Construction of Capacity-Achieving Lattice Codes: Polar Lattices

In this paper, we propose a new class of lattices constructed from polar codes, namely polar lattices, to achieve the capacity \frac{1}{2}\log(1+\SNR) of the additive white Gaussian-noise (AWGN) channel. Our construction follows the multilevel approach of Forney \textit{et al.}, where we construct a capacity-achieving polar code on each level. The component polar codes are shown to be naturally nested, thereby fulfilling the requirement of the multilevel lattice construction. We prove that polar lattices are \emph{AWGN-good}. Furthermore, using the technique of source polarization, we propose discrete Gaussian shaping over the polar lattice to satisfy the power constraint. Both the construction and shaping are explicit, and the overall complexity of encoding and decoding is $O(N\log N)$ for any fixed target error probability.Comment: full version of the paper to appear in IEEE Trans. Communication

On performance analysis and implementation issues of iterative decoding for graph based codes

There is no doubt that long random-like code has the potential to achieve good performance because of its excellent distance spectrum. However, these codes remain useless in practical applications due to the lack of decoders rendering good performance at an acceptable complexity. The invention of turbo code marks a milestone progress in channel coding theory in that it achieves near Shannon limit performance by using an elegant iterative decoding algorithm. This great success stimulated intensive research oil long compound codes sharing the same decoding mechanism. Among these long codes are low-density parity-check (LDPC) code and product code, which render brilliant performance. In this work, iterative decoding algorithms for LDPC code and product code are studied in the context of belief propagation. A large part of this work concerns LDPC code. First the concept of iterative decoding capacity is established in the context of density evolution. Two simulation-based methods approximating decoding capacity are applied to LDPC code. Their effectiveness is evaluated. A suboptimal iterative decoder, Max-Log-MAP algorithm, is also investigated. It has been intensively studied in turbo code but seems to be neglected in LDPC code. The specific density evolution procedure for Max-Log-MAP decoding is developed. The performance of LDPC code with infinite block length is well-predicted using density evolution procedure. Two implementation issues on iterative decoding of LDPC code are studied. One is the design of a quantized decoder. The other is the influence of mismatched signal-to-noise ratio (SNR) level on decoding performance. The theoretical capacities of the quantized LDPC decoder, under Log-MAP and Max-Log-MAP algorithms, are derived through discretized density evolution. It is indicated that the key point in designing a quantized decoder is to pick a proper dynamic range. Quantization loss in terms of bit error rate (BER) performance could be kept remarkably low, provided that the dynamic range is chosen wisely. The decoding capacity under fixed SNR offset is obtained. The robustness of LDPC code with practical length is evaluated through simulations. It is found that the amount of SNR offset that can be tolerated depends on the code length. The remaining part of this dissertation deals with iterative decoding of product code. Two issues on iterative decoding of\u27 product code are investigated. One is, \u27improving BER performance by mitigating cycle effects. The other is, parallel decoding structure, which is conceptually better than serial decoding and yields lower decoding latency

Achievable Information Rates for Coded Modulation with Hard Decision Decoding for Coherent Fiber-Optic Systems

We analyze the achievable information rates (AIRs) for coded modulation schemes with QAM constellations with both bit-wise and symbol-wise decoders, corresponding to the case where a binary code is used in combination with a higher-order modulation using the bit-interleaved coded modulation (BICM) paradigm and to the case where a nonbinary code over a field matched to the constellation size is used, respectively. In particular, we consider hard decision decoding, which is the preferable option for fiber-optic communication systems where decoding complexity is a concern. Recently, Liga \emph{et al.} analyzed the AIRs for bit-wise and symbol-wise decoders considering what the authors called \emph{hard decision decoder} which, however, exploits \emph{soft information} of the transition probabilities of discrete-input discrete-output channel resulting from the hard detection. As such, the complexity of the decoder is essentially the same as the complexity of a soft decision decoder. In this paper, we analyze instead the AIRs for the standard hard decision decoder, commonly used in practice, where the decoding is based on the Hamming distance metric. We show that if standard hard decision decoding is used, bit-wise decoders yield significantly higher AIRs than symbol-wise decoders. As a result, contrary to the conclusion by Liga \emph{et al.}, binary decoders together with the BICM paradigm are preferable for spectrally-efficient fiber-optic systems. We also design binary and nonbinary staircase codes and show that, in agreement with the AIRs, binary codes yield better performance.Comment: Published in IEEE/OSA Journal of Lightwave Technology, 201

