  Finite rate of innovation (FRI) is a powerful reconstruction framework
enabling the recovery of sparse Dirac streams from uniform low-pass filtered
samples. An extension of this framework, called generalised FRI (genFRI), has
been recently proposed for handling cases with arbitrary linear measurement
models. In this context, signal reconstruction amounts to solving a joint
constrained optimisation problem, yielding estimates of both the Fourier series
coefficients of the Dirac stream and its so-called annihilating filter,
involved in the regularisation term. This optimisation problem is however
highly non convex and non linear in the data. Moreover, the proposed numerical
solver is computationally intensive and without convergence guarantee.
  In this work, we propose an implicit formulation of the genFRI problem. To
this end, we leverage a novel regularisation term which does not depend
explicitly on the unknown annihilating filter yet enforces sufficient structure
in the solution for stable recovery. The resulting optimisation problem is
still non convex, but simpler since linear in the data and with less unknowns.
We solve it by means of a provably convergent proximal gradient descent (PGD)
method. Since the proximal step does not admit a simple closed-form expression,
we propose an inexact PGD method, coined as Cadzow plug-and-play gradient
descent (CPGD). The latter approximates the proximal steps by means of Cadzow
denoising, a well-known denoising algorithm in FRI. We provide local
fixed-point convergence guarantees for CPGD. Through extensive numerical
simulations, we demonstrate the superiority of CPGD against the
state-of-the-art in the case of non uniform time samples.
