Unveiling The Tree: A Convex Framework for Sparse Problems

This paper presents a general framework for generating greedy algorithms for solving convex constraint satisfaction problems for sparse solutions by mapping the satisfaction problem into one of graph traversal on a rooted tree of unknown topology. For every pre-walk of the tree an initial set of generally dense feasible solutions is processed in such a way that the sparsity of each solution increases with each generation unveiled. The specific computation performed at any particular child node is shown to correspond to an embedding of a polytope into the polytope received from that nodes parent. Several issues related to pre-walk order selection, computational complexity and tractability, and the use of heuristic and/or side information is discussed. An example of a single-path, depth-first algorithm on a tree with randomized vertex reduction and a run-time path selection algorithm is presented in the context of sparse lowpass filter design

A Branch-and-Bound Algorithm for Quadratically-Constrained Sparse Filter Design

This paper presents an exact algorithm for sparse filter design under a quadratic constraint on filter performance. The algorithm is based on branch-and-bound, a combinatorial optimization procedure that can either guarantee an optimal solution or produce a sparse solution with a bound on its deviation from optimality. To reduce the complexity of branch-and-bound, several methods are developed for bounding the optimal filter cost. Bounds based on infeasibility yield incrementally accumulating improvements with minimal computation, while two convex relaxations, referred to as linear and diagonal relaxations, are derived to provide stronger bounds. The approximation properties of the two relaxations are characterized analytically as well as numerically. Design examples involving wireless channel equalization and minimum-variance distortionless-response beamforming show that the complexity of obtaining certifiably optimal solutions can often be significantly reduced by incorporating diagonal relaxations, especially in more difficult instances. In the case of early termination due to computational constraints, diagonal relaxations strengthen the bound on the proximity of the final solution to the optimum.Texas Instruments Leadership University Consortium Progra

Statistical Communication Theory

Contains reports on one completed research project and one current research project.Joint Services Electronics Programs (U.S. Army, U.S. Navy, and U.S. Air Force) under Contract DA 28-043-AMC-02536(E)National Aeronautics and Space Administration (Grant NsG-496

Digital Signal Processing

Contains reports on one research project.U. S. Navy Office of Naval Research (Contract N00014-67-A-0204-0064)National Science Foundation (Grant GK-31353

Statistical Communication Theory

Contains reports on one completed research project and one current research project.Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 36-039-AMC-03200(E)National Aeronautics and Space Administration (Grant NsG-496)National Science Foundation (Grant GP-2495)National Aeronautics and Space Administration Grant (NsG-334

Exact and approximate polynomial decomposition methods for signal processing applications

Signal processing is a discipline in which functional composition and decomposition can potentially be utilized in a variety of creative ways. From an analysis point of view, further insight can be gained into existing signal processing systems and techniques by reinterpreting them in terms of functional composition. From a synthesis point of view, functional composition offers new algorithms and techniques with modular structure. Moreover, computations can be performed more efficiently and data can be represented more compactly in information systems represented in the context of a compositional structure. Polynomials are ubiquitous in signal processing in the form of z-transforms. In this paper, we summarize the fundamentals of functional composition and decomposition for polynomials from the perspective of exploiting them in signal processing. We compare exact polynomial decomposition algorithms for sequences that are exactly decomposable when expressed as a polynomial, and approximate decomposition algorithms for those that are not exactly decomposable. Furthermore, we identify efficiencies in using exact decomposition techniques in the context of signal processing and introduce a new approximate polynomial decomposition technique based on the use of Structured Total Least Norm (STLN) formulation.Texas Instruments Leadership University Consortium ProgramBose (Firm

A technique for measuring the plane-wave reflection coefficient of the ocean bottom

Also published as: Journal of the Acoustical Society of America 68 (1980): 602-612A new technique for the measurement of the plane-wave reflection coefficient of a horizontally stratified ocean bottom is described. It is based on the exact Hankel transform relationship between the reflection coefficient and the bottom reflected field due to a point source. The method employs a new algorithm for the numerical evaluation of the Hankel transform which is based on the "projection-slice" theorem for the two-dimensional Fourier transform. The details of the algorithm are described in the companion paper. Although the algorithm is applied to the case of an isovelocity ocean, the general theory for measuring the plane-wave reflection coefficient in a refracting ocean is developed. The technique provides information about the reflection coefficient, not only for real incident angles, _but also for complex angles, thus potentially providing substantial additional structural information about the bottom. The method is shown to yield excellent results with synthetically generated data for the cases of a hard bottom and slow isovelocity bottom.Prepared for the Office of Naval Research under Contracts N00014-77-C-0196 and N00014-75-C-0951; NR 049-328