Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels (0807.3917v5)

Abstract: A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity $I(W)$ of any given binary-input discrete memoryless channel (B-DMC) $W$. The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of $N$ independent copies of a given B-DMC $W$, a second set of $N$ binary-input channels ${W_N^{(i)}:1\le i\le N}$ such that, as $N$ becomes large, the fraction of indices $i$ for which $I(W_N^{(i)})$ is near 1 approaches $I(W)$ and the fraction for which $I(W_N^{(i)})$ is near 0 approaches $1-I(W)$. The polarized channels ${W_N^{(i)}}$ are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC $W$ with $I(W)>0$ and any target rate $R < I(W)$, there exists a sequence of polar codes ${{\mathscr C}n;n\ge 1}$ such that ${\mathscr C}_n$ has block-length $N=2^n$, rate $\ge R$, and probability of block error under successive cancellation decoding bounded as $P{e}(N,R) \le \bigoh(N^{-\frac14})$ independently of the code rate. This performance is achievable by encoders and decoders with complexity $O(N\log N)$ for each.

• The paper introduces channel polarization, transforming independent copies of a B-DMC into channels that are either nearly perfect or nearly useless.
• It designs polar codes by using high-capacity channels for transmitting information bits and freezing low-capacity ones, thereby achieving Shannon capacity.
• The paper demonstrates that encoding and SC decoding operate at O(N log N) complexity with error probabilities decaying polynomially for practical implementations.

An Overview of Channel Polarization and Polar Codes for B-DMCs

The paper by Erdal Arıkan introduces a technique known as channel polarization, which achieves capacity for binary-input discrete memoryless channels (B-DMCs) by constructing a specific technique called polar codes. This innovation represents a significant advancement in coding theory, showing that it is indeed possible to achieve the symmetric channel capacity, $I(W)$, of any B-DMC efficiently.

Main Concepts and Contributions

Channel Polarization

Polar codes are based on the phenomenon of channel polarization. This refers to the transformation of $N$ independent copies of a B-DMC $W$ into a new set of $N$ synthesized channels that exhibit either high or low capacities as $N$ grows. As a result:

• A fraction, $I(W)$, of these channels will be nearly perfect (high capacity).
• The remaining $1-I(W)$ fraction will be nearly useless (low capacity).

This polarization effect allows sending data efficiently by using high-capacity channels for transmission while ignoring those with low capacity.

Polar Coding

Leveraging the channel polarization effect, polar codes divide the $N$ channels into two groups:

1. High-capacity channels, which are used for transmitting information bits.
2. Low-capacity channels, which are set to a known value (frozen bits).

By encoding the information bits only through the channels that are near-perfect and freezing the bits on the others, polar codes can achieve reliable communication at rates up to the symmetric capacity, $I(W)$.

Computational Efficiency

One of the key strengths of polar codes is their computational efficiency:

• The encoding and decoding processes both have complexities of $O(N \log N)$.
• The successive cancellation (SC) decoder used in polar coding computes decision metrics recursively, further reducing complexity.

Key Results

1. Polarization: The paper rigorously establishes that channel polarization leads to a set of channels $\{W_N^{(i)}\}$ whose capacities tend to either 0 or 1 as $N$ increases. Specifically, the number of channels with capacities near 1 approaches $N I(W)$, and those with capacities near 0 approach $N(1-I(W))$.
2. Rate of Polarization: Theorem 2 asserts that the Bhattacharyya parameters, $Z(W_N^{(i)})$, decay exponentially with $N$ for the channels that are nearly perfect, while remaining significant for channels that are almost completely noisy. This supports the construction of efficient polar codes.
3. Performance Under Successive Cancellation: The paper proves that polar codes can achieve any rate $R < I(W)$ with an error probability that decays polynomially in $N$, specifically with $P_e \leq O(N^{-0.5})$.
4. Symmetric Channels: For symmetric B-DMCs, the performance analysis yields even stronger results, offering more refined bounds on the error probability and demonstrating that the parameter choice $A$ does not affect the performance of the symmetric channel.

Theoretical and Practical Implications

• Capacity-Achieving Codes: Practically, polar codes provide an explicit sequence of codes that achieve the Shannon capacity for any B-DMC. This is critical as it provides coding schemes that are implementable with provable performance guarantees.
• Low Complexity: The low computational complexity of encoding and decoding polar codes makes them attractive for practical applications, including scenarios that require real-time communication.
• Decoding Strategies: While the paper focuses on SC decoding, it paves the way for exploring more powerful decoding algorithms based on belief propagation or list decoding, potentially improving performance and approaching maximum-likelihood (ML) decoding error rates.

Future Directions

Several open problems and prospective enhancements arise from this foundational work:

1. Rate of Polarization: While Theorem 2 provides an order of polynomial decay, further research might quantify the exact rate of polarization more precisely, possibly identifying tighter bounds and enhancing code design strategies.
2. Design for Non-Symmetric Channels: Extending the polarization and decoding methods to non-symmetric channels or multi-dimensional channel models would open new avenues in coding theory.
3. Iterative Decoding: Exploring iterative decoding methods, like belief propagation, especially in practical implementations, could yield improvements in decoding performance and error rates.
4. Generalized Polarization Schemes: Investigating more complex polarization structures beyond the binary field, such as higher-order fields or combining multiple DMCs, may provide new insights and coding techniques for more diverse communication scenarios.

In conclusion, the paper sets a solid theoretical foundation for polar codes and channel polarization, establishing their relevance and efficacy in achieving capacity for binary-input discrete memoryless channels with practical computational benefits. This work provides substantial groundwork for subsequent developments and refinements in the field of coding theory.