# Simultaneous Correlative Interferometer Technique for Direction Finding of Signal Sources
## 3 Correlative Interferometer Method  
In this section, we first investigate the conventional CI algorithm to address the direction-finding mechanism through the phase difference. Then, we specify some problems with the existing CI algorithm that can arise when using only a single sample. Figure 1: Structure of the simultaneous CI system for 2D direction estimation. The CI algorithm can simply be performed by measuring the phase of the received signal incident on each antenna element and comparing it with the candidate phases. Specifically, if the signal received by the \(m\)-th antenna element is \(y_{m}\), the phase of \(y_{m}\) is denoted as \(r_{m}\) in this paper, i.e., \(r_{m}=\angle y_{m}\). Further, assuming that the array type and position vector are known, from (2) and (3), the candidate phase \(c_{m}(\hat{\gamma})\) for an arbitrary directional information \(\hat{\gamma}=\{\hat{\phi},\hat{\theta}\}\) can be defined as  
\[c_{m}(\hat{\gamma})=-\frac{2\pi}{\lambda}\mathbf{p}_{m}\cdot\mathbf{s}(\hat{ \gamma}), \tag{4}\]  
where the candidate directional information \(\hat{\gamma}\), in this paper, is defined as an element set in a dictionary tensor \(\mathcal{A}\). That is, \(\mathcal{A}\) has candidate sets representing pairs of the candidate azimuth and elevation angles as its elements, and, for a total range of \(A\), it can be expressed as  
\[\mathcal{A}\triangleq\{\{\hat{\phi}_{1},\hat{\theta}_{1}\},\ \{\hat{\phi}_{2},\hat{ \theta}_{2}\},\cdots,\{\hat{\phi}_{A},\hat{\theta}_{A}\}\}. \tag{5}\]  
For example, considering 1-degree interval resolution for both candidate azimuth angle \(\hat{\phi}\in(0^{\circ},360^{\circ}]\) and elevation angle \(\hat{\theta}\in[0^{\circ},90^{\circ}]\), there are 32,669 candidate directional information sets in a dictionary \(\mathcal{A}\). Then, by checking the difference between the measured received signal and all candidate phases, the direction with the smallest difference should be selected. Here, the cost function, which is the criterion for determining the smallest phase difference, can exist in various forms [22]. In this paper, we utilize the cosine function as a cost function since it has the best performance among cost functions in [22], and it is designed as  
\[J_{\text{CI}}(\hat{\gamma})=\sum_{m=1}^{M}\cos[c_{m}(\hat{\gamma})-r_{m}]. \tag{6}\]  
Interestingly, in (6), the direction with the smallest phase difference has the maximum value, and the direction with a difference of more than 90 degrees has a negative value due to the property of the cosine function. These properties help the direction-finding performance to improve even if some outliers caused by additive noise in the measured phase exist in arbitrary antennas. Now, we explain the detailed description of the CI algorithm. The conventional CI algorithm simply consists of three-steps as follows. First, the phase of the received signal \(r_{m}\) is measured, and candidate phases (4) are generated for (5). Then, the receiver calculates and stores the cost function as described in (6) for all candidate directional information. Finally, since \(J(\hat{\gamma})\) fluctuates for all \(\hat{\gamma}\), the receiver selects indices corresponding to \(K\) maximum peaks, which is the number of signal sources. After that, the direction finding of the received signal sources can be completed by replacing the selected indices with the corresponding candidate directional information in the dictionary \(\mathcal{A}\). This process can be summarized as in Algorithm 1. Figure 2 shows a simple example explaining the direction-finding mechanism of the CI algorithm when a signal of a single source is received by a certain antenna. Specifically, Figure 2 visualizes the phase of the received signal on the complex plane, which is denoted as \(r\). Here, assuming the received signal is incident at an azimuth angle of 180 degrees, the ideal phase for the corresponding direction is denoted as \(c\).