Abstract

Differential phase-shifting algorithms (DPSAs) and sum phase-shifting algorithms (SPSAs) recover directly the phase difference and the phase sum, respectively, encoded in two patterns. These algorithms can be obtained, for instance, by an appropriate combination of phase-shifting algorithms (PSAs), which makes unnecessary the previous calculation and subtraction or addition of each individual optical phase by means of conventional PSAs. A filtering process in the frequency domain is presented that allows us to obtain in a simple and elegant manner a qualitative characterization with a Fourier description of the two-stage phase-shifting evaluation that reveals possible phase shifter miscalibration errors and unexpected harmonics in the signal.

© 2010 Optical Society of America

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2008 (3)

B. Bhaduri, M. P. Kothiyal, and N. K. Mohan, “A comparative study of phase-shifting algorithms in digital speckle pattern interferometry,” Optik 119, 147-152 (2008).
[CrossRef]

N.-I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vazquez-Castillo, “Phase shifts in the Fourier spectra of phase gratings and phase grids: an application for one-shot phase-shifting interferometry,” Opt. Express 16, 19330-19341 (2008).
[CrossRef]

M. Miranda and B. V. Dorrío, “Error behavior in differential phase-shifting algorithms,” Proc. SPIE 7102, 71021B-1-71021B-9 (2008).

2006 (1)

M. R. Viotti, A. E. Dolinko, G. E. Galizzi, and G. H. Kaufmann, “A portable digital speckle pattern interferometry device to measure residual stresses using the hole drilling technique,” Opt. Lasers Eng. 44, 1052-1066 (2006).
[CrossRef]

2005 (1)

2004 (2)

2003 (1)

Q. Kemao, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42, 1721-1724 (2003).
[CrossRef]

2000 (2)

1999 (2)

K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6, 529-538 (1999).
[CrossRef]

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, 33-35 (1999).
[CrossRef]

1998 (2)

1997 (2)

1996 (1)

1995 (1)

1992 (1)

1991 (1)

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

1990 (1)

1987 (1)

1985 (1)

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350-352 (1984).

1983 (1)

Asundi, A. K.

Q. Kemao, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42, 1721-1724 (2003).
[CrossRef]

Banyard, J. E.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

Bhaduri, B.

B. Bhaduri, M. P. Kothiyal, and N. K. Mohan, “A comparative study of phase-shifting algorithms in digital speckle pattern interferometry,” Optik 119, 147-152 (2008).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000).

Brock, N.

Brohinsky, W. R.

Burke, J.

Burow, R.

Dolinko, A. E.

M. R. Viotti, A. E. Dolinko, G. E. Galizzi, and G. H. Kaufmann, “A portable digital speckle pattern interferometry device to measure residual stresses using the hole drilling technique,” Opt. Lasers Eng. 44, 1052-1066 (2006).
[CrossRef]

Dorrío, B. V.

M. Miranda and B. V. Dorrío, “Error behavior in differential phase-shifting algorithms,” Proc. SPIE 7102, 71021B-1-71021B-9 (2008).

D. Malacara-Doblado and B. V. Dorrío, “Family of detuning-insensitive phase-shifting algorithms,” J. Opt. Soc. Am. A 17, 1857-1863 (2000).
[CrossRef]

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, 33-35 (1999).
[CrossRef]

M. Miranda and B. V. Dorrío, “Error-phase compensation properties of differential phase-shifting algorithms for Fizeau fringe patterns,” in RIAO/OPTILAS 2007, Proceedings of the 2007 Iberoamerican Conference on Optics/Latinoamerican Meeting on Optics, Lasers and Applications (AIP, 2007), Vol. 992, pp. 993-998. http://riao-optilas.ifi.unicamp.br.

Eiju, T.

Elssner, K. E.

Farrant, D. I.

Fernández, J. L.

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, 33-35 (1999).
[CrossRef]

Freischlad, K.

Galizzi, G. E.

M. R. Viotti, A. E. Dolinko, G. E. Galizzi, and G. H. Kaufmann, “A portable digital speckle pattern interferometry device to measure residual stresses using the hole drilling technique,” Opt. Lasers Eng. 44, 1052-1066 (2006).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350-352 (1984).

