Abstract

The bulky form factor of traditional optical sensors limits their utility for certain applications. Flat multiplex imaging-sensor architectures face the light-gathering challenges inherent with small collection apertures. We examine a wavefront-coding approach wherein a cubic phase mask is used to increase the aperture sizes of multiplex imaging systems while maintaining the distance from the lens to the detector array. The proposed approach exploits the ability of cubic-phase-mask systems to operate over a large range of misfocus values. An exact expression for the optical transfer function of cubic-phase-mask systems is presented, and its misfocus-dependent spatial-filtering properties are described. Criteria for form-factor enhancement are assessed and trade-offs encountered in the design process are evaluated.

© 2006 Optical Society of America

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  1. J. N. Mait, R. Athale, and J. van der Gracht, "Evolutionary paths in imaging and recent trends," Opt. Express 11, 2093-2101 (2003).
    [CrossRef] [PubMed]
  2. J. Tanida, T. Kumagai, K. Yamada, and S. Miyatake, "Thin observation module by bound optics (TOMBO): concept and experimental verification," Appl. Opt. 40, 1806-1813 (2001).
    [CrossRef]
  3. J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes—different concepts and their application to ultra flat image acquisition sensors," in MOEMS and Miniaturized Systems IV, A.El-Fatatry, ed., Proc. SPIE 5346,89-100 (2004).
  4. M. W. Haney, "Performance scaling in flat imagers," Appl. Opt. 45, 2901-2910 (2006).
    [CrossRef] [PubMed]
  5. E. R. Dowski, Jr. and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995).
    [CrossRef] [PubMed]
  6. M. Somayaji and M. P. Christensen, "Form factor enhancement of imaging systems using a cubic phase mask," in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM (Optical Society of America, 2005), CMB4.
    [PubMed]
  7. D. L. Marks, D. L. Stack, D. J. Brady, and J. Van Der Gracht, "Three-dimensional tomography using a cubic phase plate extended depth-of-field system," Opt. Lett. 24, 253-255 (1999).
    [CrossRef]
  8. J. W. Goodman, "Frequency analysis of optical imaging systems," in Introduction to Fourier Optics, 2nd. ed., L.Cox and J.M.Morriss, eds., (McGraw-Hill, 1996), pp. 146-151.
  9. G. Casella and R. L. Berger, "Transformations and expectations: differentiating under an integral sign," in Statistical Inference, 2nd. ed., C.Crockett, ed. (Duxbury Press, 2002), pp. 68-69.
  10. P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, 1953).
  11. K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
    [CrossRef]
  12. H. Bartelt, J. Ojeda-Castañeda, and E. S. Enrique, "Misfocus tolerance seen by simple inspection of the ambiguity function," Appl. Opt. 23, 2693-2669 (1984).
    [CrossRef] [PubMed]
  13. W. T. Cathey and E. R. Dowski, "New paradigm for imaging systems," Appl. Opt. 41, 6080-6092 (2002).
    [CrossRef] [PubMed]
  14. S. Bradburn, E. R. Dowski Jr., and W. Thomas Cathey, "Realizations of focus invariance in optical-digital systems with wavefront coding," Appl. Opt. 36, 9157-9166 (1997).
    [CrossRef]
  15. V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F.T.Luk, ed., Proc. SPIE 5205,348-357 (2003).
  16. A. Torralba and A. Oliva, "Statistics of natural image categories," Comput. Neural Syst. 14, 391-412 (2003).
    [CrossRef]

2006 (1)

2003 (2)

A. Torralba and A. Oliva, "Statistics of natural image categories," Comput. Neural Syst. 14, 391-412 (2003).
[CrossRef]

J. N. Mait, R. Athale, and J. van der Gracht, "Evolutionary paths in imaging and recent trends," Opt. Express 11, 2093-2101 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (1)

1999 (1)

1997 (1)

1995 (1)

1984 (1)

1983 (1)

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Athale, R.

Bartelt, H.

Berger, R. L.

G. Casella and R. L. Berger, "Transformations and expectations: differentiating under an integral sign," in Statistical Inference, 2nd. ed., C.Crockett, ed. (Duxbury Press, 2002), pp. 68-69.

