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Analog radar signal design and digital signal processing —a Heisenberg nilpotent Lie group approach

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Lie Methods in Optics

Part of the book series: Lecture Notes in Physics ((LNP,volume 250))

Abstract

The notions of analog and digital radar auto- and cross-ambiguity functions are on the borderline with mathematics, physics, and electrical engineering. This paper presents the solutions of two problems of analog radar signal design: the synthesis problem (posed in 1953) and the invariance problem for ambiguity surfaces over the symplectic time-frequency plane. Both solutions are achieved via harmonic analysis on the differential principal fiber bundle over the two-dimensional polarized (resp. isotropic) cross-section with structure group isomorphic to the one-dimensional center of the simply connected real Heisenberg nilpotent Lie group. In this way, the linear oscillator representation of the three-dimensional real metaplectic group gives rise to a procedure for generating the energy-preserving linear automorphisms of any given radar ambiguity surface over the time-frequency plane by means of chirp waveforms (linear frequency modulated signals).

In the field of digital signal processing, the Whittaker-Shannon-Kotel'nikov sampling theorem also fits the framework of nilpotent harmonic analysis. The basic idea is to realize the linear Schrödinger representation by the linear lattice representation acting in a complex Hilbert space modeled on the compact Heisenberg nilmanifold, to wit, the differential principal fiber bundle over the two-dimensional compact torus I 2 with structure group isomorphic to the one-dimensional center of the reduced real Heisenberg nilpotent Lie group. In the same vein we look upon the finite Fourier transform, and finally, based on the ambiguity surface conservation principle, the paper deals in a geometric way with the phase discontinuity of Fourier and microwave optics. It follows that analog and digital signal processing, as well as Fourier optics, have a deep geometric common root in nilpotent harmonic analysis. As a mathematical by-product of this research, an identity for Laguerre functions of different orders pops up. Some of its special cases, to wit, a collection of new identities for theta constants have been explicitly calculated and numerically checked.

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Authors
  1. Walter Schempp
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J. Sánchez Mondragón K. B. Wolf

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© 1986 Springer-Verlag

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Schempp, W. (1986). Analog radar signal design and digital signal processing —a Heisenberg nilpotent Lie group approach. In: Sánchez Mondragón, J., Wolf, K.B. (eds) Lie Methods in Optics. Lecture Notes in Physics, vol 250. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16471-5_1

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  • DOI: https://doi.org/10.1007/3-540-16471-5_1

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  • Online ISBN: 978-3-540-39811-0

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