## 2 Mathematical Modeling and Theoretical Foundations
### Frequency Domain Convolutional Blind Source Separation  
The actual communication environment is more complex, and the speech information is reflected by obstacles to form multipath transmission, which propagate through multiple paths to reach the receiving end, thus obtaining the unknown observed signals. It should be noted that there are both direct and reflected waves such as obstacles in the observed signals. Due to the different multipath lengths, the observed signals at the receiver are the superposition of source signals affected by phase, time delay and multipath. Mathematical modeling of the above multipath transmission environment is performed. In the case where the source signal and channel are unknown, only a small amount of a priori knowledge of the observed signals is used to achieve an estimate of the observation matrix or source signals. Without considering the noise, the mathematical model is shown in Figure 2, and this model is referred to as convolutional blind source separation [12]. Figure 1: The three-dimensional phase diagram of [PERSON]’s chaotic motion state. (**a**) The x–y plane phase diagram; (**b**) The y–z plane phase diagram; (**c**) The x–z plane phase diagram; (**d**) The Spatial phase diagram of Chen chaos. Suppose there are \(N\) source signals \(s(t)=\left[s_{1}(t),s_{2}(t),...,s_{N}(t)\right]^{T}\) and \(M\) observation signals \(x(t)=\left[x_{1}(t),x_{2}(t),...,x_{M}(t)\right]^{T}\) (\(N>M\)) and all are discrete signals. The mathematical model for convolutional mixing of multiple speech signals is:  
\[X_{i}(t)=\sum_{j=1}^{M}\sum_{p=0}^{P-1}h_{ij}(p)\cdot S_{j}(t-p)=h_{ij}*S_{j}(t )\quad i=1,2,...,M, \tag{2}\]  
where \(h_{ij}(p)\) denotes the impulse response of the \(j\)th source signal to the \(i\)th microphone; \(S_{j}(t)\) denotes the \(j\)th source signal; \(X_{i}(t)\) denotes the \(i\)th observed signal; \(*\) denotes the convolution operation, and \(p\) denotes the order of the transfer function. That is, each observed signal is a convolutional mixture of the individual source signals and the corresponding impulse responses to them. The larger the filter order \(p\), the more complex the received observation signal. In particular, when \(p=1\), the model degenerates into a linear instantaneous mixed model. To simplify, Equation (1) is expressed in vector form as:  
\[\mathbf{X}(t)=\sum_{p=0}^{P-1}\mathbf{H}(p)\mathbf{S}(t-p). \tag{3}\]  
where \(\mathbf{H}(p)\) is the vector representation of the transfer function. In practice, the transfer function \(h_{ij}(p)\) can be approximated by a finite impulse response filter. Transform the observed signals in the time domain to the time-frequency domain through STFT:  
\[\mathbf{X}^{(k)}[n]=\sum_{k=0}^{K-1}X(n\emph{f}+t)win(t)e^{-j\omega_{k}t}\quad k =1,2,...,K, \tag{4}\]  
where \(\omega_{k}=2\pi(k-1)/K\) is the number of frequency points and \(win(t)\) is the window function--the Hamming window is usually selected because it has better low-pass characteristics so that it can better approximate the short-term speech frequency spectrum. Furthermore, \(x_{i}^{(k)}[n]\) denotes the sample of the \(n\)th random variable of \(x_{i}^{(k)}\). It should be noted that, unlike real-time \(t\), \(n\) represents a random variable in the discrete time domain, where each value of \(n\) corresponds to the STFT of each frame. For convenience, the time  
Figure 2: CBSS model for multiple source signals. symbols \(t\) and \(n\) are mostly omitted. When the length of the window function \(win(t)\) is much larger than the order \(p\) of the impulse response filter, the convolutional model can be approximated as an instantaneous model for each frequency band [28; 33]:  
\[\textbf{X}^{(k)}=\textbf{H}^{(k)}\textbf{S}^{(k)}, \tag{5}\]  
where \(\textbf{X}^{(k)}=\left[x_{1}{}^{(k)},x_{2}{}^{(k)},...,x_{M}{}^{(k)}\right]^{T}\) and \(\textbf{S}^{(k)}=\left[s_{1}{}^{(k)},s_{2}{}^{(k)},...,s_{N}{}^{(k)}\right]^{T}\) are the observation matrix and the source signal matrix in the frequency domain, respectively.