Abstract

If the best phase measurements are to be achieved, phase-stepping methods need algorithms that are (1) insensitive to the harmonic content of the sampled waveform and (2) insensitive to phase-shift miscalibration. A method is proposed that permits the derivation of algorithms that satisfy both requirements, up to any arbitrary order. It is based on a one-to-one correspondence between an algorithm and a polynomial. Simple rules are given to permit the generation of the polynomial that corresponds to the algorithm having the prescribed properties. These rules deal with the location and multiplicity of the roots of the polynomial. As a consequence, it can be calculated from the expansion of the products of monomials involving the roots. Novel algorithms are proposed, e.g., a six-sample one to eliminate the effects of the second harmonic and a 10-sample one to eliminate the effects of harmonics up to the fourth order. Finally, the general form of a self-calibrating algorithm that is insensitive to harmonics up to an arbitrary order is given.

© 1996 Optical Society of America

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References

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  1. P. Carré, “Installation et utilisation du comparateur photoélec trique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  2. J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1987), p. 414.
  3. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1988), Vol. 26, pp. 350–393.
    [CrossRef]
  4. H. P. Stahl, “Review of phase-measuring interferometry,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 704–719 (1991).
  5. M. Takeda, “Spatial carrier fringe pattern analysis and its applications to precision interferometry and profilometry: an overview,” Indust. Metrol. 1, 79–99 (1990).
    [CrossRef]
  6. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  7. K. Creath, “Phase measurement interferometry: beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 213–220 (1992).
  8. K. G. Larkin, B. F. Oreb, “A new seven-sample symmetrical phase-shifting algorithm,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1755, 2–11 (1992).
  9. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  10. J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [CrossRef]
  11. Y. Surrel, “Phase stepping: a new self-calibrating algo rithm,”Appl. Opt. 32, 3598–3600 (1993).
    [CrossRef] [PubMed]
  12. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,”J. Opt. Soc. Am. A 12, 761–768 (1995).
    [CrossRef]
  13. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,”J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  14. K. A. Stetson, W. R. Brohinsky, “Electrooptic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
    [CrossRef] [PubMed]
  15. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  16. J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” Optical Fabrication and Testing Workshop, Vol. 13 of OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1994), p. PD-4.
  17. P. de Groot, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).

1995

1993

Y. Surrel, “Phase stepping: a new self-calibrating algo rithm,”Appl. Opt. 32, 3598–3600 (1993).
[CrossRef] [PubMed]

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

1992

1990

K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,”J. Opt. Soc. Am. A 7, 542–551 (1990).
[CrossRef]

M. Takeda, “Spatial carrier fringe pattern analysis and its applications to precision interferometry and profilometry: an overview,” Indust. Metrol. 1, 79–99 (1990).
[CrossRef]

1987

1985

1983

1966

P. Carré, “Installation et utilisation du comparateur photoélec trique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Brohinsky, W. R.

Bruning, J. H.

J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1987), p. 414.

Burow, R.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoélec trique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Creath, K.

J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” Optical Fabrication and Testing Workshop, Vol. 13 of OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1994), p. PD-4.

K. Creath, “Phase measurement interferometry: beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 213–220 (1992).

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1988), Vol. 26, pp. 350–393.
[CrossRef]

de Groot, P.

P. de Groot, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).

Eiju, T.

Elssner, K.-E.

Falkenstörfer, O.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Farrant, D. I.

Freischlad, K.

Grzanna, J.

Hariharan, P.

Hibino, K.

Koliopoulos, C. L.

Larkin, K. G.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,”J. Opt. Soc. Am. A 12, 761–768 (1995).
[CrossRef]

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

K. G. Larkin, B. F. Oreb, “A new seven-sample symmetrical phase-shifting algorithm,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1755, 2–11 (1992).

Merkel, K.

Oreb, B. F.

Schmit, J.

J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” Optical Fabrication and Testing Workshop, Vol. 13 of OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1994), p. PD-4.

Schreiber, H.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Schwider, J.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

Spolaczyk, R.

Stahl, H. P.

H. P. Stahl, “Review of phase-measuring interferometry,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 704–719 (1991).

Stetson, K. A.

Streibl, N.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Surrel, Y.

Takeda, M.

M. Takeda, “Spatial carrier fringe pattern analysis and its applications to precision interferometry and profilometry: an overview,” Indust. Metrol. 1, 79–99 (1990).
[CrossRef]

Zöller, A.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Appl. Opt.

Indust. Metrol.

M. Takeda, “Spatial carrier fringe pattern analysis and its applications to precision interferometry and profilometry: an overview,” Indust. Metrol. 1, 79–99 (1990).
[CrossRef]

J. Opt. Soc. Am. A

Metrologia

P. Carré, “Installation et utilisation du comparateur photoélec trique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Eng.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Other

J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” Optical Fabrication and Testing Workshop, Vol. 13 of OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1994), p. PD-4.

