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 6 Computational complexity of algorithm I

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	P ⁻¹	O((3M)³)	\( {\displaystyle \begin{array}{c}{K}_1\left(\begin{array}{c}3D\left(M-1\right)\left(3{M}^2+M\right)\\ {}+3{D}^2M{\left(M-1\right)}^2+18{M}^2+O\left({(3M)}^3\right)\end{array}\right)\\ {}+{K}_2\left(\begin{array}{c}3\left(4M+N-2\right)\left(3{M}^2+7M+4\right)\\ {}+\left(3M+N+2\right){\left(4M+N-2\right)}^2\\ {}+3 DM\left(M-1\right)\left(3M+N-1\right)\\ {}+{\left(N-1\right)}^2\left(7M+2N-3\right)\\ {}+{\left(3M+N-1\right)}^2+3{D}^2M{\left(M-1\right)}^2\\ {}+\left(3M+4\right){\left(M-1\right)}^2+O\left({\left(N-1\right)}^3\right)\\ {}+O\left({\left(3M+3\right)}^3\right)+O\left({\left(3M+N-1\right)}^3\right)\end{array}\right)\\ {}+\left(N-1\right)\left({M}^2+2M+N-1\right)\\ {}+D\left(2{N}^2-3N+1\right)\left(M-1\right)\\ {}+\left(N-1\right){\left(3M+N-1\right)}^2\\ {}+2{D}^2\left(N-1\right){\left(M-1\right)}^2\\ {}+{\left(N-1\right)}^2\left(4M+N\right)\\ {}+O\left({(3M)}^3\right)+2\cdot O\left({\left(D\left(M-1\right)\right)}^3\right)\\ {}+2\cdot O\left({\left(M-1\right)}^3\right)+2\cdot O\left({\left(N-1\right)}^3\right)\end{array}} \)(where K₁ and K₂ denote the iteration numbers for the dimension-reduction Taylor-series iterative algorithms in the first and second stages, respectively)
	\( {\mathbf{Q}}_{\mathrm{c}}^{-1} \) and \( {\mathbf{Q}}_{\mathrm{c}}^{-1/2} \)	2 ⋅ O((D(M − 1))³)
	\( {\mathbf{Q}}_{\mathrm{c}}^{-1/2}{\boldsymbol{\Pi}}^{\perp}\left[{\mathbf{Q}}_{\mathrm{c}}^{-1/2}\overline{\boldsymbol{\Gamma}}\right]{\mathbf{Q}}_{\mathrm{c}}^{-1/2} \)	\( {\displaystyle \begin{array}{c}2{D}^2\left(N-1\right){\left(M-1\right)}^2\\ {}+2D{\left(N-1\right)}^2\left(M-1\right)+O\left({\left(N-1\right)}^3\right)\end{array}} \)
	\( {\left\{\begin{array}{c}{\left(\overline{\mathbf{F}}\left({\hat{\mathbf{w}}}_{\mathrm{f}}^{(k)}\right)\right)}^{\mathrm{T}}{\mathbf{Q}}_{\mathrm{c}}^{-1/2}{\boldsymbol{\Pi}}^{\perp}\left[{\mathbf{Q}}_{\mathrm{c}}^{-1/2}\overline{\boldsymbol{\Gamma}}\right]\\ {}\times {\mathbf{Q}}_{\mathrm{c}}^{-1/2}\overline{\mathbf{F}}\left({\hat{\mathbf{w}}}_{\mathrm{f}}^{(k)}\right)\end{array}\right\}}_{1\le k\le {K}_1} \)	\( {K}_1\left(\begin{array}{c}3{D}^2M{\left(M-1\right)}^2\\ {}+9{DM}^2\left(M-1\right)\end{array}\right) \)
