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Vahid Aref

Publications and source records attributed to Vahid Aref.

48 records · Page 3Linked to original sources
arXiv2017

Non-Uniformly Coupled LDPC Codes: Better Thresholds, Smaller Rate-loss, and Less Complexity

We consider spatially coupled low-density parity-check codes with finite smoothing parameters. A finite smoothing parameter is important for designing practical codes that are decoded using low-complexity windowed decoders. By optimizing the amount of coupling between spatial positions, we show that we can construct codes with excellent thresholds and small rate loss, even with the lowest possible smoothing parameter and large variable node degrees, which are required for low error floors. We also establish that the decoding convergence speed is faster with non-uniformly coupled codes, which we verify by density evolution of windowed decoding with a finite number of iterations. We also show that by only slightly increasing the smoothing parameter, practical codes with potentially low error floors and thresholds close to capacity can be constructed. Finally, we give some indications on protograph designs.

cs.IT↗
arXiv2016

Spatially Coupled LDPC Codes Affected by a Single Random Burst of Erasures

Spatially-Coupled LDPC (SC-LDPC) ensembles achieve the capacity of binary memoryless channels (BMS), asymptotically, under belief-propagation (BP) decoding. In this paper, we study the BP decoding of these code ensembles over a BMS channel and in the presence of a single random burst of erasures. We show that in the limit of code length, codewords can be recovered successfully if the length of the burst is smaller than some maximum recoverable burst length. We observe that the maximum recoverable burst length is practically the same if the transmission takes place over binary erasure channel or over binary additive white Gaussian channel with the same capacity. Analyzing the stopping sets, we also estimate the decoding failure probability (the error floor) when the code length is finite.

cs.IT↗
arXiv2016

Control and Detection of Discrete Spectral Amplitudes in Nonlinear Fourier Spectrum

Nonlinear Fourier division Multiplexing (NFDM) can be realized from modulating the discrete nonlinear spectrum of an $N$-solitary waveform. To generate an $N$-solitary waveform from desired discrete spectrum (eigenvalue and discrete spectral amplitudes), we use the Darboux Transform. We explain how to the norming factors must be set in order to have the desired discrete spectrum. To derive these norming factors, we study the evolution of nonlinear spectrum by adding a new eigenvalue and its spectral amplitude. We further simplify the Darboux transform algorithm. We propose a novel algorithm (to the best of our knowledge) to numerically compute the nonlinear Fourier Transform (NFT) of a given pulse. The NFT algorithm, called forward-backward method, is based on splitting the signal into two parts and computing the nonlinear spectrum of each part from boundary ($\pm\infty$) inward. The nonlinear spectrum (discrete and continuous) derived from efficiently combining both parts has a promising numerical precision. This method can use any of one-step discretization NFT methods, e.g. Crank-Nicolson, as an NFT kernel for the forward or backward part. Using trapezoid rule of integral, we use an NFT kernel (we called here Trapezoid discretization NFT) in forward-backward method which results discrete spectral amplitudes with a very good numerical precision. These algorithms, forward-backward method and Darboux transform, are used in [1],[2] for design and detection of phase-modulated 2-soliton pulses, and more recently, in [3] for design and detection of more complex pulses with 7 eigenvalues and modulation of spectral phase. For those soliton pulses, the discrete spectral amplitudes (in particular, phase) of both eigenvalues are quite precisely estimated using the forward-backward method.

math.NA↗
arXiv2015

Triggering Wave-Like Convergence of Tail-biting Spatially Coupled LDPC Codes

Spatially coupled low-density parity-check (SC-LDPC) codes can achieve the channel capacity under low-complexity belief propagation (BP) decoding, however, there is a non-negligible rate-loss because of termination effects for practical finite coupling lengths. In this paper, we study how we can approach the performance of terminated SC-LDPC codes by random shortening of tail-biting SC-LDPC codes. We find the minimum required rate-loss in order to achieve the same performance than terminated codes. We additionally study the use of tail-biting SC-LDPC codes for transmission over parallel channels (e.g., bit-interleaved-coded-modulation (BICM)) and investigate how the distribution of the coded bits between two parallel channels can change the performance of the code. We show that a tail-biting SC-LDPC code can be used with BP decoding almost anywhere within the achievable region of MAP decoding. The optimization comes with a mandatory buffer at the encoder side. We evaluate different distributions of coded bits in order to reduce this buffer length.

cs.IT↗
arXiv2015

Approaching the Rate-Distortion Limit with Spatial Coupling, Belief propagation and Decimation

