Estimating parameters in noisy low frequency exponentially damped sinusoids and exponentials

Abstract

There has been much recent interest in damped sinusoidal models, probably as a result of their relevance to magnetic resonance imaging. In [1], a model which allowed the sinusoid to decay to 0 was examined, and a Fourier coefficient estimation procedure was proposed. [2] 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 ofnonlinear least squares estimators had already been derived in [3]. 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.