Physical-Layer Cooperation in Coded OFDM Relaying Systems

Mobile communication systems nowadays require ever-increasing data rate and coverage of wide areas. One promising approach to achieve this goal is the application of cooperative communications enabled by introducing intermediate nodes known as relays to support the transmission between terminals. By processing and forwarding the receive message at the relays, the path-loss effect between the source and the destination is mitigated. One major limit factor for relay assisted communications is that a relay cannot transmit and receive using the same physical resources. Therefore, a half-duplex constraint is commonly assumed resulting in halved spectral efficiency. To combat this drawback, two-way relaying is introduced, where two sources exchange information with each. On the other hand, due to the physical limitation of the relays, e.g., wireless sensor nodes, it's not possible to implement multiple antennas at one relay, which prohibits the application of multiple-input multiple-output (MIMO) techniques. However, when treating multiple relays as a cluster, a virtual antenna array is formed to perform MIMO techniques in a distributed manner. %This thesis aims at designing efficient one-way and two-way relaying schemes. Specifically, existing schemes from the literature are improved and new schemes are developed with the emphasis on coded orthogonal frequency division multiplexing (OFDM) transmissions. Of special interest is the application of physical-layer network coding (PLNC) for two-phase two-way relaying. In this case, a network coded message is estimated from the superimposed receive signal at the relay using PLNC schemes. The schemes are investigated based on a mutual information analysis and their performance are improved by a newly proposed phase control strategy. Furthermore, performance degradation due to system asynchrony is mitigated depending on different PLNC schemes. When multiple relays are available, novel cooperation schemes allowing information exchange within the relay cluster are proposed that facilitate distributed MIMO reception and transmission. Additionally, smart signaling approaches are presented to enable the cooperation at different levels with the cooperation overhead taken into account adequately in system performance evaluation

Construction of lattices for communications and security

In this thesis, we propose a new class of lattices based on polar codes, namely polar lattices. Polar lattices enjoy explicit construction and provable goodness for the additive white Gaussian noise (AWGN) channel, \textit{i.e.}, they are \emph{AWGN-good} lattices, in the sense that the error probability (for infinite lattice coding) vanishes for any fixed volume-to-noise ratio (VNR) greater than $2\pi e$. Our construction is based on the multilevel approach of Forney \textit{et al.}, where on each level we construct a capacity-achieving polar code. We show the component polar codes are naturally nested, thereby fulfilling the requirement of the multilevel lattice construction. We present a more precise analysis of the VNR of the resultant lattice, which is upper-bounded in terms of the flatness factor and the capacity losses of the component codes. The proposed polar lattices are efficiently decodable by using multi-stage decoding. Design examples are presented to demonstrate the superior performance of polar lattices. However, there is no infinite lattice coding in the practical applications. We need to apply the power constraint on the polar lattices which generates the polar lattice codes. We prove polar lattice codes can achieve the capacity \frac{1}{2}\log(1+\SNR) of the power-constrained AWGN channel with a novel shaping scheme. The main idea is that by implementing the lattice Gaussian distribution over the AWGN-good polar lattices, the maximum error-free transmission rate of the resultant coding scheme can be arbitrarily close to the capacity \frac{1}{2}\log(1+\SNR). The shaping technique is based on discrete lattice Gaussian distribution, which leads to a binary asymmetric channel at each level for the multilevel lattice codes. Then it is straightforward to employ multilevel asymmetric polar codes which is a combination of polar lossless source coding and polar channel coding. The construction of polar codes for an asymmetric channel can be converted to that for a related symmetric channel, and it turns out that this symmetric channel is equivalent to an minimum mean-square error (MMSE) scaled $\Lambda/\Lambda'$ channel in lattice coding in terms of polarization, which eventually simplifies our coding design. Finally, we investigate the application of polar lattices in physical layer security. Polar lattice codes are proved to be able to achieve the strong secrecy capacity of the Mod-$\Lambda$ AWGN wiretap channel. The Mod-$\Lambda$ assumption was due to the fact that a practical shaping scheme aiming to achieve the optimum shaping gain was missing. In this thesis, we use our shaping scheme and extend polar lattice coding to the Gaussian wiretap channel. By employing the polar coding technique for asymmetric channels, we manage to construct an AWGN-good lattice and a secrecy-good lattice with optimal shaping simultaneously. Then we prove the resultant wiretap coding scheme can achieve the strong secrecy capacity for the Gaussian wiretap channel.Open Acces