## 1 Introduction
However, in this case, the CFO embedded in samples at differential detector \(input\) is considered to be recovered from some _unknown_ modulated symbols. Therefore, the nonlinear operation (e.g., \(M\)th power for \(M\)-PSK) [15] is required to eliminate the dependence of these datas as indicated in Equation (7) of [14]. This is because the undesired parameter CFO is now a parameter of interest, whereas the data symbols are now just nuisance parameters. For the aforementioned postcompensation step, several CFOE estimation approaches have been studied [2; 8; 10]. The full estimation scheme is powerful yet always not affordable, since the inverse tangent operation is considered to be the most complex to implement in WSNs [2]. The work of [PERSON] et al. [8] first successfully simplifies this implementation; however, a considerable performance compromise cannot be avoided. This is because the mathematical approximation \(\tan^{-1}\left(x\right)\approx 0\) is considered therein and thus the CFOE is quantified into four fixed values (in radians), i.e., \(0,\pi/2,-\pi\), and \(-\pi/2\)[10]. These phases clearly seems to be independent of the channel conditions, i.e., SNRs. Considering the more advisable trigonometric approximation \(\tan^{-1}\left(x\right)\approx x\), we recently introduced an _adaptive_ offset-term adjustment algorithm in [10]. It has the advantage of enabling the receiver to intelligently adjust the four constant phases above according to the time-varying channel conditions with respect to each packet transmission. However, a division operation is required to perform this process. This needs a complex implementation and is energy-consuming in floating-point form [16]. An efficient symbol-timing algorithm was also deeply discussed by [PERSON] et al. in [2], where the observation is necessary to be extended to three bit intervals. In this scheme, the most recent bit detection result is required in practice as shown in Equation (8) of [2]. Under this decision-feedbackmechanism, error propagation may take place. For more details about these estimators, the reader is referred to [10] and the reference therein. The main contribution of this work can be highlighted as follows. The above so-called BSDCD technique is first deduced in detail, and its receiver structure is given. Then, for the compensation processor, some particular estimators are summarized and compared. Finally, an efficient estimator with only addition operation is provided for the compensation processor. It is worth noting that the high-order modulation [17], e.g., offset quadrature phase-shift keying (O-QPSK) with half-sine pulse shape, is also employed in IEEE 802.15.4 WSNs [1]. A simple double correlation based detector has been derived in [18; 19], which however is out of the scope of this work. To the best of our knowledge, this is the first work investigating the combination of the trigonometric approximation \(\sin^{-1}\left(x\right)\approx x\) and the high chip SNR to simplify the optimal detector. The remainder of this paper is organized as follows. Section 2 concentrates on the signal model over an additive white Gaussian noise (AWGN) channel. The generalized BSDCD algorithm is explicitly proposed in Section 3, while the optimal BSDCD scheme is described in Section 4. Section 5 indicates several particular CFOE estimation schemes for the postcompensation process, and a new estimator is proposed in Section 6. Numerical results are addressed in Section 7. Finally, some conclusions and future work are offered in Section 8.