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Table 7 Complexity of algorithm II

From: On the use of calibration emitters for TDOA source localization in the presence of synchronization clock bias and sensor location errors

Stage Computational unit Computational complexity of each unit Total computational complexity
Stage 1 The same as algorithm I The same as algorithm I \( {\displaystyle \begin{array}{c}{K}_1\left(\begin{array}{c}3D\left(M-1\right)\left(3{M}^2+M\right)\\ {}+3{D}^2M{\left(M-1\right)}^2+18{M}^2+O\left({(3M)}^3\right)\end{array}\right)\\ {}+{K}_2\left(\begin{array}{c}3M\left(M-1\right)\left(3M+N-1\right)\\ {}+3\left(M+3\right)\left(M+N-2\right)\\ {}+\left(N+3\right){\left(M+N-2\right)}^2\\ {}+3M{\left(M-1\right)}^2+O\left({\left(N-1\right)}^3\right)\\ {}+2\cdot O\left({\left(M+N-2\right)}^3\right)+O(27)\end{array}\right)\\ {}+2{D}^2\left(N-1\right){\left(M-1\right)}^2+3M{\left(M-1\right)}^2\\ {}+\left(N-1\right)\left(N+M-2\right)+2{\left(N-1\right)}^2\\ {}+\left(N-1\right)\left(M-1\right)\left(2N+M-3\right)\\ {}+3 DM\left(M-1\right)\left(3M+N-1\right)\\ {}+3M\left(M-1\right)\left(3M+N-1\right)\\ {}+D\left(2N-1\right)\left(N-1\right)\left(M-1\right)\\ {}+O\left({(3M)}^3\right)+2\cdot O\left({\left(N-1\right)}^3\right)\\ {}+2\cdot O\left({\left(D\left(M-1\right)\right)}^3\right)\end{array}} \)(where K1 and K2 denote the iteration numbers for the dimension-reduction Taylor-series iterative algorithms in the first and second stages, respectively)
Stage 2 Φ 3DM(M − 1)(3M + N − 1)
\( {\left\{{\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1}\right\}}_{1\le k\le {K}_2} \) and \( {\left\{{\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1/2}\right\}}_{1\le k\le {K}_2} \) \( {K}_2\left(\begin{array}{c}3M\left(M-1\right)\left(3M+N-1\right)\\ {}+3M{\left(M-1\right)}^2+2\cdot O\left({\left(M+N-2\right)}^3\right)\end{array}\right) \)
\( {\left\{{\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1/2}\left[\begin{array}{c}{\mathbf{F}}_1\left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\\ {}{\mathbf{O}}_{\left(N-1\right)\times 3}\end{array}\right]\right\}}_{1\le k\le {K}_2} \) 3K2(N + M − 2)(M − 1)
\( {\left\{{\boldsymbol{\Pi}}^{\perp}\left({\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1/2}\left[\begin{array}{c}\boldsymbol{\Gamma} \\ {}{\mathbf{I}}_{N-1}\end{array}\right]\right)\right\}}_{1\le k\le {K}_2} \) \( {K}_2\left(\begin{array}{c}\left(N-1\right){\left(N+M-2\right)}^2\\ {}+O\left({\left(N-1\right)}^3\right)\end{array}\right) \)
\( {\left\{{\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1/2}\left[\begin{array}{c}\hat{\mathbf{r}}-\mathbf{f}\left({\hat{\mathbf{u}}}_{\mathrm{s}2}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\\ {}{\hat{\boldsymbol{\uprho}}}_{\mathrm{f}}\end{array}\right]\right\}}_{1\le k\le {K}_2} \) K2(N + M − 2)2
\( {\left\{{\hat{\mathbf{u}}}_{\mathrm{s}2}^{\left(k+1\right)}\right\}}_{1\le k\le {K}_2} \) \( {K}_2\left(\begin{array}{c}12\left(N+M-2\right)\\ {}+3{\left(N+M-2\right)}^2+O(27)\end{array}\right) \)
\( {\left({\left[\begin{array}{c}\boldsymbol{\Gamma} \\ {}{\mathbf{I}}_{N-1}\end{array}\right]}^{\mathrm{T}}{\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1}\left[\begin{array}{c}\boldsymbol{\Gamma} \\ {}{\mathbf{I}}_{N-1}\end{array}\right]\right)}^{-1} \) \( {\displaystyle \begin{array}{c}3M\left(M-1\right)\left(3M+N-1\right)\\ {}+\left(N-1\right)\left(M-1\right)\left(2N+M-3\right)\\ {}+3M{\left(M-1\right)}^2+O\left({\left(N-1\right)}^3\right)\end{array}} \)
\( {\left[\begin{array}{c}\boldsymbol{\Gamma} \\ {}{\mathbf{I}}_{N-1}\end{array}\right]}^{\mathrm{T}}{\left(\boldsymbol{\Omega} \left({\hat{\mathbf{u}}}_{\mathrm{s}2},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right)}^{-1}\left[\begin{array}{c}\hat{\mathbf{r}}-\mathbf{f}\left({\hat{\mathbf{u}}}_{\mathrm{s}2},{\hat{\mathbf{w}}}_{\mathrm{f}}\right)\\ {}{\hat{\boldsymbol{\uprho}}}_{\mathrm{f}}\end{array}\right] \) (N − 1)(N + M − 2)
\( {\hat{\boldsymbol{\uprho}}}_{\mathrm{s}2} \) (N − 1)2