Analog-to-digital conversion in compressive sampling

Compressive sampling theory is now established as an effective method for dimension reduction when the underlying signals (e.g., seismic data) are sparse with respect to some suitable basis or frame (e.g., curvelets in the case of seismic). One important problem directly related to the acquisition of analog signals is how to perform analog-to-digital conversion. This is directly related to two important issues: (i) how accurately one can acquire a signal using compressive sampling (in terms of bit depth), and (ii) how ``compressed'' is compressive sampling (in terms of the total number of bits one ends up using after acquiring the signal). We will present recent results that provide answers to these questions. Specifically, we provide an analog-to-digital conversion method that achieves nearly optimal compression using cheap analog devices with a large margin for imperfections.