Radar ambiguity functions, the Heisenberg group, and holomorphic theta series

The concept of linear Schrodinger representation of the real Heisenberg nilpotent group and its various realizations is used to link the theory of radar ambiguity functions with nilpotent harmonic analysis. This group-representation theoretic approach allows us to analyze the radar ambi- guity functions simultaneously in time and frequency. Moreover, it allows us to determine the group of all transformations that leave the radar ambiguity surfaces invariant and to specify all admissible envelope functions that belong to radar signals of the same finite energy. In particular, an investigation of the radial, i.e., S0(2, R)-invariant radar ambiguity surfaces, gives rise to an identity for Laguerre-Weber functions of different orders, which implies on its part an identity for holomorphic theta series. 1. The basic group that stands at the crossroads of quantum mechanics and radar analysis is the real Heisenberg nilpotent group A(R). Indeed, the connected, simply connected two-step nilpotent Lie group A(R) forms the group-theoretic embodiment of the Heisenberg canonical commutation relations of quantum mechanics (whence its name) which read at the Lie algebra level as follows: