Table 2 Complexity of proposed method

Gauss-Newton algorithm in (56)

Calculation of \( {t}_{0,m}\left({\widehat{\mathbf{p}}}^{(i)}\right) \)

\( {\displaystyle \begin{array}{c}{MN}^2{K}^2+\left(N+2\right){K}^2\\ {}+0.5 NK\cdot {\log}_2K\end{array}} \)

\( {I}_{\mathrm{iter}}\left(\begin{array}{c}{I}_{\mathrm{N}}\left(\begin{array}{c}{MN}^2{K}^2+{N}^2\left(3{K}^2L+2{K}^2\right)\\ {}+N\left(\begin{array}{c}{KL}^3+{KL}^2+2 KL\\ {}+{K}^2+K+0.5K\cdot {\log}_2K\end{array}\right)\\ {}+6{K}^2+4K+2{L}^2+O\left({L}^3\right)\end{array}\right)\\ {}+{I}_{\mathrm{G}\hbox{-} \mathrm{N}}N\left(\begin{array}{c}{Q}_n^2\left( MK+3\right)\\ {}+{Q}_n\left({M}^2\left({K}^2+K\right)+2 MK\right)\\ {}+{M}^2\left(3{K}^2+K\right)+M\left(2K+1\right)\\ {}+2K+O\left({Q}_n^3\right)\end{array}\right)\\ {}+M\left({N}^2{K}^2+2 NK+N\right)+\left(N+2\right){K}^2\\ {}+0.5 NK\cdot {\log}_2K+ NK\end{array}\right) \) (where I_N, I_G ‐ N, and I_iter denote the iteration numbers for Gauss-Newton algorithm and alternating minimization algorithm, respectively)

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

\( {\displaystyle \begin{array}{c}{N}^2\left(2{K}^2L+2{K}^2\right)\\ {}+N\left(\begin{array}{c}{KL}^3+{KL}^2\\ {}+ KL+K\end{array}\right)\\ {}+3{K}^2+3K+2{L}^2\end{array}} \)

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

\( {\displaystyle \begin{array}{c}{N}^2{K}^2L+ NKL\\ {}+{K}^2+K\end{array}} \)

\( {\alpha}^i{\left(\mathbf{R}\left({\widehat{\mathbf{p}}}^{(i)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{b}\right)}\right)\right)}^{-1}\mathbf{r}\left({\widehat{\mathbf{p}}}^{(i)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{b}\right)}\right) \)

O(L³)

Estimation of t₀

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

MN ² K ²

\( {{\overline{\mathbf{S}}}^{\prime}}^{\mathrm{H}}\mathbf{B}\left({\widehat{\mathbf{p}}}^{\left(\mathrm{b}\right)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{b}\right)}\right){\left(\mathbf{B}\left({\widehat{\mathbf{p}}}^{\left(\mathrm{b}\right)},{\widehat{\boldsymbol{\upmu}}}^{\left(\mathrm{b}\right)}\right)\right)}^{\mathrm{H}}{\overline{\mathbf{S}}}^{\prime } \)

(N + 2)K²

FFT algorithm

0.5NK ⋅ log₂K

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

\( \frac{{\left(\left({\widehat{\overline{\mathbf{s}}}}^{\left(\mathrm{b}\right)}\odot {\boldsymbol{\upvarphi}}_n\left({\widehat{\mathbf{p}}}^{\left(\mathrm{b}\right)},{\widehat{t}}_0^{\left(\mathrm{b}\right)}\right)\right)\otimes {\mathbf{a}}_n\left({\widehat{\mathbf{p}}}^{\left(\mathrm{b}\right)},{\widehat{\boldsymbol{\upmu}}}_n^{\left(\mathrm{b}\right)}\right)\right)}^{\mathrm{H}}{\overline{\mathbf{x}}}_n}{{\left\Vert {\mathbf{a}}_n\left({\widehat{\mathbf{p}}}^{\left(\mathrm{b}\right)},{\widehat{\boldsymbol{\upmu}}}_n^{\left(\mathrm{b}\right)}\right)\right\Vert}_2^2\cdot {\left\Vert {\widehat{\overline{\mathbf{s}}}}^{\left(\mathrm{b}\right)}\odot {\boldsymbol{\upvarphi}}_n\Big({\widehat{\mathbf{p}}}^{\left(\mathrm{b}\right)},{\widehat{t}}_0^{\left(\mathrm{b}\right)}\Big)\right\Vert}_2^2} \)

M(2K + 1) + K