Polar codes for channel and source coding
Abstract
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…
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