## 2 Materials and Methods
### Spatial Simultaneous Functioning-based Joint Design System
#### 2.2.1 Signal Model  
To investigate communication signals, we examine the MIMO JCAS system, where \(N_{t}\) transmit antennas and \(N_{r}\) receive antennas are employed at the base station (BS). This system transmits information to a single-antenna user and detects a single target in a monostatic radar configuration. Therefore, both \(N_{t}\) and \(N_{r}\) belong to the BS.  
We define \(s(n)\in\mathrm{C}\) be the \(n-\mathrm{th}\) discrete-time index, \(n=1,2,...,N\). Let \(x(n)\in\mathrm{C}\) be the \(n-\mathrm{th}\) discrete-time index, \(n=1,2,...,N\). The received communication vector at the UE can be expressed as  
\[\mathbf{y}_{c}(n)=\mathbf{h}_{c}(s(n)+x(n))+\mathbf{w}_{1}(n), \tag{2}\]  
where \(\mathbf{h}_{c}\) is the downlink communication channel and \(\mathbf{h}_{c}\in\mathrm{C}^{N_{t}\times 1}\) is assumed to be perfectly known by the BS. \(\mathbf{w}_{1}\in\mathrm{C}^{N_{t}\times 1}\) is the AWGN vector.  
For studying sensing signals, the transmitted radar signal at the BS is denoted as \(x(t)\), with \(x(n)\in\mathrm{C}\) representing its \(n-\mathrm{th}\) discrete-time index. The transmit beamforming vector is defined as \(\mathbf{f}\in\mathrm{C}^{N_{t}\times 1}\). At the JCAS BS, the reflected echo signal of the target can be expressed as  
\[\mathbf{y}_{s}(n)=\alpha(n)\mathbf{a}_{R}(\theta)\mathbf{a}_{T}^{H}(\theta) \mathbf{f}x(n)+\mathbf{w}_{2}(n), \tag{3}\]  
where \(\mathbf{a}_{T}(\theta)\) and \(\mathbf{a}_{R}(\theta)\) present the transmit and receive steering vectors, respectively, and \(\theta\) is the angle of the target. \(\alpha(n)\) denotes the reflection coefficient of the target, and \(\mathbf{w}_{2}\in\mathrm{C}^{N_{r}\times 1}\) represents the AWGN vector independent of \(\mathbf{w}_{1}\).  
Considering the BS carries out communication and sensing tasks concurrently, the transmitted communication signal and the reflected echo signal of the target may be performed simultaneously. Hence, the joint communication and sensing signals can be expressed as  
\[\mathbf{y}(n)=\mathbf{y}_{s}(n)+\mathbf{y}_{c}(n)=\mathbf{h}_{c}(s(n)+x(n))+ \alpha(n)\mathbf{a}_{R}(\theta)\mathbf{a}_{T}^{H}(\theta)\mathbf{f}x(n)+ \mathbf{w}(n), \tag{4}\]where \(\mathbf{w}(n)=\mathbf{w}_{1}(n)+\mathbf{w}_{2}(n)\) represents the zero-mean white Gaussian noise with a variance of \(\sigma^{2}\).  
In the communication side of the system, the symbol \(s(n)\) is targeted for recovery based on the known channel response \(\mathbf{h}_{c}\). In this work, the primary objective of beamforming, transmit beamforming vector \(\mathbf{f}\), or interference cancellation is to separate the desired user's signal from interference and noise. However, provided from the assumption of a single-user scenario, \(\mathbf{f}\) is excluded from the communication part. Furthermore, our research focuses on estimating the reflection coefficient, with a specific emphasis on the assumption that the target angle is accurately predicted or tracked. Within this framework, \(\mathbf{f}\) is derived as \(\mathbf{f}=\mathbf{a}_{T}(\theta)\), representing the transmit steering vector.  
Conversely, in the sensing side, the focus is on estimating the reflection coefficient \(\alpha(n)\) of the target, along with the knowledge of the probing signal \(x(n)\). We assume that the reflection coefficient of the target varies from pulse to pulse, following a Rayleigh or exponential distributed amplitude, and adheres to the Swerling II target model. Consequently, the reflection coefficient \(\alpha(n)\) follows a complex Gaussian distribution independently and identically distributed (i.i.d.) across pulses. To maintain generality, we normalize the variance of \(\alpha\) by the noise power \(\sigma^{2}\) such that \(\alpha(n)\sim\mathcal{CN}(0,1)\), where \(\mathcal{CN}(0,1)\) denotes a complex Gaussian distribution with zero mean and unit variance [9].