# Statistical Method Based Waveform Optimization in Collocated MIMO Radar Systems
## 2 System Model  
The collocated MIMO radar considered herein is adopted from [18], where a MIMO radar with \(N_{T}\) and \(N_{R}\) collocated to transmit and receive antenna elements, respectively, \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Paper** & **SINR Formulation** & **Outage Probability** & **Optimization Approach** \\ \hline
[18] & ✓ & \(\times\) & [PERSON] \& Vaidyanathan iterative method \\
[22] & ✓ & \(\times\) & Machine learning \& deep learning techniques \\
[34] & ✓ & \(\times\) & Cyclic optimization algorithm \\
[35] & ✓ [Indirectly] & \(\times\) & Minorization- maximization based method \\
[36] & ✓ & \(\times\) & Rao-based detector \\ Proposed & ✓ & ✓ & Interior-Point based approach \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Related works on collocated MIMO radars. are employed as shown in Figure 1. At inception, a finite signal \(\mathbf{f}(n)\) of dimension \(N_{T}\times 1\) duration passes through a digital to analog converter (DAC) filter, modulated and transmitted through \(N_{T}\) antenna elements. At the reflection from target, the subsequent waveform is transformed by the transfer function of both the target and clutter, i.e., \(\mathbf{T_{a}}(s)\) and \(\mathbf{C_{a}}(s)\), respectively. At the receiver end, an \(N_{R}\times 1\) waveform is acquired and thereby passed through a demodulator and analog-to-digital converter (ADC) filter to achieve a received signal \(\mathbf{r}(n)\). Thus, the receive filter \(\mathbf{H}(z)\) processes the acquired signal \(\mathbf{r}(n)\) to extract meaningful information for the existence of an unknown random target through an \(L\)th order FIR filter \(\mathbf{T}(z)\) given by  
\[\mathbf{T}(z)=\sum_{n=0}^{L}\mathbf{t}[n]z^{-n}, \tag{1}\]  
where \(\mathbf{t}[n]\sim\mathcal{CN}(\mathbf{0},\,\mathbf{R_{T}});\in\mathbb{C}^{N_ {R}\times N_{T}}\) represents the impulse response of target with covariance \(\mathbf{R_{T}}\). Similarly, the transfer function of clutter \(\mathbf{C}(z)\) is the bilateral z-Transform of the clutter impulse response, i.e., \(\mathbf{c}[n]\sim\mathcal{CN}(\mathbf{0},\,\mathbf{R_{C}});\in\mathbb{C}^{N_ {R}\times N_{T}}\) with \(\mathbf{R_{C}}\) as the covariance matrix. Hence, after ADC the received waveform \(\mathbf{r}[n]\) of dimension \(N_{R}\times 1\) is expressed as  
\[\mathbf{r}[n]=\left[\mathbf{f}[n]\ast\left[\mathbf{t}[n]+\mathbf{c}[n]\right] \right]+\mathbf{g}[n], \tag{2}\]  
where the convolution sum runs from 0 to \(L_{T}\) and \(L_{T}\) denotes the order of \(\mathbf{f}[n]\). Moreover, \(\mathbf{g}[n]\) represent the additive white noise in the receiver. Furthermore, the received baseband waveform is  
\[\mathbf{r}\triangleq\left[\mathbf{r}(0)^{T}\mathbf{r}(1)^{T}\cdots\mathbf{r} (L_{R})^{T}\right]^{T}\in{}^{N_{R}(L_{R}+1)\times 1}, \tag{3}\]  
where \(L_{R}\) denotes the order of \(\mathbf{H}(z)\); hence, the received waveform is simplified as  
\[\mathbf{r}=\mathbf{f}\times\left[\mathbf{T}+\mathbf{C}\right]+\mathbf{g}. \tag{4}\]  
Here, \(\mathbf{f}\in\,^{N_{T}(L_{T}+1)\times 1}\) and \(\mathbf{g}\in\,^{N_{R}(L_{R}+1)\times 1}\) are the accumulated vectors containing all the available terms of \(\mathbf{f}[n]\) and \(\mathbf{g}[n]\). Moreover, \(\mathbf{T}\) and \(\mathbf{C}\) in (4) are block Toeplitz matrices given in [18] and they incorporate all the multi-path vectors \(\mathbf{t}[n]\) and \(\mathbf{c}[n]\), respectively. These matrices are formulated as follows  
Figure 1: Illustration of the system model for collocated antenna MIMO Radar. \[\mathbf{T}\triangleq \begin{bmatrix}\mathbf{t}(0)&\mathbf{0}&\cdots&\mathbf{0}\\ \mathbf{t}(1)&\mathbf{t}(0)&\ddots&\vdots\\ \vdots&\mathbf{t}(1)&\ddots&\mathbf{0}\\ \mathbf{t}(L)&\vdots&\ddots&\mathbf{t}(0)\\ \mathbf{0}&\mathbf{t}(L)&\ddots&\mathbf{t}(1)\\ \vdots&\ddots&\ddots&\vdots\\ \mathbf{0}&\cdots&\mathbf{0}&\mathbf{t}(L)\\ \end{bmatrix} ;\ \mathbf{C}\triangleq