Abstract

The generalized analytical quadrature filter from a set of interferograms with arbitrary phase shifts is obtained. Both symmetrical and non symmetrical algorithms for any order are reported. The analytic expression is obtained through the convolution of a set of two-frame algorithms and expressed in terms of the combinatorial theory. Finally, the solution is applied to obtain several generalized tunable quadrature filters.

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2010 (2)

2009 (2)

2004 (1)

1997 (2)

1996 (1)

1995 (2)

1992 (1)

1987 (1)

1975 (1)

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Creath, K.

Doblado, D. M.

Eiju, T.

Estrada, J. C.

Farrant, D. I.

Gallagher, J. E.

Groot, P.

Gutiérrez-García, J. C.

Gutiérrez-García, T. A.

Han, B.

Hariharan, P.

Hernández, D. M.

Herriott, D. R.

Hibino, K.

Larkin, K. G.

Macías-Preza, J. M.

Malacara-Doblado, D.

Mosiño, J. F.

Oreb, B. F.

Phillion, D. W.

Quiroga, J. A.

Rosenfeld, D. P.

Schmit, J.

Servin, M.

Surrel, Y.

Téllez-Quiñones, A.

Wang, Z.

White, A. D.

Wyant, J. C.

Appl. Opt. (8)

J. Opt. Soc. Am. A (2)

Opt. Express (3)

Opt. Lett. (1)

Other (2)

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (Elsevier, 1990).

H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., 2007).

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Equations (125)

