We study the limits of performance of Gallager codes (low-density parity-check (LDPC) codes) over binary linear intersymbol interference (ISI) channels with additive white Gaussian noise (AWGN). Using the graph representations of the channel, the code, and the sum–product message-passing detector/decoder, we prove two error concentration theorems. Our proofs expand on previous work by handling complications introduced by the channel memory. We circumvent these problems by considering not just linear Gallager codes but also their cosets and by distinguishing between different types of message flow neighborhoods depending on the actual transmitted symbols. We compute the noise tolerance threshold using a suitably developed density evolution algorithm and verify, by simulation, that the thresholds represent accurate predictions of the performance of the iterative sum–product algorithm for finite (but large) block lengths. We also demonstrate that for high rates, the thresholds are very close to the theoretical limit of performance for Gallager codes over ISI channels. If g denotes the capacity of a binary ISI channel and if g � � � denotes the maximal achievable mutual information rate when the channel inputs are independent and identically distributed (i.i.d.) binary random variables @g � � � gA, we prove that the maximum information rate achievable by the sum–product decoder of a Gallager (coset) code is upper-bounded by g � � �. The last topic investigated is the performance limit of the decoder if the trellis portion of the sum–product algorithm is executed only once; this demonstrates the potential for trading off the computational requirements and the performance of the decoder.

The symbol-wise mutual information between the binary inputs of a channel encoder and the soft-outputs of a LogAPP decoder, i.e., the a-posteriori log-likelihood ratios (LLRs), is analyzed. This mutual information can be expressed as the expectation of a function of solely the absolute values of the a-posteriori LLRs. This result provides a simple and elegant method for computing the mutual information by simulation. As opposed to the conventional method, explicit measurements of histograms of the soft-outputs are not necessary. In fact, online estimation is possible, and bits having different statistical properties need not be treated separately. As a direct application, the computation of extrinsic information transfer (EXIT) charts is considered.

In this paper we design capacity-approaching codes for partial response channels. The codes are constructed as concatenations of inner trellis codes and outer low-density parity-check (LDPC) codes. Unlike previous constructions of trellis codes for partial response channels, we disregard any algebraic properties (e.g., the minimum distance or the run-length limit) in our design of the trellis code. Our design is purely probabilistic in that we construct the inner trellis code to mimic the transition probabilities of a Markov process that achieves a high (capacity-approaching) in-formation rate. Hence, we name it a matched information rate (MIR) design. We provide a set of 5 design rules for constructions of capacity-approaching MIR inner trellis codes. We optimize the outer LDPC code using density evolution tools specially modified to fit the superchannel consisting of the inner MIR trellis code concatenated with the partial response channel. Using this strategy, we design degree sequences of irregular LDPC codes whose noise tolerance thresh-olds are only fractions of a decibel away from the capacity. Examples of code constructions are shown for channels both with and without spectral nulls.

The performance loss due to separation of detection and decoding on the binary-input additive white Gaussian noise channel is quantified in terms of mutual information. Results are reported for both the code-division multiple-access (CDMA) channel in the large system limit and the intersymbol interference (ISI) channel. The results for CDMA rely on the replica method developed in statistical mechanics. It is shown that a previous result in [1] found for Gaussian input alphabet holds also for binary input alphabets. For the ISI channel, the performance loss is calculated via the BCJR algorithm. Comparisons are made to the capacity of separate detection and decoding using suboptimum detectors such as a decision-feedback equalizer.

This paper considers the joint iterative decoding of irregular low-density parity-check (LDPC) codes and channels with memory. It begins by introducing a new class of erasure channels with memory, known as generalizederasure channels. For these channels, a single parameter recursion for the density evolution of the joint iterative decoder is derived. This provides a necessary and sucient condition for decoder convergence, and allows the algebraic construction of sequences of LDPC degree distributions. Under certain conditions, these sequences can achieve the symmetric information rate (SIR) of the channel using only iterative decoding. Example code sequences are given for two channels, and it is conjectured that they each achieve the respective SIR. Keywords: joint iterative decoding, erasure channel, capacityachieving, LDPC codes 1.

