Projections, dissections and bandwidth expansion mappings

We address the problem of constructing explicit mappings from a k-dimensional continuous alphabet source to an n-dimensional Gaussian channel. The source is assumed to be uniformly distributed on the unit cube [0, 1)^k. The scheme considered is based on a family of piecewise linear mappings and its performance is shown to be related to specific projected lattices of Zⁿ. We study sufficient conditions for the mean squared error of such mappings to scale optimally with the signal-to-noise ratio of the channel and present an explicit construction for the case k = n-1. However, in some other cases our scheme requires the source to be uniformly distributed over a fundamental region of a specific lattice, that may be not congruent to [0, 1)^k. A dissection technique is presented in order to overcome the source support mismatch and the MSE degradation of such a transformation is analyzed. An example construction of a 2 : n expansion mapping using the dissection technique is presented and is shown to exhibit optimal scaling of the MSE with the channel SNR.