**Appendix A. Performance Analysis of the Existing SCMDE Used in Multi-Target Tracking Environments**

In this analysis, clutter is distributed with a Gamma probability density function (pdf) with the number of clutter measurements, *n*, inside the hyper-sphere volume of *Vn* as [23],

$$p(V\_n) = \frac{\rho}{(n-1)!} (\rho V\_n)^{n-1} e^{-\rho V\_n} \,\,\,\,\,\,\tag{A1}$$

where *ρ* is the clutter measurement density. In this case, the number of clutter measurements is Poisson distributed inside the volume of a hyper-sphere, *V*, such as:

$$P(m) = \frac{(\rho V)^m}{m!} e^{-\rho V}.\tag{A2}$$

When the point of interest **z***k*,*<sup>i</sup>* is a target detection, the sparsity estimate that the existing SCMDE generates is described in (30) of Section 4.

For the target cardinality |*T*| = 2 case under the assumption that the position of another target is known, the conditional pdf of the sparsity estimates for *n* = 1 becomes:

$$p\left(\hat{\gamma}\_{k,i}^{(1)}|D\right) = e^{-\rho D}\delta\left(\hat{\gamma}\_{k,i}^{(1)} - D\right) + \rho e^{-\rho \hat{\gamma}\_{k,i}^{(1)}}h\left(\hat{\gamma}\_{k,i}^{(1)} - D\right),\tag{A3}$$

where *D* = *V* (*rD*) in (31) with *rD* is the distance to the position of another target detection from the point of interest for |*T*| = 2. In (A3), *δ* is the delta function, and *h* is the Heaviside unit step function. For *n* = 2, the sparsity estimate of the existing SCMDE satisfies the following conditional pdf.

$$\begin{split} p\left(\hat{\gamma}\_{k,i}^{(2)}|D\right) &= \rho D e^{-\rho D} \delta\left(\hat{\gamma}\_{k,i}^{(2)} - \frac{D}{2}\right) + 2\rho e^{-2\rho \hat{\gamma}\_{k,i}^{(2)}} h\left(\hat{\gamma}\_{k,i}^{(2)} - \frac{D}{2}\right) \\ &+ 4\rho^2 \hat{\gamma}\_{k,i}^{(2)} e^{-2\rho \hat{\gamma}\_{k,i}^{(2)}} \left(h\left(\hat{\gamma}\_{k,i}^{(2)}\right) - h\left(\hat{\gamma}\_{k,i}^{(2)} - \frac{D}{2}\right)\right). \end{split} \tag{A4}$$

The conditional sparsity can be calculated from (A3) and (A4), and it becomes:

$$E\left\{\hat{\gamma}\_{k,i}^{(n)}|D\right\} = \begin{cases} \frac{1}{\rho}(1 - e^{-\rho D}), & n = 1\\ \frac{1}{\rho} - \frac{1}{\rho}\frac{(1 + \rho D)}{2}, & n = 2 \end{cases} \tag{A5}$$

As the target detection in (A3) and (A4) can be uniformly distributed in a hyper-sphere volume of *<sup>V</sup>*(*n*) with 0 <sup>≤</sup> *<sup>D</sup>* <sup>≤</sup> *<sup>V</sup>*(*n*), the pdf of *<sup>D</sup>* becomes:

$$p(D) = \frac{1}{V^{(n)}}.\tag{A6}$$

Then, the average sparsity estimate is calculated, and it results in:

$$E\left\{\hat{\gamma}\_{k,i}^{(n)}\right\} = \int\_{V} E\left\{\hat{\gamma}\_{k,i}^{(n)}\right\} p(D) dD = \begin{cases} \frac{1}{\rho} (1 - \frac{1 - \epsilon^{-\rho V^{(1)}}}{\rho V^{(1)}}), & n = 1\\ \frac{1}{\rho} (1 - \frac{1 - \epsilon^{-\rho V^{(2)}}}{\rho V^{(2)}} + \frac{\epsilon^{-\rho V^{(2)}}}{2}), & n = 2 \end{cases} \tag{A7}$$

Note that when the point of interest is the target detection for single target tracking environments, the original SCMDE generates the average sparsity such as:

$$E\left[\hat{\gamma}\_{k,i}^{(n)}\right] = \frac{1}{\rho} \quad for \text{ all } n. \tag{A8}$$

Similarly, the conditional pdf of the sparsity estimate under the known *D* assumption when the point of interest is a clutter detection for |*T*| = 2 can be derived as:

$$\mathbb{P}\left(\boldsymbol{\hat{\gamma}}\_{k,i}^{(1)}|\boldsymbol{D}\right) = e^{-\rho D}\delta\left(\boldsymbol{\hat{\gamma}}\_{k,i}^{(1)} - D\right) + \rho e^{-\rho \boldsymbol{\hat{\gamma}}\_{k,i}^{(1)}} \left(h\left(\boldsymbol{\hat{\gamma}}\_{k,i}^{(1)}\right) - h\left(\boldsymbol{\hat{\gamma}}\_{k,i}^{(1)} - D\right)\right) \text{ for } n = 1,\tag{A9}$$

and:

$$\begin{split} p\left(\hat{\gamma}\_{k,i}^{(2)}|D\right) = \varepsilon^{-\rho D} \delta\left(\hat{\gamma}\_{k,i}^{(2)} - \frac{D}{2}\right) + \rho D \varepsilon^{-\rho D} \delta\left(\hat{\gamma}\_{k,i}^{(2)} - \frac{D}{2}\right) + 2\rho \varepsilon^{-\rho \hat{\gamma}\_{k,i}^{(2)}} h\left(\hat{\gamma}\_{k,i}^{(2)} - \frac{D}{2}\right) \\ + 4\rho^2 \hat{\gamma}\_{k,i}^{(2)} \varepsilon^{-2\rho \hat{\gamma}\_{k,i}^{(2)}} \left(h\left(\hat{\gamma}\_{k,i}^{(2)}\right) - h\left(\hat{\gamma}\_{k,i}^{(2)} - \frac{D}{2}\right)\right) for \; n = 2. \end{split} \tag{A10}$$

Then, the average sparsity estimate for (A9) and (A10) can be obtained as:

$$E\left[\hat{\gamma}\_{k,i}^{(n)}\right] = \begin{cases} \frac{1}{\rho} - \frac{1}{\rho^2 V^{(1)}} (1 - e^{-\rho V^{(1)}}) & n = 1\\ \frac{1}{\rho} - \frac{1}{2\rho^2 V^{(2)}} (1 - e^{-\rho V^{(2)}}) & n = 2 \end{cases} \tag{A11}$$

Note that for *n* = 1 in (A11), the average sparsity estimate for |*T*| = 2 is the same as the one for single target tracking as shown in (23) of [23]. As the sparsity order *n* increases, the bias in the sparsity estimate reduces.
