  In this paper we consider the problem of localizing a set of broadband
sources from a finite window of measurements. In the case of narrowband sources
this can be reduced to the problem of spectral line estimation, where our goal
is simply to estimate the active frequencies from a weighted mixture of pure
sinusoids. There exists a plethora of modern and classical methods that
effectively solve this problem. However, for a wide variety of applications the
underlying sources are not narrowband and can have an appreciable amount of
bandwidth. In this work, we extend classical greedy algorithms for sparse
recovery (e.g., orthogonal matching pursuit) to localize broadband sources. We
leverage models for samples of broadband signals based on a union of Slepian
subspaces, which are more aptly suited for dealing with spectral leakage and
dynamic range disparities. We show that by using these models, our adapted
algorithms can successfully localize broadband sources under a variety of
adverse operating scenarios. Furthermore, we show that our algorithms
outperform complementary methods that use more standard Fourier models. We also
show that we can perform estimation from compressed measurements with little
loss in fidelity as long as the number of measurements are on the order of the
signal's implicit degrees of freedom. We conclude with an in-depth application
of these ideas to the problem of localization in multi-sensor arrays.
