Abstract

The problem of the detection and localization of a dielectric slab under a multifrequency plane-wave illumination is addressed. A linear inversion scheme, based on δ-functions, whose support represents the positions of the two slab interfaces, and on the truncated singular-value decomposition is exploited. The closed-form derivation of the amplitude of the reconstruction at the slab interface positions when the inversion scheme acts on exact model data allows us to analyze by means of analytical results the role played by the parameters of the slab, the frequency bandwidth, and the noise on the achievable probability of detection.

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  1. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1992).
  2. M. Idemen and I. Akduman, "One-dimensional profile inversion of a half-space over two-part impedance ground," IEEE Trans. Antennas Propag. 44, 933-942 (1996).
    [CrossRef]
  3. D. B. Ge, A. K. Jordan, and J. A. Kong, "Numerical inverse scattering theory for the design of planar waveguides," J. Opt. Soc. Am. A 11, 2809-2815 (1994).
    [CrossRef]
  4. R. Persico and F. Soldovieri, "One-dimensional inverse scattering with a Born model in a three-layered medium," J. Opt. Soc. Am. A 21, 35-45 (2004).
    [CrossRef]
  5. K. P. Bube and R. Burridge, "The one-dimensional inverse problem of reflection seismology," SIAM Rev. 45, 497-559 (1983).
    [CrossRef]
  6. D. L. Jaggard, Y. Kim, K. I. Schultz, and P. V. Fragos, "Experimental confirmation of nonlinear inverse scattering algorithms," J. Opt. Soc. Am. A 4, 1228-1232 (1987).
    [CrossRef]
  7. C. Song and S. Lee, "Permittivity profile inversion of a one-dimensional medium using the fractional linear transformation and the phase compensation constant," Microwave Opt. Technol. Lett. 31, 10-13 (2001).
    [CrossRef]
  8. T. Uno and S. Adachi, "Inverse scattering method for one-dimensional inhomogeneous layered media," IEEE Trans. Antennas Propag. 35, 1456-1466 (1987).
    [CrossRef]
  9. G. Leone, A. Brancaccio, and R. Pierri, "Quadratic distorted approximation for the inverse scattering of dielectric cylinders," J. Opt. Soc. Am. A 18, 600-609 (2001).
    [CrossRef]
  10. H. D. Ladouceur and A. K. Jordan, "Renormalization of an inverse-scattering theory for inhomogeneous dielectrics," J. Opt. Soc. Am. A 2, 1916-1921 (1985).
    [CrossRef]
  11. T. J. Cui and C. H. Liang, "Direct profile inversion for a half-space weakly lossy medium," J. Opt. Soc. Am. A 10, 1950-1952 (1993).
    [CrossRef]
  12. A. Gretzula and W. H. Carter, "Structural measurement by inverse scattering in the Rytov approximation," J. Opt. Soc. Am. A 2, 1958-1960 (1985).
    [CrossRef]
  13. F. C. Lin and M. A. Fiddy, "Born-Rytov controversy II. Applications to nonlinear and stochastic scattering problems in one-dimensional half-space media," J. Opt. Soc. Am. A 10, 1971-1983 (1993).
    [CrossRef]
  14. M. A. Trantanella, D. G. Dudley, and A. Nabulsi, "Beyond the Born and Rytov approximations in one-dimensional profile reconstruction," J. Opt. Soc. Am. A 12, 1469-1478 (1995).
    [CrossRef]
  15. M. A. Hooshyar, "Variational principles and the one-dimensional profile reconstruction," J. Opt. Soc. Am. A 15, 1867-1876 (1998).
    [CrossRef]
  16. R. S. Marger and N. Bleistein, "An examination of the limited aperture problem of physical optics inverse scattering," IEEE Trans. Antennas Propag. 26, 695-699 (1978).
    [CrossRef]
  17. R. Solimene and R. Pierri, "Localization of a planar perfect-electric-conducting interface embedded in a half-space," J. Opt. A, Pure Appl. Opt. 8, 10-16 (2006).
    [CrossRef]
  18. A. Liseno and R. Pierri, "Imaging perfectly conducting objects as support of induced currents: Kirchhoff approximation and frequency diversity," J. Opt. Soc. Am. A 19, 1308-1318 (2002).
    [CrossRef]
  19. M. Bertero, "Linear inverse and ill-posed problems," Adv. Electron. Electron Phys. 75, 1-120 (1989).
  20. P. Z. Peebles, Radar Principles (Wiley, 1998).
  21. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).
  22. D. M. Pozar, Microwave Engineering (Wiley, 1998).
  23. K. I. Hopcraft and P. R. Smith, "Geometrical properties of backscattered radiation and their relation to inverse scattering," J. Opt. Soc. Am. A 6, 508-516 (1989).
    [CrossRef]
  24. I. Stakgols, Boundary Value Problems of Mathematical Physics (SIAM, 2000).
  25. D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty--I," Bell Syst. Tech. J. 40, 43-64 (1961).
  26. B. R. Frieden, "Evaluation design and extrapolation methods for optical signals, based on use of the prolate functions," in Progress in Optics, E.Wolf, ed. (North-Holland, 1971), Vol. IX, pp. 311-407.
    [CrossRef]
  27. J. G. Proakis, Digital Communications (McGraw-Hill, 1995).
  28. I. S. Gradshteyn and I. M. Ryzhih, Table of Integrals, Series, and Products (Academic, 1994).

