Sampling methods for parametric non-bandlimited signals

Sampling theory has experienced a strong research revival over the past decade, which led to a generalization of Shannon's original theory and development of more advanced formulations with immediate relevance to signal processing and communications. For example, it was recently shown that it is possible to develop exact sampling schemes for a large class of non-bandlimited signals, namely, certain signals with finite rate of innovation. A common feature of such signals is that they have a parametric representation with a finite number of degrees of freedom and can be perfectly reconstructed from a finite number of samples. The goal of this thesis is to advance the sampling theory for signals of finite rate of innovation and consider its possible extensions and applications. In the first part of the thesis, we revisit the sampling problem for certain classes of such signals, including non-uniform splines and piecewise polynomials, and develop improved schemes that allow for stable and precise reconstruction in the presence of noise. Specifically, we develop a subspace approach to signal reconstruction, which converts a nonlinear estimation problem into the simpler problem of estimating the parameters of a linear model. This provides an elegant and robust framework for solving a large class of sampling problems, while offering more flexibility than the traditional scheme for bandlimited signals. In the second part of the thesis, we focus on applications of our results to certain classes of nonlinear estimation problems encountered in wideband communication systems, most notably ultra-wideband (UWB) systems, where the bandwidth used for transmission is much larger than the bandwidth or rate of information being sent. We develop several frequency domain methods for channel estimation and synchronization in UWB systems, which yield high-resolution estimates of all relevant channel parameters by sampling a received signal below the traditional Nyquist rate. We also propose algorithms that are suitable for identification of more realistic UWB channel models, where a received signal is made up of pulses with different pulse shapes. Finally, we extend our results to multidimensional signals, and develop exact sampling schemes for certain classes of parametric non-bandlimited 2-D signals, such as sets of 2-D Diracs, polygons or signals with polynomial boundaries.

Inverse problems in acoustic tomography

Ivana Jovanovic

Acoustic tomography aims at recovering the unknown parameters that describe a field of interest by studying the physical characteristics of sound propagating through the considered field. The tomographic approach is appealing in that it is non-invasive and allows to obtain a significantly larger amount of data compared to the classical one-sensor one-measurement setup. It has, however, two major drawbacks which may limit its applicability in a practical setting: the methods by which the tomographic data are acquired and then converted to the field values are computationally intensive and often ill-conditioned. This thesis specifically addresses these two shortcomings by proposing novel acoustic tomography algorithms for signal acquisition and field reconstruction. The first part of our exposition deals with some theoretical aspects of the tomographic sampling problems and associated reconstruction schemes for scalar and vector tomography. We show that the classical time-of-flight measurements are not sufficient for full vector field reconstruction. As a solution, an additional set of measurements is proposed. The main advantage of the proposed set is that it can be directly computed from acoustic measurements. It thus avoids the need for extra measuring devices. We then describe three novel reconstruction methods that are conceptually quite different. The first one is based on quadratic optimization and does not require any a priori information. The second method builds upon the notion of sparsity in order to increase the reconstruction accuracy when little data is available. The third approach views tomographic reconstruction as a parametric estimation problem and solves it using recent sampling results on non-bandlimited signals. The proposed methods are compared and their respective advantages are outlined. The second part of our work is dedicated to the application of the proposed algorithms to three practical problems: breast cancer detection, thermal therapy monitoring, and temperature monitoring in the atmosphere. We address the problem of breast cancer detection by computing a map of sound speed in breast tissue. A noteworthy contribution of this thesis is the development of a signal processing technique that significantly reduces the artifacts that arise in very inhomogeneous and absorbent tissue. Temperature monitoring during thermal therapies is then considered. We show how some of our algorithms allow for an increased spatial resolution and propose ways to reduce the computational complexity. Finally, we demonstrate the feasibility of tomographic temperature monitoring in the atmosphere using a custom-built laboratory-scale experiment. In particular, we discuss various practical aspects of time-of-flight measurement using cheap, off-the-shelf sensing devices.

EPFL2008

Power spectra of random spikes and related complex signals

Andrea Ridolfi

"Random spikes" belong to the common language used by engineers, physicists and biologists to describe events associated with time records, locations in space, or more generally, space-time events. Indeed, data and signals consisting of, or structured by, sequences of events are omnipresent in communications, biology, computer science and signal processing. Relevant examples can be found in traffic intensity and neurobiological data, pulse-coded transmission, and sampling. This thesis is concerned by random spike fields and by the complex signals described as the result of various operations on the basic event stream or spike field, such as filtering, jittering, delaying, thinning, clustering, sampling and modulating. More precisely, complex signals are obtained in a modular way by adding specific features to a basic model. This modular approach greatly simplifies the computations and allows to treat highly complex model such as the ones occurring in ultra-wide bandwidth or multipath transmissions. We present a systematic study of the properties of random spikes and related complex signals. More specifically, we focus on second order properties, which are conveniently represented by the spectrum of the signal. These properties are particularly attractive and play an important role in signal analysis. Indeed, they are relatively accessible and yet they provide important informations. Our first contribution is theoretical. As well as presenting a modular approach for the construction of complex signals, we derive formulas for the computation of the spectrum that preserve such modularity: each additional feature added to a basic model appear as a separate and explicit contribution in the corresponding basic spectrum. Moreover, these formula are very general. For instance, the basic point process is not assumed to be a homogeneous Poisson process but it can be any second order stationary process with a given spectrum. In summary, they provide very useful tools for model analysis. We then give applications of the theoretical results: spectral formulas for traffic analysis, pulse based signals used in spread spectrum communications, and randomly sampled signal.

EPFL2005

Inverse problems in acoustic tomography

Ivana Jovanovic

EPFL2008

Power spectra of random spikes and related complex signals

Andrea Ridolfi

EPFL2005