Discrete radar ambiguity problems

In this paper, we pursue the study of the radar ambiguity problem started in [Ph. Jaming, Phase retrieval techniques for radar ambiguity functions, J. Fourier Anal. Appl. 5 (1999) 313–333; G. Garrigós, Ph. Jaming, J.-B. Poly, Zéros de fonctions holomorphes et contre-exemples en théorie des radars, in: Actes des rencontres d'analyse complexe, Atlantique, Poitiers, 2000, pp. 81–104, available on http://hal.ccsd.cnrs.fr/ccsd-00007482]. More precisely, for a given function u we ask for all functions v (called ambiguity partners) such that the ambiguity functions of u and v have same modulus. In some cases, v may be given by some elementary transformation of u and is then called a trivial partner of u, otherwise we call it a strange partner. Our focus here is on two discrete versions of the problem. For the first one, we restrict the problem to functions u of the Hermite class, u=P(x)e^−x2/2, thus reducing it to an algebraic problem on polynomials. Up to some mild restriction satisfied by quasi-all and almost-all polynomials, we show that such a function has only trivial partners. The second discretization, restricting the problem to pulse type signals, reduces to a combinatorial problem on matrices of a special form. We then exploit this to obtain new examples of functions that have only trivial partners. In particular, we show that most pulse type signals have only trivial partners. Finally, we clarify the notion of trivial partner, showing that most previous counterexamples are still trivial in some restricted sense.