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Polar-Coded Modulaton

Mathis Seidl, Andreas Schenk, Clemens Stierstorfer, Johannes B. Huber

arXiv:1302.2855v1cs.IT

TL;DR

The paper addresses the limited unified treatment of binary polar coding and 2^m-ary modulation schemes such as MLC and BICM. It develops a channel-partition framework and labeling rules for optimized polar-coded modulation, then evaluates the schemes on AWGN channels. The results show that optimized BICM improves over unmodified BICM, while MLC achieves better performance than BICM.

  • Problem

    Combining binary polar codes with 2^m-ary modulation, particularly MLC and BICM, had received limited treatment despite polar coding’s established binary-channel results.

  • Method

    The paper unifies polar coding and 2^m-ary modulation using sequential and parallel binary channel partitions, then derives labeling and code-construction rules for MLC and BICM.

  • Results

    Optimized BICM polar-code construction significantly outperforms unmodified BICM, but remains below multilevel polar-code performance; Gaussian approximation is less accurate for BICM channels.

  • Takeaways & Limitations

    For polar-coded modulation, the paper concludes that MLC should be preferred over BICM when successive decoding is available.

Abstract

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A framework is proposed that allows for a joint description and optimization of both binary polar coding and $2^m$-ary digital pulse-amplitude modulation (PAM) schemes such as multilevel coding (MLC) and bit-interleaved coded modulation (BICM). The conceptual equivalence of polar coding and multilevel coding is pointed out in detail. Based on a novel characterization of the channel polarization phenomenon, rules for the optimal choice of the labeling in coded modulation schemes employing polar codes are developed. Simulation results regarding the error performance of the proposed schemes on the AWGN channel are included.

I. INTRODUCTION

The paper unifies binary polar coding with 2^m-ary PAM modulation through binary channel partitions, covering both MLC and BICM. This framework supports constellation-dependent code optimization and numerical performance comparisons.

  • Motivation: The paper addresses the previously limited treatment of combining binary polar codes with 2^m-ary digital modulation for higher spectral efficiency.It focuses on PAM-related schemes including MLC and BICM.
  • Unified framework: Channel partitions split a memoryless 2^m-ary channel into m binary-input memoryless bit channels.The resulting bit channels provide a common representation for coded modulation and polar coding.
  • Unified framework: Sequential binary partitions model dependent bit channels for MLC, whereas parallel binary partitions produce independent bit channels applicable to BICM.Both polar coding and polar-coded modulation are represented by concatenated binary partitions.
  • Optimization: The framework enables optimized constellation-dependent coding schemes by addressing the trade-off between power efficiency and spectral efficiency.The optimization applies to both MLC and BICM.
  • Evaluation: The paper develops numerical evaluation methods and compares polar-coded MLC, polar-coded BICM, and LDPC-coded modulation on the AWGN channel.The comparison with LDPC-coded modulation considers the common BICM approach.

B. Product Concatenation of SBPs

Product concatenation combines sequential binary partitions into larger partitions while preserving the mean bit-channel capacity and increasing capacity variance. Linear partitions retain a matrix-based representation through Kronecker products and permutations.

  • Construction: A product SBP is formed when a k1-SBP partitions each of the k2 independent binary channels produced by another transform.The resulting transform has order k1k2 and a fixed bit-channel order.
  • Construction: The product transform is uniquely determined by the individual SBPs because their bit channels impose a fixed order.A concatenation of two 2-SBPs produces a 4-SBP.
  • Capacity properties: Product concatenation preserves the mean value of the bit-channel capacities through the chain rule of mutual information.
  • Capacity properties: Product concatenation increases the variance of the bit-channel capacities.The variance is characterized using the variance of the first transform and the averaged variance of the second transform.
  • Linear transforms: For linear SBPs, the product remains linear and its labeling matrix is constructed with a Kronecker product and a permutation matrix.

C. Parallel Binary Partitions

Parallel binary partitions discard the sequential information transfer between bit channels and produce independent binary-input channels. Their mean capacity depends on the partition and can be lower than that of the corresponding sequential partition.

  • Definition: A parallel binary partition maps a 2^k-ary DMC to a set of independent binary-input memoryless channels.
  • Relation to SBPs: A sequential partition becomes a corresponding parallel partition when lower-index bit-channel information is discarded.The two partitions retain the same labeling rule.
  • Capacity properties: The mean capacity of a parallel partition depends on its specific transform and is generally smaller than that of the corresponding sequential partition.
  • Capacity properties: No general comparison of sequential and parallel partition variances is possible because the parallel mean capacity depends on labeling.

D. Concatenation of PBPs

The paper connects parallel-partition concatenation with BICM and relates polar-code construction to repeated sequential binary partitions. Polar coding recursively combines and splits channels, then selects high-capacity bit channels for information transmission.

