Abstract

We present an accelerated algorithm for calculating the near-field of non-uniform sparse apertures with non-uniform fast Fourier transform (NUFFT). The distances of the adjacent units in non-uniform sparse apertures are unequal and larger than half a wavelength. The near-field of apertures can be calculated by the angular spectrum method and the convolution methods, and according to the different convolution kernels, the convolution methods can be divided as the Fresnel kernel convolution and the Rayleigh-Sommerfeld kernel convolution. The Fresnel kernel is the approximation of the Rayleigh-Sommerfeld kernel in the far regions of the near-field zone. In uniform apertures, the three methods can be accelerated by fast Fourier transform (FFT). However, FFT should be replaced by NUFFT for non-uniform sparse apertures. The simulation results reveal that the Rayleigh-Sommerfeld convolution with NUFFT (RS-NUFFT) can be applied to all aperture sizes, distributions and near-field distances. After investigating the error sources in RS-NUFFT, the techniques (padding zeros for apertures, increasing sampling rate for the convolution kernel) are developed for increasing the calculation accuracy of RS-NUFFT.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
GPU-accelerated non-uniform fast Fourier transform-based compressive sensing spectral domain optical coherence tomography

Daguang Xu, Yong Huang, and Jin U. Kang
Opt. Express 22(12) 14871-14884 (2014)

References

  • View by:
  • |
  • |
  • |

  1. M. D’Urso, G. Prisco, and M. Cicolani, “Synthesis of plane wave generators via non-redundant sparse arrays,” IEEE Antennas Wirel. Propag. Lett. 8, 449–452 (2009).
    [Crossref]
  2. A. Buonanno, M. D’Urso, and G. Prisco, “Reducing complexity in indoor array testing,” IEEE Trans. Antenn. Propag. 58(8), 2781–2784 (2010).
    [Crossref]
  3. O. Bucci, M. Migliore, G. Panariello, and D. Pinchera, “Plane-wave generators: Design guidelines, achievable performances and effective synthesis,” IEEE Trans. Antenn. Propag. 61(4), 2005–2018 (2013).
    [Crossref]
  4. B. Zhang, W. Liu, and X. Gou, “Compressive sensing based sparse antenna array design for directional modulation,” IET Microw. Antennas Propag. 11(5), 634–641 (2017).
    [Crossref]
  5. G. Sun and Q. Zhu, “The Design of A Focused Sparse Microstrip Antenna Array,” in IEEE International Symposium on Antennas and Propagation (APSURSI) (IEEE, 2016), p. 515.
  6. M. Hawes and W. Liu, “Compressive sensing-based approach to the design of linear robust sparse antenna arrays with physical size constraint,” IET Microw. Antennas Propag. 8(10), 736–746 (2014).
    [Crossref]
  7. N. Ullah, H. Zhao, T. Rahim, S. Rahman, and M. MuhammadKamal, “Reduced side lobe level of sparse linear antenna array by optimized spacing and excitation amplitude using particle swarm optimization,” in 7th IEEE International Symposium on Microwave, Antenna, Propagation, and EMC Technologies (MAPE) (IEEE, 2018), p. 96.
  8. T. Shimobaba, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, “Nonuniform sampled scalar diffraction calculation using nonuniform fast Fourier transform,” Opt. Lett. 38(23), 5130–5133 (2013).
    [Crossref] [PubMed]
  9. C. Chang, J. Wu, Y. Qi, C. Yuan, S. Nie, and J. Xia, “Simple calculation of a computer-generated hologram for lensless holographic 3D projection using a nonuniform sampled wavefront recording plane,” Appl. Opt. 55(28), 7988–7996 (2016).
    [Crossref] [PubMed]
  10. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45(6), 1102–1110 (2006).
    [Crossref] [PubMed]
  11. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1996), p. 711.
  12. A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” Appl. Comput. Harmon. Anal. 2(1), 85–100 (1995).
    [Crossref]
  13. L. Greengard and J.-Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46(3), 443–454 (2004).
    [Crossref]
  14. K. K. Chan and S. Tang, “Selection of convolution kernel in non-uniform fast Fourier transform for Fourier domain optical coherence tomography,” Opt. Express 19(27), 26891–26904 (2011).
    [Crossref] [PubMed]
  15. D. Xu, Y. Huang, and J. U. Kang, “GPU-accelerated non-uniform fast Fourier transform-based compressive sensing spectral domain optical coherence tomography,” Opt. Express 22(12), 14871–14884 (2014).
    [Crossref] [PubMed]
  16. H. G. Booker and P. C. Clemmow, “The Concept of an Angular Spectrum of Plane Waves and its Relation to that of Polar Diagram and Aperture Distribution,” J. Inst. Electr. Eng. 97(3), 11–17 (1950).
  17. G. Borgiotti, “Fourier Transforms Method of Aperture Antennas,” Alta Freq. 32, 196–204 (1963).
  18. C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. (Wiley, 1997).
  19. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Field (Pergamon, 1966).
  20. E.V. Jull, Aperture Antenna and Diffraction Theory(Institution of Engineering and Technology, 1981).
  21. T.B. Hansen and A.D. Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications (Wiley-IEEE Press, 1999).
  22. M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antenn. Propag. 62(4), 1991–1999 (2014).
    [Crossref]
  23. K. Matsushima and T. Shimobaba, “Band-Limited Angular Spectrum Method for Numerical Simulation of Free-Space Propagation in Far and Near Fields,” Opt. Express 17(22), 19662–19673 (2009).
    [Crossref] [PubMed]
  24. T. Shimobaba, N. Masuda, and T. Ito, “Simple and fast calculation algorithm for computer-generated hologram with wavefront recording plane,” Opt. Lett. 34(20), 3133–3135 (2009).
    [Crossref] [PubMed]
  25. Y. Xiao, X. Tang, Y. Qin, H. Peng, W. Wang, and L. Zhong, “Nonuniform fast Fourier transform method for numerical diffraction simulation on tilted planes,” J. Opt. Soc. Am. A 33(10), 2027–2033 (2016).
    [Crossref] [PubMed]
  26. G. C. Sherman, “Application of the convolution theorem to Rayleigh’s integral formulas,” J. Opt. Soc. Am. 57(4), 546–547 (1967).
    [Crossref] [PubMed]

