Consider that a narrowband CMIMO radar is located at (x_{0}, y_{0}). In order to track Q pointlike enemy objects which carry with active oppressive jammers, a set of orthogonal and coherent pulse train signals are transmitted. In order to simplify the model analysis, we make the following assuptions:

(1)
The number and initial position of the tracked targets are known in advance as priori knwoledge;

(2)
Each target carries a selfdefense jammer that continuously transmits jamming signal to the radar, which is modeled as Gaussian white noise;

(3)
The CMIMO radar works in the SM pattern, and simultaneously transmits multiple orthogonal beams to track targets.
2.1 Radar signal model
Consider that the transmit signal for the qth target at the kth sample interval is normalized as \(s_{k,q} (t)\), and the number of transmit pulse trains in one measurement is \(L\), thus the lth pulse is [5]
$$s_{k,q,l} \left( t \right) = s_{k,q} \left[ {t^{\prime} + \left( {l  1} \right)T_{{\text{p}}} } \right]$$
(1)
where T_{p} is the pulse repetition period, and \(t^{\prime}\) is the slow time. Moreover, the lth pulse in the received baseband signal is given by
$$r_{{k,q,l}} \left( t \right) = \delta _{{k,q}} h_{{k,q,l}} \sqrt {P_{{k,q}} } s_{{k,q}} \left[ {t  \left( {l  1} \right)T_{p}  \tau _{{k,q}} } \right]e^{{  j2\pi f_{{k,q}}^{{\text{d}}} t}} + n_{{k,q,l}} \left( t \right) + j_{{k,q,l}} \left( t \right)$$
(2)
where \(\delta_{k,q}\) denotes the attenuation of signal strength due to the path loss. \(h_{k,q,l} = h_{k,q,l}^{{\text{R}}} + h_{k,q,l}^{{\text{I}}}\) is the radar crosssection (RCS), modeled as a zeromean white complex noise with variance \(\sigma_{k,q}^{2}\), denoted as \(h_{k,q,l} \sim CN(0,\sigma_{k,q}^{2} )\). \(P_{k,q}\) is the transient transmit power, the terms of \(\tau_{k,q}\) and \(f_{k,q}^{{\text{d}}}\) are the time delay and the Doppler frequency, respectively. \(n_{k,q,l} (t)\) represents the inherent environmental noise, modeled as \(n_{k,q,l} (t) \sim CN(0,\alpha_{k,q}^{2} )\), and \(j_{k,q,l} (t)\) denotes the oppressive jamming noise imposed by the jammer, distributed as \(j_{k,q,l} (t) \sim CN(0,\beta_{k,q}^{2} )\).
Aim at improving the echo signal–noise ratio (SNR), the coherent pulse accumulation technique is applied to the echo signal processing. Thus, the sampled signals of \(r_{k,q,l} (t)\) are
$${\hat{\mathbf{r}}}_{k,q,l} = {\hat{\mathbf{s}}}_{k,q,l} g_{k,q,l} + {\hat{\mathbf{n}}}_{k,q,l} + {\hat{\mathbf{j}}}_{k,q,l}$$
(3)
where \({\hat{\mathbf{s}}}_{k,q,l} \in {\mathbb{C}}^{M \times 1}\) denotes the sampling of \(s_{k,q,l} (t)\), and M indicates the sampling length. \({\hat{\mathbf{n}}}_{k,q,l} \in {\mathbb{C}}^{M \times 1}\) represents the sampling of \(n_{k,q,l} (t)\), \({\hat{\mathbf{j}}}_{k,q,l} \in {\mathbb{C}}^{M \times 1}\) is the sampling of \(j_{k,q,l} (t)\), and \(g_{k,q,l}\) dentoes the path gain coefficient, distributed as \(g_{k,q,l} \sim N(0,\gamma_{k,q}^{2} )\), with [7]
$$\gamma_{k,q}^{2} \propto P_{k,q} \sigma_{k,q}^{2} /R_{k,q}^{4}$$
(4)
where \(\propto\) is the proportional notation, and \(R_{k,q}\) is the distance from the qth target to the radar center at sample interval k.