2006 (1)

R. Solimene and R. Pierri, "Localization of a planar perfect-electric-conducting interface embedded in a half-space," J. Opt. A, Pure Appl. Opt. 8, 10-16 (2006).
[CrossRef]

2004 (1)

2002 (1)

2001 (2)

C. Song and S. Lee, "Permittivity profile inversion of a one-dimensional medium using the fractional linear transformation and the phase compensation constant," Microwave Opt. Technol. Lett. 31, 10-13 (2001).
[CrossRef]

G. Leone, A. Brancaccio, and R. Pierri, "Quadratic distorted approximation for the inverse scattering of dielectric cylinders," J. Opt. Soc. Am. A 18, 600-609 (2001).
[CrossRef]

1998 (1)

1996 (1)

M. Idemen and I. Akduman, "One-dimensional profile inversion of a half-space over two-part impedance ground," IEEE Trans. Antennas Propag. 44, 933-942 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (2)

1989 (2)

1987 (2)

D. L. Jaggard, Y. Kim, K. I. Schultz, and P. V. Fragos, "Experimental confirmation of nonlinear inverse scattering algorithms," J. Opt. Soc. Am. A 4, 1228-1232 (1987).
[CrossRef]

T. Uno and S. Adachi, "Inverse scattering method for one-dimensional inhomogeneous layered media," IEEE Trans. Antennas Propag. 35, 1456-1466 (1987).
[CrossRef]

1985 (2)

1983 (1)

K. P. Bube and R. Burridge, "The one-dimensional inverse problem of reflection seismology," SIAM Rev. 45, 497-559 (1983).
[CrossRef]

1978 (1)

R. S. Marger and N. Bleistein, "An examination of the limited aperture problem of physical optics inverse scattering," IEEE Trans. Antennas Propag. 26, 695-699 (1978).
[CrossRef]

1961 (1)

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty--I," Bell Syst. Tech. J. 40, 43-64 (1961).

Adachi, S.

T. Uno and S. Adachi, "Inverse scattering method for one-dimensional inhomogeneous layered media," IEEE Trans. Antennas Propag. 35, 1456-1466 (1987).
[CrossRef]

Akduman, I.

M. Idemen and I. Akduman, "One-dimensional profile inversion of a half-space over two-part impedance ground," IEEE Trans. Antennas Propag. 44, 933-942 (1996).
[CrossRef]

Bertero, M.

M. Bertero, "Linear inverse and ill-posed problems," Adv. Electron. Electron Phys. 75, 1-120 (1989).

Bleistein, N.

R. S. Marger and N. Bleistein, "An examination of the limited aperture problem of physical optics inverse scattering," IEEE Trans. Antennas Propag. 26, 695-699 (1978).
[CrossRef]

Brancaccio, A.

Bube, K. P.

K. P. Bube and R. Burridge, "The one-dimensional inverse problem of reflection seismology," SIAM Rev. 45, 497-559 (1983).
[CrossRef]

Burridge, R.

K. P. Bube and R. Burridge, "The one-dimensional inverse problem of reflection seismology," SIAM Rev. 45, 497-559 (1983).
[CrossRef]

Carter, W. H.

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1992).

Cui, T. J.

Dudley, D. G.

Fiddy, M. A.

Fragos, P. V.

Frieden, B. R.

B. R. Frieden, "Evaluation design and extrapolation methods for optical signals, based on use of the prolate functions," in Progress in Optics, E.Wolf, ed. (North-Holland, 1971), Vol. IX, pp. 311-407.
[CrossRef]

Ge, D. B.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhih, Table of Integrals, Series, and Products (Academic, 1994).

Gretzula, A.

Hooshyar, M. A.

