# Robust Bluetooth AoA Estimation for Indoor Localization Using Particle Filter Fusion
## 2 Array Model  
The realm of Bluetooth signal transmission through wireless channels is a labyrinth of complexity. A comprehensive understanding of the physical environment is a prerequisite for establishing a mathematical model, a task that often proves challenging. To derive a practical parameterized model, we must resort to simplifying assumptions for signal transmission, primarily encompassing array assumptions and spatial source signal assumptions. (1) Assumption of signal array:  
The array elements are independent of each other and have the same characteristics. The signal received by each array element has nothing to do with its size. It is only related to its position, and each array element contains white noise independent of the signal. (2) Assumption of spatial signal source:  
The signal propagates in a straight line in a uniform medium with the same direction, and there is no multipath effect. The signal waveform can be regarded as a beam of plane waves compared with the array. Azimuth and elevation angles are commonly used to indicate the direction of the beam coming in three dimensions. Currently, commonly used signal arrays for 3D spatial direction angle estimation generally use L-type arrays, cross-cross arrays, face arrays, etc. In this paper, we mainly estimate the spatial direction angle of the signal source based on the L-type array. Assume that \(K\) uncorrelated narrowband signals with the same carrier frequency act on the x- and y-axes of a homogeneous surface array of \(M\) array elements, and the line array is vertically formed. The isotropic array elements are aligned along the x- and y-axes in the direction of the arriving wave to form an equally spaced linear array. Among them, the spacing between the Bluetooth signal array elements is \(d\), as shown in Figure 1.  
Figure 1: Signal model of DOA estimation. \((\theta_{i},\phi_{i})\), \((i=1,\ldots,K)\) is the angle of incidence, where \(\theta_{i}\) and \(\phi_{i}\) represent the azimuth and elevation angles of the \(i\)-th signal, respectively. For narrowband signals, the time delay can be converted to a phase shift, and the time delay of an array element at position \(x_{m}(m=1,2,\ldots,M)\) concerning the first array element is  
\[\tau_{mk}=\frac{1}{c}(x_{m}\text{sin}\,\phi_{k}) \tag{1}\]  
and the spacing between every two neighboring array elements is \(d\), for which there is Equation (2):  
\[\tau_{mk}=\frac{d\text{sin}\,\phi_{k}}{c}(m-1) \tag{2}\]  
In the array, the first array element is located at the origin. Taking the position of the first array element as the reference point, at time \(t\), the spatial signal representation of the \(i\)-th signal arriving at the sensor of the reference array element is as follows:  
\[s_{i}(t)=z_{i}(t)e^{i\omega_{0}t},i=1,\ldots,K \tag{3}\]  
where \(z_{i}(t)\) denotes the \(i\)-th signal complex envelope containing the angle information of the spatial signal at the time \(t\). \(e^{i\omega_{0}t}\) represents the source carrier. When the signal is a narrowband acoustic source signal, it satisfies \(z_{i}(t-\tau)\approx z_{i}(t)\). Then \(s_{i}(t-\tau)\) can be expressed as:  
\[s_{i}(t-\tau)=z_{i}(t-\tau)e^{i\omega_{0}(t-\tau)}\approx z_{i}(t)e^{i\omega_{ 0}(t-\tau)},i=1,\ldots,K \tag{4}\]  
The vector that models the phase shift of each antenna based on AoA is generally referred to as the array streamer vector or the steering vector. \(x_{m}(t)\) is denoted as the data received by the \(m\)-th array element in the array, as shown below:  
\[x_{m}(t)=\sum_{i=1}^{K}s_{i}(t-\tau_{mi})+n_{m}(t) \tag{5}\]  
where \(\tau_{mi}\) represents the time delay from the \(i\)-th signal to the \(m\)-th array element with respect to the reference array element, and \(n_{m}(t)\) represents the additive noise of the \(m\)-th array element. From the above, the received signals of the array on the x-axis and y-axis are, respectively:  
\[ \begin{array}{c}X=A_{x}S+N_{x}\\ Y=A_{y}S+N_{y}\end{array}  \tag{6}\]  
where \(S\in\mathbb{C}^{1\times N}\) represents the array signal data.