Entropy polarization in butterfly transforms

Abstract

In signal processing, it is common to employ various transforms for analyzing or compressing real- or complex-valued signals. If the transform is chosen suitably, certain characteristics of the signal, such as spectral content or sparsity, become readily accessible by looking at the energy distribution among the coordinates of the signal in the transform domain. In contrast, in information-theoretic settings entropy replaces energy as the key parameter of interest; information is processed directly by acting on the entropy through various transforms. Here we follow the information-theoretic approach and focus on the evolution of entropy in the course of butterfly transforms over arbitrary number fields. In particular, we state conditions for entropy polarization—a phenomenon that has been useful in constructing capacity-achieving source and channel codes. We discuss the possibility of using entropy polarization as a useful tool in signal processing applications.