# A Novel Complex-Valued Blind Source Separation and Its Applications in Integrated Reception
## 2 Time Delay Mixing Model  
The linear time delay mixing model considers both the time delay and amplitude attenuation of the source signal arriving at each sensor. Considering that there are \(K\) independent far-field narrowband signals incident on an \(M\)-element uniform line array from different directions \(\theta_{1},\theta_{2},\cdots,\theta_{K}\), if the \(k\)-th target signal received by the first array element is \(s_{k}(t)\), the mixing signal received by the \(m\)-th array element is [30]  
\[x_{m}(t)=\sum_{k=1}^{K}s_{k}(t)\exp(-j2\pi f_{k}\tau_{mk})+n_{m}(t)(1\leq m\leq M,1\leq k\leq K) \tag{1}\]  
where \(t=t_{1},t_{2},\cdots,t_{N}\) is the time series, \(N\) is the array snapshot number, \(f_{k}\) is the carrier frequency of the target signal, \(\tau_{mk}\) is the time delay of the \(k\)-th signal reaching the \(m\)-th array element, and \(n_{m}(t)\) is the output noise of the \(m\)-th array element. As shown in Figure 2, for a uniform linear array, the time delay can be expressed as \(\tau_{mk}=\frac{d\sin\theta_{k}}{c}\) based on the first array element. \(d\) is the distance between adjacent elements, and \(c\) is the speed of light. The output of all array elements is expressed in matrix form as  
\[\mathbf{x}(t)=\mathbf{A}\mathbf{s}(t)+\mathbf{n}(t) \tag{2}\]  
where the observed signal is \(\mathbf{x}(t)=[x_{1}(t),x_{2}(t),\ldots,x_{M}(t)]^{T}\), the source signal is \(\mathbf{s}(t)=[s_{1}(t),s_{2}(t),\ldots,s_{K}(t)]^{T}\), and the noise is \(\mathbf{n}(t)=[n_{1}(t),n_{2}(t),\ldots,n_{M}(t)]^{T}\), \(M\geq K\). \(n_{m}(t)\) is complex-valued Gaussian white noise with zero mean. The output noise of each array element is independent of each other. The array matrix (mixing matrix) \(\mathbf{A}\) is  
\[\mathbf{A}=\left\{ \begin{array}{cccc}1&1&\cdots&1\\ e^{-j2\pi d\frac{\sin\theta_{1}}{\lambda}}&e^{-j2\pi d\frac{\sin\theta_{2}}{ \lambda}}&\cdots&e^{-j2\pi d\frac{\sin\theta_{K}}{c}}\\ \vdots&\vdots&\vdots&\vdots\\ e^{-j2\pi(M-1)d\frac{\sin\theta_{1}}{\lambda}}&e^{-j2\pi(M-1)d\frac{\sin \theta_{2}}{\lambda}}&\cdots&e^{-j2\pi(M-1)d\frac{\sin\theta_{K}}{\lambda}} \end{array} \right\} \tag{3}\]  
where \(\lambda\) is the signal wavelength. It is obvious that the array flow pattern matrix \(\mathbf{A}\) contains the DOA of the signal. The purpose of BSS is to obtain the separation matrix \(\mathbf{B}\) by some algorithm, so that  
\[\mathbf{y}(t)=\mathbf{B}\mathbf{x}(t) \tag{4}\]  
where \(\mathbf{y}(t)\) is the separated signal (estimation of the source signal). CVBSS generally has uncertainty in the amplitude, phase, and sequence. Amplitude ambiguity affects signal detection and phase ambiguity affects signal demodulation in an integrated system. Therefore, amplitude and phase correction are needed. Without considering noise, if the array is ideal, the separation matrix \(\mathbf{B}\) and the mixing matrix \(\mathbf{A}\) have the following relationship. \[\mathbf{B}\mathbf{A}=\mathbf{D}\mathbf{P}\]  
Figure 2: Far-field narrowband signal mixing model. where \(\mathbf{P}\) is the permutation matrix of dimension \(K\times K\), which represents the sequence ambiguity of CVBSS. \(\mathbf{D}=diag\big{\{}\alpha_{1},\alpha_{2},\cdots,\alpha_{K}\big{\}}\) is the diagonal matrix, which represents the amplitude and phase ambiguity of CVBSS. \(\alpha_{k}\in\mathbb{C}\) denotes the amplitude phase ambiguity between the \(i\)-th estimated signal and its corresponding source signal. When there is no amplitude phase ambiguity in this signal component, \(\alpha_{i}=1\). \(\mathbf{\hat{A}}\) is the mixing matrix estimated using the CVBSS. Then, \(\mathbf{\hat{A}}\) is the left pseudo-inverse matrix of \(\mathbf{B}\), i.e., \(\mathbf{\hat{A}}=\big{(}\mathbf{B}^{H}\mathbf{B}\big{)}^{-1}\mathbf{B}^{H}\). Replacing it into Equation (5),  
\[\mathbf{\hat{A}}=\mathbf{A}\mathbf{P}^{-1}\mathbf{D}^{-1} \tag{6}\]  
Since the inverse of the permutation matrix \(\mathbf{P}^{-1}\) is still a permutation matrix, \(\mathbf{A}\mathbf{P}^{-1}\) is swapping the positions of some columns in \(\mathbf{A}\).