Quantum limits of super-resolution of optical sparse objects via sparsity constraint

Hui Wang; Shensheng Han; Mikhail I. Kolobov

doi:10.1364/OE.20.023235

Abstract

Sparsity constraint is a priori knowledge of the signal, which means that in some properly chosen basis only a small percentage of the total number of the signal components is nonzero. Sparsity constraint has been used in signal and image processing for a long time. Recent publications have shown that the Sparsity constraint can be used to achieve super-resolution of optical sparse objects beyond the diffraction limit. In this paper we present the quantum theory which establishes the quantum limits of super-resolution for the sparse objects. The key idea of our paper is to use the discrete prolate spheroidal sequences (DPSS) as the sensing basis. We demonstrate both analytically and numerically that this sensing basis gives superior performance of super-resolution over the Fourier basis conventionally used for sensing of sparse signals. The explanation of this phenomenon is in the fact that the DPSS are the eigenfunctions of the optical imaging system while the Fourier basis are not. We investigate the role of the quantum fluctuations of the light illuminating the object, in the performance of reconstruction algorithm. This analysis allows us to formulate the criteria for stable reconstruction of sparse objects with super-resolution. Our results imply that sparsity of the object is not the only parameter which describes super-resolution achievable via sparsity constraint.

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References

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Year
Author
Publication

M. I. Kolobov (Ed.), Quantum Imaging (Springer, NY, 2006).
S. J. Bentley, Principles of Quantum Imaging: Ghost Imaging, Ghost Diffraction, and Quantum Lithography (Taylor & Francis, Boca Raton, 2010).
M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
[Crossref] [PubMed]
V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A 71, 043802 (2005).
[Crossref]
V. N. Beskrovnyy and M. I. Kolobov, “Quantum-statistical analysis of superresolution for optical systems with circular symmetry,” Phys. Rev. A 78, 043824 (2008).
[Crossref]
M. I. Kolobov, “Quantum limits of superresolution for imaging discrete subwavelength structures,” Opt. Express 16, 58–66 (2008).
[Crossref] [PubMed]
M. Elad, M. A. T. Figueiredo, and Y. Ma, “On the Role of Sparse and Redundant Representations in Image Processing,” Proc IEEE 98, 972–982 (2010).
[Crossref]
E. J. Candès, J. Romberg, and T. Tao,“Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]
E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[Crossref]
E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[Crossref]
D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]
S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009).
[Crossref]
Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35, 1148–1150 (2010).
[Crossref] [PubMed]
W. Gong and S. Han, “Super-resolution ghost imaging via compressive sampling reconstruction,” arXiv [0910.4823v1] (2009).
W. Gong and S. Han, “Super-resolution and reconstruction of far-field ghost imaging via sparcity constraints,” in the Proceedings of SPARS’11-Singal Processing with Adaptive Sparse Structured Representations, (Edinburgh, Scotland, 2011), p. 91.
[PubMed]
D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-V: The discrete case,” Bell System Tech. J. 57, 1371–1430 (1978).
D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inf. Theory 47, 2845–2862 (2001).
[Crossref]
S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Review 43, 129–159 (2001).
[Crossref]
D. F. Walls and G. J. Milburn, 2nd ed.Quantum Optics (Springer, Berlin, 2008).
[Crossref]
D. Slepian, “Some comments on Fourier analysis, uncerlainty and modeling,” SIAM Review 25, 379–393 (1983).
[Crossref]
D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[Crossref]
E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” arXiv[1203.5871v1] (2012).

2010 (2)

M. Elad, M. A. T. Figueiredo, and Y. Ma, “On the Role of Sparse and Redundant Representations in Image Processing,” Proc IEEE 98, 972–982 (2010).
[Crossref]

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35, 1148–1150 (2010).
[Crossref] [PubMed]

2009 (1)

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009).
[Crossref]

2008 (3)

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[Crossref]

V. N. Beskrovnyy and M. I. Kolobov, “Quantum-statistical analysis of superresolution for optical systems with circular symmetry,” Phys. Rev. A 78, 043824 (2008).
[Crossref]

M. I. Kolobov, “Quantum limits of superresolution for imaging discrete subwavelength structures,” Opt. Express 16, 58–66 (2008).
[Crossref] [PubMed]

2006 (3)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao,“Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[Crossref]

2005 (1)

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A 71, 043802 (2005).
[Crossref]

2001 (2)

D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inf. Theory 47, 2845–2862 (2001).
[Crossref]

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Review 43, 129–159 (2001).
[Crossref]

2000 (1)

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
[Crossref] [PubMed]

1989 (1)

D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[Crossref]

1983 (1)

D. Slepian, “Some comments on Fourier analysis, uncerlainty and modeling,” SIAM Review 25, 379–393 (1983).
[Crossref]

1978 (1)

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-V: The discrete case,” Bell System Tech. J. 57, 1371–1430 (1978).

