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Abstract
We investigate the theory of the finite Heisenberg-Weyl group in relation to the development of adaptive radar and to the construction of spreading sequences and error-correcting codes in communications. We contend that this group can form the basis for the representation of the radar environment in terms of operators on the space of waveforms. We also demonstrate, following recent developments in the theory of error-correcting codes, that the finite Heisenberg-Weyl groups provide a unified basis for the construction of useful waveforms/sequences for radar, communications, and the theory of error-correcting codes.
Highlights
The continuous Heisenberg-Weyl groups have a long history in physics [1], in the theory of radar detection [2, 3], and in signal processing
Our interest in the finite Heisenberg-Weyl groups stems from an attempt to develop an information theory of radar that is flexible enough to be applied to modern radars
We find that the theory that leads to the Kerdock codes in the multidimensional (p = 2) Heisenberg-Weyl group leads to linear frequency-modulated waveforms when applied to discrete radar
Summary
The continuous Heisenberg-Weyl groups have a long history in physics [1], in the theory of radar detection [2, 3], and in signal processing. In practical terms, the resolution of a radar is finite, by choosing a fine enough discretization in range and Doppler we can treat the radar perfectly well with the onedimensional finite Heisenberg-Weyl group This has a number of advantages, one of which is that the radar environment can be represented by a matrix acting on the space of waveforms. The m-dimensional finite Heisenberg-Weyl group provides a unifying framework for a number of important sequences significant in the construction of phase-coded radar waveforms, in communications as spreading sequences, and in the theory of error-correcting codes. It is maximal isotropic if and only if this inclusion is an equality
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