Computational Techniques for Prolate Spheroidal Wave Functions in Signal Processing

Journal of Computational Methods in Sciences and Engineering

Volumn 1, Issue 2-3, 2001, Pages 287-313

  Levitina, Tatiana V ^a,b   Brändas, Erkki J ^b  

^a RUSSIAN ACADEMY OF SCIENCES   (Russian Federation)

^b UPPSALA UNIVERSITY   (Sweden)

Author keywords

finite Fourier transform; Prolate spheroidal wave functions; signal processing

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; FUNCTIONS; WAVE FUNCTIONS;

COMPUTATIONAL SCHEMES; COMPUTATIONAL TECHNIQUE; FINITE FOURIER TRANSFORM; GENERAL SITUATION; MATHEMATICAL PHYSICS; PROLATE SPHEROIDAL WAVE FUNCTIONS; SEPARATION OF VARIABLES; ZERO ORDER;

SIGNAL PROCESSING;

EID: 2542506572     PISSN: 14727978     EISSN: None     Source Type: Journal    
DOI: 10.3233/JCM-2001-12-309     Document Type: Article

References (65)

1. P. M. Morse, and H. Feshbach, Methods of Theoretical Physics, Part 1., McGraw-Hill, New York, 1953.
2. Willard Miller, Symmetry and Separation of Variables, Reading, Mass.: Addison-Wesley Pub. Co., Advanced Book Program, 1977.
3. IV. Komarov, LI. Ponomarev, and S.Yu. Slavyanov, Spheroidal and Coulomb Spheroidal Functions [in Russian], Nauka, Moscow, 1976.
4. S. Asano and G. Yamamoto, Light scattering by a spheroidal particle,Appl. opt., 14 (1), pp. 29-49 (1975).
5. S. Asano and M. Sato, Light scattering by randomly oriented spheroidal particles,Appl. Opt., 19 (6), pp. 962-974 (1980).
6. S. Asano, Light scattering by horizontally oriented spheroidal particles, Appl. Opt., 22 (9), pp. 1390-1396 (1983).
7. N. V. Voshchinnikov and V. G. Farafonov, Optical properties of spheroidal particles, Astrophys. Space Sci., 204 (1), pp. 19-86 (1993).
8. N. V. Voshchinnikov, Electromagnetic scattering by homogeneous and coated spheroids: calculations using the separation of variables method, J. Quant. Spectrosc. Radiat. Transfer, 55, pp. 627-636 (1996).
9. V.G. Farafonov, NV. Voshchinnikov, V.V. Somsikov, Light scattering by a core-mantle spheroidal particle, Applied Optics, 35, pp. 5412-5426 (1996).
10. A.A. Abramov, A.L. Dyshko, N.B. Konyukhova, T.V. Levitina, A numerical analytic investigation of the diffraction of a plane acoustic wave by ideal prolate spheroids and triaxial ellipsoids, Comput. Maths Math. Phys., 35 (9), pp. 1103-1123 (1995).
11. A.L. Dyshko, N.B. Konyukhova, A numerical investigation of forced axisymrnetric electric oscillations of an ideally conducting prolate spheroid, Comput. Maths Math. Phys., 35 (5), pp. 593-607 (1995).
12. L. W. Li, T. S. Yeo, P. 5. Kooi and M. S. Leong, Microwave specific attenuation by oblate spheroidal raindrops: An exact analysis of TCS's in terms of spheroidal wave functions, Journal of Electromagnetic Waves and Applications, 12 (6), pp. 709-711 (1998).
13. L.P. Eisenhart, Enumeration of potentials for which one particle Schrödinger equations are separable, Phys. Review, 74, (1), pp. 87 89 (1948).
14. D. I. Abramov, I. V. Komarov, Phase method for scattering by potentials separable in spheroidal coordinates, TMF 22 (2), pp. 253-259 (1975).
15. T. V. Levitina, and E. J. Brändas, Angle Dependent Total Cross Sections and the Optical Theorem, Computersé5 Chemistry, 25(1), pp. 55 67 (2001).
16. 0. Burrau, Berechnung des Energiwertes des Wasserstoff molekul Ions (Ht) in Normaizustand, Det. Kgl. Danske Vid. Selskab., 7, pp. 1-18 (1927).
17. L. Pauling, E.B.Wilson, Introduction to Quantum Mechanics, McGraw-Hill, New York and London, 1935.
18. D. Slepian and HO. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty, I, Bell System Technical Journal, 40, pp. 43-64 (1961).
19. H.J. Landau and HO. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty, II, Bell System Technical Journal, 40, pp. 65-84 (1961).
