## 3 Generalized BSDCD Scheme  
The receiver performs BSDCD of the \(m\)th bit \(E[m]\) by forming a complex-represented statistic from sample autocorrelation operation [8]:  
\[A[m]=\sum_{k=1}^{L_{1}}r_{m,k}r_{m-1,k}^{*}=L_{1}e^{j\phi_{m}}e^{jK\omega T_{c }}+\eta(0), \tag{2}\]  
where \(L_{1}\) is the sample number, \(1\leq L_{1}\leq K\), the superscript \(*\) is complex conjugate, \(\phi_{m}\in\{0,\pi\}\) is the actual information phase corresponding to the \(m\)th transmitted bit \(E[m]\), \(\omega=2\pi f\), and \(\eta(0)\) is an integrated noise sample. Obviously, first-order differential modulation is a means of handling only one unknown invariable parameter in two adjacent bit intervals [14]. Thus, the CFOE \(\varphi\triangleq K\omega T_{c}\) embedded in autocorrelator output \(A[m]\) is a residual nuisance parameter, generating a multiplicative phase distortion \(e^{j\varphi}\) as indicated in Equation (2). It is by no means evident that this impairment, referred to as phase noise, can cause a significant performance loss if it is not wiped out. The avenue considered here is to equip the autocorrelator with a postcompensation processor. It not only is simple to implement but has the potential to be almost without performance degradation as the frequency shift increases. It is important to note that the exponential term \(e^{jK\omega T_{c}}\) is referred as the CFOE in [2], but here is \(K\omega T_{c}\). Specifically, the CFOE is first recovered based on averaged autocorrelation samples of the preamble [8]:  
\[Y=\frac{1}{(L_{2}-1)L_{3}}\sum_{m=2}^{L_{2}}\sum_{k=1}^{L_{3}}r_{m,k}r_{m-1,k}^ {*}=e^{jK\omega T_{c}}+\eta(1). \tag{3}\]  
Here, \(L_{2}\) is the observation length in bit intervals, and \(2\leq L_{2}\leq J\), where \(J=32\) is the bit number in preamble field [1]. \(L_{3}\) is referred to as the number of the autocorrelation sample, \(1\leq L_{3}\leq K\), and \(\eta(1)\) is another complex-valued noise term. Then, follow with the result reported in Equation (18) of [6], a decision rule can be intuitively written as:  
\[\widehat{\phi}_{m}=\underset{\phi_{i}}{\arg\max}\ \Xi\left(A[m],\phi_{i},Y \right), \tag{4}\]  
where \"argmax\" denotes the argument that maximizes the following function with respect to the variable of interest, i.e., \(\phi_{i}\). Here, \(\phi_{i}\in\{0,\pi\}\) is the hypothetical information phase and \(\Xi\left(A[m],\phi_{i},Y\right)\) denotes the detection metric, given by [6]  
\[\Xi\left(A[m],\phi_{i},Y\right)=Re\left\{A[m]q(Y)e^{-j\phi_{i}} \right\}. \tag{5}\]  
In Equation (5), \(q(Y)\) is the so-called quantization function [2], which can be considered as a phase coherent reference to compensate the effect of the CFO on \(A[m]\). \(Re\{x\}\) denotes the real part of \(x\). Clearly, a bit detection error is generated when \(\widehat{\phi}_{m}\
eq\phi_{m}\). Observe from Equations (2) and (5) that the quantization function \(q(Y)\) is used to undo rotation induced by CFOE in autocorrelator output \(A[m]\). Bearing in mind that the BPSK alphabet is \(\{+1,-1\}\), we can immediately arrive at another detection rule from Equation (4) in the following form:  
\[\widehat{E}[m]=\left\{ \begin{array}{ll}0,&\text{if}\quad Re\{A[m]q(Y)\}\geq 0,\\ 1,&\text{if}\quad Re\{A[m]q(Y)\}<0.\end{array} \right. \tag{6}\]  
This clearly is the receiver strategy followed in [2; 8; 10]. In fact, equip Equation (4) with respective quantization functions, and let \(truncation\)_factors_\(L_{1}=L_{3}=K\), \(L_{2}=J\); then, we can arrive at the conventional detection approaches. The details of Generalized BSDCD (GBSDCD) are presented in Algorithm 1. Note that the bits of the start-of-frame delimiter (SFD) in the first field and PHY header (PHR) in the second field of the physical layers protocol data unit (PPDU) are not considered in Algorithm 1. This simplicity is considered to make the ideas behind the detection principle apparent. In addition, the preamble field of the PPDU is composed of 32 binary zeros [1], therefore the transmitted bits \(E[m]=0\), for \(1\leq m\leq 32\).