*2.2. Observation Model*

Consider a radar network with *N* two-dimensional phased array radars (PARs) working in space, time and frequency synchronization. In order to simplify the problem, we give some moderate assumptions:


The traditional radar network system requires all of the radars in the system to radiate a target at all times. Due to the limitation of spectrum resources, communication resources, energy resources etc., multi-target tracking in a traditional radar network is inefficient. As a result, it is not necessary for all radars to work in a revisit period. Thus, we define a set of binary variables *u q <sup>i</sup>*,*<sup>k</sup>* ∈ {0, 1} to represent the radar selection index:

$$u\_{i,k}^q = \begin{cases} 1, \text{ if the } q \text{th target is traded by the } i \text{th radar at time index } k\\ 0, \text{ otherwise} \end{cases} \tag{4}$$

Assuming that all PARs in the radar network are able to extract the distance and angle information from the echo signal, then the measurement equation can be written as:

$$\mathbf{Z}\_{i,k}^{q} = \begin{cases} \mathbf{h}\_i \Big( \mathbf{X}\_k^q \Big) + \mathbf{V}\_{i,k'}^q & \text{if } u\_{i,k}^q = 1 \\ 0 & \text{if } u\_{i,k}^{q'} = 0 \end{cases} \tag{5}$$

where **Z***<sup>q</sup> <sup>i</sup>*,*<sup>k</sup>* represents the measured value, and h*<sup>i</sup>* , **X***q k* is a nonlinear transfer function with the following expression:

$$\mathbf{h}\_{i}(\mathbf{x}\_{k}^{q}) = \begin{bmatrix} R\_{i,k}^{q} \\ \theta\_{i,k}^{q} \end{bmatrix} = \begin{bmatrix} \sqrt{\left(\mathbf{x}\_{k}^{q} - \mathbf{x}\_{i}\right)^{2} + \left(y\_{k}^{q} - y\_{i}\right)^{2}} \\ \arctan\left[\frac{y\_{k}^{q} - y\_{i}}{\mathbf{x}\_{k}^{q} - \mathbf{x}\_{i}}\right] \end{bmatrix} \tag{6}$$

here, (*xi*, *yi*) denotes the *<sup>i</sup>*th radar's position, *<sup>R</sup><sup>q</sup> <sup>i</sup>*,*<sup>k</sup>* and <sup>θ</sup>*<sup>q</sup> <sup>i</sup>*,*<sup>k</sup>* are the *q*th target's distance and azimuth to radar *i*. In (5), **V***<sup>q</sup> <sup>i</sup>*,*<sup>k</sup>* is the measurement noise and can be written as **<sup>V</sup>***<sup>q</sup> <sup>i</sup>*,*<sup>k</sup>* <sup>=</sup> Δ*R<sup>q</sup> i*,*k* , Δθ*<sup>q</sup> i*,*k* T , where Δ*R<sup>q</sup> i*,*k* and Δθ*<sup>q</sup> <sup>i</sup>*,*<sup>k</sup>* are the measurement errors of distance and azimuth, respectively. Assuming that **<sup>V</sup>***<sup>q</sup> <sup>i</sup>*,*<sup>k</sup>* is zero-mean Gaussian noise with covariance **G***<sup>q</sup> i*,*k* , which can given by:

$$\mathbf{G}\_{i,k}^{q} = \begin{bmatrix} \sigma\_{\mathcal{R}\_{i,k}^{q}}^{2} & 0\\ 0 & \sigma\_{\mathcal{O}\_{i,k}^{q}}^{2} \end{bmatrix} \tag{7}$$

herein, σ<sup>2</sup> *Rq i*,*k* and σ<sup>2</sup> θ*q i*,*k* are the mean square estimation error of distance and azimuth, respectively. Both of them are related to the signal-to-noise ratio (SNR) of the echo at the current moment and can be calculated as [26]:

$$\begin{cases} \sigma\_{R\_{ik}^{\eta}} = \frac{c}{4\pi \beta\_{ik,\eta} \sqrt{\text{SNR}\_{ik}^{\eta}}}\\ \sigma\_{\theta\_{ik}^{\eta}} = \frac{\sqrt{3}\lambda}{\pi \gamma \sqrt{\text{SNR}\_{ik}^{\eta}}} \end{cases} \tag{8}$$

where SNR*<sup>q</sup> <sup>i</sup>*,*<sup>k</sup>* denotes the *<sup>i</sup>*th radar' SNR to target *<sup>q</sup>* at time index *<sup>k</sup>*. The term *<sup>c</sup>* <sup>=</sup> <sup>3</sup> <sup>×</sup> 108 m/s is the speed of light, λ and γ are the transmitted wavelength and antenna aperture, respectively. β*i*,*q*,*<sup>k</sup>* is the effective bandwidth of the *i*th radar's transmitted waveform to target *q*.

