Abstract

We present a digital integral imaging system. A Fresnel lenslet array pattern is written on a phase-only LCoS spatial light modulator device (SLM) to replace the regular analog lenslet array in a conventional integral imaging system. We theoretically analyze the capture part of the proposed system based on Fresnel wave propagation formulation. Because of pixelation and quantization of the lenslet array pattern, higher diffraction orders and multiple focal points emerge. Because of the multiple focal planes introduced by the discrete lenslets, multiple image planes are observed. The use of discrete lenslet arrays also causes some other artifacts on the recorded elemental images. The results reduce to those available in the literature when the effects introduced by the discrete nature of the lenslets are omitted. We performed simulations of the capture part. It is possible to obtain the elemental images with an acceptable visual quality. We also constructed an optical integral imaging system with both capture and display parts using the proposed discrete Fresnel lenslet array written on a SLM. Optical results when self-luminous objects, such as an LED array, are used indicate that the proposed system yields satisfactory results.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. Y. Frauel, A. Castro, and B. Javidi, “Integral imaging with large depth of field using an asymmetric phase mask,” Opt. Express 15, 10266–10273 (2007).
    [CrossRef] [PubMed]
  6. S.-W. Min, S. Jung, H. Choi, Y. Kim, J.-H. Park, and B. Lee, “Wide-viewing-angle integral three-dimensional imaging system by curving a screen and a lens array,” Appl. Opt. 44, 546–552(2005).
    [CrossRef] [PubMed]
  7. D.-H. Shin, B.-G. Lee, J. Hyun, D.-C. Hwang, and E.-S. Kim, “Curved projection integral imaging using an additional large-aperture convex lens for viewing angle improvement,” ETRI J. 31, 105–110 (2009).
    [CrossRef]
  8. M. Martinez-Corral, W. D. Furlan, and B. Javidi, “Analysis of 3-D integral imaging displays using the Wigner distribution,” J. Disp. Technol. 2, 180–185 (2006).
    [CrossRef]
  9. H. Hoshino, J. Arai, F. Okano, and I. Yuyama, “Gradient-index lens-array method based on real-time integral photography for three-dimensional images,” Appl. Opt. 37, 2034–2045(1998).
    [CrossRef]
  10. S. H. Eng, Z. Wang, and K. Alameh, “Design and optimization of programmable lens array for adaptive optics,” Proc. SPIE 6414, 64140K (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  16. T. R. Hedman, D. M. Cottrell, J. A. Davis, and R. A. Lilly, “Multiple imaging phase-encoded optical elements written as programmable spatial light modulators,” Appl. Opt. 29, 2505–2509 (1990).
    [CrossRef] [PubMed]
  17. J.-S. Jang and B. Javidi, “Three-dimensional integral imaging with electronically synthesized lenslet arrays,” Opt. Lett. 27, 1767–1769 (2002).
    [CrossRef]
  18. A. Marquez, J. C. Escalera, J. A. Davis, C. Iemmi, M. J. Yzuel, J. Campos, and S. Ledesma, “Inherent apodization of lenses encoded on liquid-crystal spatial light modulators,” Appl. Opt. 39, 6034–6039 (2000).
    [CrossRef]
  19. Y. Jeong, S. Jung, J.-H. Park, and B. Lee, “Reflection-type integral imaging scheme for displaying three-dimensional images,” Opt. Lett. 27, 704–706 (2002).
    [CrossRef]
  20. L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulation,” Opt. Eng. 43, 2557–2563(2004).
    [CrossRef]
  21. J. Jahns and S. J. Walker, “Two-dimensional array of diffractive microlenses fabricated by thin film deposition,” Appl. Opt. 29, 931–936 (1990).
    [CrossRef] [PubMed]
  22. J. Oton, M. S. Millan, and E. Perez-Cabre, “Chromatic compensation of programmable Fresnel lenses,” Opt. Express 14, 6226–6242 (2006).
    [CrossRef] [PubMed]
  23. L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).
  24. A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems Second Edition (Prentice-Hall, 1997).