2017 (1)

B. Zhang, W. Liu, and X. Gou, “Compressive sensing based sparse antenna array design for directional modulation,” IET Microw. Antennas Propag. 11(5), 634–641 (2017).
[Crossref]

2016 (2)

2014 (3)

M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antenn. Propag. 62(4), 1991–1999 (2014).
[Crossref]

D. Xu, Y. Huang, and J. U. Kang, “GPU-accelerated non-uniform fast Fourier transform-based compressive sensing spectral domain optical coherence tomography,” Opt. Express 22(12), 14871–14884 (2014).
[Crossref] [PubMed]

M. Hawes and W. Liu, “Compressive sensing-based approach to the design of linear robust sparse antenna arrays with physical size constraint,” IET Microw. Antennas Propag. 8(10), 736–746 (2014).
[Crossref]

2013 (2)

T. Shimobaba, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, “Nonuniform sampled scalar diffraction calculation using nonuniform fast Fourier transform,” Opt. Lett. 38(23), 5130–5133 (2013).
[Crossref] [PubMed]

O. Bucci, M. Migliore, G. Panariello, and D. Pinchera, “Plane-wave generators: Design guidelines, achievable performances and effective synthesis,” IEEE Trans. Antenn. Propag. 61(4), 2005–2018 (2013).
[Crossref]

2011 (1)

2010 (1)

A. Buonanno, M. D’Urso, and G. Prisco, “Reducing complexity in indoor array testing,” IEEE Trans. Antenn. Propag. 58(8), 2781–2784 (2010).
[Crossref]

2009 (3)

2006 (1)

2004 (1)

L. Greengard and J.-Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46(3), 443–454 (2004).
[Crossref]

1995 (1)

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” Appl. Comput. Harmon. Anal. 2(1), 85–100 (1995).
[Crossref]

1967 (1)

1963 (1)

G. Borgiotti, “Fourier Transforms Method of Aperture Antennas,” Alta Freq. 32, 196–204 (1963).

1950 (1)

H. G. Booker and P. C. Clemmow, “The Concept of an Angular Spectrum of Plane Waves and its Relation to that of Polar Diagram and Aperture Distribution,” J. Inst. Electr. Eng. 97(3), 11–17 (1950).

Albani, M.