	\( {\left\{{\left(\begin{array}{c}{\left(\overline{\mathbf{F}}\left({\hat{\mathbf{w}}}_{\mathrm{f}}^{(k)}\right)\right)}^{\mathrm{T}}{\mathbf{Q}}_{\mathrm{c}}^{-1/2}{\boldsymbol{\Pi}}^{\perp}\left[{\mathbf{Q}}_{\mathrm{c}}^{-1/2}\overline{\boldsymbol{\Gamma}}\right]\\ {}\times {\mathbf{Q}}_{\mathrm{c}}^{-1/2}\overline{\mathbf{F}}\left({\hat{\mathbf{w}}}_{\mathrm{f}}^{(k)}\right)+{\mathbf{P}}^{-1}\end{array}\right)}^{-1}\right\}}_{1\le k\le {K}_1} \)	K₁ ⋅ O((3M)³)
	\( {\left\{\begin{array}{c}{\left(\overline{\mathbf{F}}\left({\hat{\mathbf{w}}}_{\mathrm{f}}^{(k)}\right)\right)}^{\mathrm{T}}{\mathbf{Q}}_{\mathrm{c}}^{-1/2}{\boldsymbol{\Pi}}^{\perp}\left[{\mathbf{Q}}_{\mathrm{c}}^{-1/2}\overline{\boldsymbol{\Gamma}}\right]\\ {}\times {\mathbf{Q}}_{\mathrm{c}}^{-1/2}\left({\hat{\mathbf{r}}}_{\mathrm{c}}-\overline{\mathbf{f}}\left({\hat{\mathbf{w}}}_{\mathrm{f}}^{(k)}\right)\right)\end{array}\right\}}_{1\le k\le {K}_1} \)	3K₁DM(M − 1)
	\( {\left\{{\mathbf{P}}^{-1}\left(\hat{\mathbf{v}}-{\hat{\mathbf{w}}}_{\mathrm{f}}^{(k)}\right)\right\}}_{1\le k\le {K}_1} \)	9K₁M²
	\( {\left\{{\hat{\mathbf{w}}}_{\mathrm{f}}^{\left(k+1\right)}\right\}}_{1\le k\le {K}_1} \)	9K₁M²
	\( {\overline{\boldsymbol{\Gamma}}}^{\mathrm{T}}{\mathbf{Q}}_{\mathrm{c}}^{-1}\left({\hat{\mathbf{r}}}_{\mathrm{c}}-\overline{\mathbf{f}}\left({\hat{\mathbf{w}}}_{\mathrm{f}}\right)\right) \)	D(N − 1)(M − 1)
	\( {\hat{\boldsymbol{\uprho}}}_{\mathrm{f}} \)	(N − 1)²
Stage 2	Q⁻¹ and Q^−1/2	2 ⋅ O((M − 1)³)
	Q^−1/2Γ and Γ^TQ⁻¹Γ	(N − 1)²(M − 1) + (N − 1)(M − 1)²
	\( {\left\{{\hat{\boldsymbol{\Psi}}}_1^{(k)}\right\}}_{1\le k\le {K}_2} \) and \( {\left\{{\hat{\boldsymbol{\Psi}}}_2^{(k)}\right\}}_{1\le k\le {K}_2} \)	\( {K}_2\left(\begin{array}{c}3D\left(N-1\right)M\left(M-1\right)\\ {}+3{D}^2M{\left(M-1\right)}^2\\ {}+9{DM}^2\left(M-1\right)\\ {}+O\left({\left(3M+N-1\right)}^3\right)\end{array}\right) \)
	\( {\left[\begin{array}{cc}{\mathbf{Q}}^{-1/2}{\mathbf{F}}_1\left({\hat{\mathbf{u}}}_{\mathrm{s}1}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{s}1}^{(k)}\right)& {\mathbf{Q}}^{-1/2}{\mathbf{F}}_2\left({\hat{\mathbf{u}}}_{\mathrm{s}1}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{s}1}^{(k)}\right)\\ {}{\mathbf{O}}_{\left(3M+N-1\right)\times 3}& {\hat{\boldsymbol{\Psi}}}_1^{(k)}\end{array}\right]}_{1\le k\le {K}_2} \)	3K₂(M + 1)(M − 1)²