We propose a simulation-based method to compute the achievable information rates for general multiple-input multiple -output (MIMO) intersymbol interference (ISI) channels with inputs chosen from a finite alphabet. This method is applicable to both deterministic and stochastic channels. As an example of the stochastic MIMO ISI channels, we consider the multiantenna systems over frequency-selective fading channels, and quantify the improvement in the achievable information rates provided by the additional frequency diversity (for both ergodic and nonergodic cases). In addition, we consider the multiaccess multiantenna system and present some results on the achievable information -rate region. As for the deterministic MIMO ISI channels, we use the binary-input multitrack magnetic recording system as an example, which employs multiple write and read heads for data storage. Our results show that the multitrack recording channels have significant advantages over the single-track channels, in terms of the achievable information rates when the intertrack interference is considered. We further consider practical coding schemes over both stochastic and deterministic MIMO ISI channels, and compare their performance with the information -theoretical limits. Specifically, we demonstrate that the performance of the turbo coding/decoding scheme is only about 1.0 dB away from the information-theoretical limits at a bit-error rate of 10 5 for large interleaver lengths.

Abstract—The symmetric information rate of a modulationconstrained transmission system is the information-theoretic limit on performance under the assumption that the inputs are independent and uniformly distributed. The symmetric information rate for continuous-phase frequency-shift keying (CPFSK) over an AWGN channel may be estimated by considering the system to be a finite-state Markov channel and executing a BCJR-like algorithm. In this paper, the estimated symmetric information rate is used along with the exact expression for the 99% power bandwidth to determine the information-theoretic tradeoff between energy and spectral efficiency for CPFSK modulation. Using this tradeoff, the code rate and modulation index are jointly optimized for a particular spectral efficiency and alphabet size. Codes are then designed for the optimized system. The codes are comprised of variable nodes (which represent irregular repetition codes), check nodes (which represent single parity-check codes), and an interleaver connecting the variable and check nodes. The degree distributions of the code are optimized from the system’s EXIT chart by using linear programming. Additional details of the code design, including labeling and interleaver design, are also discussed. Simulation results show that the optimized coded systems achieve bit error rates within 0.4 dB of the informationtheoretic limits at BER = 10 −5.

For a discrete-time, binary-input Gaussian channel with finite intersymbol interference (ISI), we prove that reliable communication can be achieved if and only if E b =N0 > log 2=Gopt , for some constant Gopt that depends on the channel. To determine this constant, we consider the finite-state machine which represents the output sequences of the channel filter when driven by binary inputs. We then define Gopt as the maximum output power achieved by a simple cycle in this graph, and show that no other cycle or asymptotically long sequence can achieve an output power greater than this.

During the past five years, advances in the information-theoretic analysis of “one-dimensional (1D) ” recording channels have clarified the limits on linear densities that can be achieved by track-oriented magnetic and optical storage technologies. Channel architectures incorporating powerful codes, such as turbo codes and low-density parity-check codes, have been shown to achieve performance very close to the information-theoretic limits. As 1D track-oriented data storage technologies reach maturity, there is increasing interest in “two-dimensional (2D) ” recording technologies, such as two-dimensional optical storage (TwoDOS) and holographic storage. This paper provides an overview of some recently developed techniques for determining analytical bounds and simulation-based estimates for achievable densities of such 2D recording channels, as well as some recently proposed signal processing and coding methods that can move system performance closer to the information-theoretic limits.

Codes on sparse graphs have been shown to achieve remarkable performance in point-to-point channels with low decoding complexity. Most of the results in this area are based on experimental evidence and/or approximate analysis. The question of whether codes on sparse graphs can achieve the capacity of noisy channels with iterative decoding is still open, and has only been conclusively and positively answered for the binary erasure channel. On the other hand, codes on sparse graphs have been proven to achieve the capacity of memoryless, binary-input, output-symmetric channels with finite graphical complexity per information bit when maximum likelihood (ML) decoding is performed. In this paper, we consider transmission over finite-state channels (FSCs). We derive upper bounds on the average error probability of code ensembles with ML decoding. Based on these bounds we show that codes on sparse graphs can achieve the symmetric information rate (SIR) of FSCs, which is the maximum achievable rate with independently and uniformly distributed input sequences. In order to achieve rates beyond the SIR, we consider a simple quantization scheme that when applied to ensembles of codes on sparse graphs induces a Markov distribution on the transmitted sequence. By deriving average error probability bounds for these quantized code ensembles, we prove that they can achieve the information rates corresponding to the induced Markov distribution, and thus approach the FSC capacity. I.