A Decision Feedback Based Scheme for Slepian-Wolf Coding of sources with Hidden Markov Correlation

We consider the problem of compression of two memoryless binary sources, the correlation between which is defined by a Hidden Markov Model (HMM). We propose a Decision Feedback (DF) based scheme which when used with low density parity check codes results in compression close to the Slepian Wolf limits.Comment: Submitted to IEEE Comm. Letter

Degree Optimization and Stability Condition for the Min-Sum Decoder

The min-sum (MS) algorithm is arguably the second most fundamental algorithm in the realm of message passing due to its optimality (for a tree code) with respect to the {\em block error} probability \cite{Wiberg}. There also seems to be a fundamental relationship of MS decoding with the linear programming decoder \cite{Koetter}. Despite its importance, its fundamental properties have not nearly been studied as well as those of the sum-product (also known as BP) algorithm. We address two questions related to the MS rule. First, we characterize the stability condition under MS decoding. It turns out to be essentially the same condition as under BP decoding. Second, we perform a degree distribution optimization. Contrary to the case of BP decoding, under MS decoding the thresholds of the best degree distributions for standard irregular LDPC ensembles are significantly bounded away from the Shannon threshold. More precisely, on the AWGN channel, for the best codes that we find, the gap to capacity is 1dB for a rate 0.3 code and it is 0.4dB when the rate is 0.9 (the gap decreases monotonically as we increase the rate). We also used the optimization procedure to design codes for modified MS algorithm where the output of the check node is scaled by a constant $1/\alpha$. For $\alpha = 1.25$, we observed that the gap to capacity was lesser for the modified MS algorithm when compared with the MS algorithm. However, it was still quite large, varying from 0.75 dB to 0.2 dB for rates between 0.3 and 0.9. We conclude by posing what we consider to be the most important open questions related to the MS algorithm.Comment: submitted to ITW 0

Joint source channel coding for non-ergodic channels: the distortion signal-to-noise ratio (SNR) exponent perspective

We study the problem of communicating a discrete time analog source over a channel such that the resulting distortion is minimized. For ergodic channels, Shannon showed that separate source and channel coding is optimal. In this work we study this problem for non-ergodic channels. Although not much can be said about the general problem of transmitting any analog sources over any non-ergodic channels with any distortion metric, for many practical problems like video broadcast and voice transmission, we can gain insights by studying the transmission of a Gaussian source over a wireless channel with mean square error as the distortion measure. Motivated by different applications, we consider three different non-ergodic channel models - (1) Additive white Gaussian noise (AWGN) channel whose signal-to-noise ratio (SNR) is unknown at the transmitter; (2) Rayleigh fading multiple-input multiple-output MIMO channel whose SNR is known at the transmitter; and (3) Rayleigh fading MIMO channel whose SNR is unknown at the transmitter. The traditional approach to study these problems has been to fix certain SNRs of interest and study the corresponding achievable distortion regions. However, the problems formulated this way have not been solved even for simple setups like 2 SNRs for the AWGN channel. We are interested in performance over a wide range of SNR and hence we use the distortion SNR exponent metric to study this problem. Distortion SNR exponent is defined as the rate of decay of distortion with SNR in the high SNR limit. We study several layered transmissions schemes where the source is first compressed in layers and then the layers are transmitted using channel codes that provide variable error protection. Results show that in several cases such layered transmission schemes are optimal in terms of the distortion SNR exponent. Specifically, if the band- width expansion (number of channel uses per source sample) is b, we show that the optimal distortion SNR exponent for the AWGN channel is b and it is achievable using a superposition based layered scheme. For the L-block Rayleigh fading M x N MIMO channel the optimal exponent is characterized for b < (|N - M|+1)= min(M;N) and b > MNL2. This corresponds to the entire range of b when min(M;N) = 1 and L = 1. The results also show that the exponents obtained using layered schemes which are a small subclass of joint source channel coding (JSCC) schemes are, surprisingly, as good as and better in some cases than achievable exponent of all other JSCC schemes reported so far

A note on the rate of decay of mean squared error with snr for the awgn channel,” in WCL

We consider a Gaussian source S with K samples that has to be transmitted over T uses of an additive white Gaussian noise (AWGN) channel with Signal-to-Noise ratio (SNR) ρ. The distortion measure of interest, D(ρ), is mean-square error. We consider the case when the transmission scheme is fixed and SNR is not known at the transmitter. We are interested in characterizing the rate of decay of D(ρ) with ρ in the high SNR limit, i.e., log D(ρ) a(b) = limρ→ ∞ log ρ. An upper bound, a(b) ≤ b, is obtained by considering the case when the SNR is known at the transmitter. Here, we propose a superposition scheme that achieves an exponent a(b) = b. This therefore characterizes the optimal distortion SNR exponent of the AWGN channel. Similar results for a uniform source were presented by Santhi and Vardy [1] but they considered only integer values of b, which are greater than 1. The scheme presented here is similar to the scheme of [1] and extends the results to the Gaussian case for all b. We then consider an AWGN broadcast channel where the distortion for the weaker user is required to satisfy D(ρ2) ≤ αDmin(ρ2) for a constant α&gt; 1 and Dmin(ρ2) is the minimum possible distortion achievable at SNR ρ2 and we prove that we can obtain a decay rate of 1/ρ b 1 for the distortion of the stronger user for any α. I

LDPC Code Design for Min-Sum Based Decoding

We consider the problem of designing low-density parity-check (LDPC) codes for min-sum decoding. We wish to find the LDPC ensemble with the best asymptotic performance. In [1], [2], Amraoui and Urbanke proposed an optimization scheme where a combination of density evolution and extrinsic information transfer (EXIT) charts is used to design LDPC codes for the belief propagation decoding algorithm. In this paper, we apply a similar optimization scheme to design LDPC codes for min-sum decoding. For the AWGN channel, for the best codes that we find, the gap to capacity decreased from around 1 dB to 0.4 dB when the rate was increased from 0.3 to 0.9. We also used the optimization procedure to design codes for modified min-sum algorithm where the output of the check node is scaled by a constant 1/α. For α = 1.25, we observed that the gap to capacity was lesser for the modified min-sum algorithm when compared with the min-sum algorithm. However, it was still quite large, varying from 0.75 dB to 0.2 dB for rates between 0.3 and 0.9. I

Minimal network coding for multicast

Abstract — We give an information flow interpretation for multicasting using network coding. This generalizes the fluid model used to represent flows to a single receiver. Using the generalized model, we present a decentralized algorithm to minimize the number of packets that undergo network coding. We also propose a decentralized algorithm to construct capacity achieving multicast codes when the processing at some nodes is restricted to routing. The proposed algorithms can be coupled with existing decentralized schemes to achieve minimum cost muticast. I

Minimal network coding for multicast

Abstract — We give an information flow interpretation for multicasting using network coding. This generalizes the fluid model used to represent flows to a single receiver. Using the generalized model, we present a decentralized algorithm to minimize the number of packets that undergo network coding. We also propose a decentralized algorithm to construct capacity achieving multicast codes when the processing at some nodes is restricted to routing. The proposed algorithms can be coupled with existing decentralized schemes to achieve minimum cost muticast. I