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) = Acos(2πf r x), where the index i = 1, 2, … , 2M = 2Tf 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 a crossing exists within each period Δ = 1/2f 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

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References

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  1. C. Saloma, V. R. Daria, “Performance of a zero-crossing optical spectrum analyzer,” Opt. Lett. 18, 1468–1470 (1993).
    [CrossRef] [PubMed]
  2. V. Daria, C. Saloma, “Bandwidth and detection limit in a crossing-based spectrum analyzer,” Rev. Sci. Instrum. 68, 240–242 (1997).
    [CrossRef]
  3. C. Saloma, P. Haeberli, “Optical spectrum analysis by zero crossings,” Opt. Lett. 16, 1535–1537 (1991).
    [CrossRef] [PubMed]
  4. F. Bond, C. Cahn, “On sampling the xeros of bandwidth limited signals,” IRE Trans. Inf. Theory IT-4, 110–113 (1958).
    [CrossRef]
  5. H. Voelker, “Toward a unified theory of modulation. Part II. Zero manipulation,” Proc. IEEE 54, 735–755 (1996).
    [CrossRef]
  6. B. Logan, “Information in zero crossings of bandpass signals,” Bell Sys. Tech. J. 56, 487–510 (1977).
  7. A. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. IEEE 68, 308–328 (1980).
    [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. S. Kay, R. Sudhakar, “A zero-crossing based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. 34, 96–104 (1987).
    [CrossRef]
  10. A. Zakhor, A. Oppenheim, “Reconstruction of two-dimensional signals from level crossings,” Proc. IEEE 78, 31–55 (1990).
    [CrossRef]
  11. A. Zakhor, G. Alustad, “Two-dimensional polynomical interpolation from nonuniform samples,” IEEE Trans. Signal Process. 40, 169–175 (1992).
    [CrossRef]
  12. Y. Zeevi, E. Shlomot, “Nonuniform sampling and antialiasing in image representation,” IEEE Trans. Signal Process. 41, 1223–1229 (1993).
    [CrossRef]
  13. K. Minami, S. Kawata, S. Minami, “Zero-crossing sampling of Fourier transform interferograms and spectrum reconstruction using real-zero interpolation,” Appl. Opt. 31, 6322–6327 (1992).
    [CrossRef] [PubMed]
  14. C. Saloma, “Computational complexity and observation of physical signals,” J. Appl. Phys. 74, 5314–5319 (1993).
    [CrossRef]
  15. A. Montowski, A. Stark, Introduction to Higher Algebra (Pergamon, Oxford, 1964), pp. 364–369.
  16. J. Proakis, D. Manolakis, Digital Signal Processing: Principles, Algorithm, and Applications, 2nd ed. (Macmillan, New York, 1992), pp. 943–944.
  17. J. Hopfield, “Pattern recognition computation using action potential timing for stimulus representation,” Nature 376, 33–36 (1995).
    [CrossRef] [PubMed]
  18. C. Koch, “Computation and the single neuron,” Nature 385, 207–210 (1997).
    [CrossRef] [PubMed]
  19. C. M. Blanca, V. Daria, C. Saloma, “Spectral recovery in crossing-based spectral analysis by analytic continuation,” Appl. Opt. 35, 6417–6423 (1996).
    [CrossRef] [PubMed]
  20. M. Escobido, C. Saloma, “Detection accuracy in zero-crossing based spectrum analysis and image reconstruction,” Appl. Opt. 35, 6417–6423 (1994).
  21. S. J. Howard, “Continuation of discrete Fourier spectra using a minimum-negativity constraint,” J. Opt. Soc. Am. 7, 819–824 (1981).
    [CrossRef]
  22. S. J. Howard, “Fast algorithm for implementing the minimum-negativity constraint for Fourier spectrum extrapolation,” Appl. Opt. 25, 1670–1675 (1986).
    [CrossRef] [PubMed]
  23. W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986), pp. 24–29.
  24. G. Johhson, “Constructions of particular random processes,” Proc. IEEE 82, 270–285 (1994).
    [CrossRef]
  25. J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).
  26. L. Gammaitoni, F. Marchesoni, E. Menicella-Saetta, S. Santucci, “Multiplicative stochastic resonance,” Phys. Rev. E 49, 4878–4881 (1994).
    [CrossRef]
  27. F. O. Huck, C. Fales, N. Haylo, R. W. Samms, K. Stacey, “Image gathering and processing: information and fidelity,” J. Opt. Soc. Am. A 2, 1644–1666 (1985).
    [CrossRef] [PubMed]
  28. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, New York, 1963).
  29. B. McNamara, K. Weisenfeld, “Theory of stochastic resonance,” Phys. Rev. A 39, 4854–4869 (1989).
    [CrossRef] [PubMed]
  30. K. Weisenfeld, F. Moss, “Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDS,” Nature 373, 33–36 (1995).
    [CrossRef]
  31. L. Gammaitoni, “Stochastic resonance and the dithering effect in threshold physical systems,” Phys. Rev. E 52, 4691–4698 (1995).
    [CrossRef]
  32. A. Bulsara, L. Gammaitoni, “Tuning in to noise,” Phys. Today39–45 (March1996).
  33. M. Litong, C. Saloma, “Detection of subthreshold oscillations in sinusoid-crossing sampling,” Phys. Rev. E 57, 3579–3588 (1998).
    [CrossRef]

