Time Delay Estimation from Low Rate Samples: A Union of Subspaces Approach

Time delay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Various approaches have been proposed in the literature to identify time delays introduced by multipath environments. However, these methods either operate on the analog received signal, or require high sampling rates in order to achieve reasonable time resolution. In this paper, our goal is to develop a unified approach to time delay estimation from low rate samples of the output of a multipath channel. Our methods result in perfect recovery of the multipath delays from samples of the channel output at the lowest possible rate, even in the presence of overlapping transmitted pulses. This rate depends only on the number of multipath components and the transmission rate, but not on the bandwidth of the probing signal. In addition, our development allows for a variety of different sampling methods. By properly manipulating the low-rate samples, we show that the time delays can be recovered using the well-known ESPRIT algorithm. Combining results from sampling theory with those obtained in the context of direction of arrival estimation methods, we develop necessary and sufficient conditions on the transmitted pulse and the sampling functions in order to ensure perfect recovery of the channel parameters at the minimal possible rate. Our results can be viewed in a broader context, as a sampling theorem for analog signals defined over an infinite union of subspaces

Multichannel Sampling of Pulse Streams at the Rate of Innovation

We consider minimal-rate sampling schemes for infinite streams of delayed and weighted versions of a known pulse shape. The minimal sampling rate for these parametric signals is referred to as the rate of innovation and is equal to the number of degrees of freedom per unit time. Although sampling of infinite pulse streams was treated in previous works, either the rate of innovation was not achieved, or the pulse shape was limited to Diracs. In this paper we propose a multichannel architecture for sampling pulse streams with arbitrary shape, operating at the rate of innovation. Our approach is based on modulating the input signal with a set of properly chosen waveforms, followed by a bank of integrators. This architecture is motivated by recent work on sub-Nyquist sampling of multiband signals. We show that the pulse stream can be recovered from the proposed minimal-rate samples using standard tools taken from spectral estimation in a stable way even at high rates of innovation. In addition, we address practical implementation issues, such as reduction of hardware complexity and immunity to failure in the sampling channels. The resulting scheme is flexible and exhibits better noise robustness than previous approaches

Sparse Filter Design Under a Quadratic Constraint: Low-Complexity Algorithms

This paper considers three problems in sparse filter design, the first involving a weighted least-squares constraint on the frequency response, the second a constraint on mean squared error in estimation, and the third a constraint on signal-to-noise ratio in detection. The three problems are unified under a single framework based on sparsity maximization under a quadratic performance constraint. Efficient and exact solutions are developed for specific cases in which the matrix in the quadratic constraint is diagonal, block-diagonal, banded, or has low condition number. For the more difficult general case, a low-complexity algorithm based on backward greedy selection is described with emphasis on its efficient implementation. Examples in wireless channel equalization and minimum-variance distortionless-response beamforming show that the backward selection algorithm yields optimally sparse designs in many instances while also highlighting the benefits of sparse design.Texas Instruments Leadership University Consortium Progra

Innovation Rate Sampling of Pulse Streams with Application to Ultrasound Imaging

Signals comprised of a stream of short pulses appear in many applications including bio-imaging and radar. The recent finite rate of innovation framework, has paved the way to low rate sampling of such pulses by noticing that only a small number of parameters per unit time are needed to fully describe these signals. Unfortunately, for high rates of innovation, existing sampling schemes are numerically unstable. In this paper we propose a general sampling approach which leads to stable recovery even in the presence of many pulses. We begin by deriving a condition on the sampling kernel which allows perfect reconstruction of periodic streams from the minimal number of samples. We then design a compactly supported class of filters, satisfying this condition. The periodic solution is extended to finite and infinite streams, and is shown to be numerically stable even for a large number of pulses. High noise robustness is also demonstrated when the delays are sufficiently separated. Finally, we process ultrasound imaging data using our techniques, and show that substantial rate reduction with respect to traditional ultrasound sampling schemes can be achieved.Comment: 14 pages, 13 figure

Improved technique for design of perfect reconstruction FIR QMF banks with lossless polyphase matrices

A technique is developed for the design of analysis filters in an M-channel maximally decimated, perfect reconstruction, finite-impulse-response quadrature mirror filter (FIR QMF) bank that has a lossless polyphase-component matrix E(z). The aim is to optimize the parameters characterizing E(z) until the sum of the stopband energies of the analysis filters is minimized. There are four novel elements in the procedure reported here. The first is a technique for efficient initialization of one of the M analysis filters, as a spectral factor of an Mth band filter. The factorization itself is done in an efficient manner using the eigenfilters approach, without the need for root-finding techniques. The second element is the initialization of the internal parameters which characterize E(z), based on the above spectral factor. The third element is a modified characterization, mostly free from rotation angles, of the FIR E(z). The fourth is the incorporation of symmetry among the analysis filters, so as to minimize the number of unknown parameters being optimized. The resulting design procedure always gives better filter responses than earlier ones (for a given filter length) and converges much faste

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

Graph Signal Processing: Overview, Challenges and Applications

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing. We then summarize recent developments in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE