arXiv subjects

Krishna R. Narayanan

Publications and source records attributed to Krishna R. Narayanan.

At least 55 records · Page 3Linked to original sources

arXiv2018

A Coupled Compressive Sensing Scheme for Unsourced Multiple Access

This article introduces a novel paradigm for the unsourced multiple-access communication problem. This divide-and-conquer approach leverages recent advances in compressive sensing and forward error correction to produce a computationally efficient algorithm. Within the proposed framework, every active device first partitions its data into several sub-blocks, and subsequently adds redundancy using a systematic linear block code. Compressive sensing techniques are then employed to recover sub-blocks, and the original messages are obtained by connecting pieces together using a low-complexity tree-based algorithm. Numerical results suggest that the proposed scheme outperforms other existing practical coding schemes. Measured performance lies approximately $4.3$~dB away from the Polyanskiy achievability limit, which is obtained in the absence of complexity constraints.

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arXiv2017

Construction $π_A$ and $π_D$ Lattices: Construction, Goodness, and Decoding Algorithms

A novel construction of lattices is proposed. This construction can be thought of as a special class of Construction A from codes over finite rings that can be represented as the Cartesian product of $L$ linear codes over $\mathbb{F}_{p_1},\ldots,\mathbb{F}_{p_L}$, respectively, and hence is referred to as Construction $π_A$. The existence of a sequence of such lattices that is good for channel coding (i.e., Poltyrev-limit achieving) under multistage decoding is shown. A new family of multilevel nested lattice codes based on Construction $π_A$ lattices is proposed and its achievable rate for the additive white Gaussian channel is analyzed. A generalization named Construction $π_D$ is also investigated which subsumes Construction A with codes over prime fields, Construction D, and Construction $π_A$ as special cases.

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arXiv2017

Approaching Capacity at High-Rates with Iterative Hard-Decision Decoding

A variety of low-density parity-check (LDPC) ensembles have now been observed to approach capacity with message-passing decoding. However, all of them use soft (i.e., non-binary) messages and a posteriori probability (APP) decoding of their component codes. In this paper, we show that one can approach capacity at high rates using iterative hard-decision decoding (HDD) of generalized product codes. Specifically, a class of spatially-coupled GLDPC codes with BCH component codes is considered, and it is observed that, in the high-rate regime, they can approach capacity under the proposed iterative HDD. These codes can be seen as generalized product codes and are closely related to braided block codes. An iterative HDD algorithm is proposed that enables one to analyze the performance of these codes via density evolution (DE).

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arXiv2017

Sub-string/Pattern Matching in Sub-linear Time Using a Sparse Fourier Transform Approach

We consider the problem of querying a string (or, a database) of length $N$ bits to determine all the locations where a substring (query) of length $M$ appears either exactly or is within a Hamming distance of $K$ from the query. We assume that sketches of the original signal can be computed off line and stored. Using the sparse Fourier transform computation based approach introduced by Pawar and Ramchandran, we show that all such matches can be determined with high probability in sub-linear time. Specifically, if the query length $M = O(N^μ)$ and the number of matches $L=O(N^λ)$, we show that for $λ< 1-μ$ all the matching positions can be determined with a probability that approaches 1 as $N \rightarrow \infty$ for $K \leq \frac{1}{6}M$. More importantly our scheme has a worst-case computational complexity that is only $O\left(\max\{N^{1-μ}\log^2 N, N^{μ+λ}\log N \}\right)$, which means we can recover all the matching positions in {\it sub-linear} time for $λ<1-μ$. This is a substantial improvement over the best known computational complexity of $O\left(N^{1-0.359 μ} \right)$ for recovering one matching position by Andoni {\em et al.} \cite{andoni2013shift}. Further, the number of Fourier transform coefficients that need to be computed, stored and accessed, i.e., the sketching complexity of this algorithm is only $O\left(N^{1-μ}\log N\right)$. Several extensions of the main theme are also discussed.

