Abstract

We extend and generalize the Teager–Kaiser [in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (1993), Vol. 3, p. 149 ] and the higher-order differential energy operators [IEEE Signal Process. Lett. 2, 152 (1995) ] to a large class of operators called higher-order energy operators. We show that for AM-FM signal demodulation, the introduced partial derivative orders have to satisfy certain conditions. These operators are parameterized for local processing of AM-FM signals. The operators are illustrated using synthetic signals and a real signal from light scanning interferometry.

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References

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  1. J. F. Kaiser, "Some useful properties of Teager's energy operator," in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1993), pp. 149-152.
    [CrossRef]
  2. P. Maragos, T. F. Quatieri, and J. F. Kaiser, "Energy separation in signal modulations with applications to speech analysis," IEEE Trans. Signal Process. 41, 3024-3051 (1993).
    [CrossRef]
  3. F. Salzenstein, P. C. Montgomery, D. Montaner, and A. O. Boudraa, "Teager-Kaiser energy and higher-order operators in white-light interference microscopy for surface shape measurement," J. App. Sig. Proc. 17, 2804-2815 (2005).
  4. P. Maragos and A. Bovik, "Image demodulation using multidimensional energy separation," J. Opt. Soc. Am. A 12, 1867-1876 (1995).
    [CrossRef]
  5. F. Salzenstein, P. Montgomery, A. Benatmane, and A. O. Boudraa, "2D discrete high order energy operators for surface profiling using white light interferometry," in Proceedings of International Symposium on Signal Processing and its Applications (IEEE, 2003), pp. 601-604.
    [CrossRef]
  6. B. Santhanam and P. Maragos, "Energy demodulation of two component AM-FM signals with application to speaker separation," in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1996), p. 3518.
  7. P. Maragos and A. Potamianos, "Higher-order differential energy operators," IEEE Signal Process. Lett. 2, 152-154 (1995).
    [CrossRef]
  8. A. O. Boudraa, F. Salzenstein, and J. C. Cexus, "Two-dimensional continuous higher-order energy operators," Opt. Eng. (Bellingham) 44, 7001-7009 (2005).
    [CrossRef]
  9. A. Potamianos and P. Maragos, "A comparison of the energy operator and the Hilbert transform approach to signal and speech demodulation," Signal Process. 37, 95-120 (1994).
    [CrossRef]
  10. P. Flandrin, Temps-Fréquence (Hermès, 1998).
  11. L. Wei, C. Hamilton, and P. Chitrapu, "A generalization to the Teager-Kaiser energy function and application to resolving two closely-spaced tones," in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1995), p. 1637.
  12. P. C. Mongomery, A. Benatmane, E. Fogarassy, and J. P. Ponpon, "Large area, high resolution analysis of surface roughness of semiconductors using interference microscopy," Mater. Sci. Eng. B 91-92, 79-82 (2002).
    [CrossRef]
  13. S. S. Chim and G. S. Kino, "Correlation microscope," Opt. Lett. 15, 579-581 (1990).
    [CrossRef] [PubMed]
  14. K. G. Larkin, "Efficient nonlinear algorithm for envelope detection in white light interferometry," J. Opt. Soc. Am. A 13, 832-843 (1996).
    [CrossRef]

2005

F. Salzenstein, P. C. Montgomery, D. Montaner, and A. O. Boudraa, "Teager-Kaiser energy and higher-order operators in white-light interference microscopy for surface shape measurement," J. App. Sig. Proc. 17, 2804-2815 (2005).

A. O. Boudraa, F. Salzenstein, and J. C. Cexus, "Two-dimensional continuous higher-order energy operators," Opt. Eng. (Bellingham) 44, 7001-7009 (2005).
[CrossRef]

2002

P. C. Mongomery, A. Benatmane, E. Fogarassy, and J. P. Ponpon, "Large area, high resolution analysis of surface roughness of semiconductors using interference microscopy," Mater. Sci. Eng. B 91-92, 79-82 (2002).
[CrossRef]

1996

1995

P. Maragos and A. Bovik, "Image demodulation using multidimensional energy separation," J. Opt. Soc. Am. A 12, 1867-1876 (1995).
[CrossRef]

P. Maragos and A. Potamianos, "Higher-order differential energy operators," IEEE Signal Process. Lett. 2, 152-154 (1995).
[CrossRef]

1994

A. Potamianos and P. Maragos, "A comparison of the energy operator and the Hilbert transform approach to signal and speech demodulation," Signal Process. 37, 95-120 (1994).
[CrossRef]

1993

P. Maragos, T. F. Quatieri, and J. F. Kaiser, "Energy separation in signal modulations with applications to speech analysis," IEEE Trans. Signal Process. 41, 3024-3051 (1993).
[CrossRef]

1990

IEEE Signal Process. Lett.

