# Optimized Design for NB-LDPC-Coded High-Order CPM: Power and Iterative Efficiencies
## 3 Competitive CPM
### Design Criterion  
The power and spectral efficiencies are influenced by the choice of \(M\), \(h\), pulse shape, memory, and code rate \(R\). For example, a greater \(h\), with the remaining parameters constant, typically results in increased power efficiency but at the expense of spectral efficiency. Similarly, increasing \(M\) will increase the spectral efficiency and decrease the power efficiency. Lowering \(R\) alone can increase the coding gain but decrease the spectral efficiency. Thus, the combination of code and modulation parameters for coded CPM must be chosen carefully based on these constraints. Figure 3: Symmetry information rate (SIR) of CPM with 8M2 raised cosine (8M2 RC) using various \(h\). In practical applications, CPM demodulator complexity must be considered, which is dependent on the total number of matched filters and trellis states. The complexity could be represented as \(qM^{L}\) matched filters followed by the CPM-SISO detector with the trellis of \(PM^{L-1}\) states. All the CPM schemes subject to the constraints \(q\leq 5\), \(L<3\) and \(M\leq 8\) are analyzed in this paper to reduce the implementation complexity. In Table 1, the MSNED \(d_{\min}^{2}\) and \(B_{99\%}T_{b}\) of all candidate CPM schemes with RC and rectangle (REC) frequency pulses under the constraints (i.e., \(q\leq 5\), \(L<3\), \(M\leq 8\), and \(h<1\)) are calculated, where \(T_{b}\) is the bit period, and MSNED can be expressed by [10]:  
\[d_{\min}^{2}=\log_{2}M\underset{i}{\min}\bigg{\{}\frac{1}{T_{s}}\int_{0}^{NT_{s }}\big{[}1-\cos\phi(t,\mathbf{\chi}_{i})\big{]}dt\bigg{\}} \tag{7}\]  
with  
\[\phi(t,\mathbf{\chi}_{i})=2\pi h\underset{j}{\sum}\chi_{i,j}\int_{-\infty}^{t}g( \tau)d\tau,\mathbf{\chi}_{i}=\mathbf{\alpha}-\overline{\mathbf{\alpha}} \tag{8}\]  
where \(NT_{s}\) denotes observation symbol intervals and \(\mathbf{\chi}_{i}\) is the difference between the transmitted sequence \(\mathbf{\alpha}\) and the received sequence \(\overline{\mathbf{\alpha}}\). The values of \(\chi_{i,j}\) are obtained from \(\{0,\pm 2,\pm 4,\ldots,\pm 2(M-1)\}\). \(B_{99\%}\) can be computed indirectly by:  
\[0.99=\int_{-B_{99\%}/2}^{B_{99\%}/2}G(f)df \tag{9}\]  
where \(G(f)\) is the normalized power spectrum density of CPM and is defined as:  
\[ \begin{split} G(f)&=2\bigg{\{}\int_{0}^{LT}\Re(\tau )\,\cos 2\pi f\tau d\tau+\frac{1-\psi(jh)\cos 2\pi fT}{1+\psi^{2}(jh)-2\psi(jh) \cos 2\pi fT}\times\int_{LT}^{(L+1)T}\Re(\tau)\cos 2\pi f\tau d\tau\\ &\quad-\frac{\psi(jh)\sin 2\pi fT}{1+\psi^{2}(jh)-2\psi(jh)\cos 2 \pi fT}\times\int_{LT}^{(L+1)T}\Re(\tau)\sin 2\pi f\tau d\tau\bigg{\}}\end{split}  \tag{10}\]  
With:  
\[\psi(jh)=\sin M\pi h/M\sin\pi h \tag{11}\]  
where \(\Re(\tau)\) denotes the autocorrelation function of CPM:  
\[\Re(\tau)=\frac{1}{T}\int_{0}^{T}\prod_{k=1-L}^{\lfloor\tau/T\rfloor}\frac{1} {M}\frac{\sin 2\pi hM[q(t+\tau-kT)-q(t-kT)]}{\sin 2\pi h[q(t+\tau-kT)-q(t-kT)]}dt \tag{12}\]  
where \(\lfloor\cdot\rfloor\) is a floor rounding operator. A definite relationship between SIR and MSNED can be presented in Table 1 and Figure 3. The SIR curve of CPM with the highest MSNED converges earliest and corresponds to the lowest bound denoted as the minimum \(\mathrm{E_{s}/N_{0}}\), which can be traced back from the SIR curve by the \(R\log_{2}M\). For instance, the SIR of 8\(M\)2 RC using \(h\) = 1/2 converges earliest, followed by \(h\) = 1/3, 1/4, and 1/5, because the 8\(M\)2 RC using \(h\) = 1/2 has the highest \(d_{\min}^{2}\) of 5.56, followed by \(h\) = 1/3, 1/4, and 1/5. The computational complexity of MSNED is significantly lower than SIR, which provides convenience in designing a competitive CPM. Thus, we used \(B_{99\%}T_{b}\) and MSNED instead of SIR curves to design the suitable scheme from all candidate CPM signals under a given code rate \(R\) and spectral efficiency \(\eta\).