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 Arıkan’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-Rényi 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 ${\text { I}}_{ {f}}$ 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 ${\text { I}}_{ {f}}$ is a martingale, submartingale, or supermartingale. Represented via functional inequalities in terms of f, this condition is numerically verifiable for all ${\text { I}}_{ {f}}$ , 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].

中文翻译：

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通过布莱克威尔测量镜头的通道偏振

每个无记忆二进制输入通道 (BIC) 都可以通过其 Blackwell 测度来唯一描述，该测度是单位区间 [0, 1] 上均值为 1/2 的概率分布。相反，任何此类概率分布都定义了 BIC。在 Arıkan 的极坐标变换下 Blackwell 测度的演化是针对一般 BIC 推导出来的，类似于文献中引用的密度演化。目前的分析强调函数方程。因此，表征了各种信道泛函的演化，包括对称容量、Bhattacharyya 参数、信息密度矩、Hellinger 亲和、Gallager 可靠性函数、Hirschfeld-Gebelein-Rényi 最大相关和贝叶斯信息增益。根据对称 BIC 分解为二进制对称（子）信道（BSC），度量的演变专门用于对称 BIC，这简化了迭代计算和极坐标码的构建。已经证实，作为 Blackwell-Sherman-Stein 定理的结果，所有通道泛函 ${\text { I}}_{ {f}}$ 由于对称 BIC 类的极坐标变换，可以表示为关于每次迭代中通道极化的 Blackwell 测度的凸函数 f 的期望。此外，对于 f 无论是凸还是非凸，都建立了一个充分必要条件来确定与每个相关联的随机过程是否 ${\text { I}}_{ {f}}$ 是鞅、子鞅或超级鞅。用 f 的函数不等式表示，这个条件对于所有 ${\text { I}}_{ {f}}$ ，并且可以生成分析证明。为了展示一个这样的证明，它表明与平方最大相关参数相关的随机过程是一个超鞅，并且几乎肯定会在单位区间 [0, 1] 上收敛。