Abstract

We propose a new method for analysis of the sampling and reconstruction conditions of real and complex signals by use of the Wigner domain. It is shown that the Wigner domain may provide a better understanding of the sampling process than the traditional Fourier domain. For example, it explains how certain nonbandlimited complex functions can be sampled and perfectly reconstructed. On the basis of observations in the Wigner domain, we derive a generalization to the Nyquist sampling criterion. By using this criterion, we demonstrate simple preprocessing operations that can adapt a signal that does not fulfill the Nyquist sampling criterion. The preprocessing operations demonstrated can be easily implemented by optical means.

© 2004 Optical Society of America

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Corrections

Adrian Stern and Bahram Javidi, "Sampling in the light of Wigner distribution: errata," J. Opt. Soc. Am. A 21, 2038-2038 (2004)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-21-10-2038

References

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  1. C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 447–457 (1949).
    [CrossRef]
  2. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  3. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1980).
    [CrossRef]
  4. G. Cristobal, C. Gonzalo, J. Bescos, “Image filtering and analysis through the Wigner distribution,” in Advances in Electronics and Electron Physics Series, Vol. 80, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1991), pp. 309–397.
  5. V. Arrizon, J. Ojeda-Castaneda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A 9, 1801–1806 (1992).
    [CrossRef]
  6. J. M. Whittaker, “The Fourier theory of the cardinal functions,” Proc. - R. Soc. Edinburgh Sect. A Math. 1, 169–176 (1929).
    [CrossRef]
  7. A. Papoulis, “Pulse compression, fiber communication, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13 (1994).
    [CrossRef]
  8. F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
    [CrossRef]
  9. L. Onural, “Sampling of the diffraction field,” Appl. Opt. 39, 5929–5935 (2000).
    [CrossRef]
  10. A. Stern, B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43(1), (2004).
    [CrossRef]
  11. B. Boashash, Time–Frequency Analysis (Wiley, New York, 1992).
  12. A. V. Oppenheim, R. W. Schafer, Discrete Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  13. A. W. Lohmann, “Image rotation, Wigner distribution, and fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  14. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
    [CrossRef]
  15. Z. Zalevsky, D. Mendlovic, A. W. Lohmann, “Understanding superesolution in Wigner space,” J. Opt. Soc. Am. A 17, 2422–2430 (2000).
    [CrossRef]
  16. M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  18. L. B. Almeida, “The fractional Fourier transform and time frequency representation,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  19. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded index fibers, Wigner distribution, and the fractal Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  20. D. Dragoman, “Wigner-distribution-function representation of the coupling coefficient,” Appl. Opt. 34, 6758–6763 (1995).
    [CrossRef] [PubMed]
  21. K. Grochenig, Foundation of Time–Frequency Analysis (Birkhäuser, Boston, Mass., 2001).

2004 (1)

A. Stern, B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43(1), (2004).
[CrossRef]

2000 (3)

1996 (1)

1995 (1)

1994 (3)

1993 (1)

1992 (1)

1981 (1)

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

1980 (1)

1949 (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 447–457 (1949).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

1929 (1)

J. M. Whittaker, “The Fourier theory of the cardinal functions,” Proc. - R. Soc. Edinburgh Sect. A Math. 1, 169–176 (1929).
[CrossRef]

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time frequency representation,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Arrizon, V.

Bastiaans, M. J.

Bescos, J.

G. Cristobal, C. Gonzalo, J. Bescos, “Image filtering and analysis through the Wigner distribution,” in Advances in Electronics and Electron Physics Series, Vol. 80, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1991), pp. 309–397.

Boashash, B.

B. Boashash, Time–Frequency Analysis (Wiley, New York, 1992).

Cristobal, G.

G. Cristobal, C. Gonzalo, J. Bescos, “Image filtering and analysis through the Wigner distribution,” in Advances in Electronics and Electron Physics Series, Vol. 80, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1991), pp. 309–397.

Dorsch, R. G.

Dragoman, D.

Ferreira, C.

Gonzalo, C.

G. Cristobal, C. Gonzalo, J. Bescos, “Image filtering and analysis through the Wigner distribution,” in Advances in Electronics and Electron Physics Series, Vol. 80, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1991), pp. 309–397.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Gori, F.

