  The ringdown phase following a binary black hole merger is usually assumed to
be well described by a linear superposition of complex exponentials
(quasinormal modes). In the strong-field conditions typical of a binary black
hole merger, non-linear effects may produce mode coupling. Artificial mode
coupling can also be induced by the black hole's rotation, if the radiation
field is expanded in terms of spin-weighted spherical (rather than spheroidal)
harmonics. Observing deviations from linear black hole perturbation theory
requires optimal fitting techniques to extract ringdown parameters from
numerical waveforms, which are inevitably affected by errors. So far,
non-linear least-squares fitting methods have been used as the standard
workhorse to extract frequencies from ringdown waveforms. These methods are
known not to be optimal for estimating parameters of complex exponentials.
Furthermore, different fitting methods have different performance in the
presence of noise. The main purpose of this paper is to introduce the
gravitational wave community to modern variations of a linear parameter
estimation technique first devised in 1795 by Prony: the Kumaresan-Tufts and
matrix pencil methods. Using "test" damped sinusoidal signals in Gaussian white
noise we illustrate the advantages of these methods, showing that they have
variance and bias at least comparable to standard non-linear least-squares
techniques. Then we compare the performance of different methods on
unequal-mass binary black hole merger waveforms. The methods we discuss should
be useful both theoretically (to monitor errors and search for non-linearities
in numerical relativity simulations) and experimentally (for parameter
estimation from ringdown signals after a gravitational wave detection).
