Abstract

Gabor’s expansion of a signal into a discrete set of shifted and modulated versions of an elementary signal is introduced, and its relation to sampling of the sliding-window spectrum is shown. It is shown how Gabor’s expansion coefficients can be found as samples of the sliding-window spectrum, in which the window function is related to the elementary signal in such a way that the set of shifted and modulated elementary signals is biorthonormal to the corresponding set of window functions. The Zak transform is introduced, and its intimate relationship to Gabor’s signal expansion is demonstrated. It is shown how the Zak transform can be helpful in determining the window function that corresponds to a given elementary signal and how it can be used to find Gabor’s expansion coefficients. The continuous-time and the discrete-time cases are considered, and, by sampling the continuous frequency variable that still occurs in the discrete-time case, the discrete Zak transform and the discrete Gabor transform are introduced. It is shown how the discrete transforms enable us to determine Gabor’s expansion coefficients by a fast computer algorithm, which is analogous to the well-known fast Fourier-transform algorithm.

© 1994 Optical Society of America

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References

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  1. D. Gabor, “Theory of communication,” Proc. Inst. Electr. Eng. Part III 93, 429–457 (1946).
  2. C. W. Helstrom, “An expansion of a signal in Gaussian elementary signals,” IEEE Trans. Inf. Theory IT-12, 81–82 (1966).
    [Crossref]
  3. M. J. Bastiaans, “Gabor’s expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
    [Crossref]
  4. M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).
  5. M. J. Bastiaans, “Sampling theorem for the complex spectrogram, and Gabor’s expansion of a signal in Gaussian elementary signals,” Opt. Eng. 20, 594–598 (1981).
  6. A. J. E. M. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–396 (1981).
    [Crossref]
  7. M. J. Bastiaans, “Gabor’s signal expansion and degrees of freedom of a signal,” Opt. Acta 29, 1223–1229 (1982).
    [Crossref]
  8. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), Sec. 8.2.
  9. M. J. Bastiaans, “Gabor’s signal expansion and its relation to sampling of the sliding-window spectrum,” in Advanced Topics in Shannon Sampling and Interpolation Theory, R. J. Marks, ed. (Springer, New York, 1993), pp. 1–37.
    [Crossref]
  10. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1955), Chap. 5, Sec. 4.
  11. V. Bargmann, P. Butera, L. Girardello, J. R. Klauder, “On the completeness of the coherent states,” Rep. Math. Phys. 2, 221–228 (1971).
    [Crossref]
  12. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1927), Chaps. 20 and 21.
  13. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 16.
  14. J. Zak, “Finite translations in solid-state physics,” Phys. Rev. Lett. 19, 1385–1387 (1967).
    [Crossref]
  15. J. Zak, “Dynamics of electrons in solids in external fields,” Phys. Rev. 168, 686–695 (1968).
    [Crossref]
  16. J. Zak, “The kq-representation in the dynamics of electrons in solids,” Solid State Phys. 27, 1–62 (1972).
    [Crossref]
  17. W. Schempp, “Radar ambiguity functions, the Heisenberg group, and holomorphic theta series,” Proc. Am. Math. Soc. 92, 103–110 (1984).
    [Crossref]
  18. M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV Analysis of Operators (Academic, New York, 1978), Chap. XIII.
  19. A. J. E. M. Janssen, “Bargmann transform, Zak transform and coherent states,” J. Math. Phys. 23, 720–731 (1982).
    [Crossref]
  20. A. J. E. M. Janssen, “The Zak transform: a signal transform for sampled time-continuous signals,” Philips J. Res. 43, 23–69 (1988).
  21. H. N. Kritikos, P. T. Farnum, “Approximate evaluation of Gabor expansions,” IEEE Trans. Syst. Man Cybern. SMC-17, 978–981 (1987).
  22. M. J. Bastiaans, “Optical generation of Gabor’s expansion coefficients for rastered signals,” Opt. Acta 29, 1349–1357 (1982).
    [Crossref]
  23. M. J. Bastiaans, “On the sliding-window representation in digital signal processing,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 868–873 (1985).
    [Crossref]
  24. J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–221 (1990).
    [Crossref]
  25. L. Auslander, I. C. Gertner, R. Tolimieri, “The discrete Zak transform application to time-frequency analysis and synthesis of nonstationary signals,” IEEE Trans. Signal Process. 39, 825–835 (1991).
    [Crossref]
  26. S. Qian, K. Chen, S. Li, “Optimal biorthogonal sequence for finite discrete-time Gabor expansion,” Signal Process. 27, 177–185 (1992).
    [Crossref]
  27. S. Qian, D. Chen, “A general solution of biorthogonal analysis window functions for orthogonal-like discrete Gabor transform,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 387–389.
    [Crossref]
  28. R. S. Orr, “Finite discrete Gabor transform relations under periodization and sampling,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 395–398.
    [Crossref]
  29. R. Balart, R. S. Orr, “Computational accuracy and stability issues for the finite, discrete Gabor transform,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 403–406.
    [Crossref]
  30. R. S. Orr, “The order of computation for finite discrete Gabor transforms,” IEEE Trans. Signal Process. 41, 122–130 (1993).
    [Crossref]
  31. S. Qian, D. Chen, “Discrete Gabor transform,” IEEE Trans. Signal Process. 41, 2429–2438 (1993).
    [Crossref]
  32. I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
    [Crossref]