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

Grochenig, K.

K. Grochenig, Foundation of Time–Frequency Analysis (Birkhäuser, Boston, Mass., 2001).

Javidi, B.

A. Stern, B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43(1), (2004).
[CrossRef]

Lohmann, A. W.

Mendlovic, D.

Ojeda-Castaneda, J.

Onural, L.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Discrete Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Ozaktas, H. M.

Papoulis, A.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Discrete Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Shannon, C. E.

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 447–457 (1949).
[CrossRef]

Stern, A.

A. Stern, B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43(1), (2004).
[CrossRef]

Unser, M.

M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
[CrossRef]

Whittaker, J. M.

J. M. Whittaker, “The Fourier theory of the cardinal functions,” Proc. - R. Soc. Edinburgh Sect. A Math. 1, 169–176 (1929).
[CrossRef]

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Zalevsky, Z.

Appl. Opt. (3)

IEEE Trans. Signal Process. (1)

L. B. Almeida, “The fractional Fourier transform and time frequency representation,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

Opt. Eng. (1)

A. Stern, B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43(1), (2004).
[CrossRef]

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Proc. - R. Soc. Edinburgh Sect. A Math. (1)

J. M. Whittaker, “The Fourier theory of the cardinal functions,” Proc. - R. Soc. Edinburgh Sect. A Math. 1, 169–176 (1929).
[CrossRef]

Proc. IEEE (1)

M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
[CrossRef]

Proc. IRE (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 447–457 (1949).
[CrossRef]

Other (5)

G. Cristobal, C. Gonzalo, J. Bescos, “Image filtering and analysis through the Wigner distribution,” in Advances in Electronics and Electron Physics Series, Vol. 80, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1991), pp. 309–397.

B. Boashash, Time–Frequency Analysis (Wiley, New York, 1992).

A. V. Oppenheim, R. W. Schafer, Discrete Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

K. Grochenig, Foundation of Time–Frequency Analysis (Birkhäuser, Boston, Mass., 2001).

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Figures (4)

Fig. 1
Fig. 1

Effect of sampling in the Wigner space. (a) The space–bandwidth product of a typical real function. (b) The space–bandwidth product of the sampled function. The WD of the sampled function has nonzero values at discrete location x and is formed from infinite replicas of the WD frequency direction ν. The reconstruction is carried out by filtering the WD within the dashed horizontal lines.

Fig. 2
Fig. 2

Space–bandwidth product of a function for which the maximum frequency Bν/2 and the minimum frequency -Bν/2 do not appear at the same location x. (b) The space–bandwidth product of the sampled function. A precise reconstruction can be carried out by filtering the area within the dashed rectangle even though the Nyquist criterion is not fulfilled (Bν1/Δ).

Fig. 3
Fig. 3

Adaptation of a signal for sampling: (a) the SW(x, ν) of a signal f(x); (b) the SW(x, ν) of the f(x) sampled at a rate of 1/Δ, (c) the SW(x, ν) of f(x) and (d) the SW(x, ν) of the samples of f(x), (e) the SW(x, ν) of the Fourier transform of f(x) and (f) the SW(x, ν) of its samples, (g) the SW(x, ν) of the Fresnel transform of f(x) and (h) the SW(x, ν) of its samples, (i) the SW(x, ν) of the fractional Fourier transform of f(x) and (j) the SW(x, ν) of its samples.

Fig. 4
Fig. 4

Example of reconstruction of a signal sampled at a third of the Nyquist rate. (a) Original function f(x)=[1+0.5 cos(2πx)]rect(x/9); (b) the samples of f(x) at a sampling rate of νs=0.74 (Δ=1.35), which is three times less than the Nyquist rate. The dashed curve represents the reconstruction signal from the samples by applying a low-pass filter with cutoff frequency νs/2. The reconstruction is aliased because of Nyquist undersampling. (c) The samples of the Fresnel transform with β=0.194 (g0.194[n1.35]) of f(x). The real part of g0.194[n1.35] is denoted by dots and the imaginary part by asterisks. (d) The original function (solid curve) and the reconstructed (dashed curve) function from the Fresnel samples shown in (c). The crosses on the horizontal axis indicate the sampling locations. The reconstruction from the Fresnel samples is carried out by performing inverse Fresnel transform and low-pass filtering with a cutoff frequency slightly higher than Bx/22.2.

