Non-stationary Polarization: Proof and Application.

The polarization theory of non-stationary channels and the code construction of secure polar codes for non-stationary cases are considered in this paper. The channel transformation of Ar{\i}kan channel polarization theory is modified to non-stationary channels. As proven, the sequence of non-stationary channels can be polarized the into full noise channels and noiseless channels by the modified non-stationary channel transformation. Also it is proven that the rate of non-stationary polarization can achieve the average capacity of the initial non-stationary channels. Then for non-stationary wiretap channel model, a secure polar coding scheme is constructed by applying the non-stationary polarization. It is proven that the secure polar coding scheme achieves the average secrecy capacity of non-stationary wiretap channel model with both reliability and strong security. For wiretap channel model with non-stationary blocks, virtual bit blocks are constructed, on which a time-bit secure polar coding scheme is constructed by applying the non-stationary polarization. It is proven that the time-bit secure polar coding scheme achieves the average secrecy capacity with both reliability and strong security.