1993 (2)

R. S. Orr, “The order of computation for finite discrete Gabor transforms,” IEEE Trans. Signal Process. 41, 122–130 (1993).
[Crossref]

S. Qian, D. Chen, “Discrete Gabor transform,” IEEE Trans. Signal Process. 41, 2429–2438 (1993).
[Crossref]

1992 (1)

S. Qian, K. Chen, S. Li, “Optimal biorthogonal sequence for finite discrete-time Gabor expansion,” Signal Process. 27, 177–185 (1992).
[Crossref]

1991 (1)

L. Auslander, I. C. Gertner, R. Tolimieri, “The discrete Zak transform application to time-frequency analysis and synthesis of nonstationary signals,” IEEE Trans. Signal Process. 39, 825–835 (1991).
[Crossref]

1990 (2)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[Crossref]

J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–221 (1990).
[Crossref]

1988 (1)

A. J. E. M. Janssen, “The Zak transform: a signal transform for sampled time-continuous signals,” Philips J. Res. 43, 23–69 (1988).

1987 (1)

H. N. Kritikos, P. T. Farnum, “Approximate evaluation of Gabor expansions,” IEEE Trans. Syst. Man Cybern. SMC-17, 978–981 (1987).

1985 (1)

M. J. Bastiaans, “On the sliding-window representation in digital signal processing,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 868–873 (1985).
[Crossref]

1984 (1)

W. Schempp, “Radar ambiguity functions, the Heisenberg group, and holomorphic theta series,” Proc. Am. Math. Soc. 92, 103–110 (1984).
[Crossref]

1982 (3)

A. J. E. M. Janssen, “Bargmann transform, Zak transform and coherent states,” J. Math. Phys. 23, 720–731 (1982).
[Crossref]

M. J. Bastiaans, “Optical generation of Gabor’s expansion coefficients for rastered signals,” Opt. Acta 29, 1349–1357 (1982).
[Crossref]

M. J. Bastiaans, “Gabor’s signal expansion and degrees of freedom of a signal,” Opt. Acta 29, 1223–1229 (1982).
[Crossref]

1981 (2)

M. J. Bastiaans, “Sampling theorem for the complex spectrogram, and Gabor’s expansion of a signal in Gaussian elementary signals,” Opt. Eng. 20, 594–598 (1981).

A. J. E. M. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–396 (1981).
[Crossref]

1980 (2)

M. J. Bastiaans, “Gabor’s expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
[Crossref]

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).

1972 (1)

J. Zak, “The kq-representation in the dynamics of electrons in solids,” Solid State Phys. 27, 1–62 (1972).
[Crossref]

1971 (1)

V. Bargmann, P. Butera, L. Girardello, J. R. Klauder, “On the completeness of the coherent states,” Rep. Math. Phys. 2, 221–228 (1971).
[Crossref]

1968 (1)

J. Zak, “Dynamics of electrons in solids in external fields,” Phys. Rev. 168, 686–695 (1968).
[Crossref]

1967 (1)

J. Zak, “Finite translations in solid-state physics,” Phys. Rev. Lett. 19, 1385–1387 (1967).
[Crossref]

1966 (1)

C. W. Helstrom, “An expansion of a signal in Gaussian elementary signals,” IEEE Trans. Inf. Theory IT-12, 81–82 (1966).
[Crossref]

1946 (1)

D. Gabor, “Theory of communication,” Proc. Inst. Electr. Eng. Part III 93, 429–457 (1946).

Auslander, L.

L. Auslander, I. C. Gertner, R. Tolimieri, “The discrete Zak transform application to time-frequency analysis and synthesis of nonstationary signals,” IEEE Trans. Signal Process. 39, 825–835 (1991).
[Crossref]

Balart, R.

R. Balart, R. S. Orr, “Computational accuracy and stability issues for the finite, discrete Gabor transform,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 403–406.
[Crossref]

Bargmann, V.

V. Bargmann, P. Butera, L. Girardello, J. R. Klauder, “On the completeness of the coherent states,” Rep. Math. Phys. 2, 221–228 (1971).
[Crossref]

Bastiaans, M. J.

M. J. Bastiaans, “On the sliding-window representation in digital signal processing,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 868–873 (1985).
[Crossref]

M. J. Bastiaans, “Optical generation of Gabor’s expansion coefficients for rastered signals,” Opt. Acta 29, 1349–1357 (1982).
[Crossref]

M. J. Bastiaans, “Gabor’s signal expansion and degrees of freedom of a signal,” Opt. Acta 29, 1223–1229 (1982).
[Crossref]

M. J. Bastiaans, “Sampling theorem for the complex spectrogram, and Gabor’s expansion of a signal in Gaussian elementary signals,” Opt. Eng. 20, 594–598 (1981).