Equations (34)

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Wf(x, ν)=-fx+x2f*x-x2exp(-j2πxν)dx,
Wf(x, ν)=WF(ν, x)=-Fν+ν2F*ν-ν2exp(-j2πxν)dν,
F(ν)=-f(x)exp(-j2πxν)dx.
f(x1)f*(x2)=-Wfx1+x22, νexp[j2π(x1-x2)ν]dν,
fs(x)=f(x)nδ(x-nΔ),
Wfs(x, ν)=-fx+x2f*x-x2×nδx+x2-nΔ×lδx-x2-lΔexp(-j2πxν)dx=--Wf(x, ν)exp(-j2πxν)dν×nδ(x-nΔ)lδ(x-lΔ)exp(-j2πxν)dx=nδ(x-nΔ)-Wf(x, ν)-lδ(x-lΔ)exp[j2πx(ν-ν)]dνdx=nδ(x-nΔ)-Wf(x, ν)1Δ×kδν-ν-k1Δdν=1Δnδ(x-nΔ)kWfx, ν-k1Δ.
Wf˜s(x, ν)=1Δnδ(x-nΔ)Wf(x, ν).
f(x)g(x)=f(ax),
Wf(x, ν)Wg(x, ν)=Wf(ax, ν/a).
f(x)g(x)=-f(x)exp(-2πxν)dx.
Wf(x, ν)Wg(x, ν)=Wf(-ν, x).
gβ(x)=βjπ-f(ξ)exp[jβ(x-ξ)2]dξ.
Wf(x, ν)Wgβ(x, ν)=Wfx-πβν, ν.
max[Bν(x)]=Bν,βπBνBx,βπBx,βπ<BνBx.
f(x)=(1+0.5 cos 2πx)rect(x/9),
rect(x)=1|x|<1/21/2|x|=1/20|x|<1/2.
gp(x)=C1f(x)expjπtan ϕ×(x02+x2)exp-j2πsin ϕ xx0dx0,ϕ=p(π/2),
C1=exp-jπ sgn(sin ϕ)4ϕ2|sin ϕ|1/2.
Wf(x, ν)Wg(x, ν)=Wf(x cos ϕ-ν sin ϕ, ν cos ϕ+x sin ϕ).
max[Bν(x)]=Bν|cos ϕ|    |tan ϕ|BxBνBx|sin ϕ|    |tan ϕ|>BxBν.
g(x)=f(-x)
0<n|g(x)-αn|2<  almost everywhere.
Sϕ,g(x, ν)=-ϕ(x)g*(x-x)exp(-j2πνx)dx.
ϕ(x)=nkcn,kγ(x-nα)exp(j2πkβ),
-g(x)γ*(x-nα)exp(-j2πkβ)dx=δmδk
cn,k=-ϕ(x)g*(x-nα)exp(-j2πkβx)dx,
cn,k=Sϕ,g(nα, kβ)  n,k integers.
Wϕ,g(x, ν)=-ϕx+x2g*×x-x2ecp(-j2πxν)dx.
Wϕ,f(x, ν)=-ϕx+x2f*x-x2×exp(-j2πxν)dx=-ϕ(s)g*[-(s-t)]exp[-j4πν(s-t)]ds=exp(j4πxν)Sϕ,g(2x, 2ν),
Wϕ,f(x/2, ν/2)=exp(jπxν)Sϕ,g(x, ν).
Wϕ,fnα2, kβ2=exp(jπnkαβ)Sϕ,g(nα, kβ).
Wϕ,f(nΔ, kΩ)=exp(jπnkαβ)Sϕ,g(nα, kβ).
Gβ(ν)=G(ν)exp-jπ2βν2,
Wgβ(x, ν)=-Gβν+ν2Gβ*ν-ν2×exp(j2πxν)dν=-Gν+ν2exp-jπ2βν+ν22G*×ν-ν2expjπ2βν-ν22dν=-Gν+ν2G*ν-ν2×expj2πνx-πβνdν=Wfx-πβν, ν.

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