# An Under-Sampling Array Signal Processing Method Based on Improved Hadamard Matrix
## 2 Array Signal Processing Model Based on Compressive Sensing
### Reconstruction of The Array Signal and The Direction Estimation  
It can be seen from the above that \(\mathbf{S}=\mathbf{S}(t)\) in Equation (2) or \(\mathbf{S}=\mathbf{S}\big{(}f_{g}\big{)}\) in Equation (5) contains the amplitude information of targets in the time domain or frequency domain, and the direct estimation of the target can be achieved by solving \(\mathbf{S}\), and the targets in \(\mathbf{S}\) are sparse under the constructed grid. Obvious, either \(\mathbf{A}\) in Equation (2) or \(\mathbf{A}\big{(}f_{g},\mathbf{\theta}\big{)}\) in Equation (5) is an overcomplete array manifold. Only the direction of the corresponding target is powerful in \(\mathbf{S}\) and other bearings should be a sufficiently small value or zero, in short, \(\mathbf{S}\) is a type of sparse representation in the spatial field. According to CS model, we can regard \(\Phi\) as the measurement matrix, and \(\mathbf{A}\) as the sparse matrix, \(\mathbf{B}\) as the sensing matrix, \(\mathbf{S}\) as the sparse coefficient component to be solved, \(\mathbf{V}\) as the measurement noise.  
However, for a K-sparse reconstruction \(\mathbf{S}\) is possible by solving an optimization problem of \(min\|\mathbf{S}\|_{0}\), but solving this \(l_{0}\)-minimization is an NP hard problem which is not feasible for practical applications. One of the popular alternatives is to use the closest convex optimization e.g., [PERSON] and others pointed out that it can produce equivalent solutions to solve \(l_{1}\)-optimization, Thus, target signal \(\mathbf{S}\) can be estimated by solving the following \(l_{1}\)-optimization problem:  
\[\hat{\mathbf{S}}=min\|\mathbf{S}\|_{1}\text{s.t. }\mathbf{Y}=\mathbf{B}\mathbf{S}. \tag{10}\]  
There are many methods to solve this \(l_{1}\)-minimization problem, one such strategy is greedy algorithms. In this paper, we mainly use the Adaptive OMP algorithm to solve the \(l_{1}\)-optimization. It selects an atom \(b_{i}\in\mathbf{B}\) firstly and then projects the observed data onto the atom, obtains the residual. These can be described as:  
\[\mathbf{Y}=\mathbf{r}_{1}+\langle\mathbf{b}_{p}\mathbf{Y}\rangle\mathbf{b}_{p}. \tag{11}\]  
The algorithm selects atoms in order to make the \(l_{2}\)-norm of the residual be minimal. i.e., the algorithm requires an extensive search through all possible sets of K columns of \(\mathbf{B}\), and selects the one that has strongest correlation with the remaining part of \(\mathbf{Y}\) over and over, until the residuals are less than a set small value \(\varepsilon\), the estimation of K targets is completed, and \(min\|\mathbf{S}\|_{1}\) subject to \(\|\mathbf{B}\cdot\mathbf{S}-\mathbf{Y}\|_{2}\leq\varepsilon\) provided \(\text{spark(A)}>2\text{K}\).  
\[\mathbf{r}_{new}=\|\hat{\mathbf{B}}\cdot\hat{\mathbf{S}}-\mathbf{Y}\|_{2}\leq\varepsilon. \tag{12}\]  
The following takes the reconstruction of \(\mathbf{S}\) under the wideband signal as an example, the reconstruction result \(\hat{\mathbf{S}}\big{(}f_{g}\big{)}\) also is the frequency domain estimation of the \(g\)th narrowband target signal corresponding to target direction, and then average these \(\hat{S}\big{(}f_{g}\big{)}\) based on direction matching and save as \(\hat{S}(f)\), which is the frequency domain estimation of all potential targets. \(E(g,\theta)\) is target direction estimation at \(g\)th narrowband, it can be shown by Equation (13):  
\[E(g,\theta)=\big{|}\hat{S}\big{(}f_{g}\big{)}\big{|}^{2}. \tag{13}\]  
Thus, the total target direction estimation \(E(\theta)\) can be obtained by summing the average all narrowband spatial \(E(g,\theta)\), as Equation (14):  
\[E(\theta)=\frac{1}{G}\sum_{g=1}^{G}E(g,\theta)=\frac{1}{G}\sum_{g=1}^{G}\big{|} \hat{S}\big{(}f_{g}\big{)}\big{|}^{2}. \tag{14}\]