Discrete radar ambiguity problems

In this paper, we pursue the study of the radar ambiguity problem started in \cite{Ja,GJP}. More precisely, for a given function $u$ we ask for all functions $v$ (called \emph{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 \emph{trivial partner} of $u$ otherwise we call it a \emph{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, $u=P(x)e^{-x^2/2}$, 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 \emph{trivial partner}, showing that most previous counterexamples are still trivial in some restricted sense

Super-Resolution of Positive Sources: the Discrete Setup

In single-molecule microscopy it is necessary to locate with high precision point sources from noisy observations of the spectrum of the signal at frequencies capped by $f_c$, which is just about the frequency of natural light. This paper rigorously establishes that this super-resolution problem can be solved via linear programming in a stable manner. We prove that the quality of the reconstruction crucially depends on the Rayleigh regularity of the support of the signal; that is, on the maximum number of sources that can occur within a square of side length about $1/f_c$. The theoretical performance guarantee is complemented with a converse result showing that our simple convex program convex is nearly optimal. Finally, numerical experiments illustrate our methods.Comment: 31 page, 7 figure

Sampling Sparse Signals on the Sphere: Algorithms and Applications

We propose a sampling scheme that can perfectly reconstruct a collection of spikes on the sphere from samples of their lowpass-filtered observations. Central to our algorithm is a generalization of the annihilating filter method, a tool widely used in array signal processing and finite-rate-of-innovation (FRI) sampling. The proposed algorithm can reconstruct $K$ spikes from $(K+\sqrt{K})^2$ spatial samples. This sampling requirement improves over previously known FRI sampling schemes on the sphere by a factor of four for large $K$. We showcase the versatility of the proposed algorithm by applying it to three different problems: 1) sampling diffusion processes induced by localized sources on the sphere, 2) shot noise removal, and 3) sound source localization (SSL) by a spherical microphone array. In particular, we show how SSL can be reformulated as a spherical sparse sampling problem.Comment: 14 pages, 8 figures, submitted to IEEE Transactions on Signal Processin