M. J. Bastiaans, “Gabor’s expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
[Crossref]

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).

M. J. Bastiaans, “Gabor’s signal expansion and its relation to sampling of the sliding-window spectrum,” in Advanced Topics in Shannon Sampling and Interpolation Theory, R. J. Marks, ed. (Springer, New York, 1993), pp. 1–37.
[Crossref]

Butera, P.

V. Bargmann, P. Butera, L. Girardello, J. R. Klauder, “On the completeness of the coherent states,” Rep. Math. Phys. 2, 221–228 (1971).
[Crossref]

Chen, D.

S. Qian, D. Chen, “Discrete Gabor transform,” IEEE Trans. Signal Process. 41, 2429–2438 (1993).
[Crossref]

S. Qian, D. Chen, “A general solution of biorthogonal analysis window functions for orthogonal-like discrete Gabor transform,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 387–389.
[Crossref]

Chen, K.

S. Qian, K. Chen, S. Li, “Optimal biorthogonal sequence for finite discrete-time Gabor expansion,” Signal Process. 27, 177–185 (1992).
[Crossref]

Daubechies, I.

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[Crossref]

Farnum, P. T.

H. N. Kritikos, P. T. Farnum, “Approximate evaluation of Gabor expansions,” IEEE Trans. Syst. Man Cybern. SMC-17, 978–981 (1987).

Gabor, D.

D. Gabor, “Theory of communication,” Proc. Inst. Electr. Eng. Part III 93, 429–457 (1946).

Gertner, I. C.

L. Auslander, I. C. Gertner, R. Tolimieri, “The discrete Zak transform application to time-frequency analysis and synthesis of nonstationary signals,” IEEE Trans. Signal Process. 39, 825–835 (1991).
[Crossref]

Girardello, L.

V. Bargmann, P. Butera, L. Girardello, J. R. Klauder, “On the completeness of the coherent states,” Rep. Math. Phys. 2, 221–228 (1971).
[Crossref]

Helstrom, C. W.

C. W. Helstrom, “An expansion of a signal in Gaussian elementary signals,” IEEE Trans. Inf. Theory IT-12, 81–82 (1966).
[Crossref]

Janssen, A. J. E. M.

A. J. E. M. Janssen, “The Zak transform: a signal transform for sampled time-continuous signals,” Philips J. Res. 43, 23–69 (1988).

A. J. E. M. Janssen, “Bargmann transform, Zak transform and coherent states,” J. Math. Phys. 23, 720–731 (1982).
[Crossref]

A. J. E. M. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–396 (1981).
[Crossref]

Klauder, J. R.

V. Bargmann, P. Butera, L. Girardello, J. R. Klauder, “On the completeness of the coherent states,” Rep. Math. Phys. 2, 221–228 (1971).
[Crossref]

Kritikos, H. N.

H. N. Kritikos, P. T. Farnum, “Approximate evaluation of Gabor expansions,” IEEE Trans. Syst. Man Cybern. SMC-17, 978–981 (1987).

Li, S.

S. Qian, K. Chen, S. Li, “Optimal biorthogonal sequence for finite discrete-time Gabor expansion,” Signal Process. 27, 177–185 (1992).
[Crossref]

Orr, R. S.

R. S. Orr, “The order of computation for finite discrete Gabor transforms,” IEEE Trans. Signal Process. 41, 122–130 (1993).
[Crossref]

R. Balart, R. S. Orr, “Computational accuracy and stability issues for the finite, discrete Gabor transform,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 403–406.
[Crossref]

R. S. Orr, “Finite discrete Gabor transform relations under periodization and sampling,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 395–398.
[Crossref]

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), Sec. 8.2.

Qian, S.

S. Qian, D. Chen, “Discrete Gabor transform,” IEEE Trans. Signal Process. 41, 2429–2438 (1993).
[Crossref]

S. Qian, K. Chen, S. Li, “Optimal biorthogonal sequence for finite discrete-time Gabor expansion,” Signal Process. 27, 177–185 (1992).
[Crossref]

S. Qian, D. Chen, “A general solution of biorthogonal analysis window functions for orthogonal-like discrete Gabor transform,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 387–389.
[Crossref]

Raz, S.

J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–221 (1990).
[Crossref]

Reed, M.

M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV Analysis of Operators (Academic, New York, 1978), Chap. XIII.

Schempp, W.

W. Schempp, “Radar ambiguity functions, the Heisenberg group, and holomorphic theta series,” Proc. Am. Math. Soc. 92, 103–110 (1984).
[Crossref]

Simon, B.

M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV Analysis of Operators (Academic, New York, 1978), Chap. XIII.

Tolimieri, R.

L. Auslander, I. C. Gertner, R. Tolimieri, “The discrete Zak transform application to time-frequency analysis and synthesis of nonstationary signals,” IEEE Trans. Signal Process. 39, 825–835 (1991).
[Crossref]

von Neumann, J.