1998

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

1997

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

C. Koch, “Computation and the single neuron,” Nature 385, 207–210 (1997).
[CrossRef] [PubMed]

1996

C. M. Blanca, V. Daria, C. Saloma, “Spectral recovery in crossing-based spectral analysis by analytic continuation,” Appl. Opt. 35, 6417–6423 (1996).
[CrossRef] [PubMed]

H. Voelker, “Toward a unified theory of modulation. Part II. Zero manipulation,” Proc. IEEE 54, 735–755 (1996).
[CrossRef]

A. Bulsara, L. Gammaitoni, “Tuning in to noise,” Phys. Today39–45 (March1996).

1995

J. Hopfield, “Pattern recognition computation using action potential timing for stimulus representation,” Nature 376, 33–36 (1995).
[CrossRef] [PubMed]

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

L. Gammaitoni, “Stochastic resonance and the dithering effect in threshold physical systems,” Phys. Rev. E 52, 4691–4698 (1995).
[CrossRef]

1994

G. Johhson, “Constructions of particular random processes,” Proc. IEEE 82, 270–285 (1994).
[CrossRef]

L. Gammaitoni, F. Marchesoni, E. Menicella-Saetta, S. Santucci, “Multiplicative stochastic resonance,” Phys. Rev. E 49, 4878–4881 (1994).
[CrossRef]

M. Escobido, C. Saloma, “Detection accuracy in zero-crossing based spectrum analysis and image reconstruction,” Appl. Opt. 35, 6417–6423 (1994).

1993

Y. Zeevi, E. Shlomot, “Nonuniform sampling and antialiasing in image representation,” IEEE Trans. Signal Process. 41, 1223–1229 (1993).
[CrossRef]

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

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

1992

K. Minami, S. Kawata, S. Minami, “Zero-crossing sampling of Fourier transform interferograms and spectrum reconstruction using real-zero interpolation,” Appl. Opt. 31, 6322–6327 (1992).
[CrossRef] [PubMed]

A. Zakhor, G. Alustad, “Two-dimensional polynomical interpolation from nonuniform samples,” IEEE Trans. Signal Process. 40, 169–175 (1992).
[CrossRef]

1991

1990

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

1989

B. McNamara, K. Weisenfeld, “Theory of stochastic resonance,” Phys. Rev. A 39, 4854–4869 (1989).
[CrossRef] [PubMed]

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]

S. Kay, R. Sudhakar, “A zero-crossing based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. 34, 96–104 (1987).
[CrossRef]

1986

1985

1981

1980

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

1977

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

1958

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

Alustad, G.

A. Zakhor, G. Alustad, “Two-dimensional polynomical interpolation from nonuniform samples,” IEEE Trans. Signal Process. 40, 169–175 (1992).
[CrossRef]

Blanca, C. M.