  • PBP concatenation: Because parallel-partition output channels are independent, arbitrary permutations can be inserted between concatenated parallel partitions.
  • PBP concatenation: Concatenating a parallel partition with a sequential partition yields a degraded sequential partition whose bit-channel capacities do not sum to the original mutual information.The capacities sum to the mean value associated with the parallel partition.
  • Polar construction: The basic polar transform partitions two independent identical binary channels into two bit channels while preserving their average capacity.The construction is illustrated for N = 2.
  • Polar construction: For polar coding, the length-N construction is equivalently represented by an n-fold product concatenation of the basic 2-SBP when N = 2^n.This produces a sequential partition of the vector channel B^N.
  • Polar construction: The polar transform creates bit channels whose decoding depends on lower-index source symbols, thereby imposing a specific decoding order.
  • Polar construction: Polar coding transmits information only over the highest-capacity bit channels and fixes the remaining channels to known frozen values.This permits code rates in increments of 1/N without changing the code construction.

B. Successive Decoding

Successive cancellation decoding estimates information symbols sequentially, using previously decoded symbols and defining word error through conditional decision errors on information channels.

  • SC decoding estimates information symbols one after another, using previously decoded symbols in each successive decision.
  • The conditional error probability at index i is defined given that all previous decisions were correct.
  • The SC word error rate is formed from the error probabilities of the information channels.

C. Variance of the Bit Channel Capacities

Polarization can be tracked through the variance of bit-channel capacities: variance increases with block length toward its maximum, although its direct relation to error performance remains unestablished.

  • Polarization means that almost every bit-channel capacity approaches either 0 or 1, while the fraction of intermediate channels tends to zero.
  • The variance of bit-channel capacities represents this polarization effect as block length increases.
  • The variance sequence increases monotonically with block length and is bounded above by the maximum attained under perfect polarization.
  • Fig. 3 compares variance across BEC, BSC, and binary-input AWGN channels for multiple block lengths, with a black upper-bound curve.
  • The paper has not established an explicit relation between bit-channel-capacity variance and word or bit error performance.

A. Multilevel Coding

Multilevel coding partitions an M-ary channel into binary levels, assigns component codes and rates to those levels, and combines this construction with polar transforms through sequential binary partitions.

  • MLC partitions an M-ary channel into m binary bit levels using an m-SBP.
  • The labeling rule maps binary labels to amplitude coefficients, while each bit level receives an individual binary component code and rate.
  • Multi-stage decoding processes bit levels sequentially, passing reliability information and earlier decoding results to later levels.
  • Capacity matching assigns each level a code rate corresponding to its bit-level capacity, which can vary substantially across levels.
  • A practical drawback is the need for several comparatively short component codes with varying rates.
  • Multilevel polar coding concatenates the MLC modulation partition with polar-code partitions, selecting the most reliable bit channels as information channels and freezing the rest.
  • The modulation partition acts as the first polarization step, so labeling is chosen to maximize the resulting bit-channel-capacity variance under successive or multi-stage decoding.

C. Multilevel Polar Codes are Capacity-Achieving

Multilevel polar codes combine multilevel coding with polar component codes, achieving coded-modulation capacity asymptotically over arbitrary memoryless M-ary channels. For finite lengths, labeling strongly affects polarization, with SP labeling producing larger bit-level variance than Gray labeling for MLC.

  • Capacity achievement: Multilevel polar codes split an M-ary memoryless channel into m binary bit levels whose capacities sum to the coded-modulation capacity.Each polar component code approaches its corresponding bit-level capacity as block length increases.
  • Capacity achievement: Multilevel polar codes with multi-stage and successive-cancellation decoding achieve coded-modulation capacity for arbitrary M-ary constellations on memoryless channels.The speed-of-convergence results for a single B-DMC also apply to MLC.
  • Labeling: For finite-length codes, labeling has significant performance impact even though the asymptotic capacity result is labeling-independent.The paper selects labeling by targeting larger variance among bit-level capacities.
  • Labeling: SP labeling maximizes successive Euclidean-distance separation and is intended to create widely separated bit-level capacities.This contrasts with Gray labeling, which aims to make bit levels as independent as possible.
  • Labeling: Except at small capacities, SP labeling yields significantly larger bit-level variance than Gray labeling for ASK modulation, so SP is preferred for multilevel polar codes.The increased variance is especially pronounced relative to polar codes over a single B-DMC.
  • Parallel decoding: Under parallel decoding, SP labeling suffers serious degradation, whereas Gray-labeling variance curves remain close to those under multi-stage decoding.Consequently, the BICM setup considered here uses Gray labeling.

B. Bit-Interleaved Polar-Coded Modulation

The BICM construction fixes Gray labeling and optimizes the polar transform rather than using an interleaver. The proposed approach replaces the first polarization steps with an optimized binary transform that maximizes bit-channel variance.

