Colloquium

The circuit technology of data acquisition has introduced a high performance technique of analog-to-digital conversion based on the use of coarse quantization compensated by feedback, and called Sigma-Delta modulation.
However, while this technique enables data conversion of high resolutions in practice, its design has been mostly developed empirically and its rigorous analysis escapes from standard signal theories. The fundamental difficulty lies in the existence of a nonlinear operation (namely, the quantization) in a recursive process (physically implemented by the feedback). This prevents a tractable and explicit determination of the output in terms of the input of the system.

Partial answers to this difficult problem have been recently found as Sigma-Delta modulators have been observed to carry some new interesting mathematical properties. The state vector of the feedback system appears to systematically remain in a tile of the state space. This has been the starting point to new research developments involving mathematical tools that are very unusual to the signal processing and communications communities, while simultaneously bringing new results to applied mathematics. This includes ergodic theory, dynamical systems, as well as spectral theory.
In this talk, we give an overview on this research, including the mathematical origin of this tiling phenomenon and its consequence to the rigorous signal analysis of Sigma-Delta modulators.