Table 9 IDE steps

● Detection Step: Find indices of inactive sources:

$I^l=\left\{1\le i\le m:\left|\right.{\mathbf{a}}_i^T·\mathbf{x}-\sum _{j\ne i}^m{ŝ}_j^l{\mathbf{a}}_i^T·{\mathbf{a}}_j\left|\right.<{\epsilon }^l\right\}$

● Estimation Step: Find the following projection as the new estimate:

${\mathbf{s}}^{l+1}={\text{argmin}}_{\mathbf{s}}\sum _{i\in I^l}{s}_i^2\text{s.t.}\mathbf{x}\left(t\right)=\mathbf{A}·\mathbf{s}\left(t\right)$

The solution is derived from Karush-Kuhn-Tucker system of equations. At the (l + 1)^thiteration

$\begin{array}{cc}{\mathbf{s}}_i& ={\mathbf{A}}_i^T·\mathbf{P}\left(\mathbf{x}-{\mathbf{A}}_a·{\mathbf{s}}_a\right)\hfill \\ {\mathbf{s}}_a& ={\left({\mathbf{A}}_a^T\mathbf{P}{\mathbf{A}}_a\right)}^{-1}{\mathbf{A}}_a^T\mathbf{P}·\mathbf{x}\hfill \end{array}$

where the matrices and vectors are partitioned into inactive/active parts as A _i ,A _a ,s _i ,s _a and $\mathbf{P}={\left({\mathbf{A}}_i{\mathbf{A}}_i^T\right)}^{-1}$

● Stop after a fixed number of iterations.