Bradburn, S.

Brady, D. J.

Bräuer, A.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes—different concepts and their application to ultra flat image acquisition sensors," in MOEMS and Miniaturized Systems IV, A.El-Fatatry, ed., Proc. SPIE 5346,89-100 (2004).

Brenner, K.-H.

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Casella, G.

G. Casella and R. L. Berger, "Transformations and expectations: differentiating under an integral sign," in Statistical Inference, 2nd. ed., C.Crockett, ed. (Duxbury Press, 2002), pp. 68-69.

Cathey, W. T.

Cathey, W. Thomas

Christensen, M. P.

M. Somayaji and M. P. Christensen, "Form factor enhancement of imaging systems using a cubic phase mask," in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM (Optical Society of America, 2005), CMB4.
[PubMed]

Dannberg, P.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes—different concepts and their application to ultra flat image acquisition sensors," in MOEMS and Miniaturized Systems IV, A.El-Fatatry, ed., Proc. SPIE 5346,89-100 (2004).

Dowski, E. R.

Duparré, J.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes—different concepts and their application to ultra flat image acquisition sensors," in MOEMS and Miniaturized Systems IV, A.El-Fatatry, ed., Proc. SPIE 5346,89-100 (2004).

Enrique, E. S.

Goodman, J. W.

J. W. Goodman, "Frequency analysis of optical imaging systems," in Introduction to Fourier Optics, 2nd. ed., L.Cox and J.M.Morriss, eds., (McGraw-Hill, 1996), pp. 146-151.

Haney, M. W.

Herzig, H.-P.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes—different concepts and their application to ultra flat image acquisition sensors," in MOEMS and Miniaturized Systems IV, A.El-Fatatry, ed., Proc. SPIE 5346,89-100 (2004).

Kumagai, T.

Lohmann, A.

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Mait, J. N.

Marks, D. L.

Miyatake, S.

Ojeda-Castañeda, J.

H. Bartelt, J. Ojeda-Castañeda, and E. S. Enrique, "Misfocus tolerance seen by simple inspection of the ambiguity function," Appl. Opt. 23, 2693-2669 (1984).
[CrossRef] [PubMed]

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Oliva, A.

A. Torralba and A. Oliva, "Statistics of natural image categories," Comput. Neural Syst. 14, 391-412 (2003).
[CrossRef]

Pauca, V.

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F.T.Luk, ed., Proc. SPIE 5205,348-357 (2003).

Pelli, P.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes—different concepts and their application to ultra flat image acquisition sensors," in MOEMS and Miniaturized Systems IV, A.El-Fatatry, ed., Proc. SPIE 5346,89-100 (2004).

Plemmons, R.

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F.T.Luk, ed., Proc. SPIE 5205,348-357 (2003).

Prasad, S.

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F.T.Luk, ed., Proc. SPIE 5205,348-357 (2003).

Scharf, T.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes—different concepts and their application to ultra flat image acquisition sensors," in MOEMS and Miniaturized Systems IV, A.El-Fatatry, ed., Proc. SPIE 5346,89-100 (2004).

Schreiber, P.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes—different concepts and their application to ultra flat image acquisition sensors," in MOEMS and Miniaturized Systems IV, A.El-Fatatry, ed., Proc. SPIE 5346,89-100 (2004).

Somayaji, M.

M. Somayaji and M. P. Christensen, "Form factor enhancement of imaging systems using a cubic phase mask," in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM (Optical Society of America, 2005), CMB4.
[PubMed]

Stack, D. L.

Tanida, J.

Torgersen, T.

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F.T.Luk, ed., Proc. SPIE 5205,348-357 (2003).

Torralba, A.

A. Torralba and A. Oliva, "Statistics of natural image categories," Comput. Neural Syst. 14, 391-412 (2003).
[CrossRef]

van der Gracht, J.

Völkel, R.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes—different concepts and their application to ultra flat image acquisition sensors," in MOEMS and Miniaturized Systems IV, A.El-Fatatry, ed., Proc. SPIE 5346,89-100 (2004).