M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antenn. Propag. 62(4), 1991–1999 (2014).
[Crossref]

Balanis, C. A.

C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. (Wiley, 1997).

Booker, H. G.

H. G. Booker and P. C. Clemmow, “The Concept of an Angular Spectrum of Plane Waves and its Relation to that of Polar Diagram and Aperture Distribution,” J. Inst. Electr. Eng. 97(3), 11–17 (1950).

Borgiotti, G.

G. Borgiotti, “Fourier Transforms Method of Aperture Antennas,” Alta Freq. 32, 196–204 (1963).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1996), p. 711.

Bucci, O.

O. Bucci, M. Migliore, G. Panariello, and D. Pinchera, “Plane-wave generators: Design guidelines, achievable performances and effective synthesis,” IEEE Trans. Antenn. Propag. 61(4), 2005–2018 (2013).
[Crossref]

Buonanno, A.

A. Buonanno, M. D’Urso, and G. Prisco, “Reducing complexity in indoor array testing,” IEEE Trans. Antenn. Propag. 58(8), 2781–2784 (2010).
[Crossref]

Casaletti, M.

M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antenn. Propag. 62(4), 1991–1999 (2014).
[Crossref]

Chan, K. K.

Chang, C.

Cicolani, M.

M. D’Urso, G. Prisco, and M. Cicolani, “Synthesis of plane wave generators via non-redundant sparse arrays,” IEEE Antennas Wirel. Propag. Lett. 8, 449–452 (2009).
[Crossref]

Clemmow, P. C.

H. G. Booker and P. C. Clemmow, “The Concept of an Angular Spectrum of Plane Waves and its Relation to that of Polar Diagram and Aperture Distribution,” J. Inst. Electr. Eng. 97(3), 11–17 (1950).

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Field (Pergamon, 1966).

D’Urso, M.

A. Buonanno, M. D’Urso, and G. Prisco, “Reducing complexity in indoor array testing,” IEEE Trans. Antenn. Propag. 58(8), 2781–2784 (2010).
[Crossref]

M. D’Urso, G. Prisco, and M. Cicolani, “Synthesis of plane wave generators via non-redundant sparse arrays,” IEEE Antennas Wirel. Propag. Lett. 8, 449–452 (2009).
[Crossref]

Dutt, A.

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” Appl. Comput. Harmon. Anal. 2(1), 85–100 (1995).
[Crossref]

Endo, Y.

Ettorre, M.

M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antenn. Propag. 62(4), 1991–1999 (2014).
[Crossref]

Gou, X.

B. Zhang, W. Liu, and X. Gou, “Compressive sensing based sparse antenna array design for directional modulation,” IET Microw. Antennas Propag. 11(5), 634–641 (2017).
[Crossref]

Greengard, L.

L. Greengard and J.-Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46(3), 443–454 (2004).
[Crossref]

Hansen, T.B.

T.B. Hansen and A.D. Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications (Wiley-IEEE Press, 1999).

Hawes, M.

M. Hawes and W. Liu, “Compressive sensing-based approach to the design of linear robust sparse antenna arrays with physical size constraint,” IET Microw. Antennas Propag. 8(10), 736–746 (2014).
[Crossref]

Hirayama, R.

Huang, Y.

Ito, T.

Jull, E.V.

E.V. Jull, Aperture Antenna and Diffraction Theory(Institution of Engineering and Technology, 1981).

Kakue, T.

Kang, J. U.

Le Coq, L.

M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antenn. Propag. 62(4), 1991–1999 (2014).
[Crossref]

Lee, J.-Y.

L. Greengard and J.-Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46(3), 443–454 (2004).
[Crossref]

Liu, W.

B. Zhang, W. Liu, and X. Gou, “Compressive sensing based sparse antenna array design for directional modulation,” IET Microw. Antennas Propag. 11(5), 634–641 (2017).
[Crossref]

M. Hawes and W. Liu, “Compressive sensing-based approach to the design of linear robust sparse antenna arrays with physical size constraint,” IET Microw. Antennas Propag. 8(10), 736–746 (2014).
[Crossref]

Masuda, N.

Matsushima, K.

Migliore, M.

O. Bucci, M. Migliore, G. Panariello, and D. Pinchera, “Plane-wave generators: Design guidelines, achievable performances and effective synthesis,” IEEE Trans. Antenn. Propag. 61(4), 2005–2018 (2013).
[Crossref]

MuhammadKamal, M.

