## 2 Problem Statement  
In the wireless communication propagation model, multipath and time delay of signals are inevitable, which can be modeled as the convolutive mixture of the source signals [17]. All variables involved in the problem statement are described in Table 1. Without considering the noise, the mathematical expression of the convolutional mixture model can be expressed as  
\[\mathbf{x}(t)=\sum_{q=1}^{Q}\mathbf{A}_{q}\mathbf{s}(t) \tag{1}\]  
where \(\mathbf{s}(t)\) represents the source signal; \(\mathbf{x}(t)\) indicates the received mixed signal; \(Q\) is the number of wireless propagation paths; \(\mathbf{A}_{q}\) denotes the mixing matrix corresponding to the \(q\)th path. Considering the effect of time delay. \(\mathbf{A}_{q}\) can be expressed as  
\[\mathbf{A}_{q}= \begin{bmatrix}a_{11}^{q}\delta(t-\tau_{11}^{q})&a_{12}^{q}\delta(t -\tau_{12}^{q})&\cdots&a_{1n}^{q}\delta(t-\tau_{1n}^{q})\\ a_{21}^{q}\delta(t-\tau_{21}^{q})&a_{22}^{q}\delta(t-\tau_{22}^{q})&\cdots&a_{2 n}^{q}\delta(t-\tau_{2n}^{q})\\ \vdots&\vdots&a_{ik}^{q}\delta(t-\tau_{ki}^{q})&\vdots\\ a_{m1}^{q}\delta(t-\tau_{m1}^{q})&a_{m2}^{q}\delta(t-\tau_{m2}^{q})&\cdots&a_{ mn}^{q}\delta(t-\tau_{nm}^{q})\end{bmatrix}  \tag{2}\]  
where \(\delta(\cdot)\) represents the impulse response, \(a_{ki}^{q}\) and \(\tau_{ki}^{q}\) are the amplitude attenuation and time delay of the \(i\)th source signal to the \(k\)th sensor through the \(q\)th path, respectively. In order to obtain the sparse characteristic of the signals, the signal \(X_{k}(t,f)\) received by the \(k\)th sensor after a short-time Fourier transform (STFT) [18] can be written as  
\[X_{k}(t,f)=\sum_{i=1}^{n}\sum_{q=1}^{Q}a_{ki}^{q}S_{i}^{q}(t-\tau_{ki}^{q},f) \tag{3}\]  
Take the BPSK signal as an example; the TF expression is described as  
\[X_{k}(t,f)=\sum_{i=1}^{n}\sum_{q=1}^{Q}a_{ki}^{q}\hat{h}(f\pm f_{i})e^{\mp j2 \pi  
(\(\Delta>0\)) of the received signal. In this case, the power ratio \(P_{\alpha}(f)/P_{\beta}(f)\) of any two of the received signals \(\mathbf{X}_{\alpha}\) and \(\mathbf{X}_{\beta}\) is  
\[ \begin{split}& P_{\alpha}(f)/P_{\beta}(f)\\ &=\begin{vmatrix}\sum\limits_{i=1}^{\sum\limits_{i=1}^{Q}d_{xi}^{q }\hat{h}(f\pm f_{i})}\\ \sum\limits_{i=1}^{\sum\limits_{i=1}^{Q}Q}d_{xi}^{q}\hat{h}(f\pm f_{i}) \end{vmatrix}=\begin{vmatrix}\sum\limits_{i=1}^{Q}d_{xi}^{q}\\ \sum\limits_{i=1}^{Q}d_{xi}^{q}\end{vmatrix},\ \ f\in[f_{i}-\Delta,f_{i}+\Delta] \end{split}  \tag{6}\]  
Consequently, the short-time spectral power points of the signal \(s_{i}(t)\) will be clustered on the line with a slope of \(P_{\alpha}(f)/P_{\beta}(f)\). Figure 1 represents the scatter plot of the power ratio of the two received signals after the convolutive mixture of the four BPSK signals. Due to the sparsity of the observed signals in the power domain, the received signals exhibit linear clustering property. There are four linear clusters in Figure 1 corresponding to the four source signals, and thus the mixed signals can be effectively separated by the traditional single-source-point (SSP) detection method [10]. However, in practical communication systems, it is almost impossible to avoid TF aliasing for communication signals. When the spectrum of the source signal is overlapped, the sparsity of the observed signals in the power domain cannot be guaranteed, thus weakening the linear clustering property of the observed signals (Figure 2). The traditional SSP method will not accurately recover the source signal.