# Optimized Design for NB-LDPC-Coded High-Order CPM: Power and Iterative Efficiencies
## 6 Optimization Design for Iterative Efficiency  
LDPC decoding is an iterative detector using BP or a modified BP algorithm. Thus, the NB-LDPC-coded CPM systems have two iterative decoding structures at the receiver. The iteration choice has profound effects on iterative decoding performance. However, large numbers of iterations between demodulation and decoding are generally used to achieve excellent BER performance, which adversely increases computational complexity and iterative decoding delay. To improve iterative efficiency, we propose iteration optimization using the EXIT chart and mutual information between demodulation and decoding to achieve the suitable tradeoff for the communication reliability and iterative decoding delay.  
The optimized NB-LDPC code for the \(8M2\)RC scheme with \(\eta=0.5\) bit/s/Hz and \(R=2/3\) was investigated as an instance. The corresponding EXIT curves of the optimized NB-LDPC code with various inner iterations are shown in Figure 10. Inner iterations have no significant impact on convergence threshold because more iterations barely improve the BER performance at low \(\text{E}_{\text{b}}/\text{N}_{0}\).  
The EXIT curves for NB-LDPC code become steeper with increasing inner iterations, implying a larger iterative space of the CPM-SISO and LDPC-SISO decoder EXIT curves, resulting in an easier convergence with fewer outer iterations. However, the improved trend is no longer evident in five or more inner iterations. Hence, the optimized inner iteration of this design is set to five times.  
Figure 10: EXIT charts for optimized NB-LDPC code with various inner iterations.  
Figure 9: Iterative convergence of the NB-LDPC code for \(8M2\)RC with \(h=1/2\) using the proposed and original methods at \(\text{E}_{\text{b}}/\text{N}_{0}=1.2\) dB.  
Once the inner iterations were determined, the optimization of the outer iterations was investigated as follows. Fewer outer iterations are required to achieve a high BER performance with increasing \(\text{E}_{\text{b}}/\text{N}_{0}\) in a turbo cliff region. Thus, a limit exists on how to design suitable outer iterations for various \(\text{E}_{\text{b}}/\text{N}_{0}\). Mutual information is effective for better understanding the convergence of SISO decoders. A greater amount of mutual information means a more accurate identification of SISO decoders on information symbols. The mutual information of the outer codeword probability of the LDPC-SISO output at the \(n\)th iteration is defined as:  
\[MI(n)=\log_{2}Q-\frac{1}{N}\sum_{i=1}^{N}\sum_{q=0}^{Q-1}P_{O}^{n}(c_{q}^{i},O) \times\log_{2}\frac{1}{P_{O}^{n}(c_{q}^{i},O)} \tag{25}\]  
where \(P_{O}^{n}(c_{q}^{i},O)\) is the probability of an event wherein the \(i\)th element in the symbol vector \(\mathbf{c}\) equals \(q\). If the difference in mutual information between the current and previous iterations is extremely small, then their BER performance is comparable, and the improved performance is no longer remarkable with the following outer iterations. Therefore, an iterative stopping criterion can be estimated as:  
\[MI(n)-MI(n-1)<\varepsilon \tag{26}\]  
where \(\varepsilon\) is an extremely small value (for example, \(\varepsilon=10^{-5}\)). If the current iteration satisfies this criterion at a certain \(\text{E}_{\text{b}}/\text{N}_{0}\), it is assumed to be the optimal outer iteration, and the next iteration detection is terminated immediately.