We investigate an encoding scheme for lossy compression of a binary symmetric source based on simple spatially coupled Low-Density Generator-Matrix codes. The degree of the check nodes is regular and the one of code-bits is Poisson distributed with an average depending on the compression rate. The performance of a low complexity Belief Propagation Guided Decimation algorithm is excellent. The algorithmic rate-distortion curve approaches the optimal curve of the ensemble as the width of the coupling window grows. Moreover, as the check degree grows both curves approach the ultimate Shannon rate-distortion limit. The Belief Propagation Guided Decimation encoder is based on the posterior measure of a binary symmetric test-channel. This measure can be interpreted as a random Gibbs measure at a "temperature" directly related to the "noise level of the test-channel". We investigate the links between the algorithmic performance of the Belief Propagation Guided Decimation encoder and the phase diagram of this Gibbs measure. The phase diagram is investigated thanks to the cavity method of spin glass theory which predicts a number of phase transition thresholds. In particular the dynamical and condensation "phase transition temperatures" (equivalently test-channel noise thresholds) are computed. We observe that: (i) the dynamical temperature of the spatially coupled construction saturates towards the condensation temperature; (ii) for large degrees the condensation temperature approaches the temperature (i.e. noise level) related to the information theoretic Shannon test-channel noise parameter of rate-distortion theory. This provides heuristic insight into the excellent performance of the Belief Propagation Guided Decimation algorithm. The paper contains an introduction to the cavity method.

cs.IT↗
arXiv2015

Spatially Coupled Codes and Optical Fiber Communications: An Ideal Match?

In this paper, we highlight the class of spatially coupled codes and discuss their applicability to long-haul and submarine optical communication systems. We first demonstrate how to optimize irregular spatially coupled LDPC codes for their use in optical communications with limited decoding hardware complexity and then present simulation results with an FPGA-based decoder where we show that very low error rates can be achieved and that conventional block-based LDPC codes can be outperformed. In the second part of the paper, we focus on the combination of spatially coupled LDPC codes with different demodulators and detectors, important for future systems with adaptive modulation and for varying channel characteristics. We demonstrate that SC codes can be employed as universal, channel-agnostic coding schemes.

cs.IT↗
arXiv2013

On the Convergence Speed of Spatially Coupled LDPC Ensembles

Spatially coupled low-density parity-check codes show an outstanding performance under the low-complexity belief propagation (BP) decoding algorithm. They exhibit a peculiar convergence phenomenon above the BP threshold of the underlying non-coupled ensemble, with a wave-like convergence propagating through the spatial dimension of the graph, allowing to approach the MAP threshold. We focus on this particularly interesting regime in between the BP and MAP thresholds. On the binary erasure channel, it has been proved that the information propagates with a constant speed toward the successful decoding solution. We derive an upper bound on the propagation speed, only depending on the basic parameters of the spatially coupled code ensemble such as degree distribution and the coupling factor $w$. We illustrate the convergence speed of different code ensembles by simulation results, and show how optimizing degree profiles helps to speed up the convergence.

cs.IT↗
arXiv2012

Lossy Source Coding via Spatially Coupled LDGM Ensembles

We study a new encoding scheme for lossy source compression based on spatially coupled low-density generator-matrix codes. We develop a belief-propagation guided-decimation algorithm, and show that this algorithm allows to approach the optimal distortion of spatially coupled ensembles. Moreover, using the survey propagation formalism, we also observe that the optimal distortions of the spatially coupled and individual code ensembles are the same. Since regular low-density generator-matrix codes are known to achieve the Shannon rate-distortion bound under optimal encoding as the degrees grow, our results suggest that spatial coupling can be used to reach the rate-distortion bound, under a {\it low complexity} belief-propagation guided-decimation algorithm. This problem is analogous to the MAX-XORSAT problem in computer science.

cs.IT↗
arXiv2011

Universal Rateless Codes From Coupled LT Codes

It was recently shown that spatial coupling of individual low-density parity-check codes improves the belief-propagation threshold of the coupled ensemble essentially to the maximum a posteriori threshold of the underlying ensemble. We study the performance of spatially coupled low-density generator-matrix ensembles when used for transmission over binary-input memoryless output-symmetric channels. We show by means of density evolution that the threshold saturation phenomenon also takes place in this setting. Our motivation for studying low-density generator-matrix codes is that they can easily be converted into rateless codes. Although there are already several classes of excellent rateless codes known to date, rateless codes constructed via spatial coupling might offer some additional advantages. In particular, by the very nature of the threshold phenomenon one expects that codes constructed on this principle can be made to be universal, i.e., a single construction can uniformly approach capacity over the class of binary-input memoryless output-symmetric channels. We discuss some necessary conditions on the degree distribution which universal rateless codes based on the threshold phenomenon have to fulfill. We then show by means of density evolution and some simulation results that indeed codes constructed in this way perform very well over a whole range of channel types and channel conditions.

cs.IT↗