Channel Polarization Through the Lens of Blackwell Measures
Channel Polarization Through the Lens of Blackwell Measures
复制标题
通过布莱克威尔测量镜头观察通道偏振
DOI：
10.1109/tit.2020.3016605
复制
发表时间：
2018
影响因子：
2.5
通讯作者：
M. Raginsky
中科院分区：
文献类型：
--
作者：
Naveen Goela;M. Raginsky
每个无记忆二进制输入信道（BIC）都可以用其Blackwell测度唯一地描述，Blackwell测度是单位区间[0,1]上均值为1/2的概率分布。相反，任何这样的概率分布都定义了一个BIC。对于一般bic，推导了Arıkan极变换下Blackwell测度的演化，类似于文献中引用的密度演化。目前的分析强调的是泛函方程。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亲和度、Gallager可靠度函数、hirschfield - gebelein - r<s:1> nyi最大相关和贝叶斯信息增益。将对称信道分解为二进制对称信道（BSCs），简化了迭代计算和极坐标码的构造。根据Blackwell - sherman - stein定理，证明了所有通道泛函<inline-formula> < text -math notation="LaTeX">${\text {I}}_{{f}}$ </ text -math></inline-formula>，可以表示为一个凸函数f在每次迭代中对由于对称BICs类的极变换而引起的通道极化的Blackwell度量的期望。此外，对于f为凸或非凸，建立了一个充要条件来确定每个<inline-formula> < text -math符号="LaTeX">${\text {I}}_{{f}}$ </ text -math></inline-formula>所关联的随机过程是否为鞅、次鞅或上鞅。通过f的函数不等式表示，该条件对于所有<inline-formula> < text -math notation="LaTeX">${\text {I}}_{{f}}$ </ text -math></inline-formula>均可进行数值验证，并可生成解析证明。为了证明一个这样的证明，证明了与最大平方相关参数相关联的随机过程是一个上鞅，并且在单位区间[0,1]上几乎肯定收敛。
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 <inline-formula> <tex-math notation="LaTeX">${\text { I}}_{ {f}}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">${\text { I}}_{ {f}}$ </tex-math></inline-formula> is a martingale, submartingale, or supermartingale. Represented via functional inequalities in terms of f, this condition is numerically verifiable for all <inline-formula> <tex-math notation="LaTeX">${\text { I}}_{ {f}}$ </tex-math></inline-formula>, 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].