Bit-interleaved Coded Modulation Capacity Research Articles

Analysis and Design of QC-LDPC Coded BICM Ensembles Based on RCA Density Evolution

This paper presents density evolution (DE) techniques based on reciprocal channel approximation (RCA) for general bit-interleaved coded modulation (BICM) systems. The original RCA-based DE (RCA-DE) technique is implemented under the assumption of binary-input additive white Gaussian noise channels (BI-AWGNCs). In BICM systems, however, <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-ary modulation schemes are generally used, and they can be regarded as to be a parallel transmission of <inline-formula> <tex-math notation="LaTeX">$\log _{2} M$ </tex-math></inline-formula> bits. Thus, <inline-formula> <tex-math notation="LaTeX">$\log _{2} M$ </tex-math></inline-formula> separate bit-level channels, which are not Gaussian equivalent, need to be considered. In order to extend the conventional RCA-DE technique to BICM systems, we first establish a model of protograph BICM ensembles. Two methods are developed based on the bit error rate (BER) and the BICM capacity in order to find a corresponding BI-AWGNC that models each of the bit-level channels. Using these methods, we implement the protograph RCA-DE technique for BICM systems and show that it achieves an accurate estimation. As a practical application of the proposed RCA-DE method, we design bit interleavers in order to achieve better BICM performance in the 3GPP New Radio (NR) LDPC coding system. Numerical results show that performance gain of up to 0.3 dB is consistently achieved over a wide range of parameter values.

Bandwidth-Efficient Coded Modulation Schemes for Physical-Layer Network Coding with High-Order Modulations

This paper presents several soft decision iterative decoding schemes for physical-layer network coding (PNC) operated with coded modulation (CM) and bit-interleaved coded modulation (BICM). With respect to PNC operated with CM, we consider network coding-based channel decoding (NC-CD) and multi-user complete decoding (MUD-NC) for PNC decoding at the relay. Their BICM counterparts are XOR-based channel decoding (XOR-CD) and MUD-XOR, respectively. First, we show that, when the decoding is non-iterative, there is a gap between the BICM capacities of both XOR-CD and MUD-XOR under Gray mapping and the capacities of their CM counterparts, NC-CD, and MUD-NC. This is in contrast to the conventional point-to-point communication system, for which the BICM capacity with Gray mapping is known to be very close to the CM capacity, without the need for iterative decoding. Second, we investigate the error performance of iteratively decoded BICM XOR-CD and MUD-XOR. Extrinsic information transfer chart analysis and simulation results indicate that for these Gray-mapped BICM PNC systems, iterative decoding can achieve considerable gains over non-iterative decoding. Again, this is in contrast to the Gray-mapped BICM point-to-point communication system, for which iterative decoding provides little gain over non-iterative decoding. We further show that Gray mapping gives rise to best PNC rate for MUD-XOR and XOR-CD systems among several bits-to-symbol mappings under study. Overall, our results indicate that BICM PNC systems exhibit different decoding behavior from conventional BICM point-to-point systems. This paper serves as a first foray into the investigation of this issue.

BICMC and TD Comparative Performance Study of 16-APSK Signal Variants for DVB-S2 Systems

A comparative performance study, in terms of Bit Interleaved Coded Modulation Capacity (BICMC) and Total Degradation (TD), between Amplitude Phase Shift Keying (APSK) signal variants for the 2nd generation Digital Video Broadcasting via Satellite (DVB-S2) system is presented. Motivated by the observation that the presence of more signal points in the outer ring of 16-APSK signals appears to improve the performance in a non-linear (NL) channel, we investigate the performance of 2-14 and 3-13 APSK signals with two novel bit-to-symbol mappings as alternatives to the 4-12 APSK, which is currently proposed as the modulation scheme for the DVB-S2 standard. Performance evaluation results are obtained for an air-interface based on this standard, which includes a Non-Linear (NL) High Power Amplifier (HPA), predistortion, raised cosine Nyquist filters and Low-Density Parity-Check (LDPC) codes. Performance results have shown that the proposed 3-13 APSK signal not only achieves the best BICMC performance with higher gains occurring at high signal-to-noise ratios, but also outperforms, in terms of TD performance, all the other considered 16-ary APSK signal variants.

