# Planar Array Diagnostic Tool for Millimeter-Wave Wireless Communication Systems
## 4 Resolution via Bayesian Compressive Sensing  
For a planar antenna configuration of \(N\) elements positioned at coordinates \((x_{n}y_{n})\), \(n=1\),..., \(N\), with error-free excitations \(\alpha_{n}\), \(n=1\),..., \(N\), beaming a familiar field \(E(u,v),(where\ u=sin\theta cos\varphi\) and \(v=sin\theta sin\varphi)\), referencing a noisy case with element failure, the estimated far-field radiation pattern of (AUT) is expressed as  
\[\widetilde{E}(u_{l},v_{l})=\sum_{n=1}^{N}\beta_{n}e^{j\frac{2\pi}{\lambda}(x_{ n}u_{l}+y_{n}v_{l})}+v_{l}, \tag{10}\]  
where \((u_{l},v_{l})\) for \(l=1\),..., \(L\) is the angular location of the \(l\)-\(th\) angular sample, and \(v_{l}\) is the noise effect considered as Gaussian-distributed with zero mean and variance \(\sigma^{2}\). \(\beta_{n}\), \(n=1\),..., \(N\), is the failed excitations vector, expressed as \[\beta_{n}=\left\{ \begin{array}{ll}h\alpha_{n}&\text{with probability }\Phi\\ \alpha_{n}&\text{otherwise}\end{array} \right.,\ n=1,\ldots,N. \tag{11}\]  
\(h\in(0,1)\) is the failure factor, while \(\Phi\) is the rate of failure, and \(\alpha_{n}\) is the weighting coefficient. From knowledge of the difference in field pattern, \(W(u_{l},v_{l})=E(u_{l},v_{l})-\widetilde{E}(u_{l},v_{l})\), \(l=1,\ldots,L\), array failures can be estimated by determining the minimum \(\ell_{0}-norm\) vector  
\[\underline{Y}=\{Y_{n}=\alpha_{n}-\beta_{n};n=1,\ldots,N\}, \tag{12}\]  
which satisfies  
\[\underline{W}-\underline{\underline{\Psi}Y}=\underline{v}. \tag{13}\]  
The aim is to determine the entries of the \"failure\" vector \(\underline{W}=\{W(u_{l},v_{l});l=1,\ldots,L\},\underline{v}=\{v_{l};l=1, \ldots,L\}\) from the prior knowledge of the difference between the field samples in Equation (10) on the AUT and those of the _golden_ antenna with coefficients \(\alpha_{n}\ n=1,\ldots,N\). Against the deterministic approaches aimed at retrieving the vector \(\underline{\gamma}\) with minimum \(\ell_{0}-norm\) satisfying the condition of Equation (13), \(\underline{\underline{\Psi}}\) is the \(L\ \times\ N\) radiation measurement matrix expressed as  
\[\underline{\underline{\Psi}}=\left[ \begin{array}{ccc}e^{[j2\pi(x_{1}u_{1}+y _{1}v_{1})]}&\ldots&e^{[j2\pi(x_{N}u_{1}+y_{N}v_{1})]}\\ \vdots&\ddots&\vdots\\ e^{[j2\pi(x_{1}u_{L}+y_{1}v_{L})]}&\ldots&e^{[j2\pi(x_{N}u_{L}+y_{N}v_{L})]} \end{array} \right]. \tag{14}\]  
Hence, the BCS technique (summarized in Figure 2) can be employed to determine the sparsest solution \(\underline{\underline{Y}}\) to the problem  
\[\underline{\underline{Y}}=arg\bigg{\{}\max_{\underline{Y}}[P(\underline{Y}| \underline{W})]\bigg{\}}, \tag{15}\]  
which gives  
\[\underline{\underline{Y}}=\frac{1}{\sigma_{NP}}\bigg{[}\frac{\underline{\underline {\Psi}^{T}}\underline{\underline{\Psi}}}{\overline{\sigma_{NP}}}+diag\Big{(} \underline{\underline{\ell}}_{NP}\Big{)}\bigg{]}, \tag{16}\]  
where \(T\) is the transpose operator, and \(\sigma_{NP}\) and \(\underline{\underline{\ell}}_{NP}\) are the figures that are used to maximize the likelihood function  
\[L\Big{(}\sigma,\underline{\underline{\ell}}\Big{)}=-\frac{1}{2}\Big{[}Nlog2 \pi+log\Big{|}\underline{\underline{\Sigma}}\Big{|}+\underline{W^{T}}\, \underline{\underline{\Sigma}^{-1}}\,\underline{\underline{W}}\,\Big{]}. \tag{17}\]  
Equation (11) is computed using RVM [29], with \(\underline{\underline{\Sigma}}=\sigma+\underline{\underline{\Psi}F^{-1}} \,\underline{\underline{\Psi}^{T}}\), where \(\underline{\underline{\ell}}=diag(\underline{\underline{\ell}})\). The implementation of the BCS technique (as shown in Figure 3) is summarized in Algorithm 1. ```
a. Step 1--Array parameter selection and definition of problem: For each element (\(\boldsymbol{n}=\boldsymbol{1},\ldots\boldsymbol{N}\)), accumulate the array field at \(N\) measurement points (reference antenna) and randomly sample the AUT field. Appropriate the noise variance and threshold for the maximization of \(\boldsymbol{L}\Big{(}\sigma,\underline{\underline{\ell}}\Big{)}\). b.