Grzanna, J.

Hanayama, R.

Hariharan, P.

Hayes, J.

Helmers, H.

Hibino, K.

Huntley, J. M.

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131-150 (1997).
[CrossRef]

Kaufmann, G. H.

M. R. Viotti, A. E. Dolinko, G. E. Galizzi, and G. H. Kaufmann, “A portable digital speckle pattern interferometry device to measure residual stresses using the hole drilling technique,” Opt. Lasers Eng. 44, 1052-1066 (2006).
[CrossRef]

Kemao, Q.

Q. Kemao, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42, 1721-1724 (2003).
[CrossRef]

Koliopoulos, C. L.

Kothiyal, M. P.

B. Bhaduri, M. P. Kothiyal, and N. K. Mohan, “A comparative study of phase-shifting algorithms in digital speckle pattern interferometry,” Optik 119, 147-152 (2008).
[CrossRef]

Kuechel, M.

M. Kuechel, “Method and apparatus for phase evaluation of pattern images used in optical measurement,” US Patent 5361312, 1 November 1994.

Larkin, K. G.

Malacara, D.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, 2005).
[CrossRef]

Malacara, Z.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, 2005).
[CrossRef]

Malacara-Doblado, D.

Meneses-Fabian, C.

Merkel, K.

Millerd, J.

Miranda, M.

M. Miranda and B. V. Dorrío, “Error behavior in differential phase-shifting algorithms,” Proc. SPIE 7102, 71021B-1-71021B-9 (2008).

M. Miranda and B. V. Dorrío, “Error-phase compensation properties of differential phase-shifting algorithms for Fizeau fringe patterns,” in RIAO/OPTILAS 2007, Proceedings of the 2007 Iberoamerican Conference on Optics/Latinoamerican Meeting on Optics, Lasers and Applications (AIP, 2007), Vol. 992, pp. 993-998. http://riao-optilas.ifi.unicamp.br.

Mitsuishi, M.

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11, 337-343 (2004).
[CrossRef]

Mohan, N. K.

B. Bhaduri, M. P. Kothiyal, and N. K. Mohan, “A comparative study of phase-shifting algorithms in digital speckle pattern interferometry,” Optik 119, 147-152 (2008).
[CrossRef]

Molin, N.-E.

Nassar, N. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

North-Morris, M.

Novak, M.

Oreb, B. F.

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).

Rodriguez-Zurita, G.

Saldner, H. O.

Schwider, J.

Seah, H. S.

Q. Kemao, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42, 1721-1724 (2003).
[CrossRef]

Servín, M.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, 2005).
[CrossRef]

Spolaczyk, R.

Stetson, K. A.

Surrel, Y.

Toto-Arellano, N.-I.

Vazquez-Castillo, J. F.

Viotti, M. R.

M. R. Viotti, A. E. Dolinko, G. E. Galizzi, and G. H. Kaufmann, “A portable digital speckle pattern interferometry device to measure residual stresses using the hole drilling technique,” Opt. Lasers Eng. 44, 1052-1066 (2006).
[CrossRef]

Virdee, M. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

Warisawa, S.

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11, 337-343 (2004).
[CrossRef]

Williams, D. C.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

Wyant, J.

Appl. Opt. (7)

J. Opt. Soc. Am. A (6)

Meas. Sci. Technol. (1)

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, 33-35 (1999).
[CrossRef]

Opt. Eng. (2)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350-352 (1984).

Q. Kemao, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42, 1721-1724 (2003).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (1)

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

Opt. Lasers Eng. (2)

M. R. Viotti, A. E. Dolinko, G. E. Galizzi, and G. H. Kaufmann, “A portable digital speckle pattern interferometry device to measure residual stresses using the hole drilling technique,” Opt. Lasers Eng. 44, 1052-1066 (2006).
[CrossRef]

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131-150 (1997).
[CrossRef]

Opt. Rev. (2)

K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6, 529-538 (1999).
[CrossRef]

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11, 337-343 (2004).
[CrossRef]

Optik (1)

B. Bhaduri, M. P. Kothiyal, and N. K. Mohan, “A comparative study of phase-shifting algorithms in digital speckle pattern interferometry,” Optik 119, 147-152 (2008).
[CrossRef]

SPIE (1)

M. Miranda and B. V. Dorrío, “Error behavior in differential phase-shifting algorithms,” Proc. SPIE 7102, 71021B-1-71021B-9 (2008).