Suppose that the CMIMO radar continuously transmits \(N \le L\) pulses to the qth target at the kth sample interval, which means the number of coherent accumulation pulses in a detection is N. Then, the relevant echo signal can be denoted as
$$\begin{aligned} {\hat{\mathbf{R}}}_{k,q} & = \left( {{\hat{\mathbf{r}}}_{k,q,1} ,{\hat{\mathbf{r}}}_{k,q,2,} ...,{\hat{\mathbf{r}}}_{k,q,N} } \right) \\ & { = }{\hat{\mathbf{S}}}_{k,q} {\hat{\mathbf{G}}}_{k,q} + {\hat{\mathbf{N}}}_{k,q} + {\hat{\mathbf{J}}}_{k,q} \\ \end{aligned}$$
(5)
where
$$\left\{ {\begin{array}{*{20}l} {{\hat{\mathbf{S}}}_{k,q} = \left( {{\hat{\mathbf{s}}}_{k,q,1} ,{\hat{\mathbf{s}}}_{k,q,2} , \ldots ,{\hat{\mathbf{s}}}_{k,q,N} } \right)} \hfill \\ {{\hat{\mathbf{G}}}_{k,q} = {\text{diag}}\left( {g_{k,q,1} ,g_{k,q,2} , \ldots ,g_{k,q,N} } \right)} \hfill \\ {{\hat{\mathbf{N}}}_{k,q} = \left( {{\hat{\mathbf{n}}}_{k,q,1} ,{\hat{\mathbf{n}}}_{k,q,2} , \ldots ,{\hat{\mathbf{n}}}_{k,q,N} } \right)} \hfill \\ {{\hat{\mathbf{J}}}_{k,q} = \left( {{\hat{\mathbf{j}}}_{k,q,1} ,{\hat{\mathbf{j}}}_{k,q,2} , \ldots ,{\hat{\mathbf{j}}}_{k,q,N} } \right)} \hfill \\ \end{array} } \right.$$
(6)
In addition, to simplify the signal model, we assume that the transmit pulse waveforms are exactly the same. Thus, we have \({\hat{\mathbf{s}}}_{{k,q,1}} = {\hat{\mathbf{s}}}_{{k,q,2}} = \cdots = {\hat{\mathbf{s}}}_{{k,q,N}} = {\hat{\mathbf{s}}}_{{k,q}}\).
2.2 Target motion model
Without loss of generality, we assume that the target motion model can be described by the constant velocity (CV) model [6]. In this case, the qth target state is denoted by \({\mathbf{x}}_{k,q} = [x_{k,q} ,\dot{x}_{k,q} ,y_{k,q} ,\dot{y}_{k,q} ]^{{\text{T}}}\), where \([x_{k,q} ,y_{k,q} ]^{{\text{T}}}\) and \([\dot{x}_{k,q} ,\dot{y}_{k,q} ]^{{\text{T}}}\) represent the position and velocity at the kth sample interval in Cartesian coordinates. The target state transition model can be expressed by [7]
$${\mathbf{x}}_{k + 1,q} = {\mathbf{Fx}}_{k,q} + {\mathbf{w}}_{k,q}$$
(7)
where \({\mathbf{F}}\) denotes the state transition matrix of the CV model. The term of \({\mathbf{w}}_{k,q}\) represents an uncorrelated process noise sequence and is assumed to be a zeromean Gaussian noise with the covariance matrix \({\mathbf{Q}}_{k,q}\). Herein, \({\mathbf{F}}\) and \({\mathbf{Q}}_{k,q}\) are given by [8]
$${\mathbf{F}} = \left[ {\begin{array}{*{20}c} 1 & {T_{{\text{s}}} } \\ 0 & 1 \\ \end{array} } \right] \otimes {\mathbf{I}}_{2}$$
(8)
and
$$\begin{aligned} {\mathbf{Q}}_{k,q} & = {\mathbb{E}}\left[ {{\mathbf{w}}_{k,q} \left( {{\mathbf{w}}_{k,q} } \right)^{{\text{T}}} } \right] \\ & = \left( {\int_{0}^{{T_{{\text{S}}} }} {m_{k,q} \left[ {\begin{array}{*{20}c} {T_{{\text{s}}}  t} \\ 1 \\ \end{array} } \right]} \left[ {\begin{array}{*{20}c} {T_{{\text{s}}}  t} & 1 \\ \end{array} } \right]{\text{d}}t} \right) \otimes {\mathbf{I}}_{2} \\ & { = }m_{k,q} \left[ {\begin{array}{*{20}c} {T_{{\text{s}}}^{{3}} /3} & {T_{{\text{s}}}^{{2}} /2} \\ {T_{{\text{s}}}^{{2}} /2} & {T_{s} } \\ \end{array} } \right] \otimes {\mathbf{I}}_{2} \\ \end{aligned}$$
(9)
where \(T_{{\text{s}}}\) is the sample interval, \({\mathbf{I}}_{2}\) denotes the secondorder identity matrix, \(\otimes\) represents the Kronecker product operator, and \(m_{k,q}\) is the relevant process noise intensity [8].