P. Maragos and A. Potamianos, "Higher-order differential energy operators," IEEE Signal Process. Lett. 2, 152-154 (1995).
[CrossRef]

IEEE Trans. Signal Process.

P. Maragos, T. F. Quatieri, and J. F. Kaiser, "Energy separation in signal modulations with applications to speech analysis," IEEE Trans. Signal Process. 41, 3024-3051 (1993).
[CrossRef]

J. App. Sig. Proc.

F. Salzenstein, P. C. Montgomery, D. Montaner, and A. O. Boudraa, "Teager-Kaiser energy and higher-order operators in white-light interference microscopy for surface shape measurement," J. App. Sig. Proc. 17, 2804-2815 (2005).

J. Opt. Soc. Am. A

Mater. Sci. Eng. B

P. C. Mongomery, A. Benatmane, E. Fogarassy, and J. P. Ponpon, "Large area, high resolution analysis of surface roughness of semiconductors using interference microscopy," Mater. Sci. Eng. B 91-92, 79-82 (2002).
[CrossRef]

Opt. Eng. (Bellingham)

A. O. Boudraa, F. Salzenstein, and J. C. Cexus, "Two-dimensional continuous higher-order energy operators," Opt. Eng. (Bellingham) 44, 7001-7009 (2005).
[CrossRef]

Opt. Lett.

Signal Process.

A. Potamianos and P. Maragos, "A comparison of the energy operator and the Hilbert transform approach to signal and speech demodulation," Signal Process. 37, 95-120 (1994).
[CrossRef]

Other

P. Flandrin, Temps-Fréquence (Hermès, 1998).

L. Wei, C. Hamilton, and P. Chitrapu, "A generalization to the Teager-Kaiser energy function and application to resolving two closely-spaced tones," in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1995), p. 1637.

F. Salzenstein, P. Montgomery, A. Benatmane, and A. O. Boudraa, "2D discrete high order energy operators for surface profiling using white light interferometry," in Proceedings of International Symposium on Signal Processing and its Applications (IEEE, 2003), pp. 601-604.
[CrossRef]

B. Santhanam and P. Maragos, "Energy demodulation of two component AM-FM signals with application to speaker separation," in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1996), p. 3518.

J. F. Kaiser, "Some useful properties of Teager's energy operator," in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1993), pp. 149-152.
[CrossRef]

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Figures (21)

Fig. 1
Fig. 1

(a) Interferometric signal, (b) a profile along the optical axis, and (c) the reference surface shape.

Fig. 2
Fig. 2

Envelope detection of real signal by Ψ H k , p , m d , (a) Ψ H 4 , 2 , 3 d , (b) Ψ H 4 , 3 , 4 d , (c) Ψ H 2 , 2 , 1 d .

Fig. 3
Fig. 3

Envelope detection of real signal by Φ k m , (a) Φ 01 , (b) Φ 21 .

Fig. 4
Fig. 4

(a) Original AM signal, (b) its envelope, (c) related frequency, and (d) noisy signal.

Fig. 5
Fig. 5

Envelope detection of synthetic signal by Ψ H k , p , m d , (a) Ψ H 2 , 2 , 1 d , (b) Ψ H 4 , 2 , 3 d , (c) Ψ H 4 , 3 , 4 d .

Fig. 6
Fig. 6

Envelope detection of synthetic signal by Φ k m , (a) Φ 01 , (b) Φ 21 .

Fig. 7
Fig. 7

Frequency detection of synthetic signal by Ψ H k , p , m d , (a) Ψ H 2 , 2 , 1 d , (b) Ψ H 4 , 2 , 3 d , (c) Ψ H 4 , 3 , 4 d .

