## 8 Conclusions  
We have presented a BSDCD scheme of BPSK signals for IEEE 802.15.4 WSNs. Then, a more meaningful and practical estimate scheme, i.e., an approximate ML estimator with only linear operation, is proposed for extracting a phase reference from preamble signal. Simulations suggest that our lower complexity in implementation does not sacrifice much in terms of detection performance, which can completely meet the requirements of the IEEE 802.15.4 standard. Therefore, it is the most attractive SDCD solution of choice for WSNs, especially when it used in consumer electronics. It is important to note that our idea in [10] as well as in this work can be used for simplification of Equation (8) in [14], where arctan operation is indispensable to performing an initial CFO estimation process. Loosely speaking, these schemes can be further classified into the _hard_ receiver. It makes use of only the \(estimates\) of the nuisance parameters as if they were the true values for the noncoherent detection. In the \(soft\) receiver, the postcompensation processor is required to compute \(posteriori\)s or \(conditional\)\(probability\)\(density\) functions of the nuisance parameters embedded in the chip sequence [12]. Thus, the detection process can incorporate a statistical characterization of the nuisance parameters (instead of the estimated ones). The latter is, of course, nonimplementable in general. Furthermore, unlike those who were researching ways to use BSDCD to combat the effect of chip-error propagation [9], we add that the theory developed here is easily extended to multiple  
Figure 12: BER performance of proposed receiver versus dynamic CPO. CPO: carrier phase offset. Figure 13: PER performance of proposed receiver versus dynamic CPO. CPO: carrier phase offset. differential coherent detection (MDCD). That is to say, the extension of the proposed scheme to account for multiple-symbols situation is straightforward. As a consequence, the performance gap between ideal coherent detection and BSDCD, indicated in Figure 7, can be narrowed. Not surprisingly, multiple unknown CFOEs after DD are necessary to be initially removed. Complexity reduction techniques may be developed based on the Viterbi algorithm [27], the fast algorithm in [28] and an algorithm based on subset search [29]. Of course, owing to the reduced PER that this augmented scheme yields, a much more reasonable trade-off can be achieved and is clearly desirable for both the battery-operated transmitter and receiver, which benefit from less energy that will be consumed by retransmissions between sensor nodes [30]. Finally, the complexity and energy efficiency can be deeply analyzed with the model given in [30]. These subjects will be reported on by the authors in a forthcoming paper. This work was partially supported by the National Natural Science Foundation of China (Project Nos. 61701172, 61701059, 61701062, 61561045 and 61572114), the Natural Science Foundation of Henan Province (Project No. 162300410097), the Postdoctoral Science Foundation of University of Electronic Science and Technology of China (Project No. Y02006023601721), the Initial Scientific Research Foundation of Doctor in Henan University of Science and Technology (Project No. 13480052), the Joint Funds of the National Natural Science Foundation of China (Project No. U1404615 and U1504619), the Program for Everest Scholar Talents Development in Tibet University, the Program for Science & Technology Innovation Talents in the University of Henan Province (Educational Committee) (Project No. 17 HASTIT025), and the Program for International Science & Technology Cooperation Projects of Henan Province (Project No. 172102410072). [PERSON], [PERSON] and [PERSON] conducted the algorithm design, and [PERSON] wrote the paper and performed all the the simulations under the supervision of [PERSON] and [PERSON]. [PERSON] and [PERSON] worked on the image alignment and contributed to preparing and analyzing the simulation data.