  Sparse arrays have emerged as a popular alternative to the conventional
uniform linear array (ULA) due to the enhanced degrees of freedom (DOF) and
superior resolution offered by them. In the passive setting, these advantages
are realized by leveraging correlation between the received signals at
different sensors. This has led to the belief that sparse arrays require a
large number of temporal measurements to reliably estimate parameters of
interest from these correlations, and therefore they may not be preferred in
the sample-starved regime. In this paper, we debunk this myth by performing a
rigorous non-asymptotic analysis of the Coarray ESPRIT algorithm. This
seemingly counter-intuitive result is a consequence of the scaling of the
singular value of the coarray manifold, which compensates for the potentially
large covariance estimation error in the limited snapshot regime. Specifically,
we show that for a nested array operating in the regime of fewer sources than
sensors ($S=O(1)$), it is possible to bound the matching distance error between
the estimated and true directions of arrival (DOAs) by an arbitrarily small
quantity ($\epsilon$) with high probability, provided (i) the number of
temporal snapshots ($L$) scales only logarithmically with the number of sensors
($P$), i.e. $L=\Omega(\ln(P)/\epsilon^2)$, and (ii) a suitable separation
condition is satisfied. Our results also formally prove the well-known
empirical resolution benefits of sparse arrays, by establishing that the
minimum separation between sources can be $\Omega(1/P^2)$, as opposed to
separation $\Omega(1/P)$ required by a ULA with the same number of sensors. Our
sample complexity expression reveals the dependence on other key model
parameters such as SNR and the dynamic range of the source powers. This enables
us to establish the superior noise-resilience of nested arrays both
theoretically and empirically.