J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1955), Chap. 5, Sec. 4.

Watson, G. N.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1927), Chaps. 20 and 21.

Wexler, J.

J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–221 (1990).
[Crossref]

Whittaker, E. T.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1927), Chaps. 20 and 21.

Zak, J.

J. Zak, “The kq-representation in the dynamics of electrons in solids,” Solid State Phys. 27, 1–62 (1972).
[Crossref]

J. Zak, “Dynamics of electrons in solids in external fields,” Phys. Rev. 168, 686–695 (1968).
[Crossref]

J. Zak, “Finite translations in solid-state physics,” Phys. Rev. Lett. 19, 1385–1387 (1967).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process. (1)

M. J. Bastiaans, “On the sliding-window representation in digital signal processing,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 868–873 (1985).
[Crossref]

IEEE Trans. Inf. Theory (2)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[Crossref]

C. W. Helstrom, “An expansion of a signal in Gaussian elementary signals,” IEEE Trans. Inf. Theory IT-12, 81–82 (1966).
[Crossref]

IEEE Trans. Signal Process. (3)

R. S. Orr, “The order of computation for finite discrete Gabor transforms,” IEEE Trans. Signal Process. 41, 122–130 (1993).
[Crossref]

S. Qian, D. Chen, “Discrete Gabor transform,” IEEE Trans. Signal Process. 41, 2429–2438 (1993).
[Crossref]

L. Auslander, I. C. Gertner, R. Tolimieri, “The discrete Zak transform application to time-frequency analysis and synthesis of nonstationary signals,” IEEE Trans. Signal Process. 39, 825–835 (1991).
[Crossref]

IEEE Trans. Syst. Man Cybern. (1)

H. N. Kritikos, P. T. Farnum, “Approximate evaluation of Gabor expansions,” IEEE Trans. Syst. Man Cybern. SMC-17, 978–981 (1987).

J. Math. Anal. Appl. (1)

A. J. E. M. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–396 (1981).
[Crossref]

J. Math. Phys. (1)

A. J. E. M. Janssen, “Bargmann transform, Zak transform and coherent states,” J. Math. Phys. 23, 720–731 (1982).
[Crossref]

Opt. Acta (2)

M. J. Bastiaans, “Optical generation of Gabor’s expansion coefficients for rastered signals,” Opt. Acta 29, 1349–1357 (1982).
[Crossref]

M. J. Bastiaans, “Gabor’s signal expansion and degrees of freedom of a signal,” Opt. Acta 29, 1223–1229 (1982).
[Crossref]

Opt. Eng. (1)

M. J. Bastiaans, “Sampling theorem for the complex spectrogram, and Gabor’s expansion of a signal in Gaussian elementary signals,” Opt. Eng. 20, 594–598 (1981).

Optik (1)

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).

Philips J. Res. (1)

A. J. E. M. Janssen, “The Zak transform: a signal transform for sampled time-continuous signals,” Philips J. Res. 43, 23–69 (1988).

Phys. Rev. (1)

J. Zak, “Dynamics of electrons in solids in external fields,” Phys. Rev. 168, 686–695 (1968).
[Crossref]

Phys. Rev. Lett. (1)

J. Zak, “Finite translations in solid-state physics,” Phys. Rev. Lett. 19, 1385–1387 (1967).
[Crossref]

Proc. Am. Math. Soc. (1)

W. Schempp, “Radar ambiguity functions, the Heisenberg group, and holomorphic theta series,” Proc. Am. Math. Soc. 92, 103–110 (1984).
[Crossref]

Proc. IEEE (1)

M. J. Bastiaans, “Gabor’s expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
[Crossref]

Proc. Inst. Electr. Eng. Part III (1)

D. Gabor, “Theory of communication,” Proc. Inst. Electr. Eng. Part III 93, 429–457 (1946).

Rep. Math. Phys. (1)

V. Bargmann, P. Butera, L. Girardello, J. R. Klauder, “On the completeness of the coherent states,” Rep. Math. Phys. 2, 221–228 (1971).
[Crossref]

Signal Process. (2)

S. Qian, K. Chen, S. Li, “Optimal biorthogonal sequence for finite discrete-time Gabor expansion,” Signal Process. 27, 177–185 (1992).
[Crossref]

J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–221 (1990).
[Crossref]

Solid State Phys. (1)

J. Zak, “The kq-representation in the dynamics of electrons in solids,” Solid State Phys. 27, 1–62 (1972).
[Crossref]

Other (9)

M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV Analysis of Operators (Academic, New York, 1978), Chap. XIII.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1927), Chaps. 20 and 21.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 16.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), Sec. 8.2.

M. J. Bastiaans, “Gabor’s signal expansion and its relation to sampling of the sliding-window spectrum,” in Advanced Topics in Shannon Sampling and Interpolation Theory, R. J. Marks, ed. (Springer, New York, 1993), pp. 1–37.
[Crossref]

J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1955), Chap. 5, Sec. 4.

