Channel Polarization Through the Lens of Blackwell Measures

Abstract

Each memoryless binary-input channel (BIC) can be uniquely described by its Blackwell measure, which is a probability distribution on the unit interval $[0,1]$ with mean $1/2$. Conversely, any such probability distribution defines a BIC. The evolution of the Blackwell measure under Arikan's polar transform is derived for general BICs, and is analogous to density evolution as cited in the literature. The present analysis emphasizes functional equations. Consequently, the evolution of a variety of channel functionals is characterized, including the symmetric capacity, Bhattacharyya parameter, moments of information density, Hellinger affinity, Gallager's reliability function, the Hirschfeld-Gebelein-Renyi maximal correlation, and the Bayesian information gain. The evolution of measure is specialized for symmetric BICs according to their decomposition into binary symmetric (sub)-channels (BSCs), which simplifies iterative computations and the construction of polar codes. It is verified that, as a consequence of the Blackwell--Sherman--Stein theorem, all channel functionals $\mathrm{I}_f$ that can be expressed as an expectation of a convex function $f$ with respect to the Blackwell measure of a channel polarize in each iteration due to the polar transformation on the class of symmetric BICs. Moreover, for $f$ either convex or non-convex, a necessary and sufficient condition is established to determine whether the random process associated with each $\mathrm{I}_f$ is a martingale, submartingale, or supermartingale. Represented via functional inequalities in terms of $f$, this condition is numerically verifiable for all $\mathrm{I}_f$, and can generate analytical proofs. To exhibit one such proof, it is shown that the random process associated with the squared maximal correlation parameter is a supermartingale, and converges almost surely on the unit interval $[0,1]$.