Other (6)

M. Kuechel, “Method and apparatus for phase evaluation of pattern images used in optical measurement,” US Patent 5361312, 1 November 1994.

J. Burke, “2-D spectral error analysis for phase-shifting formulas,” in Proceedings of FRINGE '01 (4th International Workshop on Automatic Processing of Fringe Patterns) (Elsevier, 2001), pp. 199-207.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, 2005).
[CrossRef]

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000).

M. Miranda and B. V. Dorrío, “Error-phase compensation properties of differential phase-shifting algorithms for Fizeau fringe patterns,” in RIAO/OPTILAS 2007, Proceedings of the 2007 Iberoamerican Conference on Optics/Latinoamerican Meeting on Optics, Lasers and Applications (AIP, 2007), Vol. 992, pp. 993-998. http://riao-optilas.ifi.unicamp.br.

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Figures (5)

Fig. 1
Fig. 1

Comparison between the calculation of the phase difference Δ φ ( r ) and the phase sum 2 φ ( r ) + Δ φ ( r ) with two PSAs or employing just two-stage PSAs. The superscript U indicates unwrapped phase.

Fig. 2
Fig. 2

Differential sampling weights, (a) numerator and (b) denominator, for the Schwider–Hariharan GDPSA.

Fig. 3
Fig. 3

Amplitude of the differential spectra G N ( f 1 , f 2 ) and G D ( f 1 , f 2 ) for the Schwider–Hariharan GDPSA.

Fig. 4
Fig. 4

Sum sampling weights, (a) numerator and (b) denominator, for the Schwider–Hariharan GSPSA.

Fig. 5
Fig. 5

Amplitude of the sum spectra G N ( f 1 , f 2 ) and G D ( f 1 , f 2 ) for the Schwider–Hariharan GSPSA.

Equations (48)

