4,250 research outputs found
Further Results on Quadratic Permutation Polynomial-Based Interleavers for Turbo Codes
An interleaver is a critical component for the channel coding performance of
turbo codes. Algebraic constructions are of particular interest because they
admit analytical designs and simple, practical hardware implementation. Also,
the recently proposed quadratic permutation polynomial (QPP) based interleavers
by Sun and Takeshita (IEEE Trans. Inf. Theory, Jan. 2005) provide excellent
performance for short-to-medium block lengths, and have been selected for the
3GPP LTE standard. In this work, we derive some upper bounds on the best
achievable minimum distance dmin of QPP-based conventional binary turbo codes
(with tailbiting termination, or dual termination when the interleaver length N
is sufficiently large) that are tight for larger block sizes. In particular, we
show that the minimum distance is at most 2(2^{\nu +1}+9), independent of the
interleaver length, when the QPP has a QPP inverse, where {\nu} is the degree
of the primitive feedback and monic feedforward polynomials. However, allowing
the QPP to have a larger degree inverse may give strictly larger minimum
distances (and lower multiplicities). In particular, we provide several QPPs
with an inverse degree of at least three for some of the 3GPP LTE interleaver
lengths giving a dmin with the 3GPP LTE constituent encoders which is strictly
larger than 50. For instance, we have found a QPP for N=6016 which gives an
estimated dmin of 57. Furthermore, we provide the exact minimum distance and
the corresponding multiplicity for all 3GPP LTE turbo codes (with dual
termination) which shows that the best minimum distance is 51. Finally, we
compute the best achievable minimum distance with QPP interleavers for all 3GPP
LTE interleaver lengths N <= 4096, and compare the minimum distance with the
one we get when using the 3GPP LTE polynomials.Comment: Submitted to IEEE Trans. Inf. Theor
Coding theorems for turbo code ensembles
This paper is devoted to a Shannon-theoretic study of turbo codes. We prove that ensembles of parallel and serial turbo codes are "good" in the following sense. For a turbo code ensemble defined by a fixed set of component codes (subject only to mild necessary restrictions), there exists a positive number γ0 such that for any binary-input memoryless channel whose Bhattacharyya noise parameter is less than γ0, the average maximum-likelihood (ML) decoder block error probability approaches zero, at least as fast as n -β, where β is the "interleaver gain" exponent defined by Benedetto et al. in 1996
Analysis and Design of Tuned Turbo Codes
It has been widely observed that there exists a fundamental trade-off between
the minimum (Hamming) distance properties and the iterative decoding
convergence behavior of turbo-like codes. While capacity achieving code
ensembles typically are asymptotically bad in the sense that their minimum
distance does not grow linearly with block length, and they therefore exhibit
an error floor at moderate-to-high signal to noise ratios, asymptotically good
codes usually converge further away from channel capacity. In this paper, we
introduce the concept of tuned turbo codes, a family of asymptotically good
hybrid concatenated code ensembles, where asymptotic minimum distance growth
rates, convergence thresholds, and code rates can be traded-off using two
tuning parameters, {\lambda} and {\mu}. By decreasing {\lambda}, the asymptotic
minimum distance growth rate is reduced in exchange for improved iterative
decoding convergence behavior, while increasing {\lambda} raises the asymptotic
minimum distance growth rate at the expense of worse convergence behavior, and
thus the code performance can be tuned to fit the desired application. By
decreasing {\mu}, a similar tuning behavior can be achieved for higher rate
code ensembles.Comment: Accepted for publication in IEEE Transactions on Information Theor
Minimum Pseudoweight Analysis of 3-Dimensional Turbo Codes
In this work, we consider pseudocodewords of (relaxed) linear programming
(LP) decoding of 3-dimensional turbo codes (3D-TCs). We present a relaxed LP
decoder for 3D-TCs, adapting the relaxed LP decoder for conventional turbo
codes proposed by Feldman in his thesis. We show that the 3D-TC polytope is
proper and -symmetric, and make a connection to finite graph covers of the
3D-TC factor graph. This connection is used to show that the support set of any
pseudocodeword is a stopping set of iterative decoding of 3D-TCs using maximum
a posteriori constituent decoders on the binary erasure channel. Furthermore,
we compute ensemble-average pseudoweight enumerators of 3D-TCs and perform a
finite-length minimum pseudoweight analysis for small cover degrees. Also, an
explicit description of the fundamental cone of the 3D-TC polytope is given.
Finally, we present an extensive numerical study of small-to-medium block
length 3D-TCs, which shows that 1) typically (i.e., in most cases) when the
minimum distance and/or the stopping distance is
high, the minimum pseudoweight (on the additive white Gaussian noise channel)
is strictly smaller than both the and the , and 2)
the minimum pseudoweight grows with the block length, at least for
small-to-medium block lengths.Comment: To appear in IEEE Transactions on Communication
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