N. Ullah, H. Zhao, T. Rahim, S. Rahman, and M. MuhammadKamal, “Reduced side lobe level of sparse linear antenna array by optimized spacing and excitation amplitude using particle swarm optimization,” in 7th IEEE International Symposium on Microwave, Antenna, Propagation, and EMC Technologies (MAPE) (IEEE, 2018), p. 96.

Nie, S.

Oikawa, M.

Okada, N.

Panariello, G.

O. Bucci, M. Migliore, G. Panariello, and D. Pinchera, “Plane-wave generators: Design guidelines, achievable performances and effective synthesis,” IEEE Trans. Antenn. Propag. 61(4), 2005–2018 (2013).
[Crossref]

Pavone, S.

M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antenn. Propag. 62(4), 1991–1999 (2014).
[Crossref]

Peng, H.

Pinchera, D.

O. Bucci, M. Migliore, G. Panariello, and D. Pinchera, “Plane-wave generators: Design guidelines, achievable performances and effective synthesis,” IEEE Trans. Antenn. Propag. 61(4), 2005–2018 (2013).
[Crossref]

Prisco, G.

A. Buonanno, M. D’Urso, and G. Prisco, “Reducing complexity in indoor array testing,” IEEE Trans. Antenn. Propag. 58(8), 2781–2784 (2010).
[Crossref]

M. D’Urso, G. Prisco, and M. Cicolani, “Synthesis of plane wave generators via non-redundant sparse arrays,” IEEE Antennas Wirel. Propag. Lett. 8, 449–452 (2009).
[Crossref]

Qi, Y.

Qin, Y.

Rahim, T.

N. Ullah, H. Zhao, T. Rahim, S. Rahman, and M. MuhammadKamal, “Reduced side lobe level of sparse linear antenna array by optimized spacing and excitation amplitude using particle swarm optimization,” in 7th IEEE International Symposium on Microwave, Antenna, Propagation, and EMC Technologies (MAPE) (IEEE, 2018), p. 96.

Rahman, S.

N. Ullah, H. Zhao, T. Rahim, S. Rahman, and M. MuhammadKamal, “Reduced side lobe level of sparse linear antenna array by optimized spacing and excitation amplitude using particle swarm optimization,” in 7th IEEE International Symposium on Microwave, Antenna, Propagation, and EMC Technologies (MAPE) (IEEE, 2018), p. 96.

Rokhlin, V.

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” Appl. Comput. Harmon. Anal. 2(1), 85–100 (1995).
[Crossref]

Sauleau, R.

M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antenn. Propag. 62(4), 1991–1999 (2014).
[Crossref]

Shen, F.

Sherman, G. C.

Shimobaba, T.

Sun, G.

G. Sun and Q. Zhu, “The Design of A Focused Sparse Microstrip Antenna Array,” in IEEE International Symposium on Antennas and Propagation (APSURSI) (IEEE, 2016), p. 515.

Tang, S.

Tang, X.

Ullah, N.

N. Ullah, H. Zhao, T. Rahim, S. Rahman, and M. MuhammadKamal, “Reduced side lobe level of sparse linear antenna array by optimized spacing and excitation amplitude using particle swarm optimization,” in 7th IEEE International Symposium on Microwave, Antenna, Propagation, and EMC Technologies (MAPE) (IEEE, 2018), p. 96.

Valerio, G.

M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antenn. Propag. 62(4), 1991–1999 (2014).
[Crossref]

Wang, A.

Wang, W.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1996), p. 711.

Wu, J.

Xia, J.

Xiao, Y.

Xu, D.

Yaghjian, A.D.

T.B. Hansen and A.D. Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications (Wiley-IEEE Press, 1999).

Yuan, C.

Zhang, B.

B. Zhang, W. Liu, and X. Gou, “Compressive sensing based sparse antenna array design for directional modulation,” IET Microw. Antennas Propag. 11(5), 634–641 (2017).
[Crossref]

Zhao, H.

N. Ullah, H. Zhao, T. Rahim, S. Rahman, and M. MuhammadKamal, “Reduced side lobe level of sparse linear antenna array by optimized spacing and excitation amplitude using particle swarm optimization,” in 7th IEEE International Symposium on Microwave, Antenna, Propagation, and EMC Technologies (MAPE) (IEEE, 2018), p. 96.