Fig. 8
Fig. 8

Frequency detection of synthetic signal by Φ k m , (a) Φ 01 , (b) Φ 21 .

Fig. 9
Fig. 9

Envelope detection of synthetic signal with an oversampling factor d = 1 using (a) HEO, k = ( 2 , 4 ) ; (b) HEO, k = ( 4 , 8 ) .

Fig. 10
Fig. 10

Envelope detection with an oversampling factor d = 5 : (a) HEO, k = ( 2 , 4 ) ; (b) HEO, k = ( 4 , 8 ) ; d = 10 : (c) HEO, k = ( 2 , 4 ) ; (d) HEO, k = ( 4 , 8 ) .

Fig. 11
Fig. 11

Envelope detection of synthetic signal with an oversampling factor d = 1 using (a) PHEO, k = ( 2 , 4 ) ; (b) PHEO, k = ( 4 , 8 ) .

Fig. 12
Fig. 12

Envelope detection with an oversampling factor d = 5 : (a) PHEO, k = ( 2 , 4 ) ; (b) PHEO, k = ( 4 , 8 ) ; d = 10 : (c) PHEO, k = ( 2 , 4 ) ; (d) PHEO, k = ( 4 , 8 ) .

Fig. 13
Fig. 13

Envelope detection of synthetic signal with an oversampling factor d = 5 using (a) OS-HEO, k = ( 2 , 4 ) ; (b) OS-HEO, k = ( 4 , 8 ) .

Fig. 14
Fig. 14

Envelope detection of synthetic signal with an oversampling factor d = 5 using (a) OS-PHEO, k = ( 2 , 4 ) ; (b) OS-PHEO, k = ( 4 , 8 ) .

Fig. 15
Fig. 15

Frequency detection of synthetic signal with an oversampling factor d = 1 using (a) HEO, k = ( 2 , 4 ) ; (b) HEO, k = ( 4 , 8 ) .

Fig. 16
Fig. 16

Frequency detection with an oversampling factor d = 5 : (a) HEO, k = ( 2 , 4 ) ; (b) HEO, k = ( 4 , 8 ) ; d = 10 : (c) HEO, k = ( 2 , 4 ) ; (d) HEO, k = ( 4 , 8 ) .

Fig. 17
Fig. 17

Frequency detection of synthetic signal with an oversampling factor d = 5 using (a) OS-HEO, k = ( 2 , 4 ) ; (b) OS-HEO, k = ( 4 , 8 ) .

Fig. 18
Fig. 18

Frequency detection of synthetic signal with an oversampling factor d = 5 using (a) OS-PHEO, k = ( 2 , 4 ) ; (b) OS-PHEO, k = ( 4 , 8 ) .

Fig. 19
Fig. 19

Frequency detection with an oversampling factor d = 5 : (a) PHEO, k = ( 2 , 4 ) ; (b) PHEO, k = ( 4 , 8 ) ; d = 10 : (c) PHEO, k = ( 2 , 4 ) ; (d) PHEO, k = ( 4 , 8 ) .

Fig. 20
Fig. 20

Frequency detection of synthetic signal with an oversampling factor d = 1 using (a) PHEO, k = ( 2 , 4 ) ; (b) PHEO, k = ( 4 , 8 ) .

Fig. 21
Fig. 21

Surface detection using (a) HEO, k = ( 2 , 4 ) ; ε = 17.3 nm ; (b) PHEO, k = ( 2 , 4 ) ; ε = 70.2 nm ; (c) OS-HEO, k = ( 2 , 4 ) ; ε = 16.3 nm ; (d) OS-PHEO, k = ( 2 , 4 ) ; ε = 15.3 nm .

