# Simultaneous Correlative Interferometer Technique for Direction Finding of Signal Sources
## 4 Proposed Simultaneous Correlative Interferometer  
In this section, we propose a novel simultaneous CI technique to jointly utilize spatial features in multiple time-domain samples. First of all, for the proposed simultaneous CI technique, the receiver collects \(T\) samples. Here, it is assumed that the direction of the incident signal sources does not change during the \(T\) sample period. Then, the received signal matrix \(\mathbf{Y}(\in\mathbb{C}^{M\times T})\) for \(T\) time samples is expressed as  
\[\mathbf{Y}=\mathbf{AX}+\mathbf{W}, \tag{7}\]  
where \(\mathbf{Y}\) consists of \(T\) columns in which the \(t(\in\{1,\cdots,T\})\)-th column denotes the received signal vector \(\mathbf{y}_{t}(\in\mathbb{C}^{M})\) at \(t\)-th time, \(\mathbf{X}(\in\mathbb{R}^{K\times T})\) denotes the concatenate transmitted signal matrix over \(T\) time samples, and \(\mathbf{W}(\in\mathbb{C}^{M\times T})\) denotes the matrix of noise samples for \(T\) time samples where each column indicates the additive noise vector for each time sample. It is worth noting that the steering matrices \(\mathbf{A}\) in both (1) and (7) are the same as each other since we assume that the DOA angles of multiple signal sources are not changed over multiple time samples in this paper.  
Simultaneous CI technique is intended to efficiently utilize the difference between the phase of the received signal and ideal candidate phases over all time samples to average out the effects of noise. Hence, letting the \(m\)-th element of \(\mathbf{y}_{t}\) be \(y_{m,t}\) and the phase of \(y_{m,t}\) be \(r_{m,t}\)  
Figure 2: An example of the conventional CI-based DOA estimation due to additive noise (ideal DOA is 180\({}^{\circ}\); estimated DOA is 196\({}^{\circ}\)). (**a**) Effects of additive noise on phase angle of complex signal. (**b**) Effects of additional noise on cost function value estimation.  
i.e., \(r_{m,t}:=\angle y_{m,t}\), the temporal cost function \(J_{t}(\hat{\gamma})\) for an arbitrary directional information \(\hat{\gamma}\in\mathcal{A}\) at \(t\)-th sample can be defined as  
\[J_{t}(\hat{\gamma})=\sum_{m=1}^{M}\cos[c_{m}(\hat{\gamma})-r_{m,t}]. \tag{8}\]  
Finally, by adding the temporal cost functions for all samples, the cost function of simultaneous CI can be calculated as  
\[J_{\text{SCI}}(\hat{\gamma})=\sum_{t=1}^{T}J_{t}(\hat{\gamma})=\sum_{t=1}^{T} \sum_{m=1}^{M}\cos[c_{m}(\hat{\gamma})-r_{m,t}]. \tag{9}\]  
After that, DOA estimation is completed by finding \(K\) peaks in the cost function, same as the conventional CI method. The overall process of the simultaneous CI technique is summarized as Algorithm 2. The proposed simultaneous CI algorithm repeatedly performs phase comparison operation for \(T\) time samples. For example, when \(T=10\), the proposed algorithm performs 10 phase comparisons. Then, the DOA angle of signal sources is estimated through the cost function by using all computed phases collectively, as shown in line (10) of Algorithm 2 in this paper. It is worth noting that the conventional CI algorithm just averages out multiple phases of the received signal during \(T\) time samples and exploits it only once for estimating DOA of signal sources.  
```
0: Directional information dictionary \(A\), Received signal matrix \(\mathbf{Y}\), The number of signal sources \(K\)
0: Estimated DOA \(\Lambda\)
1:Initialization : \(\Lambda=\varnothing\), \(Q=\varnothing\)
2:for\(i=1,\cdots,A\)do
3:for\(t=1,\cdots,T\)do
4:for\(m=1,\cdots,M\)do
5:\(r_{m,t}=\angle y_{m,t}\)
6:\(c_{m}(\hat{\gamma}_{i})=-\frac{2\pi}{\lambda}\mathbf{p}_{m}\cdot\mathbf{s}( \hat{\gamma}_{i})\)
7:endfor
8:\(J_{t}(\hat{\gamma}_{i})=\sum\limits_{m=1}^{M}\cos[c_{m}(\hat{\gamma}_{i})-r_{ m,t}]\).
9:endfor
10:\(J_{\text{SCI}}(\hat{\gamma}_{i})=\sum\limits_{t=1}^{T}J_{t}(\hat{\gamma}_{i})\)
11:endfor
12:\(Q\leftarrow\) Indices corresponding to \(K\) peaks in \(J_{\text{SCI}}(\hat{\gamma}_{i}),\ \forall i\).
13:\(\Lambda\leftarrow\mathcal{A}(Q)\).
14:Return \(\Lambda\)
```  
**Algorithm 2** Simultaneous CI for DOA estimation.