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Bandwidth Efficient and Rate-Matched Low-Density Parity-Check Coded Modulation

Georg Böcherer, Patrick Schulte, Fabian Steiner

arXiv:1502.02733v2cs.IT

TL;DR

The paper tackles coarse rate adaptation and shaping loss in coded modulation. It combines distribution matching, systematic LDPC encoding, and bit-metric decoding for ASK and QAM, without iterative demapping. Over 1–5 bits/s/Hz, the scheme operates within 1 dB of AWGN capacity using selected DVB-S2 code-rate and constellation modes.

  • Problem

    Conventional signaling uses discrete constellation and code-rate modes, producing coarse rate adaptation and a shaping gap relative to capacity.

  • Method

    The scheme concatenates a distribution matcher with a systematic binary LDPC encoder and uses bit-metric decoding for shaped ASK signaling that extends directly to QAM.

  • Results

    Over 1–5 bits/s/Hz, the scheme operates within 1 dB of AWGN capacity; 16-ASK with rate 5/6 has a 0.61–0.69 dB gap, while 64-ASK with rate 9/10 has a 1 dB gap.

  • Takeaways & Limitations

    The results indicate that spectral efficiency can be achieved across the considered range with a small set of DVB-S2 LDPC-coded ASK modes.

Abstract

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A new coded modulation scheme is proposed. At the transmitter, the concatenation of a distribution matcher and a systematic binary encoder performs probabilistic signal shaping and channel coding. At the receiver, the output of a bitwise demapper is fed to a binary decoder. No iterative demapping is performed. Rate adaption is achieved by adjusting the input distribution and the transmission power. The scheme is applied to bipolar amplitude shift keying (ASK) constellations with equidistant signal points and it is directly applicable to two-dimensional quadrature amplitude modulation (QAM). The scheme is implemented by using the DVB-S2 low-density parity-check (LDPC) codes. At a frame error rate of 1e-3, the new scheme operates within less than 1 dB of the AWGN capacity 0.5log2(1+SNR) at any spectral efficiency between 1 and 5 bits/s/Hz by using only 5 modes, i.e., 4-ASK with code rate 2/3, 8-ASK with 3/4, 16-ASK and 32-ASK with 5/6 and 64-ASK with 9/10.

I. INTRODUCTION

The paper addresses coarse SNR-mode granularity and shaping loss by introducing probabilistic shaping with ASK constellations and DVB-S2 LDPC codes. The proposed scheme uses PAS with bit-metric decoding, avoids iterative demapping, and extends directly to QAM.

  • I. INTRODUCTION: AWGN capacity requires adapting transmission rate to SNR, but supported constellation and code-rate combinations create coarse operating modes.The cited discussion attributes the gap to finite-length LDPC coding, uniform input distributions, and resorting to available modes.
  • I. INTRODUCTION: Probabilistic amplitude shaping combines a distribution matcher with a systematic binary LDPC encoder to reduce mode granularity and remove the shaping gap.The transmitter shapes amplitudes while systematic encoding generates sign labels for ASK transmission.
  • I. INTRODUCTION: Bit-metric decoding is used at the receiver without iterative demapping, avoiding the increased complexity associated with iterative inversion of non-uniform mappings.Earlier shaping approaches cited in the paper use iterative demapping or face receiver-side inversion challenges.
  • I. INTRODUCTION: The scheme applies directly to two-dimensional QAM by mapping two real ASK symbols to one complex QAM symbol.This provides a direct extension from the one-dimensional ASK construction to complex modulation.
  • I. INTRODUCTION: Uniformly distributed inputs can be up to 1.53 dB less power efficient than Gaussian inputs on the AWGN channel, motivating probabilistic shaping.The paper contrasts uniform signaling with non-uniform distributions on equidistant signal points.

F. Bootstrap Scheme

The supplied passages describe the AWGN channel, its capacity-based reliability condition, and finite-alphabet signaling under a power constraint. They also state the decoding-error objective and introduce ASK input-distribution design.

  • F. Bootstrap Scheme: The discrete-time AWGN model uses independent zero-mean Gaussian noise terms with variance one and imposes an average input-power constraint.The channel relation and power constraint are stated for n_c channel uses.
  • F. Bootstrap Scheme: Reliable communication over AWGN is possible when the transmission rate is below the channel capacity C(P).The block error probability can be made arbitrarily small with sufficiently large block length under this condition.
  • F. Bootstrap Scheme: Finite-alphabet coded modulation chooses an alphabet and input distribution so mutual information approaches the maximum permitted by the transmission power.The supplied passages then specialize the input alphabet to ASK with 2^m signal points.

C. Optimization of the ASK Input

The paper selects Maxwell-Boltzmann input distributions under a power constraint and optimizes constellation scaling to approach capacity. The resulting shaped inputs provide increasing gains over uniform inputs, while further distribution optimization offers limited benefit.