2009 (1)

D.-H. Shin, B.-G. Lee, J. Hyun, D.-C. Hwang, and E.-S. Kim, “Curved projection integral imaging using an additional large-aperture convex lens for viewing angle improvement,” ETRI J. 31, 105–110 (2009).
[CrossRef]

2007 (2)

Y. Frauel, A. Castro, and B. Javidi, “Integral imaging with large depth of field using an asymmetric phase mask,” Opt. Express 15, 10266–10273 (2007).
[CrossRef] [PubMed]

S. H. Eng, Z. Wang, and K. Alameh, “Design and optimization of programmable lens array for adaptive optics,” Proc. SPIE 6414, 64140K (2007).
[CrossRef]

2006 (3)

2005 (1)

2004 (3)

M. Martínez-Corral, B. Javidi, R. Martínez-Cuenca, and G. Saavedra, “Integral imaging with improved depth of field by use of amplitude-modulated microlens arrays,” Appl. Opt. 43, 5806–5813 (2004).
[CrossRef] [PubMed]

X. Wang, H. Dai, K. X. Y. Liu, and J. Liu, “Characteristics of lcos phase only spatial light modulator and its applications,” Opt. Commun. 238, 269–276 (2004).
[CrossRef]

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulation,” Opt. Eng. 43, 2557–2563(2004).
[CrossRef]

2002 (2)

2000 (1)

1998 (1)

1997 (2)

1994 (1)

1990 (2)

1989 (1)

1931 (1)

1908 (1)

G. Lippmann, “La photographie intégrale,” C.R. Hebd. Seances Acad. Sci. 146, 446–451 (1908).

1881 (1)

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).

Alameh, K.

S. H. Eng, Z. Wang, and K. Alameh, “Design and optimization of programmable lens array for adaptive optics,” Proc. SPIE 6414, 64140K (2007).
[CrossRef]

Arai, J.

Asundi, A. K.

Bai, N.

Bosch, S.

Campos, J.

Carcole, E.

Castro, A.

Choi, H.

Connely, S. W.

Cottrell, D. M.

Dai, H.

X. Wang, H. Dai, K. X. Y. Liu, and J. Liu, “Characteristics of lcos phase only spatial light modulator and its applications,” Opt. Commun. 238, 269–276 (2004).
[CrossRef]

Davis, J. A.

Eng, S. H.

S. H. Eng, Z. Wang, and K. Alameh, “Design and optimization of programmable lens array for adaptive optics,” Proc. SPIE 6414, 64140K (2007).
[CrossRef]

Escalera, J. C.

Fang, Z. P.

Frauel, Y.

Furlan, W. D.

M. Martinez-Corral, W. D. Furlan, and B. Javidi, “Analysis of 3-D integral imaging displays using the Wigner distribution,” J. Disp. Technol. 2, 180–185 (2006).
[CrossRef]

Giles, M. K.

M. K. Giles, “Applications of programmable spatial light modulators,” presented at the International Conference on Lasers ’96, Society of Optical and Quantum Electronics, 2-6 Dec. 1996.

Hedman, T. R.

Hoshino, H.

Hwang, D.-C.

D.-H. Shin, B.-G. Lee, J. Hyun, D.-C. Hwang, and E.-S. Kim, “Curved projection integral imaging using an additional large-aperture convex lens for viewing angle improvement,” ETRI J. 31, 105–110 (2009).
[CrossRef]

Hyun, J.

D.-H. Shin, B.-G. Lee, J. Hyun, D.-C. Hwang, and E.-S. Kim, “Curved projection integral imaging using an additional large-aperture convex lens for viewing angle improvement,” ETRI J. 31, 105–110 (2009).
[CrossRef]

Iemmi, C.

Ives, H. E.

Jahns, J.

Jang, J.-S.

Javidi, B.

Jeong, Y.

Jun, H. A.

Jung, S.

Kim, E.-S.

D.-H. Shin, B.-G. Lee, J. Hyun, D.-C. Hwang, and E.-S. Kim, “Curved projection integral imaging using an additional large-aperture convex lens for viewing angle improvement,” ETRI J. 31, 105–110 (2009).
[CrossRef]

Kim, Y.

Ledesma, S.

Lee, B.

Lee, B.-G.