20. D. Slepian and HO. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty, III: The dimension of essentially time-and band-limited signals, Bell System Technical Journal, 41, pp. 1295-1336 (1962).
21. D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty, IV: Extensions to many dimensions; generalized prolate spheroidal functions, Bell System Technical Journal, 43, pp. 3009-3058 (1964).
22. D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty, V: The discrete case, Bell System Technical Journal, 57. pp. 1371-1430 (1978).
23. DR. Rhodes, The optimum line source for the best mean square approximation to a given radiation pattern, IEEE Trans. AP-11, pp. 440-446 (1963).
24. W.C. Barnes, Object restoration in a diffraction-limited imaging system, J.Opt. Soc. Am. 56. pp. 575-578 (1966).
25. HA. Brown, Effect of truncation on image enhancement by prolate spheroidal functions, J.Opt. Soc. Am. 59(2), pp. 228-229 (1969).
26. R. Mittra, and P.L.Ransom, Comments on "The application of prolate spheroidal wave functions to the detection and estimation of band-limited signals," Proc. of the IEEE Letters, 59, pp. 1625-1626.
27. K.M. Buckley, Spatial/Spectral Filtering with Linearly Constrained Minimum Variance Beamformers, IEEE Trans. on Acoustics, Speech and Signal Processing, 35 (3), pp. 249-266 (1987).
28. J.D. Mathews, J.K. Breakall, and U.K. Karawas, The Discrete Prolate Spheroidal Filter as a Digital Signal Processing Tool, IEEE Trans. on Acoustics, Speech and Signal Processing, 33, pp. 1471-1478 (1985).
29. R. Wilson, and M. Spann, Finite Prolate Spheroidal Sequences and Their Applications II: Image Feature Description and Segmentation, IEEE Trans. on Patern Anal. Machine Intell., 10 (2), pp. 193-203 (1988).
30. Ph. Ciblat, Ph. Loubaton, Second Order Blind Equalization: The Band Limited Case, Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, 6, pp. 3401-3404 (1998).
31. D.J. Thomson, Spectrum estimation and harmonic analysis, Proc. IEEE, 70, pp. 1055-1096 (1982).
32. D.J. Thomson, Time series analysis of Holocene climate data, Phil. Trans. R. Soc. Lond. A, 330, pp. 601-616 (1990).
33. D.J. Thomson, Quadratic-inverse spectrum estimates: applications to paleoclimatology, Phil. Trans. R. Soc. London A, 332, pp. 593-597 (1990).
34. D.B. Percival, and A.T. Walden, Spectral analysis for physical applications Multitaper and conventional univariate techniques, Cambridge University, 1993.
35. M.E. Mann, and J.M. Lees, Robust estimation of background noise and signal detection in climatic time series, Clim. Change., 33, pp. 409-445 (1996).
36. J. W. Pitton, Nonstationary Spectrum Estimation and Time-Frequency Concentration, Proc. ICASSP98 Time, Frequency, and Scale Localization, pp. 2425-2428 (1998).
37. M.R. Wall, D. Neuhauser, Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. I. Theory and application to a quantum-dynamics model,J. Chem. Phys., 102, pp. 8011-8022 (1995).
38. V.A. Mandelshtam, FDM: the filter diagonalization method for data processing in NMR experiments, Progress in Nuclear Magnetic Resonance Spectroscopy, 38. pp. 159-196 (2001).
39. D. Belkié, PA. Dando, H.S. Taylor and J. Main, Decimated Signal Diagonalization for Obtaining the Complete Eigenspectra of Large Matrices, Chem. Phys. Lett., 315, pp. 135-139 (1999).
40. D. Belkié, PA. Dando, J. Main and H.S. Taylor, Decimated Signal Diagonalization for Ion Cyclotron Resonance Mass Spectroscopy, J. Phys. Chem. A, 104. pp. 11685-11692 (2000).
41. FR. Schwab, Optimal gridding of visibility data iii radio interferometry, Indirect Imaging, Proc. of an International Symposium, ed. J. A. Roberts, Cambridge Univ. Press, pp. 333-346 (1984).
42. E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, University Press, 1931.