Hopcraft, K. I.

Idemen, M.

M. Idemen and I. Akduman, "One-dimensional profile inversion of a half-space over two-part impedance ground," IEEE Trans. Antennas Propag. 44, 933-942 (1996).
[CrossRef]

Jaggard, D. L.

Jordan, A. K.

Kim, Y.

Kong, J. A.

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1992).

Ladouceur, H. D.

Lee, S.

C. Song and S. Lee, "Permittivity profile inversion of a one-dimensional medium using the fractional linear transformation and the phase compensation constant," Microwave Opt. Technol. Lett. 31, 10-13 (2001).
[CrossRef]

Leone, G.

Liang, C. H.

Lin, F. C.

Liseno, A.

Marger, R. S.

R. S. Marger and N. Bleistein, "An examination of the limited aperture problem of physical optics inverse scattering," IEEE Trans. Antennas Propag. 26, 695-699 (1978).
[CrossRef]

Nabulsi, A.

Peebles, P. Z.

P. Z. Peebles, Radar Principles (Wiley, 1998).

Persico, R.

Pierri, R.

Pollak, H. O.

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty--I," Bell Syst. Tech. J. 40, 43-64 (1961).

Pozar, D. M.

D. M. Pozar, Microwave Engineering (Wiley, 1998).

Proakis, J. G.

J. G. Proakis, Digital Communications (McGraw-Hill, 1995).

Ryzhih, I. M.

I. S. Gradshteyn and I. M. Ryzhih, Table of Integrals, Series, and Products (Academic, 1994).

Schultz, K. I.

Slepian, D.

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty--I," Bell Syst. Tech. J. 40, 43-64 (1961).

Smith, P. R.

Soldovieri, F.

Solimene, R.

R. Solimene and R. Pierri, "Localization of a planar perfect-electric-conducting interface embedded in a half-space," J. Opt. A, Pure Appl. Opt. 8, 10-16 (2006).
[CrossRef]

Song, C.

C. Song and S. Lee, "Permittivity profile inversion of a one-dimensional medium using the fractional linear transformation and the phase compensation constant," Microwave Opt. Technol. Lett. 31, 10-13 (2001).
[CrossRef]

Stakgols, I.

I. Stakgols, Boundary Value Problems of Mathematical Physics (SIAM, 2000).

Trantanella, M. A.

Uno, T.

T. Uno and S. Adachi, "Inverse scattering method for one-dimensional inhomogeneous layered media," IEEE Trans. Antennas Propag. 35, 1456-1466 (1987).
[CrossRef]

Adv. Electron. Electron Phys. (1)

M. Bertero, "Linear inverse and ill-posed problems," Adv. Electron. Electron Phys. 75, 1-120 (1989).

Bell Syst. Tech. J. (1)

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty--I," Bell Syst. Tech. J. 40, 43-64 (1961).

IEEE Trans. Antennas Propag. (3)

T. Uno and S. Adachi, "Inverse scattering method for one-dimensional inhomogeneous layered media," IEEE Trans. Antennas Propag. 35, 1456-1466 (1987).
[CrossRef]

M. Idemen and I. Akduman, "One-dimensional profile inversion of a half-space over two-part impedance ground," IEEE Trans. Antennas Propag. 44, 933-942 (1996).
[CrossRef]

R. S. Marger and N. Bleistein, "An examination of the limited aperture problem of physical optics inverse scattering," IEEE Trans. Antennas Propag. 26, 695-699 (1978).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

R. Solimene and R. Pierri, "Localization of a planar perfect-electric-conducting interface embedded in a half-space," J. Opt. A, Pure Appl. Opt. 8, 10-16 (2006).
[CrossRef]

J. Opt. Soc. Am. A (12)

A. Liseno and R. Pierri, "Imaging perfectly conducting objects as support of induced currents: Kirchhoff approximation and frequency diversity," J. Opt. Soc. Am. A 19, 1308-1318 (2002).
[CrossRef]

D. L. Jaggard, Y. Kim, K. I. Schultz, and P. V. Fragos, "Experimental confirmation of nonlinear inverse scattering algorithms," J. Opt. Soc. Am. A 4, 1228-1232 (1987).
[CrossRef]

D. B. Ge, A. K. Jordan, and J. A. Kong, "Numerical inverse scattering theory for the design of planar waveguides," J. Opt. Soc. Am. A 11, 2809-2815 (1994).
[CrossRef]

R. Persico and F. Soldovieri, "One-dimensional inverse scattering with a Born model in a three-layered medium," J. Opt. Soc. Am. A 21, 35-45 (2004).
[CrossRef]