Woodward, P. M.

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, 1953).

Yamada, K.

Appl. Opt. (6)

Comput. Neural Syst. (1)

A. Torralba and A. Oliva, "Statistics of natural image categories," Comput. Neural Syst. 14, 391-412 (2003).
[CrossRef]

Opt. Commun. (1)

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (6)

M. Somayaji and M. P. Christensen, "Form factor enhancement of imaging systems using a cubic phase mask," in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM (Optical Society of America, 2005), CMB4.
[PubMed]

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," in Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, F.T.Luk, ed., Proc. SPIE 5205,348-357 (2003).

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes—different concepts and their application to ultra flat image acquisition sensors," in MOEMS and Miniaturized Systems IV, A.El-Fatatry, ed., Proc. SPIE 5346,89-100 (2004).

J. W. Goodman, "Frequency analysis of optical imaging systems," in Introduction to Fourier Optics, 2nd. ed., L.Cox and J.M.Morriss, eds., (McGraw-Hill, 1996), pp. 146-151.

G. Casella and R. L. Berger, "Transformations and expectations: differentiating under an integral sign," in Statistical Inference, 2nd. ed., C.Crockett, ed. (Duxbury Press, 2002), pp. 68-69.

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, 1953).

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Figures (12)

Fig. 1
Fig. 1

Conventional imaging system with a large aperture (a), is converted to a cubic-pm system with a reduced working distance (b), whose performance is compared with that of a scaled imager (c) with the reduced working distance. The setup in (b) results in increased light collection by a factor of (L 2L 1)2 over the system in (c).

Fig. 2
Fig. 2

Spatial-filtering behavior of a cubic-pm system. Left column, system with no misfocus (ψ = 0); right column, system with ψ = −70π; first row, arguments of the Fresnel integrals; second row, contribution of the Fresnel integrals to the MTF; third row, MTFs; fourth row, linear restoration filters used in the reconstruction process. The value of α was set at 70π.

Fig. 3
Fig. 3

Magnitude of the AF of (a) a conventional imaging system and (b) a cubic-pm system. The radial line in the plots has a slope of −10, corresponding to a misfocus value of −5π.

Fig. 4
Fig. 4

Relationship between the zero-crossing points of the function b(|u|) and the spatial-frequency bandwidth of the cubic-pm system as seen from the AF magnitude plot. The expression for the available spatial-frequency bandwidth given by Eq. (12) is valid as long as the radial lines fall within the eye pattern seen in the AF plot.

Fig. 5
Fig. 5

Simulated flat form-factor imaging by a cubic-pm system at various distances. The rows from top to bottom depict images captured by a 10-bit detector at dc = 0.8f, dc = 0.6f, dc = 0.4f, and dc = 0.2f. The left column shows the intermediate images formed at the image-capture plane and have been translated to counter the lateral shifts due to the phase of the OTF. The right column depicts the final images after digital reconstruction. Image scales have been normalized to enable comparison of similar object features in the scene. A reduction in working distance is accompanied by a decrease in spatial-frequency content. A 5× reduction in the working distance (dc = 0.2f) exceeded the capabilities of this configuration. The value of α was set at 70π.

Fig. 6
Fig. 6

Determination of design range. (a) Plots of b(|u|) for two different misfocus values. The curve for ψ = −70π indicates operation within the design range. The curve for ψ = −194π fails to meet the threshold criterion, indicating that the design range has been exceeded. (b) Corresponding MTFs. (c) Corresponding misfocus radial lines in the AF magnitude plot.

Fig. 7
Fig. 7

Normalized spatial-frequency bandwidth versus working distance for various configurations of a cubic-pm system. The bandwidth curves are plotted for the working distances permitted by the design range.

Fig. 8
Fig. 8

Angular resolution versus working distance for various cubic-pm configurations. The solid curves represent the angular resolution of a conventional imager with a fixed F∕#. Only those operating points that lie within the design range are shown.

