## 2 Signal Model  
The minimum shift keying (MSK) signal possesses advantages such as high spectrum utilization, strong anti-interference ability, high transmission efficiency, and simple implementation, which is an effective means for inter-station communication in networked passive detection. According to [17], the model of the baseband receiving signal containing MSK signal, in-band interferences, CFO, and noise can be expressed as follows:  
\[s(t)\ =\ c_{0}e^{j(2\pi\Delta f+\frac{t_{0}mt}{2T_{b}}+\varphi_{h})}+\sum_{m\ =\ 1}^{M}s_{m}(t)+w(t), \tag{1}\]  
where \(c_{0}\) is the signal magnitude at time \(t\), \(\Delta f\) is the CFO, \(a_{k}\) is the \(k\)th symbol, \(T_{b}\) is the symbol period, \(\varphi_{k}\) is the initial phase of the \(k\)th symbol, \(s_{m}(t)\) is the \(m\)-th in-band interference, \(M\) is the total number of in-band interferences, and \(w(t)\) is white Gaussian noise. Assume the \(M\) in-band interferences are uncorrelated with each other and are all uncorrelated with the expected signal. Without the loss of generality, we take the narrow band interferences as an example to conduct theoretical derivation and analysis. Under such circumstances, the signal model of the in-band interferences can be written as follows:  
\[s_{m}(t)\ =\ c_{m}e^{i[2\pi(f_{m}+\Delta f)t+\varphi_{m}]},1\leq m\leq M, \tag{2}\]  
where \(c_{m},f_{m}\), and \(\varphi_{m}\) are the amplitude, baseband frequency, and initial phase of the \(m\)-th in-band interference, respectively. In the MSK signal, the initial phases of any two adjacent symbols satisfy the following relationship:  
\[\varphi_{k}\ =\ \varphi_{k-1}+\frac{k\pi}{2}(a_{k-1}-a_{k}). \tag{3}\]  
We can infer from (1) and (3) that the signal phase is linear in a period of time corresponding to a segment of homopolar symbols, which shows the same characteristics as the phase of a snapshot of the array signal and lays the foundation of utilizing array signal processing algorithms in communication signal processing. In the digital domain, the baseband receiving signal can be rewritten as follows:  
\[s(n)\ =\ c_{0}e^{i(2\pi\Delta f/nT_{s}+\frac{a_{k}mT_{s}}{2T_{b}}+\varphi_{h})}+ \sum_{m\ =\ 1}^{M}c_{m}e^{i[2\pi(f_{m}+\Delta f)nT_{s}+\varphi_{m}]}+w(n), \tag{4}\]  
where \(s(n)\) is the sampled sequence of the baseband receiving signal, \(n\)(=1, 2, 3, \(\ldots\)) is the sampling number, \(T_{s}\)(=\(T_{b}/N_{s}\), \(N_{s}\) is the number of points sampled in one symbol period) is the sampling period, and \(w(n)\) is the sampled sequence of white Gaussian noise. Let \(f_{0}\ =\ a_{k}/(4T_{b})\) and \(\varphi_{0}\ =\ \varphi_{k}\), and then (4) can be rewritten as follows:  
\[s(n)\ =\ \sum_{m\ =\ 0}^{M}c_{m}e^{i[2\pi(f_{m}+\Delta f)nT_{s}+\varphi_{m}]}+ w(n), \tag{5}\]  
where \(f_{0}\) is the baseband frequency of the MSK signal. Then, let \(\Delta\varphi_{CFO}\ =\ 2\pi\Delta fT_{s}\) and \(\Delta\varphi_{m}\ =\ 2\pi f_{m}T_{s}\), and we call \(\Delta\varphi_{CFO}\) and \(\Delta\varphi_{m}\) the additional phase induced by CFO and the additional phase corresponding to the baseband frequency of each signal component, respectively. In particular, when \(m=0\), the corresponding \(\Delta\varphi_{0}\) denotes the additional phase induced by the MSK modulation. On this basis, (5) can be rewritten as follows:  
\[s(n)\ =\ \sum_{m\ =\ 0}^{M}c_{m}e^{i(n\Delta\varphi_{m}+n\Delta\varphi_{ CFO}+\varphi_{m})}+w(n). \tag{6}\]  
Based on the above signal model, we propose a data-aided method for signal detection and CFO estimation for wireless communication among stations of MPDSs in this paper. The method takes advantage of the frequency characteristic of the MSK signal and designsthe training sequence as a three-segment sequence with a structure of +**A**-**A**+**A**, where **A** represents the repeated part and we call it a sub-training sequence. To improve the readability of this paper, we summarize the proposed method in the following paragraph before explaining the key parts in detail.