Bentley, S. J.

S. J. Bentley, Principles of Quantum Imaging: Ghost Imaging, Ghost Diffraction, and Quantum Lithography (Taylor & Francis, Boca Raton, 2010).

Beskrovnyy, V. N.

V. N. Beskrovnyy and M. I. Kolobov, “Quantum-statistical analysis of superresolution for optical systems with circular symmetry,” Phys. Rev. A 78, 043824 (2008).
[Crossref]

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A 71, 043802 (2005).
[Crossref]

Candès, E. J.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao,“Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[Crossref]

E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” arXiv[1203.5871v1] (2012).

Chen, S. S.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Review 43, 129–159 (2001).
[Crossref]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Review 43, 129–159 (2001).
[Crossref]

D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inf. Theory 47, 2845–2862 (2001).
[Crossref]

D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[Crossref]

Elad, M.

M. Elad, M. A. T. Figueiredo, and Y. Ma, “On the Role of Sparse and Redundant Representations in Image Processing,” Proc IEEE 98, 972–982 (2010).
[Crossref]

Eldar, Y. C.

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35, 1148–1150 (2010).
[Crossref] [PubMed]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009).
[Crossref]

Fabre, C.

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
[Crossref] [PubMed]

Fernandez-Granda, C.

E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” arXiv[1203.5871v1] (2012).

Figueiredo, M. A. T.

M. Elad, M. A. T. Figueiredo, and Y. Ma, “On the Role of Sparse and Redundant Representations in Image Processing,” Proc IEEE 98, 972–982 (2010).
[Crossref]

Gazit, S.

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35, 1148–1150 (2010).
[Crossref] [PubMed]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009).
[Crossref]

Gong, W.

W. Gong and S. Han, “Super-resolution ghost imaging via compressive sampling reconstruction,” arXiv [0910.4823v1] (2009).

W. Gong and S. Han, “Super-resolution and reconstruction of far-field ghost imaging via sparcity constraints,” in the Proceedings of SPARS’11-Singal Processing with Adaptive Sparse Structured Representations, (Edinburgh, Scotland, 2011), p. 91.
[PubMed]

Han, S.

W. Gong and S. Han, “Super-resolution and reconstruction of far-field ghost imaging via sparcity constraints,” in the Proceedings of SPARS’11-Singal Processing with Adaptive Sparse Structured Representations, (Edinburgh, Scotland, 2011), p. 91.
[PubMed]

W. Gong and S. Han, “Super-resolution ghost imaging via compressive sampling reconstruction,” arXiv [0910.4823v1] (2009).

Huo, X.

D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inf. Theory 47, 2845–2862 (2001).
[Crossref]

Kolobov, M. I.

V. N. Beskrovnyy and M. I. Kolobov, “Quantum-statistical analysis of superresolution for optical systems with circular symmetry,” Phys. Rev. A 78, 043824 (2008).
[Crossref]

M. I. Kolobov, “Quantum limits of superresolution for imaging discrete subwavelength structures,” Opt. Express 16, 58–66 (2008).
[Crossref] [PubMed]

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A 71, 043802 (2005).
[Crossref]

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
[Crossref] [PubMed]

Ma, Y.

M. Elad, M. A. T. Figueiredo, and Y. Ma, “On the Role of Sparse and Redundant Representations in Image Processing,” Proc IEEE 98, 972–982 (2010).
[Crossref]

Milburn, G. J.

D. F. Walls and G. J. Milburn, 2nd ed.Quantum Optics (Springer, Berlin, 2008).
[Crossref]

Romberg, J.

E. J. Candès, J. Romberg, and T. Tao,“Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Saunders, M. A.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Review 43, 129–159 (2001).
[Crossref]

Segev, M.

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35, 1148–1150 (2010).
[Crossref] [PubMed]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009).
[Crossref]

Shechtman, Y.

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35, 1148–1150 (2010).
[Crossref] [PubMed]

Slepian, D.

D. Slepian, “Some comments on Fourier analysis, uncerlainty and modeling,” SIAM Review 25, 379–393 (1983).
[Crossref]

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-V: The discrete case,” Bell System Tech. J. 57, 1371–1430 (1978).