Zhong, L.

Zhu, Q.

G. Sun and Q. Zhu, “The Design of A Focused Sparse Microstrip Antenna Array,” in IEEE International Symposium on Antennas and Propagation (APSURSI) (IEEE, 2016), p. 515.

Alta Freq. (1)

G. Borgiotti, “Fourier Transforms Method of Aperture Antennas,” Alta Freq. 32, 196–204 (1963).

Appl. Comput. Harmon. Anal. (1)

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” Appl. Comput. Harmon. Anal. 2(1), 85–100 (1995).
[Crossref]

Appl. Opt. (2)

IEEE Antennas Wirel. Propag. Lett. (1)

M. D’Urso, G. Prisco, and M. Cicolani, “Synthesis of plane wave generators via non-redundant sparse arrays,” IEEE Antennas Wirel. Propag. Lett. 8, 449–452 (2009).
[Crossref]

IEEE Trans. Antenn. Propag. (3)

A. Buonanno, M. D’Urso, and G. Prisco, “Reducing complexity in indoor array testing,” IEEE Trans. Antenn. Propag. 58(8), 2781–2784 (2010).
[Crossref]

O. Bucci, M. Migliore, G. Panariello, and D. Pinchera, “Plane-wave generators: Design guidelines, achievable performances and effective synthesis,” IEEE Trans. Antenn. Propag. 61(4), 2005–2018 (2013).
[Crossref]

M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antenn. Propag. 62(4), 1991–1999 (2014).
[Crossref]

IET Microw. Antennas Propag. (2)

B. Zhang, W. Liu, and X. Gou, “Compressive sensing based sparse antenna array design for directional modulation,” IET Microw. Antennas Propag. 11(5), 634–641 (2017).
[Crossref]

M. Hawes and W. Liu, “Compressive sensing-based approach to the design of linear robust sparse antenna arrays with physical size constraint,” IET Microw. Antennas Propag. 8(10), 736–746 (2014).
[Crossref]

J. Inst. Electr. Eng. (1)

H. G. Booker and P. C. Clemmow, “The Concept of an Angular Spectrum of Plane Waves and its Relation to that of Polar Diagram and Aperture Distribution,” J. Inst. Electr. Eng. 97(3), 11–17 (1950).

J. Opt. Soc. Am. (1)

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

Opt. Express (3)

Opt. Lett. (2)

SIAM Rev. (1)

L. Greengard and J.-Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46(3), 443–454 (2004).
[Crossref]

Other (7)

C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. (Wiley, 1997).

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Field (Pergamon, 1966).

E.V. Jull, Aperture Antenna and Diffraction Theory(Institution of Engineering and Technology, 1981).

T.B. Hansen and A.D. Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications (Wiley-IEEE Press, 1999).

N. Ullah, H. Zhao, T. Rahim, S. Rahman, and M. MuhammadKamal, “Reduced side lobe level of sparse linear antenna array by optimized spacing and excitation amplitude using particle swarm optimization,” in 7th IEEE International Symposium on Microwave, Antenna, Propagation, and EMC Technologies (MAPE) (IEEE, 2018), p. 96.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1996), p. 711.