Tables (9)

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Table 1 Quantitative Results of Envelope Detection of Real Signal Using Ψ H k , p , m d and Φ k m Operators

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Table 2 Quantitative Results of Envelope Detection of Synthetic Signal Using Ψ H k , p , m d and Φ k m Operators

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Table 3 Quantitative Results of Frequency Detection of Synthetic Signal Using Ψ H k , p , m d and Φ k m Operators

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Table 4 Nonnoisy Envelope Estimation: Error Rate (%) for k = ( 2 , 4 ) with Different Oversampling Factors d and Window Sizes

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Table 5 Noisy Envelope Estimation: Error rate (%) for k = ( 2 , 4 ) ; Oversampling Factor d = 1

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Table 6 Noisy Envelope Estimation: Error Rate (%) for k = ( 2 , 4 ) , Oversampling Factor d = 5

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Table 7 Nonnoisy Frequency Estimation: Error Rate (%) for k = ( 2 , 4 ) with Different Oversampling Factors and Windows

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Table 8 Noisy Frequency Estimation: Error Rate for k = ( 2 , 4 ) ; Oversampling Factor d = 1

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Table 9 Noisy Frequency Estimation: Error Rate for k = ( 2 , 4 ) ; Oversampling Factor d = 5

Equations (69)

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Ψ [ x ( t ) ] = [ x ( 1 ) ( t ) ] 2 x ( 0 ) ( t ) x ( 2 ) ( t ) ,
Ψ [ x ( n ) ] = x 2 ( n ) x ( n + 1 ) x ( n 1 ) .
Ψ k [ x ( t ) ] = x ( 1 ) ( t ) x ( k 1 ) ( t ) x ( 0 ) ( t ) x ( k ) ( t ) .
Ψ p , q , m , l [ x ( t ) ] = x ( p ) ( t ) x ( q ) ( t ) x ( m ) ( t ) x ( l ) ( t ) .
Ψ p , q , m , l t [ x ( t ) ] = Ψ p + 1 , q , m + 1 , l [ x ( t ) ] + Ψ p , q + 1 , m , l + 1 [ x ( t ) ] ,
Ψ p , q , m , l [ x ̇ ( t ) ] = Ψ p + 1 , q + 1 , m + 1 , l + 1 [ x ( t ) ] ,
x ( p ) ( n T s ) = x ( p 1 ) ( ( n + 1 ) T s ) x ( p 1 ) ( ( n 1 ) T s ) 2 T s ,
H ( z ) = z z 1 2 T s .
X ( p ) ( z ) = X ( z ) H p ( z ) = X ( z ) 1 ( 2 T s ) p k = 0 k = p C p k ( 1 ) p k z k z k p , = 1 ( 2 T s ) p k = 0 k = p C p k ( 1 ) p k z 2 k p .
x ( p ) ( n ) = 1 ( 2 T s ) p k = 0 k = p C p k ( 1 ) p k x ( n + 2 k p ) .
