## 4 GDOP and WGDOP Calculation with Matrix Multiplication
### Proposed WGDOP Calculation with Matrix Multiplication
#### 4.2.1 Four Measurement Units  
We have the measurement matrix \(M\) given that:\[M=W^{\frac{1}{2}}HH^{\frac{1}{2}}W^{\frac{1}{2}}= \begin{bmatrix}\sqrt{k_{1}}&0&0&0 \\ 0&\sqrt{k_{2}}&0&0\\ 0&0&\sqrt{k_{3}}&0\\ 0&0&0&\sqrt{k_{4}}\end{bmatrix}\begin{bmatrix}e_{11}&e_{12}&e_{13}&1\\ e_{21}&e_{22}&e_{23}&1\\ e_{31}&e_{32}&e_{33}&e_{43}\\ e_{41}&e_{42}&e_{43}&1\end{bmatrix}  \tag{5}\]  
and following the relationship:  
\[e_{i1}^{2}+e_{i2}^{2}+e_{i3}^{2}=1,\ i=1,2,3,4 \tag{6}\]  
We have a simple form of trace and a guaranteed symmetric matrix \(M\):  
\[M=W^{\frac{1}{2}}HH^{\frac{1}{2}}W^{\frac{1}{2}}= \begin{bmatrix}2k_{1}&\sqrt{k _{1}k_{2}}B_{12}&\sqrt{k_{1}k_{3}}B_{13}&\sqrt{k_{1}k_{4}}B_{14}\\ \sqrt{k_{1}k_{2}}B_{12}&2k_{2}&\sqrt{k_{2}k_{3}}B_{23}&\sqrt{k_{2}k_{4}}B_{24 }\\ \sqrt{k_{1}k_{3}}B_{13}&\sqrt{k_{2}k_{3}}B_{23}&2k_{3}&\sqrt{k_{3}k_{4}}B_{34 }\\ \sqrt{k_{1}k_{4}}B_{14}&\sqrt{k_{2}k_{4}}B_{24}&\sqrt{k_{3}k_{4}}B_{34}&2k_{4 }\end{bmatrix}  \tag{7}\]  
where \(B_{ij}=e_{i1}e_{j1}+e_{i2}e_{j2}+e_{i3}e_{j3}+1,1\leq i<j\leq 4\). Utilizing the leaner algebra method, WGDOP can be written as:  
\[WGDOP=\sqrt{tr(H^{\frac{T}{2}}WH)^{-1}}=\sqrt{\lambda_{1}^{-1}+\lambda_{2}^{- 1}+\lambda_{3}^{-1}+\lambda_{4}^{-1}} \tag{8}\]  
Defining the following functions:  
\[p_{1}(\lambda)=\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}=trace(H^{\frac {T}{2}}WH)=trace(M) \tag{9a}\] \[p_{2}(\lambda)=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}+\lambda_{4}^{2 }=trace[(H^{\frac{T}{2}}WH)^{2}]=trace[M^{\frac{2}{2}}]\] (9b) \[p_{3}(\lambda)=\lambda_{1}^{3}+\lambda_{2}^{3}+\lambda_{3}^{3}+\lambda_{4}^{3 }=trace[(H^{\frac{T}{2}}WH)^{3}]=trace[M^{3}]\] (9c) \[p_{4}(\lambda)=\lambda_{1}\cdot\lambda_{2}\cdot\lambda_{3}\cdot\lambda_{4}= \det(H^{\frac{T}{2}}WH) \tag{9d}\]  
After verify that the measurement matrix is symmetric, we can use [PERSON]'s identity for reduction Equation (8). The WGDOP formula is thus obtained:  
\[WGDOP=\sqrt{\frac{1}{3\,p_{4}}(\frac{1}{2}\,p_{1}^{\frac{3}{2}}-\frac{3}{2}\,p _{1}p_{2}+p_{3})} \tag{10}\]  
Based on the operation rules above, we derive the WGDOP value only from calculating Equation (10). The formula values of \(p_{1}\), \(p_{2}\), \(p_{3}\) requires matrix multiplication and \(p_{4}\) requires only simple arithmetic. Because the measurement matrix \(M\) is a symmetric matrix, the components of \(M\) present a symmetric form. We can determine that the matrix of \(M^{2}\), \(M^{3}\) are symmetric matrices also, so only the upper half components of matrix \(M\), \(M^{2}\), \(M^{3}\) needs to be calculated. Then the WGDOP can be solved by the proposed matrix multiplication scheme for four measurement units. According to the above describedmatrix multiplications, we can easily find the value of the WGDOP and avoid the calculation of the matrix inverse. We summarize WGDOP formula process and present it in Figure 1.  
The steps for positioning in GPS, WSN, and cellular communication systems are listed as follows:  
1. After a mobile device receives the signals from the positioning device, we arbitrarily select \(m\) measurements among \(n\) measurements to generate different subsets, thus the \(n\) measurements are classified into \(C\) (_n_, _m_) possible subsets. 2. The optimal subset is often obtained by minimizing the WGDOP value. 3. The subset with the smallest WGDOP value is selected as the optimal measurement constellation subset: 1. First, we calculate the measurement matrix \(M\). 2. Then the matrix of \(M\), _M_\({}^{2}\), and _M_\({}^{3}\) are obtained by matrix multiplication method which is acceptable to most hardware. The determinants and trace values are calculated. 3. Finally, these parameter values can be used to calculate WGDOP value. The values of variance 1/_k_, \(i\) = 1, 2, 3, 4, can be treated as constants, which are previous assumed for the calculation of Equation (2). 4. Finally, select the optimal subset of \(m\) measurements to estimate the location solution.