A variational approach to radar signal analysis

Physical wavelets are acoustic or electromagnetic waves resulting from the emission of a time signal by an acoustic or electromagnetic source moving along an arbitrary trajectory in space. Thus they are localized solutions of the wave equation or Maxwell\´s equations. In this paper we model the emission, reflection, and reception of such wavelets and propose their application to sonar and radar. The concept of ambiguity functions is extended to ambiguity functionals, which apply in a very general setting: The transmitter, the target, and the receiver can all execute arbitrary and independent motions. In the monostatic case, when the transmitter and the receiver are identical and at rest and the target moves with a constant, radial velocity, the ambiguity functional reduces to the wideband ambiguity function, which depends on time delay and Doppler scale factor and amounts to a wavelet transform of the return. In the narrowband approximation, it reduces further to the usual ambiguity function depending on time delay and Doppler frequency shift, which amounts to a windowed Fourier transform of the return. Thus, in the setting of ambiguity functions, time-frequency analysis is a narrowband approximation to wavelet analysis, which is in turn a linear approximation to the new "physical wavelet analysis".