On Wigner-based sparse time-frequency distributions

Signals made of the superimposition of a reduced number of AM-FM components can be characterized by a time-frequency signature which consists of weighted trajectories in the plane, thus ending up with an ideal representation of their energy distribution that is intrinsically sparse. Elaborating on first studies that pioneered a compressed sensing solution to the question of approaching such an ideally localized distribution by selecting samples in the ambiguity domain and imposing sparsity in the time-frequency domain, the present paper discusses new advances aimed at achieving better performance in the construction of “cross-term-free” Wigner-type distributions. Improved optimization schemes are first proposed, that both speed up the computation and prove more versatile to accommodate for side constraints such as positivity. A special attention is then paid to the choice of the necessary measurements in the ambiguity plane (in fixed or adapted geometries), emphasizing the key role played by the Heisenberg minimum area, regardless of the signal complexity.