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EURASIP Journal on Wireless Communications and Networking

Table 1 The soft information processing of each node in Fig. 3

From: RSSD-based 3-D localization of an unknown radio transmitter using weighted least square and factor graph

SI(a,b)

Inputs

Outputs

SI(Pi,pi)

(\(\widehat {p}_{i}\), 0)

\(\left ({m_{{P_{i}},{p_{i}}}},\sigma _{{P_{i}},{p_{i}}}^{2}\right)\)

SI(pi,Dt)

\(\left ({m_{{P_{i}},{p_{i}}}},\sigma _{{P_{i}},{p_{i}}}^{2}\right)\)

\(\left ({m_{{p_{i}},{D_{t}}}},\sigma _{{p_{i}},{D_{t}}}^{2}\right)\)

SI(Dt,Rt)

\(\left ({m_{{p_{i}},{D_{t}}}},\sigma _{{p_{i}},{D_{t}}}^{2}\right)\), \(\left ({m_{{p_{j}},{D_{t}}}},\sigma _{{p_{j}},{D_{t}}}^{2}\right)\)

\({m_{{D_{t}},{R_{t}}}} = {m_{{p_{i}},{D_{t}}}} - {m_{{p_{j}},{D_{t}}}}\), \(\sigma _{{D_{t}},{R_{t}}}^{2} = \sigma _{{p_{i}},{D_{t}}}^{2} + \sigma _{{p_{j}},{D_{t}}}^{2}\)

SI(Rt,At)

\(\left ({m_{{D_{t}},{R_{t}}}}, \sigma _{{D_{t}},{R_{t}}}^{2}\right)\)

\({m_{{R_{t}},{A_{t}}}} = {m_{{D_{t}},{R_{t}}}}\), \(\sigma _{{R_{t}},{A_{t}}}^{2} = \sigma _{{D_{t}},{R_{t}}}^{2}\)

SI(At,x)

\(\left ({m_{{R_{t}},{A_{t}}}},\sigma _{{R_{t}},{A_{t}}}^{2}\right)\), \(({m_{{y},{A_{t}}}},\sigma _{{y},{A_{t}}}^{2})\phantom {\dot {i}\!}\)

\({m_{{A_{t}},x}} = \left (1- k_{y,t}^{'}{m_{y,{A_{t}}}} - k_{z,t}^{'}{m_{z,{A_{t}}}}- k_{r,t}^{'}{m_{{R_{t}},{A_{t}}}}\right)/k_{x,t}^{'}\),

  

\(\sigma _{_{{A_{t}},x}}^{2} = \left (k_{y,t}^{'2}\sigma _{y,{A_{t}}}^{2} + k_{z,t}^{'2}\sigma _{z,{A_{t}}}^{2} + k_{r,t}^{'2}\sigma _{{R_{t}},{A_{t}}}^{2}\right)/k_{x,t}^{'2}\)

SI(At,y)

\(\left ({m_{{R_{t}},{A_{t}}}},\sigma _{{R_{t}},{A_{t}}}^{2}\right)\), \(\left ({m_{{x},{A_{t}}}},\sigma _{{x},{A_{t}}}^{2}\right)\phantom {\dot {i}\!}\)

\({m_{{A_{t}},y}} = \left (1- k_{x,t}^{'}{m_{x,{A_{t}}}} - k_{z,t}^{'}{m_{z,{A_{t}}}} - k_{r,t}^{'}{m_{{R_{t}},{A_{t}}}}\right)/k_{y,t}^{'}\),

  

\(\sigma _{_{{A_{t}},y}}^{2} = \left (k_{x,t}^{'2}\sigma _{x,{A_{t}}}^{2} + k_{z,t}^{'2}\sigma _{z,{A_{t}}}^{2} + k_{r,t}^{'2}\sigma _{{R_{t}},{A_{t}}}^{2}\right)/k_{y,t}^{'2}\)

SI(At,z)

\(\left ({m_{{R_{t}},{A_{t}}}},\sigma _{{R_{t}},{A_{t}}}^{2}\right)\), \(\left ({m_{{x_{t}},{A_{t}}}},\sigma _{{x_{t}},{A_{t}}}^{2}\right)\)

