The estimation and analysis of signals that have polynomial phase and constant or time-varying amplitudes with the addititve noise is considered in this dissertation.Much work has been undertaken on this problem over the last decade or so, and there are a number of estimation schemes available. The fundamental problem when trying to estimate the parameters of these type of signals is the nonlinear characterstics of the signal, which lead to computationally difficulties when applying standard techniques such as maximum likelihood and least squares. When considering only the phase data, we also encounter the well known problem of the unobservability of the true noise phase curve. The methods that are currently most popular involve differencing in phase followed by regression, or nonlinear transformations. Although these methods perform quite well at high signal to noise ratios, their performance worsens at low signal to noise, and there may be significant bias. One of the biggest problems to efficient estimation of these models is that the majority of methods rely on sequential estimation of the phase coefficients, in that the highest-order parameter is estimated first, its contribution removed via demodulation, and the same procedure applied to estimation of the next parameter and so on. This is clearly an issue in that errors in estimation of high order parameters affect the ability to estimate the lower order parameters correctly. As a result, stastical analysis of the parameters is also difficult. In thie dissertation, we aim to circumvent the issues of bias and sequential estiamtion by considering the issue of full parameter iterative refinement techniques. ie. given a possibly biased initial estimate of the phase coefficients, we aim to create computationally efficient iterative refinement techniques to produce stastically efficient estimators at low signal to noise ratios. Updating will be done in a multivariable manner to remove inaccuracies and biases due to sequential procedures. Stastical analysis and extensive simulations attest to the performance of the schemes that are presented, which include likelihood, least squares and bayesian estimation schemes. Other results of importance to the full estimatin problem, namely when there is error in the time variable, the amplitude is not constant, and when the model order is not known, are also condsidered.

La méthode d'intégration produit a été proposée pour résoudre des équations linéaires de Fredholm de deuxième espèce singulières dont la solution exacte est régulière, au moins continue. Dans ce travail on adapte cette méthode à des équations dont la solution est juste intégrable. On étudie également son extension au cas non linéaire posé dans l'espace des fonctions intégrables. Ensuite, on propose une autre manière de mettre en oeuvre la méthode d'intégration produit : on commence par linéariser l'équation par une méthode de type Newton puis on discrétise les itérations de Newton par la méthode d'intégration produit / The product integration method has been proposed for solving singular linear Fredholm equations of the second kind whose exact solution is smooth, at least continuous. In this work, we adapt this method to the case where the solution is only integrable. We also study the nonlinear case in the space of integrable functions. Then, we propose a new version of the method in the nonlinear framework : we first linearize the eqaution by a Newton type method and then discretize the Newton iterations by the product integration method