Fig. 9
Fig. 9

Product of gain in photon count and loss of spatial-frequency resolution for various baseline aperture sizes, α = 70π, and F∕2. As the working distance is reduced, the gain in photon count exceeds the loss of spatial-frequency resolution up to the trend reversal point highlighted by the star-shaped symbols.

Fig. 10
Fig. 10

Plot of normalized spatial-frequency bandwidth versus working distance for various baseline aperture sizes, α = 70π, and F∕2. The star-shaped symbols denote the points at which the product of gain in photon count and loss of spatial-frequency resolution is at a maximum for a given baseline aperture size. The dashed curve connecting these symbols is described analytically by Eq. (21).

Fig. 11
Fig. 11

Plot of angular resolution versus working distance for various baseline aperture sizes, α = 70π, and F∕2. The star-shaped symbols denote the points at which the product of gain in photon count and loss of spatial-frequency resolution is at a maximum for a given baseline aperture size. The dashed curve connecting these markers is described analytically by Eq. (22). The dotted curve represents a conventional imager.

Fig. 12
Fig. 12

Horizontal (a) and diagonal (b) slices along the two-dimensional MTF of an F∕2.6 cubic-pm system with α = 70π, showing signal degradation along the two directions. Plot (a) is shown using the normalized rectangular frequency coordinates (u, v) with v = 0, and plot (b) is displayed using the normalized polar coordinates (ρ, ϕ) such that ρ = (u 2 + v 2)1∕2 and ϕ = tan−1 (v/u). Here ϕ = π∕4 rad.

Equations (35)

