# Robust Bluetooth AoA Estimation for Indoor Localization Using Particle Filter Fusion
## 3 Methods  
### MVDR  
As one of the core tasks of array signal processing technology, beam forming can realize functions such as beam steering, interference and noise suppression, target identification and positioning, etc. The MVDR beamforming proposed by [PERSON] is a commonly used algorithm [25], which weights and sums the output of the array and focuses the gain in one direction. The appropriate filter coefficients are chosen to minimize the average power at the array's output under the constraint of the desired signal without distortion. Figure 2 shows the basic structure of the algorithm.  
Assuming that the desired signal is undistorted, appropriate filter coefficients must be selected to minimize the average power of the array output. The MVDR weight optimization problem can be expressed as:  
\[\underset{w}{\text{min}}w^{H}R_{i+n}w\text{ s.t.}w^{H}a_{0}=1 \tag{12}\]  
The above MVDR optimization problem can be solved using the Lagrange multiplier method as follows:  
\[w_{opt}=\frac{R_{i+n}^{-1}a_{0}}{a_{0}^{H}R_{i+n}^{-1}a_{0}} \tag{13}\]  
where \(w=(w_{1},\dots w_{M})^{T}\) is the complex weighting vector of the beamformer and \(R_{i+n}^{-1}\) and \(a_{0}\) represent the ideal interference plus noise covariance matrix and the precise desired signal steering vector. In practice, they are usually unavailable, so the sampled covariance matrix \(\bar{R}^{-1}\) and the nominal guiding vector \(\overline{a}_{0}\) based on the array geometry computed are  
Figure 2: Beamforming basic structure.  
commonly used instead [26], and in this case, the MVDR beamformer is also referred to as the sampled covariance matrix inversion (SMI) beamformer [11]:  
\[w_{SMI}=a\widetilde{R}^{-1}\overline{a}_{0}=\frac{\widetilde{R}^{-1}\overline{a }_{0}}{\overline{a}_{0}^{H}\widetilde{R}^{-1}\overline{a}_{0}} \tag{14}\]  
\(a=\frac{1}{2\overline{a}_{0}^{H}}\) The output sequence of the beamformer is:  
\[y(k)=w_{SMI}^{H}x(k) \tag{15}\]  
The output SINR of the beamformer is:  
\[\begin{array}{l}SINR\triangleq\frac{E\left[w^{H}x_{i}(k)x_{i}^{H}(k)w\right] }{E\left[w^{H}x_{i_{x}(k)x_{i+n}^{H}(k)w}\right]}\\ =\frac{w^{H}R_{SW}}{w^{H}R_{i+w}w}=\frac{\sigma_{0}^{2}\left|w^{H}a_{0}\right| }{w^{H}R_{i+w}w},\end{array} \tag{16}\]  
\(\sigma_{0}^{2}=E\left[s_{i}(k)s_{i}(k)^{H}\right]\) is the expected signal power. The array pattern of the Capon beamformer represents the response relationship between the complex weighted vector and the incident signal in a given direction.  
Also known as the beam response, it can be expressed as:  
\[B(\theta)=w^{H}a(\theta) \tag{17}\]  
The overall execution process steps of the MVDR algorithm, designed with a systematic approach, are as follows:  
1. [label=(0)]
2. Conduct \(L\) snapshot observations of the signal source at time \(t\) and use the formula \(R_{k}=\frac{1}{L}\frac{i}{L}\ G_{ki}G_{ki}^{H}\) to construct a covariance matrix from the \(2M-1\) signal data received by the array.
3. Calculate the inverse matrix or pseudo-inverse matrix of the covariance matrix to represent the relationship between signals.
4. Calculate the weight vector, which is the inverse matrix (or pseudo-inverse matrix) of the covariance matrix and the received signal.
5. Sort according to the size of the eigenroots, take the eigenvectors corresponding to the first \(K\) larger eigenvalues to form the signal subspace, and the remaining eigenvectors are the noise subspace.
6. Change \((\theta,\phi)\), and calculate the spectral function according to the formula to find the position of the maximum value, and then determine the source's azimuth angle and elevation angle.