  The reconstruction of multipolar acoustic or electromagnetic sources from
their far-field signature plays a crucial role in numerous applications. Most
of the existing techniques require dense multi-frequency data at the Nyquist
sampling rate. The availability of a sub-sampled grid contributes to the null
space of the inverse source-to-data operator, which causes significant imaging
artifacts. For this purpose, additional knowledge about the source or
regularization is required. In this letter, we propose a novel two-stage
strategy for multipolar source reconstruction from sub-sampled sparse data that
takes advantage of the sparsity of the sources in the physical domain. The data
at the Nyquist sampling rate is recovered from sub-sampled data and then a
conventional inversion algorithm is used to reconstruct sources. The data
recovery problem is linked to a spectrum recovery problem for the signal with
the \textit{finite rate of innovations} (FIR) that is solved using an
annihilating filter-based structured Hankel matrix completion approach (ALOHA).
For an accurate reconstruction, a Fourier inversion algorithm is used. The
suitability of the approach is supported by experiments.
