## 2 Methodology of Time-Frequency Separation Neural Network  
### Basic Blind Source Separation Problem and Typical Solutions  
Blind source separation (BSS) originated from the \"cocktail party problem\", which is initially a term related to speech recognition, aiming to separate a target individual signal from multi-channel speech mixtures. Theoretically, the aim of BSS is to seek out the estimated source signals using only the observed signals from the recording devices, without any prior information of the source signals or mixing parameters of the channels. Blind source separation problems can be categorized into four types based on the hypothetical mixing models involved, as shown in Figure 1. This article mainly focuses on the instantaneous linear mixing model, which is illustrated as follows:  
Supposing there are \(n\) unknown source signals, which are statistically independent of each other, after passing through a certain linear system, they are mixed and then captured by \(m\) devices, resulting in \(m\) mixed signals in the linear instantaneous mixing model. The relationship between the mixed signals and the source signals can be expressed as  
\[\mathbf{x}(t)=\mathbf{As}(t)+\mathbf{n}(t) \tag{1}\]  
The matrix, \(\mathbf{A}=\left[a_{ij}\right]_{m\times n}(i=1,2,\ldots,m;j=1,2,\ldots,n)\), is called the mixing matrix, \(a_{ij}\) representing the unknown mixing coefficients, and \(\mathbf{n}(t)\) is the additive white noise in the mixing process. Then, BSS is used to obtain estimates of the source signals from the mixed signals \(\mathbf{y}(t)=\left[y_{1}(t),y_{2}(t),\ldots,y_{n}(t)\right]^{T}=\overset{ \wedge}{\mathbf{s}}(t)\). If matrix A is invertible, namely that the number of observations is equivalent to or larger than the number of sources, this objective can be completed with a separation matrix \(\mathbf{W}=\mathbf{A}^{-1}\). \[\mathbf{y}(t)=\mathbf{Wx}(t)=\mathbf{WAs}(t)=\overset{\wedge}{\mathbf{s}}(t) \tag{2}\]  
Therefore, the BSS problem can be summarized as seeking a separation matrix W or projection vectors to linearly combine the mixed noisy signals and obtain estimated signals that are as mutually independent as possible, as shown in Figure 1. One of the major assumptions related to BSS is the statistical independence of the primary sources, which leads to the solution of the independent component analysis (ICA) method. According to the central limit theorem, under the hypothesis of independent and identically distributed variables, the characteristics of mixed signals are closer to a Gaussian distribution than  
Figure 1: Categories of blind source separation problem and diagram of instantaneous linear mixing model. those of individual source signals. Then non-Gaussianity metrics such as negentropy and kurtosis can be utilized as measures of statistical independence in BSS. When negentropy is used, as shown in Equation (3), the separation vector \(\mathbf{W}^{*}\) with a maximal metric indicates the highest probability of independent sources. \[\left\{ \begin{array}{l}\mathbf{W}^{*}=\underset{\mathbf{W}}{\operatorname{ argmax}}\{H(\mathbf{y}_{Gauss})-H(\mathbf{y}(t))\}^{2}\approx\underset{\mathbf{W}}{ \operatorname{argmax}}\mathcal{C}\{E[G(\mathbf{y}_{Gauss})]-E[G(\mathbf{y}(t ))]\}^{2}\\ H(\mathbf{y})=-\int p(\mathbf{y})\log p(\mathbf{y})d\mathbf{y}\end{array} \right. \tag{3}\]  
where \(\mathbf{y}_{Gauss}\) is a random vector composed of \(n\) Gauss variables, which has the same mean values and covariance matrices as variable \(\mathbf{y}(t)=\mathbf{W}\mathbf{x}(t)\). \(H(\cdot)\) is the negentropy, rigorously defined as Equation (3). In addition, the derivative of \(H(\cdot)\) is the cumulative distribution function (CDF) of observed variables, which should be monotonically increasing in the range of \([0,1]\). Accordingly, there are also three typical approximate estimate functions for \(H\), \(G_{1}(\mathbf{y})=\mathbf{y}^{3}/4\), \(G_{2}(\mathbf{y})=1-\exp\bigl{(}-\mathbf{y}^{2}/2\bigr{)}\), \(G_{3}(\mathbf{y})=1/a\log\cosh(a\mathbf{y})\), \(1\leq a\leq 2\).