  In this paper, a sparse-based method for the estimation of the parameters of
multidimensional ($R$-D) modal (harmonic or damped) complex signals in noise is
presented. The problem is formulated as $R$ simultaneous sparse approximations
of multiple 1-D signals. To get a method able to handle large size signals
while maintaining a sufficient resolution, a multigrid dictionary refinement
technique is associated with the simultaneous sparse approximation problem. The
refinement procedure is proved to converge in the single $R$-D mode case. Then,
for the general multiple modes $R$-D case, the signal tensor model is
decomposed in order to handle each mode separately in an iterative scheme. The
proposed method does not require an association step since the estimated modes
are automatically "paired". We also derive the Cram\'er-Rao lower bounds of the
parameters of modal $R$-D signals. The expressions are given in compact form in
the single $R$-D mode case. Finally, numerical simulations are conducted to
demonstrate the effectiveness of the proposed method.
