In this dissertation we model complex signals by approximating the phase of the signal as a finite-order polynomial in time. Further, we model the logarithm of the time-varying amplitude of the signal as a finite-order polynomial as well. We refer to a signal that has this form as an Exponential Polynomial Signal (EPS). We derive a computationally efficient algorithm to estimate all of the unknown parameters in this model when a noisy version of an EPS is observed. The estimates obtained from this algorithm can be used to initialize an iterative Maximum-Likelihood (ML) estimation algorithm.; The initial sub-optimal estimation algorithm and the iterative ML algorithm are derived and analyzed in this thesis. The statistical analysis is based upon a finite-order Taylor expansion of the Mean-Squared Error (MSE) of the estimate about the variance of the additive noise. This perturbation analysis gives a method of predicting the MSE of the estimate for any choice of the signal parameters. Also several properties are observed to be common in both the sub-optimal estimate and the ML estimate. The MSE from the perturbation analysis is compared with the MSE from a Monte Carlo simulation and the Cramer-Rao Bound (CRB).; A special case of Exponential Polynomial Signals (EPSs) are signals with constant-amplitude and polynomial phase. For this special case, it is shown that the algorithm (or delay) parameters can be chosen off-line to minimize the first-order approximation of the MSE of the estimate. We also show that by modifying the Discrete Polynomial Transform (DPT) algorithm a reduction of the first-order approximation of the MSE can be obtained.; We examine the application of our algorithms to EPS data with a low Signal-to-Noise Ratio (SNR), to data obtained from non-EPS models, and finally, to seismic data obtained in Northern California.
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