Abstract

We introduce the concept of a metamaterial aperture, in which an underlying reference mode interacts with a designed metamaterial surface to produce a series of complex field patterns. The resonant frequencies of the metamaterial elements are randomly distributed over a large bandwidth (18–26 GHz), such that the aperture produces a rapidly varying sequence of field patterns as a function of the input frequency. As the frequency of operation is scanned, different subsets of metamaterial elements become active, in turn varying the field patterns at the scene. Scene information can thus be indexed by frequency, with the overall effectiveness of the imaging scheme tied to the diversity of the generated field patterns. As the quality (Q-) factor of the metamaterial resonators increases, the number of distinct field patterns that can be generated increases—improving scene estimation. In this work we provide the foundation for computational imaging with metamaterial apertures based on frequency diversity, and establish that for resonators with physically relevant Q-factors, there are potentially enough distinct measurements of a typical scene within a reasonable bandwidth to achieve diffraction-limited reconstructions of physical scenes.

© 2013 Optical Society of America

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2013

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[CrossRef]

2012

S. Larouche, Y. J. Tsai, T. Tyler, N. M. Jokerst, and D. R. Smith, “Infrared metamaterial phase holograms,” Nat. Mater. 11, 450–454 (2012).
[CrossRef]

D. R. Jackson, C. Caloz, and T. Itoh, “Leaky-wave antennas,” Proc. IEEE 100, 2194–2206 (2012).
[CrossRef]

K. Afrooz, A. Abdipour, and F. Martin, “Broadband bandpass filter using open complementary split ring resonator based on metamaterial unit-cell concept,” Microw. Opt. Technol. Lett. 54, 2832–2835 (2012).
[CrossRef]

2011

C. Rockstuhl, C. Menzel, S. Muhlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[CrossRef]

2010

B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper, “Scalar and tensor holographic artificial impedance surfaces,” IEEE Trans. Antennas Propag. 58, 3212–3221 (2010).
[CrossRef]

C. F. Cull, D. A. Wikner, J. N. Mait, M. Mattheiss, and D. J. Brady, “Millimeter-wave compressive holography,” Appl. Opt. 49, E67–E82 (2010).
[CrossRef]

W. Freese, T. Kampfe, E. B. Kley, and A. Tunnermann, “Design of binary subwavelength multiphase level computer generated holograms,” Opt. Lett. 35, 676–678 (2010).
[CrossRef]

R. Calderbank, S. Howard, and S. Jafarpour, “Construction of a large class of deterministic sensing matrices that satisfy a statistical isometry property,” IEEE J. Sel. Top. Signal Process. 4, 358–374 (2010).
[CrossRef]

R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hegde, “Model-based compressive sensing,” IEEE Trans. Inf. Theory 56, 1982–2001 (2010).
[CrossRef]

2009

R. P. Liu, X. M. Yang, J. G. Gollub, J. J. Mock, T. J. Cui, and D. R. Smith, “Gradient index circuit by waveguided metamaterials,” Appl. Phys. Lett. 94, 073506 (2009).
[CrossRef]

Q. Cheng, H. F. Ma, and T. J. Cui, “Broadband planar Luneburg lens based on complementary metamaterials,” Appl. Phys. Lett. 95, 181901 (2009).
[CrossRef]

J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18, 1395–1408 (2009).
[CrossRef]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[CrossRef]

2008

A. Sutinjo, M. Okoniewski, and R. H. Johnston, “Radiation from fast and slow traveling waves,” IEEE Antennas Propag. Mag. 50(4), 175–181 (2008).
[CrossRef]

J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
[CrossRef]

Q. Cheng, R. P. Liu, J. J. Mock, T. J. Cui, and D. R. Smith, “Partial focusing by indefinite complementary metamaterials,” Phys. Rev. B 78, 121102 (2008).
[CrossRef]

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

T. H. Hand, J. Gollub, S. Sajuyigbe, D. R. Smith, and S. A. Cummer, “Characterization of complementary electric field coupled resonant surfaces,” Appl. Phys. Lett. 93, 212504 (2008).
[CrossRef]

2007

J. M. Bioucas-Dias and M. A. Figueiredo, “A new twIst: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

M. Elad, “Optimized projections for compressed sensing,” IEEE Trans. Signal Process. 55, 5695–5702 (2007).
[CrossRef]

R. G. Baraniuk, “Compressive sensing [lecture notes],” IEEE Signal Process. Mag. 24(4), 118–121 (2007).
[CrossRef]

2006

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

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444, 597–600 (2006).
[CrossRef]