Stark, P. B.

D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[Crossref]

Szameit, A.

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35, 1148–1150 (2010).
[Crossref] [PubMed]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009).
[Crossref]

Tao, T.

E. J. Candès, J. Romberg, and T. Tao,“Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[Crossref]

Wakin, M. B.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[Crossref]

Walls, D. F.

D. F. Walls and G. J. Milburn, 2nd ed.Quantum Optics (Springer, Berlin, 2008).
[Crossref]

Bell System Tech. J. (1)

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-V: The discrete case,” Bell System Tech. J. 57, 1371–1430 (1978).

IEEE Signal Process. Mag. (1)

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[Crossref]

IEEE Trans. Inf. Theory (4)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inf. Theory 47, 2845–2862 (2001).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao,“Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[Crossref]

Opt. Express (2)

M. I. Kolobov, “Quantum limits of superresolution for imaging discrete subwavelength structures,” Opt. Express 16, 58–66 (2008).
[Crossref] [PubMed]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009).
[Crossref]

Opt. Lett. (1)

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35, 1148–1150 (2010).
[Crossref] [PubMed]

Phys. Rev. A (2)

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A 71, 043802 (2005).
[Crossref]

V. N. Beskrovnyy and M. I. Kolobov, “Quantum-statistical analysis of superresolution for optical systems with circular symmetry,” Phys. Rev. A 78, 043824 (2008).
[Crossref]

Phys. Rev. Lett. (1)

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
[Crossref] [PubMed]

Proc IEEE (1)

M. Elad, M. A. T. Figueiredo, and Y. Ma, “On the Role of Sparse and Redundant Representations in Image Processing,” Proc IEEE 98, 972–982 (2010).
[Crossref]

SIAM J. Appl. Math. (1)

D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[Crossref]

SIAM Review (2)

D. Slepian, “Some comments on Fourier analysis, uncerlainty and modeling,” SIAM Review 25, 379–393 (1983).
[Crossref]

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Review 43, 129–159 (2001).
[Crossref]

Other (6)

D. F. Walls and G. J. Milburn, 2nd ed.Quantum Optics (Springer, Berlin, 2008).
[Crossref]

W. Gong and S. Han, “Super-resolution ghost imaging via compressive sampling reconstruction,” arXiv [0910.4823v1] (2009).

W. Gong and S. Han, “Super-resolution and reconstruction of far-field ghost imaging via sparcity constraints,” in the Proceedings of SPARS’11-Singal Processing with Adaptive Sparse Structured Representations, (Edinburgh, Scotland, 2011), p. 91.
[PubMed]

M. I. Kolobov (Ed.), Quantum Imaging (Springer, NY, 2006).

S. J. Bentley, Principles of Quantum Imaging: Ghost Imaging, Ghost Diffraction, and Quantum Lithography (Taylor & Francis, Boca Raton, 2010).

E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” arXiv[1203.5871v1] (2012).

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Fig. 1 Optical one-dimensional far-field imaging scheme of discrete sparse objects.

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Fig. 2 Coherences γ(K, W) between the spike and the DPSS basis, and Γ(K, W) between the Fourier and the DPSS basis. The graphs (a) and (c) show γ(K, W) and Γ(K, W) as functions of K for fixed value of W. The black lines correspond to the extreme values of coherence γ = 1 and

γ = \sqrt{K}

. The graphs (b) and (d) show the coherences as functions of W for fixed number of pixels K.

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Fig. 3 Reconstruction of an original sparse object with super-resolution via sparsity constraint using the Fourier and the DPSS sensing bases. The columns (1)–(3) correspond to three different values of the space-bandwidth product c = 30, 20, 10, providing the super-resolution factor SF = 2.7, 4, 7, respectively. The row (a) shows the absolute value of the spatial Fourier spectrum of the object together with the size of the aperture in the Fourier plane. The row (b) gives the original object (black dots) and its diffraction-limited image (red circles). The row (c) shows the original object and the reconstructed object using the Fourier sensing basis. The row (d) shows the original object and the reconstructed object using the DPSS sensing basis.

