  This paper is concerned with the problem of frequency estimation from
multiple-snapshot data. It is well-known that ESPRIT (and spatial-smoothing
ESPRIT in presence of coherent sources or given limited snapshots) can locate
the true frequencies if either the number of snapshots or the signal-to-noise
ratio (SNR) approaches infinity. In this paper, we analyze the nonasymptotic
performance of ESPRIT and spatial-smoothing ESPRIT with finitely many snapshots
and finite SNR. We show that the absolute frequency estimation error of ESPRIT
(or spatial-smoothing ESPRIT) is bounded from above by $C\frac{\max(\sigma,
\sigma^2)}{\sqrt{L}}$ with overwhelming probability, where $\sigma^2$ denotes
the Gaussian noise variance, $L$ is the number of snapshots and $C$ is a
coefficient independent of $L$ and $\sigma^2$, if and only if the true
frequencies can be localized by ESPRIT (or spatial-smoothing ESPRIT) without
noise or with infinitely many snapshots. Our results are obtained by deriving
new matrix perturbation bounds and generalizing the classical Schur product
theorem, which may be of independent interest. Extensions to MUSIC and SS-MUSIC
are also made. Numerical results are provided corroborating our analysis.
