This notebook currently outlines a highly suspect approach to tracking multiple objects using range-Doppler measurements, it currently contains no code.
Remark : I prattle on about "Big-$O$" notation in this notebook because I think it looks fancy. I take $f = O(g)$ to be roughly analagous to $f \leq g$, an upper bound on $f$. In similar respects, $f = \Omega(g)$ is a lower bound on $f$, $f \geq g $. Lastly, $f = \Theta(g)$ means $f = O(g)$ and $f = \Omega(g)$, this is hard to guarantee which is why I guess it is usually used for worst case notation.
If all values within a $MxN$ range-Doppler map, $R_{D}$, greater than some threshold, $\epsilon$ , are to be considered possible targets, then the algorithm to determine their map coordinates will be $\Omega(MN)$.
Remark : There are linear time peak finding algorithms but they can only identify a single local peak in a 2D grid. There are also operators in image processing which can identify local maxima, but I can't speak for them.
$R_{D}$ amounts to real, positive valued matrix. The maximum value can be found in all $N$ columns, using $M$ compares per column. Similarly the maximum can be found in $M$ rows using $N$ compares per row. $$ \therefore T(M, N) = 2 M N = \Theta(MN)$$
Remark : This is an example of spurious rigour.
This will identify $K_{i}$ many targets within a map for a sensor $s_{i}$. Since the sensors have distinct locations and view different portions of the field, in general $K_{i} \neq K_{j}$. Luckily if less than $4$ sensors do not detect this object it cannot be located anyway, so as before we will assume all sensors see all $K$ objects.
Multilateration in general requires solving $2 \mathbf{A}^{T} \mathbf{A} \cdot \mathbf{p} = \mathbf{A}^{T}\mathbf{b}$. This is definately less than $O(n^{3})$, which is the cost of directly computing $(\mathbf{A}^{T}\mathbf{A})_{nxn}^{-1}$. For $K_{T}$ objects with an assosciated track, this section will require $O(K_{T}n^{3})$ additions.
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