A Simple Approximation for the Bit-Interleaved Coded Modulation Capacity

The generalized mutual information (GMI) is an achievable rate for bit-interleaved coded modulation (BICM) and is highly dependent on the binary labeling of the constellation. The BICM-GMI, sometimes called the BICM capacity, can be evaluated numerically. This approach, however, becomes impractical when the number of constellation points and/or the constellation dimensionality grows, or when many different labelings are considered. A simple approximation for the BICM-GMI based on the area theorem of the demapper's extrinsic information transfer (EXIT) function is proposed. Numerical results show the proposed approximation gives good estimates of the BICM-GMI for labelings with close to linear EXIT functions, which includes labelings of common interest, such as the natural binary code, binary reflected Gray code, etc. This approximation is used to optimize the binary labeling of the 32-APSK constellation defined in the DVB-S2 standard. Gains of approximately 0.15 dB are obtained.

Signal Shaping for BICM at Low SNR

The generalized mutual information (GMI) of bit-interleaved coded modulation (BICM) systems, sometimes called the BICM capacity, is investigated at low signal-to-noise ratio (SNR). The combinations of input alphabet, input distribution, and binary labeling that achieve the Shannon limit <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${- 1.59}\;{\rm dB}$</tex></formula> are completely characterized. The main conclusion is that a BICM system with probabilistic shaping achieves the Shannon limit at low SNR if and only if it can be represented as a zero-mean linear projection of a hypercube. Hence, probabilistic shaping offers no extra degrees of freedom to optimize the low-SNR BICM-GMI, in addition to what is provided by geometrical shaping. The analytical conclusions are confirmed by numerical results, which also show that for a fixed input alphabet, probabilistic shaping can improve the BICM-GMI in the low and medium SNR range.

Rotated Multi-D Constellations in Rayleigh Fading: Mutual Information Improvement and Pragmatic Approach for Near-Capacity Performance in High-Rate Regions

This paper studies the mutual information improvement attained by rotated multidimensional (multi-D) constellations via a unitary precoder G in Rayleigh fading. At first, based on the symmetric cut-off rate of the N-D signal space, we develop a design criterion with regard to the precoder G. It is then demonstrated that the use of rotated constellations in only a reasonably low dimensional signal space can significantly increase the mutual information in high-rate regimes. Based on parameterizations of unitary matrices, we then construct good unitary precoder G in 4-D signal space using a simple optimization problem, which involves only four real variables and it is applicable to any modulation scheme. To further illustrate the potential of multi-D constellation and to show the practical use of mutual information improvement, we propose a simple yet powerful bit-interleaved coded modulation (BICM) scheme in which a (multi-D) mapping technique employed in a multi-D rotated constellation is concatenated with a short-memory high-rate convolutional code. By using extrinsic information transfer (EXIT) charts, it is shown that the proposed technique provides an exceptionally good error performance. In particular, both EXIT chart analysis and simulation results indicate that a turbo pinch-off and a bit error rate around 10-6 happen at a signal-to-noise ratio that is well below the coded modulation and BICM capacities using traditional signal sets. For example, with code rates ranging from 2/3 to 7/8, the proposed system can operate 0.82 dB-2.93 dB lower than the BICM capacity with QPSK and Gray labeling. The mutual information gain offered by rotated constellations can be therefore utilized to design simple yet near Shannon limit systems in the high-rate regions.

Optimal Alphabets and Binary Labelings for BICM at Low SNR

Optimal binary labelings, input distributions, and input alphabets are analyzed for the so-called bit-interleaved coded modulation (BICM) capacity, paying special attention to the low signal-to-noise ratio (SNR) regime. For 8-ary pulse amplitude modulation (PAM) and for 0.75 bit/symbol, the folded binary code results in a higher capacity than the binary reflected gray code (BRGC) and the natural binary code (NBC). The 1 dB gap between the additive white Gaussian noise (AWGN) capacity and the BICM capacity with the BRGC can be almost completely removed if the input symbol distribution is properly selected. First-order asymptotics of the BICM capacity for arbitrary input alphabets and distributions, dimensions, mean, variance, and binary labeling are developed. These asymptotics are used to define first-order optimal (FOO) constellations for BICM, i.e. constellations that make BICM achieve the Shannon limit $-1.59 \tr{dB}$. It is shown that the $\Eb/N_0$ required for reliable transmission at asymptotically low rates in BICM can be as high as infinity, that for uniform input distributions and 8-PAM there are only 72 classes of binary labelings with a different first-order asymptotic behavior, and that this number is reduced to only 26 for 8-ary phase shift keying (PSK). A general answer to the question of FOO constellations for BICM is also given: using the Hadamard transform, it is found that for uniform input distributions, a constellation for BICM is FOO if and only if it is a linear projection of a hypercube. A constellation based on PAM or quadrature amplitude modulation input alphabets is FOO if and only if they are labeled by the NBC; if the constellation is based on PSK input alphabets instead, it can never be FOO if the input alphabet has more than four points, regardless of the labeling.

