## 3 SWIFTNET: Protocol for Fast-Reactive Monitoring
### Compressed Sensing
#### 3.1.1 Signal Recovery in Sparse Domain  
In general there are the following classes of sparse signal recovering methods. The first class solves \(\tilde{b}\) with the smallest \(l_{0}\) norm:  
\[\text{min}_{\tilde{b}\in\mathbb{R}^{n}}\left\|\tilde{b}\right\|_{l_{0}}\text{ subject to }y_{k}=\Phi\Psi\tilde{b} \tag{7}\]  
Solving Equation (7) above is intractable [42]. There are other faster algorithms designed to solve this set of problem e.g., using the Smoothed \(l_{0}\) (SL0) norm method proposed by [43]. Their experimental results revealed that the proposed algorithm is about two to three orders of magnitude faster than the state-of-the-art interior-point Linear Programming (LP) solvers such as the Basis Pursuit (BP) [44, 45]. The second class seeks to solve the \(l_{1}\)-minimization problem (also known as the Basis Pursuit problem) through linear programming. For example, using this second class approach, the result from [18] asserts that, if \(x\) is sufficiently sparse, recovery via \(l_{1}\)-minimization is possible. Normally,we would like to sample all the \(n\) coefficients of \(x\), however, we can only get to sample a subset of these, such that, we can recover the data through Equation (8) given below:  
\[y_{k}=\left\langle x,\psi_{k}\right\rangle,\;k\in M \tag{8}\]  
where \(M\subset\{\)1,..., \(N\}\) is a cardinality constraint \(M<N\). Thus, the reconstruction of \(x\) is either through a greedy algorithm, as proposed in [46], or using \(l_{1}\)-norm minimization. The proposed reconstruction of \(x\) through \(l_{1}\)-norm minimization is given by \(x^{*}\) = \(\Psi b^{*},\) where \(b^{*}\) is the solution to the convex optimization problem (\(\left\|b\right\|_{l_{1}}:=\sum_{i}\left|b_{i}\right|\)) [41]:  
\[\text{min}_{\tilde{b}\in\mathbb{R}^{n}}\left\|\tilde{b}\right\|_{l_{1}}\text{ subject to }y_{k}=\left\langle\psi_{k},\Psi\tilde{b}\right\rangle,\forall k\in M \tag{9}\]  
From Equation (9), minimizing \(l_{1}\) subject to linear equality constraints can be solved using LP. This leads to the following _Theorem 1_, see proof in [47]:  
Fix \(x\in\mathbb{R}^{n}\) and suppose that the coefficient sequence \(b\) of \(x\) in the basis \(\Psi\) is \(K\)-sparse. Select \(m\) measurements in the \(\Phi\) domain uniformly at random. Then if,  
\[m\geq C\cdot\mu^{2}(\Phi,\Psi)\cdot K\cdot\text{log}\;n \tag{10}\]  
for some positive constant \(C\), the solution to Equation (10) is exact with overwhelming probability [41], see proof of _Theorem 1_ in [47]. With the concept of sparsity and incoherency in mind, the problem of CS is therefore reduced to: (1) finding a stable measurement matrix \(\Phi\) such that the important information embedded in any \(K\)-sparse signal can be recovered easily and (2) finding an algorithm that can recover the \(x\) signal in Equation (4) from any \(m\approx K\) measurements. It follows that:  
* No significant information is lost by measuring any set of \(m\) coefficients that is far less than the signal size. If \(\mu(\Phi,\Psi)\) is close or equal to one, the order of \(K\)log \(n\) samples is enough. * The role of incoherency becomes transparent _i.e._, the smaller the coherency, the fewer the samples that will be needed for reconstruction. * We just run the algorithm; if the signal is well sparse, exact reconstruction is possible, _i.e._, the signal \(x\) is exactly recoverable from set \(m\) by minimizing a convex function, without assuming knowledge of the non-zero coordinates of \(b\), the amplitudes or locations, which are unknown _a priori_. In real-world applications, the signal will be invariably corrupted by some level of noise; however, a small perturbation in the signal should cause small perturbation in the recovery process.