G. Sun and Q. Zhu, “The Design of A Focused Sparse Microstrip Antenna Array,” in IEEE International Symposium on Antennas and Propagation (APSURSI) (IEEE, 2016), p. 515.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1 The geometry of the calculation model with two parallel planes orthogonal to the z-axis. The plane with scattered points located at z = 0 presents the non-uniform sparse aperture plane. The plane with grids located at z = d is the near-field destination plane. The lines connecting the points on the two planes present the contribution of all the elements to a point on the near-field destination plane.
Fig. 2
Fig. 2 Schematic diagram of the accelerated near-field algorithm of non-uniform sparse apertures.
Fig. 3
Fig. 3 The applicable distance of the three methods.
Fig. 4
Fig. 4 The Errors of the three methods with different aperture sizes and near-field distance.
Fig. 5
Fig. 5 The accuracy of the NUFFTs with different kernel functions. The x-axis is the linear wavenumber.
Fig. 6
Fig. 6 The Errors of the RS-NUFFT with different aperture distributions. (a) uniform aperture. (b) Gaussian aperture. (c) random aperture.
Fig. 7
Fig. 7 The Errors of the RS-NUFFT with different average sampling rates in non-uniform sparse apertures. (a) The average sampling rate is λ. (b) The average sampling rate is 2λ.(c) The average sampling rate is 4λ.
Fig. 8
Fig. 8 The Errors of the RS-NUFFT with different zero-padding ratio.
Fig. 9
Fig. 9 The Error of the RS-NUFFT with the sampling rate λ/2 of the convolution kernel. (a) The average sampling rate is λ (maximum value 2.2091). (b) The average sampling rate is 2λ (maximum value 1.7204). (c) The average sampling rate is 4λ (maximum value 2.2937).
Fig. 10
Fig. 10 The Error of the RS-NUFFT with the sampling rate λ/4 of the convolution kernel. (a) The average sampling rate is λ (maximum value 0.5849). (b) The average sampling rate is 2λ (maximum value 0.74). (c) The average sampling rate is 4λ (maximum value 0.8569).
Fig. 11
Fig. 11 The computation complexity of the Direct Calculation, the RS-NUFFT, the RS-NUFFT with the double zero padding and the over resampling interval of 1/4 wavelengths when the sparsity ratio is set as 2 and 64, respectively.
Fig. 12
Fig. 12 The computation time of the RS-NUFFT and the Direct Calculation, as aperture size increases. The focused near-field results are listed on the right. The first row, from the top to the bottom is in sizes of 20λ,40λ,60λ. The second row, from the top to the bottom is in sizes of 80λ, 100λ, 120λ. The third row, the top is in the size of 140λ.
Fig. 13
Fig. 13 The positions of aperture units. The amplitude of the units is related to the size of the square and the color is related to the phase (unit in degree).
Fig. 14
Fig. 14 (a) and (b) are the amplitude and phase of the destination near-field. The curves in (c) are the amplitude (up subplot) and phase (down subplot) calculated by the RS-NUFFT and the Direct Calculation along the diagonal.

Tables (1)

Tables Icon

Table 1 Computation time of the two kernel functions.

Equations (16)

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

E( x,y ) | z=d = [ r×[ z ^ ×E( x ' , y ' ) | z=0 ] ( 1+jkr ) e jkr r 3 ] d x ' d y '
E( x,y ) | z=d = [ r×[ z ^ ×E( x ' , y ' ) | z=0 ] k dj e jk ( x x ' ) 2 + ( y y ' ) 2 2d ] d x ' d y '
E( x,y ) | z=d = [ E( x ' , y ' ) | z=0 e j( k x x ' + k y y ' ) d x ' d y ' ] e j k z ·d d k x d k y
E( x,y ) | z=d =IFFT2( FFT2( E equal ( x ' , y ' ) | z=0 )×FFT2( d(1+jkr) r 3 e jkr ) )
E( x,y ) | z=d =IFFT2( NUFFT2( E sparse ( x ' , y ' ) | z=0 )×FFT2( d(1+jkr) r 3 e jkr ) )
E( x,y ) | z=d =IFFT2( FFT2( E equal ( x ' , y ' ) | z=0 )×FFT2( k dj e jk(d+ ( x 2 + y 2 ) 2d ) ) )
E( x,y ) | z=d =IFFT2( NUFFT2( E equal ( x ' , y ' ) | z=0 )×FFT2( k dj e jk(d+ ( x 2 + y 2 ) 2d ) ) )
E( x,y ) | z=d =IFFT2( FFT2( E equal ( x ' , y ' ) | z=0 ) e j k z d )
E( x,y ) | z=d =IFFT2( NUFFT2( E sparse ( x ' , y ' ) | z=0 ) e j k z d )
O( 3N log 2 N )
O( 2N log 2 N )
O( N log 2 N )
Error= ( E 1 ˜ E 0 ˜ ) 2 N ( max( E 0 ˜ ) ) 2 ×100
MES=100 | x ref x NUFFT | 2 x ref 2
ME S dB =20×lg( MES )
O= O 1 { pWM+R( 2N1 ) log 2 [R( 2N1 )] +( 2N1 ) }+ O 2 [2( 2N1 ) log 2 ( 2N1 )]+ O 3 ( 2N1 )

Metrics