We consider the asymptotic behavior of the polarization process in the large block-length regime when transmission takes place over a binary-input memoryless symmetric channel W. In particular, we study the asymptotics of the cumulative distribution P(Zn ≤ z), where {Zn} is the Bhattacharyya process associated with W, and its dependence on the rate of transmission. On the basis of this result, we characterize the asymptotic behavior, as well as its dependence on the rate, of the block error probability of polar codes using the successive cancellation decoder. This refines the original asymptotic bounds by Arıkan and Telatar. Our results apply to general polar codes based on l×l kernel matrices. We also provide asymptotic lower bounds on the block error probability of polar codes using the maximum a posteriori (MAP) decoder. The MAP lower bound and the successive cancellation upper bound coincide when l = 2, but there is a gap for l > 2.
Rate-Dependent Analysis of the Asymptotic Behavior of Channel Polarization
Seyed Hamed Hassani,R. Mori,Toshiyuki TANAKA,R. Urbanke
Published 2011 in IEEE Transactions on Information Theory
ABSTRACT
PUBLICATION RECORD
- Publication year
2011
- Venue
IEEE Transactions on Information Theory
- Publication date
2011-10-02
- Fields of study
Mathematics, Computer Science
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CLAIMS
CONCEPTS
- bhattacharyya process
The sequence of Bhattacharyya parameters generated by repeated channel polarization for a given channel.
Aliases: Zn process, Bhattacharyya parameter process
- binary-input memoryless symmetric channel
A communication channel with binary inputs, memoryless use across channel instances, and symmetric behavior across the input symbols.
Aliases: BMS channel, binary-input symmetric channel
- block error probability
The probability that at least one symbol or bit in a decoded codeword block is incorrect.
Aliases: block error rate
- l×l kernel matrix
The base transformation matrix of size l by l used to define a generalized polar-coding construction.
Aliases: kernel matrix, l by l kernel
- maximum a posteriori decoder
A decoder that selects the most probable codeword or bit assignment given the received channel output.
Aliases: MAP decoder
- polar codes
Error-correcting codes constructed by channel polarization using a kernel matrix and a decoding rule.
Aliases: polar coding
- rate of transmission
The coding rate parameter that specifies how many information bits are sent per channel use.
Aliases: transmission rate, code rate
- successive cancellation decoder
A polar-code decoder that decides bits sequentially using previously decoded bits as side information.
Aliases: SC decoder
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