  • C. Optimization of the ASK Input: Maximizing entropy under the power constraint yields a Maxwell-Boltzmann input distribution for each constellation scaling.The parameter enforcing the power constraint can be found efficiently by bisection.
  • C. Optimization of the ASK Input: The mutual information is optimized over constellation scaling because it is unimodal, allowing efficient solution by the golden section method.The maximizing scaling and corresponding distribution are denoted by ∆♣ and PX♣.
  • C. Optimization of the ASK Input: 1.33 dB shaping gain is achieved for 64-ASK at a rate of 5 bits per channel use compared with a uniformly distributed input.For increasing rates and constellation sizes, the shaping gain approaches the upper bound of approximately 1.53 dB.
  • C. Optimization of the ASK Input: The shaped input PX♣ is within 0.1 dB of capacity at the operating points in Table I, although it is generally suboptimal.The optimal distribution can instead be approximated numerically with the Blahut-Arimoto algorithm.
  • C. Optimization of the ASK Input: Reliable transmission requires rates no larger than the average mutual information between input and output; exceeding it leaves error probability bounded away from zero.The mutual information terms are calculated from the marginal input distributions, motivating a transmitter with the desired marginals.
  • C. Optimization of the ASK Input: The transmitter is developed to realize the marginal input distributions needed to approach the rates associated with the optimized shaped inputs.The paper explicitly frames this as the next transmitter-design step.

A. Preliminaries

The proposed construction factors ASK symbols into amplitudes and signs, then uses systematic binary encoding to generate sign labels from shaped amplitudes. PAS transmits scaled amplitude-sign products while relying on approximately uniform redundant bits.

  • A. Preliminaries: The input is factored as amplitude A = |X♣| and sign S = sign(X♣), with symmetry making the sign uniformly distributed and independent of the amplitude.This factorization supports separate amplitude shaping and sign generation.
  • A. Preliminaries: A systematic generator matrix [Ik|P] copies k data bits and produces n−k redundant bits through the parity matrix P.The identity matrix Ik preserves the data-bit portion of the codeword.
  • A. Preliminaries: Redundant bits are assumed approximately uniform because each is a modulo-two sum of several data bits, including for a broad class of non-uniform data distributions.This is the uniform check bit assumption.
  • A. Preliminaries: PAS uses a rate (m−1)/m binary code to label 2^m-1 amplitudes with m−1 bits and generate nc sign labels from the amplitude labels.For nc channel uses, the code length is n = ncm and dimension is k = nc(m−1).
  • A. Preliminaries: PAS maps iid shaped amplitudes to binary labels, generates signs with the parity matrix, converts labels to signs, and transmits scaled products Xi = Ai · Si.The signs are deterministic functions of the amplitudes, so the transmitted symbols are correlated.
  • A. Preliminaries: PAS assumes equality in its marginal-distribution conditions, while numerical results support that assumption and deviations should prompt checking it.A uniformly distributed independent input bit can make a generated sign label exactly uniform.
  • A. Preliminaries: PAS is a special case of bootstrap shaping that uses uniformly distributed redundancy bits directly as sign labels in the same block.It is also described as a probabilistic version of shell mapping.
  • A. Preliminaries: The sign operation preserves the amplitude-selected power shell because multiplying amplitudes by ±1 does not change power.This connects amplitude-source shaping with the transmitted signal’s power constraint.

C. Optimal Operating Points

Reliable operation is identified where the PAS transmission-rate curve intersects the mutual-information curve. These operating points are evaluated across ASK constellation sizes, with rate adaptation addressed separately.

  • Reliable communication at rate R is achievable only when H(A) < I(AS; Y).
  • For 4-, 8-, 16-, 32-, and 64-ASK, Fig. 4 displays mutual information as solid curves and transmission rate as dashed curves.
  • As SNR increases, mutual information saturates at m bits while transmission rate saturates at m −1 bits.
  • The curve-crossing points indicate optimal operation for block length nc →∞, whereas other transmission rates require rate adaptation.

D. PAS for Higher Code Rates

Extended PAS supports code rates above (m−1)/m by assigning a fraction of sign bits to uniformly distributed data. A distribution matcher replaces the amplitude source, creating a binary interface while approximating the desired amplitude distribution.

  • PAS for Higher Code Rates: A fraction γ of sign bits carries uniformly distributed data, while amplitude-label bits and those sign bits are jointly encoded by a rate-c code.
  • Distribution Matching: The matcher maps kc uniformly distributed input bits to nc amplitudes that approximately follow the desired distribution PA.
  • Distribution Matching: A distribution matcher provides a binary interface, approximately emulates a DMS output, and is invertible.
  • Distribution Matching: The CCDM is fixed-length and avoids buffer overflow, synchronization loss, and error propagation associated with variable-length matchers.
  • Distribution Matching: As nc →∞, the matcher’s normalized informational divergence approaches zero and its rate approaches H(A).
  • Distribution Matching: For finite nc, performance evaluations use the CCDM’s actual rate kc/nc.

