Rate-Dependent Analysis of the Asymptotic Behavior of Channel Polarization

For a binary-input memoryless symmetric channel $W$, we consider the asymptotic behavior of the polarization process in the large block-length regime when transmission takes place over $W$. In particular, we study the asymptotics of the cumulative distribution $\mathbb{P}(Z_n \leq z)$, where $\{Z_n\}$ is the Bhattacharyya process defined from $W$, and its dependence on the rate of transmission. On the basis of this result, we characterize the asymptotic behavior, as well as its dependence on the rate, of the block error probability of polar codes using the successive cancellation decoder. This refines the original bounds by Ar{\i}kan and Telatar. Our results apply to general polar codes based on $\ell \times \ell$ kernel matrices. We also provide lower bounds on the block error probability of polar codes using the MAP decoder. The MAP lower bound and the successive cancellation upper bound coincide when $\ell=2$, but there is a gap for $\ell>2$.