Shannon sampling theory/data compression

Big question

How to compress continuous data, such as music recordings, most efficiently?

Basic idea

The crucial link between continuous representations of information, such as music, and discrete representations of information, such as sequences of numbers, is provided by a mathematical discipline called sampling theory. The basic Shannon sampling theorem, for example, shows that if a music signal is bandlimited to 20KHz, then it suffices to record its amplitudes 40000 times per second (those numbers are stored in a computer file) and from those numbers, it then, in principle, possible to reconstruct the music perfectly (within the precision to which the amplitudes were measured. I develop and use sampling theory for two purposes:

1. I generalized Shannon sampling theory to time-varying bandlimits by using functional analytic tools that generalize Fourier theory. This improves sampling efficiency because it allows one to sample and reconstruct signals at time-varying rates that are adjusted to the time-varying properties of individual signals. I hold U.S. patent #6531971 on a resulting new data compression method. A recent result with my student Yufang Hao adds a very valuable new tool to that method, namely a completely adapted frequency filtering method. My student James "Chuck" Bronson works on the implementation.
2. I am generalizing sampling theory to the sampling and reconstruction of fields on spacetime and also spacetime itself. This provides mathematical tools for my project I) above.

Selected publications

• Y. Hao, A. Kempf, Generalized Shannon Sampling Method reduces the Gibbs Overshoot in the Approximation of a Step Function, accepted for publication in J. Conc. & Appl. Math.
• Y. Hao, A. Kempf, Filtering, sampling and reconstruction with time-varying bandwidth, IEEE Signal Processing Letters, 17, pp. 241-244 (2010)
• R. Martin, A. Kempf, Approximation of Bandlimited Functions on a Non-Compact Manifold by Bandlimited Functions on Compact Submanifolds, Sampl. Theor. in Sign. and Image Processing, 7, 281 (2008)
• Y. Hao, A. Kempf, On a Non-Fourier Generalization of Shannon Sampling Theory, IEEE Information Theory CWIT 2007, 193 (2007)