P. de Groot, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).

K. Creath, “Phase measurement interferometry: beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 213–220 (1992).

K. G. Larkin, B. F. Oreb, “A new seven-sample symmetrical phase-shifting algorithm,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1755, 2–11 (1992).

J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1987), p. 414.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1988), Vol. 26, pp. 350–393.
[CrossRef]

H. P. Stahl, “Review of phase-measuring interferometry,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 704–719 (1991).

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Figures (5)

Fig. 1
Fig. 1

Characteristic diagram for an arbitrary phase shift δ for the special case of j = 4: The numbers represent harmonics (m). The dots show the locations of the roots of the characteristic polynomial. The coefficients of the polynomial are those of an algorithm that is insensitive, up to the jth order, to the harmonics in the fringe signal. Next to each root is indicated the order of the harmonic whose effects are canceled by the presence of this root in the polynomial. The circle is the unit circle of the complex plane.

Fig. 2
Fig. 2

Same diagram as for Fig. 1, but with a phase shift of δ = π/3. Again, the numbers represent harmonics, and the dots their associated roots. Now, only five roots are required to cancel the same number of harmonics in the fringe signal. In this configuration, some roots cancel the effects of two harmonics simultaneously. This diagram is characteristic of the DFT algorithm.

Fig. 3
Fig. 3

Characteristic diagram for the special case of N = 6 for the (N + 1)-bucket algorithm proposed by Surrel.11 The circled dot indicates a double root, and the presence of a double root provides insensitivity to a phase-shift miscalibration when the signal is a perfect sine.

Fig. 4
Fig. 4

Characteristic diagram for the special case of j = 4 of the minimal algorithm that involves the lowest number of intensity values and is insensitive to the harmonic content of the intensity profile up to the jth order, even in the presence of a phase-shift miscalibration. The circled dots indicate double roots.

Fig. 5
Fig. 5

Characteristic diagram of the WDFT algorithm. The algorithm has the same properties as that associated with the diagram in Fig. 4 (minimal algorithm), but its coefficients have a much simpler analytic form in the general case. The WDFT algorithm corresponds to the discrete Fourier transform of a set of intensities that extend over two signal periods and are windowed by a symmetrical triangle function.

Tables (2)

Tables Icon

Table 1 Sensitivity of the DFT Algorithm to Harmonics as Deduced with Eq. (21) a

Tables Icon

Table 2 Comparison of the Present and Fourier-Transform Approaches to the Cancellation of Algorithmic Sensitivity to Harmonics m

Equations (74)