D.-H. Shin, B.-G. Lee, J. Hyun, D.-C. Hwang, and E.-S. Kim, “Curved projection integral imaging using an additional large-aperture convex lens for viewing angle improvement,” ETRI J. 31, 105–110 (2009).
[CrossRef]

Li, X.

Lilly, R. A.

Lippmann, G.

G. Lippmann, “La photographie intégrale,” C.R. Hebd. Seances Acad. Sci. 146, 446–451 (1908).

Liu, J.

X. Wang, H. Dai, K. X. Y. Liu, and J. Liu, “Characteristics of lcos phase only spatial light modulator and its applications,” Opt. Commun. 238, 269–276 (2004).
[CrossRef]

Liu, K. X. Y.

X. Wang, H. Dai, K. X. Y. Liu, and J. Liu, “Characteristics of lcos phase only spatial light modulator and its applications,” Opt. Commun. 238, 269–276 (2004).
[CrossRef]

Marquez, A.

Martinez-Corral, M.

M. Martinez-Corral, W. D. Furlan, and B. Javidi, “Analysis of 3-D integral imaging displays using the Wigner distribution,” J. Disp. Technol. 2, 180–185 (2006).
[CrossRef]

Martínez-Corral, M.

Martínez-Cuenca, R.

Millan, M. S.

Min, S.-W.

Nawab, S. H.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems Second Edition (Prentice-Hall, 1997).

Okano, F.

Ong, L. S.

Onural, L.

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulation,” Opt. Eng. 43, 2557–2563(2004).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems Second Edition (Prentice-Hall, 1997).

Oton, J.

Park, J.-H.

Perez-Cabre, E.

Rayleigh, L.

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).

Saavedra, G.

Shin, D.-H.

D.-H. Shin, B.-G. Lee, J. Hyun, D.-C. Hwang, and E.-S. Kim, “Curved projection integral imaging using an additional large-aperture convex lens for viewing angle improvement,” ETRI J. 31, 105–110 (2009).
[CrossRef]

Walker, S. J.

Wang, X.

X. Wang, H. Dai, K. X. Y. Liu, and J. Liu, “Characteristics of lcos phase only spatial light modulator and its applications,” Opt. Commun. 238, 269–276 (2004).
[CrossRef]

Wang, Z.

S. H. Eng, Z. Wang, and K. Alameh, “Design and optimization of programmable lens array for adaptive optics,” Proc. SPIE 6414, 64140K (2007).
[CrossRef]

Willsky, A. S.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems Second Edition (Prentice-Hall, 1997).

Yuyama, I.

Yzuel, M. J.

Zhao, L.

Appl. Opt. (9)

H. Hoshino, H. A. Jun, F. Okano, and I. Yuyama, “Real-time pickup method for a three-dimensional image based on the integral photography,” Appl. Opt. 36, 1598–1603 (1997).
[CrossRef] [PubMed]

M. Martínez-Corral, B. Javidi, R. Martínez-Cuenca, and G. Saavedra, “Integral imaging with improved depth of field by use of amplitude-modulated microlens arrays,” Appl. Opt. 43, 5806–5813 (2004).
[CrossRef] [PubMed]

S.-W. Min, S. Jung, H. Choi, Y. Kim, J.-H. Park, and B. Lee, “Wide-viewing-angle integral three-dimensional imaging system by curving a screen and a lens array,” Appl. Opt. 44, 546–552(2005).
[CrossRef] [PubMed]

H. Hoshino, J. Arai, F. Okano, and I. Yuyama, “Gradient-index lens-array method based on real-time integral photography for three-dimensional images,” Appl. Opt. 37, 2034–2045(1998).
[CrossRef]

L. Zhao, N. Bai, X. Li, L. S. Ong, Z. P. Fang, and A. K. Asundi, “Efficient implementation of a spatial light modulator as a diffractive optical microlens array in a digital Shack–Hartmann wavefront sensor,” Appl. Opt. 45, 90–94 (2006).
[CrossRef] [PubMed]

J. Campos, E. Carcole, and S. Bosch, “Diffraction theory of Fresnel lenses encoded in low resolution devices,” Appl. Opt. 33, 162–174 (1994).
[CrossRef] [PubMed]