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

Δ l 3 = ( 1 u c ) ( λ f 2 L 2 ) .
θ 3 = ( 1 u c ) ( λ f 2 L 2 ) ( 1 f 1 ) .
P ( x ) = { ( 1 / 2 ) exp [ j ( ψ x 2 + α x 3 ) ] | x | 1 , α 20 0 otherwise ,
ψ = π L 2 4 λ ( 1 f 1 d o 1 d c ) .
H ( u , ψ ) = { ( π 24 αu ) 1 / 2 exp ( j 2 α u 3 ) exp ( j 2 ψ 2 u 3 α ) × 1 2 { C [ b ( u ) ] C [ a ( u ) ] + j S [ b ( u ) ] j S [ a ( u ) ] } 0 < | u | 1 1 u = 0 .
a ( u ) = ( 12 αu π ) 1 / 2 ( ψ 3 α ( 1 | u | ) ) ,
b ( u ) = ( 12 αu π ) 1 / 2 ( ψ 3 α + ( 1 | u | ) ) .
| H ( u , ψ ) | = ( π 48 α | u | ) 1 / 2 { [ C ( b ( | u | ) ) C ( a ( | u | ) ) ] 2 + [ S ( b ( | u | ) ) S ( a ( | u | ) ) ] 2 } 1 / 2 .
H ( u , ψ ) = { ( π 24 α | u | ) 1 / 2 exp ( j 2 α u 3 ) exp ( j 2 ψ 2 u 3 α ) 0 < | u | 1 1 u = 0 .
A ( u , ν ) = | u | 1 1 | u | P m ( x + u ) P m * ( x u ) exp ( j 2 πν x ) d x ,
H ( u , ψ ) = | u | 1 1 | u | P m ( x + u ) P m * ( x u ) exp ( j 4 u ψ x ) d x .
H ( u , ψ ) = A ( u , 2 u ψ / π ) .
u c = 1 | ψ | 3 α , α α min .
a πL 2 2 12 λ L 1 [ ( f 1 / L 1 - ( f 2 / L 2 ) ] ( 1 - f 1 f 2 ) . ( f 1 / L 1 ) > ( f 2 / L 2 ) .
u max = 1 3 ( 1 + ψ 3 α ) .
| ψ | 3 α [ 3 π ( 9 α t 4 ) 2 ] 1 / 3 .
α min = ( 3 4 t ) 2 π .
p ( 1 + 4 λ ( F / # ) π L 2 { 3 α [ 3 π ( 9 αt 4 ) 2 ] 1 / 3 } ) 1 .
L 2 4 λ ( F / # ) π ( p 1 p ) { 3 α [ 3 π ( 9 αt 4 ) 2 ] 1 / 3 } .
F / # π L 2 4 λ ( 1 p p ) { 3 α [ 3 π ( 9 αt 4 ) 2 ] 1 / 3 } 1 .
p opt = { 3 π L 2 2 π L 2 + 24 αλ ( F / # ) α π L 2 / 24 λ ( F / # ) 1 otherwise .
u opt = { 1 3 + π L 2 36 αλ ( F / # ) α π L 2 / 24 λ ( F / # ) 1 otherwise .
θ 3 ( opt ) = { [ 24 αλ ( F / # ) π L 2 ] θ 2 α π L 2 / 24 λ ( F / # ) θ 2 otherwise ,
α = 2 π ξ λ ,
H ( u , v ) = A ( u , v ) exp { j k [ W ( x + u , y + v ) W ( x u , y v ) ] } d x d y A ( 0 , 0 ) d x d y ,
H ( u , ψ ) = ( 1 / 2 ) | u | 1 1 | u | exp { j [ ψ ( ( x + u ) 2 ( x u ) 2 ) + α ( ( x + u ) 3 ( x u ) 3 ) ] } d x ,
H ( u , ψ ) = ( 1 / 2 ) exp [ j ( 2 α u 3 2 ψ 2 u 3 α ) ] | u | 1 1 | u | × exp [ j π 2 ( 12 αu π ) ( x + ψ 3 α ) 2 ] d x .
H ( u , ψ ) = ( 1 / 2 ) exp [ j ( 2 α u 3 2 ψ 2 u 3 α ) ] × ( π 12 αu ) 1 / 2 a ( u ) b ( u ) exp ( j π 2 τ 2 ) d τ ,
H ( u , ψ ) = ( π 48 α u ) 1 / 2 exp [ j ( 2 α u 3 2 ψ 2 u 3 α ) ] × [ 0 b ( u ) cos ( π 2 τ 2 ) d τ + j 0 b ( u ) sin ( π 2 τ 2 ) d τ 0 a ( u ) cos ( π 2 τ 2 ) d τ j 0 a ( u ) sin ( π 2 τ 2 ) d τ ] .
H ( u , ψ ) = ( π 48 αu ) 1 / 2 exp ( j 2 α u 3 ) exp ( j 2 ψ 2 u 3 α ) × { C ( b ( u ) ) C ( a ( u ) ) + j S ( b ( u ) ) - j S ( a ( u ) ) } .
lim u 0 H ( u , ψ ) = lim u 0 d d u { ( π 48 α ) 1 / 2 exp [ j ( 2 α u 3 2 ψ 2 u 3 α ) ] a ( u ) b ( u ) exp ( j π 2 τ 2 ) d τ } d d u ( u ) 1 / 2 .
lim u 0 H ( u , ψ ) = 2 ( π 48 α ) 1 / 2 lim u 0 ( u ) 1 / 2 { exp [ j ( 2 α u 3 2 ψ 2 u 3 α ) ] d d u a ( u ) b ( u ) exp ( j π 2 τ 2 ) d τ + [ a ( u ) b ( u ) exp ( j π 2 τ 2 ) d τ ] × d d u exp [ j ( 2 α u 3 2 ψ 2 u 3 α ) ] } .
lim u 0 H ( u , ψ ) = lim u 0 exp [ j ( 2 α u 3 2 ψ 2 u 3 α ) ] × { exp [ j π 2 b ( u ) 2 ] [ u + 1 2 ( ψ 3 α + ( 1 | u | ) ) ] exp [ j π 2 a ( u ) 2 ] × [ + u + 1 2 ( ψ 3 α ( 1 | u | ) ) ] } .
lim u 0 H ( u , ψ ) = 1.
H ( u , ψ ) = ( π 24 α u ) 1 / 2 exp ( j 2 α u 3 ) exp ( j 2 ψ 2 u 3 α ) × 1 2 { C [ b ( u ) ] C [ a ( u ) ] + j S [ b ( u ) ] j S [ a ( u ) ] } ,

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