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Fig. 4 The influence of the quantum fluctuations on the reconstruction of sparse objects via sparsity constraint using DPSS sensing basis. In the column (1) black dots show the original object a(m) and the colored circles - ten random realizations of the reconstructed images a^(r)(m). The column (2) gives the absolute value of the corresponding spatial spectra |f(ξ)| together with the size of the diaphragm. The Shannon number is equal to S = 12.7. The rows (a)–(c) correspond to an object reconstructed with Q = 13 sensing coefficients, while the rows (d)–(f) to another object, reconstructed with Q = 19 sensing coefficients. The SNR are: 10¹² in (a) and (d), 10⁸ in (b) and (e), and 10⁴ in (c) and (f).

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Tables (1)

Table 1 Statistical results of stable reconstructions.

View Table

Equations (38)

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

F (ξ) = \sqrt{\frac{c}{2 π}} \int_{- 1}^{1} a (s) e^{- ics ξ} d s,

a (m) = \frac{1}{\sqrt{Δ}} \int_{Δ (m - 1 / 2)}^{Δ (m + 1 / 2)} a (s) d s .

f (ξ) = \sqrt{W} \sum_{m = - M}^{M} a (m) e^{- 2 π i W ξ m},

\sum_{m^{'} = - M}^{M} \frac{sin [2 π W (m - m^{'})]}{π (m - m^{'})} ψ_{μ} (m^{'}) = λ_{μ} ψ_{μ} (m),

\sum_{m = - M}^{M} ψ_{μ} (m) ψ_{ν} (m) = λ_{μ} \sum_{m = - \infty}^{\infty} ψ_{μ} (m) ψ_{ν} (m) = λ_{μ} δ_{μ ν} .

\int_{- 1}^{1} \frac{sin [π K W (ξ - ξ^{'})]}{sin [π W (ξ - ξ^{'})]} Ψ_{μ} (ξ^{'}) d ξ^{'} = λ_{μ} Ψ_{μ} (ξ),

\int_{- 1}^{1} Ψ_{μ} (ξ) Ψ_{ν} (ξ) d ξ = λ_{μ} \int_{- 1 / 2 W}^{1 / 2 W} Ψ_{μ} (ξ) Ψ_{ν} (ξ) d ξ = λ_{μ} δ_{μ ν} .

\sum_{m = - M}^{M} ψ_{μ} (m) e^{- 2 π i W ξ m} = {(- i)}^{μ} \frac{\sqrt{λ_{μ}}}{\sqrt{W}} Ψ_{μ} (ξ),

φ_{μ} (m) = {\begin{matrix} \frac{1}{\sqrt{λ_{μ}}} ψ_{μ} (m) & | m | \leq M, \\ 0 & | m | > M, \end{matrix}

Φ_{μ} (ξ) = {\begin{matrix} \frac{1}{\sqrt{λ_{μ}}} Ψ_{μ} (ξ) & | ξ | \leq 1, \\ 0 & | ξ | > 1, \end{matrix}

a (m) = \sum_{μ = 0}^{2 M} a_{μ} φ_{μ} (m),

f (ξ) = \sum_{μ = 0}^{2 M} f_{μ} Φ_{μ} (ξ) .

f_{μ} = {(- i)}^{μ} \sqrt{λ_{μ}} a_{μ} .

a_{μ}^{(r)} = \frac{f_{μ}}{{(- i)}^{μ} \sqrt{λ_{μ}}} = a_{μ} .

a (m) = \sum_{μ = 1}^{K} c_{μ}^{'} ϕ_{μ}^{'} (m),

a (m) = \sum_{μ = 1}^{K} c_{μ} ϕ_{μ} (m),

c_{μ} = \sum_{ν = 1}^{K} A_{μ ν} c_{ν}^{'},

A_{μ ν} = \sum_{m = - M}^{M} ϕ_{μ} (m) ϕ_{ν}^{'} (m),

γ = \sqrt{K} max_{1 \leq μ, ν \leq K} | 〈 ϕ_{μ}, ϕ_{ν}^{'} 〉 |,

〈 ϕ_{μ}, ϕ_{ν}^{'} 〉 = \sum_{m = - M}^{M} ϕ_{μ} (m) ϕ_{ν}^{'} (m) .

φ_{μ} (W, m) = {(- 1)}^{μ} φ_{K - μ - 1} (\frac{1}{2} - W, K - m - 1) .

f_{μ} = \int_{- 1}^{1} f (ξ) Φ_{μ} (ξ) d ξ .

min {‖ a (m) ‖}_{1} subject to c_{μ} = \sum_{ν = 1}^{K} A_{μ ν} c_{ν}^{'},

\frac{Δ E}{E} = \frac{{(\sum_{m = - M}^{M} {| a^{(r)} (m) - a (m) |}^{2})}^{1 / 2}}{\sum_{m = - M}^{M} {| a (m) |}^{2}} .

