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Frames, Irregular Sampling, and a Wavelet Auditory Model

  • Chapter
Nonuniform Sampling

Abstract

A Wavelet Auditory Model (WAM) is constructed in terms of wavelet frames and an irregular sampling algorithm for Fourier frames. Its theoretical effectiveness is demonstrated in the context of speech coding, and its original formulation is found in [8–9]. The presentation of WAM in this chapter emphasizes its underlying mathematical ideas, and, in particular, develops the notions from the theory of frames and irregular sampling that arise naturally in constructing WAM.

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References

  1. J. J. Benedetto. Real Variable and Integration. Stuttgart, Teubner, 1976.

    Book  MATH  Google Scholar 

  2. J. J. Benedetto. Irregular Sampling and Frames. In C. Chui, Ed., Wavelets: A Tutorial in Theory and Applications, Academic Press, Boston, 1992, pp. 445–507.

    Google Scholar 

  3. J. J. Benedetto. Harmonic Analysis and Applications. CRC Press Inc., Boca Raton, FL, 1997.

    Google Scholar 

  4. J. J. Benedetto. Frames, Sampling, and Seizure Prediction. In Ka-Sing Lau, Ed., Advances in Wavelets, Springer Verlag, New York, 1999.

    Google Scholar 

  5. J. J. Benedetto and P. J. S. G. Ferreira, Eds., Modern Sampling Theory: Mathematics and Applications. Birkhäuser, Boston, 2001.

    MATH  Google Scholar 

  6. J. J. Benedetto and M.W. Frazier, Eds., Wavelets: Mathematics and Applications. CRC Press Inc., Boca Raton, FL, 1994.

    MATH  Google Scholar 

  7. J. J. Benedetto and W. Heller. Irregular Sampling and Frames. Mat. Note, 10(Supp. 1):103–125, 1990.

    MathSciNet  Google Scholar 

  8. J. J. Benedetto and A. Teolis. A Wavelet Auditory Model and Data Compression. Applied and Computational Harmonic Analysis, 1:3–28, 1993.

    Article  MATH  Google Scholar 

  9. J. J. Benedetto and A. Teolis. Nonlinear Methods and Apparatus for Coding and Decoding Acoustic Signals with Data Compression and Noised Suppression using Cochlear Filters, Wavelet Analysis, and Irregular Sampling Reconstructions. U. S. Patent 5, 388, 182, 1995.

    Google Scholar 

  10. J. J. Benedetto and O. M. Treiber. Wavelet Frames: Multiresolution Analysis and Extension Principles. In L. Debnath, Ed., Wavelet Transforms and Time-Frequency Signal Analysis, Birkhäuser, Boston, 1999.

    Google Scholar 

  11. J. J. Benedetto and H. C. Wu. A Multidimensional Irregular Sampling Algorithm and Applications. ICASSP, Phoenix, Arizona, 1999.

    Google Scholar 

  12. J. J. Benedetto and H. C. Wu. Fourier Frame Theoretic Multidimensional Irregular Sampling, to appear.

    Google Scholar 

  13. A. Beurling. Collected Works Volume II. Birkhäuser, Boston, 1989.

    Google Scholar 

  14. A. Beurling and P. Malliavin. On Fourier Transforms of Measures with Compact Support. Acta Math., 107:291–309, 1962.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Beurling and P. Malliavin. On the Closure of Characters and the Zeros of Entire Functions. Acta Math., 118:79–93, 1967.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Childers et al. The Past, Present, and Future of Speech Processing. IEEE Signal Processing Magazine, 24-48, May 1998.

    Google Scholar 

  17. O. Christensen, Finite-Dimensional Approximation of the Inverse Frame Operator. Journal of Fourier Analysis and Applications, 6:79–91, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Cohen. Application of an Auditory Model to Speech Recognition. J. Acoust. Soc. Amer., 85:2623–2629, 1989.

    Article  Google Scholar 

  19. D. Colella. Detection of Signals in 1/f-Noise Using Wavelets, MITRE TR, 1993.

    Google Scholar 

  20. I. Daubechies. Ten Lectures on Wavelets. Philadelphia, Pennsylvania: CBMS/NSF Series on Applied Math SIAM Publ., no. 61, 1992.

    Book  MATH  Google Scholar 

  21. I. Daubechies, A. Grossmann, and Y. Meyer. Painless Nonorthogonal Expansions. J. Math. Phsyics, 27:1271–1283, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. J. Duffin and A. C. Schaeffer. A Class of Nonharmonic Fourier Series. Trans. Amer. Math. Soc., 72:341–366, 1952.

    Article  MathSciNet  MATH  Google Scholar 

  23. H. G. Feichtinger and K.-H. Gröchenig. Theory and Practice of Irregular Sampling. In J. J. Benedetto and M W. Frazier, Eds., Wavelets: Mathematics and Applications, CRC Press Inc, Boca Raton, FL, 1994.

