Re-proving Channel Polarization Theorems: An Extremality and Robustness Analysis

The general subject considered in this thesis is a recently discovered coding technique, polar coding, which is used to construct a class of error correction codes with unique properties. In his ground-breaking work, Ar{\i}kan proved that this class of codes, called polar codes, achieve the symmetric capacity --- the mutual information evaluated at the uniform input distribution ---of any stationary binary discrete memoryless channel with low complexity encoders and decoders requiring in the order of $O(N\log N)$ operations in the block-length $N$. This discovery settled the long standing open problem left by Shannon of finding low complexity codes achieving the channel capacity. Polar coding settled an open problem in information theory, yet opened plenty of challenging problems that need to be addressed. A significant part of this thesis is dedicated to advancing the knowledge about this technique in two directions. The first one provides a better understanding of polar coding by generalizing some of the existing results and discussing their implications, and the second one studies the robustness of the theory over communication models introducing various forms of uncertainty or variations into the probabilistic model of the channel.Comment: Preview of my PhD Thesis, EPFL, Lausanne, 2014. For the full version, see http://people.epfl.ch/mine.alsan/publication

Universal Polar Decoding with Channel Knowledge at the Encoder

Polar coding over a class of binary discrete memoryless channels with channel knowledge at the encoder is studied. It is shown that polar codes achieve the capacity of convex and one-sided classes of symmetric channels

Polarization as a novel architecture to boost the classical mismatched capacity of B-DMCs

We show that the mismatched capacity of binary discrete memoryless channels can be improved by channel combining and splitting via Ar{\i}kan's polar transformations. We also show that the improvement is possible even if the transformed channels are decoded with a mismatched polar decoder.Comment: Submitted to ISIT 201

Extremality Properties for Gallager's Random Coding Exponent

We describe certain extremality properties for Gallager's reliability function E-0 for binary input symmetric DMCs. In particular, we show that amongst such DMC's whose E-0(rho(1)) has a given value for a given rho(1), the BEC and BSC have the largest and smallest value of the derivative of E-0(rho(2)) for any rho(2) >= rho(1). As the random coding exponent is obtained by tracing the map rho -> (E-0'(rho), E-0(rho) - rho E-0'(rho)) this conclusion includes as a special case the results of [1]. Furthermore, we show that amongst channels W with a given value of E-0(rho) for a given rho the BEC and BSC are the most and least polarizing under Arikan's polar transformations in the sense that their polar transforms W+ and W- has the largest and smallest difference in their E-0 values

A Lower Bound on Achievable Rates by Polar Codes with Mismatch Polar Decoding

In this paper we show that mismatched polar codes over symmetric B-DMCs symmetrized under the same permutation can achieve rates of at least I (W, V) bits whenever I (W, V) > 0, where W denotes the communication channel, V the mismatched channel used in the code design including both the encoder and decoder, and I(W, V) = Sigma(y) Sigma(x is an element of{0,1}) 1/2W(y vertical bar x) log(2) V(y vertical bar x)/1/2V(y vertical bar 0) + 1/2V(y vertical bar 1)