# Statistical Method Based Waveform Optimization in Collocated MIMO Radar Systems
## 2 System Model
\begin{bmatrix}\mathbf{c}(0)&\mathbf{c}(-1)&\cdots& \mathbf{c}(-L_{T})\\ \mathbf{c}(1)&\mathbf{c}(0)&\ddots&\vdots\\ \vdots&\ddots&\ddots&\mathbf{c}(0)\\ \vdots&\ddots&\ddots&\vdots\\ \mathbf{c}(L_{R})&\mathbf{c}(L_{R}-1)&\cdots&\mathbf{c}(L)\\ \end{bmatrix} \]  
Hence, the receiving filter output, i.e., \(y\) formulates as  
\[y=\mathbf{h}\mathbf{r}=\mathbf{h}\mathbf{T}\mathbf{f}+\mathbf{h}\mathbf{C} \mathbf{f}+\mathbf{h}\mathbf{g}, \tag{5}\]  
where vector \(\mathbf{h}\) of dimension \(1\times N_{R}(L_{R}+1)\) includes the impulse response of \(\mathbf{H}(z)\). The first term in (5) represents the desired signal, the second term is the clutter received, while the third term is noise. Using (5), the SINR (denoted for notation simplicity as \(\eta\)) is a function of \(\mathbf{f}\) and \(\mathbf{h}\), defined as:  
\[\eta(\mathbf{f},\mathbf{h})\triangleq\frac{|\mathbf{h}\mathbf{T}\mathbf{f}|^{2 }}{|\mathbf{h}\mathbf{g}|^{2}+|\mathbf{h}\mathbf{C}\mathbf{f}|^{2}}. \tag{6}\]  
Next, we use a Kronecker-structured correlation matrix formulation in which the transmit correlation \(\mathbf{R}_{tx}\) and receive correlation \(\mathbf{R}_{rx}\) for both target \(\mathbf{R}_{T}\) and clutter \(\mathbf{R}_{C}\) are represent as [32]:  
\[\mathbf{R}_{T} = \mathbf{R}_{Ttx}^{T}\otimes\mathbf{R}_{Trx}, \tag{7}\] \[\mathbf{R}_{C} = \mathbf{R}_{Ctx}^{T}\otimes\mathbf{R}_{Crx}. \tag{8}\]  
Furthermore, whitening transformation of \(\mathbf{T}\) and \(\mathbf{C}\) are \(\mathbf{T}=\mathbf{R}_{Ttx}^{\frac{1}{2}}\bar{\mathbf{T}}\mathbf{R}_{Ttx}^{ \frac{1}{2}}\) and \(\mathbf{C}=\mathbf{R}_{Crx}^{\frac{1}{2}}\bar{\mathbf{C}}\mathbf{R}_{Ctx}^{ \frac{1}{2}}\), respectively. Here, elements of \(\bar{\mathbf{T}}\) and \(\bar{\mathbf{C}}\) are independent and identically distributed (i.i.d.) \(\mathcal{CN}(0,\,1)\). Hence, the SINR reformulates to  
\[\eta(\mathbf{f},\mathbf{h})\triangleq\frac{|\mathbf{h}\mathbf{R}_{Ttx}^{\frac {1}{2}}\mathbf{T}\mathbf{R}_{Ttx}^{\frac{1}{2}}\mathbf{f}|^{2}}{(\mathbf{h} \mathbf{g})(\mathbf{h}\mathbf{g})^{T}+|\mathbf{h}\mathbf{R}_{Crx}^{\frac{1}{2 }}\bar{\mathbf{C}}\mathbf{R}_{Ctx}^{\frac{1}{2}}\mathbf{f}|^{2}}. \tag{9}\]  
Next, \(\bar{\mathbf{T}}\) and \(\bar{\mathbf{C}}\) are vectorized as \(\bar{\mathbf{t}}=\text{vec}(\bar{\mathbf{T}})\) and \(\bar{\mathbf{c}}=\text{vec}(\bar{\mathbf{C}})\), respectively, with dimensions \(N_{R}(L_{R}+1)N_{T}(L_{T}+1)\times 1\). To further simplify, we express the composite channel and waveform terms of target and clutter parts of the above expression as \(\bar{\mathbf{T}}\mathbf{R}_{Ttx}^{\frac{1}{2}}\mathbf{f}=(\mathbf{I}\otimes( \mathbf{R}_{\mathbf{Ttx}}^{\frac{1}{2}}\mathbf{f})^{\mathbf{T}})\bar{\mathbf{t}}\) and \(\bar{\mathbf{C}}\mathbf{R}_{Ctx}^{\frac{1}{2}}=(\mathbf{I}\otimes(\mathbf{R}_ {\mathbf{Ctx}}^{\frac{1}{2}}\mathbf{f})^{\mathbf{T}})\bar{\mathbf{c}}\), respectively. Such a transformation enables us to express the SINR in a quadratic form given by  
\[\eta(\mathbf{f},\mathbf{h})\triangleq\frac{\|\bar{\mathbf{t}}\|_{\mathbf{A}}^ {2}}{\sigma^{2}\|\mathbf{h}\|^{2}+\|\mathbf{c}\|_{\mathbf{B}}^{2}}, \tag{10}\]  
where the weight matrices of target and clutter components are \(\mathbf{A}\) and \(\mathbf{B}\), respectively, while \(\mathbf{A}=\mathbf{a}^{H}\mathbf{a}\), and the vectors are \(\mathbf{B}=\mathbf{b}^{H}\mathbf{b}\), and \(\mathbf{a}=\mathbf{h}\mathbf{R}_{Ttx}^{\frac{1}{2}}(\mathbf{I}\otimes( \mathbf{R}_{\mathbf{Ttx}}^{\frac{1}{2}}\mathbf{f})^{\mathbf{T}})\); \(\mathbf{b}=\mathbf{h}\mathbf{R}_{Crx}^{\frac{1}{2}}(\mathbf{I}\otimes( \mathbf{R}_{\mathbf{Ctx}}^{\frac{1}{2}}\mathbf{f})^{\mathbf{T}})\). Hence, such formulation makes both \(\mathbf{A}\) and \(\mathbf{B}\) unit-rank Hermitian matrices.