The Zeros of Entire Functions: Theory and Engineering Applications

Proceedings of the IEEE

Volumn 68, Issue 3, 1980, Pages 308-328

  Requicha, Aristides A G ^a  

^a UNIVERSITY OF ROCHESTER   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

MATHEMATICAL TECHNIQUES;

EID: 0018995309     PISSN: 00189219     EISSN: 15582256     Source Type: Journal    
DOI: 10.1109/PROC.1980.11644     Document Type: Letter

References (77)

1. C. V. Baker, “A technique for the analytical synthesis of shaped beams for arrays,” IEEE Trans. Antennas Propagat., vol. AP-17, pp. 803-805, Nov. 1969.
2. I. Bar-David, “An implicit sampling theorem for bounded bandlimited functions,” Inform. Contr., vol. 24, pp. 36-44, Jan. 1974.
3. “Sample functions of a Gaussian process cannot be recovered from their zero crossings,” IEEE Trans. Inform. Theory, vol. IT-21, pp. 86-87, Jan. 1975.
4. V. Barcilon and G. C. Temes, “Optimum impulse response and the van der Maas function,” IEEE Trans. Circuit Theory, vol. CT-19, pp. 336-342, July 1972.
5. R. D. Barnard, “On the spectral properties of single-sideband angle-modulated signals,” Bell Syst. Tech. J., vol. 43, pp. 2811-2838, Nov. 1964.
6. E. Bedrosian, “The analytic signal representation of modulated waveforms,” Proc. IRE, vol. 50, pp. 2071-2076, Oct. 1962.
7. Yu. K. Belyaev, “Analytic random processes,” Theory Prob. Appl., vol. 4, pp. 402-409, 1959.
8. H. S. Black, Modulation Theory. New York: D. Van Nostrand, 1953.
9. R. P. Boas, Jr., “Remarks on a theorm of B. Lewitan,” Mat. Sb. T.5 (47), no. 1, pp. 185-187, 1939.
10. Entire Functions. New York: Academic Press, 1954.
11. S. Bochner, Lectures on Fourier Integrals. Princeton, NJ: Princeton Univ. Press, 1959.
12. N. Bom and B. W. Conoly, “Zero-crossing shift as a detection method,” J. Acoust. Soc. Amer., vol. 47, pp. 1408-1411, May 1970.
13. F. E. Bond and C. R. Cahn, “On sampling the zeros of bandwidth limited signals,” IRE Trans. Inform. Theory, vol. IT-4, pp. 110-113, Sept. 1958.
14. W. S. Brainerd and L. H. Landweber, Theory of Computation. New York: Wiley, 1974.
15. L. de Branges, Hilbert Spaces of Entire Functions. Englewood Cliffs, NJ: Prentice-Hall, 1968.
16. R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, “The phase problem,” Proc. Roy. Soc, Ser. A, vol. 350, pp. 191-192, 1976.
17. M. L. Cartwright, Integral Functions. Cambridge, England: Cambridge Univ. Press, 1956.
18. J. Clarke, “Steering of zeros in the directional pattern of a linear array,” IEEE Trans. Antennas Propagat., vol. AP-16, pp. 267-268, Mar. 1968.
19. H. Cramer and M. R. Leadbetter, Stationary and Related Stochastic Processes. New York: Wiley, 1968.
20. D. E. N. Davies, “Independent angular steering of each zero of the directional pattern for a linear array,” IEEE Trans. Antennas Propagat., vol. AP-15, Mar. 1967.
21. J. Dugundji, “Envelopes and pre-envelopes of real waveforms,” IRE Trans. Inform. Theory, vol. IT-4, pp. 53-57, Mar. 1958.
22. I. J. Good, “The loss of information due to clipping a waveform,” Inform. Contr., vol. 10, pp. 220-222, May 1967.
23. S. J. Haavik, “The conversion of the zeros of noise,” M.Sc. thesis, Univ. Rochester, Rochester, NY, 1966.
24. A. S. B. Holland, Introduction to the Theory of Entire Functions. New York: Academic Press, 1973.
25. R. F. Hyneman, “A technique for the synthesis of line-source antenna patterns having specified sidelobe behavior,” IEEE Trans. Antennas Propagat., vol. AP-16, pp. 430-435, July 1968.