Distribution of Max-Log Metrics for QAM-Based BICM in Fading Channels

In this letter we derive closed-form expressions for the probability density functions (PDFs) of the bits' reliability metrics (L-values) in bit-interleaved coded modulation (BICM) transmission over fully-interleaved fading channels. The expressions are valid for the relevant case of quadrature amplitude modulation (QAM) with Gray mapping when the metrics are calculated via the so-called max-log approximation. Using the developed expressions, the performance of coded BICM transmissions is efficiently evaluated, i.e., without resorting to otherwise required two-dimensional numerical integration. The BICM capacity for different fading channels and constellation sizes is also evaluated.

Distribution of L-values in gray-mapped M2-QAM: closed-form approximations and applications

In this paper we develop closed form approximations for the probability density function (PDF) of the reliability metrics (L-values) in bit-interleaved coded modulation (BICM). The expressions are valid for M2-ary quadrature amplitude modulations (M2-QAM) with binary reflected Gray mapping when the metrics are calculated using the so-called max-log approximation. Based on the developed expressions, we also propose two simple Gaussian mixture approximations that are analytically tractable. We apply our developments to efficiently calculate the BICM capacity, and to develop bounds on the coded bit-error rate when a convolutional code is used. The coded performance of a hybrid automatic repeat request (HARQ) based on constellation rearrangement is also evaluated.

Bit-Interleaved Coded Modulation Revisited: A Mismatched Decoding Perspective

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We revisit the information-theoretic analysis of bit-interleaved coded modulation (BICM) by modeling the BICM decoder as a mismatched decoder. The mismatched decoding model is well defined for finite, yet arbitrary, block lengths, and naturally captures the channel memory among the bits belonging to the same symbol. We give two independent proofs of the achievability of the BICM capacity calculated by Caire <etal/>, where BICM was modeled as a set of independent parallel binary-input channels whose output is the bitwise log-likelihood ratio. Our first achievability proof uses typical sequences, and shows that due to the random coding construction, the interleaver is not required. The second proof is based on the random coding error exponents with mismatched decoding, where the largest achievable rate is the generalized mutual information. Moreover, the generalized mutual information of the mismatched decoder coincides with the infinite-interleaver BICM capacity. We show that the error exponent—and hence the cutoff rate—of the BICM mismatched decoder is upper-bounded by that of coded modulation and may thus be lower than in the infinite-interleaved model; for binary reflected Gray mapping in Gaussian channels the loss in error exponent is small. We also consider the mutual information appearing in the analysis of iterative decoding of BICM with extrinsic information transfer (EXIT) charts: if the symbol metric has knowledge of the transmitted symbol, EXIT mutual information admits a representation as a pseudo-generalized mutual information, which is in general not achievable. A different symbol decoding metric, for which the extrinsic side information refers to the hypothesized symbol, induces a generalized mutual information lower than the coded modulation capacity. In this case, perfect extrinsic side information turns the mismatched-decoder error exponent into that of coded modulation. </para>

Asymptotically optimal mappings for BICM with M-PAM and M2-QAM

The asymptotic optimality of particular binary labelling rules is proved for M-PAM and M2-QAM constellations in the wideband regime in terms of the so-called parallel-decoding capacity, also known as bit-interleaved coded modulation capacity. Shannon&apos;s ultimate limit of −1.59 dB can be achieved with bit-interleaved coded modulation.

Bit-Interleaved Coded Modulation in the Wideband Regime

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The wideband regime of bit-interleaved coded modulation (BICM) in Gaussian channels is studied. The Taylor expansion of the coded modulation capacity for generic signal constellations at low signal-to-noise ratio (SNR) is derived and used to determine the corresponding expansion for the BICM capacity. Simple formulas for the minimum energy per bit and the wideband slope are given. BICM is found to be suboptimal in the sense that its minimum energy per bit can be larger than the corresponding value for coded modulation schemes. The minimum energy per bit using standard Gray mapping on <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$M$</tex></formula></emphasis>-PAM or <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$M^2$</tex> </formula></emphasis>-QAM is given by a simple formula and shown to approach <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">${-}$</tex> </formula></emphasis> 0.34 dB as <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$M$</tex></formula></emphasis> increases. Using the low SNR expansion, a general tradeoff between power and bandwidth in the wideband regime is used to show how a power loss can be traded off against a bandwidth gain. </para>