E. Jarauta, M. A. G. Laso, T. Lopetegi, F. Falcone, M. Beruete, J. D. Baena, A. Marcotegui, J. Bonache, J. Garcia, R. Marques, and F. Martin, “Novel microstrip backward coupler with metamaterial cells for fully planar fabrication techniques,” Microw. Opt. Technol. Lett. 48, 1205–1209 (2006).
[CrossRef]

E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

D. L. Donoho, “For most large underdetermined systems of equations, the minimal l(1)-norm near-solution approximates the sparsest near-solution,” Commun. Pure Appl. Math. 59, 907–934 (2006).
[CrossRef]

2005

U. Levy, H. C. Kim, C. H. Tsai, and Y. Fainman, “Near-infrared demonstration of computer-generated holograms implemented by using subwavelength gratings with space-variant orientation,” Opt. Lett. 30, 2089–2091 (2005).
[CrossRef]

E. J. Candesand and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[CrossRef]

2004

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef]

J. Martel, R. Marques, F. Falcone, J. D. Baena, F. Medina, F. Martin, and M. Sorolla, “A new LC series element for compact bandpass filter design,” IEEE Microw. Wirel. Compon. Lett. 14, 210–212 (2004).
[CrossRef]

2003

F. Martin, J. Bonache, F. Falcone, M. Sorolla, and R. Marques, “Split ring resonator-based left-handed coplanar waveguide,” Appl. Phys. Lett. 83, 4652–4654 (2003).
[CrossRef]

2001

B. E. Usevitch,” A tutorial on modern lossy wavelet image compression: foundations of JPEG 2000,” IEEE Signal Process. Mag. 18(5), 22–35 (2001).
[CrossRef]

1979

W. Menzel, “New traveling-wave antenna in microstrip,” AEU Int. J. Electron. Commun. 33, 137–140 (1979).

Abdipour, A.

K. Afrooz, A. Abdipour, and F. Martin, “Broadband bandpass filter using open complementary split ring resonator based on metamaterial unit-cell concept,” Microw. Opt. Technol. Lett. 54, 2832–2835 (2012).
[CrossRef]

Afrooz, K.

K. Afrooz, A. Abdipour, and F. Martin, “Broadband bandpass filter using open complementary split ring resonator based on metamaterial unit-cell concept,” Microw. Opt. Technol. Lett. 54, 2832–2835 (2012).
[CrossRef]

Averitt, R. D.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444, 597–600 (2006).
[CrossRef]

Baena, J. D.

E. Jarauta, M. A. G. Laso, T. Lopetegi, F. Falcone, M. Beruete, J. D. Baena, A. Marcotegui, J. Bonache, J. Garcia, R. Marques, and F. Martin, “Novel microstrip backward coupler with metamaterial cells for fully planar fabrication techniques,” Microw. Opt. Technol. Lett. 48, 1205–1209 (2006).
[CrossRef]

J. Martel, R. Marques, F. Falcone, J. D. Baena, F. Medina, F. Martin, and M. Sorolla, “A new LC series element for compact bandpass filter design,” IEEE Microw. Wirel. Compon. Lett. 14, 210–212 (2004).
[CrossRef]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef]

Balanis, C. A.

C. A. Balanis, in Modern Antenna Handbook (Wiley, 2008), Chap. 7.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

Baraniuk, R. G.

R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hegde, “Model-based compressive sensing,” IEEE Trans. Inf. Theory 56, 1982–2001 (2010).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

R. G. Baraniuk, “Compressive sensing [lecture notes],” IEEE Signal Process. Mag. 24(4), 118–121 (2007).
[CrossRef]

Beruete, M.

E. Jarauta, M. A. G. Laso, T. Lopetegi, F. Falcone, M. Beruete, J. D. Baena, A. Marcotegui, J. Bonache, J. Garcia, R. Marques, and F. Martin, “Novel microstrip backward coupler with metamaterial cells for fully planar fabrication techniques,” Microw. Opt. Technol. Lett. 48, 1205–1209 (2006).
[CrossRef]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef]

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. Figueiredo, “A new twIst: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

Bonache, J.

E. Jarauta, M. A. G. Laso, T. Lopetegi, F. Falcone, M. Beruete, J. D. Baena, A. Marcotegui, J. Bonache, J. Garcia, R. Marques, and F. Martin, “Novel microstrip backward coupler with metamaterial cells for fully planar fabrication techniques,” Microw. Opt. Technol. Lett. 48, 1205–1209 (2006).
[CrossRef]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef]

F. Martin, J. Bonache, F. Falcone, M. Sorolla, and R. Marques, “Split ring resonator-based left-handed coplanar waveguide,” Appl. Phys. Lett. 83, 4652–4654 (2003).
[CrossRef]

Brady, D.

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[CrossRef]

Brady, D. J.