2.3 Measurement model
According to the receive signal model in (2) and (5), the conditional probability density function (PDF) \(p({\hat{\mathbf{R}}}_{k,q} {{\varvec{\upxi}}}_{k,q} )\) is given by
$$p\left( {\left. {{\hat{\mathbf{R}}}_{{k,q}} } \right{\mathbf{\xi }}_{{k,q}} } \right) \propto \exp \left\{ {  \frac{1}{{\delta _{{k,q}}^{2} }}} \right.\int {r_{{k,q,l}}^{{}} \left( t \right)  \alpha _{{k,q}}^{{}} h_{{k,q,l}}^{{}} } \left. {\sqrt {P_{{k,q}} } {\text{ }}s_{{k,q}} \left( {t  \tau _{{k,q}} } \right)e^{{  j2\pi f_{{k,q}}^{{\text{d}}} t}}  j_{{k,q,l}}^{{}} \left( t \right){\text{d}}t} \right\}$$
(10)
where \({{\varvec{\upxi}}}_{k,q}^{{}} = [R_{k,q}^{{}} ,f_{k,q}^{{\text{d}}} ,\theta_{k,q}^{{}} ]^{{\text{T}}}\). Moreover, by adopting the maximum likelihood (ML) estimate method [24], the ML estimate of \({{\varvec{\upxi}}}_{k,q}^{{}}\) can be calculated as
$$\begin{aligned} \left\{ {{\hat{\mathbf{\xi }}}_{k,q}^{{}} } \right\}_{{{\text{ML}}}} & = \arg \left\{ {\mathop {\max }\limits_{{{{\varvec{\upxi}}}_{k,q} }} \left[ {\ln p\left( {\left. {{\hat{\mathbf{R}}}_{k,q} } \right{{\varvec{\upxi}}}_{k,q} } \right)} \right]} \right\} \\ & = \arg \left\{ {\mathop {\max }\limits_{{{{\varvec{\upxi}}}_{k,q} }} \left[ {\sum\limits_{l = 1}^{L} {\left {\int {r_{k,q,l}^{{}} \left( t \right) \times s_{k,q}^{ * } \left( {t  \tau_{k,q}^{{}} } \right)} } \right.} } \right.} \right.\left. {\left. {\left. { \times e^{{{  j2\pi f_{k,q}{\text{d}}} t}} dt} \right^{2} } \right]} \right\} \\ \end{aligned}$$
(11)
Therefore, the target information can be extracted from the receive signal, e.g., the timedelay, the Doppler frequency and the bearing angle. The measurement model can be expressed as
$${\mathbf{z}}_{k,q} = h\left( {{\mathbf{x}}_{k,q} } \right) + {{\varvec{\upnu}}}_{k,q}$$
(12)
where \(h({\mathbf{x}}_{k,q} ) = [R_{k,q} ,\dot{R}_{k,q} ,\theta_{k,q} ]^{{\text{T}}}\), \(R_{k,q}\), \(\dot{R}_{k,q}\) and \(\theta_{k,q}\) denote the range, radial velocity, and bearing angle of the qth target at the kth sample interval, respectively, i.e., given by
$$\left\{ {\begin{array}{*{20}l} {R_{k,q} = \sqrt {\left( {x_{k,q}  x_{0} } \right)^{2} + \left( {y_{k,q}  y_{0} } \right)^{2} } } \hfill \\ {\dot{R}_{k,q} = \left( {\dot{x}_{k,q} ,\dot{y}_{k,q} } \right)\left( \begin{gathered} x_{k,q}  x_{0} \hfill \\ y_{k,q}  y_{0} \hfill \\ \end{gathered} \right)/R_{k,q} } \hfill \\ {\theta_{k,q} = \arctan {{\left( {y_{k,q}  y_{0} } \right)} \mathord{\left/ {\vphantom {{\left( {y_{k,q}  y_{0} } \right)} {\left( {x_{k,q}  x_{0} } \right)}}} \right. \kern\nulldelimiterspace} {\left( {x_{k,q}  x_{0} } \right)}}} \hfill \\ \end{array} } \right.$$
(13)
The measurement noise \({{\varvec{\upnu}}}_{k,q} \sim N({\mathbf{0}},{\mathbf{\Re }}_{k,q} )\), with \({\mathbf{\Re }}_{k,q} = {\text{diag}}(\sigma_{{R_{k,q} }}^{2} ,\sigma_{{\dot{R}_{k,q} }}^{2} ,\sigma_{{\theta_{k,q} }}^{2} )\). In addition, the elements in \({\mathbf{\Re }}_{k,q}\) are given by [9]
$$\left\{ {\begin{array}{*{20}l} {\sigma_{{R_{k,q} }}^{2} \propto \left( {\alpha_{k,q} P_{k,q} \left {h_{k,q} } \right^{2} \beta_{k,q}^{2} } \right)^{  1} } \hfill \\ {\sigma_{{\dot{R}_{k,q} }}^{2} \propto \left( {\alpha_{k,q} P_{k,q} \left {h_{k,q} } \right^{2} T_{k,q}^{2} } \right)^{  1} } \hfill \\ {\sigma_{{\theta_{{_{k,q} }} }}^{2} \propto \left( {\alpha_{k,q} P_{k,q} \left {h_{k,q} } \right^{2} /B_{{\text{w}}} } \right)^{  1} } \hfill \\ \end{array} } \right.$$
(14)
where \(\beta_{k,q}\) denotes the effective bandwidth, \(T_{k,q}\) is the effective time width, and \(B_{{\text{w}}}\) is the nulltonull beam width of receive antennas [25]. Moreover, it should be noted that all the elements in (14) are inversely linear with \(P_{k,q}\) [12], and thus, the measurement covariance can be rewritten as \({\mathbf{\Re }}_{k,q} = P_{k,q}^{  1} {{\varvec{\upchi}}}_{k,q}\). In this case, it is theoretically possible to achieve higher tracking accuracy by increasing the transmit power allocation for a certain target.