Polar codes for channel and source coding

The two central topics of information theory are the compression and the transmission of data. Shannon, in his seminal work, formalized both these problems and determined their fundamental limits. Since then the main goal of coding theory has been to find practical schemes that approach these limits. Polar codes, recently invented by Arikan, are the first "practical" codes that are known to achieve the capacity for a large class of channels. Their code construction is based on a phenomenon called "channel polarization". The encoding as well as the decoding operation of polar codes can be implemented with O(N log N) complexity, where N is the blocklength of the code. We show that polar codes are suitable not only for channel coding but also achieve optimal performance for several other important problems in information theory. The first problem we consider is lossy source compression. We construct polar codes that asymptotically approach Shannon's rate-distortion bound for a large class of sources. We achieve this performance by designing polar codes according to the "test channel", which naturally appears in Shannon's formulation of the rate-distortion function. The encoding operation combines the successive cancellation algorithm of Arikan with a crucial new ingredient called "randomized rounding". As for channel coding, both the encoding as well as the decoding operation can be implemented with O(N log N) complexity. This is the first known "practical" scheme that approaches the optimal rate-distortion trade-off. We also construct polar codes that achieve the optimal performance for the Wyner-Ziv and the Gelfand-Pinsker problems. Both these problems can be tackled using "nested" codes and polar codes are naturally suited for this purpose. We further show that polar codes achieve the capacity of asymmetric channels, multi-terminal scenarios like multiple access channels, and degraded broadcast channels. For each of these problems, our constructions are the first known "practical" schemes that approach the optimal performance. The original polar codes of Arikan achieve a block error probability decaying exponentially in the square root of the block length. For source coding, the gap between the achieved distortion and the limiting distortion also vanishes exponentially in the square root of the blocklength. We explore other polar-like code constructions with better rates of decay. With this generalization, we show that close to exponential decays can be obtained for both channel and source coding. The new constructions mimic the recursive construction of Arikan and, hence, they inherit the same encoding and decoding complexity. We also propose algorithms based on message-passing to improve the finite length performance of polar codes. In the final two chapters of this thesis we address two important problems in graphical models related to communications. The first problem is in the area of low-density parity-check codes (LDPC). For practical lengths, LDPC codes using message-passing decoding are still the codes to beat. The current analysis, using density evolution, evaluates the performance of these algorithms on a tree. The tree assumption corresponds to using an infinite length code. But in practice, the codes are of finite length. We analyze the message-passing algorithms for this scenario. The absence of tree assumption introduces correlations between various messages. We show that despite this correlation, the prediction of the tree analysis is accurate. The second problem we consider is related to code division multiple access (CDMA) communication using random spreading. The current analysis mainly focuses on the information theoretic limits, i.e., using Gaussian input distribution. However in practice we use modulation schemes like binary phase-shift keying (BPSK), which is far from being Gaussian. The effects of the modulation scheme cannot be analyzed using traditional tools which are based on spectrum of large random matrices. We follow a new approach using tools developed for random spin systems in statistical mechanics. We prove a tight upper bound on the capacity of the system when the user input is BPSK. We also show that the capacity depends only on the power of the spreading sequences and is independent of their exact distribution.

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