S. Qian, D. Chen, “A general solution of biorthogonal analysis window functions for orthogonal-like discrete Gabor transform,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 387–389.
[Crossref]

R. S. Orr, “Finite discrete Gabor transform relations under periodization and sampling,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 395–398.
[Crossref]

R. Balart, R. S. Orr, “Computational accuracy and stability issues for the finite, discrete Gabor transform,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 403–406.
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

Fourier transform d ¯ (t, ω)/T in the case of a Gaussian elementary signal.

Fig. 2
Fig. 2

Zak transform g ˜ (t, ω) in the case of a Gaussian elementary signal.

Fig. 3
Fig. 3

Gaussian elementary signal g(t) (dashed curve) and its corresponding window function Tw(t) (solid curve).

Equations (133)

Equations on this page are rendered with MathJax. Learn more.

φ ¯ ( ω ) = φ ( t ) exp ( - j ω t ) d t ;
φ ( ω ) = 1 2 π φ ¯ ( ω ) exp ( j ω t ) d ω .
g ( t ) = 2 1 / 4 exp [ - π ( t / T ) 2 ] ,
g m k ( t ) = g ( t - m T ) exp ( j k Ω t ) ,
φ ( t ) = m k a m k g m k ( t )
φ ¯ ( ω ) = m k a m k g ¯ m k ( ω ) ,
g ¯ m k ( ω ) = g ¯ ( ω - k Ω ) exp ( - j m ω T ) .
g n l * ( t ) g m k ( t ) d t = d n - m , l - k = d m - n , k - l * .
d m k = T ( - 1 ) m k exp [ - 1 2 π ( m 2 + k 2 ) ] ,
a m k = φ ( t ) w m k * ( t ) d t ,
w m k ( t ) = w ( t - m T ) exp ( j k Ω t ) .
w n l * ( t ) g m k ( t ) d t = δ n - m δ l - k ,
m k w m k * ( t 1 ) g m k ( t 2 ) = δ ( t 1 - t 2 ) ;
w ( t ) = m k c m k g m k ( t ) ,
δ m δ k = w * ( t ) g m k ( t ) d t ,
δ m δ k = [ n l c n l * g n l * ( t ) ] g m k ( t ) d t .
δ m δ k = n l c n l * g n l * ( t ) g m k ( t ) d t
δ m δ k = n l c n l * d n - m , l - k = n l c n l * d m - n , k - l * .
n l c n l d m - n , k - l = δ m δ k ,
d ¯ ( t , ω ) = m k d m k exp [ - j ( m ω T - k Ω t ) ]
d m k = 1 2 π T Ω d ¯ ( t , ω ) exp [ j ( m ω T - k Ω t ) ] d t d ω ,
c ¯ ( t , ω ) d ¯ ( t , ω ) = 1.
1 T d ¯ ( t , ω ) = θ 3 ( Ω t ) θ 3 ( ω T ) + θ 3 ( Ω t ) θ 2 ( ω T ) + θ 2 ( Ω t ) θ 3 ( ω T ) - θ 2 ( Ω t ) θ 2 ( ω T ) ,
θ 3 ( x ) = θ 3 [ x ; exp ( - 2 π ) ] = m exp ( - 2 π m 2 ) exp ( j 2 m x ) ,
θ 2 ( x ) = θ 2 [ x ; exp ( - 2 π ) ] = m exp [ - 2 π ( m + 1 2 ) 2 ] × exp [ j ( 2 m + 1 ) x ] .
1 T d ¯ ( 1 2 T + m T , 1 2 Ω + k Ω ) = θ 3 2 ( 0 ) [ 1 - 2 θ 2 ( 0 ) θ 3 ( 0 ) - θ 2 2 ( 0 ) θ 3 2 ( 0 ) ] .
z ¯ ( t , ω ) = m k δ ( t - 1 2 T - m T ) 2 π δ ( ω - 1 2 Ω - k Ω ) z .
z m k = ( - 1 ) m + k z ,
φ ˜ ( t , ω ) = m φ ( t + m T ) exp ( - j m ω T ) ;
T φ ˜ ( t , ω ) = exp ( j ω t ) k φ ¯ ( ω + k ω ) exp ( j k Ω t ) .