T. R. Hedman, D. M. Cottrell, J. A. Davis, and R. A. Lilly, “Multiple imaging phase-encoded optical elements written as programmable spatial light modulators,” Appl. Opt. 29, 2505–2509 (1990).
[CrossRef] [PubMed]

A. Marquez, J. C. Escalera, J. A. Davis, C. Iemmi, M. J. Yzuel, J. Campos, and S. Ledesma, “Inherent apodization of lenses encoded on liquid-crystal spatial light modulators,” Appl. Opt. 39, 6034–6039 (2000).
[CrossRef]

J. Jahns and S. J. Walker, “Two-dimensional array of diffractive microlenses fabricated by thin film deposition,” Appl. Opt. 29, 931–936 (1990).
[CrossRef] [PubMed]

C.R. Hebd. Seances Acad. Sci. (1)

G. Lippmann, “La photographie intégrale,” C.R. Hebd. Seances Acad. Sci. 146, 446–451 (1908).

ETRI J. (1)

D.-H. Shin, B.-G. Lee, J. Hyun, D.-C. Hwang, and E.-S. Kim, “Curved projection integral imaging using an additional large-aperture convex lens for viewing angle improvement,” ETRI J. 31, 105–110 (2009).
[CrossRef]

J. Disp. Technol. (1)

M. Martinez-Corral, W. D. Furlan, and B. Javidi, “Analysis of 3-D integral imaging displays using the Wigner distribution,” J. Disp. Technol. 2, 180–185 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

X. Wang, H. Dai, K. X. Y. Liu, and J. Liu, “Characteristics of lcos phase only spatial light modulator and its applications,” Opt. Commun. 238, 269–276 (2004).
[CrossRef]

Opt. Eng. (1)

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulation,” Opt. Eng. 43, 2557–2563(2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Philos. Mag. (1)

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).

Proc. SPIE (1)

S. H. Eng, Z. Wang, and K. Alameh, “Design and optimization of programmable lens array for adaptive optics,” Proc. SPIE 6414, 64140K (2007).
[CrossRef]

Other (2)

M. K. Giles, “Applications of programmable spatial light modulators,” presented at the International Conference on Lasers ’96, Society of Optical and Quantum Electronics, 2-6 Dec. 1996.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems Second Edition (Prentice-Hall, 1997).

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Figures (20)

Fig. 1
Fig. 1

3 × 5 Lenslet array phase profile on the SLM, each lens has f = 43.3 mm . There are equal number of unused pixels both at left and right edges.

Fig. 2
Fig. 2

Illustration of a quadratic phase function and its sampled and quantized version. Vertical axis shows the phase, mod 2 π , while the horizontal axis shows the spatial extent of the function.

Fig. 3
Fig. 3

Multiple focal points and higher diffraction orders. The focal points are shown by small circles. Dashed lines show the converging waves toward multiple focal points from a single lenslet. Solid lines show the converging waves toward higher diffraction orders at the fundamental focal plane. (Not all lines are shown in order not to clutter the drawing.)

Fig. 4
Fig. 4

Capture setup.

Fig. 5
Fig. 5

Display setup.

Fig. 6
Fig. 6

Pixelated and quantized lens with f = 14.43 mm . Sampling period is 8 μm , λ = 532 nm , and array dimension is 120 × 120 pixels.

Fig. 7
Fig. 7

Magnitude square of the cross section of the field due to the pixelated and quantized lenslet, with f = 14.43 mm , under plane wave illumination. The SLM is on the left. The bright areas indicate the multiple focal points and higher diffraction orders. (For visual purposes, we adjusted the brightness of the figure by stretching the contrast. The adjustment is given by the small graph. The horizontal axis represent the original gray values of the pixels, whereas the vertical axis is the modified gray value. We used a similar enhancement procedure also in Figs. 9, 11, 12, 13, 14.)

Fig. 8
Fig. 8

Sampled lens with f = 43.3 mm . Sampling period is 8 μm , λ = 532 nm , and array dimension is 360 × 360 pixels.

Fig. 9
Fig. 9

Magnitude square of the cross section of the field due to the pixelated and quantized lenslet, with f = 43.3 mm , under plane wave illumination. The bright areas indicate the multiple focal points and higher diffraction orders. The brightest area on the right is the fundamental focal point. (For visual purposes, we adjusted the brightness of the figure.)

