Tutorial

Tutorial Presenter

Nguyen T. Thao, The City College of New York, USA nguyen@ccny.cuny.edu

Tutorial Topic:

The Linear Algebra of Nonuniform Sampling

Aim and scope:

While Shannon sampling theorem has a simple justification by Fourier transform argument, nonuniform sampling ­­­­poses some major mathematical difficulties. Although nonuniform sampling can be simply viewed as a linear transformation of bandlimited continuous-time signals into discrete-time signals, the infinite dimensionality of the problem together with the lack of time invariance in this linear approach has made its theory mostly reserved to applied mathematicians. The growing interest in event-based signal processing is now moving the theory of nonuniform sampling to the foreground.

The aim of this tutorial is not only to show what the fundamental results are on this topic, but also to intuitively motivate them from scratch. In this process, an objective will be to give engineers access to a more general view on signals that goes beyond the commonly known mathematics of signal processing, is inherited from functional analysis and operator theory in mathematics, and is necessary to understand the issues of reconstruction from nonuniform samples. We will start with the simple finite dimensional case of periodic signals, where sampling and reconstruction boil down to commonly known linear algebra based on matrix manipulations. From this simple context, it will be easy to see finite dimensional issues that can be extended to infinite dimensions, as well as identify issues that are specific to infinite dimensions. Attendees will be thus progressively introduced to difficult but necessary notions in infinite dimensional signal spaces such as frames. Equipped with a well-defined framework of analysis, we will then see how methodic answers can be given to a number of questions such as the problem of generalized sampling (the sampling of derivatives being just one example), the robustness of reconstruction to sample errors, sampling and reconstruction under analog circuit constraints, and sliding-window processing for real-time reconstruction implementations (time-varying FIR filter type). Nguyen T. Thao received the engineering degrees from Ecole Polytechnique, France in 1984 and Ecole Nationale Superieure des Telecommunications, France in 1985, the M.Sc. degree from Princeton University in 1986 and the Ph.D. degree from Columbia University in 1993 in electrical engineering.

His career combines positions in industry including Thomson-CSF, France (GaAs A/D converters, from 1986 to 1989) and HP Laboratories, Palo Alto, CA (digital image processing, from 1998 to 1999), as well as positions in academia including Hong Kong University of Science of Technology in the Department of Electrical and Electronic Engineering (from 1993 to 1997) and City College of the City University of New York in the Department of Electrical Engineering (2000 to present). His research interest is the mathematical analysis of analog-to-digital conversion.

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