43. J.A. Stratton, Ph. M. Morse, L.J. Chu, J.D.C.Little, and F.J. Corbato, Spheroidal Wave Functions, John Wiley, New York, 1956.
44. C.Flammer, Spheroidal wave functions, Stanford Univ. Press, Stanford, 1957.
45. J.F. Kaiser, Digital filters, in F. F. Kuo and J. F. Kaiser, eds., System Analysis by Digital Computer, John Wiley, New York, pp. 218-285, 1966.
46. J. F. Kaiser, Nonrecursive digital filter design using the Io SINH window function, Proc. IEEE mt. Symp.Circuits Syst., pp. 20-23, (1974).
47. Zh. Lin, R.W. McCallum, Computation and Performance of the Prolate-Spheroidal Wave Function Window in Spectral Estimation, Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, 5, pp. 2976-2978 (1996).
48. AT. Walden, Accurate approximation of a 0-th order discrete prolate spheroidal sequence for filtering and data tapering. Signal Processing, 18, pp. 341-348 (1989).
49. B. Bell, D.B. Percival, D.B. and AT. Walden, Calculating Thomson's Spectral Multitapers by Inverse Iteration, Journal of Computational and Graphical Statistics, 2 (1), pp. 119-130 (1993).
50. R. Wilson, Finite Prolate Spheroidal Sequences and Their Applications I: Generation and Properties, IEEE Trans. Patern. And Machine Intell., 9 (6), pp. 787-795 (1987).
51. L. Baker, C mathematical function handbook, McGraw-Hill, Inc., New York, 1992.
52. T. A. Beu and R.I.Campeanu, Prolate radial spheroidal wave functions, Comput. Phys. Comm., 30, pp. 177-185 (1983).
53. J. Caldwell, Computation of eigenvalues of spheroidal harmonics using relaxation, J. Phys. A, 21, pp. 3685-3693 (1988).
54. B. J. King, R. V. Baier, and S. Hanish, A Fortran computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives, NRL Report No. 7012, Naval Res. Lab., Washington, D. C., 1970.
55. B. J. King and A. L. van Buren, A Fortran computer program for calculating the prolate and oblate angle functions of the first kind and their first and second derivatives, NRL Report No. 7161, Naval Res. Lab., Washington, D. C., 1970.
56. B. P. Sinha and R. H. MacPhie, On the computation of the prolate spheroidal radial functions of the second kind, J. Math. Phys., 16, pp. 2378-2381 (1975).
57. N. Tugbay and E. Panayirci, An efficient algorithm for generation of prolate spheroidal wave functions, Bull. Tech. Univ. Istanbul, 36. pp. 563-577 (1983).
58. A. L. van Buren, R. V. Baier, S. Hanish, and B. J. King, Calculation of spheroidal wave functions, J. Acoust. Soc. Amer., 51, pp. 414-416 (1972).
59. A. A. Abramov, A. L. Dyshko, N. B. Konyukhova, T. V. Pak, and B. S. Pariiskii, Evaluation of prolate spheroidal function by solving the corresponding differential equations, U.S.S.R. Comput. Math. and Math. Phys., 24 (1), pp. 1-11(1984).
60. A. A. Abramov, A. L. Dyshko, N. B. Konyukhova, and T. V. Levitina, Computation of radial wave functions for spheroids and triaxial ellipsoids by the modified phase function method, U.S.S.R. Comput. Math. and Math. Phys., 31 (2), pp. 25-42 (1991).
61. D.W. Lozier and F.W.J. Olver, in the American Mathematical Society Proceedings of Symposia in Applied Mathematics "Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics", 1994.
62. D.I.Kitoroage, N.B.Konyukhova, B.S. Pariiskii, A modified phase function method for problems concerning bound states of particles [in Russian], Soobshch. Prikl.Mat., CC RAS, Moscow, 1987.
63. N. B. Konyukhova, I. B. Staroverova, Modification of the Phase Method for Singular SelfAdjoint Sturm-Liouville Problems, Comput. Maths. Math. Phys., 37 (10), pp. 1143-1160 (1997).
64. M. V. Fedorjuk, Asymptotic Analysis. Linear Ordinary Differential Equations, Springer Verlag, 1993.
65. A. A. Abramov, On Boundary Conditions in Singular Points of Systems of Linear Ordinary Differential Equations, U.S.S.R. Comp. Maths. Math. Phys. 11, pp. 275-278 (1971).