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s ( r , φ , α ) = k = 0 a k ( r ) cos { k [ φ ( r ) + α ] } = k = 0 a k ( r ) 2 { e j k [ φ ( r ) + α ] + e j k [ φ ( r ) + α ] } ,
φ ( r ) = arctan N 1 { s m ( r , φ , α m ) } D 1 { s m ( r , φ , α m ) } = arctan m = 1 M n m s m ( r , φ , α m ) m = 1 M d m s m ( r , φ , α m ) = arctan m = 1 M h N ( u 1 m ) s m ( r , φ , α m ) m = 1 M h D ( u 1 m ) s m ( r , φ , α m ) ,
t ( r , φ + Δ φ , β ) = l = 0 b l ( r ) cos { l [ φ ( r ) + Δ φ ( r ) + β ] } = l = 0 b l ( r ) 2 { e j l [ φ ( r ) + Δ φ ( r ) + β ] + e j l [ φ ( r ) + Δ φ ( r ) + β ] } ,
φ ( r ) + Δ φ ( r ) = arctan N 2 { t p ( r , φ + Δ φ , β p ) } D 2 { t p ( r , φ + Δ φ , β p ) } = arctan p = 1 P n p t p ( r , φ + Δ φ , β p ) p = 1 P d p t p ( r , φ + Δ φ , β p ) = arctan p = 1 P h N ( u 2 p ) t p ( r , φ + Δ φ , β p ) p = 1 P h D ( u 2 p ) t p ( r , φ + Δ φ , β p ) ,
Δ φ ( r ) = arctan D 1 { s m ( r , φ , α m ) } N 2 { t p ( r , φ + Δ φ , β p ) } N 1 { s m ( r , φ , α m ) } D 2 { t p ( r , φ + Δ φ , β p ) } N 1 { s m ( r , φ , α m ) } N 2 { t p ( r , φ + Δ φ , β p ) } + D 1 { s m ( r , φ , α m ) } D 2 { t p ( r , φ + Δ φ , β p ) } ,
2 φ ( r ) + Δ φ ( r ) = arctan D 1 { s m ( r , φ , α m ) } N 2 { t p ( r , φ + Δ φ , β p ) } + N 1 { s m ( r , φ , α m ) } D 2 { t p ( r , φ + Δ φ , β p ) } N 1 { s m ( r , φ , α m ) } N 2 { t p ( r , φ + Δ φ , β p ) } + D 1 { s m ( r , φ , α m ) } D 2 { t p ( r , φ + Δ φ , β p ) } ,
arctan C N C D = arctan m = 1 M p = 1 P s m ( r , φ , α m ) t p ( r , φ + Δ φ , β p ) g N ( u 1 m , u 2 p ) m = 1 M p = 1 P s m ( r , φ , α m ) t p ( r , φ + Δ φ , β p ) g D ( u 1 m , u 2 p ) .
C N , D = | m = 1 M p = 1 P s m ( r , φ , α m ) t p ( r , φ + Δ φ , β p ) g N , D ( α m , β p ) e j 2 π f 1 u 1 e j 2 π f 2 u 2 | ( f 1 , f 2 ) = ( 0 , 0 ) = FT [ s m ( r , φ , α m ) t p ( r , φ + Δ φ , β p ) ] ( f 1 , f 2 ) = ( 0 , 0 ) FT [ g N , D ( α m , β p ) ] ( f 1 , f 2 ) = ( 0 , 0 ) = k = l = a k b l 2 [ G N , D * ( k f s , l f t ) e j [ k φ + l ( φ + Δ φ ) ] + G N , D * ( k f s , l f t ) e j [ k φ + l ( φ + Δ φ ) ] ] ,
arctan C N C D = arctan { k = 1 l = 1 a k b l | G N ( k f s , l f t ) | cos [ k φ + l ( φ + Δ φ ) γ N ( k f s , l f t ) ] + k = 1 l = 1 a k b l | G N ( k f s , l f t ) | cos [ k φ + l ( φ + Δ φ ) γ N ( k f s , l f t ) ] } { k = 1 l = 1 a k b l | G D ( k f s , l f t ) | cos [ k φ + l ( φ + Δ φ ) γ D ( k f s , l f t ) ] + k = 1 l = 1 a k b l | G D ( k f s , l f t ) | cos [ k φ + l ( φ + Δ φ ) γ D ( k f s , l f t ) ] } 1 .
