K. G. Larkin, “Efficient demodulator for bandpass sampled AM signals,” Electron. Lett. 32(2) (1996).

[Crossref]

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).

[Crossref]

A. C. Bovik, P. Maragos, “Conditions for positivity of an energy operator,” IEEE Trans. Signal Process. 42, 469–471 (1994).

[Crossref]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).

[Crossref]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).

[Crossref]

P. Sandoz, G. Tribillon, “Profilometry by zero-order interference fringe identification,” J. Mod. Opt. 40, 1691–1700 (1993).

[Crossref]

H.-H. Liu, P.-H. Cheng, J. Wang, “Spatially coherent white-light interferometer based on a point fluorescent source,” Opt. Lett. 18, 678–680 (1993).

[Crossref]
[PubMed]

P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).

[Crossref]
[PubMed]

P. de Groot, L. Deck, “Three-dimensional imaging by sub-Nyquist sampling of white-light interferograms,” Opt. Lett. 18, 1462–1464 (1993).

[Crossref]
[PubMed]

T. Dresel, G. Häusler, H. Venzke, “Three dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).

[Crossref]
[PubMed]

S. S. C. Chim, G. S. Kino, “Three-dimensional image realization in interference microscopy,” Appl. Opt. 31, 2550–2553 (1992).

[Crossref]
[PubMed]

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

[Crossref]

B. F. Alexander, K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30, 1320–1331 (1991).

[Crossref]

B. L. Danielson, C. Y. Boisrobert, “Absolute optical ranging using low coherence interferometry,” Appl. Opt. 30, 2975–2979 (1991).

[Crossref]
[PubMed]

G. Schulz, K.-E. Elssner, “Errors in phase-measurement interferometry with high numerical apertures,” Appl. Opt. 30, 4500–4506 (1991).

[Crossref]
[PubMed]

S. S. C. Chim, G. S. Kino, “Phase measurements using the Mirau correlation microscope,” Appl. Opt. 30, 2197–2201 (1991).

[Crossref]
[PubMed]

G. S. Kino, S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).

[Crossref]
[PubMed]

B. S. Lee, T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784–3788 (1990).

[Crossref]
[PubMed]

S. S. C. Chim, G. S. Kino, “Correlation microscope,” Opt. Lett. 15, 579–581 (1990).

[Crossref]
[PubMed]

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

[Crossref]

R. P. Loce, R. E. Jodoin, “Sampling theorem for geometric moment determination and its application to a laser beam position detector,” Appl. Opt. 29, 3835–3843 (1990).

[Crossref]
[PubMed]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

[Crossref]

S. C. Pohlig, “Signal duration and the Fourier transform,” Proc. IEEE 68, 629–630 (1980).

[Crossref]

P. Carré, “Installation et utilisation du comparateur photoélectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).

[Crossref]

B. F. Alexander, K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30, 1320–1331 (1991).

[Crossref]

R. E. Bogner, A. G. Constantinides, Introduction to Digital Filtering (Wiley, New York, 1975).

A. C. Bovik, P. Maragos, “Conditions for positivity of an energy operator,” IEEE Trans. Signal Process. 42, 469–471 (1994).

[Crossref]

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

O. Brigham, The Fast Fourier Transform, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988).

D. K. Cohen, P. J. Caber, C. P. Brophy, “Rough surface profiler and method,” U.S. patent5,133,601 (July28, 1992).

P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).

[Crossref]
[PubMed]

D. K. Cohen, P. J. Caber, C. P. Brophy, “Rough surface profiler and method,” U.S. patent5,133,601 (July28, 1992).

P. Carré, “Installation et utilisation du comparateur photoélectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).

[Crossref]

S. S. C. Chim, G. S. Kino, “Three-dimensional image realization in interference microscopy,” Appl. Opt. 31, 2550–2553 (1992).

