In relation to the study of instantaneous frequency, HHT and the EMD algorithm in signal analysis people have been trying to find solutions of the eigenfunction problem: Find f (t) = ρ(t)e such that Hf = −if, ρ(t) ≥ 0 and θ′(t) ≥ 0, a.e., where Hf is Hilbert transform of f. This article serves as a survey on some recent studies, and presents some new results as well. In the survey part we first review the systematic study on the unimodular case, and then give a detailed account on a fundamental class of non-unimodular solutions, called H-atoms, in terms of starlike functions in one complex variable. As new result we construct certain circular monocomponents that do not fall into the category of H-atoms but of the form ρ(t)e, where ρ(t) ≥ 0, and eis some Fourier atom, as well as those of the form ρ(s)e, where e is one on the line induced from some Fourier atom under Cayley transform.