VI. BIT-METRIC DECODING

The receiver estimates transmitted codewords from channel outputs using binary decoding rather than symbol-metric decoding. PAS labels amplitudes and signs separately, while a single binary code jointly handles the bit levels.

  • The receiver estimates X^nc from Y^nc using a bit-metric decoder.
  • For 2^m-ASK, PAS represents each amplitude with m−1 bits and each sign with one bit.
  • Because the labeling is one-to-one, selecting a length-m label B uniquely selects a signal point.
  • The mutual-information chain rule motivates multistage decoding, which successively estimates bit levels using previous estimates.
  • Bit-interleaved coded modulation instead jointly encodes all bit levels with one binary code of block length mnc.

B. Bit-Metric Decoding

Bit-metric decoding derives soft information for each labeled bit and uses it for binary decoding. The resulting bit channels depend on labeling and interference from other bit levels.

  • A soft-demapper calculates soft information for each bit level, incorporating channel likelihood and a-priori information.
  • The bit metric L_i is sufficient for estimating B_i from Y, so I(B_i; Y) = I(B_i; L_i).
  • Bit-metric decoding transforms the channel into m parallel binary-input channels pLi|Bi.
  • Different bit levels have different channels, motivating parity-matrix optimization for bit-metric decoding.
  • Other bit levels act as interference because their distribution influences each bit-level channel’s transition probabilities.
  • The achievable rate can vary with the labeling of signal points, as illustrated by the two 8-ASK labelings in Table II.

C. Optimizing the Labeling

The scheme optimizes bit-level labeling because irregular LDPC codes transmit coded bits over different bit-channels. An optimized bit-mapper improves energy efficiency relative to a random mapping.

  • Different bit levels have different channels, and the labeling of signal points changes the achievable bit-metric decoding rate.
  • Iterative demapping or multilevel decoding offers negligible gain in the considered scenario and is not worth the added complexity.
  • 0.6 dB: The optimized bit-mapper is more energy efficient than a randomly chosen bit-mapper.
  • The implementation uses DVB-S2 LDPC codes with four distinct variable-node degrees and places uniformly distributed B1 bits at the codeword end.
  • The optimization searches (m −1)! bit-level interleavers; for 64-ASK, this requires choosing among 120 possibilities.

VIII. RATE ADAPTION

Rate adaption selects amplitude distributions and constellation scaling so the transmission rate matches the achievable bit-metric decoding rate. Practical finite-length operation backs off from the ideal crossing point to reduce errors.

  • The optimal operating point is where the transmission-rate curve R crosses the achievable-rate curve RBMD.
  • The scaling is then selected to obtain the same error probability as a reference operating point.
  • Increasing rate back-off reduces FER for the rate 2/3 DVB-S2 LDPC code, from 1.1 × 10−1 to 2.8 × 10−3.
  • With γ = 0, the feasible transmission-rate range is between 0 and m −1.
  • For a target rate, the amplitude distribution is adjusted so that H(Aλ) + γ equals the desired transmission rate.

C. Adaption for Universal Codes

For universal codes, rate adaption can use rate back-off rather than repeated Monte Carlo searches for constellation scaling. The experiments apply this procedure across operating points using DVB-S2 codes.

  • Universal codes permit constellation scaling to be adapted from the rate back-off, assuming FER depends only on RBMD − R.
  • Because the code rate is constant, FER is determined by the mutual information I(X; Y) for uniformly distributed binary inputs.
  • The evaluation uses Monte Carlo simulation, DVB-S2 LDPC codes, 100 decoding iterations, and optimized bit-mappers.
  • The same constellation-and-code-rate mode is operated over ranges of SNR and spectral efficiency using the rate-adaption procedure.

1) Spectral Efficiency:

The evaluated scheme remains close to AWGN capacity across spectral efficiencies from 1 to 5 bits/s/Hz. The conclusion extends the approach to higher-dimensional QAM while identifying future short-blocklength and MIMO work.

  • Spectral Efficiency: 1 to 5 bits/s/Hz: The scheme operates within 1 dB of AWGN capacity.
  • Spectral Efficiency: 0.61 to 0.69 dB: The 5/6 code with 16-ASK has this capacity gap, while 64-ASK with rate 9/10 has a 1 dB gap.
  • Spectral Efficiency: The effective rate slightly decreases for larger 32-ASK and 64-ASK constellations; for example, target rate 5.00 becomes R = 4.98.
  • Rate: In-block error propagation can occur because one corrupted dematcher input symbol may produce multiple corrupted output bit errors.
  • Conclusions: The proposed scheme achieves 2–10 bit/s/Hz within 1 dB of complex-baseband AWGN capacity using DVB-S2 codes and QAM constellations up to 4096 points.
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