Fig. 10
Fig. 10

Array of lenslets consisting of pixelated lenslets with f = 14.4 mm . Total size is 360 × 360 pixels. Each lenslet in the array has the same properties defined as in Fig. 6.

Fig. 11
Fig. 11

Magnitude square of the cross section of the field due to the array of lenslets consisting of sampled lenslets, with f = 14.4 mm , under plane wave illumination. Bright areas indicate the multiple focal points and higher diffraction orders. The brightest areas on the right are the fundamental focal points corresponding to each lenslet. (For visual purposes, we adjusted the brightness of the figure.)

Fig. 12
Fig. 12

Image of the absolute value of d 3 [ n ] . There are nine elemental images due to nine lenslets of the letter “A”. There is an artifact due to the out-of-focus images introduced by the multiple focal length properties of the lenslets. However, the visibility is still good. (For visual purposes, we adjusted the brightness of the figure by stretching the contrast using a procedure similar to the one described in Fig. 7.)

Fig. 13
Fig. 13

Image of the absolute value of d 4 [ n ] . The elemental images, which are depicted inside the rectangles, of the letter “A” are seen together with the higher diffraction orders between the elemental images. A zoomed-in version of the central elemental image is given in Fig. 14. We observe a similar artifact at the background. However, the visibility of elemental images are now degraded significantly due to this artifact. This is because of the smaller-size elemental images with less power. (For visual purposes, we adjusted the brightness of the figure.)

Fig. 14
Fig. 14

(a) Zoomed-in elemental image corresponding to the central part of Fig. 13. (b) Zoomed-in elemental image corresponding to the image right to the central part of Fig. 13. (For visual purposes, we adjusted the brightness of the figure.)

Fig. 15
Fig. 15

Experimental setup.

Fig. 16
Fig. 16

Top view of the optical setup: upper rectangle shows the capture part and lower square shows the display part. In between, a small rectangle shows the diffuser, which acts as a capture and display device, on the elemental images plane. The object is behind the white cardboard on the right before the projector lens. The cardboard prevents the light from the LED array to spread everywhere. The vertical dashed line after the projector lens shows the object plane. Dashed lines with the arrows show the optical path. The small diffuser after the mirror is used to show that the image at the calculated reconstruction distance is real.

Fig. 17
Fig. 17

LED array that we used as the object. We put a black mask over the inner LEDs to form a (mirror image) “C” shaped object.

Fig. 18
Fig. 18

An image of the LED array on the object plane: the object is first imaged onto this plane by a projector lens to control both the depth and the size of the object.

Fig. 19
Fig. 19

Optically captured elemental images.

Fig. 20
Fig. 20

Optical reconstruction.

Equations (45)