It can be seen that under the same conditions of other parameters, the higher the β*i*,*q*,*<sup>k</sup>* in (8), the smaller the measurement error of distance. In addition, the amount of radar data samples from the illuminated targets is also related to the transmitted signal bandwidth. Given the oversampling ratio ρ ≥ 1, the *i*th radar's sampling frequency on the *q*th target at time index *k* is *f<sup>s</sup> <sup>i</sup>*,*<sup>k</sup>* = ρβ*i*,*q*,*<sup>k</sup>* [25]. Then, given the observation area *V* of radar network, the number of the *q*th target's from *i*th radar can be calculated as:

$$N\_{i,q,k} = \mu\_{i,k}^{\rho} \frac{\rho \beta\_{i,q,k}}{c} VM \tag{9}$$

From Equation (8), we can conclude that the measurement error of distance and azimuth is inversely proportional to the SNR of the echo. According to the radar equation, if the beams are unbiased with target when the *i*th radar irradiate target *q* at time index *k*, the echo SNR of a single pulse, can be expressed as:

$$\text{SNR}\_{i,q,k}^{s} = \frac{P\_{\text{I}}G\_{\text{I}}G\_{\text{I}}\sigma^{q}\lambda^{2}G\_{\text{RP}}}{\left(4\pi\right)^{3}kT\_{\text{O}}B\_{\text{I}}F\_{\text{I}}\left(\mathbb{R}\_{i}^{q}\right)^{4}}\tag{10}$$

where *P*<sup>t</sup> denotes the transmitted power of radar; *G*<sup>t</sup> is the transmit antenna gain; *G*<sup>r</sup> is the receive antenna gain; σ*<sup>q</sup>* is the radar cross section (RCS) of the target *q*; *G*RP, *T*<sup>o</sup> and *F*<sup>r</sup> are the processing gain, noise temperature and noise coefficient of the radar receiver, respectively; *k* is the Boltzmann constant; *<sup>B</sup>*<sup>r</sup> is the bandwidth of the radar receiver-matched filter, and *<sup>R</sup><sup>q</sup> <sup>i</sup>* is the distance from the *i*th radar to target *q*.

During the dwell time of a single irradiation to the target, the radar can receive several reflection pulses from the target. Since the radar has known its own emission parameters, all of the target reflections can be accumulated by coherent accumulation technology to improve the SNR of the echo. Suppose *T<sup>d</sup> <sup>i</sup>*,*q*,*<sup>k</sup>* represents the dwell time of the *i*th radar's irradiation on target *q* at time index *k*, and *T*<sup>r</sup> represents the pulse repetition period of radar, then the number of coherent accumulated pulses can be given by:

$$m\_{i,q,k} = \frac{T\_{i,q,k}^d}{T\_r} \tag{11}$$

Assuming that coherent accumulation is ideal, the SNR obtained after *ni*,*q*,*<sup>k</sup>* pulses can be written as:

$$\text{SNR}\_{i,q,k}^{\text{CI}} = n\_{i,q,k} \text{SNR}\_{i,q,k}^{s} \tag{12}$$

When there is an angle difference 'α*<sup>q</sup> <sup>i</sup>* between the true azimuth of target *q* and the beam pointing of the *i*th radar, the echo SNR after coherent accumulation can be expressed as:

$$\text{SNR}\_{i,k}^{q} = \text{SNR}\_{i,q,k}^{\text{CI}} \exp\left(-4\ln(2)\frac{\left(\overleftarrow{\alpha}\_{i}^{q}\right)^{2}}{\theta\_{3\text{dB}}^{2}}\right) \tag{13}$$

where θ3dB denotes 3dB antenna beam width. Substitute Equations (10)–(12) into Equation (13), then we can obtain:

$$\text{SNR}\_{i,k}^{q} = \frac{T\_{i,q,k}^{d}}{T\_r} \frac{P\_\text{t} \text{G}\_\text{t} \text{G}\_\text{r} \sigma^q \lambda^2 \text{G}\_\text{RP}}{(4\pi)^3 kT\_\text{o} B\_\text{r} F\_\text{r} \left(\text{R}\_i^q\right)^4} \exp\left(-4\ln(2) \frac{\left(\overline{a}\_i^q\right)^2}{\partial\_{3\text{dB}}^2}\right) \tag{14}$$