G. Leone, A. Brancaccio, and R. Pierri, "Quadratic distorted approximation for the inverse scattering of dielectric cylinders," J. Opt. Soc. Am. A 18, 600-609 (2001).
[CrossRef]

H. D. Ladouceur and A. K. Jordan, "Renormalization of an inverse-scattering theory for inhomogeneous dielectrics," J. Opt. Soc. Am. A 2, 1916-1921 (1985).
[CrossRef]

T. J. Cui and C. H. Liang, "Direct profile inversion for a half-space weakly lossy medium," J. Opt. Soc. Am. A 10, 1950-1952 (1993).
[CrossRef]

A. Gretzula and W. H. Carter, "Structural measurement by inverse scattering in the Rytov approximation," J. Opt. Soc. Am. A 2, 1958-1960 (1985).
[CrossRef]

F. C. Lin and M. A. Fiddy, "Born-Rytov controversy II. Applications to nonlinear and stochastic scattering problems in one-dimensional half-space media," J. Opt. Soc. Am. A 10, 1971-1983 (1993).
[CrossRef]

M. A. Trantanella, D. G. Dudley, and A. Nabulsi, "Beyond the Born and Rytov approximations in one-dimensional profile reconstruction," J. Opt. Soc. Am. A 12, 1469-1478 (1995).
[CrossRef]

M. A. Hooshyar, "Variational principles and the one-dimensional profile reconstruction," J. Opt. Soc. Am. A 15, 1867-1876 (1998).
[CrossRef]

K. I. Hopcraft and P. R. Smith, "Geometrical properties of backscattered radiation and their relation to inverse scattering," J. Opt. Soc. Am. A 6, 508-516 (1989).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

C. Song and S. Lee, "Permittivity profile inversion of a one-dimensional medium using the fractional linear transformation and the phase compensation constant," Microwave Opt. Technol. Lett. 31, 10-13 (2001).
[CrossRef]

SIAM Rev. (1)

K. P. Bube and R. Burridge, "The one-dimensional inverse problem of reflection seismology," SIAM Rev. 45, 497-559 (1983).
[CrossRef]

Other (8)

I. Stakgols, Boundary Value Problems of Mathematical Physics (SIAM, 2000).

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1992).

B. R. Frieden, "Evaluation design and extrapolation methods for optical signals, based on use of the prolate functions," in Progress in Optics, E.Wolf, ed. (North-Holland, 1971), Vol. IX, pp. 311-407.
[CrossRef]

J. G. Proakis, Digital Communications (McGraw-Hill, 1995).

I. S. Gradshteyn and I. M. Ryzhih, Table of Integrals, Series, and Products (Academic, 1994).

P. Z. Peebles, Radar Principles (Wiley, 1998).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

D. M. Pozar, Microwave Engineering (Wiley, 1998).

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Figures (8)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Comparison among max f ̃ ( z ) (solid curve), A f ( z 1 ) (dashed curve), and A f ( z ̃ 2 ) (dotted-dashed curve): k 1 min = 9 π m 1 , k 1 max = 12 π m 1 , z 1 = 0.2 m , D = [ 0.05 , 1.5 ] m . [A] Varying thickness d and fixed ϵ 2 = 4 ϵ 0 . [B] Varying ϵ 2 and fixed d = 0.1 m .

Fig. 3
Fig. 3

Maximum of f ̃ ( z ) does not necessarily occur in z 1 : k 1 min = 9 π , k 1 max = 12 π , z 1 = 0.2 , D = [ 0.05 , 1.5 ] m , ϵ 2 = 9 ϵ 0 , and three different values of d.

Fig. 4
Fig. 4

Comparison between P D returned by Eq. (29) (solid curves) and numerical simulations (curves with circles): P FA = 10 5 , k 1 min = 9 π m 1 , k 1 max = 12 π m 1 , z 1 = 0.2 m , D = [ 0.05 , 1.5 ] m . (Top two panels) γ 0 = 10 1 ; (bottom two panels) γ 0 = 10 0.5 . [A] Varying thickness d and fixed ϵ 2 = 4 ϵ 0 . [B] Varying ϵ 2 and fixed d = 0.1 m .

Fig. 5
Fig. 5

Role of ϵ 2 r in the localization of the interfaces. The arrows indicate the positions the interfaces should have in the reconstruction if properly localized. D = [ 0 , 3 ] cm . [A] d = 2 ϵ 1 ϵ 2 Ω π . [B] d chosen according to Eq. (35).