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I k = a ( x , y ) + b ( x , y ) cos [ α t k + φ ( x , y ) ]   For  k = 1 and 2,
tan [ φ ( x , y ) ] = N I 2 D I 2 = cos ( α / 2 ) [ 1 1 ] I 2 sin ( α / 2 ) [ 1    1 ]   I 2 = cos ( α / 2 ) sin ( α / 2 ) [ I 1 I 2 I 1 + I 2 ] ,
H ( ω ) = 2 sin [ ( ω α ) / 2 ] .
N D = cos ( α / 2 ) sin ( α / 2 ) [ 1 1 ] [ 1    1 ]   .
tan ( φ 1 ) = N 1   I n D 1   I n ,  and   tan ( φ 2 ) = N 2   I m D 2   I m ,
h 1 ( t ) = ( D 1 + i   N 1 ) δ n   and   h 2 ( t ) = ( D 2 + i   N 2 )     δ m ,
δ n = [ δ ( t ) , δ ( t α ) , ... δ ( t n α ) ] T   and   δ m = [ δ ( t ) , δ ( t α ) , ... δ ( t m α ) ] T .
h ( t ) = [ D 1 δ n + i   N 1 δ n ] [ D 2 δ m + i   N 2 δ m ]                                       = [ D 1 D 2 N 1 N 2 ] δ n + m 1 + i [ N 1 D 2 + D 1 N 2 ] δ n + m 1 ,
tan [ φ ( x , y ) ] = N   I n + m 1 D   I n + m 1 = [ N 1 D 2 + D 1 N 2 ]   I n + m 1 [ D 1 D 2 N 1 N 2 ]   I n + m 1 .
N D = ( N 1 D 1 ) ( N 2 D 2 ) = N 1 D 2 + D 1 N 2 D 1 D 2 N 1 N 2 .
( N θ D θ ) = ( cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ) ( N D ) = ( N cos ( θ ) D sin ( θ ) N sin ( θ ) + D cos ( θ ) ) ,
( N θ D θ ) = ( cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ) ( cos ( α / 2 ) [ 1 1 ] sin ( α / 2 ) [ 1    1 ] ) = ( [ cos ( θ + α / 2 ) cos ( θ α / 2 ) ] [ sin ( θ + α / 2 ) sin ( θ α / 2 ) ] ) .
N θ D θ = [ cos ( θ + α / 2 ) , cos ( θ α / 2 ) ] [ sin ( θ + α / 2 ) , sin ( θ α / 2 ) ] .
tan ( φ ) = N θ I 2 D θ I 2 = cos ( θ + α / 2 ) I 1 cos ( θ α / 2 ) I 2 sin ( θ + α / 2 ) I 1 sin ( θ α / 2 ) I 2 ,
H r ( ω ) = 2 exp ( i θ ) sin [ ( ω α ) / 2 ] .
N α / 2 D α / 2 = [ cos α 1 ] [ sin α 0 ] .
tan ( φ k ) = N k I 2 D k I 2 = [ cos α k 1 ] I 2 [ sin α k 0 ] I 2 = I 1 cos α k I 2 I 1 sin α k .
tan ( φ ) = k = 1 M b k I k k = 1 M a k I k = [ b 1 b 2 ... b M ]   I M [ a 1 a 2 ... a M ]   I M = N   I M D   I M ,
P ( x ) = k = 1 M ( a k + i b k ) x k 1 .
P ( x ) = k = 1 M 1 ( x k A k i B k ) .
α k = tan 1 ( B k / A k ) .
H ( ω ) = k = 1 M 1 H k ( ω α k ) .
h ( t ) = 1 [ H 1 ( ω ) H 2. ( ω ) ... H M 1 ( ω ) ] = h 1 ( t ) h 2 ( t ) ... h M 1 ( t ) = Ω k = 1 M 1 h k ( t ) .
h k ( t ) = ( D k + i   N k ) δ 2 .
h ( t ) = Ω k = 1 M 1 [ ( D k + i   N k ) δ 2 ] = ( D + i   N ) δ M ,
N D = Ω k = 1 M 1 ( N k D k ) = ( N k D k ) ( N k D k ) ( N k D k )   ...   ( N M 1 D M 1 ) .
