Abstract
A class of operators on L 2 (R 2) are introduced which, when acting on rank-one tensors, yield ambiguity functions and their generalizations. The L 2 — synthesis problem by functions in the range of these operators is solved. The same approach applied to functions whose Fourier transforms are rank-one tensors yields analogous results for wideba.nd ambiguity functions (wavelet transforms).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Auslander, L., Tolimieri, R., Radar ambiguity functions and group theory, SIAM J. Math. Anal., Vol. 16 No. 3, (1985), pp. 577–601.
Bastians, M. J., The Wigner distribution function applied to optical signals and systems, Optics Comm. No. 25, (1978), pp. 26–30.
Daubechies, I., The wavelet transform, time-frequency localization and signal analysis, to appear in IEEE Trans. Information Theory.
Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. Pure and Applied Math, Vol. 41 (1988), pp. 909–996.
Feig, E., Computational methods with the ambiguity function, IBM RC 13140 (1987).
Feig, E., Estimating interesting portions of ambiguity functions, Workshop on Signal Processing, Institute for Mathematics and its Applications, Minneapolis, Minnesota, (1988).
Feig, E. and Greenleaf, F., Inversion of an integral transform associated with tomography in radar detection, Inverse Problems vol 2, (1986), pp. 405–411.
Feig, E. and Grünbaum, F. A., Tomographic methods in range-Doppler radar, Inverse Problems Vol. 2, (1986), pp. 185–195.
Feig, E. and Micchelli, C. A., Least-squares synthesis by Generalized Ambiguity Functions, IBM RC, (1989).
Grossmann, A. and Morlet, J., Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15, (1984), pp. 723–736.
Grossmann, A., Morlet, J. and Paul, T., Transforms associated to square integrable group representations, I, General results, J. Math. Physics 26, (1985), pp. 2473–2479.
Knight, W. C., Pridham, R. G., and Kay, S. M., Digital signal processing for sonar, Proc. IEEE 69, (1982), pp. 1451–1506.
Meyer, Y., Principe d’incertitude, bases hilbertiennes et algébres d’operateurs, Seminaire Bourbaki 662, (1985–86).
Naparst, H., Radar signal processing for a dense target environment, PhD thesis, U. California, Berkeley, (1988).
Rihaczek, A. W., Principles of High Resolution Radar, McGraw-Hill, (1962).
Schempp, W., Radar reception and nilpotent harmonic analysis I., C. R. Math. Rep. Acad. Sci. Canada 4, (1982), pp. 43–48.
Schempp, W., Radar reception and nilpotent harmonic analysis II., C. R. Math. Rep. Acad. Sci. Canada 4, (1982), pp. 139–144.
Schempp, W., Radar reception and nilpotent harmonic analysis III., C. R. Math. Rep. Acad. Sci. Canada 4, (1982), pp. 219–224.
Schempp, W., Radar reception and nilpotent harmonic analysis IV., C. R. Math. Rep. Acad. Sci. Canada 4, (1982), pp. 287–292.
Schempp, W., On the Wigner quasi-probability distribution function I, C. R. Math Rep. Acad. Sci. Canada 4, (1982), pp. 353–358.
Schempp, W., On the Wigner quasi-probability distribution function II, C. R. Math Rep. Acad. Sci. Canada, 5, (1983) pp. 3–8.
Schempp, W., On the Wigner quasi-probability distribution function III, C. R. Math Rep. Acad. Sci. Canada, 5, (1983) pp. 35–40.
Sussman, S. M., Least squares synthesis of radar ambiguity functions, IRE Trans. Information Th., (1962) pp. 246–254.
Tolimieri, R., Winograd, S., Computing the ambiguity surface, IEEE-ASSP 33, No. 5, (1985), pp. 1239–1245.
Wigner, E. P., On the quantum correction for thermodynamics and equilibrium, Physics Rev. 40, (1932), pp. 749–759.
Wilcox, C. H. The synthesis problem for radar ambiguity functions, MRC Technical Report 157, Mathematics Research Center, U. S. Army, University of Wisconsin (1960).
Wolf, J. D., Lee, G. M., and Suyo, C. E., Radar waveform synthesis by mean-squared optimization techniques, IEEE Trans. Aerospace and Elect. Systems, (1969) pp. 611–619.
Woodward, P. M., Probability and Information Theory with Applications to Radar, New York-London, Pergamon Press (1953).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1989 Birkhäuser Verlag Basel
About this chapter
Cite this chapter
Feig, E., Micchelli, C.A. (1989). L2-Synthesis by Ambiguity Functions. In: Multivariate Approximation Theory IV. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 90. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7298-0_16
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7298-0_16
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-7300-0
Online ISBN: 978-3-0348-7298-0
eBook Packages: Springer Book Archive