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h γ ( x ) = exp ( j γ x T x ) ,
h [ n ] = exp ( j γ n T V T V n ) ,
h γ ( x ) = exp ( j γ x 2 ) ·
a k = 1 L 0 L ϕ ( u ) exp ( j 2 π k L u ) d u ,
ψ ( x ) = ϕ ( u ) | u = x 2 = k a k exp ( j 2 π k L x 2 ) = k a k exp ( j π λ f k x 2 ) .
ψ s ( x ) = [ p ( x ) ψ ( x ) n δ ( x n X ) ] * s ( x ) .
L A s ( x ) = { [ p ( x ) ψ ( x ) * r δ ( x r x 0 ) ] n δ ( x n X ) } * s ( x ) ,
q ( x ) = L A s ( x ) * h χ ( x ) = η L A s ( η ) h χ ( x η ) d η ,
q ( x ) = { [ p ( x ) ψ ( x ) * r δ ( x r x 0 ) ] n δ ( x n X ) } * h χ ( x ) * s ( x ) .
q ( x ) = k { [ P ( x λ f k ) * x 0 k n δ ( x n k x 0 ) ] * [ r c k , r ( x ) δ ( x r x 0 ) ] } * s ( x ) ,
t ( x , z ) = i t i δ ( x x i , z z i ) ,
h L P i ( x ) = t i h α i ( x ) * s ( x ) * δ ( x x i ) = t i [ h α i ( x η ) s ( η ) d η ] * δ ( x x i ) = t i [ h α i ( x ) exp ( j 2 α i x η ) h α i ( η ) s ( η ) d η ] * δ ( x x i ) = t i h α i ( x ) [ h α i ( η ) s ( η ) exp ( j 2 α i x η ) d η ] * δ ( x x i ) = { t i h α i ( x ) [ H α i ( x λ z i ) * S ( x λ z i ) ] } * δ ( x x i ) = [ t i h α i ( x ) V ( x ) ] * δ ( x x i ) = t i h α i ( x x i ) V ( x x i ) ,
h L P D i [ n ] = h L P i ( x ) | x = n X ,
L A D [ n ] = [ p ( x ) ψ ( x ) * r δ ( x r x 0 ) ] x = n X ·
q i ( x ) = { n h L P D i [ n ] L A D [ n ] δ ( x n X ) } * s ( x ) * h β ( x ) ,
q i ( x ) = { n h L P D i [ n ] L A D [ n ] h β ( x n X ) } * s ( x ) ·
q i ( x ) = P i ( x λ g ) * s ( x ) * [ k a k H γ k ( x λ g ) ] * Ψ ( x ) * ϒ i ( x ) ·
ϒ i ( x ) = r c ( x ) { { v ( x λ g ) exp [ j 2 β ( x i r x 0 ) x ] } * δ [ x ( 1 + g z i ) r x 0 + g z i x i ] } ,
Ψ ( x ) = x g n δ ( x n x g ) ,
q ( x ) = i q i ( x ) ,
I ( x ) = | q ( x ) | 2 = | i q i ( x ) | 2 ·
I ( x ) i { | Γ ( x ) | 2 * ( λ z i ) 2 s ( z i g x ) * x g 2 n δ ( x n x g ) * r δ [ x ( 1 + g z i ) r x 0 + g z i x i ] } .
I ( x ) = i { | P i ( x λ g ) | 2 * r δ [ x ( 1 + g z i ) r x 0 + g z i x i ] } .
d i [ n ] = IDFT { DFT { h [ n ] } H χ i [ k ] } ,
d 1 [ n ] = IDFT { DFT { t [ n ] } H α [ k ] } , d 2 [ n ] = d 1 [ n ] w L A [ n ] , d 3 [ n ] = IDFT { DFT { d 2 [ n ] } H β 1 [ k ] } , d 4 [ n ] = IDFT { DFT { d 2 [ n ] } H β 2 [ k ] } ·
[ p ( η ) ψ ( η ) * r δ ( η r x 0 ) ] n δ ( η n X ) h χ ( x η ) d η ·
r p ( η r x 0 ) ψ ( η r x 0 ) n δ ( η n X ) h χ ( x η ) d η ,
r p ( σ ) ψ ( σ ) n δ ( σ + r x 0 n X ) h χ ( x σ r x 0 ) d σ = r p ( σ ) ψ ( σ ) n δ ( σ + r x 0 n X ) h χ ( x ) exp [ j 2 χ x ( σ + r x 0 ) ] exp [ j χ ( σ + r x 0 ) 2 ] d σ ·