Bond, F.

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

Bulsara, A.

A. Bulsara, L. Gammaitoni, “Tuning in to noise,” Phys. Today39–45 (March1996).

Cahn, C.

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

Chamberlain, J.

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).

Daria, V.

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

C. M. Blanca, V. Daria, C. Saloma, “Spectral recovery in crossing-based spectral analysis by analytic continuation,” Appl. Opt. 35, 6417–6423 (1996).
[CrossRef] [PubMed]

Daria, V. R.

Escobido, M.

Fales, C.

Flannery, B.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986), pp. 24–29.

Gammaitoni, L.

A. Bulsara, L. Gammaitoni, “Tuning in to noise,” Phys. Today39–45 (March1996).

L. Gammaitoni, “Stochastic resonance and the dithering effect in threshold physical systems,” Phys. Rev. E 52, 4691–4698 (1995).
[CrossRef]

L. Gammaitoni, F. Marchesoni, E. Menicella-Saetta, S. Santucci, “Multiplicative stochastic resonance,” Phys. Rev. E 49, 4878–4881 (1994).
[CrossRef]

Gavriely, A.

Haeberli, P.

Haylo, N.

Hopfield, J.

J. Hopfield, “Pattern recognition computation using action potential timing for stimulus representation,” Nature 376, 33–36 (1995).
[CrossRef] [PubMed]

Howard, S. J.

Huck, F. O.

Johhson, G.

G. Johhson, “Constructions of particular random processes,” Proc. IEEE 82, 270–285 (1994).
[CrossRef]

Kawata, S.

Kay, S.

S. Kay, R. Sudhakar, “A zero-crossing based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. 34, 96–104 (1987).
[CrossRef]

Koch, C.

C. Koch, “Computation and the single neuron,” Nature 385, 207–210 (1997).
[CrossRef] [PubMed]

Litong, M.

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

Logan, B.

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

Manolakis, D.

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

Marchesoni, F.

L. Gammaitoni, F. Marchesoni, E. Menicella-Saetta, S. Santucci, “Multiplicative stochastic resonance,” Phys. Rev. E 49, 4878–4881 (1994).
[CrossRef]

McNamara, B.

B. McNamara, K. Weisenfeld, “Theory of stochastic resonance,” Phys. Rev. A 39, 4854–4869 (1989).
[CrossRef] [PubMed]

Menicella-Saetta, E.

L. Gammaitoni, F. Marchesoni, E. Menicella-Saetta, S. Santucci, “Multiplicative stochastic resonance,” Phys. Rev. E 49, 4878–4881 (1994).
[CrossRef]

Minami, K.

Minami, S.

Montowski, A.

A. Montowski, A. Stark, Introduction to Higher Algebra (Pergamon, Oxford, 1964), pp. 364–369.

Moss, F.

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

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, New York, 1963).

Oppenheim, A.

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

Press, W.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986), pp. 24–29.

Proakis, J.

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

Requicha, A.

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

Saloma, C.

Samms, R. W.

Santucci, S.

L. Gammaitoni, F. Marchesoni, E. Menicella-Saetta, S. Santucci, “Multiplicative stochastic resonance,” Phys. Rev. E 49, 4878–4881 (1994).
[CrossRef]

Shamai, S.

Shlomot, E.

Y. Zeevi, E. Shlomot, “Nonuniform sampling and antialiasing in image representation,” IEEE Trans. Signal Process. 41, 1223–1229 (1993).
[CrossRef]

Stacey, K.

Stark, A.

A. Montowski, A. Stark, Introduction to Higher Algebra (Pergamon, Oxford, 1964), pp. 364–369.

Sudhakar, R.

S. Kay, R. Sudhakar, “A zero-crossing based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. 34, 96–104 (1987).
[CrossRef]

Teukolsky, S.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986), pp. 24–29.

Vetterling, W.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986), pp. 24–29.

Voelker, H.

H. Voelker, “Toward a unified theory of modulation. Part II. Zero manipulation,” Proc. IEEE 54, 735–755 (1996).
[CrossRef]

Weisenfeld, K.

