## 6 Proposed Divisor-Free CFOE Estimator  
The starting point of our estimation approach is the realization that the modulo-\(2\pi\) reduced phase of the measurement \(Y\) can also be expressed in form of \(\sin^{-1}\left(x\right)\), i.e., Equation (9) can be rewrote as follows:  
\[\angle Y= \begin{cases}\sin^{-1}\left(\frac{Im(Y)}{|Y|}\right),&\text{if}\quad Re (Y)\geq 0\\ -\pi-\sin^{-1}\left(\frac{Im(Y)}{|Y|}\right),&\text{if}\quad Re(Y)<0\end{cases} . \tag{14}\]  
Observe from Equation (14) that the absolute value of the magnitude divided argument of the inverse sine function is never more than 1 because length of any right-angle side is never more than that of the hypotenuse. Then, we can immediately simplify Equation (14) as follows:  
\[\angle Y\approx \begin{cases}\frac{Im(Y)}{|Y|},&\text{if}\quad Re(Y)\geq 0,\\ -\pi-\frac{Im(Y)}{|Y|},&\text{if}\quad Re(Y)<0.\end{cases}  \tag{15}\]  
Note that the mathematical approximation \(\sin^{-1}\left(x\right)\approx x\), for small \(x\), is involved in Equation (15). Assuming further that high SNR, known to be useful in estimation problems [26], keeps the magnitude of the non-Gaussian noise term \(\eta(1)\) in Equation (3) sufficiently small, we get  
\[|Y|=|e^{I\kappa\
u T_{e}}+\eta(1)|\approx 1. \tag{16}\]  
It is worth noting that this simplification is irrespective of the frequency offset being estimated. Substituting Equation (16) into (15), we can finally improve a CFOE estimator as follows:  
\[\widehat{\varphi}_{new}\triangleq \begin{cases}Im(Y),&\text{if}\quad Re(Y)\geq 0,\\ -\pi-Im(Y),&\text{if}\quad Re(Y)<0.\end{cases}  \tag{17}\]  
Note that Equation (14) can be rewritten in the following form:  
\[\angle Y= \begin{cases}\sin^{-1}\left(\frac{Im(Y)}{|Y|}\right),&\text{if}\quad Re (Y)\geq 0,\\ \pi-\sin^{-1}\left(\frac{Im(Y)}{|Y|}\right),&\text{if}\quad Re(Y)<0.\end{cases}  \tag{18}\]  
Then, Equation (17) can also be given by:  
\[\widehat{\varphi}_{new}\triangleq \begin{cases}Im(Y),&\text{if}\quad Re(Y)\geq 0,\\ \pi-Im(Y),&\text{if}\quad Re(Y)<0.\end{cases}  \tag{19}\]  
We provide some interpretations for the receiver whose estimator is expressed by Equation (19). * Equation (19) are obtained from approximations made to a structure motivated by DAML estimation scheme in Equation (14) or Equation (18), and a simple adaptive offset terms \(Im(Y)\) is provided. This enables the receiver to intelligently adjust the estimated CPOE value according to the time-varying channel conditions with respect to each packet transmission. * The complex observation space is only divided into two subspaces as shown in Figure 3. They can be easily distinguished from one another just with the sign of the real part of \(Y\). Thus, the absolute value operation of real and imaginary parts of \(Y\) in Equation (12) is not required in this case. An useful assumption, namely, the asymptotic or high chip SNR, is considered for simplification of Equation (15). Thus, the nonlinear division operation in Equation (12) is avoided. * No limitation on the CFOE range is required for the mathematical approximation \(\sin^{-1}\left(x\right)\approx x\) in our estimator. That is to say, it appears to be a full-range estimator, and the corresponding receiver is a full-range detector, the same feature as the receiver with Equation (12). By the approximation \(|Y|\approx 1\) valid at high chip SNR as illustrated in Figure 4, the result in Equation (19) may be shown to approximately hold for arbitrary SNRs, which is also the same as Equation (12). In a word, the CFO and chip SNR conditions do not need a priori knowledge for the mathematical approximation involved in Equation (19). As a result, an excellent detection performance is achieved at the receiver, which can completely meet the requirements of the IEEE 802.15.4 standard. This will be confirmed by the numerical results shown in the following section. * Clearly, the estimation error \(e\triangleq\widehat{\varphi}-\varphi\) is only introduced by the mathematical approximation if the noise is not considered.