26. R. F. Hyneman and R. M. Johnson, “A technique for the synthesis of shaped-beam radiation patterns with approximately equal-percentage ripple,” IEEE Trans. Antennas Propagat., vol. AP-15, pp. 736-743, Nov. 1967.
27. A. J. Jerri, “The Shannon sampling theorem-its various extensions and applications: A tutorial review,” Proc. IEEE, vol. 65, pp. 1565-1596, Nov. 1977.
28. L. R. Kahn, “Compatible single sideband,” Proc. IRE, vol. 49, pp. 1503-1527, Oct. 1961.
29. Ya. I. Khurgin and V. P. Yakovlev, “Progress in the Soviet Union on the theory and applications of bandlimited functions,” Proc. IEEE, vol. 65, pp. 1005-1029, July 1977.
30. H. J. Landau, “Sampling, data transmission, and the Nyquist Rate,” Proc. IEEE, vol. 55, pp. 1701-1706, Oct. 1967.
31. R. D. Larsen, “Spectral representations for quantized chemical signals,” P. R. Griffiths, Ed., Transform Techniques in Chemistry. New York: Plenum Press, 1977, pp. 333-353.
32. B. Ya. Levin, Distribution of Zeros of Entire Functions. Providence, RI: American Mathematical Society, 1964.
33. N. Levinson, Gap and Density Theorems. New York: American Mathematical Society, 1940.
34. J. C. R. Licklider and I. Pollack, “Effects of differentiation, integration and infinite peak clipping upon the intelligibility of speech,” J. Acoust. Soc. Amer., vol. 20, pp. 42-51, Jan. 1948.
35. G. B. Lockhart, “Studies of bandwidth reduction techniques applicable to amplitude-modulated signals,” Ph.D. dissertation, Imperial Col. of Science and Technology, London, U.K., 1970.
36. B. F. Logan, Jr., “Information in the zero crossings of bandpass signals,” Bell Syst. Tech. J., vol. 56, pp. 487-510, Apr. 1977.
37. “Theory of analytic modulation systems,” Bell Syst. Tech. J., vol. 57, pp. 491-576, Mar. 1978.
38. B. F. Logan and M. R. Schroeder, “A solution to the problem of compatible single-sideband transmission,” IRE Trans. Inform. Theory, vol. IT-8, pp. S252-S259, Sept. 1962.
39. P. Marcou and J. Daguet, “New methods of speech transmission,” C. Cherry, Ed., Information Theory. London: Butterworth, 1956, pp. 231-244.
40. M. Marden, Geometry of Polynomials. Providence, RI: American Mathematical Society, 1966.
41. A. I. Markushevich, Entire Functions. New York: American Elsevier, 1966.
42. E. Masry, “The recovery of distorted bandlimited stochastic processes,” IEEE Trans. Inform. Theory, vol. IT-19, pp. 398-403, July 1973.
43. L. R. Morris, “The role of zero crossings in speech recognition and processing,” Ph.D. dissertation, Imperial Col. of Science and Technology, London, U.K., 1970.
44. E. L. O'Neill and A. Walther, “The question of phase in image formation,” Opt. Acta, vol. 10, pp. 33-40, Jan. 1963.
45. R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain. New York: Amer. Math. Soc, 1934.
46. A. Pfluger, “Uber ganze funktionen ganzer ordnung,” Comment. Math. Helv., vol. 18, pp. 177-203, 1946.
47. M. Plancherel and G. Polya, “Fonctions entières et intégrales de Fourier multiples,” Comment. Math. Helv., vol. 10, pp. 110-163, 1938.
48. G. Pólya, “Remarks on characteristic functions,” Proc. 1st Berkeley Symp. Mathematical Statistics and Probability, Univ. of California Press, 1949.
49. K. H. Powers, “The compatibility problem in single-sideband transmission,” Proc. IRE, vol. 48, pp. 1431-1435, Aug. 1960.
50. R. G. Pridham and R. A. Mucci, “Digital interpolation beamforming for low-pass and bandpass signals,” Proc. IEEE, vol. 67, pp. 904-919, June 1979.
51. A. A. G. Requicha, “Contributions to a zero-based theory of bandlimited signals,” Ph.D. dissertation, Univ. of Rochester, Rochester, NY, 1970.
