## 3 RSS-Based DoA Estimation for Two-Row ESPAR Antenna
### PPCC-MCP Algorithm  
PPCC algorithm which allows DoA estimation based on ESPAR antenna radiation patterns measured in an anechoic chamber relies on a cross-correlation coefficient between the measured radiation patterns and RSS values recorded for each directional antenna radiation pattern [18]. The cross-correlation coefficient for the ESPAR antenna having 12 directional beams has the following form:  
\[\Gamma(\varphi)=\frac{\sum_{n=1}^{12}(P(V_{max}^{n},\varphi)\;Y(V_{max}^{n})) }{\sqrt{\sum_{n=1}^{12}P(V_{max}^{n},\varphi)^{2}}\sqrt{\sum_{n=1}^{12}Y(V_{max }^{n})^{2}}} \tag{1}\]  
where \(V_{max}^{n}\) is ESPAR antenna's steering vector that allows to create \(n\)-th directional radiation pattern, \(P(V_{max}^{n},\varphi)\) are directional ESPAR antenna radiation patterns measured during the calibration phase in an anechoic chamber in the azimuth plane for \(0^{\circ}\leq\varphi<360^{\circ}\), \(Y(V_{max}^{n})\) are RSS values measured at the antenna output port for all 12 directional radiation patterns during the DoA estimation phase for an unknown RF signal, and \(\Gamma(\varphi)\) is the cross-correlation coefficient, which have its highest value associated with the estimated direction of arrival angle \(\phi\)[18]. In practical implementations, the angular step precision \(\Delta\varphi=1^{\circ}\) is commonly used during the calibration phase, and therefore, the cross-correlation coefficient can be written in a convenient vector form as [19]:  
\[g=\frac{\sum_{n=1}^{12}(p^{n}Y(V_{max}^{n}))}{\sqrt{\sum_{n=1}^{12}(p^{n}\circ p ^{n})}\sqrt{\sum_{n=1}^{12}Y(V_{max}^{n})^{2}}} \tag{2}\]  
where vector \(p^{n}=\left[p_{1}^{n},\;p_{2}^{n},\cdots,\;p_{1}^{n}\right]^{T}\), where the superscript \({}^{T}\) is the vector transpose operator, contains \(I=360\) measured discreet values of \(P(V_{max}^{n},\varphi)\) and '\(\circ\)' denotes the element-wise product of vectors. In result, the cross-correlation coefficient \(g=\left[\Gamma(\varphi_{1}),\;\Gamma(\varphi_{2}),\;\ldots,\Gamma(\varphi_{I} )\right]^{T}\) is also a vector with \(I=360\) entries being discretized values of the correlation coefficient \(\Gamma(\varphi)\) for every considered value of \(\varphi\) in \(\varphi=\left[\varphi_{1},\;\varphi_{2},\;\ldots,\;\varphi_{I}\right]^{T}= \left[0^{\circ},\;1^{\circ},\;\ldots,\;359^{\circ}\right]^{T}\) and \(\phi\) corresponds now to the maximum value of \(g\). One should note, however, that as only a single calibration plane at \(\theta=90^{\circ}\) is used to measure directional ESPAR antenna radiation patterns in an anechoic chamber during the calibration phase, cross-correlation operation Equation (2) will produce the most accurate results when RF signals impinging on the antenna arrive from \(\theta=90^{\circ}\) vertical angle, which is aligned with the calibration plane. The results will significantly deteriorate for lower \(\theta\) angles as the shape of ESPAR antenna radiation patterns gradually changes for \(\theta>90^{\circ}\)[23; 24]. To further extend PPCC algorithm and improve its accuracy in situations when \(\theta\) angles of RF signals impinging on the antenna are different than \(90^{\circ}\), PPCC-MCP algorithm, which allows to incorporate higher number of calibration planes, has been introduced [24]. It is based on the following cross-correlation coefficient:  
\[g\theta=\frac{\sum_{n=1}^{12}\left(p_{\theta}^{n}\;Y(V_{max}^{n})\right)}{ \sqrt{\sum_{n=1}^{12}\left(p_{\theta}^{n}\circ p_{\theta}^{n}\right)}\sqrt{ \sum_{n=1}^{12}Y(V_{max}^{n})^{2}}} \tag{3}\]  
where, for the considered number of \(M\) calibration planes Equation (3), vector \(p_{\theta}^{n}=\left[\;\left(p_{\theta_{1}}^{n}\right)^{T},\;\left(p_{\theta_{ 2}}^{n}\right)^{T},\;\cdots,\;\left(p_{\theta_{M}}^{n}\right)^{T}\;\right]^{T}\) of length \(I\cdot M\) contains ESPAR antenna's radiation pat tern values measured at \(\{\theta_{1},\ \theta_{2},\ \ldots,\theta_{M}\}\) angles in elevation direction and vector \(g_{\theta}\) of length _I-M_ contains PPCC-MCP cross-correlation coefficient in a vector form.