A new DOA-based factor graph geolocation technique for detection of unknown radio wave emitter position using the first-order Taylor series approximation

Kahar Aziz, Muhammad Reza; Anwar, Khoirul; Matsumoto, Tad

doi:10.1186/s13638-016-0683-4

EURASIP Journal on Wireless Communications and Networking

Table 1 The operations required for each node in the DOA-based TS FG

From: A new DOA-based factor graph geolocation technique for detection of unknown radio wave emitter position using the first-order Taylor series approximation

Nodes	Means, variances
	Inputs	Outputs
\(D_{\theta _{i}} \rightarrow N_{\theta _{i}}\)	\(\hat {\theta }_{i}\) samples	\(m_{\hat {\theta }_{i}},\sigma ^{2}_{\hat {\theta }_{i}}\)
\(N_{\theta _{i}} \rightarrow C_{\theta _{i}}\)	\(m_{\hat {\theta }_{i}},\sigma ^{2}_{\hat {\theta }_{i}}\)	\(m_{\hat {\theta }_{i}},\sigma ^{2}_{\hat {\theta }_{i}}\)
\(C_{\theta _{i}}\rightarrow \Delta y_{i}\)	\(m_{i},{\sigma _{i}^{2}}\)	\(m_{i} \cdot \tan \left (m_{\hat {\theta }_{i}}\right), \)
	\(m_{\hat {\theta }_{i}},\sigma _{\hat {\theta }_{i}}^{2}\)	\({\sigma _{i}^{2}} \cdot \tan ^{2}\left (m_{\hat {\theta }_{i}}\right) +{m_{i}^{2}} \cdot \sigma _{\hat {\theta _{i}}}^{2} \cdot \sec ^{4}\left (m_{\hat {\theta }_{i}}\right)\)
		\(+ {\sigma _{i}^{2}} \cdot \sigma _{\hat {\theta }_{i}}^{2} \cdot \sec ^{4}\left (m_{\hat {\theta }_{i}}\right)\)
\(C_{\theta _{i}}\rightarrow \Delta x_{i}\)	\(m_{i}, {\sigma _{i}^{2}}\)	\(m_{i} \cdot \cot \left (m_{\hat {\theta }_{i}}\right),\)
	\(m_{\hat {\theta }_{i}}, \sigma _{\hat {\theta }_{i}}^{2}\)	\({\sigma _{i}^{2}} \cdot \cot ^{2}\left (m_{\hat {\theta }_{i}}\right) +{m_{i}^{2}} \cdot \sigma _{\hat {\theta }_{i}}^{2} \cdot \csc ^{4}\left (m_{\hat {\theta }_{i}}\right)\)
		\(+ {\sigma _{i}^{2}} \cdot \sigma _{\hat {\theta }_{i}}^{2} \cdot \csc ^{4}\left (m_{\hat {\theta }_{i}}\right)\)
\(C_{\theta _{i}} \leftarrow \Delta x_{i} \rightarrow A_{\theta _{i}}\)	\(m_{i},{\sigma _{i}^{2}}\)	\(m_{i},{\sigma _{i}^{2}}\)
\(C_{\theta _{i}} \leftarrow \Delta y_{i} \rightarrow B_{\theta _{i}}\)
\(\Delta x_{i} \leftarrow A_{\theta _{i}} \rightarrow x\)	\(m_{i},{\sigma ^{2}_{i}}\)	\(X_{i}-m_{i},{\sigma _{i}^{2}}\)
\(\Delta y_{i} \leftarrow B_{\theta _{i}} \rightarrow y\)	\(m_{i},{\sigma _{i}^{2}}\)	\(Y_{i}-m_{i},{\sigma _{i}^{2}}\)
\(x \rightarrow A_{\theta _{i}}\)	\(m_{j},{\sigma _{j}^{2}}\)	\({\sigma _{i}^{2}} \sum _{j\neq i} \frac {m_{j}}{{\sigma _{j}^{2}}}, {\sigma _{i}^{2}}=\frac {1}{\sum _{j\neq i} \frac {1}{{\sigma ^{2}_{j}}}}\)
\(y \rightarrow B_{\theta _{i}}\)	j≠i
x and y	\(m_{i},{\sigma _{i}^{2}}\)	\(\sigma ^{2} \sum _{i} \frac {m_{i}}{{\sigma _{i}^{2}}}, \sigma ^{2}=\frac {1}{\sum _{i} \frac {1}{{\sigma ^{2}_{i}}}}\)

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