# Optimized Design for NB-LDPC-Coded High-Order CPM: Power and Iterative Efficiencies
## 3 Competitive CPM  
### Sir  
The SIR is defined as the mutual information rate under the assumption that the input information is independent and identically distributed. To design an excellent coded CPM, the SIR is generally applied to design the best CPM scheme with the lowest \(\text{E}_{\text{s}}/\text{N}_{0}\) required to achieve SIR. Given that CPM is a time invariant finite state machine (FSM) with complex signal outputs, the CPM system over an AWGN channel can be viewed as a finite state Markov channel (FSMC, Figure 2), and the SIR can be calculated using the algorithm developed in [31; 32; 33; 34].  
In Figure 2, \(\mathbf{x}_{1}^{N}\) and \(\mathbf{v}_{1}^{N}\) denote input information sequence and modulated sequence, respectively, and \(\mathbf{y}_{1}^{N}\) is the output information sequence disturbed by AWGN sequence \(\mathbf{w}_{1}^{N}\). Then, on the basis of the definition of channel capacity, the mutual information rate between \(\mathbf{x}_{1}^{N}=(x_{1},x_{2},\ldots,x_{N})\) and \(\mathbf{y}_{1}^{N}=(y_{1},y_{2},\ldots,y_{N})\) can be estimated as:  
\[C=\underset{N\rightarrow\infty}{\lim}\frac{1}{N}I(\mathbf{x}_{1}^{N},\mathbf{ y}_{1}^{N})=\underset{N\rightarrow\infty}{\lim}\frac{1}{N}\left[H(\mathbf{x}_{1}^{N})-H( \mathbf{x}_{1}^{N}|\mathbf{y}_{1}^{N})\right]=\text{log}_{2}M-\underset{N \rightarrow\infty}{\lim}\frac{1}{N}H(\mathbf{x}_{1}^{N}|\mathbf{y}_{1}^{N}) \tag{1}\]  
where \(H(\cdot)\) is an entropy function and \(\mathbf{x}_{1}^{N}\) is independently and uniformly distributed. The expression \(H(\mathbf{x}_{1}^{N}|\mathbf{y}_{1}^{N})\) must also be calculated. \(\mathbf{s}_{0}^{N}\) is defined as the state transition sequence of the CPM. \(\mathbf{s}_{0}^{N}\) is a Markov random process and can only be related to \(\mathbf{x}_{1}^{N}\); thus:  
Figure 1: Non-binary low-density parity-check (NB-LDPC)-coded high-order continuous phase modulation (CPM) transmitter and receiver.  
Figure 2: CPM model over an additive white Gaussian noise (AWGN) channel.  
\[H(\mathbf{x}_{1}^{N}|\mathbf{y}_{1}^{N})=H(\mathbf{s}_{0}^{N}|\mathbf{y}_{1}^{N}) =\sum_{i=1}^{N}H(s_{i}|s_{i-1},\mathbf{y}_{1}^{N})=-\sum_{i=1}^{N}p(s_{i}|s_{i-1 },\mathbf{y}_{1}^{N})\log_{2}p(s_{i}|s_{i-1},\mathbf{y}_{1}^{N}). \tag{2}\]  
In computing \(p(s_{i}|s_{i-1},\mathbf{y}_{1}^{N})\), [PERSON] rule transforms it to:  
\[p(s_{i}|s_{i-1},\mathbf{y}_{1}^{N})=\frac{p(s_{i},\mathbf{y}_{1}^{N}|s_{i-1})}{ \sum_{s_{i}}p(s_{i},\mathbf{y}_{1}^{N}|s_{i-1})}. \tag{3}\]  
\(\gamma\) and \(\beta\) are introduced using the Bahl-Cocke-Jelinek-Raviv (BCJR)-like algorithm to estimate \(p(s_{i},\mathbf{y}_{1}^{N}|s_{i-1})\), and they are defined as follows [34]:  
\[\gamma_{i}(s_{i-1},s_{i})=p(s_{i},y_{i}|s_{i-1})=p(y_{i}|s_{i-1},s_{i})p(s_{i} |s_{i-1})=p(y_{i}|v_{i})p(x_{i})=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\!\left(- \frac{\|y_{i}-v_{i}\|^{2}}{2\sigma^{2}}\right)\!p(x_{i}) \tag{4}\]  
\[\beta_{i-1}(s_{i-1})=p(\mathbf{y}_{i}^{N}|s_{i-1})=\sum_{s_{i}}p(s_{i},\mathbf{ y}_{i}^{N}|s_{i-1})=\sum_{s_{i}}p(\mathbf{y}_{i+1}^{N}|s_{i})p(s_{i},y_{i}|s_{i-1 })=\sum_{s_{i}}\beta_{i}(s_{i})\gamma_{i}(s_{i-1},s_{i}) \tag{5}\]  
where \(\beta_{N}(s_{N})\) is initialized as an equally likely state. Then, combining (4) and (5), \(p(s_{i}|s_{i-1},\mathbf{y}_{1}^{N})\) is rewritten as:  
\[p(s_{i}|s_{i-1},\mathbf{y}_{1}^{N})=\frac{\overline{\beta_{i}}(s_{i})\gamma_{ i}(s_{i-1},s_{i})}{\sum_{s_{i}}\overline{\beta_{i}}(s_{i})\gamma_{i}(s_{i-1},s_{i })}, \tag{6}\]  
with \(\overline{\beta_{i}}(s_{i})=\beta_{i}(s_{i})/\sum_{s_{i}}\beta_{i}(s_{i})\).  
In Figure 3, the simulated SIR of CPM signals with 8M2 raised cosine (8M2 RC) using various modulation index \(h\) (\(h=p/q,p\), and \(q\) are the co-primes.) are shown, where 8M2 RC denotes a particular CPM family with \(M=8\), memory length \(L=2\), and RC frequency pulse.