# An LDPC-RS Concatenation and Decoding Scheme to Lower the Error Floor for FTN Signaling
## 2 FTN Signaling with Turbo Equalization
### LDPC and RS Code  
Let \(N\) be the block length of the LDPC code and \(K\) be the data length; then, \(M=N-K\) is the parity length. Let \(\mathbf{d}=(d_{1},d_{2},\ldots,d_{K})\) be the data and \(\mathbf{G}\) be the generation matrix of \(K\)-by-\(N\) size. The code word \(\mathbf{c}=\mathbf{d}\mathbf{G}\) has the property \(\mathbf{c}\mathbf{H}^{T}=0\), where \(\mathbf{H}\) is the parity matrix of \(M\)-by-\(N\). The code is called low-density parity check code, a feature of which is that parity matrix \(\mathbf{H}\) is sparse; namely, there are many zeros and few ones in \(\mathbf{H}\). There are binary and non-binary LDPC codes, and only binary codes are discussed in this work. LDPC codes can be classified into two types from the construction of parity matrix \(\mathbf{H}\), as regular and irregular codes. The number of ones in a row (column) of \(\mathbf{H}\) is called the row (column) degree. Regular codes have the same row degree for each row and the same column degree for each column. Row \(i\)'s degree indicates how many variable nodes are connected to check node \(i\), and column \(j\)'s degree indicates how many check nodes are connected to variable node \(j\). Among soft decision-decoding methods, belief propagation (BP) is a promising decoding scheme for the LDPC and other codes that need soft decisions and iterative decoding. The BP decoder uses message passing and iteration to propagate soft information among check and variable nodes. Since BP involves many nonlinear calculations, the min-sum (MS) algorithm was proposed. This algorithm uses the minimum value of soft information from relevant variable nodes to approximate the result of nonlinear functions that the BP decoder performs, thus reducing computation complexity and latency significantly [20]. Although the MS decoder requires fewer computations, the approximated results always have a greater absolute value than those of BP, which causes performance degradation. To mitigate performance loss, the normalized min-sum (NMS) scheme [21] and its variant were studied [22]; moreover, a neural network-aided NMS approach was proposed in [23]. The BP decoding scheme can be described as Algorithm 1. Figure 1: FTN signaling with turbo equalization. ```
Define: Set of variable nodes connected to check node \(i\): \(N(i)=\{j:h_{ij}=1\}\). Set of check nodes connected to variable node \(j\): \(M(j)=\{i:h_{ij}=1\}\). Soft information from variable node \(j\) to check node \(i\): \(Z_{ij}\). Soft information from check node \(i\) to variable node \(j\): \(L_{ij}\). Input: Soft bit information: \(\mathbf{L}=(l_{1},l_{2},\ldots,l_{n})\). Initialization:\(Z_{ij}=l_{j}\). Iteration: For each \(i,j\), the following steps are performed until the stopping criterion is satisfied. (1) Row operation: For each check node \(i\), calculate \[L_{i}=\prod_{j\in N(i)}\tanh(\frac{Z_{ij}}{2}),\] (12) and for messages from \(i\) to each relevant variable node \(j\), \[L_{ij}=2\tanh^{-1}(L_{i}/\tanh(\frac{Z_{ij}}{2})).\] (13) (2) Column operation: For each variable node \(j\), \[Z_{j}=l_{j}+\sum_{i\in M(j)}L_{ij},\] (14) and the message passed from \(j\) to each relevant check node \(i\) is \[Z_{ij}=Z_{j}-L_{ij}.\] (15) (3) Stopping criterion:  Make hard decision \(\mathbf{b}=(b_{1},b_{2},\ldots,b_{n})\) from \(Z_{j}\) as \[b_{i}= \begin{cases}0,&\text{if }Z_{j}>0\\ 1,&\text{if }Z_{j}\leq 0.\end{cases} \] (16) If \(\mathbf{b}\mathbf{H}^{T}=\mathbf{0}\) (no error) or the maximum iteration limit has been reached, stop the decoder, otherwise continue. Output: The decoding result \(\mathbf{b}\). ```  
**Algorithm 1** BP Decoding of LDPC Code. RS code is defined with finite field \(\mathbb{GF}(2^{p})\), where \(p\) is a positive integer, and all calculations are performed with this field. RS(\(n\),\(k\)) code has code word length \(n\) and \(k\) data symbols, each symbol having \(p\) bits. It can correct up to \(\lfloor(n-k)/2\rfloor\) errors.