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

B. McNamara, K. Weisenfeld, “Theory of stochastic resonance,” Phys. Rev. A 39, 4854–4869 (1989).
[CrossRef] [PubMed]

Zakhor, A.

A. Zakhor, G. Alustad, “Two-dimensional polynomical interpolation from nonuniform samples,” IEEE Trans. Signal Process. 40, 169–175 (1992).
[CrossRef]

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

Zeevi, Y.

Y. Zeevi, E. Shlomot, “Nonuniform sampling and antialiasing in image representation,” IEEE Trans. Signal Process. 41, 1223–1229 (1993).
[CrossRef]

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

Appl. Opt.

Bell Sys. Tech. J.

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

IEEE Trans. Acoust. Speech Signal Process.

S. Kay, R. Sudhakar, “A zero-crossing based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. 34, 96–104 (1987).
[CrossRef]

IEEE Trans. Signal Process.

A. Zakhor, G. Alustad, “Two-dimensional polynomical interpolation from nonuniform samples,” IEEE Trans. Signal Process. 40, 169–175 (1992).
[CrossRef]

Y. Zeevi, E. Shlomot, “Nonuniform sampling and antialiasing in image representation,” IEEE Trans. Signal Process. 41, 1223–1229 (1993).
[CrossRef]

IRE Trans. Inf. Theory

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

J. Appl. Phys.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature

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

J. Hopfield, “Pattern recognition computation using action potential timing for stimulus representation,” Nature 376, 33–36 (1995).
[CrossRef] [PubMed]

C. Koch, “Computation and the single neuron,” Nature 385, 207–210 (1997).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. A

B. McNamara, K. Weisenfeld, “Theory of stochastic resonance,” Phys. Rev. A 39, 4854–4869 (1989).
[CrossRef] [PubMed]

Phys. Rev. E

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

L. Gammaitoni, “Stochastic resonance and the dithering effect in threshold physical systems,” Phys. Rev. E 52, 4691–4698 (1995).
[CrossRef]

L. Gammaitoni, F. Marchesoni, E. Menicella-Saetta, S. Santucci, “Multiplicative stochastic resonance,” Phys. Rev. E 49, 4878–4881 (1994).
[CrossRef]

Phys. Today

A. Bulsara, L. Gammaitoni, “Tuning in to noise,” Phys. Today39–45 (March1996).

Proc. IEEE

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

G. Johhson, “Constructions of particular random processes,” Proc. IEEE 82, 270–285 (1994).
[CrossRef]

H. Voelker, “Toward a unified theory of modulation. Part II. Zero manipulation,” Proc. IEEE 54, 735–755 (1996).
[CrossRef]

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

Rev. Sci. Instrum.

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

Other

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986), pp. 24–29.

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).

A. Montowski, A. Stark, Introduction to Higher Algebra (Pergamon, Oxford, 1964), pp. 364–369.

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

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, New York, 1963).

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Figures (11)

Fig. 1
Fig. 1

Plots of the NMSE versus the f s produced by NF-based reconstruction, CS interpolation, and SH substitution. The parameters used were s(t) = 1.0cos(2πf s x), r(x) = 2.5cos(2π256x), T = 1, 2M = 512, and 0 ≤ f s ≤ 255. Each SC is located with an accuracy of 1 in 1000 (N = 1000).

Fig. 2
Fig. 2

Procedure for recovering the high-frequency components of an all-positive signal s(x) of bandwidth W. These components are lost because of low-pass filtering (f r = M/ T < W/2), where T is the sampling period. The signal is assumed to be nonnegative within the sampling period T.

Fig. 3
Fig. 3

(a) Uniformly sampled representation of the test signal (256 data points): s(x) =cos200(x - 1) +cos200(x + 1), where -3.5 ≤ x ≤ 3.5 (T = 7). (b) Modulus Fourier spectrum (256 components) of the plot of Fig. 2(a) showing that s(x) has an effective bandwidth of W = 88/7.

