2 and an amplitude of A. If A ≥ |fa(t)| within the sampling period T, then a crossing exists within each time interval Δ = 1/2W, and a total of 2WT = 2M sinusoid crossings are detected, where M is a positive integer. The cut-off frequency for sampling is W = ±M/T. In a crossing detector, a trade-off exists between the size of Δ and the accuracy with which a crossing can be located within it because the detector has a finite response time. Low-accuracy detection of the crossing positions degrades the detection limit of the detector and results in a computed Fourier spectrum that contains spurious wideband frequencies. We show however that, if fa(t) has a known compact support within T, then sampling at a frequency of W < B/2 may still be possible because the correct fa(t) spectrum can be recovered from the aliased spectrum by means of analytic continuation. The technique is demonstrated for an interferogram test signal in both the absence and presence of additive Gaussian noise.

© 1996 Optical Society of America

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References

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  1. C. Saloma, V. Daria, “Performance of a zero-crossing optical spectrum analyzer,” Opt. Lett. 18, 1468–1470 (1993).
    [CrossRef] [PubMed]
  2. C. Saloma, P. Haeberli, “Optical spectrum analysis from zero crossings,” Opt. Lett. 16, 1535–1537 (1991).
    [CrossRef] [PubMed]
  3. C. Saloma, “Computational complexity and the observation of physical signals,” J. Appl. Phys. 74, 5314–5319 (1993).
    [CrossRef]
  4. C. Saloma, M. Escobido, “Detection accuracy in zero-crossing-based spectrum analysis and image reconstruction,” Appl. Opt. 33, 7617–7621 (1994).
    [CrossRef] [PubMed]
  5. S. Howard, “Method for continuing Fourier spectra given by the fast Fourier transform,” J. Opt. Soc. Am. A 71, 1 (1981).
  6. P. Jansson, ed., Deconvolution with Applications to Spectroscopy (Academic, Tokyo, 1984), p. 261.
  7. A. Montowski, A. Stark, Introduction to Higher Algebra (Pergamon, Oxford, England, 1964), p. 364.
  8. R. Bracewell, The Fourier Transform and its Applications (Prentice-Hall, Englewood Cliffs, New Jersey, 1988), p. 104.
  9. J. Proakis, D. Manolakis, Digital Signal Processing: Principles, Algorithm, and Applications, 2nd ed. (Macmillan, New York, 1992), pp. 943–944.
  10. W. Press, W. Vetterling, S. Teukolsky, B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 41–43.
  11. C. Saloma, “Reconstructing the Hartley intensity spectrum from its sinusoidal crossings,” Opt. Lett. 20, 1–3 (1995).
    [CrossRef] [PubMed]
  12. C. Saloma, “Wavelet transform analysis directly from sinusoid crossings,” Phys. Rev. E 53, 1962–1966 (1996).
    [CrossRef]

1996 (1)

C. Saloma, “Wavelet transform analysis directly from sinusoid crossings,” Phys. Rev. E 53, 1962–1966 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (2)

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

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

1991 (1)

1981 (1)

S. Howard, “Method for continuing Fourier spectra given by the fast Fourier transform,” J. Opt. Soc. Am. A 71, 1 (1981).

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (Prentice-Hall, Englewood Cliffs, New Jersey, 1988), p. 104.

Daria, V.

Escobido, M.

Flannery, B.

W. Press, W. Vetterling, S. Teukolsky, B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 41–43.

Haeberli, P.

Howard, S.

S. Howard, “Method for continuing Fourier spectra given by the fast Fourier transform,” J. Opt. Soc. Am. A 71, 1 (1981).

Manolakis, D.

J. Proakis, D. Manolakis, Digital Signal Processing: Principles, Algorithm, and Applications, 2nd ed. (Macmillan, New York, 1992), pp. 943–944.

Montowski, A.

A. Montowski, A. Stark, Introduction to Higher Algebra (Pergamon, Oxford, England, 1964), p. 364.

Press, W.

W. Press, W. Vetterling, S. Teukolsky, B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 41–43.

Proakis, J.

J. Proakis, D. Manolakis, Digital Signal Processing: Principles, Algorithm, and Applications, 2nd ed. (Macmillan, New York, 1992), pp. 943–944.

Saloma, C.

Stark, A.

A. Montowski, A. Stark, Introduction to Higher Algebra (Pergamon, Oxford, England, 1964), p. 364.

Teukolsky, S.