r p ( σ ) ψ ( σ ) n δ ( σ n X ) h χ ( x ) exp ( j 2 χ x σ ) exp ( j 2 χ x r x 0 ) h χ ( σ ) exp ( j 2 χ r x 0 σ ) exp ( j χ r 2 x 0 2 ) d σ = k a k p ( σ ) h γ k ( σ ) h χ ( σ ) n δ ( σ n X ) r h χ ( x r x 0 ) exp ( j 2 χ r x 0 σ ) exp ( j 2 π x λ z σ ) d σ ,
F σ x λ z { k a k p ( σ ) h γ k ( σ ) h χ ( σ ) n δ ( σ n X ) r h χ ( x r x 0 ) exp ( j 2 χ r x 0 σ ) } ,
F ( ν ) = F x ν { f ( x ) } = f ( x ) exp ( j 2 π ν x ) d x f ( x ) = F ν x 1 { F ( ν ) } = F ( ν ) exp ( j 2 π ν x ) d ν .
k a k { P ( x λ z ) * H γ k ( x λ z ) * H χ ( x λ z ) * λ z X n δ ( x n λ z X ) * r h χ ( x r x 0 ) δ ( x r x 0 ) }
k a k { P ( x λ f k ) * H γ k ( x λ f k ) * H γ k ( x λ f k ) * λ f k X n δ ( x n λ f k X ) * r h γ k ( x r x 0 ) δ ( x r x 0 ) } ,
q ( x ) = k { [ P ( x λ f k ) * x 0 k n δ ( x n k x 0 ) ] * [ r c k , r ( x ) δ ( x r x 0 ) ] } * s ( x ) ,
q i ( x ) = { h β ( x ) n h L P D i [ n ] L A D [ n ] h β ( n X ) exp ( j 2 β x n X ) } * s ( x ) ,
u [ n ] = h L P D i [ n ] L A D [ n ] h β ( n X ) ,
F ( ν ^ ) | ν ^ = ν X = F c ( ν ) * 1 X k δ ( ν k 1 X ) ,
q i ( x ) = [ h β ( x ) U c ( x λ g ) ] * x g n δ ( x n x g ) * s ( x ) ,
h β ( x ) U c ( x λ g ) = h β ( x ) h L P i ( η ) L A ( η ) h β ( η ) exp ( j 2 π x λ g η ) d η = h β ( x ) h L P i ( η ) [ p ( η ) ψ ( η ) * r δ ( η r x 0 ) ] h β ( η ) exp ( j 2 π x λ g η ) d η = h β ( x ) [ t i h α i ( η x i ) V ( η x i ) ] [ r p ( η r x 0 ) ψ ( η r x 0 ) ] h β ( η ) exp ( j 2 π x λ g η ) d η = h β ( x ) t i h α i ( η ) exp ( j 2 α i η x i ) exp ( j α i x i 2 ) V ( η x i ) [ r p ( η r x 0 ) ψ ( η r x 0 ) ] h β ( η ) exp ( j 2 π x λ g η ) d η ·
h β ( x ) U c ( x λ g ) = r h β ( x ) t i σ h α i ( σ + r x 0 ) exp [ j 2 α i ( σ + r x 0 ) x i ] exp ( j α i x i 2 ) p ( σ ) ψ ( σ ) [ h β ( σ ) exp ( j 2 β σ r x 0 ) exp ( j β r 2 x 0 2 ) ] exp ( j 2 β x r x 0 ) exp ( j 2 π x λ g σ ) V ( σ + r x 0 x i ) d σ ,
h β ( x ) U c ( x λ g ) = r h β ( x ) t i h α i ( σ ) exp ( j 2 α i σ r x 0 ) exp ( j α i r 2 x 0 2 ) exp ( j 2 α i σ x i ) exp ( j 2 α i x i r x 0 ) exp ( j α i x i 2 ) p ( σ ) ψ ( σ ) h β ( σ ) exp ( j 2 β σ r x 0 ) exp ( j β r 2 x 0 2 ) exp ( j 2 β x r x 0 ) exp ( j 2 π x λ g σ ) V ( σ + r x 0 x i ) d σ .
h β ( x ) U c ( x λ g ) = h θ i ( σ ) p ( σ ) ψ ( σ ) [ r c ( x ) V ( σ + r x 0 x i ) exp [ j 2 σ ( θ i r x 0 α i x i ) ] ] exp ( j 2 π x λ g σ ) d σ ,
F σ x λ g { [ h θ i ( σ ) p ( σ ) ] [ k a k h γ k ( σ ) ] [ r c ( x ) V ( σ + r x 0 x i ) exp [ j 2 σ ( θ i r x 0 α i x i ) ] ] } ·
ϒ i ( x ) = r c ( x ) { { v ( x λ g ) exp [ j 2 β ( x i r x 0 ) x ] } * δ [ x ( 1 + g z i ) r x 0 + g z i x i ] } ,
q i ( x ) = P i ( x λ g ) * s ( x ) * [ k a k H γ k ( x λ g ) ] * x g n δ ( x n x g ) * ϒ i ( x ) ·

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