Analog Error Correction via Tube PackingJanuary 2013
Topics: Probability and Statistics, Signal Processing
Analog error correction in the form of bandwidth expansion Shannon-Kotelnikov maps allow us to directly map a given source symbol to multiple channel inputs thereby combining source and channel coding into a single vector-valued mapping function. These joint source-channel codes have the potential to approach Shannon's optimal performance theoretically achievable (OPTA) with minimal delay and complexity, but currently there are no theoretical constructions for how to find them. This is particularly the case for dimensions greater than two. This paper proposes a framework for analog coding based on densely packing (hyper)tubes inside of (hyper)spheres whereby the center line of the hypertube is a bounded 1-D manifold corresponding to the image of our desired encoder map. We show for dimensions two, three, and four that this construction method provides extremely simple yet powerful source-channel codes that roughly follow the slope of the OPTA curve in the low SNR region. The tube packing density decreases with increasing dimension which may explain the increasing gap to OPTA as the dimension grows.