| The problem of characterizing propagating wave-fields and tracking the energy sources by means of spatially distributed sensors arises in many engineering disciplines such as biomedical sensing and imaging, radar imaging, wireless communication, etc. The enormous amount of data obtained by the sensors is often squeezed by correlating different sensor measurements. This produces stable statistical quantities pertinent to the random wave-field of interest. Interpreting such statistics and assessing temporal and spatial spectral density of the random wave-field is the subject of spectral analysis and estimation.; The underlying mathematic relates to the so-called moment problem---a subject with a history going back to the 1890's. The moment problem has become intertwined with a number of other disciplines of great engineering as well as mathematical significance---analytic interpolation theory, information theory, passive circuit design, etc. Modem nonlinear spectral analysis techniques typically rely on second order statistics of the temporal or spatial stochastic process and exploit the eigenstructure of the associated Toeplitz covariance matrix. An innovative extension of such techniques to deal with generalized statistics is pursued in this thesis. These statistics can be abstracted in the form of a state covariance of a general linear filter. The dynamics of such a filter, which may be partially dictated by the sensing apparatus, can be designed so as to enhance resolution and robustness of the overall spectral estimator.; The applications of such techniques in ultrasound focused surgery and synthetic aperture radar imaging are discussed in details. It is shown how the new spectral analysis tools can be applied to measure tissue temperature from pulse-echo radio frequency signals obtained by standard diagnostic ultrasound equipments. Also it is shown how such tools can be applied for image reconstruction in synthetic aperture radar imaging to remove side-lobe artifacts and reduce speckle noise. |
