## 2 Materials and Methods
### Spatial Simultaneous Functioning-based Joint Design System
#### 2.2.2 Interference Cancellation Technique  
To detect the communication symbol amidst other signals considered as unwanted, such as target echoes and noise, we can express the signal model as  
\[\mathbf{y}(n)=\mathbf{h}_{c}(s(n)+x(n))+\mathbf{w}_{c}(n), \tag{5}\]  
where \(\mathbf{w}_{c}(n)=\alpha(n)\mathbf{a}_{R}(\theta)\mathbf{a}_{T}^{H}(\theta) \mathbf{f}x(n)+\mathbf{w}(n)\). Note that both the random variable \(\alpha\) and vector \(w(n)\) are white Gaussian distributed and are independent of each other. Thus, their linear combination \(w_{c}(n)\) is also Gaussian distributed with zero-mean and a covariance matrix  
\[\mathbf{R}_{c}=\mathbf{E}\{\mathbf{g}x(n)x(n)^{H}\mathbf{g}^{H}\}+\sigma^{2} \mathbf{I}=\mathbf{g}\mathbf{g}^{H}+\sigma^{2}\mathbf{I}, \tag{6}\]  
where \(\mathbf{g}=\mathbf{a}_{R}(\theta)\mathbf{a}_{T}^{H}(\theta)\mathbf{f}\). It is recommended that \(\mathbf{w}_{c}(n)\) is color Gaussian distributed, which results in correlated communication subchannels. The received signals can be whitened with this known covariance matrix, and the communication channel is decomposed into several independent subchannels as [9]  
\[\mathbf{Q}\mathbf{y}(n)=\mathbf{Q}\mathbf{h}_{c}(s(n)+x(n))+\mathbf{w}_{n}(n), \tag{7}\]  
where \(\mathbf{Q}=\mathbf{R}_{c}^{-1/2}\) is the whitening matrix and \(\mathbf{w}_{n}(n)\sim\mathcal{CN}(0,1)\) is a standard Gaussian white noise matrix.  
The combination of communication signal model is provided by  
\[\tilde{y}_{c}(n)=\mathbf{Q}\mathbf{y}(n)=\mathbf{Q}\mathbf{h}_{c}(s(n)+x(n))+ \mathbf{w}_{n}(n). \tag{8}\]  
To minimize the decoding error, the maximum likelihood detector is used, noting that \(\mathbf{w}_{n}(n)\) again follows the Gaussian distribution and can be treated as an AWGN for the useful signal. In this case, the maximum likelihood detection problem is formulated using least square (LS) problem as [9]  
\[min_{s(n)}|\tilde{y}_{c}(n)-\mathbf{Q}\mathbf{h}_{c}(s(n)+x(n))|^{2}, \tag{9}\]  
\[\text{subject to }\,s(n)\in\mathcal{A}\]  
where \(\mathcal{A}\) is the alphabet set of communication symbols. This problem can be solved by searching for a minimizer \(\hat{s}\) in the alphabet \(\mathcal{A}\). Because \(s(n)\) is sufficient, this design scheme provides optimal performance for communication. When obtaining \(\hat{s}(n)\), the communication signal can be recovered and subtracted from the mixed reception to obtain  
\[\tilde{\mathbf{y}}(n)=\mathbf{y}(n)-\mathbf{h}_{c}(\hat{s}(n)+x(n))=\alpha(n) \mathbf{a}_{R}(\theta)\mathbf{a}_{T}^{H}(\theta)\mathbf{f}x(n)+\mathbf{w}(n)+ \Delta\mathbf{y}_{s}. \tag{10}\]  
where \(\Delta\mathbf{y}_{s}=\mathbf{h}_{c}(s(n)-\hat{s}(n))\).  
The remaining task is then to estimate \(\alpha(n)\). It can be observed that this equation is a standard Bayesian linear model, with \(\alpha(n)\) subject to the prior distribution \(\mathcal{CN}(0,1)\). When omitting the residual \(\Delta\mathbf{y}_{s}\), a minimum mean square error (MMSE) estimator can be constructed as  
\[\hat{\alpha}(n)=\mathbf{k}_{\alpha}^{H}(\mathbf{k}_{\alpha}^{H}\mathbf{k}_{ \alpha}+\sigma^{2}\mathbf{I})^{-1}\tilde{\mathbf{y}}_{s}(n), \tag{11}\]  
where \(\mathbf{k}_{\alpha}=\mathbf{a}_{R}(\theta)\mathbf{a}_{T}^{H}(\theta)\mathbf{ f}x(n)\). This may not be true of MMSE estimators because it cancels the residual error and results in a suboptimal performance [9].