[Crossref]
[PubMed]

S. S. C. Chim, G. S. Kino, “Phase measurements using the Mirau correlation microscope,” Appl. Opt. 30, 2197–2201 (1991).

[Crossref]
[PubMed]

G. S. Kino, S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).

[Crossref]
[PubMed]

S. S. C. Chim, G. S. Kino, “Correlation microscope,” Opt. Lett. 15, 579–581 (1990).

[Crossref]
[PubMed]

D. K. Cohen, P. J. Caber, C. P. Brophy, “Rough surface profiler and method,” U.S. patent5,133,601 (July28, 1992).

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).

[Crossref]

R. E. Bogner, A. G. Constantinides, Introduction to Digital Filtering (Wiley, New York, 1975).

K. Creath, “Calibration of numerical aperture effects in interferometric microscope objectives,” Appl. Opt. 28, 3333–3338 (1989).

[Crossref]
[PubMed]

There are a number of review papers and book chapters that consider the ever increasing range of phase-shifting algorithms. One of the more recent is K. Creath, “Temporal phase-measurement methods,” in Interferogram Analysis: Digital Processing Techniques for Fringe Pattern Measurement, D. W. Robinson, G. T. Reid (Institute of Physics, Bristol, UK, 1993).

M. Davidson, “Method and apparatus for using a two beam interference microscope for inspection of integrated circuits and the like,” U.S. patent4,818,110 (April4, 1989).

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).

[Crossref]

G. Schulz, K.-E. Elssner, “Errors in phase-measurement interferometry with high numerical apertures,” Appl. Opt. 30, 4500–4506 (1991).

[Crossref]
[PubMed]

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

[Crossref]
[PubMed]

W. H. Press, S. A. Teulolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).

[Crossref]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometer: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).

[Crossref]
[PubMed]

D. A. Zweig, R. E. Hufnagel, “A Hilbert transform algorithm for fringe-pattern analysis,” in Advanced Optical Manufacturing and Testing, L. R. Baker, P. B. Reid, G. M. Sanger, eds., Proc. SPIE1333, 295–302 (1990).

[Crossref]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).

[Crossref]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).

[Crossref]

J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Albuquerque, N.M., 1990 (IEEE, New York, 1990), pp. 381–384.

[Crossref]

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).

[Crossref]

S. S. C. Chim, G. S. Kino, “Three-dimensional image realization in interference microscopy,” Appl. Opt. 31, 2550–2553 (1992).

[Crossref]
[PubMed]

S. S. C. Chim, G. S. Kino, “Phase measurements using the Mirau correlation microscope,” Appl. Opt. 30, 2197–2201 (1991).

[Crossref]
[PubMed]

S. S. C. Chim, G. S. Kino, “Correlation microscope,” Opt. Lett. 15, 579–581 (1990).

[Crossref]
[PubMed]

G. S. Kino, S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).

[Crossref]
[PubMed]

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993).

K. G. Larkin, “Efficient demodulator for bandpass sampled AM signals,” Electron. Lett. 32(2) (1996).

[Crossref]

C. J. R. Sheppard, K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).

[Crossref]
[PubMed]

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).

[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]

A. C. Bovik, P. Maragos, “Conditions for positivity of an energy operator,” IEEE Trans. Signal Process. 42, 469–471 (1994).

[Crossref]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).

[Crossref]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).

[Crossref]

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).

[Crossref]

B. F. Alexander, K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30, 1320–1331 (1991).

[Crossref]

I. Pitas, A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Kluwer, Boston, 1990).

S. C. Pohlig, “Signal duration and the Fourier transform,” Proc. IEEE 68, 629–630 (1980).

[Crossref]

W. H. Press, S. A. Teulolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).

[Crossref]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).

[Crossref]

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).

[Crossref]

B. E. A. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).

[Crossref]

P. Sandoz, G. Tribillon, “Profilometry by zero-order interference fringe identification,” J. Mod. Opt. 40, 1691–1700 (1993).