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I ( φ ) = m = α m exp ( i m φ ) ,
I ( φ ) = I 0 ( 1 + γ cos φ ) = I 0 + I 0 γ 2 exp ( i φ ) + I 0 γ 2 exp [ ( i φ ) ] ,
α 1 = α 1 = I 0 γ 2 .
I ( φ + δ ) = m = [ α m exp ( i m φ ) ] exp ( i m δ ) = m = β m ( φ ) exp ( im δ ) .
φ * = tan 1 [ k = 0 M 1 b k I ( φ + k δ ) k = 0 M 1 a k I ( φ + k δ ) ] .
φ * = arg [ S ( φ ) ] ,
S ( φ ) = k = 0 M 1 c k I ( φ + k δ ) ,
S ( φ ) = m = { α m exp ( i m φ ) k = 0 M 1 c k [ exp ( i m δ ) ] k } = m = { α m exp ( imp φ ) P [ exp ( i m δ ) ] } ,
P ( x ) = k = 0 M 1 c k x k .
S ( φ ) = m = γ m exp ( i m φ ) ,
γ m = α m P [ exp ( i m δ ) ] .
P [ exp ( i m δ ) ] = 0 , m = j , j + 1 , , 1 , 0 , 2 , 3 , , j .
P ( x ) r = j r 1 j ( x ξ r ) ,
δ = 2 π j + 2 = 2 π N .
P ( x ) k = j k 1 j ( x ζ k ) ,
k = j k 1 j ( x ζ k ) = x N 1 x ζ = ζ 1 1 ( ζ 1 x ) N 1 ζ 1 x = ζ 1 [ 1 + ζ 1 x + ζ 2 x 2 + + ζ ( N 1 ) x N 1 ] .
P N ( x ) = 1 + ζ 1 x + ζ 2 x 2 + + ζ ( N 1 ) x N 1 = ζ x N 1 x ζ = k = 0 N 1 ζ k x k .
a k = cos ( 2 π k N ) , b k = sin ( 2 π k N ) ,
γ 1 = α 1 P N [ exp ( i 2 π / N ) ] = α 1 P N ( ζ ) = N α 1 ,
γ m = α m P N ( ζ m ) .
m = N ± 1 + p N ,
δ = δ ( 1 + ɛ ) = δ + δ ɛ ,
γ m = α m P [ exp ( i m δ ) ] .
P exp ( i m δ ) = P [ exp ( i m δ ) ] + ( δ δ ) [ P [ exp ( i m δ ) ] i m × exp ( i m δ ) + O ( ɛ 2 ) = P [ exp ( i m δ ) ] + i m ɛ δ exp ( i m δ ) × P [ exp ( i m δ ) ] + O ( ɛ 2 ) = P [ exp ( i m δ ) ] + i m ɛ δ D P [ exp ( i m δ ) ] + O ( ɛ 2 ) ,
P ( x ) = d d x [ P ( x ) ]
D = x . d d x .
γ m = α m { P [ exp ( i m δ ) ] + i m ɛ δ D P [ exp ( i m δ ) ] } + O ( ɛ 2 ) .
D P ( x ) = k = 0 M 1 k c k x k .
S ( φ ) = γ 1 exp ( i φ ) + y 1 exp ( i φ ) ,
γ 1 = α 1 [ P N ( ζ ) + i ɛ δ D P N ( ζ ) ] = α 1 [ N + i ɛ δ N ( N 1 ) 2 ] , γ 1 = α 1 [ P N ( ζ 1 ) i ɛ δ D P N ( ζ 1 ) ] = α 1 [ ɛ δ N exp ( i 2 π / N ) 2 sin ( 2 π / N ) ] .
P N ( ζ ) = N , D P N ( ζ ) = N ( N 1 ) 2 , P N ( ζ 1 ) = 0 , D P N ( ζ 1 ) = N ζ ζ 1 ζ = Ni exp ( i 2 π / N ) 2 sin ( 2 π / N ) .
S ( φ ) = α 1 N exp ( i φ ) ( 1 + i ɛ δ ( N 1 ) 2 + ɛ δ exp { i [ 2 φ ( 2 π / N ) ] } 2 sin ( 2 π / N ) ) .
Δ φ = φ * φ = π ɛ N { N 1 sin [ 2 φ ( 2 π / N ) ] sin ( 2 π / N ) } ,
tan φ = { [ I ( 0 ) I ( N ) ] / 2 } cot ( 2 π / N ) n = 1 N 1 I ( n ) sin ( 2 π n / N ) { [ I ( 0 ) + I ( N ) ] / 2 } + n = 1 N 1 I ( n ) cos ( 2 π n / N ) ,
P ( x ) = 1 2 [ 1 + i cot ( 2 π / N ) ] + ζ 1 x + ζ 2 x 2 + + ζ ( N 1 ) x N 1 + 1 2 [ 1 i cot ( 2 π / N ) ] x N , = ζ 1 ζ ζ 1 + ζ 1 x + ζ 2 x 2 + + ζ ( N 1 ) x ( N 1 ) + ζ ζ ζ 1 x N ,
P ( x ) = P N ( x ) x ζ 1 ζ ζ 1 ,
d d δ P [ exp ( im δ ) ] = im exp ( i m δ ) P [ exp ( i m δ ) ] = im D P [ exp ( i m δ ) ] .
d 2 d δ 2 = P [ exp ( i m δ ) ] = ( i m D ) 2 P [ exp ( i m δ ) ] ,
d k d δ k = P [ exp ( i m δ ) ] = ( i m D ) k P [ exp ( i m δ ) ] .
