Bilayer Low-Density Parity-Check Codes for Decode-and-Forward in Relay Channels

This paper describes an efficient implementation of binning for the relay channel using low-density parity-check (LDPC) codes. We devise bilayer LDPC codes to approach the theoretically promised rate of the decode-and-forward relaying strategy by incorporating relay-generated information bits in specially designed bilayer graphical code structures. While conventional LDPC codes are sensitively tuned to operate efficiently at a certain channel parameter, the proposed bilayer LDPC codes are capable of working at two different channel parameters and two different rates: that at the relay and at the destination. To analyze the performance of bilayer LDPC codes, bilayer density evolution is devised as an extension of the standard density evolution algorithm. Based on bilayer density evolution, a design methodology is developed for the bilayer codes in which the degree distribution is iteratively improved using linear programming. Further, in order to approach the theoretical decode-and-forward rate for a wide range of channel parameters, this paper proposes two different forms bilayer codes, the bilayer-expurgated and bilayer-lengthened codes. It is demonstrated that a properly designed bilayer LDPC code can achieve an asymptotic infinite-length threshold within 0.24 dB gap to the Shannon limits of two different channels simultaneously for a wide range of channel parameters. By practical code construction, finite-length bilayer codes are shown to be able to approach within a 0.6 dB gap to the theoretical decode-and-forward rate of the relay channel at a block length of $10^5$ and a bit-error probability (BER) of $10^{-4}$. Finally, it is demonstrated that a generalized version of the proposed bilayer code construction is applicable to relay networks with multiple relays.Comment: Submitted to IEEE Trans. Info. Theor

Universal relaying for the interference channel

Abstract—This paper considers a Gaussian relay-interference channel and introduces a generalized hash-and-forward relay strategy, where the relay sends out a bin index of its quantized observation, and the receivers first decode the relay quantization codeword to a list, then use the list to help decode the respective messages from the transmitters. The main advantage of the proposed approach is in a scenario where the relay observes a linear combination of the transmitted signals and broadcasts a common relay message through a digital relay link of fixed rate to help both receivers of the interference channel. We show that when compared to the achievable rates with interference treated as noise, generalized hash-and-forward can provide one bit of rate improvement for every relay bit for both users at the same time in an asymptotic regime where the background noises go down to zero. The proposed approach is universal, in contrast to the compress-and-forward or amplify-and-forward strategies which are not asymptotically optimal for multiple users simultaneously, if at all. I

Parity Forwarding for Relay Networks

In this dissertation, we introduce a relay protocol for multiple-relay networks called parity forwarding. The relay channel is a classic multiuser information theory problem introduced in 1971, modeling a network of three nodes: a source, a destination, and a relay node. The relay has no message of its own and assists the source to communicate to the destination. Of the two main coding techniques for the relay channel, decode-and-forward (DF) and compress-and-forward (CF), our focus is to understand the DF scheme for single- and multiple-relay channels. For the single-relay channel, we present an interpretation of the classic DF strategy from a linear-coding perspective. Identifying binning in the classic DF strategy as parity bit generation, we devise ensembles of low-density parity-check (LDPC) codes, called bilayer LDPC codes, to incorporate the parity bits generated by the relay in the decoding process at the destination. We develop code design techniques and optimize the parameters of the bilayer LDPC code structure to show that bilayer LDPC codes can approach the theoretical DF rate. Inspired by the relation between binning and parity bits, we introduce the parity forwarding protocol to improve on the achievable DF rate in a multiple-relay network. For a two-relay network, we show that the previous multihop DF protocol can be improved if the relays forward parity messages and receivers (relays or the destination)use a joint decoding scheme. Depending on the relation between relay messages and decoded messages at each relay, different parity forwarding protocols are possible. In this thesis, we present a structured characterization of a class of parity forwarding protocols for multiple-relay networks. We propose a tree structure to describe the relations between messages. Using this tree structure, we derive a closed-form expression for the parity forwarding rate in a relay network with an arbitrary number of relays. Finally, examples of new types of degraded multiple-relay networks are presented for which the parity forwarding protocol achieves capacity.Ph

The Causal Relationship Between Saving and Growth : Case of Colombia, Mexico, Sweden and the UK

This paper uses a vector error correction model (VECM) and the unrestricted vector autoregressive (VAR) model combined with Granger causality in order to determine the casual direction between saving and growth. The countries being studied are Colombia, Mexico, Sweden and the UK. First, the stationarity of the variables are tested by employing an Augmented Dickey-Fuller test. Secondly, the long-run relationship is tested through a Johansen cointegrating system. For those countries that have cointegrated variables the VECM model is used. For those lacking such a long-run relationship a standard VAR model is used and further analysis is done through Granger non-causality testing. The results from the cointegration analysis suggests a long-run relationship for Sweden and the UK. This was further attested to by the VECM analysis. For Colombia saving only Granger-caused growth in the short-run and for Mexico no relationship was found as the Granger-causality test showed no causal link in any direction

Bilayer ldpc codes for the relay channel

Abstract — This paper describes a methodology for efficient implementation of binning and block-Markov coding for the relay channel using powerful features of low-density parity-check (LDPC) codes. We devise bilayer LDPC codes to approach the theoretically promised rate of the decode-and-forward relaying strategy by incorporating relay-generated random linear paritybits in a specially designed bilayer graphical code structure. Bilayer density evolution is devised as a novel extension of the standard density evolution algorithm to analyze the performance of the proposed bilayer LDPC code. Based on this bilayer density evolution technique, an EXIT-chart-based code design method using linear programming is developed. While conventional LDPC codes are sensitively tuned to operate efficiently at a certain channel parameter, the proposed bilayer LDPC code is capable of working at two different channel parameters, the signal-to-noise ratio (SNR) at the relay and the SNR at the destination. In this paper, for specific channel parameters, it is demonstrated that a bilayer LDPC code can approach the theoretical decode-and-forward rate of the relay channel within a 0.19 dB gap to the source-relay channel capacity and a 0.34 dB gap to the relay-destination channel capacity. I