Fig. 4
Fig. 4

SC-sampled representation obtained as a series of 256 pairs of FS coefficients (2M = 256, f r = 128/7, N = 2048, NMSE = 0.000108, F = 1, S = 1.01305, and Q = 1.00512). (b) Undersampled representation obtained by use of f r = 30/7 = 4.286 (NMSE = 0.0179, F = 0.9993, S = 0.9843, and Q = 0.9832). (c) Recovered representation obtained after 10 iterations (NMSE = 0.0075, F = 1, S = 0.9889, and Q = 0.9907).

Fig. 5
Fig. 5

Plots of the NMSE versus the f r : (a) Original sampled signal. (b) Recovered signal. The plots correspond to σ2 = 0 (crosses), σ2 = 0.0001 (open circles), σ2 = 0.001 (filled circles), and σ2 = 0.01 (squares). Note that the bandwidth of s(x) is W = 2f r = 90/7 ≈ 12.86.

Fig. 6
Fig. 6

Plots of the NMSE versus q for the original low-pass-filtered data [open squares (f r = 5)], the recovered data (filled squares), and the data obtained by use of f r = 45/7, which satisfies the Nyquist sampling criterion (circles). The number of partitions N within Δ = 7/70 is N = 2 q .

Fig. 7
Fig. 7

Plots of the NMSE versus the numbers of iterations, illustrating that the use of more iterations beyond 10 no longer affects the results of the recovery (f r = 35/7, 2M = 70, and N = 1000). Three values of the variance of the additive Gaussian noise are considered: σ2 = 0 (crosses), σ2 = 0.001 (open circles), and σ2 = 0.01 (filled circles).

Fig. 8
Fig. 8

Plots of the NMSE of the recovered signal versus 2b for the case in which σ2 = 0. The recovered signal is given by the FS of the (M + b) pair of FS coefficients, where M = 30 (squares), M = 35 (filled circles), and M = 40 (open circles). The bandwidth of the test signal is 90/7.

Fig. 9
Fig. 9

Plots of the fidelity F versus the f r : (a) Original sampled signal. (b) Recovered signal. The plots correspond to σ2 = 0 (crosses), σ2 = 0.0001 (open circles), σ2 = 0.001 (filled circles), and σ2 = 0.01 (squares).

Fig. 10
Fig. 10

Plots of the structural content S versus the f r : (a) Original sampled signal. (b) Recovered signal. The plots correspond to σ2 = 0 (crosses), σ2 = 0.0001 (open circles), σ2 = 0.001 (filled circles), and σ2 = 0.01 (squares).

Fig. 11
Fig. 11

Plots of the quality Q versus the f r : (a) Original sampled signal. (b) Recovered signal. The plots correspond to σ2 = 0 (crosses), σ2 = 0.0001 (open circles), σ2 = 0.001 (filled circles), and σ2 = 0.01 (squares).

Equations (10)

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

u k = n = 0 M   A n cos 2 π T   nk + n = 0 M   B n sin 2 π T   nk ,
v k = n = M + 1 b   A n cos 2 π T   nk + n = M + 1 b   B n sin 2 π T   nk ,
A M + p = - k s k - A M + p cos 2 π T M + p k cos 2 π T M + p k k cos 2 2 π T M + p k ,
B M + p = - k s k - B M + p sin 2 π T M + p k sin 2 π T M + p k k sin 2 2 π T M + p k .
A init M + p = - k   u k cos 2 π T M + p k k cos 2 2 π T M + p k ,
B init M + p = - k   u k sin 2 π T M + p k k sin 2 2 π T M + p k ,
s x i + s x i s x i + s x i ,
s x = n = - M M   S n exp jn 2 π f o x = m = - M M   S n Z n = Z - M S - M + S - M + 1 Z + + S m Z 2 M = Z - M i 2 M Z - Z i = Z - m p Z = Z - M a 0 + a 1 Z + + a 2 M Z 2 M ,
a M + n = - 1 M + n k = 1 2 M   d k a M + n - k ,
E = E t = 1 / 2 | δ r t | = π f r δ tA | sin π p i / N | = π A / N | sin π p i / N | = B | sin π p i / N | ,

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