## 2 MIMO Radar Principle
In the current work, the cell-averaging constant false alarm rate (CA-CFAR) was adopted to correct the adverse effect caused by false alarm detection. The advantages of CFAR are that it can adjust the detection threshold adaptively according to the characteristics of the reference window and, besides, the false alarm probability is low compared to other algorithms. There are many CFAR detection algorithms, such as the element averaging method and the element maximum method. Figure 2 illustrates the classic CA-CFAR algorithm process. The protection unit plays a role in preventing the target energy from leaking to the sides of the unit, which is generally used in the single-target case. There are \(n\) signals on both sides of the protection unit, used as the reference unit, and \(Y\) is called the detection unit. Clutter level estimation \(Z\) is obtained by processing the reference unit via the CFAR processor. After being multiplied with the normalized factor \(T\), it enters into the comparator for comparison. If the result is higher than \(T\times\)z, it indicates that the target is found; if it is below, the target is not found. \(K\) distribution is a recognized hybrid model that reflects the radar clutter characteristics. The probability density function of the \(K\) distribution is shown in (4):  
\[f_{X}(x;a,v)=\frac{2}{a\Gamma(v+1)}\Big{(}\frac{x}{2a}\Big{)}^{v+1}\times K_{v }\Big{(}\frac{x}{a}\Big{)}U(x) \tag{4}\]  
\(U(x)\) represents the unit step function and \(K_{v}(\cdot)\) is the \(v\) order Bessel function of the second kind. \(v\) and \(a\) denote the shape coefficient and scaling factor, respectively. For a variable \(\theta\) that is uniformly distributed between 0 and \(2\pi\), the characteristic function \(C_{Y}\) (\(u\);\(v\)) can be defined as:  
\[C_{Y}(u,v)=E_{X,\theta}[\exp(jXu\cos(\theta))]=E_{X}[J_{0}(Xu)]=\bigg{(}\frac{ 2(v+1)}{2(v+1)+u^{2}}\bigg{)}^{v+1} \tag{5}\]  
\(J_{0}\) (\(\cdot\)) and \(E[\cdot]\) represent the zero-order Bessel function expected operation. When the second Bessel function approaches infinity, the characteristic function \(C_{Y}\) (\(u\);\(v\)) has a limited value:  
\[\lim_{v\rightarrow\infty}C_{Y}(u;v)=\exp\bigg{(}-\frac{u^{2}}{2}\bigg{)} \tag{6}\]  
Figure 2: Cell-averaging constant false alarm rate (CA-CFAR) algorithm flowchart. There are \(n\) unit signals at each side of the protection unit. The CFAR processor will process the input signal and transmit the clutter estimation to the multiplier for target detection. If the limit can be expressed as (7), then we can get (8):  
\[\lim_{v\rightarrow\infty}f_{X}(x;a,v)\to f_{X}(x) \tag{7}\]  
\[E_{X}[J_{0}(Xu)]=\int_{0}^{\infty}f_{X}(x)J_{0}(xu)\mathrm{d}x=\exp(-\frac{u^{2 }}{2}) \tag{8}\]  
Therefore, when the order tends to infinity, the distribution is changed into Rayleigh distribution. \[y_{bs}=\Phi_{R}^{H}H_{g}\Phi_{T}s_{bs}+n_{bs} \tag{3}\]0  
In particular, if we set \(v=n+0.5\), then the second kind of Bessel function can be expressed as:  
\[y_{bs}=\Phi_{R}^{H}H_{g}\Phi_{T}s_{bs}+n_{bs} \tag{3}\]1  
Assuming \(G(y;a,i)\) is the Gamma distribution of \(a\) and \(i\):  
\[y_{bs}=\Phi_{R}^{H}H_{g}\Phi_{T}s_{bs}+n_{bs} \tag{3}\]2  
If we plug (10) into (4), we obtain:  
\[y_{bs}=\Phi_{R}^{H}H_{g}\Phi_{T}s_{bs}+n_{bs} \tag{3}\]3  
\[y_{bs}=\Phi_{R}^{H}H_{g}\Phi_{T}s_{bs}+n_{bs} \tag{3}\]4  
When we integrate \(x\) on both sides of (12), we obtain (14). \[y_{bs}=\Phi_{R}^{H}H_{g}\Phi_{T}s_{bs}+n_{bs} \tag{3}\]5  
According to (13), every element in the sequence \(\{\alpha_{i}=2,\,\ldots,\,n+2\}\) is non-negative, which means the variables of Rayleigh distribution can be considered as a mixture of several two-dimensional \(\chi^{2}\) distribution variables.