Table 1 Complexity of proposed method

Gauss-Newton algorithm in (27)

\( \mathbf{C}\left({\widehat{\mathbf{p}}}^{(i)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{a}\right)}\right) \)

MN²K² + N²K

\( {I}_{\mathrm{iter}}\left(\begin{array}{c}{I}_{\mathrm{N}}\left(\begin{array}{c}{MN}^2{K}^2+{N}^2K+{N}^2\left(N-1\right)\\ {}+{N}^2\left({L}^2+2L\right)+N\left({L}^2+3L\right)\\ {}+O\left({N}^3\right)+O\left({L}^3\right)\end{array}\right)\\ {}+{I}_{\mathrm{G}\hbox{-} \mathrm{N}}N\left(\begin{array}{c}{Q}_n^2\left( MK+3\right)\\ {}+{Q}_n\left({M}^2\left({K}^2+K\right)+2 MK\right)\\ {}+{M}^2\left(3{K}^2+K\right)+M\left(2K+1\right)\\ {}+2K+O\left({Q}_n^3\right)\end{array}\right)\\ {}+M\left({N}^2{K}^2+N\left(2K+1\right)\right)\\ {}+N\left({K}^2+K\right)+O\left({K}^3\right)\end{array}\right) \) (where I_N, I_G ‐ N, and I_iter denote the iteration numbers for Gauss-Newton algorithm and alternating minimization algorithm, respectively)

Eigen decomposition of \( \mathbf{C}\left({\widehat{\mathbf{p}}}^{(i)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{a}\right)}\right) \)

O(N³)

\( {\mathbf{V}}_N\left({\widehat{\mathbf{p}}}^{(i)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{a}\right)}\right) \)

N²(N − 1)

\( \mathbf{H}\left({\widehat{\mathbf{p}}}^{(i)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{a}\right)}\right) \)

\( {\displaystyle \begin{array}{c}{N}^2\left({L}^2+L\right)\\ {}+N\left({L}^2+2L\right)\end{array}} \)

\( \mathbf{h}\left({\widehat{\mathbf{p}}}^{(i)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{a}\right)}\right) \)

N²L + NL

\( {\alpha}^i{\left(\operatorname{Re}\left\{\mathbf{H}\left({\widehat{\mathbf{p}}}^{(i)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{a}\right)}\right)\right\}\right)}^{-1}\cdot \operatorname{Re}\left\{\mathbf{h}\left({\widehat{\mathbf{p}}}^{(i)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{a}\right)}\right)\right\} \)

O(L³)

Estimation of \( \overline{\mathbf{s}} \)

\( \mathbf{B}\left({\widehat{\mathbf{p}}}^{\left(\mathrm{a}\right)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{a}\right)}\right) \)

MN ² K ²

\( \mathbf{B}\left({\widehat{\mathbf{p}}}^{\left(\mathrm{a}\right)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{a}\right)}\right){\left(\mathbf{B}\left({\widehat{\mathbf{p}}}^{\left(\mathrm{a}\right)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{a}\right)}\right)\right)}^{\mathrm{H}} \)

NK ²

Eigen decomposition of \( \mathbf{B}\left({\widehat{\mathbf{p}}}^{\left(\mathrm{a}\right)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{a}\right)}\right){\left(\mathbf{B}\left({\widehat{\mathbf{p}}}^{\left(\mathrm{a}\right)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{a}\right)}\right)\right)}^{\mathrm{H}} \)

O(K³)

Gauss-Newton algorithm in (33)

\( {\mathbf{g}}_n\left({\widehat{\boldsymbol{\upmu}}}_n^{(i)}\right) \)

\( {\displaystyle \begin{array}{c}{Q}_n^2+2{M}^2{K}^2\\ {}+M\left(K+1\right)+2K\end{array}} \)

\( {\mathbf{G}}_n\left({\widehat{\boldsymbol{\upmu}}}_n^{(i)}\right) \)

\( {\displaystyle \begin{array}{c}{Q}_n\left(\begin{array}{c}{M}^2\left({K}^2+K\right)\\ {}+ MK\end{array}\right)\\ {}+{M}^2\left({K}^2+K\right)+ MK\end{array}} \)

\( {\displaystyle \begin{array}{c}{Q}_n^2\left( MK+2\right)\\ {}+{Q}_n MK+O\left({Q}_n^3\right)\end{array}} \)

Estimation of β

\( \frac{{\left(\left({\widehat{\overline{\mathbf{s}}}}^{\left(\mathrm{a}\right)}\odot {\boldsymbol{\upgamma}}_n\left({\widehat{\mathbf{p}}}^{\left(\mathrm{a}\right)}\right)\right)\otimes {\mathbf{a}}_n\left({\widehat{\mathbf{p}}}^{\left(\mathrm{a}\right)},{\widehat{\boldsymbol{\upmu}}}_n^{\left(\mathrm{a}\right)}\right)\right)}^{\mathrm{H}}{\overline{\mathbf{x}}}_n}{{\left\Vert {\mathbf{a}}_n\left({\widehat{\mathbf{p}}}^{\left(\mathrm{a}\right)},{\widehat{\boldsymbol{\upmu}}}_n^{\left(\mathrm{a}\right)}\right)\right\Vert}_2^2\cdot {\left\Vert {\widehat{\overline{\mathbf{s}}}}^{\left(\mathrm{a}\right)}\odot {\boldsymbol{\upgamma}}_n\left({\widehat{\mathbf{p}}}^{\left(\mathrm{a}\right)}\right)\right\Vert}_2^2} \)

M(2K + 1) + K