Algorithm 4. TLI—Tangent Lines Intersections Model Algorithm

Data Input

$m\ge 2$ is the number of receivers. $C_k\left(x_k,y_k\right)$, $k=1,2,\dots ,m$, is the planar position of each receiver. $r_k,k=1,2,\dots ,m$, is the signal range of each receiver to an emitter.

Procedure

1: for each $k=1,2,3,\dots ,m$

2: $\forall k,j:1,2,\dots ,m/k\ne j$, by (d-6) and (d-7), for each receiver position $C_k$, used as pole-points, compute all combinations of polar-points $P_{kj}\left(x_{pkj},y_{pkj}\right)$ and $Q_{kj}\left(x_{qkj},y_{qkj}\right)$ with the respective receiver at position $C_j$ and signal range $r_j$.

3: Stores the points $P_{kj}$ and $Q_{kj}$ in the set $R$ $\left(R\leftarrow R{\displaystyle \cup }\left\{P_{kj},Q_{kj}\right\}\right)$.

4: For each corresponding $P_{kj}$ and $Q_{kj}$ polar-points, by (d-6), computes the respective two tangent lines equation $t_{Pkj}$ and $t_{Qkj}$ that passes by each $C_k$.

5: end for

6: For all $t_{Pkj}$ and $t_{Qkj}$ tangent-lines, by (d-3), computes the intersections points, $\left(x_{+},y_{+}\right)$, among all tangent-lines. Store these intersections points in $S$.

7: Apply (d-8), find the convex hull polygon for all polar-points in $R$. The obtained polygon is the minimal convex polygon that involves all interest points in $R$. This polygon constitutes our ROI.

8: Exclude from $S$ all intersections points among all tangent-lines on the boundary, or out, of the ROI, called bad intersections points.

9: Apply (d-9), compute the location estimation, $E_{xy}\left(E_x,E_y\right)$, of the emitter.

Information Output

10: Emitter location estimation $E_{xy}\left(E_x,E_y\right)$.