Tarek A. Lahlou

Tarek A. Lahlou, Alan V. Oppenheim

This paper presents two novel regularization methods motivated in part by the geometric significance of biorthogonal bases in signal processing applications. These methods, in particular, draw upon the structural relevance of orthogonality and biorthogonality principles and are presented from the perspectives of signal processing, convex programming, continuation methods and nonlinear projection operators. Each method is specifically endowed with either a homotopy or tuning parameter to facilitate tradeoff analysis between accuracy and numerical stability. An example involving a basis comprised of real exponential signals illustrates the utility of the proposed methods on an ill-conditioned inverse problem and the results are compared to standard regularization techniques from the signal processing literature.

Tarek A. Lahlou, Thomas A. Baran

This primary purpose of this paper is to succinctly state a number of verifiable and tractable sufficient conditions under which a particular class of conservative signal processing structures may be readily used to solve a companion class of constraint satisfaction problems using both synchronous and asynchronous implementation protocols. In particular, the mentioned class of structures is shown to have desirable convergence and robustness properties with respect to various uncertainties involving communication and processing delays. Essential ingredients to the arguments herein involve blending together functional composition methods, conservation principles, asynchronous signal processing implementation protocols, and methods of homotopy. Numerical experiments complement the theoretical presentation and connections to optimization theory are made.

Tarek A. Lahlou, Anuran Makur

In this paper, we study the problem of transient signal analysis. A signal-dependent algorithm is proposed which sequentially identifies the countable sets of decay rates and expansion coefficients present in a given signal. We qualitatively compare our method to existing techniques such as orthogonal exponential transforms generated from orthogonal polynomial classes. The presented algorithm has immediate utility to signal processing applications wherein the decay rates and expansion coefficients associated with a transient signal convey information. We also provide a functional interpretation of our parameter extraction method via signal approximation using monomials over the unit interval from the perspective of biorthogonal constraint satisfaction.