  Given a stable SISO LTI system $G$, we investigate the problem of estimating
the $\mathcal{H}_\infty$-norm of $G$, denoted $||G||_\infty$, when $G$ is only
accessible via noisy observations. Wahlberg et al. recently proposed a
nonparametric algorithm based on the power method for estimating the top
eigenvalue of a matrix. In particular, by applying a clever time-reversal
trick, Wahlberg et al. implement the power method on the top left $n \times n$
corner $T_n$ of the Toeplitz (convolution) operator associated with $G$. In
this paper, we prove sharp non-asymptotic bounds on the necessary length $n$
needed so that $||T_n||$ is an $\varepsilon$-additive approximation of
$||G||_\infty$. Furthermore, in the process of demonstrating the sharpness of
our bounds, we construct a simple family of finite impulse response (FIR)
filters where the number of timesteps needed for the power method is
arbitrarily worse than the number of timesteps needed for parametric FIR
identification via least-squares to achieve the same $\varepsilon$-additive
approximation.
