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LT codes [38] were originally designed for the BEC [11], where each packet is either perfectly re-ceived, or lost for example owing to network-congestion at one of the routers. The LT codes transmit an appropriately designed number of redundant packets in order to fill these packet erasure events.

However, when LT codes are employed for transmission over wireless channels, which impose both fading and inter-symbol interference, in addition to packet erasures, each LT codeword might become corrupted, which results in catastrophic inter-packet error propagation during LT decoding [182]. As illustrated in Fig. 3.16, the errors of the first packet after the first LT decoding cycle are propagated to the second and third packets after five decoding cycles. As a result, more and more decoding errors appear, when the number of decoded packets increases. In order to mitigate the deleterious effects of error propagation, LT codes have been frequently combined with classic physical-layer FEC codes [182, 183]. The idea of combining classic FEC codes with LT codes by directly amalgamating them was proposed in [40], where systematically concatenated parity bits were incorporated in order to create the family of SLT codes.

The soft-output LT decoding process is based on the classic concept of LDPC decoding [37]. Given a LT code C, a generator matrix G of C is a matrix whose rows generate all the elements of C, i.e., if G = (g₁ g₂ ... g_k)^T, then every codeword w of C can be uniquely represented as

G = c₁g₁+ c₂g₂+ ... + c_kg_k= cG, (3.16) where we have c = (c₁ c₂ ... c_k). Then the Parity Check Matrix (PCM) H of a LT code, which checks a codeword c is in C if and only if the following matrix-vector product satisfies Hc = 0, which is calculated similarly to procedures of a classic LDPC code. More specifically, a LT code’s generator

3.4.1. Iterative Decoding Aided Systematic Luby Transform Codes 103

Table 3.5: Major rateless code contributions and their applications in cooperative wireless communi-cations.

2002 Luby [38] Introduced LT codes, the first rateless erasure codes.

2004 Shokrollahi [39, 175] Introduced Raptor codes, an extension of LT-codes with linear time encoding and decoding.

2005 MacKay [170] Presented Fountain codes that are ‘record-breaking’ sparse-graph codes for channels subjected to packet erasures.

2006 Castura and Mao [166] Proposed a framework for communication over fading channels utilizing rateless codes.

2007 Castura and Mao [167, 168]

Proposed a coding framework for wireless relay channels based on rateless codes.

Puducheri et al. [169] Designed distributed LT codes for wireless relaying networks.

Nguyen et al. [40] Presented systematic LT codes and their soft decoding.

2008 Nikjah and Beaulieu [173, 174]

Studied the average throughput of DF half-duplex as well as of DF full-duplex rateless coded relaying schemes under a specific peak power constraint and an average power constraint.

Yuan and Ping [176] Proposed a family of systematic rateless codes that are capacity-approaching on a binary erasure channel, regardless of the chan-nel’s erasure rate.

2009 Liu and Lim [130] Discussed cooperative protocols based on Fountain codes in a relay network.

Bonello [177, 178] Proposed reconfigurable rateless codes, that are capable of varying the block length as well as adaptively modifying their encoding strategy.

Cheng et al. [179] Designed Raptor codes for binary-input AWGN channels using the mean-LLR-EXIT chart approach presented.

2010 Gong et al. [180] Considered the code design for a half-duplex 4-node joint relay system with two sources, one relay, and one destination, employing superposition coding and Raptor coding.

Fan et al. [181] Studied the performance limits and design principles of rateless codes over MIMO fading channels.

3.4.1. Iterative Decoding Aided Systematic Luby Transform Codes 104

Figure 3.16: An example of error propagation in the LT hard-decoding process, when the LT decoder receive packets corrupted by channel noise ( c°Nguyen et al. [40]).

Figure 3.17: Tree-based representation of LT codes

3.4.1. Iterative Decoding Aided Systematic Luby Transform Codes 105 matrix G_{(K×N )} is partitioned into two matrices, namely A and B having a size of (K × K) and (K × M ) elements, respectively. If A is a non-singular matrix, then the PCM is calculated as [37]

H_{(M ×N )}= [(B^T · (A^T)⁻¹)_{(M ×K)}|I_{(M ×M )}], (3.17) where I_{(M ×M )} is (M × M )-element identity matrix.

An LT PCM can be represented by a classic Tanner graph [171]. To elaborate a little further, the filled circles and the filled squares of Fig. 3.17 represent the variable nodes - or synonymously the information nodes - and the parity check nodes of the LT code, respectively, while the horizontal lines connected to the variable nodes correspond to the intrinsic information provided by the channel’s output. Let us assume that the circular node at the top of Fig. 3.17 represents the k^th variable node in the block of N single-bit LT encoded packets, which is also termed as a root node. The root node receives extrinsic parity-check information from the specific check nodes that are directly connected to it at the tree-level immediately below it, as seen in Fig. 3.17. Similarly, these check nodes also receive extrinsic information from the specific variable nodes they are directly connected to at the next level down in Fig. 3.17, etc. The dotted lines in Fig. 3.17 indicate that the above process is repeated further by expanding the tree. The number of connections associated with a variable node of the LT code - excluding the line representing the intrinsic information - indicates the column weight of this particular message node, while the number of connections associated with an LT check node represent the corresponding row weight. The column weight and row weight of the LT PCM are related to the degree distribution of LT packets. The LT decoding process is implemented in the same way as the classic LDPC decoding procedure. Initially, the LT decoder’s soft values are set to a value corresponding to the demodulator’s soft output. The decoder’s soft values of R_i,j^a and Q^a_i,j, which denote the LLRs - as detailed in Section 2.2.4.4 of Chapter 2 - are then passed from the check nodes to the variable nodes and vice versa, which are then iteratively updated after each decoding iteration as follows [40]:

The LT decoder outputs its tentative hard decision values after each iteration and then checks whether the product of the corresponding codeword and the transpose of the PCM H is equal to zero, i.e whether a legitimate codeword was produced. If not, the LT decoding process will be continued in an iterative fashion, until the output codeword becomes legitimate or the maximum affordable number of iterations is exhausted.

In order to improve the achievable performance, the LT code’s degree distribution, which will be detailed below, is created by expanding its (K × N )-element generator matrix A with the aid of attaching a unity matrix having a size of (K × K), resulting in a systematic code [40]. For example, if we have a generator matrix G_K×N = [I_K×K|A_K×M], where (N = K + M ), as shown in Fig. 3.18, then the PCM H is calculated as

H = [A^T|I⁰]. (3.19)