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arXiv2017

Group Testing using left-and-right-regular sparse-graph codes

We consider the problem of non-adaptive group testing of $N$ items out of which $K$ or less items are known to be defective. We propose a testing scheme based on left-and-right-regular sparse-graph codes and a simple iterative decoder. We show that for any arbitrarily small $ε>0$ our scheme requires only $m=c_εK\log \frac{c_1N}{K}$ tests to recover $(1-ε)$ fraction of the defective items with high probability (w.h.p) i.e., with probability approaching $1$ asymptotically in $N$ and $K$, where the value of constants $c_ε$ and $\ell$ are a function of the desired error floor $ε$ and constant $c_1=\frac{\ell}{c_ε}$ (observed to be approximately equal to 1 for various values of $ε$). More importantly the iterative decoding algorithm has a sub-linear computational complexity of $\mathcal{O}(K\log \frac{N}{K})$ which is known to be optimal. Also for $m=c_2 K\log K\log \frac{N}{K}$ tests our scheme recovers the \textit{whole} set of defective items w.h.p. These results are valid for both noiseless and noisy versions of the problem as long as the number of defective items scale sub-linearly with the total number of items, i.e., $K=o(N)$. The simulation results validate the theoretical results by showing a substantial improvement in the number of tests required when compared to the testing scheme based on left-regular sparse-graphs.

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arXiv2016

Adaptive Compute-and-Forward with Lattice Codes Over Algebraic Integers

We consider the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all principal ideal domains (PID). A novel scheme called adaptive compute-and-forward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. This is enabled by generalizing the famous Construction A from PID to other rings of imaginary quadratic integers which may not form PID and by showing such the construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive compute-and-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely.

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arXiv2016

Joint Source-Channel Decoding of Polar Codes for Language-Based Source

We exploit the redundancy of the language-based source to help polar decoding. By judging the validity of decoded words in the decoded sequence with the help of a dictionary, the polar list decoder constantly detects erroneous paths after every few bits are decoded. This path-pruning technique based on joint decoding has advantages over stand-alone polar list decoding in that most decoding errors in early stages are corrected. In order to facilitate the joint decoding, we first propose a construction of dynamic dictionary using a trie and show an efficient way to trace the dictionary during decoding. Then we propose a joint decoding scheme of polar codes taking into account both information from the channel and the source. The proposed scheme has the same decoding complexity as the list decoding of polar codes. A list-size adaptive joint decoding is further implemented to largely reduce the decoding complexity. We conclude by simulation that the joint decoding schemes outperform stand-alone polar codes with CRC-aided successive cancellation list decoding by over 0.6 dB.

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arXiv2014

Asynchronous Physical-Layer Network Coding with Quasi-Cyclic Codes

Communication in the presence of bounded timing asynchronism which is known to the receiver but cannot be easily compensated is studied. Examples of such situations include point-to-point communication over inter-symbol interference (ISI) channels and asynchronous wireless networks. In these scenarios, although the receiver may know all the delays, it is often not be an easy task for the receiver to compensate the delays as the signals are mixed together. A novel framework called interleave/deinterleave transform (IDT) is proposed to deal with this problem. It is shown that the IDT allows one to design the delays so that quasi-cyclic (QC) codes with a proper shifting constraint can be used accordingly. When used in conjunction with QC codes, IDT provides significantly better performance than existing schemes relying solely on cyclic codes. Two instances of asynchronous physical-layer network coding, namely the integer-forcing equalization for ISI channels and asynchronous compute-and-forward, are then studied. For integer-forcing equalization, the proposed scheme provides improved performance over using cyclic codes. For asynchronous compute-and-forward, the proposed scheme shows that there is no loss in the achievable information due to delays which are integer multiples of the symbol duration. Further, the proposed approach shows that delays introduced by the channel can sometimes be exploited to obtain higher information rates than those obtainable in the synchronous case. The proposed IDT can be thought of as a generalization of the interleaving/deinterleaving idea proposed by Wang et al. which allows the use of QC codes thereby substantially increasing the design space.

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arXiv2014

Lattices over Eisenstein Integers for Compute-and-Forward

In this paper, we consider the use of lattice codes over Eisenstein integers for implementing a compute-and-forward protocol in wireless networks when channel state information is not available at the transmitter. We extend the compute-and-forward paradigm of Nazer and Gastpar to decoding Eisenstein integer combinations of transmitted messages at relays by proving the existence of a sequence of pairs of nested lattices over Eisenstein integers in which the coarse lattice is good for covering and the fine lattice can achieve the Poltyrev limit. Using this result, we show that both the outage performance and error-correcting performance of nested lattice codebooks over Eisenstein integers surpasses lattice codebooks over integers considered by Nazer and Gastpar with no additional computational complexity.

