Economical sampling of parametric signals

This thesis proposes architectures and algorithms for digital acquisition of parametric signals. It furthermore provides bounds for the performance of these systems in the presence of noise. Our simple acquisition circuitry and low sampling rate enable accurate parameter estimation to be achieved economically. In present practice, sampling and estimation are not integrated: the sampling device does not take advantage of the parametric model, and the estimation assumes that noise in the data is signal-independent additive white Gaussian noise. We focus on estimating the timing information in signals that are linear combinations of scales and shifts of a known pulse. This signal model is well-known in a variety of disciplines such as ultra-wideband signaling, neurobiology, etc. The signal is completely determined by the amplitudes and shifts of the summands. The delays determine a subspace that contains the signals, so estimating the shifts is equivalent to subspace estimation. By contrast, conventional sampling theory yields a least-squares approximation to a signal from a fixed shift-invariant subspace of possible reconstructions. Conventional acquisition takes samples at a rate higher than twice the signal bandwidth. Although this may be feasible, there is a trade-off between power, accuracy, and speed. Under the signal model of interest, when the pulses are very narrow, the number of parameters per unit time—the rate of innovation—is much lower than the Fourier bandwidth. There is thus potential for much lower sampling rate so long as nonlinear reconstruction algorithms are used. We present a new sampling scheme that takes simultaneous samples at the outputs of multiple channels. This new scheme can be implemented with simple circuitry and has a successive approximation property that can be used to detect undermodeling. In many regimes our algorithms provide better timing accuracy and resolution than conventional systems. Our new analytical and algorithmic techniques are applied to previously proposed systems, and it is shown that all the systems considered have super-resolution properties. Finally, we consider the same parameter estimation problem when the sampling instances are perturbed by signal-independent timing noise. We give an iterative algorithm that achieves accurate timing estimation by exploiting knowledge of the pulse shape. Thesis Supervisor: Vivek K Goyal Title: Assistant Professor Acknowledgments Throughout my academic training I have been blessed with the presence of many wonderful teachers, friends, and supporters. Without their help, I will either have lost my sanity or failed to finish my PhD! A comprehensive list of those to whom I owe their support and generosity will by itself comprise a larger volume than this thesis; hence I can only offer a small list and apologize to those who are not mentioned on this list. I would like to begin by thanking my family for all their unconditional love, limitless support, and patience. I could not have asked for a better Father, Mother, nor Brother! I thank my advisor Vivek K Goyal for being my last advisor, and for being a great advisor. I am flattered to be his first PhD graduate and hope to do him proud. I would also like to thank Moe Z. Win, Sanjoy K. Mitter and Denny M. Freeman for serving on my thesis committee, and for their encouragements through the worst times at MIT. My work has benefited from collaborations, discussions, and suggestions from many researchers. Among others, I am grateful for the continuing collaboration with Martin Vetterli, Andrea Ridolfi, and Kannan Ramchandran. I would not have been half the student I was without the advising of Sekhar Tatikonda, Vincent Chan, Muriel Medard, Bob Gallager, Greg Wornell, and Vahid Tarokh. My stay at LIDS was thanks to Vincent Chan and Muriel Medard, I would like to thank the support staff at RLE especially Eric J. Strattman. I also give Marilyn Pierce and the staff members at the EECS Graduate Office special thanks for their support and patience as I fail again and again to submit anything on time. In the last part of my graduate career Joel S. Schindall and Manish Bhardwaj have given generous advice and encouragement, leading well into my job search and the completion of my thesis. During my stay in Boston I have made many friends, starting with my roomates who tolerated all my cooking and coffee roasting: Ariel, Abby, Justin, Chris Brown, Shauna, Alex Feldman, and Chris Fang-Yen. I probably will not be here today if not for the great care of my teammates Mark Abramson, Shane Mulrooney, Zach Hartwig, and Eric Fleming. I probably will not have maintained my sanity if not for the bike rides and bike geeking with Chris Dever, Lodrina Cherne, Eric Sakalowsky, and Julie Monagle. I owe Mario, Tricia and Ian Valenti (or is it The Valentii) special thanks for all the good times and support in the bad times. I will miss the drinks and musings about the virtues of Belgian versus Swiss chocolate with Tom Schouwenaars, Hadley Yates, Charles Swannack-Willenborg, Nicole Willenborg-Swannack, Lav Varshney, Guy Weichenberg, Raj Rao, Michael J. Neely, Constantine Caramanis, Raul Blazquez-Fernandez, and Hayden Marcollo. I owe Trevor Meyerowitz a big thanks for the patient read and re-read of large portions of my thesis, and to Kush and Lav Varshney, Manish, Mitra, Mario, and Adam for reviewing and commenting on my thesis defense preparation. I am grateful for old friends Paul Twohey, Dan Tabion, Jay Freyensee, and the occasional but very wise advising of Eric A. Jacobsen. My stays in Switzerland have been very productive and enjoyable thanks to the hospitality of Nils Rinaldi and his family, Guillermo, Thibaut, Aslan, Thomas, Robin, Irena, Emre, Rüdiger, Cyril, and Jocelyn Plantefol. Je vous adresse mes plus vifs remerciements. Finally, I thank Sarah Ann Slavoff for her wonderful company and support through the final stages of the writing and defense of this thesis. My work at MIT was supported by the MIT Presidential Fellowship, the Alan T. Waterman Award, Grant No. CCR-0139398, and the Analog Devices Corporation.