    Google Scholar 

  24. S. Greenberg. Acoustic Transduction in the Auditory Periphery. Journal of Phonet., 16:3–18, 1988.

    Google Scholar 

  25. K.-H. Gröchenig. A Discrete Theory of Irregular Sampling. Linear Algebra and its Applications, 193:129–150, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. L. Harrison. Frames and Irregular Sampling from a Computational Perspective. Ph.D. thesis. Dept. of Math., Univ. of Maryland, College Park, Maryland, May 1998.

    Google Scholar 

  27. K. Hofiman. Banach Spaces of Analytic Functions. Prentice Hall, Inc., Englewood Cliffs, NJ, 1962.

    Google Scholar 

  28. S. Jaffard. A Density Criterion for Frames of Complex Exponentials. Michigan Math. J., 38:339–348, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Landau. Necessary Density Condition for Sampling and Interpolation of Certain entire Functions. Acta Math., 117:37–52, 1967.

    Article  MathSciNet  MATH  Google Scholar 

  30. N. Levinson. Gap and Density Theorems. Amer. Math. Soc., Colloquium Publications, Providence, RI, 1940.

    Google Scholar 

  31. B. Logan. Hilbert Transform of a Function having a Bounded Integral and a Bounded Derivative. SIAM J. Math. Analysis, 14:247–248, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Mallat and Z. Zhang. Matching Pursuits with Time-Frequency Dictionaries. IEEE Trans. Signal Processing, 41:3397–3415, 1993.

    Article  MATH  Google Scholar 

  33. F. A. Marvasti. A Unified Approach to Zero-Crossing and Nonuniform Sampling of Single and Multidimensional Systems. Nonuniform, Oak Park, IL 60304, 1987.

    Google Scholar 

  34. I. Morishita and A. Yajima. Analysis and Simulation of Networks of Mutually Inhibiting Neurons. Kybernetik, 11:154–165, 1972.

    Article  MATH  Google Scholar 

  35. R. E. A. C. Paley and N. Wiener. Fourier Transforms in the Complex Domain. Amer. Math. Soc., Colloquium Publications, Providence, RI, 1934.

    Google Scholar 

  36. H. Pollard. Completeness Theorems of Paley-Wiener Type. Annals of Math., 45:738–739, 1944.

    Article  MathSciNet  MATH  Google Scholar 

  37. P. Porcelli. Linear Spaces of Analytic Functions. Rand McNally, Chicago, 1966.

    MATH  Google Scholar 

  38. R. Redheffer. Completeness of Sets of Complex Exponentials. Advances in Math., 24:1–63, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  39. S. Shamma. Speech Processing in the Auditory System. I. The Representation of Speech Sounds in the Responses of the Auditory Nerve. J. Acoust. Soc. Amer., 78:1612–1621, 1985.

    Article  Google Scholar 

  40. S. Shamma. Speech Processing in the Aauditory System. II. Lateral Inhibition and the Central Processing of Speech Evoked Activity in the Auditory Nerve. J. Acoust. Soc. Amer., 78:1622–1632, 1985.

    Article  Google Scholar 

  41. S. Shamma. Auditory Cortical Representation of Complex Acoustic Spectra as Inferred from the Ripple Analysis Method. Network: Computation in Neural Systems, 7:439–476, 1996.

    Article  MATH  Google Scholar 

  42. E. Stein and G. Weiss. Fourier Analysis on Euclidean Spaces. Princeton, Princeton University Press, NJ, 1971.

    MATH  Google Scholar 

  43. T. Strohmer. Computationally Attractive Reconstruction of Band-Limited Images from Irregular Samples. IEEE Trans. on Image Processing, 6:540–548, 1997.

    Article  Google Scholar 

  44. A. Teolis and J. J. Benedetto. Local Frames and Noise Reduction. Signal Processing, 45:369–387, 1995.

    Article  MATH  Google Scholar 

  45. A. Waibel and K. F. Lee, Eds., Readings in Speech Recognition. San Mateo, Morgan Kaufmann Publishers, Inc., CA, 1990.

    Google Scholar 

  46. X. Yang, K. Wang, and S. A. Shamma. Auditory Representations of Acoustic Signals. IEEE Trans. Inform. Theory, 38(2), 824–839, March 1992.

    Article  Google Scholar 

  47. K. Yao and J. Thomas. On Some Stability and Interpolatory Properties of Nonuniform Sampling Expansions. IEEE Trans. Circuit Theory, 14:404–408, 1967.

    Article  Google Scholar 

  48. R. M. Young. An Introduction to Nonharmonic Fourier Series. Academic Press, New York, 1980.

    MATH  Google Scholar 

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Benedetto, J.J., Scott, S. (2001). Frames, Irregular Sampling, and a Wavelet Auditory Model. In: Marvasti, F. (eds) Nonuniform Sampling. Information Technology: Transmission, Processing, and Storage. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1229-5_14

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  • DOI: https://doi.org/10.1007/978-1-4615-1229-5_14

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-5451-2

  • Online ISBN: 978-1-4615-1229-5

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