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arXiv2014

Lattices from Codes for Harnessing Interference: An Overview and Generalizations

In this paper, using compute-and-forward as an example, we provide an overview of constructions of lattices from codes that possess the right algebraic structures for harnessing interference. This includes Construction A, Construction D, and Construction $π_A$ (previously called product construction) recently proposed by the authors. We then discuss two generalizations where the first one is a general construction of lattices named Construction $π_D$ subsuming the above three constructions as special cases and the second one is to go beyond principal ideal domains and build lattices over algebraic integers.

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arXiv2014

Multistage Compute-and-Forward with Multilevel Lattice Codes Based on Product Constructions

A novel construction of lattices is proposed. This construction can be thought of as Construction A with codes that can be represented as the Cartesian product of $L$ linear codes over $\mathbb{F}_{p_1},\ldots,\mathbb{F}_{p_L}$, respectively; hence, is referred to as the product construction. The existence of a sequence of such lattices that are good for quantization and Poltyrev-good under multistage decoding is shown. This family of lattices is then used to generate a sequence of nested lattice codes which allows one to achieve the same computation rate of Nazer and Gastpar for compute-and-forward under multistage decoding, which is referred to as lattice-based multistage compute-and-forward. Motivated by the proposed lattice codes, two families of signal constellations are then proposed for the separation-based compute-and-forward framework proposed by Tunali \textit{et al.} together with a multilevel coding/multistage decoding scheme tailored specifically for these constellations. This scheme is termed separation-based multistage compute-and-forward and is shown having a complexity of the channel coding dominated by the greatest common divisor of the constellation size (may not be a prime number) instead of the constellation size itself.

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arXiv2012

Code Design for the Noisy Slepian-Wolf Problem

We consider a noisy Slepian-Wolf problem where two correlated sources are separately encoded (using codes of fixed rate) and transmitted over two independent binary memoryless symmetric channels. The capacity of each channel is characterized by a single parameter which is not known at the transmitter. The goal is to design systems that retain near-optimal performance without channel knowledge at the transmitter. It was conjectured that it may be hard to design codes that perform well for symmetric channel conditions. In this work, we present a provable capacity-achieving sequence of LDGM ensembles for the erasure Slepian-Wolf problem with symmetric channel conditions. We also introduce a staggered structure which enables codes optimized for single user channels to perform well for symmetric channel conditions. We provide a generic framework for analyzing the performance of joint iterative decoding, using density evolution. Using differential evolution, we design punctured systematic LDPC codes to maximize the region of achievable channel conditions. The resulting codes are then staggered to further increase the region of achievable parameters. The main contribution of this paper is to demonstrate that properly designed irregular LDPC codes can perform well simultaneously over a wide range of channel parameters.

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arXiv2011

Joint Source-Channel Coding with Correlated Interference

We study the joint source-channel coding problem of transmitting a discrete-time analog source over an additive white Gaussian noise (AWGN) channel with interference known at transmitter.We consider the case when the source and the interference are correlated. We first derive an outer bound on the achievable distortion and then, we propose two joint source-channel coding schemes. The first scheme is the superposition of the uncoded signal and a digital part which is the concatenation of a Wyner-Ziv encoder and a dirty paper encoder. In the second scheme, the digital part is replaced by the hybrid digital and analog scheme proposed by Wilson et al. When the channel signal-tonoise ratio (SNR) is perfectly known at the transmitter, both proposed schemes are shown to provide identical performance which is substantially better than that of existing schemes. In the presence of an SNR mismatch, both proposed schemes are shown to be capable of graceful enhancement and graceful degradation. Interestingly, unlike the case when the source and interference are independent, neither of the two schemes outperforms the other universally. As an application of the proposed schemes, we provide both inner and outer bounds on the distortion region for the generalized cognitive radio channel.