\({m_{{A_{t}},z}} = \left (1- k_{x,t}^{'}{m_{x,{A_{t}}}} - k_{y,t}^{'}{m_{y,{A_{t}}}}- k_{r,t}^{'}{m_{{R_{t}},{A_{t}}}}\right)/k_{z,t}^{'}\),

  

\(\sigma _{_{{A_{t}},z}}^{2} = \left (k_{x,t}^{'2}\sigma _{x,{A_{t}}}^{2} + k_{y,t}^{'2}\sigma _{y,{A_{t}}}^{2} + k_{r,t}^{'2}\sigma _{{R_{t}},{A_{t}}}^{2}\right)/k_{z,t}^{'2}\)

SI(x,At)

\(\left ({m_{{A_{l}},x}},\sigma _{{A_{l}},x}^{2}\right)\)

\({m_{x,{A_{t}}}} = \sigma _{_{x,{A_{t}}}}^{2}\left (\sum \limits _{l \ne t}^{n} {\frac {{{m_{{A_{l}},x}}}}{{\sigma _{{A_{l}},x}^{2}}}}\right)\), \(\sigma _{x,{A_{t}}}^{2} = 1/\left (\sum \limits _{l \ne t}^{n} {\frac {1}{{\sigma _{{A_{l}},x}^{2}}}} \right)\)

SI(y,At)

\(\left ({m_{{A_{l}},y}},\sigma _{{A_{l}},y}^{2}\right)\)

\({m_{y,{A_{t}}}} = \sigma _{_{y,{A_{t}}}}^{2}\left (\sum \limits _{l \ne t}^{n} {\frac {{{m_{{A_{l}},y}}}}{{\sigma _{{A_{l}},y}^{2}}}}\right)\), \(\sigma _{y,{A_{t}}}^{2} = 1/\left (\sum \limits _{l \ne t}^{n} {\frac {1}{{\sigma _{{A_{l}},y}^{2}}}}\right)\)

SI(z,At)

\(\left ({m_{{A_{l}},z}},\sigma _{{A_{l}},z}^{2}\right)\)

\({m_{z,{A_{t}}}} = \sigma _{_{z,{A_{t}}}}^{2}\left (\sum \limits _{l \ne t}^{n} {\frac {{{m_{{A_{l}},z}}}}{{\sigma _{{A_{l}},z}^{2}}}}\right)\), \(\sigma _{z,{A_{t}}}^{2} = 1/\left (\sum \limits _{l \ne t}^{n} {\frac {1}{{\sigma _{{A_{l}},z}^{2}}}}\right)\)

SI(x)

\(\left ({m_{{A_{t}},x}},\sigma _{{A_{t}},x}^{2}\right)\)

\({m_{x}} = \sigma _{x}^{2} \cdot \left (\sum \limits _{t = 1}^{n} {\frac {{{m_{{A_{t}},x}}}}{{\sigma _{{A_{t}},x}^{2}}}} \right)\), \(\sigma _{x}^{2} = 1/\left (\sum \limits _{t = 1}^{n} {\frac {1}{{\sigma _{{A_{t}},x}^{2}}}} \right)\)

SI(y)

\(\left ({m_{{A_{t}},y}},\sigma _{{A_{t}},y}^{2}\right)\)

\({m_{y}} = \sigma _{y}^{2} \cdot \left (\sum \limits _{t = 1}^{n} {\frac {{{m_{{A_{t}},y}}}}{{\sigma _{{A_{t}},y}^{2}}}} \right)\), \(\sigma _{y}^{2} = 1/\left (\sum \limits _{t = 1}^{n} {\frac {1}{{\sigma _{{A_{t}},y}^{2}}}} \right)\)

SI(z)

\(\left ({m_{{A_{t}},z}},\sigma _{{A_{t}},z}^{2}\right)\)

\({m_{z}} = \sigma _{z}^{2} \cdot \left (\sum \limits _{t = 1}^{n} {\frac {{{m_{{A_{t}},z}}}}{{\sigma _{{A_{t}},z}^{2}}}} \right)\), \(\sigma _{z}^{2} = 1/\left (\sum \limits _{t = 1}^{n} {\frac {1}{{\sigma _{{A_{t}},z}^{2}}}} \right)\)

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