Abstract
We intent to investigate the links between the representation of groups in the Hilbert space and the analysis of signal waveforms as measurement tools, that is, in the realm of radar or sonar, as means of extracting information (range, velocity, and so on) about a target. The starting point is Woodward’s theory of the radar measurement [1], where he introduced the concept of ambiguïty function; it is now widely recognised that it is essentially a coefficient function for a representation of the group of Weyl-Heisenberg [2]. The first natural extension of this concept deals with the determination of high velocities and accelerations [3]; the extension for high velocities, more specifically Doppler effects not reducible to frequency shifts, leads naturally to the affine group, which has been much studied in the framework of wavelets [4, 5] and of the time-frequency representations of real signals [6, 7]. Here we try, to sketch the most general extensions of this approach in relation with the problem of the choice of an optimal waveform in a given context: is it possible to shape the transmitted signal in order to get the “best” achievable measurement?
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References
Woodward (P.) —Probability and Information Theory with Applications to Radar Artech House, 1980.
Schempp (W.) — Harmonic Analysis on the Heisenberg nilpotent Lie Group with Applications to Signal Theory — Longman, 1986.
Kelly (E.J.), Wishner (R.P.) — Matched-filter Theory for High-Velocity, Accelerating Targets — IEEE Trans. Military Electronics, January 1965.
Grossmann (A), Morlet (J) — Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape — SIAM J. Math. Anal. vol. 15, n°4, july 1984.
Combes (J.M), Grossmann A), Tchamichian (Ph), ed. —Wavelets. Springer Verlag, 1989.
Jourdain (G) — Synthèse de signaux certains dont on connaît la fonction d’ambiguïté de type Woodward ou de type en compression — Ann. Télécomm. 32, 19–23, 1977.
Bertrand (J), Bertrand (P), Ovarlez (J.P.) — Compression d’impulsion en large bande — Douzième colloque GRETSI, 1989.
Glauber (R.) — Coherent and Incoherent States of the Radiation Field. Phys. Rev. vol.131, n°6, pp. 2766–2788, 1963.
Ville (J.) — Théorie et application de la notion de signal analytique. Câble et transmission, vol. 2, n°1, 1948.
Berthon (A.) — Operator Groups and Ambiguity Functions in Signal Processing, in [5],
Kailath (T.) - RKHS Approach to Detection and Estimation Problems Part I: Deterministic Signals in Gaussian Noise. IEEE Trans. IT-17, Nr.5, September 1971.
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© 1991 Birkhäuser Boston
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Berthon, A. (1991). Groups and Signals. In: Bonnard, B., Bride, B., Gauthier, JP., Kupka, I. (eds) Analysis of Controlled Dynamical Systems. Progress in Systems and Control Theory, vol 8. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3214-8_6
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DOI: https://doi.org/10.1007/978-1-4612-3214-8_6
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