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arXiv2011

Spatially-Coupled Codes and Threshold Saturation on Intersymbol-Interference Channels

Recently, it has been observed that terminated low-density-parity-check (LDPC) convolutional codes (or spatially-coupled codes) appear to approach capacity universally across the class of binary memoryless channels. This is facilitated by the "threshold saturation" effect whereby the belief-propagation (BP) threshold of the spatially-coupled ensemble is boosted to the maximum a-posteriori (MAP) threshold of the underlying constituent ensemble. In this paper, we consider the universality of spatially-coupled codes over intersymbol-interference (ISI) channels under joint iterative decoding. More specifically, we empirically show that threshold saturation also occurs for the considered problem. This can be observed by first identifying the EXIT curve for erasure noise and the GEXIT curve for general noise that naturally obey the general area theorem. From these curves, the corresponding MAP and the BP thresholds are then numerically obtained. With the fact that regular LDPC codes can achieve the symmetric information rate (SIR) under MAP decoding, spatially-coupled codes with joint iterative decoding can universally approach the SIR of ISI channels. For the dicode erasure channel, Kudekar and Kasai recently reported very similar results based on EXIT-like curves.

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arXiv2011

Universal Codes for the Gaussian MAC via Spatial Coupling

We consider transmission of two independent and separately encoded sources over a two-user binary-input Gaussian multiple-access channel. The channel gains are assumed to be unknown at the transmitter and the goal is to design an encoder-decoder pair that achieves reliable communication for all channel gains where this is theoretically possible. We call such a system \emph{universal} with respect to the channel gains. Kudekar et al. recently showed that terminated low-density parity-check convolutional codes (a.k.a. spatially-coupled low-density parity-check ensembles) have belief-propagation thresholds that approach their maximum a-posteriori thresholds. This was proven for binary erasure channels and shown empirically for binary memoryless symmetric channels. It was conjectured that the principle of spatial coupling is very general and the phenomenon of threshold saturation applies to a very broad class of graphical models. In this work, we derive an area theorem for the joint decoder and empirically show that threshold saturation occurs for this problem. As a result, we demonstrate near-universal performance for this problem using the proposed spatially-coupled coding system.

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arXiv2010

On Multiple Decoding Attempts for Reed-Solomon Codes: A Rate-Distortion Approach

One popular approach to soft-decision decoding of Reed-Solomon (RS) codes is based on using multiple trials of a simple RS decoding algorithm in combination with erasing or flipping a set of symbols or bits in each trial. This paper presents a framework based on rate-distortion (RD) theory to analyze these multiple-decoding algorithms. By defining an appropriate distortion measure between an error pattern and an erasure pattern, the successful decoding condition, for a single errors-and-erasures decoding trial, becomes equivalent to distortion being less than a fixed threshold. Finding the best set of erasure patterns also turns into a covering problem which can be solved asymptotically by rate-distortion theory. Thus, the proposed approach can be used to understand the asymptotic performance-versus-complexity trade-off of multiple errors-and-erasures decoding of RS codes. This initial result is also extended a few directions. The rate-distortion exponent (RDE) is computed to give more precise results for moderate blocklengths. Multiple trials of algebraic soft-decision (ASD) decoding are analyzed using this framework. Analytical and numerical computations of the RD and RDE functions are also presented. Finally, simulation results show that sets of erasure patterns designed using the proposed methods outperform other algorithms with the same number of decoding trials.

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arXiv2010

LDPC Code Design for Transmission of Correlated Sources Across Noisy Channels Without CSIT

We consider the problem of transmitting correlated data after independent encoding to a central receiver through orthogonal channels. We assume that the channel state information is not known at the transmitter. The receiver has access to both the source correlation and the channel state information. We provide a generic framework for analyzing the performance of joint iterative decoding, using density evolution. Using differential evolution, we design punctured systematic LDPC codes to maximize the region of achievable channel conditions, with joint iterative decoding. The main contribution of this paper is to demonstrate that properly designed LDPC can perform well simultaneously over a wide range of channel parameters.

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arXiv2010

On the Queueing Behavior of Random Codes over a Gilbert-Elliot Erasure Channel

This paper considers the queueing performance of a system that transmits coded data over a time-varying erasure channel. In our model, the queue length and channel state together form a Markov chain that depends on the system parameters. This gives a framework that allows a rigorous analysis of the queue as a function of the code rate. Most prior work in this area either ignores block-length (e.g., fluid models) or assumes error-free communication using finite codes. This work enables one to determine when such assumptions provide good, or bad, approximations of true behavior. Moreover, it offers a new approach to optimize parameters and evaluate performance. This can be valuable for delay-sensitive systems that employ short block lengths.

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