Radar Ambiguity Functions, the Heisenberg Group, and Holomorphic Theta Series

The concept of linear Schrödinger 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 ambiguity 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: [X,Y}=T, [X,T] = 0, [F,T]=0. The vectors {X, Y, T} are generators of the real Heisenberg nilpotent Lie algebra n with one-dimensional center RT and the exponential mapping exp: n —► A(R) is a global diffeomorphism that carries Lebesgue measure on n to Haar measure on i(R). On the other hand, let H(f,g;x,y)= f f\t + ±x)g(t \x)e2^dt Jr denote the cross-correlation function of two radar signal pulses which are reflected by a moving target. Here /, g are the complex envelope functions of the signals involved. These envelopes are assumed to belong to the Schwartz space S(R) of rapidly decreasing complex-valued smooth functions on the real line R. Moreover, x denotes the time delay, y the Doppler frequency shift and H(f,g; -, ■) the radar cross-ambiguity function. This function plays an important rôle in radar analysis and design (see Woodward [10]). In particular, H(f,g; -, ■) allows us to formulate Received by the editors June 9, 1983. Presented to the American Mathematical Society at the 797th Meeting, College Park, October 21, 1982, University of Maryland.. 1980 Mathematics Subject Classification. Primary 22E25; Secondary 22E27, 22E30, 33A75, 43A80, 60G35, 81D05, 94A12. 1 Supported by the Deutsche Forschungsgemeinschaft in Bonn. ©1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use