C. F. Cull, D. A. Wikner, J. N. Mait, M. Mattheiss, and D. J. Brady, “Millimeter-wave compressive holography,” Appl. Opt. 49, E67–E82 (2010).
[CrossRef]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[CrossRef]

D. J. Brady, Optical Imaging and Spectroscopy (Wiley-OSA, 2009).

Calderbank, R.

R. Calderbank, S. Howard, and S. Jafarpour, “Construction of a large class of deterministic sensing matrices that satisfy a statistical isometry property,” IEEE J. Sel. Top. Signal Process. 4, 358–374 (2010).
[CrossRef]

Caloz, C.

D. R. Jackson, C. Caloz, and T. Itoh, “Leaky-wave antennas,” Proc. IEEE 100, 2194–2206 (2012).
[CrossRef]

Candes, E. J.

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Candès, E. J.

E. J. Candès, “Compressive sampling,” in Proceedings of the International Congress of Mathematicians, Madrid, August22–30, 2006 (invited lectures, 2006).

Candesand, E. J.

E. J. Candesand and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[CrossRef]

Cevher, V.

R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hegde, “Model-based compressive sensing,” IEEE Trans. Inf. Theory 56, 1982–2001 (2010).
[CrossRef]

Chan, W. L.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Charan, K.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Chen, H. T.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444, 597–600 (2006).
[CrossRef]

Cheng, Q.

Q. Cheng, H. F. Ma, and T. J. Cui, “Broadband planar Luneburg lens based on complementary metamaterials,” Appl. Phys. Lett. 95, 181901 (2009).
[CrossRef]

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
[CrossRef]

Q. Cheng, R. P. Liu, J. J. Mock, T. J. Cui, and D. R. Smith, “Partial focusing by indefinite complementary metamaterials,” Phys. Rev. B 78, 121102 (2008).
[CrossRef]

Chipouline, A.

C. Rockstuhl, C. Menzel, S. Muhlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[CrossRef]

Choi, K.

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[CrossRef]

Colburn, J. S.

B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper, “Scalar and tensor holographic artificial impedance surfaces,” IEEE Trans. Antennas Propag. 58, 3212–3221 (2010).
[CrossRef]

Cui, T. J.

Q. Cheng, H. F. Ma, and T. J. Cui, “Broadband planar Luneburg lens based on complementary metamaterials,” Appl. Phys. Lett. 95, 181901 (2009).
[CrossRef]

R. P. Liu, X. M. Yang, J. G. Gollub, J. J. Mock, T. J. Cui, and D. R. Smith, “Gradient index circuit by waveguided metamaterials,” Appl. Phys. Lett. 94, 073506 (2009).
[CrossRef]

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
[CrossRef]

Q. Cheng, R. P. Liu, J. J. Mock, T. J. Cui, and D. R. Smith, “Partial focusing by indefinite complementary metamaterials,” Phys. Rev. B 78, 121102 (2008).
[CrossRef]

Cull, C. F.

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R. P. Liu, X. M. Yang, J. G. Gollub, J. J. Mock, T. J. Cui, and D. R. Smith, “Gradient index circuit by waveguided metamaterials,” Appl. Phys. Lett. 94, 073506 (2009).
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J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
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Figures (7)

Fig. 1.
Fig. 1.

Single-source aperture operating as a transceiver, shown here as a discretized set of array elements. The array’s radiation pattern, U 0 , is computed from the contribution of all array elements. Also shown is U 0 projected onto a plane.

Fig. 2.
Fig. 2.

(A) Exploded view of the metaimager (showing two parallel plates above and below a dielectric supporting a cylindrical guided wave (ii). One plate serves as the ground plane (i), while the other is patterned with complementary metamaterial elements (iii). (B) We model each element as a dipole, depicted using the functional composite structure. The angle θ is defined as the angle between the dipole moment and the vector pointing from the dipole to a location at the scene.

Fig. 3.
Fig. 3.

Magnitude (A and D) and phase (B and E) distributions of a center-fed 50 × 50 dipole array operating at 20 GHz (top row) and 26 GHz (bottom row). All dipoles are oriented along the x axis. The corresponding radiation patternsare shown as well, both as a 3D beam pattern and projected onto a plane at z = 1.3 m (C and F).

Fig. 4.
Fig. 4.

Average mutual coherence μ g of the canonical ( H C ) and wavelets-based ( H W ) measurement matrix as a function of (A) array elements’ Q -factors and (B) number of alternating sources for a constant Q -factor of 200. The location of the six alternating sources is shown in the inset.

Fig. 5.
Fig. 5.