arctan C N C D = arctan | G N ( f s , f t ) | cos [ 2 φ + Δ φ γ N ( f s , f t ) ] + | G N ( f s , f t ) | cos [ Δ φ γ N ( f s , f t ) ] | G D ( f s , f t ) | cos [ 2 φ + Δ φ γ D ( f s , f t ) ] + | G D ( f s , f t ) | cos [ Δ φ γ D ( f s , f t ) ] .
arctan C N C D = arctan | G N ( f s , f t ) | cos [ Δ φ γ N ( f s , f t ) ] | G D ( f s , f t ) | cos [ Δ φ γ D ( f s , f t ) ] .
arctan C N C D = arctan | G N ( f s , f t ) | sin [ Δ φ Δ φ 0 ] | G D ( f s , f t ) | cos [ Δ φ Δ φ 0 ] .
arctan C N C D = arctan sin [ Δ φ Δ φ o ] cos [ Δ φ Δ φ 0 ] .
Δ φ ( r ) = arctan C N C D + Δ φ 0 .
g N ( u 1 , u 2 ) = m = 1 M p = 1 P n m , p δ ( u 1 u 1 m , u 2 u 2 p ) ,
g D ( u 1 , u 2 ) = m = 1 M p = 1 P d m , p δ ( u 1 u 1 m , u 2 u 2 p ) ,
G N ( f 1 , f 2 ) = m = 1 M p = 1 P n m , p exp ( j 2 π u 1 m f 1 ) exp ( j 2 π u 2 p f 2 ) = m = 1 M p = 1 P n m , p exp ( j α m f 1 f s ) exp ( j β p f 2 f t ) ,
G D ( f 1 , f 2 ) = m = 1 M p = 1 P d m , p exp ( j 2 π u 1 m f 1 ) exp ( j 2 π u 2 p f 2 ) = m = 1 M p = 1 P d m , p exp ( j α m f 1 f s ) exp ( j β p f 2 f t ) ,
m = 1 M p = 1 P ( d m , p + j n m , p ) exp ( j α m ) exp ( j β p ) = 0.
m = 1 M p = 1 P n m , p = m = 1 M p = 1 P sin ( α m β p ) ,
m = 1 M p = 1 P d m , p = m = 1 M p = 1 P cos ( α m β p ) .
Δ φ ( r ) = arctan m = 1 M p = 1 P s m t p sin ( α m β p ) m = 1 M p = 1 P s m t p cos ( α m β p ) + Δ φ 0 .
arctan C N C D = arctan | G N ( f s , f t ) | cos [ 2 φ + Δ φ γ N ( f s , f t ) ] | G D ( f s , f t ) | cos [ 2 φ + Δ φ γ D ( f s , f t ) ] .
arctan C N C D = arctan | G N ( f s , f t ) | sin [ 2 φ + Δ φ Δ φ 0 ] | G D ( f s , f t ) | cos [ 2 φ + Δ φ Δ φ 0 ] .
arctan C N C D = arctan sin [ 2 φ + Δ φ Δ φ o ] cos [ 2 φ + Δ φ Δ φ 0 ] .
2 φ ( r ) + Δ φ ( r ) = arctan C N C D + Δ φ 0 .
m = 1 M p = 1 P n m , p = m = 1 M p = 1 P sin ( β p + α m ) ,
m = 1 M p = 1 P d m , p = m = 1 M p = 1 P cos ( β p + α m ) .
2 φ ( r ) + Δ φ ( r ) = arctan m = 1 M p = 1 P s m t p sin ( β p + α m ) m = 1 M p = 1 P s m t p cos ( β p + α m ) + Δ φ 0 .
m = 1 M p = 1 P n m , p = m = 1 M p = 1 P d m , p = 0
( φ + Δ φ ) φ = arctan p = 1 P h N ( u 2 p ) t p m = 1 M h D ( u 1 m ) s m m = 1 M h N ( u 1 m ) s m p = 1 P h D ( u 2 p ) t p ± m = 1 M h N ( u 1 m ) s m p = 1 P h N ( u 2 p ) t p + m = 1 M h D ( u 1 m ) s m p = 1 P h D ( u 2 p ) t p = arctan p = 1 P sin β p δ ( u 2 u 2 p ) t p m = 1 M cos α m δ ( u 1 u 1 m ) s m m = 1 M sin α m δ ( u 1 u 1 m ) s m p = 1 P cos β p δ ( u 2 u 2 p ) t p ± m = 1 M sin α m δ ( u 1 u 1 m ) s m p = 1 P sin β p δ ( u 2 u 2 p ) t p + m = 1 M cos α m δ ( u 1 u 1 m ) s m p = 1 P cos β p δ ( u 2 u 2 p ) t p .
α m E = α m + E α m = α m + q = 1 ε q α m q q π q 1 ,
