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

Wang, Ding; Yin, Jiexin; Chen, Xin; Jia, Changgui; Wei, Fushan

doi:10.1186/s13634-019-0629-1

EURASIP Journal on Advances in Signal Processing

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

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