## 2 System Model
### Radar Model  
MIMO radar systems are capable of simultaneously transmitting multiple distinct probing signals, thereby increasing the degrees of freedom (DoFs) available. In this study, we concentrate on the design of beamforming patterns for MIMO radar within a DFRC framework [33; 34]. The beamforming strategy plays a crucial role in enhancing detection performance and improving the accuracy of target localization. Consider a target located in the far-field direction denoted by \(\theta\). The synthesized received signal for the \(l\)-th snapshot can be expressed as  
\[t(\theta,l)=\mathbf{a}_{T}^{T}(\theta)\mathbf{x}_{l},l=1,2,\ldots,L \tag{5}\]  
where \(\mathbf{a}_{T}(\theta)\), the \(B\)-dimensional transmit steering vector, can be represented as  
\[\mathbf{a}_{T}(\theta)=\left[1,e^{\beta\pi d\sin\theta/\lambda},\ldots,e^{ \beta 2\pi(B-1)d\sin\theta/\lambda}\right]^{T} \tag{6}\]  
with \(d\) being the inter-element spacing, and \(\lambda\) being the carrier wavelength. Here, the energy of probing signal in the \(\theta\) direction, known as the transmitting beampattern, can be expressed as  
\[P(\theta)=\mathbf{a}_{T}^{H}(\theta)\mathbf{R}\mathbf{a}_{T}(\theta) \tag{7}\]  
where \(\mathbf{R}=\mathbf{X}\mathbf{X}^{H}/L\) is the covariance matrix of transmitting signals, directly affecting the spatial distribution characteristics of the beampattern. Therein, the transmitting beampattern also reflects the spatial energy distribution. Given that each array element has equal transmit energy, the covariance matrix \(\mathbf{R}\) generally satisfies  
\[R_{\text{bb}}=\frac{P_{T}}{B},\mathbf{b}=1,\cdots,B \tag{8}\]  
where \(R_{\text{bb}}\) denotes the \((\text{b},\text{b})\) entry of the covariance matrix \(\mathbf{R}\), and \(P_{T}\) represents the average transmission power allocated per antenna. In previous work [35], the cost function minimizes the squared error between the desired and actual beampatterns, aiming to closely approximate the target beampattern while reducing cross-correlation sidelobes. Namely, this cost function includes terms for beampattern matching and cross-correlation reduction, and can be given as  
\[ \begin{split}\mathbb{J}_{b}&(\mathbf{R},\alpha)=\frac{1 }{K}\sum_{k=1}^{K}\omega_{k}\Big{|}\mathbf{a}_{T}^{H}(\theta_{k})\mathbf{R} \mathbf{a}_{T}(\theta_{k})-\alpha\phi(\theta_{k})\Big{|}^{2}\\ &+\frac{2\omega_{c}}{\tilde{K}(\tilde{K}-1)}\sum_{p=1}^{\tilde{K} -1}\sum_{q=p+1}^{\tilde{K}}\Big{|}\mathbf{a}_{T}^{H}(\tilde{\theta}_{p}) \mathbf{R}\mathbf{a}_{T}(\tilde{\theta}_{q})\Big{|}^{2}\end{split}  \tag{9}\]  
where \(\omega_{k}\geq 0\) is the weighting coefficient for the \(k\)-th point, \(\tilde{K}\) is the number of targets, \(\alpha\) is the scaling parameter that needs to be optimized, \(\omega_{c}\) denotes the weight of the cross-correlation term, and \(\phi(\theta)\) represents the given desired beampattern. Here, \(\theta_{k}\) denotes the angle of interest for the \(k\)-th point in the beampattern design, while \(\theta_{p}\) and \(\theta_{q}\) in Equation (9) represent angles corresponding to different targets or interference directions. Each \(\theta\) is a distinct direction in space where the radar beampattern is evaluated. Furthermore, when imposing appropriate constraints on the covariance matrix, we would discuss the requirements for low cross-correlation beampatterns. According to discussion in [35], this process must ensure that the covariance matrix is semi-positive definite. Assuming that each diagonal entry is set to a consistent antenna power, the problem of optimizing the covariance matrix can subsequently be formulated as follows  
\[ \begin{split}\min_{\mathbf{R},\alpha}&\mathbb{J}_{b }(\mathbf{R},\alpha)\\ \text{s.t.}&\mathbf{R}\geq 0\\ R_{bb}&=\frac{P_{T}}{B},\quad b=1,\cdots,B\end{split}  \tag{10}\]  
To address this optimization problem involving variables \((\mathbf{R},\alpha)\), the SQP method can solve Problem Equation (10) efficiently within polynomial time [35].