  We present different computational approaches for the rapid extraction of the
signal parameters of discretely sampled damped sinusoidal signals. We compare
time- and frequency-domain-based computational approaches in terms of their
accuracy and precision and computational time required in estimating the
frequencies of such signals, and observe a general trade-off between precision
and speed. Our motivation is precise and rapid analysis of damped sinusoidal
signals as these become relevant in view of the recent experimental
developments in cavity-enhanced polarimetry and ellipsometry, where the
relevant time scales and frequencies are typically within the $\sim1-10\,\mu$s
and $\sim1-100$MHz ranges, respectively. In such experimental efforts,
single-shot analysis with high accuracy and precision becomes important when
developing experiments that study dynamical effects and/or when developing
portable instrumentations. Our results suggest that online, running-fashion,
microsecond-resolved analysis of polarimetric/ellipsometric measurements with
fractional uncertainties at the $10^{-6}$ levels, is possible, and using a
proof-of-principle experimental demonstration we show that using a
frequency-based analysis approach we can monitor and analyze signals at kHz
rates and accurately detect signal changes at microsecond time-scales.
