Groups and Signals
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 , 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 . The first natural extension of this concept deals with the determination of high velocities and accelerations ; 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?
KeywordsLinear Frequency Modulation Signal Waveform Ambiguity Function Affine Group Narrowband Signal
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