Fig. 6
Fig. 6

Role of background losses: k 1 min = 9 π m 1 , k 1 max = 12 π m 1 , z 1 = 0.1 m , ϵ 1 = ϵ 0 , ϵ 2 = 4 ϵ 0 , D = [ 0.05 , 1.5 ] m , P FA = 10 4 , γ 0 = 10 1 . (Solid curves) P D returned by Eq. (37); (lines with circles) numerical simulations. (Top panel) σ 1 = 10 1.4 S m ; (bottom panel) σ 1 = 10 1 S m .

Fig. 7
Fig. 7

Role of the slab losses: k 1 min = 9 π m 1 , k 1 max = 12 π m 1 , z 1 = 0.1 m , ϵ 1 = ϵ 0 , ϵ 2 = 4 ϵ 0 , D = [ 0.05 , 1.5 ] m , P FA = 10 5 , γ 0 = 10 1 . (Solid curves) P D returned by Eq. (42); (curves with circles) numerical simulations. (Top panel) σ 2 = 10 2 S m ; (bottom panel) σ 2 = 10 1 S m .

Fig. 8
Fig. 8

Case of a buried void layer. Comparison between the numerical evaluated P D (curve with circles) and the one returned by Eq. (45) (solid curve): f min = 675 MHz , f max = 900 MHz , z 1 = 0.1 m , D = [ 0.05 , 1.5 ] m , P FA = 10 5 . (Panels on the left side) ϵ 1 = 4 ϵ 0 ; (panels on the right side) ϵ 1 = 8 ϵ 0 .

Tables (1)

Tables Icon

Table 1 Estimation of the Slab’s Parameters

Equations (69)