γ m = α m [ k = 0 ( i m ɛ δ D ) k P k ! ] [ exp ( i m δ ) ] ,
D P ( x ) = x P ( x ) , D 2 P ( x ) = x [ P ( x ) + x P ( x ) ] = x P ( x ) + x 2 P ( x ) , D 3 P ( x ) = x [ P ( x ) + x P ( x ) + 2 x P ( x ) + x 2 P ( x ) ] , D 4 P ( x ) = x P ( x ) + 3 x 2 P ( x ) + x 3 P ( x ) , ,
tan φ * = 2 ( I 1 I 3 ) I 0 + 2 I 2 I 4 .
P ( x ) = 1 + 2 i x + 2 x 2 2 i x 3 x 4 = ( x 1 ) ( x + 1 ) ( x + i ) 2 .
tan φ * = I 0 + 4 ( I 1 I 3 ) + I 4 I 0 + 2 I 1 6 I 2 + 2 I 3 + I 4 ,
P ( x ) = 1 i + ( 2 + 4 i ) x 6 x 2 + ( 2 4 i ) x 3 + ( 1 + i ) x 4 = ( 1 + i ) ( x 1 ) ( x i ) 3 .
tan φ * = 3 I 1 + I 2 I 4 I 5 I 0 I 1 + I 2 + 2 I 3 + I 4 I 5 I 6 .
P ( x ) = 1 ( 1 i 3 ) x + ( 1 + i 3 ) x 2 + 2 x 3 + ( 1 i 3 ) x 4 ( 1 + i 3 ) x 5 x 6 = ( x 1 ) ( x + 1 ) ( x ζ 1 ) ( x ζ 2 ) ( x ζ 2 ) 2 ,
tan φ * = I 0 7 ( I 2 I 4 ) I 6 4 I 1 8 I 3 + 4 I 5 ,
P ( x ) = i + 4 x 7 i x 2 8 x 3 + 7 i x 4 + 4 x 5 i x 6 = i ( x 1 ) ( x + 1 ) ( x + i ) 4 .
P ( x ) = i ( x 1 ) ( x + 1 ) 2 ( x + i ) 2 = i + ( 2 i ) x + ( 2 + 2 i ) x 2 + ( 2 + 2 i ) x 3 + ( 2 i ) x 4 i x 5 .
α m = 0 , | m | 3 ,
S ( φ ) = α 1 P ( i ) exp ( i φ ) ,
φ * = φ π 4 .
φ * = tan 1 [ I 0 I 1 + 2 I 2 + 2 I 3 I 4 I 5 2 ( I 1 I 2 + I 3 + I 4 ) ] .
φ * = tan 1 [ I 0 3 I 2 + 3 I 4 I 6 2 ( I 1 + 2 I 3 I 5 ) ] ,
P ( x ) = i 2 x + 3 i x 2 + 4 x 3 3 i x 4 2 x 5 + i x 6 = i [ ( x 1 ) ( x + 1 ) ( x + i ) ] 2 ,
S ( φ ) = α 1 P ( i ) exp ( i φ ) = 16 i α 1 exp ( i φ ) ,
φ * = φ π 2 .
P ( x ) = 2 ζ ( x 1 ) ( x ζ 1 ) 2 ( x ζ 2 ) 2 ( x + 1 ) 2 ( x ζ 2 ) 2 ,
φ * = tan 1 [ 3 ( I 0 3 I 1 3 I 2 + I 3 + 6 I 4 + 6 I 5 + I 6 3 I 7 3 I 8 I 9 ) I 0 I 1 7 I 2 11 I 3 6 I 4 + 6 I 5 + 11 I 6 + 7 I 7 + I 8 I 9 ] .
P ( x ) = 2 ( x 1 ) ( x ζ 1 ) 2 ( x ζ 2 ) 2 ( x + 1 ) 2 ( x ζ 2 ) 3 ,
P ( x ) = [ P N ( x ) ] 2 = 1 + 2 ζ 1 x + 3 ζ 2 x 2 + + ( N 1 ) ζ 2 x N 2 + N ζ x N 1 + ( N 1 ) x N + ( N 2 ) ζ 1 x N + 1 + + 2 ζ 3 x 2 N 3 + ζ 2 x 2 N 2 = k = 0 2 N 1 ζ k t k x k ,
S ( φ ) = k = 0 2 N 1 t k ζ k I k = k = 0 2 N 1 ( t k I k ) exp { i [ 2 π ( 2 k ) 2 N ] } .
φ * = tan 1 [ k = 1 N 1 k ( I k 1 I 2 N k 1 ) sin ( 2 π k N ) N I N 1 + k = 1 N 1 k ( I k 1 + I 2 N k 1 ) cos ( 2 π k N ) ] .
φ * = φ 2 π N .
φ * = tan 1 [ ( I 0 I 6 ) 3 ( I 2 I 4 ) 4 I 3 2 ( I 1 + I 5 ) ] ,
F 1 ( m 2 π ) = i F 2 ( m 2 π )
d F 1 d ν ( m 2 π ) = i d F 2 d ν ( m 2 π )
f 1 ( t ) = k = 0 M 1 a k δ ( t t k ) , f 2 ( t ) = k = 0 M 1 b k δ ( t t k ) ,
F 1 ( ν ) = 1 f 1 ( ν ) , F 2 ( ν ) = 1 f 2 ( ν ) ,
f 1 ( φ ) = k = 0 M 1 a k δ ( φ k δ ) , f 2 ( φ ) = k = 0 M 1 b k δ ( φ k δ ) ,
ω = ν 2 π .
g ( φ ) = f 1 ( φ ) + i f 2 ( φ ) .
G ( ν ) = 1 g ( ν ) = F 1 ( ν ) + i F 2 ( ν ) , = k = 0 M 1 ( a k + i b k ) exp ( i 2 π ν k δ ) = P [ exp ( i 2 π ν δ ) ] = P [ exp ( i ω δ ) ] ,

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