β p E = β p + E β p = β p + r = 1 χ r β p r r π r 1 ,
G N E ( f 1 , f 2 ) = m = 1 M p = 1 P n m , p exp ( j α m f 1 f s ) ( 1 j q = 1 ε q α m q q π q 1 f 1 ) exp ( j β p f 2 f t ) ( 1 j r = 1 χ r β p r r π r 1 f 2 ) ,
G D E ( f 1 , f 2 ) = m = 1 M p = 1 P d m , p exp ( j α m f 1 f s ) ( 1 j q = 1 ε q α m q q π q 1 f 1 ) exp ( j β p f 2 f t ) ( 1 j r = 1 χ r β p r r π r 1 f 2 ) ,
G N , D E ( f 1 , f 2 ) = G N , D ( f 1 , f 2 ) + q = 1 f 1 f s q q ( j π ) q 1 q G N , D ( f 1 , f 2 ) f 1 q + r = 1 χ r f 2 f t r r ( j π ) r 1 r G N , D ( f 1 , f 2 ) f 2 r + q = 1 r = 1 ε q χ r q r f 1 f 2 f s q f t r ( j π ) ( q 1 ) ( r 1 ) r f 2 r ( q G N , D ( f 1 , f 2 ) f 1 q ) .
[ G N ( f 10 , f 20 ) + j G D ( f 10 , f 20 ) ] + q = 1 ( j π q ) q 1 ε q f 10 q + 1 [ | q G N ( f 1 , f 2 ) f 1 q | ( f 10 , f 20 ) + j | q G D ( f 1 , f 2 ) f 1 q | ( f 10 , f 20 ) ] + r = 1 ( j π r ) r 1 χ r f 20 r + 1 [ | r G N ( f 1 , f 2 ) f 2 r | ( f 10 , f 20 ) + j | r G D ( f 1 , f 2 ) f 2 r | ( f 10 , f 20 ) ] + q = 1 r = 1 ε q χ r f 1 f 2 f 10 q f 20 r q r ( j π ) ( q 1 ) ( r 1 ) [ | r f 2 r ( q G N ( f 1 , f 2 ) f 1 q ) | ( f 10 , f 20 ) + j | r f 2 r ( q G D ( f 1 , f 2 ) f 1 q ) | ( f 10 , f 20 ) ] = 0 ;
| q G N ( f 1 , f 2 ) f 1 q | ( f 10 , f 20 ) + j | q G D ( f 1 , f 2 ) f 1 q | ( f 10 , f 20 ) = 0 ,
| r G N ( f 1 , f 2 ) f 2 r | ( f 10 , f 20 ) + j | r G D ( f 1 , f 2 ) f 2 r | ( f 10 , f 20 ) = 0 ,
r f 2 r | ( q G N ( f 1 , f 2 ) f 1 q ) | ( f 10 , f 20 ) + j | r f 2 r ( q G D ( f 1 , f 2 ) f 1 q ) | ( f 10 , f 20 ) = 0.
| q G N , D ( f 1 , f 2 ) f 1 q | ( k f 10 , l f 20 ) = 0 for k > 1 , l > 1 and q > 0 ,
| r G N , D ( f 1 , f 2 ) f 2 r | ( k f 10 , l f 20 ) = 0 for k > 1 , l > 1 and r > 0 ,
| r f 2 r ( q G N , D ( f 1 , f 2 ) f 1 q ) | ( k f 10 , l f 20 ) = 0 for k > 1 , l > 1 , q > 0 and r > 0.
( φ + Δ φ ) φ = arctan ( 2 s 3 s 1 s 5 ) ( 2 t 2 2 t 4 ) ( 2 s 2 2 s 4 ) ( 2 t 3 t 1 t 5 ) ± ( 2 s 2 2 s 4 ) ( 2 t 2 2 t 4 ) + ( 2 s 3 s 1 s 5 ) ( 2 t 3 t 1 t 5 ) ,
G N ( f 1 , f 2 ) = 2 j [ sin ( π 2 f 1 f s ) sin ( π 2 f 2 f t ) sin ( π 2 f 1 f s ) cos ( π f 2 f t ) + cos ( π f 1 f s ) sin ( π 2 f 2 f t ) ] ,
G D ( f 1 , f 2 ) = 1 cos ( π f 1 f s ) cos ( π f 2 f t ) 4 sin ( π 2 f 1 f s ) sin ( π 2 f 2 f t ) + cos ( π f 1 f s ) cos ( π f 2 f t ) ,
G N ( f 1 , f 2 ) = 2 j [ sin ( π 2 f 1 f s ) sin ( π 2 f 2 f t ) + sin ( π 2 f 1 f s ) cos ( π f 2 f t ) + cos ( π f 1 f s ) sin ( π 2 f 2 f t ) ] ,
G D ( f 1 , f 2 ) = 1 cos ( π f 1 f s ) cos ( π f 2 f t ) + 4 sin ( π 2 f 1 f s ) sin ( π 2 f 2 f t ) + cos ( π f 1 f s ) cos ( π f 2 f t ) .

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