N D = Ω k = 1 M 1 { [ cos ( θ k + α k / 2 ) , cos ( θ k α k / 2 ) ] [ sin ( θ k + α k / 2 ) , sin ( θ k α k / 2 ) ] } .
H ( ω ) = ( 2 ) M 1 exp ( σ i ) k = 1 M 1 sin [ ( ω α k ) / 2 ]     ,  where    σ = k = 1 M 1 θ k .
H ( ω ) = ( 2 ) M 1 k = 1 M 1 sin [ ( ω α k ) / 2 ] .
N D = Ω k = 1 M 1 { cos ( α k / 2 ) [ 1 1 ] sin ( α k / 2 ) [ 1    1 ]   } .
N r D r = Ω k = 1 M 1 { [ cos α k 1 ] [ sin α k 0 ] } .
N D = [ cos α 1 , 1 ] [ sin α 1 , 0 ] [ cos α 2 , 1 ] [ sin α 2 , 0 ] = [ sin α 2 , 0 ] [ cos α 1 , 1 ] + [ cos α 2 , 1 ] [ sin α 1 , 0 ] [ sin α 1 , 0 ] [ sin α 2 , 0 ] [ cos α 1 , 1 ] [ cos α 2 , 1 ] ,
N D = [ sin ( α 1 + α 2 ) , sin ( α 1 ) sin ( α 2 ) , 0 ] [ cos ( α 1 + α 2 ) , cos ( α 1 ) + cos ( α 2 ) , 1 ] .
N π / 2 D π / 2 = [ cos ( α 1 + α 2 ) , cos ( α 1 ) cos ( α 2 ) , 1 ] [ sin ( α 1 + α 2 ) , sin ( α 1 ) sin ( α 2 ) , 0 ] .
N D = [ sin ( α ) , sin ( α ) , 0 ] [ cos ( α ) , cos ( α ) + 1 , 1 ] .
tan ( φ ) = sin ( α ) ( I 1 I 2 ) cos ( α ) ( I 1 I 2 ) + ( I 2 I 3 ) .
tan ( φ ) = cos ( α ) ( I 1 I 2 ) ( I 2 I 3 ) sin ( α ) ( I 1 I 2 ) ,
N D = [ sin ( α 1 + α 2 ) , sin ( α 1 ) sin ( α 2 ) , 0 ] [ cos ( α 1 + α 2 ) , cos ( α 1 ) + cos ( α 2 ) , 1 ] [ cos α 3 , 1 ] [ sin α 3 , 0 ] = [ b 1 , b 2 , b 3 , b 4 ] [ a 1 , a 2 , a 3 , a 4 ] .
b 1 = cos ( α 1 + α 2 + α 3 ) ,
b 2 = cos ( α 1 + α 2 ) + cos ( α 1 + α 3 ) + cos ( α 2 + α 3 ) ,
b 3 = cos ( α 1 ) cos ( α 2 ) cos ( α 3 ) , b 4 = 1 ,
a 1 = sin ( α 1 + α 2 + α 3 ) ,
a 2 = sin ( α 1 + α 2 ) + sin ( α 1 + α 3 ) + sin ( α 2 + α 3 ) ,
a 3 = sin ( α 1 ) sin ( α 2 ) sin ( α 3 ) , a 4 = 0.
N D = [ cos ( α 1 + α 2 ) , cos ( α 1 + α 2 ) + cos ( α 1 ) + cos ( α 2 ) , 1 cos ( α 1 ) cos ( α 2 ) , 1 ] [ sin ( α 1 + α 2 ) , sin ( α 1 + α 2 ) + sin ( α 1 ) + sin ( α 2 ) , sin ( α 1 ) sin ( α 2 ) , 0 ] ,
N D = [ sin ( α 1 + α 2 ) , sin ( α 1 + α 2 ) sin ( α 1 ) sin ( α 2 ) , sin ( α 1 ) + sin ( α 2 ) , 0 ] [ cos ( α 1 + α 2 ) , cos ( α 1 + α 2 ) cos ( α 1 ) cos ( α 2 ) , 1 + cos ( α 1 ) + cos ( α 2 ) , 1 ] .
N D = [ cos ( α ) , 1 , cos ( α ) , 1 ] [ sin ( α ) , 0 , sin ( α ) , 0 ] .
tan ( φ ) = N I D I = cos ( α ) ( I 1 I 3 ) I 2 + I 4 sin ( α ) ( I 1 I 3 ) .
tan ( φ ) = N I D I = sin ( α ) ( I 1 I 3 ) cos ( α ) ( I 1 I 3 ) I 2 + I 4
N D = Ω k = 1 4 { [ cos α k , 1 ] [ sin α k , 0 ] } = [ b 1 , b 2 , b 3 , b 4 , b 5 ] [ a 1 , a 2 , a 3 , a 4 , a 5 ] .
