arXiv subjects
Publications and source records attributed to Erdal Arıkan.
This note is a written and extended version of the Shannon Lecture I gave at 2019 International Symposium on Information Theory. It gives an account of the original ideas that motivated the development of polar coding and discusses some new ideas for exploiting channel polarization more effectively in order to improve the performance of polar codes.
We consider the evolution of variance of entropy (varentropy) in the course of a polar transform operation on binary data elements (BDEs). A BDE is a pair $(X,Y)$ consisting of a binary random variable $X$ and an arbitrary side information random variable $Y$. The varentropy of $(X,Y)$ is defined as the variance of the random variable $-\log p_{X|Y}(X|Y)$. A polar transform of order two is a certain mapping that takes two independent BDEs and produces two new BDEs that are correlated with each other. It is shown that the sum of the varentropies at the output of the polar transform is less than or equal to the sum of the varentropies at the input, with equality if and only if at least one of the inputs has zero varentropy. This result is extended to polar transforms of higher orders and it is shown that the varentropy decreases to zero asymptotically when the BDEs at the input are independent and identially distributed.
This paper proposes a high-throughput energy-efficient Successive Cancellation (SC) decoder architecture for polar codes based on combinational logic. The proposed combinational architecture operates at relatively low clock frequencies compared to sequential circuits, but takes advantage of the high degree of parallelism inherent in such architectures to provide a favorable tradeoff between throughput and energy efficiency at short to medium block lengths. At longer block lengths, the paper proposes a hybrid-logic SC decoder that combines the advantageous aspects of the combinational decoder with the low-complexity nature of sequential-logic decoders. Performance characteristics on ASIC and FPGA are presented with a detailed power consumption analysis for combinational decoders. Finally, the paper presents an analysis of the complexity and delay of combinational decoders, and of the throughput gains obtained by hybrid-logic decoders with respect to purely synchronous architectures.
Polar coding was conceived originally as a technique for boosting the cutoff rate of sequential decoding, along the lines of earlier schemes of Pinsker and Massey. The key idea in boosting the cutoff rate is to take a vector channel (either given or artificially built), split it into multiple correlated subchannels, and employ a separate sequential decoder on each subchannel. Polar coding was originally designed to be a low-complexity recursive channel combining and splitting operation of this type, to be used as the inner code in a concatenated scheme with outer convolutional coding and sequential decoding. However, the polar inner code turned out to be so effective that no outer code was actually needed to achieve the original aim of boosting the cutoff rate to channel capacity. This paper explains the cutoff rate considerations that motivated the development of polar coding.
A packing lemma is proved using a setting where the channel is a binary-input discrete memoryless channel $(\mathcal{X},w(y|x),\mathcal{Y})$, the code is selected at random subject to parity-check constraints, and the decoder is a joint typicality decoder. The ensemble is characterized by (i) a pair of fixed parameters $(H,q)$ where $H$ is a parity-check matrix and $q$ is a channel input distribution and (ii) a random parameter $S$ representing the desired parity values. For a code of length $n$, the constraint is sampled from $p_S(s) = \sum_{x^n\in {\mathcal{X}}^n} ϕ(s,x^n)q^n(x^n)$ where $ϕ(s,x^n)$ is the indicator function of event $\{s = x^n H^T\}$ and $q^n(x^n) = \prod_{i=1}^nq(x_i)$. Given $S=s$, the codewords are chosen conditionally independently from $p_{X^n|S}(x^n|s) \propto ϕ(s,x^n) q^n(x^n)$. It is shown that the probability of error for this ensemble decreases exponentially in $n$ provided the rate $R$ is kept bounded away from $I(X;Y)-\frac{1}{n}I(S;Y^n)$ with $(X,Y)\sim q(x)w(y|x)$ and $(S,Y^n)\sim p_S(s)\sum_{x^n} p_{X^n|S}(x^n|s) \prod_{i=1}^{n} w(y_i|x_i)$. In the special case where $H$ is the parity-check matrix of a standard polar code, it is shown that the rate penalty $\frac{1}{n}I(S;Y^n)$ vanishes as $n$ increases. The paper also discusses the relation between ordinary polar codes and random codes based on polar parity-check matrices.