# Optimized Design for NB-LDPC-Coded High-Order CPM: Power and Iterative Efficiencies
## 1 Introduction  
Continuous phase modulation (CPM) has the advantages of a continuous phase, constant envelope, high spectrum utilization, and excellent bandwidth efficiencies [1; 2]. These characteristics are more notable for satellite systems with less power and fewer bandwidth resources, such as satellite navigation [3; 4; 5], satellite mesh networks [6; 7], and satellite communication [8; 9]. To further enhance the power and bandwidth efficiencies, coded CPM with symbol mapping using iterative decoding was presented. Considering the excellent properties of CPM and the serially concatenated principles [10; 11], some particularly attractive solutions have become the focus of research, such as the convolutional [12; 13], turbo [14; 15; 16], and low-density parity-check (LDPC) coded CPM [17; 18; 19; 20]. As the serially concatenated CPM schemes in terms of convergence threshold are difficult to approach the Shannon limit, the non-binary (NB)-LDPC [21] code is considered the outer code, which has been the subject of numerous studies due to its excellent error correction capability. Unlike the traditional bit-interleaved coded modulation (BICM) systems, the interleaver of a high-order CPM works at the symbol level, which always yields a lower convergence threshold than the bit level [22]. The NB-LDPC code and high-order CPM have the same alphabet size, so the conversion information loss from bit to symbol caused by symbol mapping need not be considered [23]. To design and optimize coded CPM, previous works mainly focused on approaching the minimum signal-to-noise ratio (SNR) with a certain symmetry information rate (SIR) for a particular CPM with large block codes using large numbers of iterations [19]. This approach significantly limits the comprehensive analysis of a family of CPM schemes and is unsuitable for real-time applications due to excessive code length and decoding delay. Fortunately, [PERSON] filled the gap and designed a suitable CPM using SIR among all CPM candidates [8], however, the SIR should be calculated for each CPM, which is a difficult task. Similarly, [PERSON] used a systematic search procedure among all the candidates to identify the appropriate CPM scheme [20], achieving a targeted spectral efficiency and error rate, which results in considerable inconvenience in the CPM parameter design. [PERSON] derives an asymptotic analysis and optimization of coded CPM systems using both unstructured and protograph-based LDPC codes ensembles [24]. A practical system model with global interleaving is proposed for the optimization of degree distribution of the LDPC codes, which systematically solves the optimal system design of cascade irregular LDPC-CPM [25]. [PERSON] propose a two-stage decoding scheme mainly based on parity check matrix transform, which can efficiently improve the bit error rate performance [26]. All of the above schemes optimize the LDPC-CPM system to some extent, but they all adopt binary coding, the design and optimization of NB-LDPC-CPM system are still lacking. In summary, the major limitations of coded CPM optimization include CPM parameter design, decoding delay, the positive feedback problem in iterative decoding and the optimization of NB-LDPC. This paper designed and optimized an NB-LDPC-coded high-order CPM to pursue high power efficiency and improve iterative efficiency under given spectral efficiency \(\eta\) and code rate \(R\) for practical considerations. Firstly, given the definite relationship between the SIR of CPM and minimum squared normalized Euclidean distance (MSNED), the MSNED and the 99% double-sided power bandwidth (\(B_{99\%}\)) are adopted to design a competitive CPM for a given \(R\) and \(\eta\). Secondly, the code can be designed through a three-step method based on extrinsic information transfer (EXIT) and entropy theory when the CPM is determined. Thirdly, an extrinsic information operation is proposed in this paper to address the positive feedback problem in iterative detection and decoding, which problem may worsen bit error rate (BER) performance.