Bit-Interleaved Coded Modulation

The principle of coding in the signal space follows directly from Shannon's analysis of waveform Gaussian channels subject to an input constraint. The early design of communication systems focused separately on modulation, namely signal design and detection, and error correcting codes, which deal with errors introduced at the demodulator of the underlying waveform channel. The correct perspective of signal-space coding, although never out of sight of information theorists, was brought back into the focus of coding theorists and system designers by Imai's and Ungerbock's pioneering works on coded modulation. More recently, powerful families of binary codes with a good tradeoff between performance and decoding complexity have been (re-)discovered. Bit-Interleaved Coded Modulation (BICM) is a pragmatic approach combining the best out of both worlds: it takes advantage of the signal-space coding perspective, whilst allowing for the use of powerful families of binary codes with virtually any modulation format. BICM avoids the need for the complicated and somewhat less flexible design typical of coded modulation. As a matter of fact, most of today's systems that achieve high spectral efficiency such as DSL, Wireless LANs, WiMax and evolutions thereof, as well as systems based on low spectral efficiency orthogonal modulation, feature BICM, making BICM the de-facto general coding technique for waveform channels. The theoretical characterization of BICM is at the basis of efficient coding design techniques and also of improved BICM decoders, e.g., those based on the belief propagation iterative algorithm and approximations thereof. In this text, we review the theoretical foundations of BICM under the unified framework of error exponents for mismatched decoding. This framework allows an accurate analysis without any particular assumptions on the length of the interleaver or independence between the multiple bits in a symbol. We further consider the sensitivity of the BICM capacity with respect to the signal-to-noise ratio (SNR), and obtain a wideband regime (or low-SNR regime) characterization. We review efficient tools for the error probability analysis of BICM that go beyond the standard approach of considering infinite interleaving and take into consideration the dependency of the coded bit observations introduced by the modulation. We also present bounds that improve upon the union bound in the region beyond the cutoff rate, and are essential to characterize the performance of modern randomlike codes used in concatenation with BICM. Finally, we turn our attention to BICM with iterative decoding, we review extrinsic information transfer charts, the area theorem and code design via curve fitting. We conclude with an overview of some applications of BICM beyond the classical coherent Gaussian channel.

Improved decoding of LDPC coded modulations

A coded modulation belief propagation (CMBP) decoder is proposed for decoding LDPC codes with multilevel modulations. The decoder takes into account statistical dependencies among bits originating in the same symbol, providing better performance than the marginal BP (MBP) decoder. Asymptotically it converges to MAP decoding. The CMBP decoder is based on a single-level coding (SLC) scheme and does not suffer from practical disadvantages of multi-level coding (MLC) schemes. Furthermore, the CMBP decoder can close the capacity gap of the bit interleaved coded modulation (BICM) SLC scheme. The BICM capacity gap increases when the modulation size increases and in scenarios where gray mapping is not possible.

Signalling constellations for power-efficient bit-interleaved coded modulation schemes

Bit-interleaved coded modulation (BICM) is a bandwidth-efficient coding technique consisting of serial concatenation of binary error-correcting coding, bit-by-bit interleaving, and high-order modulation. The author addresses the problem of finding the signal sets that are most suitable for designing power-efficient BICM schemes over an additive white Gaussian noise channel. To this end, the expression of the BICM capacity limit is exploited and evaluated for several 8- and 16-ary constellations. It is also shown that the bit error probability curves of the modulation schemes without coding can be used to determine the most attractive constellations. Finally, the bit error rate performance of some BICM schemes made up of turbo codes and various signal sets is investigated to illustrate the theoretical results.

Signal constellations for bit-interleaved coded modulation

Bit-interleaved coded modulation (BICM) is a bandwidth-efficient coding technique consisting of serial concatenation of binary error-correcting coding, bit-by-bit interleaving, and high-order modulation. BICM is capable of achieving excellent error performance provided that powerful codes, such as for example turbo codes or low-density parity-check (LDPC) codes, are employed. We address the problem of finding the signal sets that are the most suitable ones for designing power-efficient BICM schemes over an additive white Gaussian noise (AWGN) channel. To this end, we exploit the expression of the BICM capacity limit, and evaluate it for several 8- and 16-ary constellations. The bit-error rate (BER) performance of some BICM schemes made up of turbo codes and various signal sets is also investigated by computer simulations so as to illustrate the theoretical results. We show that, for spectral efficiencies of practical interest, the most attractive signal sets are those for which Gray mapping is possible, provided that their symbol error rate performance is "sufficiently close" to the optimum. This explains why some constellations having a simple structure, such as 8-PSK and 16-QAM, perform very well when combined with a powerful code. At the same time, the constellations displaying optimal error performance without coding are, generally, not of interest for BICM.