[BGJam] a parâtre dans Applied and Computational Harmonic Analysis. Discrete radar ambiguity problems

Abstract: In this paper, we pursue the study of the radar ambiguity problem started in [Jam4, GJP]. 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 rst one, we restrict the problem to functions u of the Hermite class, u = P (x)ex2=2, thus reducing it to an algebraic problem on polynomials. Up to some mild restriction satised 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, re-duces to a combinatorial problem on matrices of a special form. We then exploit this to obtain new examples of functions that have only trivial part-ners. 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