T φ ( t + m T ) exp ( - j m ω T ) = exp ( j ω t ) k φ ¯ ( ω + k Ω ) exp ( j k Ω t )
φ ˜ ( t + m T , ω + k Ω ) = φ ˜ ( t , ω ) exp ( j m ω T ) .
φ ( t + m T ) = 1 Ω Ω φ ˜ ( t , ω ) exp ( j m ω T ) d ω ,
1 2 π T Ω φ ˜ ( t , ω ) 2 d t d ω = 1 T φ ( t ) 2 d t .
δ m δ k = g ( τ ) w m k * ( τ ) d τ ,
1 = m k [ g ( τ ) w * ( τ - m T ) exp ( - j k Ω τ ) d τ ] × exp [ - j ( m ω T - k Ω t ) ] .
1 = m ( g ( τ ) w * ( τ - m T ) { k exp [ - j k Ω ( τ - t ) ] } d τ ) × exp ( - j m ω T )
1 = m { g ( τ ) w * ( τ - m T ) [ T n δ ( τ - t - n T ) ] d τ } × exp ( - j m ω T ) .
1 = T m n [ g ( τ ) w * ( τ - m T ) δ ( τ - t - n T ) d τ ] × exp ( - j m ω T ) ,
1 = T m n g ( t + n T ) w * ( t + [ n - m ] T ) exp ( - j m ω T ) .
1 = T n g ( t + n T ) exp ( - j n ω T ) × { m w * ( t + [ n - m ] T ) exp [ j ( n - m ) ω T ] } = T [ n g ( t + n T ) exp ( - j n ω T ) ] × [ m w ( t + m T ) exp ( - j m ω T ) ] * ,
T g ˜ ( t , ω ) w ˜ * ( t , ω ) = 1.
g ˜ ( t , ω ) = 2 1 / 4 exp [ - π ( t / T ) 2 ] θ 3 [ π ζ * ; exp ( - π ) ] ,
θ 3 [ z ; exp ( - π ) ] = m exp ( - π m 2 ) exp ( j 2 m z )
T w ˜ ( t , ω ) = 1 g ˜ * ( t , ω ) = 2 - 1 / 4 exp [ π ( t / T ) 2 ] 1 θ 3 [ π ζ ; exp ( - π ) ] ,
1 θ 3 [ π ζ ; exp ( - π ) ] = ( K 0 π ) 3 / 2 [ c 0 + 2 m = 1 ( - 1 ) m c m cos ( 2 π m ζ ) ] ,
c m = n = 0 ( - 1 ) n exp [ - π ( n + 1 2 ) ( 2 m + n + 1 2 ) ]
T w ( t + m T ) = 2 - 1 / 4 exp [ π ( t / T ) 2 ] ( K 0 π ) - 3 / 2 ( - 1 ) m c m × exp [ 2 π m ( t / T ) ] ,
T w ( t ) = 2 - 1 / 4 exp [ π ( t / T ) 2 ] ( K 0 π ) - 3 / 2 p + 1 2 | t T | ( - 1 ) p × exp [ - π ( p + 1 2 ) 2 ] .
T w ( t ) = 2 - 1 / 4 ( K 0 π ) - 3 / 2 ( - 1 ) m exp { π [ ( t T ) 2 - ( m + 1 2 ) 2 ] } × p = m ( - 1 ) p - m exp { - π [ ( p + 1 2 ) 2 - ( m + 1 2 ) 2 ] } ,
T w ( t ) 2 - 1 / 4 ( K 0 π ) - 3 / 2 ( - 1 ) m × exp { π [ ( t T ) 2 - ( m + 1 2 ) 2 ] } ,
T w ( ± [ r + 1 2 ] T ) 2 - 1 / 4 ( K 0 π ) - 3 / 2 ( - 1 ) r ,
z ˜ ( t , ω ) = m k ( - 1 ) m δ ( t - 1 2 T - m T ) × 2 π δ ( ω - 1 2 Ω - k Ω ) z ,
z ( t ) = z T m ( - 1 ) m δ ( t - 1 2 T - m T )
1 T w ( t ) 2 d t = 1 2 π T Ω w ˜ ( t , ω ) 2 d t d ω 1 2 π T Ω 1 g ˜ ( t , ω ) 2 d t d ω .
a m k = φ ( t ) w m k * ( t ) d t ;
φ ( t ) = m k a m k g m k ( t ) ;
a ¯ ( t , ω ) = m k a m k exp [ - j ( m ω T - k Ω t ) ] .
a ¯ ( t , ω ) = m k [ φ ( τ ) w * ( τ - m T ) exp ( - j k Ω τ ) d τ ] × exp [ - j ( m ω T - k Ω t ) ] ,
a ¯ ( t , ω ) = m ( φ ( τ ) w * ( τ - m T ) { k exp [ - j k Ω ( τ - t ) ] } d τ ) × exp ( - j m ω T ) .
a ¯ ( t , ω ) = m { φ ( τ ) w * ( τ - m T ) [ T k δ ( τ - t - k T ) ] d τ } × exp ( - j m ω T )
a ¯ ( t , ω ) = T m k [ φ ( τ ) w * ( τ - m T ) δ ( τ - t - k T ) d τ ] × exp ( - j m ω T ) .
a ¯ ( t , ω ) = T m k φ ( t + k T ) w * ( t + [ k - m ] T ) × exp ( - j m ω T ) ,
