# Minimum Variance Distortionless Response--Hanbury Brown and Twiss Sound Source Localization
## 2 MVDR-HBT Localization Principle
### Principle of Narrowband Sound Source Localization Using MVDR-HBT  
Assuming that the expected signal received by the microphone array is \(\mathbf{p}_{s}\), the interference signal is \(\mathbf{p}_{i}\), and the noise signal is \(\mathbf{p}_{n}\), the beamforming receiver receives signal \(\mathbf{p}=\mathbf{p}_{s}+\mathbf{p}_{i}+\mathbf{p}_{n}\), which is a coupling of the expected signal, interference signal, and noise signal. \(\mathbf{p}_{s}\) is the target signal that we expect to receive and process, \(\mathbf{p}_{i}\) comes from external interference to the signal, and \(\mathbf{p}_{n}\) is caused by the inherent uncertainty of the signal itself. Therefore, the output beamforming is given by  
\[\mathbf{y}=\mathbf{w}^{H}\,\mathbf{p}=\mathbf{w}^{H}(\mathbf{p}_{s}+\mathbf{p}_{i}+\mathbf{p}_{n})=\mathbf{y}_{ s}+\mathbf{y}_{i}+\mathbf{y}_{n} \tag{1}\]  
where \(\mathbf{y}_{s}\) represents the interfering signal component, \(\mathbf{y}_{i}\) represents the noise signal component, \(\mathbf{y}_{n}\) represents the desired signal component, \(\mathbf{w}=\left[\mathbf{w}_{1},\mathbf{w}_{2},\ldots,\mathbf{w}_{n}\right]^{T}\) represents the weight vector, \(T\) represents the transpose, and \(H\) represents the conjugate transpose. By combining and rearranging the \(\mathbf{p}_{i}\) and \(\mathbf{p}_{n}\) terms in Equation (1) to form \(\mathbf{p}_{in}\), Equation (2) can be obtained:  
\[\mathbf{y}=\mathbf{w}^{H}(\mathbf{p}_{s}+\mathbf{p}_{in})=\mathbf{y}_{s}+\mathbf{y}_{in} \tag{2}\]  
Based on Equation (2), the power of interference and noise signals can be obtained as shown below:  
\[E\Big{\{}\left|\mathbf{y}_{in}\right|^{2}\Big{\}}=E\bigg{\{}\left|\mathbf{w}^{H}\mathbf{ p}_{in}\right|^{2}\bigg{\}}=\mathbf{w}^{H}E\Big{\{}\Big{|}\mathbf{p}_{in}\mathbf{p}_{in}^{H} \Big{|}\Big{\}}\mathbf{w}=\mathbf{w}^{H}\mathbf{R}_{in}\mathbf{w} \tag{3}\]  
where \(\mathbf{R}_{in}=E\big{\{}\big{|}\mathbf{p}_{in}\mathbf{p}_{in}^{H}\big{|}\big{\}}\) represents the covariance matrix of the interference signal and the noise signal. After satisfying the matrix constraint \(\mathbf{w}^{H}\mathbf{a}(\theta_{s})=1\), in order to minimize the interference and noise, we establish the objective optimization equation as follows:  
\[ \begin{cases}\min\mathbf{w}^{H}\mathbf{R}_{in}\mathbf{w}\\ \textit{subject to }\ \mathbf{w}^{H}\mathbf{a}(\theta_{s})\ =\ 1\end{cases}  \tag{4}\]  
where \(\mathbf{a}(\theta_{s})\) refers to the steering vector of the target signal. The essence of MVDR is to solve the weight coefficients of each array element. By using the Lagrange multiplier method to solve for the optimal value of the weight vector, the following expression can be obtained:  
\[f(\mathbf{w},\!\lambda)=\mathbf{w}^{H}\mathbf{R}_{in}+\lambda\Big{(}\mathbf{w}^{H}\mathbf{a}( \theta_{s})-1\Big{)}+\lambda^{*}\Big{(}\mathbf{w}^{H}\mathbf{a}(\theta_{s})-1\Big{)} \tag{5}\]  
Figure 1: Schematic diagram of MVDR-HBT sound field positioning. After taking the derivative of Equation (5) and setting it equal to 0, we can obtain \(\mathbf{w}^{H}=-\lambda\mathbf{a}^{H}(\theta_{s})\mathbf{R}_{in}^{-1}\). Substituting into Equation (5), we have  
\[\lambda=-\Big{(}\mathbf{a}^{H}(\theta_{s})\mathbf{R}_{in}^{-1}\mathbf{a}(\theta_{s})\Big{)}^ {-1} \tag{6}\]  
Combining Equation (6), the weighted vector of MVDR can be obtained as follows:  
\[\mathbf{w}_{MVDR}=\frac{\mathbf{R}_{in}^{-1}\mathbf{a}(\theta_{s})}{\mathbf{a}^{H}(\theta_{s} )\mathbf{R}_{in}^{-1}\mathbf{a}(\theta_{s})} \tag{7}\]  
To ensure the desired distortion-free signal, the smaller the interference signal and the noise signal, the better, i.e., the covariance matrix of the interference signal and the noise signal needs to be obtained. However, under actual working conditions, the signal received by the beamformer is a coupling of the desired, interference, and noise signals, and it is difficult to separate the interference signal from the noise signal. For this reason, the covariance matrix of the interference signal and the noise signal cannot be obtained.