N D = [ sin ( α 1 + α 2 ) , sin ( α 1 ) sin ( α 2 ) , 0 ] [ cos ( α 1 + α 2 ) , cos ( α 1 ) + cos ( α 2 ) , 1 ] [ sin ( α 3 + α 4 ) , sin ( α 3 ) sin ( α 4 ) , 0 ] [ cos ( α 3 + α 4 ) , cos ( α 3 ) + cos ( α 4 ) , 1 ] ,
b 1 = sin ( α 1 + α 2 + α 3 + α 4 ) ,
b 2 = sin ( α 1 + α 2 + α 3 ) sin ( α 1 + α 2 + α 4 ) sin ( α 1 + α 3 + α 4 ) sin ( α 2 + α 3 + α 4 ) ,
b 3 = sin ( α 1 + α 2 ) + sin ( α 1 + α 3 ) + sin ( α 1 + α 4 )                                                  + sin ( α 2 + α 3 ) + sin ( α 2 + α 4 ) + sin ( α 3 + α 4 ) ,
b 4 = sin ( α 1 ) sin ( α 2 ) sin ( α 3 ) sin ( α 4 ) , b 5 = 0 ,
a 1 = cos ( α 1 + α 2 + α 3 + α 4 ) ,
a 2 = cos ( α 1 + α 2 + α 3 ) cos ( α 1 + α 2 + α 4 ) cos ( α 1 + α 3 + α 4 ) cos ( α 2 + α 3 + α 4 ) ,
a 3 = cos ( α 1 + α 2 ) + cos ( α 1 + α 3 ) + cos ( α 1 + α 4 )                                                             + cos ( α 2 + α 3 ) + cos ( α 2 + α 4 ) + cos ( α 3 + α 4 ) ,
a 4 = cos ( α 1 ) cos ( α 2 ) cos ( α 3 ) cos ( α 4 )   and  a 5 = 1.
tan ( φ ) = N I 5 D I 5 = 2 sin ( α ) [ 0 , 1 , 0 , 1 0 ] I 5 [ 1 , 0 , 2 , 0 , 1 ] I 5 = 2 sin ( α ) ( I 2 I 4 ) I 1 + 2 I 3 I 5 .
tan ( φ ) = N I 5 D I 5 = sin ( 2 α ) ( I 1 I 3 ) 2 sin ( α ) ( I 2 I 4 ) cos ( 2 α ) ( I 1 I 3 ) + 2 cos ( α ) ( I 2 I 4 ) ( I 3 I 5 ) .
N D = Ω k = 1 2 { cos ( α k / 2 ) [ 1 , 1 ] sin ( α k / 2 ) [ 1 ,    1 ]   } = [ sin [ ( α 1 + α 2 ) / 2 ] , 0 , sin [ ( α 1 + α 2 ) / 2 ] ] [ cos [ ( α 1 + α 2 ) / 2 ] , 2 cos [ ( α 1 α 2 ) / 2 ] , cos [ ( α 1 + α 2 ) / 2 ] ] ,
tan ( φ ) = [ sin ( α / 2 ) , 0 , sin ( α / 2 ) ] I 3 [ cos ( α / 2 ) , 2 cos ( α / 2 ) , cos ( α / 2 ) ] I 3 = tan ( α / 2 ) I 1 I 3 I 1 + 2 I 2 I 3 .
N D = Ω k = 1 3 { cos ( α k / 2 ) [ 1 1 ] sin ( α k / 2 ) [ 1    1 ]   } = [ b 1 , b 2 , b 2 , b 1 ] [ a 1 , a 2 , a 2 , a 1 ] .
b 1 = cos [ ( α 1 + α 2 + α 3 ) / 2 ] , a 1 = sin [ ( α 1 + α 2 + α 3 ) / 2 ] ,
b 2 = cos [ ( α 1 + α 2 α 3 ) / 2 ] + cos [ ( α 1 α 2 + α 3 ) / 2 ] + cos [ ( α 1 + α 2 + α 3 ) / 2 ] ,
a 2 = sin [ ( α 1 + α 2 α 3 ) / 2 ] + sin [ ( α 1 α 2 + α 3 ) / 2 ] + sin [ ( α 1 + α 2 + α 3 ) / 2 ] .
b 1 = cos [ ( α 2 + α 3 ) / 2 ] , b 2 = 2 cos [ ( α 2 α 3 ) / 2 ] + cos [ ( α 2 + α 3 ) / 2 ]
a 1 = a 2 = sin [ ( α 2 + α 3 ) / 2 ] .
N D = Ω k = 1 4 { cos ( α k / 2 ) [ 1 1 ] sin ( α k / 2 ) [ 1    1 ]   } = [ b 1 b 2 0 b 2 , b 1 ] [ a 1 a 2 a 3 a 2 a 1 ] ,