a ¯ ( t , ω ) = T k φ ( t + k T ) exp ( - j k ω T ) × { m w ( t + [ k - m ] T ) exp [ - j ( k - m ) ω T ] } * = T [ k φ ( t + k T ) exp ( - j k ω T ) ] × [ m w ( t + m T ) exp ( - j m ω T ) ] * .
a ¯ ( t , ω ) = T φ ˜ ( t , ω ) w ˜ * ( t , ω ) .
φ ˜ ( t , ω ) = n [ m k a m k g ( t + n T - m T ) exp ( j k Ω t ) ] × exp ( - j n ω T ) .
φ ˜ ( t , ω ) = m k a m k exp [ - j ( m ω T - k Ω t ) ] × { n g ( t + [ n - m ] T ) exp [ - j ( n - m ) ω T ] } = { m k a m k exp [ - j ( m ω T - k Ω t ) ] } × [ n g ( t + n T ) exp ( - j n ω T ) ] ,
φ ˜ ( t , ω ) = a ¯ ( t , ω ) g ˜ ( t , ω ) .
z ¯ ( t , ω ) g ˜ ( t , ω ) = 0.
m k z m k g m k ( t ) = 0 ,
φ ¯ ( θ ) = n φ [ n ] exp ( - j θ n ) ;
φ [ n ] = 1 2 π 2 π φ ¯ ( θ ) exp ( j θ n ) d θ ,
a m k = n φ [ n ] w m k * [ n ] ,
w m k [ n ] = w [ n - m N ] exp ( j k Θ n )
φ [ n ] = m k = N a m k g m k [ n ] ,
a ¯ ( n , θ ) = m k = N a m k exp [ - j ( m θ N - k Θ n ) ] ,
a m k = 1 2 π n = N Θ a ¯ ( n , θ ) exp [ j ( m θ N - k Θ n ) ] d θ ,
φ ˜ ( n , θ ) = m φ [ n + m N ] exp ( - j m θ N ) .
φ ˜ ( n + m N , θ + k Θ ) = φ ˜ ( n , θ ) exp ( j m θ N ) .
φ [ n + m N ] = 1 Θ Θ φ ˜ ( n , θ ) exp ( j m θ N ) d θ ,
a ¯ ( n , θ ) = N φ ˜ ( n , θ ) w ˜ * ( n , θ ) ,
φ ˜ ( n , θ ) = a ¯ ( n , θ ) g ˜ ( n , θ ) .
g [ n ] = g ( n T / N ) = 2 1 / 4 exp [ - π ( n / N ) 2 ]             ( N odd ) ,
g [ n ] = g ( [ n + 1 2 ] T / N ) = 2 1 / 4 exp { - π [ ( n + 1 2 ) / N ] 2 }             ( N even ) .
g ˜ ( n , θ ) = 2 1 / 4 exp [ - π ( n / N ) 2 ] θ 3 [ π ζ * ; exp ( - π ) ]             ( with ζ = θ Θ + j n N ) ,
g ˜ ( n , θ ) = 2 1 / 4 exp { - π [ ( n + 1 2 ) / N ] 2 } θ 3 [ π ζ * ; exp ( - π ) ]             ( with ζ = θ Θ + j n + 1 2 N )
N w [ n ] = 2 - 1 / 4 exp [ π ( n / N ) 2 ] ( K 0 π ) - 3 / 2 × p + 1 2 | n N | ( - 1 ) p exp [ - π ( p + 1 2 ) 2 ] ,
N w [ n ] = 2 - 1 / 4 ( K 0 π ) - 3 / 2 ( - 1 ) m × exp { π [ ( n N ) 2 - ( m + 1 2 ) 2 ] } p = m ( - 1 ) p - m × exp { - π [ ( p + 1 2 ) 2 - ( m + 1 2 ) 2 ] } ,
N w [ n ] 2 - 1 / 4 ( K 0 π ) - 3 / 2 ( - 1 ) m × exp { π [ ( n N ) 2 - ( m + 1 2 ) 2 ] } ,
N w [ ± { ( r + 1 2 ) N - 1 2 } ] 2 - 1 / 4 ( K 0 π ) - 3 / 2 ( - 1 ) r × exp { π [ ( r + 1 2 - 1 2 N ) 2 - ( r + 1 2 ) 2 ] } = 2 - 1 / 4 ( K 0 π ) - 3 / 2 ( - 1 ) r exp { π [ ( 1 2 N ) 2 - ( r + 1 2 ) / N ] } 2 - 1 / 4 ( K 0 π ) - 3 / 2 ( - 1 ) r exp [ - π ( r / N ) ] .
a ¯ ( n , θ ) = m k = N a m k exp [ - j ( m θ N - k Θ n ) ] ,
a ¯ [ n , l ] = a ¯ ( n , Θ M l ) = m k = N a m k × exp { - j [ m ( 2 π / M ) l - k ( 2 π / N ) n ] } ;
A m k = 1 M N n = N l = M a ¯ [ n , l ] × exp { j [ m ( 2 π / M ) l - k ( 2 π / N ) n ] } .
A m k = r a m + r M , k ,
a ¯ [ n , l ] = m = M k = N A m k × exp { - j [ m ( 2 π / M ) l - k ( 2 π / N ) n ] } .
