We study Low-Density Parity-Check (LDPC) codes with iterative decoding on block-fading (BF) Relay Channels. We consider two users that employ coded cooperation, a variant of decode-and-forward with a smaller outage probability than the latter. An outage probability analysis for discrete constellations shows that full diversity can be achieved only when the coding rate does not exceed a maximum value that depends on the level of cooperation. We derive a new code structure by extending the previously published full-diversity root-LDPC code, designed for the BF point-to-point channel, to exhibit a rate-compatibility property which is necessary for coded cooperation. We estimate the asymptotic performance through a new density evolution analysis and the word error rate performance is determined for finite length codes. We show that our code construction exhibits near-outage limit performance for all block lengths and for a range of coding rates up to 0.5, which is the highest possible coding rate for two cooperating users.
Deep Dive into Low-Density Graph Codes for slow fading Relay Channels.
We study Low-Density Parity-Check (LDPC) codes with iterative decoding on block-fading (BF) Relay Channels. We consider two users that employ coded cooperation, a variant of decode-and-forward with a smaller outage probability than the latter. An outage probability analysis for discrete constellations shows that full diversity can be achieved only when the coding rate does not exceed a maximum value that depends on the level of cooperation. We derive a new code structure by extending the previously published full-diversity root-LDPC code, designed for the BF point-to-point channel, to exhibit a rate-compatibility property which is necessary for coded cooperation. We estimate the asymptotic performance through a new density evolution analysis and the word error rate performance is determined for finite length codes. We show that our code construction exhibits near-outage limit performance for all block lengths and for a range of coding rates up to 0.5, which is the highest possible codi
When communicating over fading channels, Word Error Rate (WER) performances as well as power savings are dramatically improved through transmit diversity, i.e., transmitting signals carrying the same information over different paths in time, frequency or space. Recently, a new network protocol called Cooperative Communication [11], [26], [32], [40], [41] yields transmit diversity using single-antenna devices in a multi-user environment by taking advantage of the broadcast nature of wireless transmission.
The most elementary example of a cooperative network is the relay channel, introduced by van der Meulen [31]. In a relay channel, a relay helps the source in transmitting its data to a destination by relaying the messages sent by the source so that the received energy at the destination is increased. This relay channel can be generalized to a cooperative Multiple Access Channel (MAC) [26], where two users transmitting data to a single receiver cooperate by alternately being the relay for the other user, as indicated in Fig. 1. Further generalization to more users is possible, but this will not be discussed here for simplicity. A challenging channel model is the BF [3] frequency non-selective Single-Input Single-Output (SISO) channel. When the fading gain is constant over a codeword and no cooperation is used, the resulting word error rate curve (displaying the logarithm of the error rate versus the average signal-to-noise ratio (SNR) in dB) has the same high-SNR slope as for uncoded transmission: the corresponding diversity order1 equals one. The potential diversity increase brought by cooperative techniques allows to save much transmit energy at a given error rate. BF channels are a realistic model for a number of channels affected by slowly varying fading and flat fading is assumed in order to isolate the effect of cooperative diversity.
The specific task of the relay is determined by the strategy or protocol. In the case Decode and Forward (DF), the relay first decodes and then re-encodes the message before sending it to the destination. A variant of DF is coded cooperation, where the relay decodes the message received from the source, and then transmits additional parity bits of the message, resulting in a more spectral efficient strategy [22], compared to a traditional DF protocol. Instead of SNR accumulation (logarithmic rise of mutual information with received power from the relay) at the destination, we get information accumulation (linear rise of mutual information with received power from the relay) [46]. It has been shown in [23] that the outage probability [3], [33] of coded cooperation for half-duplex BF channels is smaller than for repetition-based protocols.
Moreover, the concept of coded cooperation can be used in more complex strategies, such as Amplify-Decode-Forward [2], where the relay can choose between DF and AF. So finally, replacing the decode-and-repeat part in any protocol by this more intelligent “information adding” strategy improves the outage probability performance. As a consequence, constructing a nearoutage channel code for a coded cooperation scenario results in a competitive error-correcting code in terms of error-rate performance vs. SNR for a given rate R.
Up till now, coded cooperation has mainly been implemented using rate-compatible convolutional codes [22]. The main drawback of these codes is that the WER increases with the logarithm of block length to the power d where d is the diversity order [6], [7]. The WER of practical near-outage codes should be independent of the block length in order to approach the outage probability limit [16], [17]. The solution is to use capacity-achieving codes, for example LDPC codes [36]. LDPC codes designed for the special case of a cooperative channel have been reported for the Gaussian channel by Razaghi and Yu [34], [35] and by Chakrabarti et al. [9].
For the block-fading channel however, there is still a lack of a near-outage LDPC code. Hu et al. [20] also designed LDPC codes for the Gaussian relay channel, whereafter they applied this random LDPC code to a BF relay channel. Unfortunately, a random code does not perform very well on a BF relay channel, because it has not the structure to achieve full diversity, as shown by Boutros et al. [5] and as will be explained in the rest of the paper.
In Section III, this paper analyzes the outage probability for binary phase shift keying (BPSK) modulations and derives a coding rate limitation that is necessary for the protocol to have diversity two, valid for all discrete alphabets. Deriving a code structure for coded cooperation will be treated in the second part of the paper. The aim of coded cooperation is to send a codeword over two independent fading paths and the relay must be able to decode after receiving the first part of the codeword. An error-correcting code must therefore exhibit two properties: full-diversity and rate-compatibility. This paper derives a new code structu
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