b 1 = sin [ ( α 1 + α 2 + α 3 + α 4 ) / 2 ] , a 1 = cos [ ( α 1 + α 2 + α 3 + α 4 ) / 2 ] ,
b 2 = sin [ ( α 1 + α 2 + α 3 α 4 ) / 2 ] + sin [ ( α 1 + α 2 α 3 + α 4 ) / 2 ]                                        + sin [ ( α 1 α 2 + α 3 + α 4 ) / 2 ] + sin [ ( α 1 + α 2 + α 3 + α 4 ) / 2 ] ,
a 2 = cos [ ( α 1 + α 2 + α 3 α 4 ) / 2 ] cos [ ( α 1 + α 2 α 3 + α 4 ) / 2 ]                                           cos [ ( α 1 α 2 + α 3 + α 4 ) / 2 ] cos [ ( α 1 + α 2 + α 3 + α 4 ) / 2 ] ,
a 3 = 2 cos [ ( α 1 + α 2 α 3 α 4 ) / 2 ] + 2 cos [ ( α 1 α 2 + α 3 α 4 ) / 2 ]                                                                                  + 2 cos [ ( α 1 + α 2 + α 3 α 4 ) / 2 ] .
tan ( φ ) = N I 5 D I 5 = [ cos ( α ) 2 0 2 , cos ( α ) ] I 5 sin ( α )     [ 1 0 2 0 1 ] I 5 = cos ( α ) ( I 1 I 5 ) + 2 ( I 2 I 4 ) sin ( α ) ( I 1 2 I 3 + I 5 ) .
tan ( φ ) = N I 5 D I 5 = [ 0 2 0 2 , 0 ] I 5 [ 1 0 2 0 1 ] I 5 = 2 ( I 2 I 4 ) I 1 + 2 I 3 I 5 .
N D = [ sin ( 3 α / 2 ) sin ( 3 α / 2 ) + 3 sin ( α / 2 ) 0 sin ( 3 α / 2 ) 3 sin ( α / 2 ) sin ( 3 α / 2 ) ] [ cos ( 3 α / 2 ) cos ( 3 α / 2 ) 3 cos ( α / 2 ) 6 sin ( α / 2 ) cos ( 3 α / 2 ) 3 cos ( α / 2 ) cos ( 3 α / 2 ) ] .
tan ( φ ) = N I 5 D I 5 = [-1    4    0    -4    1] I 5 [-1    -2    6    -2    -1] I 5 = I 1 4 I 2 + 4 I 4 I 5 I 1 + 2 I 2 6 I 3 + 2 I 4 + I 5 .
W 3 4 = { a , b , c , d } 3 4 = { ( a , b , c ) , ( a , b , d ) , ( a , c , d ) , ( b , c , d ) } .
W 2 4 = { a , b , c , d } 2 4 = { ( a , b ) , ( a , c ) , ( a , d ) , ( b , c ) , ( b , d ) , ( c , d ) } .
r = 0 n C n r = r = 0 n n ! ( n r ) ! r ! n = 2 n ,
Σ W 3 4 = Σ { a , b , c , d } 3 4 = { ( a + b + c ) , ( a + b + d ) , ( a + c + d ) , ( b + c + d ) } .
cos [ Σ { a , b , c , d } 3 4 ] = { cos ( a + b + c ) , cos ( a + b + d ) , cos ( a + c + d ) , cos ( b + c + d ) } .
σ + ( a , b , c , d ) = ( σ + a , σ + b , σ + c , σ + d ) .
σ + Σ { a , b , c , d } 3 4 = { ( σ + a + b + c ) , ( σ + a + b + d ) , ( σ + a + c + d ) , ( σ + b + c + d ) } .
Σ cos ( σ + Σ W 3 4 ) = cos ( σ + a + b + c ) + cos ( σ + a + b + d ) + cos ( σ + a + c + d ) + cos ( σ + b + c + d ) .
N D = Ω k = 1 M 1 { [ cos α k 1 ] [ sin α k 0 ] } = [ b 1 b 2 b 3 ... b M ] [ a 1 a 2 a 3 ... a M ] .
b 1 = sin [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M 1 M 1 ] = sin ( σ ) ,
