Channel Polarization through the Lens of Blackwell Measures
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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. Viewing each BIC through the lens of its Blackwell measure, this paper provides a unified framework for analyzing the evolution of a variety of channel functionals under Arikan's polar transform. These include the symmetric capacity, Bhattacharyya parameter, moments of information density, Hellinger affinity, Gallager's reliability function, and the Bayesian information gain. An explicit general characterization is derived for the evolution of the Blackwell measure under Arikan's polar transform. The evolution of the Blackwell measure is specified for symmetric BICs based on their decomposition into binary symmetric (sub)-channels (BSCs). As a byproduct, a simple algorithm is designed and simulated for computing the successive polarization of symmetric BICs. It is shown that all channel functionals that can be expressed as an expectation of a convex function with respect to the Blackwell measure of the channel polarize on the class of symmetric BICs. For this broad class, a necessary and sufficient condition is established which determines whether the bounded random process associated to a channel functional is a martingale, submartingale, or supermartingale. This condition is numerically verifiable for all known channel functionals.
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