In this correspondence, we investigate in a comprehensive fashion a one-layer coding/shaping sche... more In this correspondence, we investigate in a comprehensive fashion a one-layer coding/shaping scheme resembling a perfectly cooperated multiple-access system. At the transmitter, binary data are encoded by either single-level or multilevel codes. The coded bits are first randomly interleaved and then entered into a signal mapper. At each time, the signal mapper accepts as input multiple binary digits and delivers as output an amplitude signal, where the input are first independently mapped into 2-PAM signals (possibly having different amplitudes) and then superimposed to form the output. The receiver consists of an iterative decoding/demapping algorithm with an entropy-based stopping criterion. In the special cases when all the 2-PAM signals have equal amplitudes, based on an irregular trellis, we propose an optimal soft-input-soft-output (SISO) demapping algorithm with quadratic rather than exponential complexity. In the general cases, when multilevel codes are employed, we propose power-allocation strategies to facilitate the iterative decoding/dempaping algorithm. Using the unequal power-allocations and the Gaussian-approximation-based suboptimal demapping algorithm (with linear complexity), coded modulation with high bandwidth efficiency can be implemented.
We study the limits of performance of Gallager codes (low-density parity-check (LDPC) codes) over... more 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 C denotes the capacity of a binary ISI channel and if C i i d denotes the maximal achievable mutual information rate when the channel inputs are independent and identically distributed (i.i.d.) binary random variables (C i i d C), we prove that the maximum information rate achievable by the sum-product decoder of a Gallager (coset) code is upper-bounded by C i i d . 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.
In this paper, we design capacity-approaching codes for partial response channels. The codes are ... more 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) information rate. Hence, we name it a matched information rate (MIR) design. We provide a set of five 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 thresholds are only fractions of a decibel away from the capacity. Examples of code constructions are shown for channels both with and without spectral nulls.
We study the limits of performance of Gallager codes (low-density parity-check (LDPC) codes) over... more 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 C denotes the capacity of a binary ISI channel and if C i i d denotes the maximal achievable mutual information rate when the channel inputs are independent and identically distributed (i.i.d.) binary random variables (C i i d C), we prove that the maximum information rate achievable by the sum-product decoder of a Gallager (coset) code is upper-bounded by C i i d . 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.
In this paper, we design capacity-approaching codes for partial response channels. The codes are ... more 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) information rate. Hence, we name it a matched information rate (MIR) design. We provide a set of five 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 thresholds 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 last decade has seen the progressive, but in many ways difficult, reconnection of a divided E... more The last decade has seen the progressive, but in many ways difficult, reconnection of a divided Europe. Europeanisation, National Identities and Migration concentrates on the changes in collective identities resulting from European integration and the Eastern enlargement of the EU. This overall process particularly includes the restoration and reorganisation of the European system of nation-states, alongside the redefinition and intensification of national identities and reciprocal boundary constructions.
In this correspondence, we investigate in a comprehensive fashion a one-layer coding/shaping sche... more In this correspondence, we investigate in a comprehensive fashion a one-layer coding/shaping scheme resembling a perfectly cooperated multiple-access system. At the transmitter, binary data are encoded by either single-level or multilevel codes. The coded bits are first randomly interleaved and then entered into a signal mapper. At each time, the signal mapper accepts as input multiple binary digits and delivers as output an amplitude signal, where the input are first independently mapped into 2-PAM signals (possibly having different amplitudes) and then superimposed to form the output. The receiver consists of an iterative decoding/demapping algorithm with an entropy-based stopping criterion. In the special cases when all the 2-PAM signals have equal amplitudes, based on an irregular trellis, we propose an optimal soft-input-soft-output (SISO) demapping algorithm with quadratic rather than exponential complexity. In the general cases, when multilevel codes are employed, we propose power-allocation strategies to facilitate the iterative decoding/dempaping algorithm. Using the unequal power-allocations and the Gaussian-approximation-based suboptimal demapping algorithm (with linear complexity), coded modulation with high bandwidth efficiency can be implemented.
We study the limits of performance of Gallager codes (low-density parity-check (LDPC) codes) over... more 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 C denotes the capacity of a binary ISI channel and if C i i d denotes the maximal achievable mutual information rate when the channel inputs are independent and identically distributed (i.i.d.) binary random variables (C i i d C), we prove that the maximum information rate achievable by the sum-product decoder of a Gallager (coset) code is upper-bounded by C i i d . 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.
In this paper, we design capacity-approaching codes for partial response channels. The codes are ... more 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) information rate. Hence, we name it a matched information rate (MIR) design. We provide a set of five 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 thresholds are only fractions of a decibel away from the capacity. Examples of code constructions are shown for channels both with and without spectral nulls.
We study the limits of performance of Gallager codes (low-density parity-check (LDPC) codes) over... more 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 C denotes the capacity of a binary ISI channel and if C i i d denotes the maximal achievable mutual information rate when the channel inputs are independent and identically distributed (i.i.d.) binary random variables (C i i d C), we prove that the maximum information rate achievable by the sum-product decoder of a Gallager (coset) code is upper-bounded by C i i d . 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.
In this paper, we design capacity-approaching codes for partial response channels. The codes are ... more 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) information rate. Hence, we name it a matched information rate (MIR) design. We provide a set of five 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 thresholds 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 last decade has seen the progressive, but in many ways difficult, reconnection of a divided E... more The last decade has seen the progressive, but in many ways difficult, reconnection of a divided Europe. Europeanisation, National Identities and Migration concentrates on the changes in collective identities resulting from European integration and the Eastern enlargement of the EU. This overall process particularly includes the restoration and reorganisation of the European system of nation-states, alongside the redefinition and intensification of national identities and reciprocal boundary constructions.
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