Q k [ x ( n ) ] = 1 ( 2 T s ) k × [ i = 0 i = p j = 0 j = q C p i C q j ( 1 ) k ( i + j ) x ( n + 2 i p ) x ( n + 2 j q ) i = 0 i = m j = 0 j = l C m i C l j ( 1 ) k ( i + j ) x ( n + 2 i m ) x ( n + 2 j l ) ] .
Ψ H k , p , m d [ x ( n ) ] = 1 2 [ x ( n + p ) x ( n + q ) + x ( n p ) x ( n q ) ( x ( n + m ) x ( n + l ) + x ( n m ) x ( n l ) ) ] ,
Ψ H k , p , m d [ x ( n ) ] = 1 2 [ x ( n + m ) x ( n + l ) + x ( n m ) x ( n l ) ( x ( n + p ) x ( n + q ) + x ( n p ) x ( n q ) ) ] .
x ( p ) ( t 0 ) = A Ω p cos ( Ω t 0 + α + π 2 p ) .
Ψ H k , p , m [ x ( t 0 ) ] = A 2 Ω k 2 [ cos ( ( p q ) π 2 ) cos ( ( m l ) π 2 ) ] .
Ψ H k , p , m [ A cos ( Ω t 0 + α ) ] = A 2 Ω k sin ( π 4 c ) sin ( π 4 b ) ,
c = p q + m l = 2 ( m + p k ) ,
b = q p + m l = 2 ( m p ) .
Ψ H k , p , m [ x ̇ ( t 0 ) ] = Ψ H k , p , m [ A Ω cos ( Ω t 0 + α + π 2 ) ] ,
= A 2 Ω k + 2 sin ( π 4 c ) sin ( π 4 b ) .
Ψ H k , p , m [ x ̇ ( t 0 ) ] Ψ H k , p , m [ x ( t 0 ) ] = A 2 Ω k + 2 sin [ ( π 4 ) c ] sin [ ( π 4 ) b ] A 2 Ω k sin [ ( π 4 ) c ] sin [ ( π 4 ) b ] = Ω 2 .
Ψ H 2 k , p 1 , m 1 [ A cos ( Ω t 0 + α ) ] = A 2 Ω 2 k sin ( π 4 c 1 ) sin ( π 4 b 1 ) ,
c 1 = p 1 q 1 + m 1 l 1 = 2 ( m 1 + p 1 2 k ) ,
b 1 = q 1 p 1 + m 1 l 1 = 2 ( m 1 p 1 ) .
Ψ H k , p , m 2 [ x ( t 0 ) ] Ψ H 2 k , p 1 , m 1 [ x ( t 0 ) ] = A 4 Ω 2 k sin 2 [ ( π 4 ) c ] sin 2 [ ( π 4 ) b ] A 2 Ω 2 k sin [ ( π 4 ) c 1 ] sin [ ( π 4 ) b 1 ] , = A 2 sin 2 [ ( π 4 ) c ] sin 2 [ ( π 4 ) b ] sin [ ( π 4 ) c 1 ] sin [ ( π 4 ) b 1 ] .
Ψ H k , p , m 2 [ x ( t 0 ) ] Ψ H 2 k , p 1 , m 1 [ x ( t 0 ) ] A 2 .
ω 2 ( t ) Ψ H k , p , m [ x ̇ ( t ) ] Ψ H k , p , m [ x ( t ) ] ,
a 2 ( t ) Ψ H k , p , m 2 [ x ( t ) ] Ψ H 2 k , p 1 , m 1 [ x ( t ) ] .
x ( n + p ) x ( n + q ) = A 2 2 ( cos ( 2 Ω n + Ω k + 2 θ ) + cos ( p q ) Ω ) .
Ψ H k , p , m d [ x ( n ) ] = A 2 2 ( cos ( p q ) Ω cos ( m l ) Ω ) , = A 2 sin ( b Ω 2 ) sin ( c Ω 2 ) ,
b = 2 ( m p ) , c = 2 ( m + p k ) ,
Ψ H k , p , m d [ x ( n ) ] = A 2 sin ( ( m p ) Ω ) sin ( ( m + p k ) Ω ) .
Ψ H k , p , m d [ x ( n ) ] = A 2 2 ( cos ( m l ) Ω cos ( p q ) Ω ) ,
= A 2 sin ( ( p m ) Ω ) sin ( ( m + p k ) Ω ) .
x 1 ( n ) A [ cos ( Ω n + θ ) cos ( Ω ( n 1 ) + θ ) ] ,
= 2 A sin ( Ω 2 ) sin ( Ω ( n 0.5 ) + θ ) .
Ψ H k , p , m d [ x 1 ( n ) ] = 4 A 2 sin 2 ( Ω 2 ) sin ( b Ω 2 ) sin ( c Ω 2 ) .
Ψ H k , p , m d [ x 1 ( n ) ] A 2 Ω 2 sin ( b Ω 2 ) sin ( c Ω 2 ) .
Ω ̂ 2 = Ψ H k , p , m d [ x 1 ( n ) ] Ψ H k , p , m d [ x ( n ) ] ,
A ̂ = Ψ H k , p , m d [ x ( n ) ] sin [ ( m p ) Ω ] sin [ ( m + p k ) Ω ] .
