# Optimized Design for NB-LDPC-Coded High-Order CPM: Power and Iterative Efficiencies
## 4 Code Design and Advantages
### Additional Advantages  
The investigation into NB-LDPC-coded high-order CPM is particularly significant for coded modulation systems. In addition to the excellent properties in terms of continuous phase, rapidly decaying spectrum side lobes, and constant envelope, this scheme has the following additional advantages:  
1. Each edge of the binary LDPC code in the Tanner graph carries bit messages, but the NB-LDPC code carries \(Q\)-ary symbol messages, thus, short girths are avoided in the Tanner graph. This reduces the influence of short girths and stopping set on decoding convergence. Therefore, the BP algorithm becomes closer to the maximum likelihood decoding algorithm. The NB-LDPC code, as an outer code in coded modulation systems, provides an alternative solution in enhancing BER performance in practical applications. Figure 6: EXIT chart of the optimized NB-LDPC-coded high-order CPM system with 8M2 RC under five inner iterations. In comparison with the traditional BICM, the interleaver of the NB-LDPC-coded high-order CPM works at the symbol level, which always yields a lower convergence threshold than bit level. This advantage is rather significant in a serial concatenation [22]. 3. As the NB-LDPC code and CPM select the uniform \(M\)-ary in the investigated systems, the symbol mapping issue that is likely to result in conversion information loss from bit to symbol may be ignored. This phenomenon usually occurs in the case of \(M>q\). Thus, more possible input code symbols exist between the current and next phase states in the trellis diagrams. For an example of the CPM scheme with 8\(M\)2 RC using \(h=1/2\), the corresponding transfer diagram of the phase states using Gray and natural mappings is shown in Figure 7.  
In this case, eight input code symbols and two phase states of 0 and \(\pi\) are presented. Each transfer between the adjacent phase states exists in four possible input code symbols. Information loss in an inappropriate symbol mapping would occur when the MAP-like algorithm in CPM-SISO is used [25], that is:  
\[P_{I}(u^{k,j};O)=H_{u^{j}}\sum_{u:\;U^{j}_{k}=u^{j}}\mathbf{P}_{I}(\mathbf{u}^{ k};O)\prod_{i=1,j\
eq j}^{\log_{2}M}P_{I}(u^{k,j},I) \tag{20}\]  
With:  
\[\mathbf{P}_{I}(\mathbf{u}^{k};O)=H_{u}\overline{H_{u}}\sum_{e:\;u(e)=u}A_{k-1 }[s^{S}(e)]\mathbf{P}_{I}[\mathbf{c}^{k};I]B_{k}[s^{E}(e)] \tag{21}\]  
where \(H_{u}\), \(\overline{H_{u}}\), and \(H_{u^{j}}\) are normalization constants; \(\mathbf{P}_{I}(\mathbf{c};I)\) is the inner codeword probability of the input; \(\mathbf{P}_{I}(\mathbf{u};I)\) is the inner information probability of the input; \(\mathbf{P}_{I}(\mathbf{u};O)\) denotes the inner information probability of the output; and \(A_{k}(\cdot)\) and \(B_{k}(\cdot)\) are obtained through forward and backward recursions, respectively [28]. On the basis of (20), knowing the identities of the other bits is necessary in deciding on a certain bit. For instance, it is assumed that current and next phase states are 0 and \(\pi\), separately, there are four possible input code symbols (001, 101, 011, 111) when the natural mapping is used as shown in Figure 7. It is hard to decide whether the first bit is 0 or 1 when the other bits are either 01 or 11. In the Gray mapping case, the decoder can make the appropriate decision on the first bit when the couples of other two mapped bits are different. Consequently, symbol mapping must be chosen carefully for BICM systems with high-order CPM. Figure 7: Transfer diagram of the phase states of 8\(M\)2 RC with \(h=1/2\) using Gray and natural mappings. NB-LDPC code has many advantages in comparison to binary coding, however, the decoding complexity restricts its development. Since the decoding algorithm of LDPC code is at the symbol level, the decoding complexity will increase rapidly with the increase of \(Q\). In the check-node update of BP decoding, an asymptotical complexity of \(O\) (Q'2) for log-decoding or \(O\) (Q log2(Q)) for fast fourier transform (FFT) decoding is present.