	\( {\left\{{\boldsymbol{\Pi}}^{\perp}\left(\left[\begin{array}{c}{\mathbf{Q}}^{-1/2}\boldsymbol{\Gamma} \\ {}{\hat{\boldsymbol{\Psi}}}_2^{(k)}\end{array}\right]\right)\right\}}_{1\le k\le {K}_2} \)	\( {K}_2\left(\begin{array}{c}{\left(N-1\right)}^2\left(7M+2N-3\right)\\ {}+\left(N-1\right){\left(4M+N-2\right)}^2\\ {}+O\left({\left(N-1\right)}^3\right)\end{array}\right) \)
	\( {\left[\begin{array}{c}{\mathbf{Q}}^{-1/2}\left(\hat{\mathbf{r}}-\mathbf{f}\left({\hat{\mathbf{u}}}_{\mathrm{s}1}^{(k)},{\hat{\mathbf{w}}}_{\mathrm{s}1}^{(k)}\right)\right)\\ {}{\hat{\boldsymbol{\Psi}}}_2^{(k)}{\hat{\boldsymbol{\uprho}}}_{\mathrm{f}}+{\hat{\boldsymbol{\Psi}}}_1^{(k)}\left({\hat{\mathbf{w}}}_{\mathrm{f}}-{\hat{\mathbf{w}}}_{\mathrm{s}1}^{(k)}\right)\end{array}\right]}_{1\le k\le {K}_2} \)	K₂((3M + N − 1)² + (M − 1)²)
	\( {\left[\begin{array}{c}{\hat{\mathbf{u}}}_{\mathrm{s}1}^{\left(k+1\right)}\\ {}{\hat{\mathbf{w}}}_{\mathrm{s}1}^{\left(k+1\right)}\end{array}\right]}_{1\le k\le {K}_2} \)	\( {K}_2\left(\begin{array}{c}3\left(4M+N-2\right)\left(3{M}^2+7M+4\right)\\ {}+3\left(M+1\right){\left(4M+N-2\right)}^2\\ {}+O\left({\left(3M+3\right)}^3\right)\end{array}\right) \)
	\( {\left({\boldsymbol{\Gamma}}^{\mathrm{T}}{\mathbf{Q}}^{-1}\boldsymbol{\Gamma} +{\hat{\boldsymbol{\Psi}}}_2^{\mathrm{T}}{\hat{\boldsymbol{\Psi}}}_2\right)}^{-1} \)	(N − 1)²(3M + N − 1) + O((N − 1)³)
	\( {\displaystyle \begin{array}{c}{\boldsymbol{\Gamma}}^{\mathrm{T}}{\mathbf{Q}}^{-1}\left(\hat{\mathbf{r}}-\mathbf{f}\left({\hat{\mathbf{u}}}_{\mathrm{s}1},{\hat{\mathbf{w}}}_{\mathrm{s}1}\right)\right)\\ {}+{\hat{\boldsymbol{\Psi}}}_2^{\mathrm{T}}{\hat{\boldsymbol{\Psi}}}_1\left({\hat{\mathbf{w}}}_{\mathrm{f}}-{\hat{\mathbf{w}}}_{\mathrm{s}1}\right)+{\hat{\boldsymbol{\Psi}}}_2^{\mathrm{T}}{\hat{\boldsymbol{\Psi}}}_2{\hat{\boldsymbol{\uprho}}}_{\mathrm{f}}\end{array}} \)	\( {\displaystyle \begin{array}{c}\left(N-1\right){\left(3M+N-1\right)}^2\\ {}+\left(N-1\right)\left(4M+N-2\right)\end{array}} \)
	\( {\hat{\boldsymbol{\uprho}}}_{\mathrm{s}1} \)	(N − 1)²

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