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E S ( ω ) = Γ ( ω , ϵ 1 , ϵ 2 , ϵ 3 , z 1 , z 2 ) exp ( j 2 k 1 z 1 ) ,
Γ ( ω , ϵ 1 , ϵ 2 , ϵ 3 , z 1 , z 2 ) = Γ 12 + Γ 23 exp [ 2 j k 1 θ ( ϵ 2 , z 1 , z 2 ) ϵ 1 ] 1 + Γ 12 Γ 23 exp [ 2 j k 1 θ ( ϵ 2 , z 1 , z 2 ) ϵ 1 ] ,
Γ 12 = ϵ 1 ϵ 2 ( z 1 ) 1 ϵ 1 ϵ 2 ( z 1 ) + 1 ,
Γ 23 = ϵ 2 ( z 2 ) ϵ 3 1 ϵ 2 ( z 2 ) ϵ 3 + 1 ,
θ ( ϵ 2 , z 1 , z 2 ) = z 1 z 2 ϵ 2 ( z ) d z .
Γ ( ω , ϵ 1 , ϵ 2 , ϵ 3 , z 1 , z 2 ) = Γ 12 + ( 1 Γ 12 2 ) Γ 23 n = 0 ( Γ 12 ) n Γ 23 n exp [ j 2 ( n + 1 ) k 1 θ ( ϵ 2 , z 1 , z 2 ) ϵ 1 ] .
Γ ( ω , ϵ 1 , ϵ 2 , ϵ 3 , z 1 , z 2 ) Γ 12 + ( 1 Γ 12 2 ) Γ 23 exp [ j 2 k 1 θ ( ϵ 2 , z 1 , z 2 ) ϵ 1 ] ,
E s ( k 1 ) Γ 12 exp ( j 2 k 1 z 1 ) + ( 1 Γ 21 2 ) Γ 23 exp ( j 2 k 1 z ̃ 2 ) ,
E s ( k 1 ) = z min z max f ( z ) exp ( j 2 k 1 z ) d z ,
h ( z z ¯ ) = n = 0 N T u n * ( z ¯ ) u n ( z ) = exp [ j 2 k 1 av ( z z ¯ ) ] n = 0 N T ϕ n ( z z av ) ϕ n ( z ¯ z av ) λ n ,
h ( z z ¯ ) exp [ 2 j k 1 av ( z z ¯ ) ] Ω π sinc [ Ω ( z z ¯ ) ] ,
f ̃ ( z ) = Ω π exp ( 2 j k 1 av z ) { Γ 12 exp ( 2 j k 1 av z 1 ) sinc [ Ω ( z z 1 ) ] + ( 1 Γ 12 2 ) Γ 23 n = 0 ( Γ 12 ) n Γ 23 n exp { 2 j k 1 av [ z 1 + ( n + 1 ) θ ϵ 1 ] } sinc { Ω [ z z 1 ( n + 1 ) θ ϵ 1 ] } } .
η ( z ) = n = 0 N T η , ν n σ n u n ( z ) .
R η ( z , z ) = E [ η ( z ) , η * ( z ) ] = γ 0 2 exp [ 2 j k 1 av ( z z ) ] n = 0 N T ϕ n ( z z av ) ϕ n ( z z av ) λ n σ n 2 ,
R η ( z z ) exp [ 2 j k 1 av ( z z ) ] γ 0 2 Ω π 2 sinc [ Ω ( z z ) ] .
var η = γ 0 2 Ω π 2 .
f ̃ ( z ) + η ( z ) = exp ( j 2 k 1 av z ) { A f ( z ) exp [ j θ f ( z ) ] + A η ( z ) exp [ j θ η ( z ) ] } ,
p f ( r ) = r var η exp ( r 2 + A f 2 ) 2 var η I 0 ( r A f var η ) ,
p η ( r ) = r var η exp r 2 2 var η ,
P FA = A th p η ( r ) d r = exp ( A th 2 2 var η ) ,
P D = A th p f ( r ) d r = Q ( A f var η , A th var η ) ,
A th = γ 0 π Ω ln ( 1 P FA ) ,
R e { f ̃ ( z 1 ) } = Ω π Γ 12 + 1 Γ 12 2 2 π Γ 12 θ ϵ 1 × { tan 1 [ Γ 12 Γ 23 sin ( 2 k 1 min θ ϵ 1 ) 1 + Γ 12 Γ 23 cos ( 2 k 1 min θ ϵ 1 ) ] tan 1 [ Γ 12 Γ 23 sin ( 2 k 1 max θ ϵ 1 ) 1 + Γ 12 Γ 23 cos ( 2 k 1 max θ ϵ 1 ) ] } ,
I m { f ̃ ( z 1 ) } = 1 Γ 12 2 2 π Γ 12 θ ϵ 1 { ln [ 1 1 + 2 Γ 12 Γ 23 cos ( 2 k 1 min θ ϵ 1 ) + Γ 12 2 Γ 23 2 ] ln [ 1 1 + 2 Γ 12 Γ 23 cos ( 2 k 1 max θ ϵ 1 ) + Γ 12 2 Γ 23 2 ] } ,
f ̃ ( z 1 ) = A f ( z 1 ) = R e { f ̃ ( z 1 ) } 2 + I m { f ̃ ( z 1 ) } 2 .
f ̃ ( z ̃ 2 ) = A f ( z ̃ 2 ) = R e { f ̃ ( z ̃ 2 ) } 2 + I m { f ̃ ( z ̃ 2 ) } 2 ,
