Channel polarization with higher - order memory

Abstract

Channel coding is one of the most fundamental problems regarding the nature of communication. Polar codes, developed by Ar kan, were the rst demonstration of practical channel codes that provably achieve theoretical limits in a wide range of communication scenarios. Based on simple channel transformations which are called channel combining and splitting, Ar kan developed the idea of channel polarization which resulted in polar codes. In this dissertation we focus on the unique properties of polar codes and channel polarization. First, we investigate the channel-speci c construction of polar codes, a point which discriminates polar codes from Reed-Muller (RM) codes. We obtain results showing the inherent e ect of the underlying channel on the construction of polar codes and provide a bridge between polar and RM codes. Our results easily extend to obtaining a characterization for the rate of polarization as well. Next, we consider practical uses of polar codes for fading channels. We design a bit-interleaved polar-coded modulation scheme (BIPCM) by deriving a low complexity code-construction method and designing a lower complexity successive cancellation list decoder (SCLD). We compare the resultant BIPCM system with the existing solutions and show that it provides signi cant performance advantages. Finally, we generalize the channel polarization idea by changing channel combining and splitting operations. By introducing a memory order in the channel combining process we obtain a class of codes, including the original ones, that are parametrized by the memory order. We show that the new family of polar codes achieve the theoretical limits as well and they can also be used with lower complexity by increasing the memory order. We thereby complement Ar kan's conjecture that channel polarization is in fact a general phenomenon.