W. Press, W. Vetterling, S. Teukolsky, B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 41–43.

Vetterling, W.

W. Press, W. Vetterling, S. Teukolsky, B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 41–43.

Appl. Opt. (1)

J. Appl. Phys. (1)

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

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

S. Howard, “Method for continuing Fourier spectra given by the fast Fourier transform,” J. Opt. Soc. Am. A 71, 1 (1981).

Opt. Lett. (3)

Phys. Rev. E (1)

C. Saloma, “Wavelet transform analysis directly from sinusoid crossings,” Phys. Rev. E 53, 1962–1966 (1996).
[CrossRef]

Other (5)

P. Jansson, ed., Deconvolution with Applications to Spectroscopy (Academic, Tokyo, 1984), p. 261.

A. Montowski, A. Stark, Introduction to Higher Algebra (Pergamon, Oxford, England, 1964), p. 364.

R. Bracewell, The Fourier Transform and its Applications (Prentice-Hall, Englewood Cliffs, New Jersey, 1988), p. 104.

J. Proakis, D. Manolakis, Digital Signal Processing: Principles, Algorithm, and Applications, 2nd ed. (Macmillan, New York, 1992), pp. 943–944.

W. Press, W. Vetterling, S. Teukolsky, B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 41–43.

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Figures (4)

Fig. 1
Fig. 1

(a) Interferogram test signal (T = 1), (b) real part of the Fourier spectrum (f = m), (c) imaginary part of the Fourier spectrum (f = m), and (d) the modulus spectrum (B = 64). The spectra were computed from 256 sinusoidal crossings (W = 128). Each crossing within a time interval of Δ = 1/256 was located with an accuracy of 1 in 216 (q = 16).

Fig. 2
Fig. 2

NMSE plots for different values of the variance σ where σ2 is the variance of the additive Gaussian noise: (a) the NMSE versus q (the partition number is 2 q ), (b) the NMSE versus the bit number b (i.e., the number of digital outputs) of an amplitude-sampling AD converter. The test signal is the interferogram shown in Fig. 1(a).

Fig. 3
Fig. 3

(a) Plots of the truncated spectrum (open squares, frequency of W = 22 = 1/2Δ) and the spectrum recovered with the analytic-continuation technique (filled squares), and (b) the dependence of the NMSE on the value of the reference frequency W. A relative decrease in the NMSE value was gained after analytic continuation was applied to the truncated spectrum of the cut-off frequency W.

Fig. 4
Fig. 4

Dependence of the NMSE on the frequency W for the case of a noisy signal (additive Gaussian noise σ = 0.05). The plot formed by the filled squares represents the NMSE values that were obtained after analytic continuation was applied to the truncated spectrum of the cut-off frequency W.

Equations (9)

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f ( t ) = m = M M c ( m ) exp ( j m 2 π f 0 t ) = m = M M c ( m ) Z m = Z M [ c ( M ) + c ( M + 1 ) Z + + c ( M ) Z 2 M ] = Z M i = 1 2 M ( Z Z i ) = Z M p ( Z ) = Z M [ a ( 0 ) + a ( 1 ) Z + a ( 2 ) Z 2 + + a ( 2 M ) Z 2 M ] ,
a ( M + m ) = 1 ( M + m ) n = 1 M + m s n a ( M + m n ) ,
NMSE = k = 0 M 1 | f a ( k ) f ( k ) | 2 k = 0 M 1 | f ( k ) | 2 ,
f ( k Δ ) = A ( 0 ) 2 + m = 1 2 M 1 [ A ( m ) cos ( π m k M ) + B ( m ) sin ( π m k M ) ] ,
f u ( k Δ ) = A ( 0 ) 2 + m = 1 β 1 [ A ( m ) cos ( π m k M ) + B ( m ) sin ( π m k M ) ] .
f e ( k Δ ) = f u ( k Δ ) + m = β 2 M 1 [ A ( m ) cos ( π m k M ) + B ( m ) sin ( π m k M ) ] .
ϕ = k < k 1 , k > k 2 | f e ( k Δ ) | 2 ,
k < k 1 , k > k 2 | f e ( k Δ ) | cos ( π m k M ) = 0 , m = β , β + 1 , , 2 M 1 ,
k < k 1 , k > k 2 | f e ( k Δ ) | sin ( π m k M ) = 0 , m = β , β + 1 , , 2 M 1 ,

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