[Crossref]

F. G. Stremler, Introduction to Communication Systems (Addison-Wesley, Reading, Mass., 1982).

W. H. Press, S. A. Teulolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

P. Sandoz, G. Tribillon, “Profilometry by zero-order interference fringe identification,” J. Mod. Opt. 40, 1691–1700 (1993).

[Crossref]

I. Pitas, A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Kluwer, Boston, 1990).

W. H. Press, S. A. Teulolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

A. D. Whalen, Detection of Signals in Noise (Academic, New York, 1971), p. 200.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

[Crossref]

D. A. Zweig, R. E. Hufnagel, “A Hilbert transform algorithm for fringe-pattern analysis,” in Advanced Optical Manufacturing and Testing, L. R. Baker, P. B. Reid, G. M. Sanger, eds., Proc. SPIE1333, 295–302 (1990).

[Crossref]

P. A. Flourney, R. W. McClure, G. Wyntjes, “White-light interferometric thickness gauge,” Appl. Opt. 11, 1907–1915 (1972).

[Crossref]

T. Dresel, G. Häusler, H. Venzke, “Three dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).

[Crossref]
[PubMed]

B. S. Lee, T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784–3788 (1990).

[Crossref]
[PubMed]

S. S. C. Chim, G. S. Kino, “Phase measurements using the Mirau correlation microscope,” Appl. Opt. 30, 2197–2201 (1991).

[Crossref]
[PubMed]

G. S. Kino, S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).

[Crossref]
[PubMed]

S. S. C. Chim, G. S. Kino, “Three-dimensional image realization in interference microscopy,” Appl. Opt. 31, 2550–2553 (1992).

[Crossref]
[PubMed]

B. L. Danielson, C. Y. Boisrobert, “Absolute optical ranging using low coherence interferometry,” Appl. Opt. 30, 2975–2979 (1991).

[Crossref]
[PubMed]

K. Creath, “Calibration of numerical aperture effects in interferometric microscope objectives,” Appl. Opt. 28, 3333–3338 (1989).

[Crossref]
[PubMed]

G. Schulz, K.-E. Elssner, “Errors in phase-measurement interferometry with high numerical apertures,” Appl. Opt. 30, 4500–4506 (1991).

[Crossref]
[PubMed]

C. J. R. Sheppard, K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).

[Crossref]
[PubMed]

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

[Crossref]
[PubMed]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometer: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).

[Crossref]
[PubMed]

P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).

[Crossref]
[PubMed]

N. Bareket, “Undersampling errors in measuring the moments of images aberrated by turbulence,” Appl. Opt. 18, 3064–3069 (1979).

[Crossref]
[PubMed]

R. P. Loce, R. E. Jodoin, “Sampling theorem for geometric moment determination and its application to a laser beam position detector,” Appl. Opt. 29, 3835–3843 (1990).

[Crossref]
[PubMed]

B. Bushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985).

[Crossref]

K. G. Larkin, “Efficient demodulator for bandpass sampled AM signals,” Electron. Lett. 32(2) (1996).

[Crossref]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).

[Crossref]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).

[Crossref]

A. C. Bovik, P. Maragos, “Conditions for positivity of an energy operator,” IEEE Trans. Signal Process. 42, 469–471 (1994).

[Crossref]

P. Sandoz, G. Tribillon, “Profilometry by zero-order interference fringe identification,” J. Mod. Opt. 40, 1691–1700 (1993).

[Crossref]

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).

[Crossref]

P. Carré, “Installation et utilisation du comparateur photoélectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).

[Crossref]

B. E. A. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).

[Crossref]

B. F. Alexander, K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30, 1320–1331 (1991).

[Crossref]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

[Crossref]

P. de Groot, L. Deck, “Three-dimensional imaging by sub-Nyquist sampling of white-light interferograms,” Opt. Lett. 18, 1462–1464 (1993).