[\hat{a} (m), {\hat{a}}^{†} (m^{'})] = δ_{m m^{'}} .

[\hat{f} (ξ), {\hat{f}}^{†} (ξ^{'})] = δ (ξ - ξ^{'}) .

\hat{f} (ξ) = \sqrt{W} \sum_{m = - \infty}^{\infty} \hat{a} (m) e^{- 2 π i W ξ m} .

φ_{μ} (m) = {\begin{matrix} \frac{1}{\sqrt{λ_{μ}}} ψ_{μ} (m) & | m | \leq M, \\ 0 & | m | > M, \end{matrix} χ_{μ} (m) = {\begin{matrix} 0 & | m | \leq M, \\ \frac{1}{\sqrt{1 - λ_{μ}}} ψ_{μ} (m) & | m | > M . \end{matrix}

Φ_{μ} (ξ) = {\begin{matrix} \frac{1}{\sqrt{λ_{μ}}} Ψ_{μ} (ξ) & | ξ | \leq 1, \\ 0 & | ξ | > 1, \end{matrix} X_{μ} (ξ) = {\begin{matrix} 0 & | ξ | \leq 1, \\ \frac{1}{\sqrt{1 - λ_{μ}}} Ψ_{μ} (ξ) & | ξ | > 1 . \end{matrix}

\hat{a} (m) = \sum_{μ = 0}^{2 M} {\hat{a}}_{μ} φ_{μ} (m) + \sum_{μ = 0}^{2 M} {\hat{b}}_{μ} χ_{μ} (m) + {\hat{a}}_{⊥} (m),

\hat{f} (ξ) = \sum_{μ = 0}^{2 M} {\hat{f}}_{μ} Φ_{μ} (ξ) + \sum_{μ = 0}^{2 M} {\hat{g}}_{μ} X_{μ} (ξ) + {\hat{f}}_{⊥} (ξ) .

{\hat{f}}_{μ} = {(- i)}^{μ} (\sqrt{λ_{μ}} {\hat{a}}_{μ} + \sqrt{1 - λ_{μ}} {\hat{b}}_{μ}) .

{\hat{a}}_{μ}^{(r)} = {\hat{a}}_{μ} + \sqrt{\frac{1 - λ_{μ}}{λ_{μ}}} {\hat{b}}_{μ},

α_{μ} = a_{μ} + δ α_{μ}, β_{μ} = δ β_{μ} .

δ α_{μ} = δ X_{μ}^{α} + i δ Y_{μ}^{α}, δ β_{μ} = δ X_{μ}^{β} + i δ Y_{μ}^{β} .

〈 δ X_{μ}^{n} δ X_{μ^{'}}^{n} 〉 = 〈 δ Y_{μ}^{n} δ Y_{μ^{'}}^{n} 〉 = \frac{1}{4} δ_{μ μ^{'}}, n = α, β .

SNR = \frac{{〈 \hat{N} 〉}^{2}}{〈 {(Δ \hat{N})}^{2} 〉},

SNR = 〈 \hat{N} 〉 = \sum_{m = - M}^{m = M} 〈 {\hat{a}}^{†} (m) \hat{a} (m) 〉 .

K = 21	SF = 2	4	6	8	10

P = 1	100%	100%	19%	2%	0%
P = 2	94%	25%	2%	0%	0%
P = 3	78%	4%	0%	0%	0%
P = 4	0%	0%	0%	0%	0%
P = 5	0%	0%	0%	0%	0%

K = 41	SF = 2	4	6	8	10

P = 1	100%	100%	6%	2%	2%
P = 2	99%	80%	5%	1%	1%
P = 3	98%	44%	1%	0%	0%
P = 4	82%	11%	0%	0%	0%
P = 5	76%	3%	0%	0%	0%

K = 61	SF = 2	4	6	8	10

P = 1	100%	100%	100%	37%	8%
P = 2	99%	89%	28%	2%	0%
P = 3	91%	64%	5%	0%	0%
P = 4	85%	33%	0%	0%	0%
P = 5	82%	14%	0%	0%	0%

K = 81	SF = 2	4	6	8	10

P = 1	100%	100%	100%	55%	7%
P = 2	99%	95%	48%	6%	5%
P = 3	95%	70%	7%	0%	0%
P = 4	94%	50%	0%	0%	0%
P = 5	84%	25%	0%	0%	0%

Abstract

References

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