## 6 Conclusions and Future Work  
This paper proposes two simple schemes to quickly generate bit LLR for coded MPSK signals: one for the phase noncoherent channel and the other for the Rayleigh fading channels. Our analysis shows that, when compared with the traditional optimal  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Scheme** & \((\bullet)(\bullet)\) or \((\bullet)/(\bullet)\) & \((\bullet)\pm(\bullet)\) & \(|\bullet|\) & \((\bullet)^{*}\) & \((\bullet)^{2}\) & In\((\bullet)\) & \(I_{0}(\bullet)\) & **CSI** & \(e^{(\bullet)}\) & **Re** \\ \hline (21) & 4164 & 3000 & 2048 & 1024 & 5120 & \(\backslash\) & \(\backslash\) & \(\backslash\) & 64 & \(\backslash\) \\ \hline (23) & 64 & 990 & 64 & 64 & 64 & \(\backslash\) & \(\backslash\) & \(\backslash\) & \(\backslash\) & \(\backslash\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Complexity comparison of the LLR calculation under the Rayleigh fading channel.  
Figure 13: Detection performance for LDPC coded IEEE 802.15.4c MPSK signals under dynamic CPO channel, wherein LLR given in (16) is considered. MS algorithm is adopted for decoding, where \(\rho=1\). **(a)** BER performance; **(b)** SER performance; **(c)** PER performance.  
method, the scheme in this paper greatly reduces the implementation complexity. The proposed algorithm is applied to IEEE 802.15.4 WSNs to verify its effectiveness. The simulation results show that under both phase noncoherent and Rayleigh fading channels, the proposed simplified algorithm can depict an appropriate performance loss.  
There are several directions remaining for future research. First, it is pointed out that we focus all our attention on end-to-end communication. The extension to distributed detection is straightforward and worthy of further study. Furthermore, all of the analysis is tailored uniquely for a slow changing channel, and the extension to more complex channel case, e.g., fast changing fading channel, is a challenging problem. In addition, only the CPO is considered to characterize the effect of the channel transmission; the channel with carrier frequency offset is also worthy of further study.  
Conceptualization, B.X. and G.Z.; methodology, G.Z. and C.H.; software, G.Z. and H.L.; validation, G.Z. and H.L.; formal analysis, H.L.; investigation, B.X. and G.Z.; resources, B.X.; data curation, H.L.; writing--original draft preparation, C.M. and H.L.; writing--review and editing, B.X. and G.Z.; visualization, C.M. and H.L.; supervision, B.X. and G.Z.; project administration, B.X.; and funding acquisition, G.Z. and C.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China grant numbers [61701172 and 41605122], the Henan Province Soft Science Research grant number [222400410440], the Key Laboratory of Middle Atmosphere and Global environment Observation grant number [LAGEO-2021-04], the backbone teacher training program of HAUST, the Natural Science Foundation of Henan Province grant number [162300410097], and the Scientific and Technological Innovation Team of Colleges and Universities in Henan Province grant number [20 IRTSTHN018].  
Not applicable.  
We thank the National Natural Science Foundation of China for the financial support.  
The authors declare no conflicts of interest.