Binary intersymbol interference channels: gallager codes, density evolution, and code performance bounds

Kavcic, A.; Ma, Xiao; Mitzenmacher, Michael

doi:10.1109/tit.2003.813563

Cited by 164 publications

(90 citation statements)

References 45 publications

(126 reference statements)

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“…As we will state more precisely later, the marginal probability density of a message sent over a randomly chosen edge in the FG converges almost surely to a deterministic probability density that depends on the BICM ensemble parameters for large block lengths, provided that the FG is sufficiently local (e.g., the maximum degree of each node is bounded). The proof of this concentration result for BICM-ID closely follows the footsteps of analogous proofs for LDPC codes over BIOS channels [96,97], for LDPC codes over binary-input ISI channels [60], and for linear codes in multiple access Gaussian channels [23]. By way of illustration, consider a BICM ensemble based on the concatenation of a binary convolutional code with a modulation signal set through an interleaver, whose FG is shown in Figure 5.2.…”

Section: Density Evolutionmentioning

confidence: 81%

“…Moreover, the joint probability density of the message vector at a given iteration is a function of the (random) interleaver and of the (random) scrambling sequence. Fortunately, a powerful set of rather general results come to rescue [23,60,96,97]. As we will state more precisely later, the marginal probability density of a message sent over a randomly chosen edge in the FG converges almost surely to a deterministic probability density that depends on the BICM ensemble parameters for large block lengths, provided that the FG is sufficiently local (e.g., the maximum degree of each node is bounded).…”

Section: Density Evolutionmentioning

confidence: 99%

“…As n → ∞, however, one can show that the probability that the neighborhood N (µ k → c i ) is cycle-free for finite can be made arbitrarily close to 1. Averaged over all possible interleavers, the probability that the neighborhood has cycles is bounded as [23,60] 14) as n → ∞ and for some positive α independent of n. The key element in the proof is the locality of the FG, both for the code and for the modulator. Notice that this result is reminiscent of Equation (4.56) in Section 4, which concerned the probability that d bits randomly distributed over n bits belong to different modulation symbols.…”

Section: Density Evolutionmentioning

confidence: 99%

“…The saddlepoint approximation corresponding to the average pairwise error probability PEP(d) of QPSK with codeword length N in Nakagami-m f fading is 60) where…”

Section: Theorem 414 ([73])mentioning

confidence: 99%

“…As done in [23,60], we decode the convolutional code using a windowed BCJR algorithm. This algorithm produces the extrinsic information output for symbol c i by operating over a trellis finite window centered around the symbol position i.…”

Section: Density Evolutionmentioning

confidence: 99%

See 4 more Smart Citations

Bit-Interleaved Coded Modulation

Fàbregas

Martinez

Caire

2007

FNT in Communications and Information Theory

186

165

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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 Ungerböck'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. List of Abbreviations, Acronyms, and SymbolsRatio between average bit energy and noise spectral densityReliability function at rate R E bicm 0 (ρ, s) BICM random coding exponent E cm 0 (ρ) CM random coding exponent ...

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Section: Density Evolutionmentioning

confidence: 81%

Section: Density Evolutionmentioning

confidence: 99%

Section: Density Evolutionmentioning

confidence: 99%

“…The saddlepoint approximation corresponding to the average pairwise error probability PEP(d) of QPSK with codeword length N in Nakagami-m f fading is 60) where…”

Section: Theorem 414 ([73])mentioning

confidence: 99%

Section: Density Evolutionmentioning

confidence: 99%

See 3 more Smart Citations

Bit-Interleaved Coded Modulation

Fàbregas

Martinez

Caire

2007

FNT in Communications and Information Theory

186

165

View full text Add to dashboard Cite

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High-throughput VLSI Implementations of Iterative Decoders and Related Code Construction Problems

Nagarajan

Jayakumar

Khatri

et al. 2007

J VLSI Sign Process Syst Sign Im

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We describe an efficient, fully-parallel Network of Programmable Logic Array (NPLA)-based realization of iterative decoders for structured LDPC codes. The LDPC codes are developed in tandem with the underlying VLSI implementation technique, without compromising chip design constraints. Two classes of codes are considered: one, based on combinatorial objects derived from difference sets and generalizations of non-averaging sequences, and another, based on progressive edge-growth techniques. The proposed implementation reduces routing congestion, a major issue not addressed in prior work. The operating power, delay and chip-size of the circuits are estimated, indicating that the proposed method significantly outperforms presently used standard-cell based architectures. The described LDPC designs can be modified to accommodate widely different requirements, such as those arising in recording systems, as well as wireless and optical data transmission devices.

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