φ ˜ ( n , θ ) = m φ [ n + m N ] exp ( - j m θ N ) ,
φ ˜ [ n , l ] = φ ˜ ( n , Θ M l ) = m φ [ n + m N ] exp [ - j m ( 2 π / M ) l ] ;
ϕ [ n + m N ] = 1 M l = M φ ˜ [ n , l ] exp [ j m ( 2 π / M ) l ] .
ϕ [ n ] = r φ [ n + r M N ]
φ ˜ [ n , l ] = m = M ϕ [ n + m N ] exp [ - j m ( 2 π / M ) l ] .
a ¯ [ n , l ] = N φ ˜ [ n , l ] w ˜ * [ n , l ] .
A m k = 1 M N n = N l = M N φ ˜ [ n , l ] w ˜ * [ n , l ] × exp { j [ m ( 2 π / M ) l - k ( 2 π / N ) n ] } ,
A m k = 1 M n = N l = M { r = M ϕ [ n + r N ] exp [ - j r ( 2 π / M ) l } × w ˜ * [ n , l ] exp { j [ m ( 2 π / M ) l - k ( 2 π / N ) n ] } .
A m k = n = N r = M ϕ [ n + r N ] × { 1 M l = M w ˜ [ n , l ] exp [ j ( r - m ) ( 2 π / M ) l ] } * × exp [ - j k ( 2 π / N ) n ,
A m k = n = N r = M ϕ [ n + r N ] W * [ n + r N - m N ] × exp [ - j k ( 2 π / N ) n ] .
W m k [ n ] = W [ n - m N ] exp ( j k Θ n ) = W [ n - m N ] exp [ j k ( 2 π / N ) n ] ,
A m k = n = N r = M ϕ [ n + r N ] W m k * [ n + r N ] ,
A m k = n = M N ϕ [ n ] W m k * [ n ] .
φ ˜ [ n , l ] = a ¯ [ n , l ] g ˜ [ n , l ] .
ϕ [ n ] = 1 M l = M a ¯ [ n , l ] g ˜ [ n , l ] ,
ϕ [ n ] = 1 M l = M ( m = M k = N A m k exp { - j [ m ( 2 π / M ) l - k ( 2 π / N ) n ] } ) g ˜ [ n , l ] .
ϕ [ n ] = m = M k = N A m k { 1 M l = M g ˜ [ n , l ] exp [ - j m ( 2 π / M ) l ] } × exp [ j k ( 2 π / N ) n ]
ϕ [ n ] = m = M k = N A m k G [ n - m N ] exp [ j k ( 2 π / N ) n ] .
G m k [ n ] = G [ n - m N ] exp [ j k ( 2 π / N ) n ] ,
ϕ [ n ] = m = M k = N A m k G m k [ n ] .
a m k = n φ [ n ] w m k * [ n ]
A m k = n = M N ϕ [ n ] W m k * [ n ] .
φ [ n ] = r φ r [ n ] ,
φ r [ n ] = { φ [ n ] for 0 n - r K K - 1 0 elsewhere .
a m k = n ( r φ r [ n ] ) w m k * [ n ] = r ( r φ r [ n ] w m k * [ n ] ) = r a m k r ,
a m k r = n φ r [ n ] w m k * [ n ]
T w ˜ * ( t , ω ) g ˜ ( t , ω ) = 1 ,
m k w m k * ( t 1 ) g m k ( t 2 ) = δ ( t 2 - t 1 ) .
m k w m k * ( t 1 ) g m k ( t 2 ) = m k w * ( t 1 - m T ) exp ( - j k Ω t 1 ) × g ( t 2 - m T ) exp ( j k Ω t 2 ) .
m k [ 1 Ω Ω w ˜ ( t 1 , ω ) exp ( - j m ω T ) d ω ] * × exp ( - j k Ω t 1 ) g ( t 2 - m T ) exp ( j k Ω t 2 ) .
k exp [ j k Ω ( t 2 - t 1 ) ] 1 Ω Ω w ˜ * ( t 1 , ω ) × [ m g ( t 2 - m T ) exp ( j m ω T ) ] d ω
k exp [ j k Ω ( t 2 - t 1 ) ] 1 Ω Ω w ˜ * ( t 1 , ω ) g ˜ ( t 2 , ω ) d ω .
T n δ ( t 2 - t 1 - n T ) 1 Ω Ω w ˜ * ( t 1 , ω ) g ˜ ( t 2 , ω ) d ω
T n δ ( t 2 - t 1 - n T ) 1 Ω Ω w ˜ * ( t 1 , ω ) g ˜ ( t 1 + n T , ω ) d ω .
n δ ( t 2 - t 1 - n T ) 1 Ω Ω T w ˜ * ( t 1 , ω ) g ˜ ( t 1 , ω ) exp ( j n ω T ) d ω ,
n δ ( t 2 - t 1 - n T ) 1 Ω Ω exp ( j n ω T ) d ω .
n δ ( t 2 - t 1 - n T ) δ n
δ ( t 2 - t 1 ) .

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