# 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)