  • Optimization choices: With Gray labeling fixed, BICM polar-code optimization can target either the interleaver or the polar code itself.Prior work found improvement from partial exhaustive interleaver search over random interleaving.
  • Unpermuted construction: The proposed BICM approach uses no interleaver and connects a length-mN polar code directly to the Gray-labeled parallel bit-channel transform.The construction assumes m is a power of two to use Arıkan’s standard construction.
  • Optimized transform: Optimization replaces the standard first m-SBP polarization steps with an m-SBP τ that maximizes the bit-channel variance after Gray labeling.The remaining polar transform is retained after this optimized first stage.
  • Labeling transform: For one-dimensional constellations, natural or SP labeling and binary-reflected Gray labeling can be related by a bijective linear transform.The paper represents both labelings as binary matrices and describes their conversion through a binary matrix.
  • QAM labeling: For square QAM constellations, SP labeling can likewise be converted into Gray labeling through a linear transform related to the length-2 polar generator matrix.The relation is proven in the appendix.

2) QAM Constellations:

For QAM-related constructions, the optimized BICM transform uses a linear transform that supports successive reversal of Gray-labeled symbols. This yields an optimized length-8 example for 16-ASK and shares its labeling rule with the optimal MLC code.

  • QAM labeling: The relation between SP and Gray labeling for square QAM is established through a linear transform involving G2.G2 is the generator matrix of a length-2 polar code.
  • Successive decoding: The transform T^m maps independent binary channels to ordered binary channels and can be successively decoded like the polar transform.Its construction yields SP labeling when concatenated with the Gray-labeled channel transform.
  • Decoding operation: For Gray-labeled symbols, x=[u0⊕u1,u1⊕u2,...,u_m−2⊕u_m−1,u_m−1], enabling successive recovery of u from reliability values and previously estimated bits.The first component is obtained by combining all x components; later components use independent equations and known earlier estimates.
  • Optimized code: The optimized BICM construction uses generator matrix P_4,2·(G2⊗T^4) for a length-8 code with 16-ASK modulation.The permutation P_4,2 is applied to the input vector before encoding.
  • Relation to MLC: The optimized BICM polar code and the optimal SP-labeled multilevel code encode identical binary source symbols into identical transmission symbols.Their bit-metric decoding differs: BICM uses parallel decoding, while MLC uses successive decoding.

VI. SIMULATION RESULTS

The simulations evaluate polar-coded modulation over AWGN using density evolution and Monte Carlo, comparing labelings, MLC, BICM, and DVB-T2. SP-labeled multilevel polar codes outperform Gray-labeled and BICM alternatives, while retaining lower decoding complexity than concatenated LDPC+BCH coding.

  • Evaluation method: Density evolution approximates the bit-level capacities, component-code error probabilities, and maximum achievable rate for each Eb/N0 value.The procedure models component channels as Gaussian channels and applies Gaussian-approximated density evolution.
  • MLC results: 16-ASK simulations show a large performance loss for Gray labeling compared with SP labeling under successive-cancellation decoding.The comparison covers multiple block lengths and uses rate-versus-SNR plots.
  • BICM results: Optimized BICM polar-code construction significantly improves performance over unmodified BICM, but remains below multilevel polar codes.The remaining gap is attributed to BICM's suboptimality and additional Gaussian-approximation error for BICM channels.
  • Complexity: Multilevel polar codes use single-step, non-iterative decoding with fewer information-combining operations than the concatenated DVB-T2 approach.This provides a reduced computational-complexity comparison between the decoding architectures.
  • Conclusion: The study concludes that MLC should be preferred over BICM for polar-coded modulation when successive decoding is available.The conclusion follows the observed BICM degradation relative to the multilevel approach.

APPENDIX A PROOF OF EQUATION (20)

The appendix establishes the stated labeling transformation for square QAM by applying two linear transforms. It shows that a set-partitioned labeling can be converted into Gray labeling through separable row and column transformations.

  • Label representation: The square-QAM labels are represented as binary tuples containing naturally labeled row and column indices.The first and last m bits encode the row and column indices, respectively.
  • Set-partitioning transform: Applying G2⊗I_m produces labels whose first m bits are the component-wise modulo-2 sums of the row and column bits.The resulting labeling is identified as a set-partitioning labeling.
  • Inverse transform: Because G2⊗I_m is self-inverse, applying it to the set-partitioned constellation recovers the original natural row-and-column representation.This reversibility enables the subsequent Gray-labeling transform.
  • Gray labeling: Applying I_2⊗T_m then independently Gray-labels the row and column indices, yielding a Gray-labeled square-QAM constellation.The appendix therefore connects the set-partitioned and Gray-labeled constellations through simple linear transforms.
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