[Crossref]
[PubMed]

S. S. C. Chim, G. S. Kino, “Correlation microscope,” Opt. Lett. 15, 579–581 (1990).

[Crossref]
[PubMed]

H.-H. Liu, P.-H. Cheng, J. Wang, “Spatially coherent white-light interferometer based on a point fluorescent source,” Opt. Lett. 18, 678–680 (1993).

[Crossref]
[PubMed]

S. C. Pohlig, “Signal duration and the Fourier transform,” Proc. IEEE 68, 629–630 (1980).

[Crossref]

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

D. A. Zweig, R. E. Hufnagel, “A Hilbert transform algorithm for fringe-pattern analysis,” in Advanced Optical Manufacturing and Testing, L. R. Baker, P. B. Reid, G. M. Sanger, eds., Proc. SPIE1333, 295–302 (1990).

[Crossref]

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Neat: cleverly effective in character or execution. The Macquarie Dictionary, 2nd ed. (The Macquarie Library, Macquarie University, Sydney, Australia1991).

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).

[Crossref]

M. Davidson, “Method and apparatus for using a two beam interference microscope for inspection of integrated circuits and the like,” U.S. patent4,818,110 (April4, 1989).

Zygo product brochure, “New view 100:3D imaging surface structure analyzer,” Zygo Corporation, Middlefield, Conn. (1993).

F. G. Stremler, Introduction to Communication Systems (Addison-Wesley, Reading, Mass., 1982).

R. E. Bogner, A. G. Constantinides, Introduction to Digital Filtering (Wiley, New York, 1975).

O. Brigham, The Fast Fourier Transform, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993).

Analytically, Eq. (20) is an approximate solution of the linear demodulation problem g(z) = a+ c(z)cos(2πu0z+ α) using five equally spaced measurements to solve for cand having only second-order errors [related to the derivatives (∂c/∂z)2and ∂2c/∂z2]. The other algorithms have various first-order errors.

D. K. Cohen, P. J. Caber, C. P. Brophy, “Rough surface profiler and method,” U.S. patent5,133,601 (July28, 1992).

Closely related to Savitsky–Golay, or digital smoothing polynomial, filters. A LSF over a symmetrical domain causes many off-diagonal elements in the matrix equation to be zero, and hence matrix inversion is trivial. The resulting kernels have even or odd symmetry. See the following reference for more information.

W. H. Press, S. A. Teulolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Typical peak detection algorithms involve a combination of samples in both numerator and denominator of a quotient resembling Eq. (23). Linear combinations are equivalent to correlation or convolution with a kernel function. In the Fourier domain the transformed signal is multiplied by the kernel transform. If the kernel is suitably chosen yet still satisfies the LSF criteria, then zeros (of the transform) can occur at certain frequencies and these frequencies are thus removed from the signal. Essentially filtering and peak detection have been combined into one. Equation (23) resembles the filtered signal quotient analyzed in detail in Refs. 22 and 23.

I. Pitas, A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Kluwer, Boston, 1990).

A. D. Whalen, Detection of Signals in Noise (Academic, New York, 1971), p. 200.

J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Albuquerque, N.M., 1990 (IEEE, New York, 1990), pp. 381–384.

[Crossref]

There are a number of review papers and book chapters that consider the ever increasing range of phase-shifting algorithms. One of the more recent is K. Creath, “Temporal phase-measurement methods,” in Interferogram Analysis: Digital Processing Techniques for Fringe Pattern Measurement, D. W. Robinson, G. T. Reid (Institute of Physics, Bristol, UK, 1993).

The nonlinear filter defined by relation (20a) can be represented by a bandpass filter (equivalent to a finite-difference filter) followed by a quadratic Volterra series filter. The bandpass filter has the same form as that of F1shown in Figs. 2 and 3 when the sampling is the nominal four per period. Interestingly the Hilbert envelope detector can be expressed as a quadratic filter with many terms, but strangely truncation to the first few terms does not give relation (20a).