b 2 = ( 1 ) sin [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M 2 M 1 ] = ( 1 ) k = 1 M 1 sin ( σ α k ) ,
b M = ( 1 ) M + 1 sin [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M M M 1 ] = ( 1 ) M + 1 sin ( 0 ) = 0 ,
σ = k = 1 M 1 α k .
b k = ( 1 ) k + 1 sin [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M k M 1 ]   ;  For  1 k M .
a 1 = cos [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M 1 M 1 ] = cos ( σ ) ,
a 2 = ( 1 ) 2 cos [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M 2 M 1 ] = ( 1 ) 2 k = 1 M 1 cos ( σ α k ) ,
a M = ( 1 ) M cos [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M M M 1 ] = ( 1 ) M cos ( 0 ) = ( 1 ) M .
a k = ( 1 ) k cos [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M k M 1 ]   ;  For  1 k M .
tan ( φ ) = N I M D I M = k = 1 M { ( 1 ) k + 1 sin [ Σ { α 1 , α 2 , ... α M 1 } M k M 1 ] }     I k k = 1 M { ( 1 ) k cos [ Σ { α 1 , α 2 , ... α M 1 } M k M 1 ] }     I k ,
tan ( φ ) = N I M D I M = k = 1 M { ( 1 ) k + 1 cos [ Σ { α 1 , α 2 , ... α M 1 } M k M 1 ] }     I k k = 1 M { ( 1 ) k + 1 sin [ Σ { α 1 , α 2 , ... α M 1 } M k M 1 ] }     I k .
N D = Ω k = 1 M 1 { cos ( α k / 2 ) [ 1 1 ] sin ( α k / 2 ) [ 1    1 ]   } = [ b 1 b 2 b 3 ... b M ] [ a 1 a 2 a 3 ... a M ] .
b 1 = sin [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } 0 M 1 ] = b M = sin ( σ / 2 ) ,
a 1 = ( 1 ) cos [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } 0 M 1 ] = a M = cos ( σ / 2 ) ,
b ( m + 1 ) / 2 = ( 1 ) ( m + 3 ) / 2 sin [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } ( M 1 ) / 2 M 1 ] = 0.
b k = ( 1 ) k + 1 sin [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } k 1 M 1 ]   ;  For  1 k M ,
a k = ( 1 ) k cos [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } k 1 M 1 ]   ;  For  1 k M .
tan ( φ ) = N I M D I M = k = 1 M { ( 1 ) k + 1 sin [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } k 1 M 1 ] }       I k k = 1 M { ( 1 ) k cos [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } k 1 M 1 ] }       I k .
tan ( φ ) = N I M D I M = k = 1 M { ( 1 ) k + 1 cos [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } k 1 M 1 ] }       I k k = 1 M { ( 1 ) k + 1 sin [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } k 1 M 1 ] }       I k .
P ( x ) = k = 1 5 ( a k + i b k ) x k 1 = ( x 1 ) ( x + 1 ) [ x + cos ( α ) + i sin ( α ) ] [ x cos ( α ) + i sin ( α ) ] .