R e { f ̃ ( z ̃ 2 ) } = Ω π cos ( 2 k 1 av θ ϵ 1 ) sinc ( Ω θ ϵ 1 ) Γ 12 + Ω π ( 1 Γ 12 2 ) Γ 23 + ( 1 Γ 12 2 ) Γ 23 2 π θ ϵ 1 × { tan 1 [ Γ 12 Γ 23 sin ( 2 k 1 max θ ϵ 1 ) 1 + Γ 12 Γ 23 cos ( 2 k 1 max θ ϵ 1 ) ] tan 1 [ Γ 12 Γ 23 sin ( 2 k 1 min θ ϵ 1 ) 1 + Γ 12 Γ 23 cos ( 2 k 1 min θ ϵ 1 ) ] } ,
I m { f ̃ ( z ̃ 2 ) } = Ω π sin ( 2 k 1 av θ ϵ 1 ) sinc ( Ω θ ϵ 1 ) Γ 12 ( 1 Γ 12 2 ) Γ 23 2 π θ ϵ 1 { ln [ 1 1 + 2 Γ 12 Γ 23 cos ( 2 k 1 min θ ϵ 1 ) + Γ 12 2 Γ 23 2 ] ln [ 1 1 + 2 Γ 12 Γ 23 cos ( 2 k 1 max θ ϵ 1 ) + Γ 12 2 Γ 23 2 ] } .
P D = Q ( A f ( z 1 ) var η , A th var η ) .
f ̃ ( z ) A f ( z 1 ) exp ( j θ f ) exp ( j 2 k 1 av z ) sinc [ Ω ( z z 1 ) ] .
A f ( z 1 ) Γ 12 Ω π ,
d = n π k 1 av ϵ 1 ϵ 2 .
A f ( z ̃ 2 ) ( 1 Γ 12 2 ) Γ 12 Ω π ,
P D 2 = Q ( A f ( z ̃ 2 ) var η , A th var η ) .
d > ϵ 1 ϵ 2 ( 1 0.13 Ω ( 1 Γ 12 2 ) + π 2 Ω ) .
A f loss 1 ( z 1 ) = exp ( 2 α 1 z 1 ) A f ( z 1 ) ,
P D = Q ( A f loss 1 ( z 1 ) var η , A th var η ) .
Γ = Γ 12 ( 1 Γ 12 2 ) Γ 12 n = 0 ( Γ 12 ) 2 n exp [ j 2 ( n + 1 ) × k 1 ϵ 2 ϵ 1 d ] exp [ 2 α 2 ( n + 1 ) d ] ,
R e { f ̃ ( z 1 ) } = Ω π Γ 12 1 Γ 12 2 2 π d Γ 12 ϵ 2 ϵ 1 { tan 1 [ Γ 12 2 exp ( 2 α 2 d ) sin ( 2 k 1 max d ϵ 2 ϵ 1 ) 1 Γ 12 2 exp ( 2 α 2 d ) cos ( 2 k 1 max d ϵ 2 ϵ 1 ) ] tan 1 [ Γ 12 2 exp ( 2 α 2 d ) sin ( 2 k 1 min d ϵ 2 ϵ 1 ) 1 Γ 12 2 exp ( 2 α 2 d ) cos ( 2 k 1 min d ϵ 2 ϵ 1 ) ] } ,
I m { f ̃ ( z 1 ) } = 1 Γ 12 2 2 π d Γ 12 ϵ 2 ϵ 1 { ln [ 1 1 2 Γ 12 2 exp ( 2 α 2 d ) cos ( 2 k 1 min d ϵ 2 ϵ 1 ) + Γ 12 4 exp ( 4 α 2 d ) ] ln [ 1 1 2 Γ 12 2 exp ( 2 α 2 d ) cos ( 2 k 1 max d ϵ 2 ϵ 1 ) + Γ 12 4 exp ( 4 α 2 d ) ] } ,
A f loss 2 ( z 1 ) = R e { f ̃ ( z 1 ) } 2 + I m { f ̃ ( z 1 ) } 2 .
P D = Q ( A f loss 2 ( z 1 ) var η , A th var η ) .
E S ( k 1 ) = ( 1 Γ as 2 ) exp [ 2 j ( k 1 j α 1 ) z 1 ] Γ ,
A f void ( z 1 ) = ( 1 Γ as 2 ) exp ( 2 α 1 z 1 ) A f ( z 1 ) ,
P D = Q ( A f void ( z 1 ) var η , A th var η ) .
u n σ n 2 = L L u n ,
u n ( z ) σ n 2 = z min z max u n ( z ) exp [ 2 j k 1 a v ( z z ) ] sin [ Ω ( z z ) ] z z d z ,
u ̃ n ( s ) σ n 2 = Δ Δ u ̃ n ( s ) sin [ Ω ( s s ) ] s s d s ,
u z ( z ) = ϕ ( z z a v , c ) exp [ 2 j k 1 a v ( z z a v ) ] λ n ( c ) ,
σ n = λ n π ,
ν n = 1 σ n L u n .
R η ( z , z ) = E [ η ( z ) , η * ( z ) ] ,
R η ( z , z ) = n = 0 N T m = 0 N T u n ( z ) u m * ( z ) σ n σ m E [ η 0 , v n , η 0 , v m * ] .
E [ η 0 , v n , η 0 , v m * ] = γ 0 2 δ n m ,
R ζ ( z , z ) = n = 0 N T γ 0 2 u n ( z ) u n * ( z ) σ n σ n .
