There has been much recent interest in damped sinusoidal models, probably as a result of their relevance to magnetic resonance imaging. In \cite{about2010}, a model which allowed the sinusoid to decay to $0$ was examined, and a Fourier coefficient estimation procedure was proposed. \cite{Quinn2014} noted that in order for any asymptotic theory to be available, the decay should not be allowed to complete, and examined the asymptotic behavior of a Fourier coefficient procedure based on this assumption, for which the asymptotic behavior of nonlinear least squares estimators had already been derived in \cite{BoBri}. In this paper, we consider the problem of estimating the frequency and damping factor when the frequency is so low that only a finite number of periods appear in the data. Additionally, we consider a Fourier technique for estimating the damping factor in a noisy real exponential.