Image of the author holding reflective construction tools (A) is discretized to resolution-limited pixels (B). The discretized scene is then illuminated by a center-fed aperture with element Q -factor of 750. Reconstruction by matrix inversion is shown in (C) and (D) for no noise and 10 dB SNR, respectively, using 600 measurements. Improved results are obtained when the scene’s sparsity is used as a prior in a compressive-sensing algorithm in the canonical basis (E) and the Haar wavelets basis (F). When high Q -factors are unattainable, we can switch between various source locations. Compressive-sensing reconstructions in the canonical and wavelets basis are shown in (G) and (H), respectively, for an aperture with a Q -factor of 200 fed from six source locations.

Fig. 6.
Fig. 6.

MSE for a 2D scene reconstruction as a function of (A) array elements’ Q -factor and (B) number of alternating sources.

Fig. 7.
Fig. 7.

(A and B) Fields radiating from the aperture illuminate a 3D STL scene. (C) Discretized volumetric scattering density. (D) Reconstruction with no noise. (E) Reconstruction with 5 dB SNR.

Equations (32)

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

U A ( r ¯ A ) = I f T A ( r ¯ A , r ¯ f ) .
U 0 ( r ¯ s ) = S U A ( r ¯ A ) z G ( r ¯ S , r ¯ A ) d 2 r ¯ A ,
2 U T + ( β 0 + Δ β ( r ¯ S ) ) 2 U T = 0 ,
2 U T + β 0 2 U T = 2 β 0 Δ β ( r ¯ S ) U T .
2 β 0 Δ β ( r ¯ S ) U 0 ( r ¯ S ) = f ( r ¯ S ) U 0 ( r ¯ S ) .
U S ( r ¯ A ) = V G ( r ¯ A , r ¯ S ) f ( r ¯ S ) U 0 ( r ¯ S ) d 3 r ¯ S .
g = S U S ( r ¯ A ) T A ( r ¯ f , r ¯ A ) d 2 r ¯ A .
z G ( r ¯ S , r ¯ A ) = G ( r ¯ S , r ¯ A ) D ( r ¯ S , r ¯ A ) ,
G ( r ¯ S , r ¯ A ) = z G ( r ¯ S , r ¯ A ) D 1 ( r ¯ S , r ¯ A ) .
g = V f ( r ¯ S ) U 0 ( r ¯ S ) Z U 0 ( r ¯ S ) D 1 ( r ¯ A , r ¯ S ) d z d 3 r ¯ S .
g = j β 0 V f ( r ¯ S ) U 0 2 ( r ¯ S ) d 3 r ¯ S .
f γ , η = Δ y / 2 Δ y / 2 Δ x / 2 Δ x / 2 f ( x S γ Δ x , y S η Δ y ) d x S d y S ,
f ( r ¯ S ) f ˜ ( r ¯ S ) = γ η f γ , η σ Δ x Δ y ( x S γ Δ x , y S η Δ y ) ,
g = j β 0 r ¯ S U 0 2 ( r ¯ S ) f ( r ¯ S ) .
g = [ h 1 h 2 h M ] [ f 1 f 2 f M ] ,
h m = [ U 0 ( m ) ] 2
g k = r ¯ S [ U 0 k ( r ¯ S ) ] 2 f ( r ¯ S ) .
[ g 1 g 2 g N ] = [ h 11 h 12 h 1 M h 21 h 22 h N 1 h N M ] [ f 1 f 2 f M ]
g = H f ,
G = H ˜ T H ˜ ,
μ g = i j | G i j | 2 M ( M 1 ) ,
MSE = 1 M i = 1 M | f ^ i f i | 2 .
U GW ( r ¯ A ) = J f ( r ¯ f ) H 0 1 ( β GW | r ¯ A r ¯ f | ) ,
m ¯ ( r ¯ A ) = α ( r ¯ A ) · U GW ( r ¯ A ) .
α ( r ¯ A ) = F ω 2 ω 2 ω 0 ( r ¯ A ) 2 + j ω γ ( r ¯ A ) .
U 0 ( r ¯ S ) r ¯ A Z 0 β 0 ω m ¯ ( r ¯ A ) 4 π R exp ( j β 0 R ) sin ( θ ) ,
i | f i | 2 = 1 ,
σ n 2 = σ g 2 / ( 10 SNR / 10 ) ,
f ^ = arg min f ( g H f 2 + Γ f 1 ) ,
f ^ W = arg min f W ( g H Ψ W f W 2 + Γ f W 1 ) ,
f ( r ¯ S ) = | A ( r ¯ S ) n ^ S ( r ¯ S ) · r ¯ S | ,
U 0 ( R 2 ) = R 1 R 2 U 0 ( R 1 ) exp ( j β 0 | R 2 R 1 | ) .

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