R e { f ̃ ( z ) } = Ω π { Γ 12 cos [ 2 k 1 av ( z z 1 ) ] sinc [ Ω ( z z 1 ) ] + ( 1 Γ 12 2 ) Γ 23 n = 0 ( Γ 12 ) n Γ 23 n cos { 2 k 1 av [ z z 1 ( n + 1 ) θ ϵ 1 ] } sin { Ω [ z z 1 ( n + 1 ) θ ϵ 1 ] } Ω [ z z 1 ( n + 1 ) θ ϵ 1 ] } ,
I m { f ̃ ( z ) } = Ω π { Γ 12 sin [ 2 k 1 av ( z z 1 ) ] sinc [ Ω ( z z 1 ) ] + ( 1 Γ 12 2 ) Γ 23 n = 0 ( Γ 12 ) n Γ 23 n sin { 2 k 1 av [ z z 1 ( n + 1 ) θ ϵ 1 ] } sin { Ω [ z z 1 ( n + 1 ) θ ϵ 1 ] } Ω [ z z 1 ( n + 1 ) θ ϵ 1 ] } .
sin α cos β = sin ( α + β ) + sin ( α β ) 2 ,
sin α sin β = cos ( α β ) cos ( α + β ) 2
R e { f ̃ ( z ) } = Ω π { Γ 12 cos [ 2 k 1 av ( z z 1 ) ] sinc [ Ω ( z z 1 ) ] + ( 1 Γ 12 2 ) Γ 23 2 Ω n = 0 ( Γ 12 ) n Γ 23 n × sin { 2 k 1 max [ z z 1 ( n + 1 ) θ ϵ 1 ] } sin { 2 k 1 min [ z z 1 ( n + 1 ) θ ϵ 1 ] } [ z z 1 ( n + 1 ) θ ϵ 1 ] } ,
I m { f ̃ ( z ) } = Ω π { Γ 12 sin [ 2 k 1 av ( z z 1 ) ] sinc [ Ω ( z z 1 ) ] + ( 1 Γ 12 2 ) Γ 23 2 Ω n = 0 ( Γ 12 ) n Γ 23 n × cos { 2 k 1 min [ z z 1 ( n + 1 ) θ ϵ 1 ] } cos { 2 k 1 max [ z z 1 ( n + 1 ) θ ϵ 1 ] } [ z z 1 ( n + 1 ) θ ϵ 1 ] } .
R e { f ̃ ( z 1 ) } = Ω π { Γ 12 + ( 1 Γ 12 2 ) Γ 23 2 Ω n = 0 ( Γ 12 ) n Γ 23 n × sin { 2 k 1 max [ ( n + 1 ) θ ϵ 1 ] } ( n + 1 ) θ ϵ 1 sin { 2 k 1 min [ ( n + 1 ) θ ϵ 1 ] } ( n + 1 ) θ ϵ 1 } ,
I m { f ̃ ( z 1 ) } = Ω π { ( 1 Γ 12 2 ) Γ 23 2 Ω n = 0 ( Γ 12 ) n Γ 23 n × cos { 2 k 1 max [ ( n + 1 ) θ ϵ 1 ] } ( n + 1 ) θ ϵ 1 cos { 2 k 1 min [ ( n + 1 ) θ ϵ 1 ] } ( n + 1 ) θ ϵ 1 } ,
R e { f ̃ ( z 1 ) } = Ω π { Γ 12 ( 1 Γ 12 2 ) 2 Ω Γ 12 θ ϵ 1 ( n = 1 ( Γ 12 ) n Γ 23 n sin { 2 k 1 max n θ ϵ 1 } n n = 1 ( Γ 12 ) n Γ 23 n sin { 2 k 1 min n θ ϵ 1 } n ) } ,
I m { f ̃ ( z 1 ) } = Ω π { ( 1 Γ 12 2 ) 2 Ω Γ 12 θ ϵ 1 ( n = 1 ( Γ 12 ) n Γ 23 n cos { 2 k 1 max n θ ϵ 1 } n n = 1 ( Γ 12 ) n Γ 23 n cos { 2 k 1 min n θ ϵ 1 } n ) } .
n = 1 p n sin n x n = tan 1 ( p sin x 1 p cos x ) ,
n = 1 p n cos n x n = ln ( 1 1 2 p cos x + p 2 ) ,
R e { f ̃ ( z ̃ 2 ) } = Ω π { Γ 12 cos ( 2 k 1 av θ ϵ 1 ) sinc ( Ω θ ϵ 1 ) + ( 1 Γ 12 2 ) Γ 23 + ( 1 Γ 12 2 ) Γ 23 2 Ω ( n = 1 ( Γ 12 ) n Γ 23 n sin { 2 k 1 max n θ ϵ 1 } n θ ϵ 1 n = 1 ( Γ 12 ) n Γ 23 n sin { 2 k 1 min n θ ϵ 1 } n θ ϵ 1 ) } ,
I m { f ̃ ( z ̃ 2 ) } = Ω π { Γ 12 sin ( 2 k 1 av θ ϵ 1 sinc ( Ω θ ϵ 1 ) ) + ( 1 Γ 12 2 ) Γ 23 2 Ω ( n = 1 ( Γ 12 ) n Γ 23 n cos { 2 k 1 max n θ ϵ 1 } n θ ϵ 1 n = 1 ( Γ 12 ) n Γ 23 n cos { 2 k 1 min n θ ϵ 1 } n + 1 θ ϵ 1 ) } ,

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