N D = { 2 sin ( α ) [ 0 1 0 1 0 ] [ 1 0 2 0 1 ] } { cos ( β / 2 ) [ 1 1 ] sin ( β / 2 ) [ 1 1 ] } { sin ( β / 2 ) [ 1 1 ] cos ( β / 2 ) [ 1 1 ] } .
H ( ω ) = ( 2 ) 6 sin ( ω )     [ sin ( ω ) sin ( α ) ] [ sin ( ω ) sin ( β ) ] .
N D = [ 1 0 +  4 sin( α ) sin( β 0  4 sin( α ) sin( β 0 1 ] [ 0 2 sin( β ) 2 sin( α 0 4 sin( α + 4sin( β 0 2 sin( β ) 2 sin( α 0 ] .
tan ( φ ) = N I 7 D     I 7 = ( I 1 I 7 ) + [ 3 + 4 sin ( α ) sin ( β ) ] ( I 3 I 5 ) 2 [ sin ( α ) + sin ( β ) ] ( I 2 2 I 4 + I 6 ) .
tan ( φ ) = ( I 1 I 7 ) + 3 ( I 3 I 5 ) 2 sin ( α ) ( I 2 2 I 4 + I 6 ) .
tan ( φ ) = I 1 3 I 3 + 3 I 5 I 7 2 I 2 4 I 4 + 2 I 6 .
tan ( φ ) = I 1 5 I 3 + 5 I 5 I 7 3 I 2 6 I 4 + 3 I 6 .
tan ( φ ) = ( I 1 I 7 ) + [ 3 + 4 sin 2 ( α ) ] ( I 3 I 5 ) 4 sin ( α ) ( I 2 2 I 4 + I 6 ) = ( I 1 I 7 ) [ 5 2 cos ( 2 α ) ] ( I 3 I 5 ) 4 sin ( α ) ( I 2 2 I 4 + I 6 ) ,
tan ( φ ) = I 1 7 I 3 + 7 I 5 I 7 4 ( I 2 2 I 4 + I 6 ) .
N D = lim α 1 0 Ω k = 1 4 { [ cos α 1 ] [ sin α 0 ] } [ cos ( α 1 ) 1 ] [ sin ( α 1 ) 0 ] .
H ( ω ) = ( 2 ) 5 exp ( 2 α i ) sin ( ω / 2 ) sin 4 [ ( ω α ) / 2 ] .
b k = ( 1 ) k + 1 sin [ Σ { α , α , α , α } 5 k 4 ] = ( 1 ) k + 1 ( 4 5 k ) sin [ ( 5 k ) α ]   ;  For  1 k 5 ,
a k = ( 1 ) k cos [ Σ { α , α , α , α } 5 k 4 ] = ( 1 ) k ( 4 5 k ) cos [ ( 5 k ) α ]   ;  For  1 k 5.
N D = [ cos ( 4 α ) ,   cos ( 4 α ) 4 cos ( 3 α ) ,   4 cos ( 3 α ) + 6 cos ( 2 α ) ,   4 cos ( α ) 6 cos ( 2 α ) ,   4 cos ( α ) + 1 ,   1 ] [ sin ( 4 α ) ,   sin ( 4 α ) 4 sin ( 3 α ) ,   4 sin ( 3 α ) + 6 sin ( 2 α ) ,   4 sin ( α ) 6 sin ( 2 α ) ,   4 sin ( α ) , 0 ] .
tan ( φ ) = N I 6 D I 6 = [ 1 1 6 6 1 1 ] I 6 [ 0 4 4 4 4 0 ] I 6 = I 1 I 2 6 I 3 + 6 I 4 + I 5 I 6 4 ( I 2 I 3 I 4 + I 5 ) .
N = sin ( α ) ( I 1 I 11 ) 2 sin ( 2 α ) ( I 2 I 10 ) sin ( 3 α ) ( I 3 I 9 )                                            + 4 sin ( 2 α ) ( I 4 I 8 ) + [ 3 sin ( 3 α ) + 5 sin ( α ) ] ( I 5 I 7 ) ,
D = cos ( α ) ( I 1 + I 11 ) + 2 cos ( 2 α ) ( I 2 + I 10 ) + [ cos(3 α ) - 4 cos( α ) ] ( I 3 + I 9 )             - [ 4 cos(2a)  +  4 ] ( I 4 + I 8 ) [ cos(3a)   3cos(a) ] ( I 5 + I 7 ) + [  4cos(2a) + 8 ] I 6 .
N D = 3   [  1,  2,  0, -4, -5,  0,  5,  4,  0, -2, -1] [-1,   2,   6,   4,  -5, -12,  -5,   4,   6,   2,  -1] .

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