# On the Trade-Off between the Main Parameters of Planar Antenna Arrays
## 3 Sparse Array Synthesis
### The Two Steps of IDEA  
In the first step, for each iteration \(k\), the new excitation vector \(\mathbf{a}^{k}\in\mathcal{C}^{N_{k}}\) is obtained solving the problem:  
\[minimize \|\mathbf{w}^{k}\circ\mathbf{a}^{k}\|_{1} \tag{12}\] \[subject\;\;to |\mathbf{F}\mathbf{a}^{k}|<\mathbf{M}_{UP}\] (13) \[\mathbf{F}_{0}\mathbf{a}^{k}=1 \tag{14}\]  
where \(\mathbf{a}^{k}=[a_{1}^{k};\cdots;a_{N_{k}}^{k}]\) and \(\mathbf{w}^{k}=[w_{1}^{k};\cdots;w_{N_{k}}^{k}]\) with  
\[w_{n}^{k}=\frac{1}{\max(|a_{n}^{k-1}|,\mu)} \tag{15}\]  
and \(\mathbf{w}^{k}\circ\mathbf{a}^{k}=[a_{1}^{k}w_{1}^{k};\cdots;a_{N_{k}}^{k}w_{N_ {k}}^{k}]\) is the Hadamard entry-wise product of the two vectors \(\mathbf{w}^{k}\) and \(\mathbf{a}^{k}\), \(\|\mathbf{q}\|_{1}\) stands for \(\ell_{1}\) norm of \(\mathbf{q}\) vector, given by \(\sum_{k}|q_{k}|\), \(\mathbf{M}_{UP}\) is column vector sampling the upper mask in the aforementioned directions, \(\mathbf{F}\) is a matrix sampling the array factor (1), \(\mathbf{F}_{0}\) is a row vector, sampling the radiation operator in the direction of the desired maximum of the focused beam, and \(\mu\) is a constant, usually chosen equal to a small fraction of the maximum of \(|\mathbf{a}|\), to limit the weighting for entries close to zero. The problem in Equations (12)-(14) is a re-weighted L1 minimization where the weighting in Equation (15) is similar to the one proposed in Reference [48], and it is used because it tends to force the lower entries of the unknown vector \(\mathbf{a}\) to zero. Its choice has shown excellent sparsification capabilities: when the algorithm has converged, the \(\ell_{1}\) norm of the vector \(\mathbf{w}^{k}\circ\mathbf{a}^{k}\) tends to the number of radiators with an excitation greater than the chosen threshold [48]. The second step of IDEA is the one in which \"inflation\" and \"deflation\" actually occur. Each one of the \(N_{k}\) sources obtained from the previous step, is split into a set of \(P\) sources of coordinates \((\hat{x}_{n,p}^{k};\hat{y}_{n,p}^{k})\), placed at the vertexes of a regular polygon of radius \(\delta\) with the centre in \((x_{n};y_{n})\):  
\[(\hat{x}_{n,p}^{k};\hat{y}_{n,p}^{k})=(x_{n}^{k}+\delta\cos(\phi_{p}+\phi_{n,0 });y_{n}^{k}+\delta\sin(\phi_{p}+\phi_{n,0})) \tag{16}\]where \(\phi_{p}=2p\pi/P\) and \(\phi_{n,0}\) is an offset angle, which can be chosen randomly. The output of this inflation is an increased number of sources, from \(N_{k}\) to \(N_{k}P\). These new \(N_{k}P\) sources can well approximate the field of the starting \(N_{k}\) sources, and it is possible to demonstrate that the approximation error is proportional to \((\beta\delta)^{2}\). Once the sources have been inflated, we can solve the following problem for each one of them:  
\[minimize \|\tilde{\mathbf{w}}^{k}\circ\mathbf{\tilde{a}}^{k}\|_{1} \tag{17}\] \[subject\ to |\tilde{\mathbf{F}}\mathbf{\tilde{a}}^{k}|<\mathbf{M}_{UP}\] (18) \[\tilde{\mathbf{F}}_{\mathbf{0}}\mathbf{\tilde{a}}^{k}=1 \tag{19}\]  
where \(\mathbf{\tilde{a}}^{k}\) is the vector of the \(N_{k}P\) excitations, \(\tilde{\mathbf{F}}\) is the radiation operator for the sources of coordinates \((\tilde{x}_{n,p};\tilde{y}_{n,p})\), \(\mathbf{\tilde{A}}_{\mathbf{0}}\) is a row vector, sampling the radiation operator in the direction of the maximum of the focused beam and \(\tilde{\mathbf{w}}^{k}=[\tilde{w}_{1,1}^{k};\cdots;\tilde{w}_{N_{k},p}^{k}]\) where  
\[\tilde{w}_{n,p}^{k}=\frac{1}{\max(|a_{n}^{k}|,\mu)}. \tag{20}\]  
Once we have solved the problem in Equations(17)-(19), we can \"deflate\" the \(N_{k}\) polygons: we can find the positions and excitations of the new \(N_{k}\) sources, of coordinates \((x_{n}^{k+1};y_{n}^{k+1})\) and excitation \(a_{n}^{k+1}\), that best approximate each set of \(P\) sources obtained by Equations (17)-(19).