Φ k m [ x ( n ) ] = x ( n ) x ( n + k ) x ( n m ) x ( n + m + k ) .
Ω ̂ 2 = Φ k m [ x 1 ( n ) ] Φ k m [ x ( n ) ] ,
A ̂ = Φ k m [ x ( n ) ] sin ( m Ω ) sin ( ( m + k ) Ω ) .
sin ( b Ω 2 ) sin ( c Ω 2 ) 1 .
b Ω = ( 2 k 1 + 1 ) π , c Ω = ( 2 k 2 + 1 ) π .
2 ( m p ) Ω = ( 2 k 1 + 1 ) π , 2 ( m + p k ) Ω = ( 2 k 2 + 1 ) π .
m p m + p k = 2 k 1 + 1 2 k 2 + 1 .
A 2 Ψ H k , p , m d [ x ( n ) ] .
T s = 2 k 1 + 1 4 ( m p ) ν 0 = 1 4 ν 0 ,
k = 2 p = 2 , q = 0 , m = 1 , l = 1 ,
k = 4 p = 2 , q = 2 , m = 3 , l = 1 ,
k = 4 p = 3 , q = 1 , m = 4 , l = 0 .
Ψ H k , p , m d [ x ( n ) ] = A 2 sin ( ( 2 1 ) Ω ) sin ( ( 2 + 1 2 ) Ω ) = A 2 sin 2 Ω .
Φ 2 [ x ( n ) ] = x 2 ( n ) x ( n 1 ) x ( n + 1 ) .
Ψ H k , p , m d [ x ( n ) ] = 1 2 ( x 2 ( n 1 ) + x 2 ( n + 1 ) x ( n 2 ) x ( n ) x ( n + 2 ) x ( n ) ) ,
= 1 2 ( Φ 2 [ x ( n 1 ) ] + Φ 2 [ x ( n + 1 ) ] ) .
Φ 2 [ x ( n ) ] = Φ 2 [ x ( n 1 ) ] + Φ 2 [ x ( n + 1 ) ] 2 .
Ψ P H k , τ , p , m [ x ( t ) ] = 1 2 [ x ( p ) ( t + τ 2 ) x ( q ) ( t τ 2 ) + x ( p ) ( t τ 2 ) x ( q ) ( t + τ 2 ) ( x ( m ) ( t + τ 2 ) x ( l ) ( t τ 2 ) + x ( m ) ( t τ 2 ) x ( l ) ( t + τ 2 ) ) ] .
γ x y ( τ ; t ) = x ( t τ 2 ) y ( t + τ 2 ) .
Φ P H k , τ , p , m [ x ( t ) ] = γ x ( p ) x ( q ) ( τ ; t ) + γ x ( p ) x ( q ) ( τ ; t ) γ x ( m ) x ( n ) ( τ ; t ) γ x ( m ) x ( n ) ( τ ; t ) ,
Ψ P H k , τ , p , m [ x ( t 0 ) ] = A 2 Ω τ k cos ( Ω τ τ ) sin ( π 4 c ) sin ( π 4 b ) ,
Ψ P H k , τ , p , m [ x ̇ ( t 0 ) ] = A 2 Ω τ k + 2 cos ( Ω τ τ ) sin ( π 4 c ) sin ( π 4 b ) .
Ψ P H k , τ , p , m [ x ̇ ( t 0 ) ] Ψ P H k , τ , p , m [ x ( t 0 ) ] = A 2 ω τ k + 2 cos ( Ω τ τ ) sin [ ( π 4 ) c ] sin [ ( π 4 ) b ] A 2 Ω τ k cos ( Ω τ τ ) sin [ ( π 4 ) c ] sin [ ( π 4 ) b ] , = Ω τ 2 ,
Ψ P H k , τ , p , m 2 [ x ( t 0 ) ] Ψ P H 2 k , τ , p 1 , m 1 [ x ( t 0 ) ] = A 4 Ω τ 2 k cos 2 ( Ω τ τ ) sin 2 [ ( π 4 ) c ] sin 2 [ ( π 4 ) b ] A 2 Ω τ 2 k cos ( Ω τ τ ) sin [ ( π 4 ) c 1 ] sin [ ( π 4 ) b 1 ] = A 2 cos ( Ω τ τ ) sin 2 [ ( π 4 ) c ] sin 2 [ ( π 4 ) b ] sin [ ( π 4 ) c 1 ] sin [ ( π 4 ) b 1 ] .
τ = n π 2 Ω τ , n integer Ψ P H k , τ , p , m = Ψ P H k , 0 .
ω τ 2 ( t ) Ψ P H k , τ , p , m [ x ̇ ( t ) ] Ψ P H k , τ , p , m [ x ( t ) ] ,
a ̂ τ 2 ( t ) 1 cos ( ω τ ( t ) τ ) Ψ P H k , τ , p , m 2 [ x ( t ) ] Ψ P H 2 k , τ , p 1 , m 1 [ x ( t ) ] .
ω ̂ ( t ) = 1 T 0 T ω τ ( t ) d τ .
a ( t ) ̂ = 1 T 0 T a ̂ τ ( t ) d τ .

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