52. “Digital filtering techniques for broadband beamforming,” SACLANT ASW Res. Center, La Spezia, Italy, Tech. Memo. 179, July 1972.
53. “Representations of rigid solid objects,” Production Automation Project, Univ. Rochester, Rochester, NY, Tech. Memo. 29, 1979.
54. A. A. G. Requicha and H. B. Voelcker, “Design of nonrecursive filters by specification of frequency-domain zeros,” IEEE Trans. Audio Electroacoust, vol. AU-18, pp. 464-470, Dec. 1970.
55. S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J., vol. 24, pp. 46-156, Jan. 1945.
56. G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, and M. W. L. Wheeler, “The phase problem in scattering phenomena: the zeros of entire functions and their significance,” Proc. Roy. Soc, Series A, vol. 360, pp. 25-45, 1978.
57. R. W. Schafer and L. R. Rabiner, “A digital signal processing approach to interpolation,” Proc. IEEE, vol. 61, pp. 692-702, June 1973.
58. L. Schwartz, Mathematics for the Physical Sciences. Reading, MA: Addison-Wesley, 1966.
59. A. Sekey, “Simulation of real-zero interpolation by the BLODIB compiler,” in Proc. Polytech. Inst Brooklyn Int. Symp. Computer Processing in Communications, pp. 473-489, 1969.
60. “A computer simulation study of real-zero interpolation,” IEEE Trans. Audio Electro-acoust., vol. AU-18, pp. 43-54, Mar. 1970.
61. D. Slepian, “On Bandwidth,” Proc. IEEE, vol. 64, pp. 292-300, Mar. 1976.
62. T. T. Taylor, “Design of line-source antennas for narrow beam-width and low side lobes,” IRE Trans. Antennas Propagat., vol. AP-3, pp. 16-28, Jan. 1955.
63. V. N. Teterev, “The connection between the distribution functions for the zeros in a harmonic signal and its frequence characteristic in the case of extreme amplitude limiting,” Telecommun. Radio Eng., vol. 19, part l, pp. 62-64, July 1965.
64. E. C. Titchmarsh, The Theory of Functions. London, England: Oxford Univ. Press, 2nd ed., 1939.
65. H. B. Voelcker, “Demodulation of single-sideband signals via envelope detection,” IEEE Trans. Commun., vol. COM-14, pp. 22-40, Feb. 1966.
66. “Toward a unified theory of modulation,” Proc. IEEE, vol. 54, pp. 340-383, Mar. 1966, and pp. 735-755, May 1966.
67. “Generation of digital signalling waveforms,” IEEE Trans. Commun., vol. COM-16, pp. 81-93, Feb. 1968.
68. “Method and apparatus for intepolation and conversion of signals specified by real and complex zeros,” U.S. Patent 3510640, May 1970.
69. “Zero-crossing properties of angle-modulated signals,” IEEE Trans. Commun., vol. COM-20, pp. 307-315, June 1972.
70. H. B. Voelcker and A. A. G. Requicha, “Clipping and signal determinism: Two algorithms requiring validation,” IEEE Trans. Commun., vol. COM-21, pp. 738-744, June 1973.
71. “bandlimited random-real-zero signals,” IEEE Trans. Commun., vol. COM-21, pp. 933-936, Aug. 1973.
72. A. Walther, “The question of phase retrieval in optics,” Opt. Acta, vol. 10, pp. 41-49, Jan. 1963.
73. N. Wiener, The Fourier Integral and Certain of its Applications. New York: Dover, 1958.
74. E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurements of the degree of coherence?” Proc. Phys. Soc., vol. 80, part 6, pp. 1269-1272, 1962.
75. J. L. Yen, “On nonuniform sampling of bandwidth-limited signals,” IRE Trans. Circuit Theory, vol. CT-3, pp. 251-257, Dec. 1956.
76. M. Zakai, “bandlimited functions and the sampling theorem,” Inform. Contr., vol. 8, pp. 143-158, 1965.
77. A. Zygmund, Trigonometric Interpolation. Chicago, IL: Univ. of Chicago Press, 1950.